research article modeling and chaotic dynamics of the

Research ArticleModeling and Chaotic Dynamics of the Laminated CompositePiezoelectric Rectangular Plate

Minghui Yao Wei Zhang and D M Wang

College of Mechanical Engineering Beijing University of Technology Beijing 100124 China

Correspondence should be addressed to Minghui Yao ymhbjuteducn

Received 12 October 2013 Accepted 27 December 2013 Published 2 March 2014

Academic Editor Rongni Yang

Copyright copy 2014 Minghui Yao et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper investigates the multipulse heteroclinic bifurcations and chaotic dynamics of a laminated composite piezoelectricrectangular plate by using an extended Melnikov method in the resonant case According to the von Karman type equationsReddyrsquos third-order shear deformation plate theory andHamiltonrsquos principle the equations of motion are derived for the laminatedcomposite piezoelectric rectangular plate with combined parametric excitations and transverse excitation The method of multiplescales and Galerkinrsquos approach are applied to the partial differential governing equation Then the four-dimensional averagedequation is obtained for the case of 1 3 internal resonance and primary parametric resonance The extended Melnikov method isused to study the Shilnikov typemultipulse heteroclinic bifurcations and chaotic dynamics of the laminated composite piezoelectricrectangular plate The necessary conditions of the existence for the Shilnikov type multipulse chaotic dynamics are analyticallyobtained From the investigation the geometric structure of the multipulse orbits is described in the four-dimensional phasespace Numerical simulations show that the Shilnikov type multipulse chaotic motions can occur To sum up both theoreticaland numerical studies suggest that chaos for the Smale horseshoe sense in motion exists for the laminated composite piezoelectricrectangular plate

1 Introduction

The need for high-speed light-weight and energy-savingstructures in the aerospace and aviation industry has ledto the composite materials instead of traditional materi-als Additional requirements for multifunctionality activevibration shape control vibration suppression and acousticcontrol have made the development of smart and intelligentstructures A piezoelectric composite laminate is composedof piezoelectric layers which are embedded in laminatedcomposite structures or are boned on the surface of struc-turesThe direct and converse piezoelectric effects are used tosuppress the transient vibration and to control the deforma-tion shape and buckling of the structures Such lightweightflexible structures generate large deformations geometricalnonlinearity and structural instability when piezoelectriccomposite laminates are subjected to the coupling betweenthe mechanical and electrical loads Therefore it is neces-sary to study geometrically nonlinear effects on dynamiccharacteristics of structures in order to accurately design

and effectively control vibrations of piezoelectric compositelaminate structures It is very important to investigate thelarge amplitude nonlinear vibrations of smart structures withpiezoelectric materials in order to achieve and predict thedesired performance of the systems

Recently the studies on dynamics of composite structureswith piezoelectric materials have made some progress Tzouet al [1] used spatially distributed orthogonal piezoelectricactuators to perform the distributed structural control ofelastic shell They utilized a gain factor and a spatially dis-tributed mode actuator function to describe modal feedbackfunctions Purekar et al [2] presented phased array filterswith piezoelectric sensors to detect damage in isotropicplates and adopted wave propagation to describe platedynamics Ishihara and Noda [3] took into account theeffect of transverse shear to analyze the dynamic behaviorof the laminate composed of fiber-reinforced laminae andpiezoelectric layers constituting a symmetric cross-ply lam-inate rectangular plate with simply supported edges Oh [4]considered snap-through thermopiezo-elastic behaviors to

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 345072 19 pageshttpdxdoiorg1011552014345072

2 Mathematical Problems in Engineering

examine the buckling bifurcation and sling-shot buckling ofactive piezo-laminated plates Lee et al [5] employed third-order shear deformation theory and nonlinear finite elementto canvass deflection suppression characteristics of laminatedcomposite shell structures with smart material laminaePanda andRay [6] exploited the first-order shear deformationtheory and the three-dimensional finite element method todelve into the open-loop and closed-loop nonlinear dynamicsof functionally graded plates with the piezoelectric fiber-reinforced composite material under the thermal environ-ment Dumir et al [7] used the extendedHamiltonrsquos principleto derive the coupled nonlinear equations of motion and theboundary conditions for buckling and vibration of symmet-rically laminated hybrid angle-ply piezoelectric panels underin-plane electrothermomechanical loading Yao and Zhang[8] employed the third-order shear deformation plate theoryto explore the bifurcations and chaotic dynamics of the four-edge simply supported laminated composite piezoelectricrectangular plate in the case of the 1 2 internal resonances

The global bifurcations and chaotic dynamics of high-dimensional nonlinear systems have been at the forefrontof nonlinear dynamics for the past two decades There aretwo ways of solutions on Shilnikov type chaotic dynamics ofhigh-dimensional nonlinear systems One is Shilnikov typesingle-pulse chaotic dynamics and the other is Shilnikovtype multipulse chaotic dynamics Most researchers focusedon Shilnikov type single-pulse chaotic dynamics of high-dimensional nonlinear systems Much research in this fieldhas concentrated on Shilnikov type single-pulse chaoticdynamics of thin plate structures Feng and Sethna [9]utilized the global perturbation method to study the globalbifurcations and chaotic dynamics of the thin plate underparametric excitation and obtained the conditions in whichthe Shilnikov type homoclinic orbits and chaos can occurTien et al [10] applied the Melnikov method to investigatethe global bifurcation and chaos for the Smale horseshoesense of a two-degree-of-freedom shallow arch subjectedto simple harmonic excitation for the case of 1 2 internalresonance Malhotra and Sri Namachchivaya [11] employedthe averaging method and Melnikov technique to canvassthe local global bifurcations and chaotic motions of a two-degree-of-freedom shallow arch subjected to simple har-monic excitation for the case of 1 1 internal resonance Theglobal bifurcations and chaotic dynamics were investigatedby Zhang [12] for the simply supported rectangular thinplates subjected to the parametrical-external excitation andthe parametrical excitation Yeo and Lee [13] made use ofthe global perturbation technique to examine the globaldynamics of an imperfect circular plate for the case of1 1 internal resonance and obtained the criteria for chaoticmotions of homoclinic orbits and heteroclinic orbits Yu andChen [14] adopted the global perturbationmethod to explorethe global bifurcations of a simply supported rectangularmetallic plate subjected to a transverse harmonic excitationfor the case of 1 1 internal resonance

While most of studies are on the Shilnikov typesingle-pulse global bifurcations and chaotic dynamics ofhigh-dimensional nonlinear systems there are researchersinvestigating the Shilnikov type multipulse homoclinic and

heteroclinic bifurcations and chaotic dynamics So far thereare two theories of the Shilnikov type multipulse chaoticdynamics One is the extended Melnikov method and theother theory is the energy phase method Much achievementis made in the former theory of high-dimensional nonlinearsystems In 1996 Kovacic and Wettergren [15] used a modi-fiedMelnikovmethod to investigate the existence of the mul-tipulse jumping of homoclinic orbits and chaotic dynamicsin resonantly forced coupled pendula Furthermore Kaperand Kovacic [16] studied the existence of several classes ofthe multibump orbits homoclinic to resonance bands forcompletely integral Hamiltonian systems subjected to smallamplitude Hamiltonian and damped perturbations Camassaet al [17] presented a new Melnikov method which is calledthe extended Melnikov method to explore the multipulsejumping of homoclinic and heteroclinic orbits in a class ofperturbed Hamiltonian systems Until recently Zhang andYao [18] introduced the extended Melnikov method to theengineering field They came up with a simplification of theextended Melnikov method in the resonant case and utilizedit to analyze the Shilnikov type multipulse homoclinic bifur-cations and chaotic dynamics for the nonlinear nonplanaroscillations of the cantilever beam

The study on the second theory of the Shilnikov typemul-tipulse chaotic dynamics was stated by Haller and Wiggins[19] They presented the energy phase method to investigatethe existence of the multipulse jumping homoclinic andheteroclinic orbits in perturbed Hamiltonian systems Upto now few researchers have made use of the energy phasemethod to study the Shilnikov type multipulse homoclinicand heteroclinic bifurcations and chaotic dynamics of high-dimensional nonlinear systems in engineering applicationsMalhotra et al [20] used the energy-phase method to inves-tigate multipulse homoclinic orbits and chaotic dynamics forthe motion of flexible spinning discs Yu and Chen [21] madeuse of the energy-phase method to examine the Shilnikovtype multipulse homoclinic orbits of a harmonically excitedcircular plate

This paper focuses on the Shilnikov typemultipulse orbitsand chaotic dynamics for a simply supported laminatedcomposite piezoelectric rectangular plate under combinedparametric excitations and transverse load Based on thevon Karman type equations and Reddyrsquos third-order sheardeformation plate theory Hamiltonrsquos principle is employedto obtain the governing nonlinear equations of the lami-nated composite piezoelectric rectangular plate with com-bined parametric excitation and transverse load We applyGalerkinrsquos approach and the method of multiple scales to thepartial differential governing equations to obtain the four-dimensional averaged equation for the case of 1 3 internalresonance and primary parametric resonance From theaveraged equation the theory of normal form is used tofind the explicit formulas of normal form We study theheteroclinic bifurcations of the unperturbed system and thecharacteristic of the hyperbolic dynamics of the dissipativesystem respectively Finally we employ the extended Mel-nikov method to analyze the Shilnikov type multipulse orbitsand chaotic dynamics in the laminated composite piezoelec-tric plate In this paper the extended Melnikov function

Mathematical Problems in Engineering 3

x

a

y

b

qcosΩ3t

qxcosΩ1t

qycosΩ2t

Figure 1 The model of a laminated composite piezoelectric rectangular plate is given

can be simplified in the resonant case and does not dependon the perturbation parameter We have used the extendedMelnikovmethod to investigate heteroclinic bifurcations andmultipulse chaotic dynamics of the laminated compositepiezoelectric plate under the case of 1 3 internal resonancesThe analysis indicates that there exist the Shilnikov typemultipulse jumping orbits in the perturbed phase space forthe averaged equations We present the geometric structureof the multipulse orbits in the four-dimensional phase spaceThe results from numerical simulation also show that thechaotic motion can occur in the motion of the laminatedcomposite piezoelectric plate which verifies the analyticalprediction The Shilnikov type multipulse orbits are discov-ered from the results of numerical simulation In summaryboth theoretical and numerical studies demonstrate thatchaos for the Smale horseshoe sense in themotion existsThispaper demonstrates how to employ the extended Melnikovmethod to analyze the Shilnikov type multipulse heteroclinicbifurcations and chaotic dynamics of high-dimensional non-linear systems in engineering applications

The laminated composite piezoelectric rectangular platesare widely applied in space stations satellite solar panelssensors and actuators for the active control of structuresand so on In this paper we have investigated the multipulseglobal bifurcations and chaotic dynamics of a laminated com-posite piezoelectric rectangular plate by using an extendedMelnikov method and numerical simulations in detail Wehave understood nonlinear vibration characteristics of alaminated composite piezoelectric rectangular plate Ourtheoretical results can be used to solve some engineeringproblems Since these smart structures are generally lightweight and relatively large structural flexibility laminatedcomposite piezoelectric rectangular plates can induce largevibration deformation during the rapid deployment In orderto eliminate or suppress large vibration and chaotic motiontheoretical results can help optimize the design of thestructural parameters of laminated composite piezoelectricrectangular plates Therefore the theoretical studies on themultipulse global bifurcations and chaotic dynamics of lam-inated composite piezoelectric rectangular plates play a very

important role in applications in aerospace and mechanicalengineering

2 Equations of Motion andPerturbation Analysis

We consider a four-edge simply supported laminated com-posite piezoelectric rectangular plate where the length thewidth and the thickness are denoted by 119886 119887 and ℎ respec-tively The laminated composite piezoelectric rectangularplate is subjected to in-plane excitation transverse excita-tion and piezoelectric excitation as shown in Figure 1 Weconsider the laminated composite piezoelectric rectangularplate as regular symmetric cross-ply laminates with 119899 layerswith respect to principal material coordinates alternativelyoriented at 0∘ and 90∘ to the laminated coordinate axes Someof layers are made of the piezoelectric materials as actuatorsand the other layers are made of fiber-reinforced compositematerials It is assumed that different layers of the symmetriccross-ply composite laminated piezoelectric rectangular plateare perfectly clung to each other and piezoelectric actuatorlayers are embedded in the plate The fiber direction ofodd-numbered layers is the 119909-direction of the laminate Thefiber direction of even-numbered layers is the 119910-directionof the laminate Simply supported plate with immovableedges satisfies the symmetry requirement that eliminatesthe coupling between bending and extension However thedisplacement of 119909 is free to move at the edge of 119910 = 0 andthe displacement of 119910 is free to move at the edge of 119909 = 0Therefore the membrane stress is smaller and there exists thecoupling between bending and extension A Cartesian coor-dinate system 119874119909119910119911 is located in the middle surface of thecomposite laminated piezoelectric rectangular plate Assumethat (119908 V 119906) and (119908

0 V0 1199060) describe the displacements of

an arbitrary point and a point in the middle surface of thecomposite laminated piezoelectric rectangular plate in the119909 119910 and 119911 directions respectively It is also assumed thatin-plane excitations of the composite laminated piezoelectricrectangular plate are loaded along the 119910-direction at 119909 = 0


and the 119909-direction at 119910 = 0 with the form of 1199020+ 119902119909cosΩ1119905

and 1199021+119902119910cosΩ2119905 respectively Transverse excitation loaded

to the composite laminated piezoelectric rectangular plate isexpressed as 119902 = 119902

3cosΩ3119905The dynamic electrical loading is

represented by 119864119911= 119864119911cosΩ4119905

Considering Reddyrsquos third-order shear deformationdescription of the displacement field we have

119906 (119909 119910 119911 119905) = 1199060(119909 119910 119905) + 119911120601

119909(119909 119910 119905)

minus 1199113

4

3ℎ2

(120601119909+

1205971199080

120597119909

)

(1a)

V (119909 119910 119911 119905) = V0(119909 119910 119905) + 119911120601

119910(119909 119910 119905)

minus 1199113

4

3ℎ2

(120601119910+

1205971199080

120597119910

)

(1b)

119908 (119909 119910 119911 119905) = 1199080(119909 119910 119905) (1c)

where (1199060 V0 1199080) are the deflection of a point on the middle

surface (119906 V 119908) are the displacement components along the(119909 119910 119911) coordinate directions and 120601

119909and 120601

119910represent the

rotation components of normal to the middle surface aboutthe 119910 and 119909 axes respectively

The nonlinear strain-displacement relations are assumedto have the following form

120576119909119909

=

120597119906

120597119909

+

1

2

(

120597119908

120597119909

)

2

120576119909119911

=

1

2

(

120597119906

120597119911

+

120597119908

120597119909

)

120576119909119910

=

1

2

(

120597119906

120597119910

+

120597V120597119909

+

120597119908

120597119909

120597119908

120597119910

)

120576119910119910

=

120597V120597119910

+

1

2

(

120597119908

120597119910

)

2

120576119910119911

=

1

2

(

120597V120597119911

+

120597119908

120597119910

)

120576119911119911

=

120597119908

120597119911

(2)

Stress constitutive relations are presented as follows

120590119894119895= 120590119904

119894119895119896119897

120576119896119897minus 119890119894119895119896119864119896 (119894 119895 119896 119897 = 119909 119910 119911) (3)

where120590119894119895and 120576119896119897denote themechanical stresses and strains in

extended vector notation 120590119904119894119895119896119897

represents the elastic stiffnesstensor 119864

119896stands for the electric field vector and 119890

119894119895is the

piezoelectric tensorAccording to Hamiltonrsquos principle the nonlinear gov-

erning equations of motion in terms of generalized dis-placements (119906

0 V0 1199080 120601119909 120601119910) for the composite laminated

piezoelectric rectangular plate are given in the previousstudies as follows [8]

11986011

1205972

1199060

1205971199092

+ 11986066

1205972

1199060

1205971199102

+ (11986012

+ 11986066)

1205972V0

120597119909120597119910

+ 11986011

1205971199080

120597119909

1205972

1199080

1205971199092

+ 11986066

1205971199080

120597119909

1205972

1199080

1205971199102

+ (11986012

+ 11986066)

1205971199080

120597119910

1205972

1199080

120597119909120597119910

= 11986800+ 1198691

120601119909minus 11988811198683

1205970

120597119909

(4a)

11986066

1205972V0

1205971199092

+ 11986022

1205972V0

1205971199102

+ (11986021

+ 11986066)

1205972

1199060

120597119909120597119910

+ 11986066

1205971199080

120597119910

1205972

1199080

1205971199092

+ 11986022

1205971199080

120597119910

1205972

1199080

1205971199102

+ (11986021

+ 11986066)

1205971199080

120597119909

1205972

1199080

120597119909120597119910

= 1198680V0+ 1198691

120601119910minus 11988811198683

1205970

120597119910

(4b)

11986066

1205971199080

120597119909

1205972

1199060

1205971199102

minus 119867221198882

1

1205974

1199080

1205971199104

+ 1198881(211986566

+ 11986512

minus 2119867661198881minus 119867121198881)

1205973

120601119910

1205971199101205971199092

+ 1198881(11986522

minus 119867221198881)

1205973

120601119910

1205971199103

minus 119867111198882

1

1205974

1199080

1205971199094

+ 11986011

1205971199080

120597119909

1205972

1199060

1205971199092

+ (119865441198882

2

minus 2119863441198882+ 11986044)

120597120601119910

120597119910

+ 1198881(11986521

+ 211986566

minus 119867211198881minus 2119867661198881)

1205973

120601119909

1205971199102

120597119909

minus 1198882

1

(11986721

+ 411986766

+ 11986712)

1205974

1199080

1205971199102

1205971199092

+ (11986044

minus 119873119875

119910

cos (Ω4119905) + 119865

441198882

2

minus 2119863441198882)

1205972

1199080

1205971199102

minus

120597119873119875

119910

120597119910

cos (Ω2119905)

1205971199080

120597119910

+ (11986021

+ 411986066

+ 11986012)

times

1205971199080

120597119909

1205971199080

120597119910

1205972

1199080

120597119910120597119909

+ 1198881(11986511

minus 119867111198881)

1205973

120601119909

1205971199093

+ (11986021

+ 11986066)

1205971199080

120597119910

1205972

1199060

120597119910120597119909

+ 11986021

1205971199060

120597119909

1205972

1199080

1205971199102

times 11986066

1205971199080

120597119910

1205972V0

1205971199092

+ 11986022

1205971199080

120597119910

1205972V0

1205971199102


+

1

2

(11986012

+ 211986066) (

1205971199080

120597119910

)

2

1205972

1199080

1205971199092

+ 11986022

1205972

1199080

1205971199102

120597V0

120597119910

+ (11986012

+ 11986066)

1205971199080

120597119909

1205972V0

120597119910120597119909

+

1

2

(11986021

+ 211986066)

1205972

1199080

1205971199102

times (

1205971199080

120597119909

)

2

+

3

2

11986011(

1205971199080

120597119909

)

2

1205972

1199080

1205971199092

+ 11986011

1205972

1199080

1205971199092

1205971199060

120597119909

+ 11986012

1205972

1199080

1205971199092

120597V0

120597119910

+ 211986066

1205972

1199080

120597119910120597119909

120597V0

120597119909

+ 211986066

1205972

1199080

120597119910120597119909

1205971199060

120597119910

+

3

2

11986022(

1205971199080

120597119910

)

2

1205972

1199080

1205971199102

+ (11986055

+ 119902119909cos (Ω

1119905) minus 119873

119875

119909

cos (Ω4119905)

+ 119865551198882

2

minus 2119863551198882)

1205972

1199080

1205971199092

minus

120597119873119875

119909

120597119909

cos (Ω3119905)

120597119908

120597119909

+ (119865551198882

2

minus 2119863551198882+ 11986055)

120597120601119909

120597119909

minus 119902 cos (Ω3119905) + 119896119891

1205971199080

120597119905

= 11986800minus 1198882

1

1198686(

1205972

0

1205971199092

+

1205972

0

1205971199102

)

+ 11988811198683(

1205970

120597119909

+

120597V0

120597119910

) + 11988811198694(

120597

120601119909

120597119909

+

120597

120601119910

120597119910

)

(4c)

(11986311

minus 2119865111198881+ 119867111198882

1

)

1205972

120601119909

1205971199092

+ (11986366

minus 2119865661198881+ 119867661198882

1

)

1205972

120601119909

1205971199102

minus 1198881(11986511

minus 119867111198881)

1205973

1199080

1205971199093

minus (119865551198882

2

minus 2119863551198882+ 11986055)

1205971199080

120597119909

+ (11986312

+ 11986366

+ 119867661198882

1

minus 2119865661198881+ 119867121198882

1

minus 2119865121198881)

times

1205972

120601119910

120597119910120597119909

minus 1198881(211986566

+ 11986512

minus 2119867661198881minus 119867121198881)

times

1205973

1199080

1205971199102

120597119909

+ (2119863551198882minus 11986055

minus 119865551198882

2

) 120601119909

= 11986910+ 1198702

120601119909minus 11988811198694

1205970

120597119909

(4d)

(11986366

minus 2119865661198881+ 119867661198882

1

)

1205972

120601119910

1205971199092

minus 1198881(11986521

+ 211986566

minus 119867211198881minus 2119867661198881)

1205973

1199080

1205971199101205971199092

+ (119867211198882

1

+ 11986366

+ 11986321

minus 2119865211198881+ 119867661198882

1

minus 2119865661198881)

times

1205972

120601119909

120597119910120597119909

+ (119867221198882

1

+ 11986322

minus 2119865221198881)

1205972

120601119910

1205971199102

minus 1198881(11986522

minus 119867221198881)

1205973

1199080

1205971199103

minus (119865441198882

2

minus 2119863441198882+ 11986044)

times

1205971199080

120597119910

+ (2119863441198882minus 119865441198882

2

minus 11986044) 120601119910

= 1198691V0+ 1198702

120601119910minus 11988811198694

1205970

120597119910

(4e)

The simply supported boundary conditions of the com-posite laminated piezoelectric rectangular plate can be repre-sented as follows [8 22]

119909 = 0 119908 = 0 120601119910= 0 119873

119909119910= 0 119872

119909119909= 0

(5a)

119909 = 119886 119908 = 0 120601119910= 0 119873

119909119910= 0 119872

119909119909= 0

(5b)

119910 = 0 119908 = 0 120601119909= 0 119873

119909119910= 0 119872

119910119910= 0

(5c)

119910 = 119887 119908 = 0 120601119909= 0 119873

119909119910= 0 119872

119910119910= 0

(5d)

int

ℎ

0

119873119909119909

1003816100381610038161003816119909=0

119889119911 = minusint

ℎ

0

(1199020+ 119902119909cosΩ1119905) 119889119911 (5e)

int

ℎ

0

119873119910119910

10038161003816100381610038161003816119910=0

119889119911 = minusint

ℎ

0

(1199021+ 119902119910cosΩ2119905) 119889119911 (5f)

The boundary condition (5f) includes the influence of thein-plane load We consider complicated nonlinear dynamicsof the composite laminated piezoelectric rectangular plate inthe first two modes of 119906

0 V0 1199080 120601119909 and 120601

119910 It is desirable

that we select an appropriate mode function to satisfy theboundary conditionThus we can rewrite 119906

0 V0 1199080 120601119909 and

120601119910in the following forms

1199060= 11990601(119905) cos 120587119909

2119886

cos120587119910

2119887

+ 11990602(119905) cos 3120587119909

2119886

cos120587119910

2119887

(6a)

V0= V1(119905) cos

120587119910

2119887

cos 1205871199092119886

+ V2(119905) cos

120587119910

2119887

cos 31205871199092119886

(6b)

1199080= 1199081(119905) sin 120587119909

119886

sin120587119910

119887

+ 1199082(119905) sin 3120587119909

119886

sin120587119910

119887

(6c)

120601119909= 1206011(119905) cos 120587119909

119886

sin120587119910

119887

+ 1206012(119905) cos 3120587119909

119886

sin120587119910

119887

(6d)

120601119910= 1206013(119905) cos

120587119910

119887

sin 120587119909

119886

+ 1206014(119905) cos

120587119910

119887

sin 3120587119909

119886

(6e)

Bymeans of the Galerkin method substituting (6a) (6b)(6c) (6d) (6e) into (4a) (4b) (4c) (4d) (4e) integrating


and neglecting all inertia terms in (4a) (4b) (4d) and (4e)we obtain the expressions of 119906

01 11990602 V1 V2 1206011 1206012 1206013 and

1206014via 1199081and 119908

2as follows

11990601

= 11989611199082

1

+ 11989621199082

2

+ 119896311990811199082 (7a)

11990602

= 11989641199082

1

+ 11989651199082

2

+ 119896611990811199082 (7b)

V1= 11989671199082

1

+ 11989681199082

2

+ 119896911990811199082 (7c)

V2= 119896101199082

1

+ 119896111199082

2

+ 1198961211990811199082 (7d)

1206011= 119896191199081 120601

2= 119896201199082 (7e)

1206013= 119896211199081 120601

4= 119896221199082 (7f)

where the coefficients presented in (7a) (7b) (7c) (7d) (7e)(7f) can be found in the previous studies [8]

In order to obtain the dimensionless governing equationsof motion we introduce the transformations of the variablesand parameters

119906 =

1199060

119886

V =

V0

119887

119908 =

1199080

ℎ

120601119909

= 120601119909 120601

119910

= 120601119910 119909 =

119909

119886

119910 =

119910

119887

119902 =

1198872

119864ℎ3

119902 119902119909

=

1198872

119864ℎ3

119902119909 119902

119910

=

1198872

119864ℎ3

119902119910

119905 = 1205872

(

119864

119886119887120588

)

12

119905 Ω119894=

1

1205872

(

119886119887120588

119864

)

12

Ω119894

(119894 = 1 2)

119860119894119895=

(119886119887)12

119864ℎ2

119860119894119895 119861

119894119895=

(119886119887)12

119864ℎ3

119861119894119895

119863119894119895=

(119886119887)12

119864ℎ4

119863119894119895 119864

119894119895=

(119886119887)12

119864ℎ5

119864119894119895

119865119894119895=

(119886119887)12

119864ℎ6

119865119894119895 119867

119894119895=

(119886119887)12

119864ℎ8

119867119894119895

119868119894=

1

(119886119887)(119894+1)2

120588

119868119894

(8)

For simplicity we drop the overbar in the followinganalysis Substituting (5a) (5b) (5c) (5d) (5e) (5f)ndash(8) into(4c) and applying the Galerkin procedure we obtain thegoverning equations of motion of the composite laminatedpiezoelectric rectangular plate for the dimensionless as fol-lows

1+ 12058311+ 1205962

1

1199081

+ (1198862cosΩ1119905 + 1198863cosΩ2119905 minus 1198864cosΩ4119905) 1199081

+ 11988651199082

1

1199082+ 11988661199082

2

1199081+ 11988671199083

1

+ 11988681199083

2

= 1198911cosΩ3119905

(9a)

2+ 12058322+ 1205962

2

1199082

+ (1198872cosΩ1119905 + 1198873cosΩ2119905 + 1198874cosΩ4119905) 1199082

+ 11988751199082

2

1199081+ 11988761199082

1

1199082+ 11988771199083

2

+ 11988781199083

1

= 1198912cosΩ3119905

(9b)

where the coefficients presented in (9a) (9b) are given in theprevious studies [8]

The above equations include the cubic terms in-planeexcitation transverse excitation and piezoelectric excitationEquation (9a) (9b) can describe the nonlinear transverseoscillations of the composite laminated piezoelectric rectan-gular plate We only study the case of primary parametricresonance and 1 3 internal resonances In this resonant casethere are the following resonant relations

1205962

1

=

1205962

9

+ 1205761205901 120596

2

2

= 1205962

+ 1205761205902

Ω3= 120596 Ω

1= Ω2= Ω4=

2120596

3

1205962asymp 31205961

(10)

where 1205901and 120590

2are two detuning parameters

The method of multiple scales [23] is employed to (9a)(9b) to find the uniform solutions in the following form

1199081(119905 120576) = 119909

10(1198790 1198791) + 120576119909

11(1198790 1198791) + sdot sdot sdot (11a)

1199082(119905 120576) = 119909

20(1198790 1198791) + 120576119909

21(1198790 1198791) + sdot sdot sdot (11b)

where 1198790= 119905 1198791= 120576119905

Substituting (10) and (11a) (11b) into (9a) (9b) andbalancing the coefficients of corresponding powers of 120576 onthe left-hand and right-hand sides of equations the four-dimensional averaged equations in the Cartesian form areobtained as follows

1= minus

1

2

12058311199091minus

1

2

12059011199092+

1

4

(1198862+ 1198863minus 1198864) 1199092

minus

3

2

11988671199092(1199092

1

+ 1199092

2

) minus

1

2

11988651199094(1199092

1

minus 1199092

2

)

minus 11988661199092(1199092

3

+ 1199092

4

) + 1198865119909111990921199093

(12a)

2= minus

1

2

12058311199092+

1

2

12059011199091+

1

4

(1198862+ 1198863minus 1198864) 1199091

+

3

2

11988671199091(1199092

1

+ 1199092

2

) +

1

2

11988651199093(1199092

1

minus 1199092

2

)

+ 11988661199091(1199092

3

+ 1199092

4

) + 1198865119909111990921199094

(12b)

3= minus

1

2

12058321199093minus

1

6

12059021199094minus

1

3

11988761199094(1199092

1

+ 1199092

2

)

minus

1

2

11988771199094(1199092

3

+ 1199092

4

) minus

1

6

11988781199092(31199092

1

minus 1199092

2

)

(12c)


4= minus

1

2

12058321199094+

1

6

12059021199093+

1

3

11988761199093(1199092

1

+ 1199092

2

)

+

1

2

11988771199093(1199092

3

+ 1199092

4

) +

1

6

11988781199091(1199092

1

minus 31199092

2

) minus

1

12

1198912

(12d)

3 Computation of Normal Form

In order to assist the analysis of the Shilnikov type multipulseorbits and chaotic dynamics of the laminated compositepiezoelectric rectangular plate it is necessary to reduce theaveraged equation (12a) (12b) (12c) (12d) to a simplernormal form It is found that there are 119885

2oplus 1198852and 119863

4

symmetries in the averaged equation (12a) (12b) (12c) (12d)without the parameters Therefore these symmetries are alsoheld in normal form

We take into account the excitation amplitude 1198912as a

perturbation parameter Amplitude 1198912can be considered as

an unfolding parameter when the Shilnikov type multipulseorbits are investigated Obviously when we do not considerthe perturbation parameter (12a) (12b) (12c) (12d) become

1= minus

1

2

12058311199091+ (1198910minus

1

2

1205901)1199092minus

3

2

11988671199092(1199092

1

+ 1199092

2

)

minus

1

2

11988651199094(1199092

1

minus 1199092

2

)

minus 11988661199092(1199092

3

+ 1199092

4

) + 1198865119909111990921199093

(13a)

2= minus

1

2

12058311199092+ (1198910+

1

2

1205901)1199091+

3

2

11988671199091(1199092

1

+ 1199092

2

)

+

1

2

11988651199093(1199092

1

minus 1199092

2

)

+ 11988661199091(1199092

3

+ 1199092

4

) + 1198865119909111990921199094

(13b)

3= minus

1

2

12058321199093minus

1

6

12059021199094minus

1

3

11988761199094(1199092

1

+ 1199092

2

)

minus

1

2

11988771199094(1199092

3

+ 1199092

4

) minus

1

6

11988781199092(31199092

1

minus 1199092

2

)

(13c)

4= minus

1

2

12058321199094+

1

6

12059021199093+

1

3

11988761199093(1199092

1

+ 1199092

2

)

+

1

2

11988771199093(1199092

3

+ 1199092

4

) +

1

6

11988781199091(1199092

1

minus 31199092

2

)

(13d)

where 1198910= (14)(119886

2+ 1198863minus 1198864)

Executing the Maple program given by Zhang et al [24]the nonlinear transformation used here is given as follows

1199091= 1199101minus

1

4

11988671199103

1

+

31198865(1205902minus 6)

1205902

2

1199102

1

1199103minus 119886611991011199102

3

minus 119886611991011199102

4

minus

31198865(1205903

2

minus 181205902

2

+ 2161205902minus 1296)

1205904

2

1199102

2

1199103

+

1198865(61205902

2

minus 721205902+ 432)

1205903

2

119910111991021199104

(14a)

1199092= 1199102+

3

2

11988671199103

2

+

3

4

11988671199102

1

1199102+

31198865

1205902

1199102

1

1199104

minus

31198865(1205902

2

minus 121205902+ 72)

1205903

2

1199102

2

1199104minus

61198865(1205902minus 6)

1205902

2

119910111991021199103

(14b)

1199093= 1199103minus

1198878

1205902

1199103

1

+

31198878(1205902

2

minus 121205902+ 72)

1205903

2

11991011199102

2

minus

1

3

1198876119910111991021199104

(14c)

1199094= 1199104+

1198878(1205903

2

minus 181205902

2

+ 2161205902minus 1296)

1205904

2

1199103

2

minus

31198878(1205902minus 6)

1205902

2

1199102

1

1199102+

1

3

1198876119910111991021199103

(14d)

Substituting (14a) (14b) (14c) (14d) into (13a) (13b)(13c) (13d) yields a simpler 3rd-order normal form with theparameters for averaged equation (12a) (12b) (12c) (12d) asfollows

1199101= minus1205831

1199101+ (1 minus 120590

1) 1199102 (15a)

1199102= 12059011199101minus 1205831

1199102+ 11988661199101(1199102

3

+ 1199102

4

) +

3

2

11988671199103

1

(15b)

1199103= minus1205832

1199103minus 12059021199104minus

1

3

11988761199102

1

1199104minus

1

2

11988771199104(1199102

3

+ 1199102

4

) (15c)

1199104= 12059021199103minus 1205832

1199104+

1

3

11988761199102

1

1199103+

1

2

11988771199103(1199102

3

+ 1199102

4

) minus 1198912

(15d)

where the coefficients are 1205831

= (12)1205831 1205832

= (12)1205832 1205902=

(16)1205902 and 119891

2

= (112)1198912 respectively

Further let

1199103= 119868 cos 120574 119910

4= 119868 sin 120574 (16)

Substituting (16) into (15a) (15b) (15c) (15d) yields

1199101= minus1205831

1199101+ (1 minus 120590

1) 1199102 (17a)

1199102= 12059011199101minus 1205831

1199102+ 119886611991011198682

+

3

2

11988671199103

1

(17b)

119868 = minus1205832

119868 minus 1198912

sin 120574 (17c)

119868 120574 = 1205902119868 +

1

3

11988761199102

1

119868 +

1

2

11988771198683

minus 1198912

cos 120574 (17d)

In order to get the unfolding of (17a) (17b) (17c) (17d) alinear transformation is introduced

[

1199101

1199102

] = radic3

radic10038161003816100381610038161198866

1003816100381610038161003816

radic10038161003816100381610038161198876

1003816100381610038161003816

[

1 minus 1205901

0

1205831

1] [

1199061

1199062

] (18)


Substituting (18) into (17a) (17b) (17c) (17d) and can-celing nonlinear terms including the parameter 120590

1yield the

unfolding as follows

1= 1199062 (19a)

2= minus120583119906

1minus 12058331199062+ 12057811199063

1

+ 119886611990611198682

(19b)

119868 = minus1205832

119868 minus 1198912

sin 120574 (19c)

119868 120574 = 1205902119868 + 11988661199062

1

119868 + 12057221198683

minus 1198912

cos 120574 (19d)

where 120583 = 1205832

1

minus 1205901(1 minus 120590

1) 1205833= 21205831

1205781= 91198866119886721198876and

1205722= (12)119887

7

The scale transformations to be introduced into (19a)(19b) (19c) (19d) are

1205832

997888rarr 1205761205832

1205833997888rarr 120576120583

3 119891

2

997888rarr 1205761198912

1205781997888rarr 1205781 120572

2997888rarr 1205722 119886

6997888rarr 1198866

(20)

Then normal form (19a) (19b) (19c) (19d) can berewritten in the form with the perturbations

1=

120597119867

1205971199062

+ 1205761198921199061= 1199062 (21a)

2= minus

120597119867

1205971199061

+ 1205761198921199062= minus120583119906

1+ 12057811199063

1

+ 119886611990611198682

minus 12057612058331199062 (21b)

119868 =

120597119867

120597120574

+ 120576119892119868

= minus1205761205832

119868 minus 1205761198912

sin 120574 (21c)

119868 120574 = minus

120597119867

120597119868

+ 120576119892120574

= 1205902119868 + 12057221198683

+ 11988661198681199062

1

minus 1205761198912

cos 120574 (21d)

where the Hamiltonian function119867 is of the form

119867(1199061 1199062 119868 120574) =

1

2

1199062

2

+

1

2

1205831199062

1

minus

1

4

12057811199064

1

minus

1

2

11988661198682

1199062

1

minus

1

2

12059021198682

minus

1

4

12057221198684

(22)

and 1198921199061 1198921199062 119892119868 and 119892

120574 are the perturbation terms inducedby the dissipative effects

1198921199061= 0 119892

1199062= minus12058331199062

119892119868

= minus1205832

119868 minus 1198912

sin 120574 119892120574

= minus1198912

cos 120574(23)

4 Heteroclinic Bifurcations ofUnperturbed System

In this section we focus on studying the nonlinear dynamicscharacteristic of the unperturbed system When 120576 = 0 itcan be known that system from (21a) (21b) (21c) (21d) isan uncoupled two-degree-of-freedom nonlinear system Thevariable 119868 appears in the subspace (119906

1 1199062) of (21a) (21b)

(21c) (21d) as a parameter since 119868 = 0 Consider the first twodecoupled equations of (21a) (21b) (21c) (21d)

1= 1199062 (24a)

2= minus120583119906

1+ 12057811199063

1

+ 119886611990611198682

(24b)

Since 1205781

gt 0 (24a) (24b) can exhibit the heteroclinicbifurcations It is obvious from (24a) (24b) that when 120583 minus

11988661198682

lt 0 the only solution to (24a) (24b) is the trivial zerosolution (119906

1 1199062) = (0 0) which is the saddle point On the

curve defined by 120583 = 11988661198682 that is

11986812

= plusmn[

1205832

1

minus 1205901(1 minus 120590

1)

1198866

]

12

(25)

the trivial zero solution bifurcates into three solutionsthrough a pitchfork bifurcation which are given by 119902

0= (0 0)

and 119902plusmn(119868) = (119861 0) respectively where

119861 = plusmn

1

1205781

[1205832

1

minus 1205901(1 minus 120590

1) minus 11988661198682

]

12

(26)

From the Jacobian matrix evaluated at the nonzerosolutions it can be found that the singular point 119902

0is the

center point and the singular points 119902plusmn(119868) are saddle points It

is observed that 119868 and 120574 actually represent the amplitude andphase of vibrations Therefore we assume that 119868 ge 0 and (25)becomes

1198681= 0 119868

2= [

1205832

1

minus 1205901(1 minus 120590

1)

1198866

]

12

(27)

such that for all 119868 isin [1198681 1198682] (24a) (24b) have two hyperbolic

saddle points 119902plusmn(119868) which are connected by a pair of

heteroclinic orbits 119906ℎplusmn

(1198791 119868) that is lim

1198791rarrplusmninfin

119906ℎ

plusmn

(1198791 119868) =

119902plusmn(119868) Thus in the full four-dimensional phase space the set

defined by

119872 = (119906 119868 120574) | 119906 = 119902plusmn(119868) 1198681lt 119868 lt 119868

2 0 le 120574 lt 2120587 (28)

is a two-dimensional invariant manifoldFrom the results obtained by Feng et al [9ndash11] it is

known that two-dimensional invariant manifold 119872 is nor-mally hyperbolic The two-dimensional normally hyperbolicinvariant manifold 119872 has the three-dimensional stable andunstable manifolds represented as 119882

119904

(119872) and 119882119906

(119872)respectively The existence of the heteroclinic orbit of (24a)(24b) to 119902

plusmn(119868) = (119861 0) indicates that 119882119904(119872) and 119882

119906

(119872)

intersect nontransversally along a three-dimensional mani-fold denoted by Γ which can be written as

Γ = (119906 119868 120574) | 119906 = 119906ℎ

plusmn

(1198791 119868) 1198681lt 119868 lt 119868

2

120574 = int

1198791

0

119863119868119867(119906ℎ

plusmn

(1198791 119868) 119868) 119889119904 + 120574

0

(29)

We analyze the dynamics of the unperturbed system of(21a) (21b) (21c) (21d) restricted to 119872 Considering theunperturbed system of (21a) (21b) (21c) (21d) restricted to119872 yields

119868 = 0 (30a)

119868 120574 = 119863119868119867(119902plusmn(119868) 119868) 119868

1lt 119868 lt 119868

2 (30b)


where

119863119868119867(119902plusmn(119868) 119868) = minus

120597119867 (119902plusmn(119868) 119868)

120597119868

= 1205902119868 + 12057221198683

+ 11988661198681199062

1

(31)

From the results obtained by Feng et al [9ndash11] it is knownthat if 119863

119868119867(119902plusmn(119868) 119868) = 0 119868 = constant is called a periodic

orbit and if 119863119868119867(119902plusmn 119868) = 0 119868 = constant is known as a

circle of the singular points Any value of 119868 isin [1198681 1198682] at which

119863119868119867(119902plusmn 119868) = 0 is a resonant value 119868 and these singular points

are resonant singular points We denote a resonant value by119868119903such that

119863119868119867(119902plusmn 119868) = 120590

2119868119903+ 12057221198683

119903

+ 11988661198681199031199062

1

= 0 (32)

Then we obtain

119868119903= plusmn

12059021205781+ 1198866[1205832

1

minus 1205901(1 minus 120590

1)]

1198862

6

minus 12057221205781

12

(33)

The geometric structure of the stable and unstable mani-folds of 119872 in the full four-dimensional phase space for theunperturbed system of (21a) (21b) (21c) (21d) is given inFigure 2 Since 120574 represents the phase of the oscillationswhen119868 = 119868119903 the phase shift Δ120574 of oscillations is defined by

Δ120574 = 120574 (+infin 119868119903) minus 120574 (minusinfin 119868

119903) (34)

The physical interpretation of the phase shift is the phasedifference between the two end points of the orbit In thesubspace (119906

1 1199062) there exists a pair of heteroclinic orbits

connecting to saddle points Therefore the homoclinic orbitin the subspace (119868 120574) is in fact a heteroclinic connecting inthe full four-dimensional space (119906

1 1199062 119868 120574) The phase shift

denotes the difference of the value 120574 when a trajectory leavesand returns to the basin of attraction of 119872 We will use thephase shift in subsequent analysis to obtain the condition forthe existence of the Shilnikov typemultipulse orbitThephaseshift will be calculated later in the heteroclinic orbit analysis

We consider the heteroclinic orbits of (24a) (24b) Let1205761= 120583 minus 119886

61198682 and 120583

3= 1205762 (24a) (24b) can be rewritten as

1= 1199062 (35a)

2= minus12057611199061+ 12057811199063

1

minus 12057612057621199062 (35b)

Set 120576 = 0 (35a) (35b) is a system with the Hamiltonianfunction

119867(1199061 1199062) =

1

2

1199062

2

+

1

2

12057611199062

1

minus

1

4

12057811199064

1

(36)

When 119867 = 0 there is a heteroclinic loop Γ0 which

consists of the two hyperbolic saddle points 119902plusmn(119868) and a pair

of heteroclinic orbits 119906plusmn(1198791) In order to calculate the phase

shift and the extended Melnikov function it is necessary to

obtain the equations of a pair of heteroclinic orbits which aregiven as follows

1199061(1198791) = plusmnradic

1205761

1205781

tanh(

radic21205761

2

1198791) (37a)

1199062(1198791) = plusmn

1205761

radic21205781

sech2 (radic21205761

2

1198791) (37b)

We turn our attention to the computation of the phaseshift Substituting the first equation of (37a) (37b) into thefourth equation of the unperturbed system of (21a) (21b)(21c) (21d) and integrating yield

120574 (1198791) = 1205961199031198791minus

1198866radic21205761

1205781

tanh(

radic21205761

2

1198791) + 1205740 (38)

where 120596119903= 1205902+ 12057221198682

+ 119886612057611205781

At 119868 = 119868119903 there is 120596

119903equiv 0 Therefore the phase shift may

be expressed as

Δ120574 = [minus

21198866radic21205761

1205781

]

119868=119868119903

= minus

21198866

1205781

radic2 [1205832

1

minus 1205901(1 minus 120590

1) minus 11988661198682

119903

]

(39)

5 Existence of Multipulse Orbits

After obtaining detailed information on the nonlineardynamic characteristics of the subspace (119906

1 1199062) for the

unperturbed system from (21a) (21b) (21c) (21d) the nextstep is to examine the effects of small perturbation terms (0 lt

120576 ≪ 1) on the unperturbed system from (21a) (21b) (21c)(21d) The extended Melnikov method developed by Kovacicet al [15ndash17] is utilized to discover the existence of the multi-pulse orbits and chaotic dynamics of the nonlinear vibrationfor the laminated composite piezoelectric rectangular plateWe start by studying the influence of such small perturbationson themanifold119872The objective of the research is to identifythe parameter regions where the existence of the multipulseorbits is possible in the perturbed phase space The main aimis to verify whether these parameters satisfy the transversalitycondition of multipulse chaotic dynamics It will be shownthat thesemultipulse orbits can occur in theHamilton systemwith dissipative perturbations if the parameters meet thetransversality condition The existence of such multipulseorbits provides a robust mechanism for the existence ofthe complicated dynamics in the perturbed system In thissection the emphasis is put on the application aspects of theextended Melnikov method to (21a) (21b) (21c) (21d)

51 Dissipative Perturbations of the Homoclinic Loop Weanalyze dynamics of the perturbed system and the influenceof small perturbations on 119872 Based on the analysis byKovacic et al [15ndash17] we know that 119872 along with its stableand unstable manifolds is invariant under small sufficientlydifferentiable perturbations It is noticed that 119902

plusmn(119868) in (24a)


M

I

120574

times

u2

u1

(a)

0 1205742120587

I = I1

I = Ir

I = I2

(b)

Figure 2 The geometric structure of manifolds119872119882119904(119872) and119882119906

(119872) is given in the full four-dimensional phase space

(24b) maintains the characteristic of the hyperbolic singularpoint under small perturbations in particular 119872 rarr 119872

120576

Therefore we obtain

119872 = 119872120576= (119906 119868 120574) | 119906 = 119902

plusmn(119868) 1198681lt 119868 lt 119868

2 0 le 120574 lt 2120587

(40)

Considering the last two equations of (21a) (21b) (21c)(21d) yields

119868 = minus1205832

119868 minus 1198912

sin 120574 (41a)

120574 = 1205902+ 12057221198682

+ 11988661199062

1

minus

1198912

cos 120574119868

(41b)

It is known from the above analysis that the last two equationsof (21a) (21b) (21c) (21d) are of a pair of pure imaginaryeigenvalues Therefore the resonance can occur in (41a)(41b) Also introduce the scale transformations

1205832

997888rarr 1205761205832

119868 = 119868119903+ radic120576ℎ

1198912

997888rarr 1205761198912

1198791997888rarr

1198791

radic120576

(42)

Substituting the above transformations into (41a) (41b)yields

ℎ = minus120583

2

119868119903minus 1198912

sin 120574 minus radic1205761205832

ℎ (43a)

120574 = minus

2120575

1205781

119868119903ℎ minus radic120576(

120575

1205781

ℎ2

+

1198912

119868119903

cos 120574) (43b)

where 120575 = 1198862

6

minus 12057221205781

When 120576 = 0 (43a) (43b) become

ℎ = minus120583

2

119868119903minus 1198912

sin 120574 (44a)

120574 = minus

2120575

1205781

119868119903ℎ (44b)

The unperturbed system from (44a) (44b) is a Hamiltonsystem with the function

119863(ℎ 120574) = minus120583

2

119868119903120574 + 1198912

cos 120574 +

120575

1205781

119868119903ℎ2

(45)

The singular points of (44a) (44b) are given as

1198750= (0 120574

119888) = (0 minus arcsin(

1205832

119868119903

1198912

))

1198760= (0 120574

119904) = (0 120587 + arcsin(

1205832

119868119903

1198912

))

(46)

Based on the characteristic equations evaluated at the twosingular points119875

0and119876

0 we can know the stabilities of these

singular points Therefore it is known that the singular point1198750is a center pointThe singular point119876

0is a saddle which is

connected to itself by a homoclinic orbit The phase portraitof system for (44a) (44b) is shown in Figure 3(a)

It is found that for the sufficiently small parameter 120576the singular point 119876

0remains a hyperbolic singular point

119876120576of the saddle stability type For small perturbations the

singular point 1198750becomes a hyperbolic sink 119875

120576 The phase

portrait of the perturbed system from (43a) (43b) is depictedin Figure 3(b)

Using the function (45) at ℎ = 0 and substituting 120574119904in

(46) into (45) the estimate of the basin of the attractor for120574min is obtained as

120574min minus

1198912

1205832

119868119903

cos 120574min = 120587 + arcsin1205832

119868119903

1198912

+

radic119891

2

2

minus 1205832

2

1198682

119903

1205832

119868119903

(47)

Define an annulus 119860120576near 119868 = 119868

119903as

119860120576= (1199061 1199062 119868 120574) | 119906

1= 119861 119906

2= 0

1003816100381610038161003816119868 minus 119868119903

1003816100381610038161003816lt radic120576119862 120574 isin 119879

119871

(48)

where 119862 is a constant and is sufficiently large so that theunperturbed orbit is enclosed within the annulus

It is noticed that the three-dimensional stable and unsta-ble manifolds of 119860

120576 denoted as119882119904(119860

120576) and119882

119906

(119860120576) are the

subsets of the manifolds 119882119904(119872120576) and 119882

119906

(119872120576) respectively

We will indicate that for the perturbed system the saddlefocus 119875

120576on 119860120576has the multipulse orbits which come out

of the annulus 119860120576and can return to the annulus in the full

four-dimensional spaceThese orbits which are asymptotic tosome invariant manifolds in the slow manifold119872

120576 leave and


h

120574

P0

120574min 120574c 120574s

Q0

(a)

h

120574Q120576

P120576

(b)

Figure 3 Dynamics on the normally hyperbolic manifold is described (a) the unperturbed case (b) the perturbed case

enter a small neighborhood of 119872120576multiple times and finally

return and approach an invariant set in119872120576asymptotically as

shown in Figure 4 In Figure 4 this is an example of the three-pulse jumping orbit which depicts the formation mechanismof the multipulse orbits

52 The 119896-Pulse Melnikov Function Most researchers fo-cused on Shilnikov type single-pulse chaotic dynamics ofthe high-dimensional nonlinear systems from the thin platestructures in the past There exist multipulse chaotic dynam-ics in the practical engineering systems The extended Mel-nikov method is a kind of theory which can be used to inves-tigate the multipulse jumping orbits in the high-dimensionalnonlinear systems Since the theory on multipulse chaoticdynamics is very esoteric and abstract it is difficult to beextended to solve the engineering problems Up to nowfew researchers have made use of the extended Melnikovmethod to study the Shilnikov type multipulse homoclinicand heteroclinic bifurcations and chaotic dynamics of high-dimensional nonlinear systems in engineering applications

The extended Melnikov method was first presented byKovacic et al [15ndash17] which is an extension of the global per-turbation method developed by Feng et al [9ndash11] Camassaet al [17] gave the detailed procedure of mathematical proofon the extended Melnikov method which unifies severaldisjoint perturbation theoretical methods This method canbe also utilized to detect the Shilnikov typemultipulse homo-clinic or heteroclinic orbits to the slow manifolds of four-dimensional near-integrable Hamilton systems or higher-dimensional nonlinear systems The extended Melnikovfunction is different from the usual Melnikov function anddescribes slow dynamics of the multipulse orbits on thehyperbolic manifold

The key of the extended Melnikov method is how tocalculate the extended Melnikov methodThe extended Mel-nikov function is computed by a recursion procedure fromthe usual 1-pulseMelnikov function and depends on the smallperturbation parameter 120576 through a logarithmic functionwhich calculates the asymptotic in the particularly delicatesmall 120576 limit In this paper the extended Melnikov function

120574

|u|

h

120574c 120574c + 3Δ120574

p120576qc

Figure 4 The Shilnikov type three-pulse orbits are obtained

can be simplified in the resonant case and does not dependon the perturbation parameter We have used the extendedMelnikovmethod to investigate heteroclinic bifurcations andmultipulse chaotic dynamics of the laminated compositepiezoelectric rectangular plate

We use the extended Melnikov method described byKovacic et al [15ndash17] to find the Shilnikov type multipulseorbits for nonlinear vibration for the laminated compositepiezoelectric rectangular plate We search for the multipulseexcursions to find the nondegenerate zeroes of the extendedMelnikov function119872

119896(120576 119868 120574

0 1205832

) with the certain combina-tion of parameters 120576 119868 120574

0 and 120583

2

which we name the 119896-pulseMelnikov function

It is important to obtain the detailed expression of the 119896-pulse Melnikov function We compute the 1-pulse Melnikovfunction based on the formula obtained by Kovacic et al[15ndash17] at the resonant case 119868 = 119868

119903 The 1-pulse Melnikov

function 1198721(120576 119868119903 1205740 1205832

) coincides with the standard Mel-nikov function 119872(119868

119903 1205740 1205832

) The 1-pulse Melnikov function119872(119868119903 1205740 1205832

) on both heteroclinic manifolds 119882119904

(119872) and119882119906

(119872) is given as follows

119872(119868119903 1205740 1205832

1205781 1198866 1205761)

= int

+infin

minusinfin

⟨n (119901ℎ

(119905)) g (119901ℎ (119905) 1205832

0)⟩ 1198891198791


= int

+infin

minusinfin

(

120597119867

1205971199061

1198921199061+

120597119867

1205971199062

1198921199062+

120597119867

120597119868

119892119868

+

120597119867

120597120574

119892120574

)1198891198791

= minus

2radic21205833

31205781

12057632

1

minus 2radic211988661205832

1198682

119903

12057612

1

1205781

minus 1198912

119868119903[cos(120574

0minus 1198866

radic21205761

1205781

) minus cos(1205740+ 1198866

radic21205761

1205781

)]

(49)

Based on the results given by Kovacic et al [15ndash17] it is known that the 119896-pulse Melnikov function119872119896(120576 119868119903 1205740 1205832

) (119896 = 1 2 ) is defined as

119872119896(120576 119868119903 1205740 1205832

)

=

119896minus1

sum

119895=0

119872(119868119903 119895Δ120574 (119868

119903) + Γ119895(120576 119868119903 1205740 1205832

) + 1205740 1205832

)

(50)

where

Γ119895(120576 119868119903 1205740 1205832

) =

Ω (1199090(119868119903) 119868119903)

120582 (119868119903)

119895

sum

119903=1

log100381610038161003816100381610038161003816100381610038161003816

120589 (119868119903)

120576119872119903(120576 119868119903 1205740 1205832

)

100381610038161003816100381610038161003816100381610038161003816

(51)

for 119895 = 1 119896 minus 1 and Γ0(120576 119868119903 1205740 1205832

) = 0It is noticed that the angle function Γ

119895(120576 119868119903 1205740 1205832

) isthe complex formula where 119872

119896(120576 119868119903 1205740 1205832

) appears as theargument of a logarithm When resonance occurs theperiodic orbit corresponding to the value 119868

119903degenerates

into a circle of equilibria Under this case there existsΓ119895(120576 119868119903 1205740 1205832

) = 0 (119895 = 0 1 119896 minus 1) Based on theexpression obtained by Kovacic et al [15ndash17] the 119896-pulseMelnikov function can be written as follows

119872119896(119868119903 1205740 1205832

1205781 1198866 1205761)

=

119896minus1

sum

119895=0

119872(119868119903 1205740+ 119895Δ120574 (119868

119903) 1205832

1205781 1198866 1205761)

= minus1198912

119868119903[cos(120574

0minus

1198866radic21205761

1205781

)

minus cos(1205740+

1198866radic21205761

1205781

)]

minus

2radic21205833

31205781

12057632

1

minus 2radic211988661205832

1198682

119903

12057612

1

1205781

minus 1198912

119868119903[cos(120574

0minus

1198866radic21205761

1205781

minus

21198866radic21205761

1205781

)

minus cos(1205740+

1198866radic21205761

1205781

minus

21198866radic21205761

1205781

)]

minus

2radic21205833

31205781

12057632

1

minus 2radic211988661205832

1198682

119903

12057612

1

1205781

+ sdot sdot sdot

minus 1198912

119868119903[cos(120574

0minus

1198866radic21205761

1205781

minus 2 (119896 minus 1) 1198866

radic21205761

1205781

)

minus cos(1205740+

1198866radic21205761

1205781

minus 2 (119896 minus 1)

1198866radic21205761

1205781

)]

minus

2radic21205833

31205781

12057632

1

minus 2radic211988661205832

1198682

119903

12057612

1

1205781

= minus1198912

119868119903[cos(120574

0minus

1198866radic21205761

1205781


1198866radic21205761

1205781

)

minus cos(1205740+

1198866radic21205761

1205781

)]

minus

2radic21198961205833

31205781

12057632

1

minus 2radic211989611988661205832

1198682

119903

12057612

1

1205781

(52)

If we set Δ120574 = minus21198866(radic212057611205781) and 120574

119896minus1= 1205740+ (119896 minus

1)(Δ1205742) (52) can be rewritten as follows

119872119896(119868119903 1205740 1205832

1205781 1198866 1205761)

= 119872119896(119868119903 120574119896minus1

minus (119896 minus 1)

Δ120574

2

1205832

1205781 1198866 1205761)

= minus1198912

119868119903[cos(120574

119896minus1+

1

2

119896Δ120574)

minus cos(120574119896minus1

minus

1

2

119896Δ120574)]

+

11989612058331205761

31198866

Δ120574 + 1205832

1198682

119903

(119896Δ120574)

= 21198912

119868119903sin 120574119896minus1

sin(

1

2

119896Δ120574)

+

12058331205761

31198866

(119896Δ120574) + 1205832

1198682

119903

(119896Δ120574)

(53)

Based on Proposition 31 given by Kovacic et al [15ndash17]the nonfolding condition is always satisfied in the resonantcase We obtain the following two conditions

10038161003816100381610038161003816100381610038161003816100381610038161003816

(12) 119896Δ120574

sin ((12) 119896Δ120574)

(12058331205761+ 311988661205832

1198682

119903

)

311988661198912

119868119903

10038161003816100381610038161003816100381610038161003816100381610038161003816

lt 1

1

2

119896Δ120574 = 119899120587 119899 = 0 plusmn1 plusmn2

(54)

The main aim of the following analysis focuses onidentifying simple zeroes of the 119896-pulse Melnikov functionDefine a set that contains all such simple zeroes to be

119885119899

minus

= (119868119903 120574119896minus1

1205832

1205781 1198866 1205761) | 119872119896= 0119863

1205740

119872119896

= 0 (55)


The 119896-pulse Melnikov function has two simple zeroes in theinterval 120574

119896minus1isin [0 120587]

120574119896minus11

= minus arcsin(12) 119896Δ120574

sin ((12) 119896Δ120574)

(12058331205761+ 311988661205832

1198682

119903

)

(311988661198912

119868119903)

120574119896minus12

= 120587 minus 120574119896minus11

(56)

53 Geometric Structure of Multipulse Orbits Based on theaforementioned analysis we obtain the following conclu-sions When the parameters of 119896 120583

3 1205761 1205832

1198866 and 119891

2

satisfy condition (54) the 119896-pulse Melnikov function (53)has simple zeroes at 120574

119896minus1= 120574119896minus11

and 120574119896minus1

= 120574119896minus12

=

120587 minus 120574119896minus11

For 119894 = 1 or 119894 = 2 when the 119895-pulse Melnikovfunction 119872

119895(119868119903 1205740119894

1205832

1205781 1198866 1205761) has no simple zeroes the

stable and unstable manifolds119882119904(119872120576) and119882

119906

(119872120576) intersect

transversely along a symmetric pair of the two-dimensional119896-pulse surfaces sum

1205832120578111988661205761

plusmn120576(120574119896minus1119894

) This signifies that thepresence of the Shilnikov type 119899-pulse orbits leads to chaoticdynamics in the sense of the Smale horseshoes for thenonlinear motion for the laminated composite piezoelectricrectangular plate In the phase space of the unperturbedsystem from (21a) (21b) (21c) (21d) this symmetric pair ofthe two-dimensional 119896-pulse surfaces breaks down smoothlyonto a pair of limiting 119896-pulse surfaces sum1205832 12057811198866 1205761

plusmn0

(120574119896minus1119894

)parametrized by (37a) (37b) and (38) with 119868 = 119868

119903 1205740

=

120574119896minus1119894

minus (119896 minus 1)(Δ1205742) + 119895Δ120574 and an arbitrary ℎ The sign in(49) is determined by the sign of the corresponding 119895-pulseMelnikov function119872

119895(119868119903 1205740119894

1205832

1205781 1198866 1205761)

From the discussion given by Kovacic et al [15ndash17] itis easily found that for 120574

0119894

= 120574119896minus1119894

minus (119896 minus 1)(Δ1205742) + 119895Δ120574

(119894 = 1 or 119894 = 2) the values of the 119895-pulse Melnikovfunctions 119872

119895(119868119903 1205740119894

1205832

1205781 1198866 1205761) are not zero for all 119895 =

1 119896 minus 1 and all 119895 have the same sign It is knownthat this sign is negative for 120574

01

and positive for 12057402

Therefore the 119896-pulse heteroclinic surfaces sum1205832 12057811198866 1205761

plusmn120576(120574119896minus11

)

and sum1205832120578111988661205761

plusmn120576(120574119896minus12

) indeed exist and the limiting 119896-pulsesurfaces sum

1205832120578111988661205761

plusmn0

(120574119896minus11

) and sum1205832120578111988661205761

plusmn0

(120574119896minus12

) also existwhen 120576 = 0 Since the regions enclosed by the stable andunstable heteroclinic manifolds119882119904(119872) and119882

119906

(119872) are bothconvex and the normal vector

n = ((minus1205831199061+ 12057811199063

1

+ 11988661198682

1199061) minus1199062 0 0) (57)

is known to point out of these manifolds it demonstratesthat the orbits forming each of the surfaces sum1205832 12057811198866 1205761

plusmn0

(120574119896minus11

)

are parametrized by (37a) (37b) and (38) with the alter-nating signs and the orbits forming each of the surfacessum1205832120578111988661205761

plusmn0

(120574119896minus12

) are parametrized by (37a) (37b) and (38)with the same signs

For the parameter 1205832

= 120583 there exist119873minus1 orbit segments119874119894(120583) (119894 = 2 119873) on the annulus119872 where the end points

of the segments 119874119894(120583) are 119889

119894(120583) and 119888

119894(120583) respectively The

trajectories of (44a) (44b) on the segments119874119894(120583) travel from

the end points 119889119894(120583) to 119888

119894(120583) in forward time Therefore the

end points 119889119894(120583) and 119888

119894(120583) are respectively referred to as

the departure and landing points of the heteroclinic jumpingΓ119894 In addition the line 120574 = 120574

0119894

(119868119903 120583) minus Δ120574

minus

(119868119903) transversely

intersects the segments 119874119894(120583) at the end point 119888

119894(120583) for 119894 =

2 119873 while the line 120574 = 1205740119894

(119868119903 120583) + Δ120574

+


intersects the segments119874119894+1

(120583) at the end point 119889119894+1

(120583)when119894 = 1 119873 minus 1 For all 119894 = 2 119873 minus 1 the difference inthe coordinates ℎ of two end points 119888

119894(120583) and 119889

119894+1(120583) is zero

namely

ℎ (119888119894(120583)) minus ℎ (119889

119894+1(120583)) = 0 (58)

For each 119894 = 2 119873 minus 1 one of the heteroclinic orbitsrepresented by Γ

119894and contained in the limiting surfaces

sum1205832120578111988661205761

0

(1205740119894

) at the value 120583 = 120583 connects two intersectionpoints 119888

119894(120583) and 119889

119894+1(120583) Therefore a heteroclinic orbit Γ

1

on the limiting surfaces sum1205832 12057811198866 12057610

(12057401

) connects the certainpoint 119888

1(120583) on the annulus 119872 to the end point 119889

2(120583) on the

segment 1198742(120583) It is also known that a heteroclinic orbit Γ

119873

on the limiting surfaces sum1205832120578111988661205761

0

(1205740119873

) connects the endpoint 119888

119873(120583) on the segments 119874

119873(120583) to the certain point

119889119873+1

(120583) on the annulus119872 According to the study of Kovacicet al [15ndash17] there exists an 119899-bump singular transitionorbit or a modified 119873-bump singular transition orbit The3-bump jumping orbit depicted in Figure 5 consists of theheteroclinic orbits Γ

119894( 119894 = 1 2 3) on the limiting surfaces

sum1205832120578111988661205761

0

(1205740119894

) (119894 = 1 2 3) at the parameter 120583 = 120583 and theorbit segments 119874

1(120583) and 119874

2(120583) of (44a) (44b) It is known

from the above analysis that the orbit segments 119874119894(120583) (119894 =

2 119873) intersect transversely with the lines 120574 = 1205740119894

(119868119903 120583) +

Δ120574+

(119868119903) and 120574 = 120574

0119894

(119868119903 120583) minus Δ120574

minus

(119868119903)

The 2-bump singular surface shown in Figure 6 is com-posed of two single-pulse singular intersection surfacessum1205832120578111988661205761

0

(120574119896minus11

) andsum1205832120578111988661205761

0

(120574119896minus12

)This surface connectsthe singular points of (44a) (44b) that lie on the line 120574 = 120574

01

minus

Δ120574minus to those of (44a) (44b) that lie on the line 120574 = 120574

01

minusΔ120574+

on the annulus119872We obtain a countable infinity of the singular heteroclinic

jumping orbits as follows Each orbit starts along one branchof the manifold 119882(119876

0) of the saddle 119876

0on the annulus 119872

Then the singular heteroclinic jumping orbit departs fromthe annulus 119872 goes along one of the singular 119896-pulse orbitsΓ119896 and lands back at a point on the separatrix that connects

the saddle1198760to itself on the annulus119872 After traveling along

the separatrix for a while the singular heteroclinic jumpingorbit takes off again along the singular 119897-pulse orbit Γ

119897and

continues such process Eventually the singular heteroclinicjumping orbit lands back on the separatrix

Therefore it is concluded that the multipulse orbits of(21a) (21b) (21c) (21d) consist of several portions of the slowtime scale on the hyperbolic manifold119872

120576andmany fast time

scale heteroclinic pulses leaving from the manifold 119872120576 and

these multipulse heteroclinic orbits form a consecutive andrecurrence process

6 Numerical Results of Chaotic Motions

Based on the above qualitative analysis for the multipulseorbits and chaotic dynamics of the laminated composite


120574

|u|h

Γ3

Γ2

Γ1

M

O2O3

12057403 + Δ120574+ 12057402 + Δ120574+ 12057401 + Δ120574+12057403 minus Δ120574minus 12057402 minus Δ120574minus 12057401 minus Δ120574minus

Figure 5 The 3-bump orbit with the single-pulse is depicted

piezoelectric rectangular plate the conditions of the chaoticmotion under the sense of the Smale horses are obtainedTheheteroclinic bifurcations of (12a) (12b) (12c) (12d) appearwhen 120578

1gt 0 Therefore the above theoretical analysis

is focused on the situation which there exist heteroclinicbifurcations in (12a) (12b) (12c) (12d) The parameter 120578

1

is the combination of the parameters 1198866 1198867 and 119887

6 where

1205781

= (911988661198867)21198876 In this section we have only performed

numerical simulations of the multipulse chaotic motionsof the laminated composite piezoelectric rectangular plateunder heteroclinic bifurcations in order to further verify thetheoretical analysis Consequently the parameters 119886

6 1198867 and

1198876are chosen to satisfy 120578

1gt 0

We choose the averaged equation (12a) (12b) (12c) (12d)to conduct numerical simulations A numerical approachthrough the computer software Matlab is utilized to explorethe existence of the Shilnikov type multipulse chaoticmotions in the laminated composite piezoelectric rectangularplate Based on the above qualitative analysis it is found thatthe damping coefficients 120583

1 1205832and transverse excitation 119891

2

play an important role in the multipulse chaotic motionsof the laminated composite piezoelectric rectangular plateIn addition the parameters 119886

2and 119886

3are related to the in-

plane excitation in the119909-direction and the in-plane excitationin the 119910-direction respectively The parameter 119886

4is the

piezoelectric excitation which reflects the characteristics ofthe piezoelectric material Hence the parameters 120583

1 1198862 1198864

and 1198912are selected as the controlling parameters to discover

the law for complicated nonlinear dynamics of the laminatedcomposite piezoelectric rectangular plate

We begin to draw bifurcation diagrams of the parameters1198912 1205831 1198862 and 119886

4 Bifurcation diagrams describe the vibration

law of the modal displacements 1199091and 119909

3 respectively when

the parameters 1198912 1205831 1198862 and 119886

4change in a certain region

We draw bifurcation diagrams according to the rules of theRunge-Kutta algorithm and the Poincare map theory For theperiodic motions Poincare map is of several separate pointsFor a chaotic motion the Poincare map consists of a numberof points on the limited Poincare sectionTherefore it can beobserved that chaotic motion and periodic motion of nonlin-ear systemappear frombifurcation diagramsThe chaotic and

120574

|u|h

M

Γsum (12057401 12057402)

12057402 + Δ120574+ 12057401 + Δ120574+

12057402 minus Δ120574minus 12057401 minus Δ120574minus

Figure 6 The 2-pulse singular surfaces sum(12057401

12057402

) are depicted

periodic responses can be identified by several conventionalcriteria such as phase portraits and Poincare map Basedon the response law of bifurcation diagrams phase portraitsand Poincare map are utilized to further verify the existenceof the chaotic motions and the periodic motions In orderto compare the influence of these parameters 119891

2 1205831 1198862

and 1198864on nonlinear vibration in the laminated composite

piezoelectric rectangular plate we choose the same initialconditions to carry out numerical simulation

Figure 7 illustrates the bifurcation diagram of the lam-inated composite piezoelectric rectangular plate when theexcitation 119891

2varies in the interval 119891

2= 2 sim 200

Other parameters and initial conditions are chosen as 1205901=

183 1205902

= 197 1205831

= 02 1205832

= 02 1198862

= 230 1198863

=

120 1198864= 130 119886

5= minus101 119886

6= minus203 119886

7= minus205 119887

6=

407 1198877= minus308 119887

8= 109 119909

10= minus001 119909

20= minus005 119909

30=

minus001 11990940

= minus001 Figures 7(a) and 7(b) represent thebifurcation diagram on the plane (119909

1 1198912) and (119909

3 1198912) respec-

tively It is observed from Figure 7 that the excitation 1198912

is an important parameter that influences on the nonlineardynamic responses of the laminated composite piezoelectricrectangular plate Figure 7 shows that the chaotic motionof the laminated composite piezoelectric rectangular plateappears first followed by a periodic motion of that With theincrease of excitation 119891

2 Figure 7 presents the following law

chaotic motionrarrmulti-period motionWe study the impact of the damping parameter on the

nonlinear dynamic responses of the laminated compositepiezoelectric rectangular plate Figure 8 is the bifurcationdiagramof the laminated composite piezoelectric rectangularplate with the damping coefficient 120583

1 The figure demon-

strates that system is beginning to enter into the region ofthe chaoticmotion then appears the periodicmotionwindowand finally comes into the region of the chaotic motion againas the damping coefficient 120583


1=

001 sim 07 Other parameters and initial conditions are thesame as those in Figure 7 when excitation is chosen as 119891

2=

827 Figures 8(a) and 8(b) describe the nonlinear motion ofthe laminated composite piezoelectric rectangular plate on


50 100 150minus2

minus15

minus1

minus05

0

05

1

x1

f2

(a)

20 40 60 80 100 120 140 160 180

minus3

minus25

minus2

minus15

x3

f2

(b)

Figure 7 The bifurcation diagram is obtained for the excitation 1198912

= 2sim200 and initial conditions 11990910

= minus001 11990920

= minus005 11990930

=

minus001 11990940

= minus001 (a) the bifurcation diagram on the plane (1199091

1198912

) (b) the bifurcation diagram on the plane (1199093

1198912

)

0 01 02 03 04 05 06 07minus25

minus2

minus15

minus1

minus05

0

05

1

15

x1

1205831

(a)

0 01 02 03 04 05 06 07minus32

minus3

minus28

minus26

minus24

minus22

minus2

minus18

x1

1205831

x3

1205831

x3

(b)

Figure 8 The bifurcation diagram is obtained for the damping coefficient 1205831

= 001sim07 the excitation 1198912

= 827 and initial conditions11990910

= minus001 11990920

= minus005 11990930

= minus001 11990940


1205831

) (b) the bifurcation diagram on theplane (119909

3

1205831

)

the planes (1199091 1205831) and (119909

3 1205831) respectively as well as the

impact of the damping coefficient 1205831on the system

Figure 9 portraysthe bifurcation diagram for the lami-nated composite piezoelectric rectangular plate when the in-plane excitation 119886

2in the 119909-direction varies in the interval

1198862= 2sim65 Other parameters and initial conditions remain

the same as those in Figure 8 when the damping coefficient1205831is selected as 120583

1= 02 Figures 9(a) and 9(b) display

the bifurcation diagram on the plane (1199091 1198862) and (119909

3 1198862)

respectively Figure 9 presents that the beginning movementof the system is the periodic motion then the system appearsthe chaotic motion With the increase of the excitation1198862 Figure 9 shows the following evolution law periodic

motionrarr chaotic motion

Figure 10 indicates the bifurcation diagram for the lam-inated composite piezoelectric rectangular plate when thepiezoelectric excitation 119886

4varies from 119886

4= 2 to 119886

4= 120

Other parameters and initial conditions remain the sameas those in Figure 9 when the in-plane excitation 119886

2is

chosen as 1198862

= 23 Figures 10(a) and 10(b) demonstratethe bifurcation diagram on the planes (119909

1 1198864) and (119909

3 1198864)

respectively It is observed from Figure 10 that the piezoelec-tric excitation 119886

4has a significant influence on the compli-

cated nonlinear dynamic behaviors of the laminated com-posite piezoelectric rectangular plate As the piezoelectricexcitation 119886

4increases Figure 10 reveals the following

law chaotic motionrarrmultiperiod motionrarr one-periodmotionrarrmultiperiod motionrarr chaotic motion


0 10 20 30 40 50 60

minus3

minus2

minus1

0

1

x1

a2

(a)

0 10 20 30 40 50 60minus34

minus32

minus3

minus28

minus26

minus24

minus22

minus2

minus18

a2

x3

(b)

Figure 9 The bifurcation diagram is obtained for the in-plane excitation 1198862

= 2sim65 the damping coefficients 1205831

= 02 and 1205832

= 02 theexcitation 119891

2

= 827 and initial conditions 11990910

= minus001 11990920

= minus005 11990930

= minus001 11990940

= minus001 (a) the bifurcation diagram on the plane(1199091

1198862


1198862

)

0 20 40 60 80 100 120minus25

minus2

minus15

minus1

minus05

0

05

1

x1

a4

(a)

0 20 40 60 80 100 120

minus35

minus3

minus25

minus2

a4

x3

(b)

Figure 10 The bifurcation diagram is obtained for the piezoelectric excitation 1198864


= 827 the damping coefficients1205831

= 02 and 1205832

= 02 the in-plane excitations 1198862

= 23 and 1198863


= minus001 11990920

= minus005 11990930

= minus001 11990940

=

minus001 (a) the bifurcation diagram on the plane (1199091

1198864


1198864

)

Based on the above bifurcation diagram the excitations1198912 1198862 1198864 and the damping coefficient 120583

1are selected as

specific values in order to find themultipulse chaoticmotionsof the laminated composite piezoelectric rectangular plateFigure 11 indicates existence of themultipulse chaotic motionof the laminated composite piezoelectric rectangular platewhen the excitation 119891

2is 827 In this case the chosen

parameters and initial conditions are the same as those inFigure 7 Figures 11(a) and 11(b) are the three-dimensionalphase portrait in the space (119909

1 1199092 1199093) and the Poincare

map on the plane (1199091 1199092) respectively Figure 11 shows that

the excitation 1198912has a noticeable effect on the existence of

the multipulse chaotic motions on the laminated compositepiezoelectric rectangular plate

Besides the excitations 1198912 1198862 1198864and the damping coef-

ficient 1205831 the multipulse chaotic motions of the laminated

composite piezoelectric rectangular plate also depend onother parameters Figure 12 is obtained when the parametersand initial conditions are chosen as 120590

1= 1437 120590

2=

1142 1205831= 02 120583

2= 02 119886

2= 300 119886

3= 750 119886

4= 450

1198865

= minus1166 1198866

= 1227 1198912

= 1227 1198867

= minus268 1198876

=

minus22 1198877

= minus969 1198878

= minus2232 11990910

= minus108 11990920

= 0511990930

= minus001 11990940

= 916 Comparing with Figures 11and 12 it is found that there are differences in the phaseportrait and the Poincare map respectively From thethree-dimensional phase portrait in Figure 12 we can see thatthere exists obvious multipulse jumping phenomenon Thethree-dimensional phase portrait is composed of the four


minus1minus050051

minus05

0

05

minus25

minus2

minus15

minus1

x1

x2

x3

(a)

minus15 minus1 minus05 0 05 1 15 2

minus1

minus05

0

05

1

x1

x2

(b)

Figure 11 The multipulse chaotic motion is obtained when 1198912

= 827 and 1205831

= 02 (a) the phase portrait in the three-dimensional space(1199091

1199092

1199093

) (b) Poincare map on the plane (1199091

1199092

)

minus50

5

minus50

5minus6

minus4

minus2

0

2

4

6

x1x2

x3

(a)

minus10 minus5 0 5 10

minus10

minus5

0

5

10

x1

x2

(b)


= 1437 1205902

= 1142 1198862

= 300 1198863

= 750 1198864

= 450 1198865

= minus1166 1198866

=

1227 1198912

= 1227 1198867

= minus268 1198876

= minus22 1198877

= minus969 1198878

= minus2232 11990910

= minus108 11990920

= 05 11990930

= minus001 11990940

= 916 (a) the phase portrait inthe three-dimensional space (119909

1

1199092

1199093


1199092

)

regionsThe different regions are connected by themultipulseorbits

In the following numerical simulations several differ-ent sets of parameters and initial conditions are givenin order to investigate the different shapes of the multi-pulse chaotic motion Figure 13 demonstrates the multipulsechaotic response in the laminated composite piezoelectricrectangular plate for 119891

2= 9238 Some parameters and

initial conditions are chosen as 1205901= 361 120590

2= 313 119886

5=

minus1501 1198878

= 409 11990910

= minus001 11990920

= minus009 11990930

=

minus005 11990940

= minus005 In this case other parameters are thesame as those in Figure 7 From Figure 13 we can see thatthere is another shape for the multipulse chaotic motion Itis found that the shapes of these two phenomena depicted inFigures 12 and 13 are completely different From the three-dimensional phase portrait in Figure 13 it is found thatmultipulse jumping phenomenon is more prominent

7 Conclusions

In this paper the nonlinear vibrations of the laminated com-posite piezoelectric rectangular plate are studied by applyingthe theories of the global bifurcations and chaotic dynamicsfor high-dimensional nonlinear systems The multipulseheteroclinic orbits and chaotic dynamics are investigatedusing the extended Melnikov method for the case where theaveraged equations have one nonsemisimple double zero anda pair of pure imaginary eigenvaluesThe extendedMelnikovmethod can be applied to study the Shilnikov typemultipulseheteroclinic bifurcations and chaotic dynamics of high-dimensional nonlinear systems in engineering applicationsAnalysis of themultipulse heteroclinic orbits in the laminatedcomposite piezoelectric rectangular plate demonstrates thatsuch an analysis is a typical singular perturbation problemin which there are two different time scales Dynamics onthe hyperbolic manifold 119872

120576are of the slow time scale and


minus50

5

minus10

0

10

minus6

minus4

minus2

0

2

x1x2

x3

(a)

minus5 0 5 10

minus10

minus5

0

5

10

x1

x2

(b)


= 361 1205902

= 313 1198862

= 230 1198863

= 120 1198864

= 130 1198865

= minus1501 1198866

=

minus203 1198912

= 9238 1198867

= minus205 1198876

= 407 1198877

= minus308 1198878

= 409 11990910

= minus001 11990920

= minus009 11990930

= minus005 11990940

= minus005 (a) the phase portrait inthe three-dimensional space (119909

1

1199092

1199093


1199092

)

the multipulse jumping orbits taking off from this manifoldare of the fast time scale It is shown that the transfer ofenergy between the two different modes occurs throughthe multipulse jumping orbits The studies have led to thefollowing conclusions

(1) There exist the Shilnikov type multipulse chaoticmotions in nonlinear vibration of the laminated compositepiezoelectric rectangular plate The geometric interpretationof the 119896-pulse Melnikov function is a signed distance mea-sured along the normal to a heteroclinic manifold whichgives the more delicate local estimates near the hyperbolicmanifold In the resonant case the 119896-pulse extended Mel-nikov function 119872

119896(120576 119868 120574

0 1205832

) does not depend on the smallperturbation parameter 0 lt 120576 ≪ 1 and the nonfolding condi-tion is automatically satisfied resulting in the angle functionΓ119895(120576 119868119903 1205740 1205832

) (119895 = 0 1 119896 minus 1) being zero Therefore thecomputing procedure of the extendedMelnikov function canbe simplified

(2) In order to verify the theoretical predictions numer-ical simulation is used to examine the bifurcations andchaotic motions of the laminated composite piezoelectricrectangular plate Several types of the bifurcation diagramsare obtained when the transverse excitation 119891

2 the in-plane

excitation 1198862 the piezoelectric excitation 119886

4 and the damping

coefficient 1205831are chosen as several different kinds of control

parameters Based on the bifurcation diagrams the nonlinearcomplicated dynamic behavior of the laminated compositepiezoelectric rectangular plate is controlled by varying theexcitations 119891

2 1198862 1198864and the damping coefficient 120583

1 respec-

tively Therefore the excitations 1198912 1198862 1198864and the damping

coefficient 1205831have important influence on the nonlinear

dynamics responses of the laminated composite piezoelectricrectangular plate

(3) There exist different shapes of the chaotic motionsin the nonlinear oscillations of the laminated compositepiezoelectric rectangular plate under different excitationsparameters and initial conditions It is found from numerical

simulations that the shapes of the chaotic motions are com-pletely different From the three-dimensional phase portraitsin Figures 12 and 13 it is found that there exist obviousmultipulse jumping phenomena Therefore parameters andinitial conditions impact the shapes of the multipulse chaoticmotions

(4)There existmultipulse chaoticmotions in the averagedequations It is well known that the multipulse chaoticmotions in the averaged equations can lead to the multi-pulse amplitude modulated chaotic vibrations in the originalsystem under certain conditions Therefore the multipulseamplitudemodulated chaoticmotions occur in the laminatedcomposite piezoelectric rectangular plate

In summary both theoretical and numerical studiessuggest that chaos for the Smale horseshoe sense in nonlinearmotion of the simply supported laminated composite piezo-electric rectangular plate exists

Conflict of Interests

The authors declare that there is no conflict of interests in thispaper

Acknowledgments

The authors gratefully acknowledge the support of theNational Natural Science Foundation of China (NNSFC)through Grant nos 11172009 11372015 10872010 1129015210732020 and 11072008 the National Science Foundationfor Distinguished Young Scholars of China (NSFDYSC)through Grant no 10425209 the Funding Project for Aca-demic Human Resources Development in Institutions ofHigher Learning under the Jurisdiction of Beijing Munici-pality (PHRIHLB) the Foundation of Beijing University ofTechnology through Grant no X4001015201301 the PhDPrograms Foundation of Beijing University of Technology(DPFBUT) through Grant no 52001015200701


References

[1] H S Tzou J P Zhong and J J Hollkamp ldquoSpatially distributedorthogonal piezoelectric shell actuators theory and applica-tionsrdquo Journal of Sound and Vibration vol 177 no 3 pp 363ndash378 1994

[2] A S Purekar D J Pines S Sundararaman and D E AdamsldquoDirectional piezoelectric phased array filters for detectingdamage in isotropic platesrdquo Smart Materials and Structures vol13 no 4 pp 838ndash850 2004

[3] M Ishihara and N Noda ldquoControl of mechanical deformationof a laminate by piezoelectric actuator taking into account thetransverse shearrdquo Archive of Applied Mechanics vol 74 no 1-2pp 16ndash28 2004

[4] I K Oh ldquoThermopiezoelastic nonlinear dynamics of activepiezolaminated platesrdquo Smart Materials and Structures vol 14no 4 pp 823ndash834 2005

[5] S J Lee J N Reddy and F Rostam-Abadi ldquoNonlinear finiteelement analysis of laminated composite shells with actuatinglayersrdquo Finite Elements in Analysis and Design vol 43 no 1 pp1ndash21 2006

[6] S Panda and M C Ray ldquoActive constrained layer dampingof geometrically nonlinear vibrations of functionally gradedplates using piezoelectric fiber-reinforced compositesrdquo SmartMaterials and Structures vol 17 no 2 Article ID 025012 2008

[7] P C Dumir P Kumari and S Kapuria ldquoAssessment of thirdorder smeared and zigzag theories for buckling and vibration offlat angle-ply hybrid piezoelectric panelsrdquoComposite Structuresvol 90 no 3 pp 346ndash362 2009

[8] M H Yao and W Zhang ldquoMulti-Pulse chaotic motions ofhigh-dimension nonlinear system for a laminated compositepiezoelectric rectangular platerdquoMeccanica 2013

[9] Z C Feng and P R Sethna ldquoGlobal bifurcations in the motionof parametrically excited thin platesrdquo Nonlinear Dynamics vol4 no 4 pp 389ndash408 1993

[10] W M Tien N S Namachchivaya and A K Bajaj ldquoNon-linear dynamics of a shallow arch under periodic excitationmdashI1 2 internal resonancerdquo International Journal of Non-LinearMechanics vol 29 no 3 pp 349ndash366 1994

[11] N Malhotra and N Sri Namachchivaya ldquoChaotic motion ofshallow arch structures under 1 1 internal resonancerdquo Journalof Engineering Mechanics vol 123 no 6 pp 620ndash627 1997

[12] W Zhang ldquoGlobal and chaotic dynamics for a parametricallyexcited thin platerdquo Journal of Sound and Vibration vol 239 no5 pp 1013ndash1036 2001

[13] M H Yeo and W K Lee ldquoEvidences of global bifurcations ofan imperfect circular platerdquo Journal of Sound and Vibration vol293 no 1-2 pp 138ndash155 2006

[14] W Yu and F Chen ldquoGlobal bifurcations of a simply supportedrectangular metallic plate subjected to a transverse harmonicexcitationrdquo Nonlinear Dynamics vol 59 no 1-2 pp 129ndash1412010

[15] G Kovacic and T A Wettergren ldquoHomoclinic orbits in thedynamics of resonantly driven coupled pendulardquo Zeitschrift furAngewandte Mathematik und Physik vol 47 no 2 pp 221ndash2641996

[16] T J Kaper and G Kovacic ldquoMulti-bump orbits homoclinic toresonance bandsrdquo Transactions of the American MathematicalSociety vol 348 no 10 pp 3835ndash3887 1996

[17] R Camassa G Kovacic and S K Tin ldquoA Melnikov methodfor homoclinic orbits with many pulsesrdquo Archive for RationalMechanics and Analysis vol 143 no 2 pp 105ndash193 1998

[18] W Zhang and M H Yao ldquoTheories of multi-pulse globalbifurcations for high-dimensional systems and application tocantilever beamrdquo International Journal of Modern Physics B vol22 no 24 pp 4089ndash4141 2008

[19] G Haller and S Wiggins ldquoN-pulse homoclinic orbits inperturbations of resonant hamiltonian systemsrdquo Archive forRationalMechanics and Analysis vol 130 no 1 pp 25ndash101 1995

[20] N Malhotra N Sri Namachchivaya and R J McDonaldldquoMultipulse orbits in the motion of flexible spinning discsrdquoJournal of Nonlinear Science vol 12 no 1 pp 1ndash26 2002

[21] W Yu and F Chen ldquoOrbits homoclinic to resonances in aharmonically excited and undamped circular platerdquoMeccanicavol 45 no 4 pp 567ndash575 2010

[22] J N ReddyMechanics of LaminatedComposite Plates and ShellsSpringer New York NY USA 2004

[23] AHNayfeh andDTMookNonlinearOscillations JohnWileyamp Sons New York NY USA 1979

[24] W Zhang F Wang and J W Zu ldquoComputation of normalforms for high dimensional non-linear systems and applicationto non-planar non-linear oscillations of a cantilever beamrdquoJournal of Sound and Vibration vol 278 no 4-5 pp 949ndash9742004

Submit your manuscripts athttpwwwhindawicom

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MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


examine the buckling bifurcation and sling-shot buckling ofactive piezo-laminated plates Lee et al [5] employed third-order shear deformation theory and nonlinear finite elementto canvass deflection suppression characteristics of laminatedcomposite shell structures with smart material laminaePanda andRay [6] exploited the first-order shear deformationtheory and the three-dimensional finite element method todelve into the open-loop and closed-loop nonlinear dynamicsof functionally graded plates with the piezoelectric fiber-reinforced composite material under the thermal environ-ment Dumir et al [7] used the extendedHamiltonrsquos principleto derive the coupled nonlinear equations of motion and theboundary conditions for buckling and vibration of symmet-rically laminated hybrid angle-ply piezoelectric panels underin-plane electrothermomechanical loading Yao and Zhang[8] employed the third-order shear deformation plate theoryto explore the bifurcations and chaotic dynamics of the four-edge simply supported laminated composite piezoelectricrectangular plate in the case of the 1 2 internal resonances

The global bifurcations and chaotic dynamics of high-dimensional nonlinear systems have been at the forefrontof nonlinear dynamics for the past two decades There aretwo ways of solutions on Shilnikov type chaotic dynamics ofhigh-dimensional nonlinear systems One is Shilnikov typesingle-pulse chaotic dynamics and the other is Shilnikovtype multipulse chaotic dynamics Most researchers focusedon Shilnikov type single-pulse chaotic dynamics of high-dimensional nonlinear systems Much research in this fieldhas concentrated on Shilnikov type single-pulse chaoticdynamics of thin plate structures Feng and Sethna [9]utilized the global perturbation method to study the globalbifurcations and chaotic dynamics of the thin plate underparametric excitation and obtained the conditions in whichthe Shilnikov type homoclinic orbits and chaos can occurTien et al [10] applied the Melnikov method to investigatethe global bifurcation and chaos for the Smale horseshoesense of a two-degree-of-freedom shallow arch subjectedto simple harmonic excitation for the case of 1 2 internalresonance Malhotra and Sri Namachchivaya [11] employedthe averaging method and Melnikov technique to canvassthe local global bifurcations and chaotic motions of a two-degree-of-freedom shallow arch subjected to simple har-monic excitation for the case of 1 1 internal resonance Theglobal bifurcations and chaotic dynamics were investigatedby Zhang [12] for the simply supported rectangular thinplates subjected to the parametrical-external excitation andthe parametrical excitation Yeo and Lee [13] made use ofthe global perturbation technique to examine the globaldynamics of an imperfect circular plate for the case of1 1 internal resonance and obtained the criteria for chaoticmotions of homoclinic orbits and heteroclinic orbits Yu andChen [14] adopted the global perturbationmethod to explorethe global bifurcations of a simply supported rectangularmetallic plate subjected to a transverse harmonic excitationfor the case of 1 1 internal resonance

While most of studies are on the Shilnikov typesingle-pulse global bifurcations and chaotic dynamics ofhigh-dimensional nonlinear systems there are researchersinvestigating the Shilnikov type multipulse homoclinic and

heteroclinic bifurcations and chaotic dynamics So far thereare two theories of the Shilnikov type multipulse chaoticdynamics One is the extended Melnikov method and theother theory is the energy phase method Much achievementis made in the former theory of high-dimensional nonlinearsystems In 1996 Kovacic and Wettergren [15] used a modi-fiedMelnikovmethod to investigate the existence of the mul-tipulse jumping of homoclinic orbits and chaotic dynamicsin resonantly forced coupled pendula Furthermore Kaperand Kovacic [16] studied the existence of several classes ofthe multibump orbits homoclinic to resonance bands forcompletely integral Hamiltonian systems subjected to smallamplitude Hamiltonian and damped perturbations Camassaet al [17] presented a new Melnikov method which is calledthe extended Melnikov method to explore the multipulsejumping of homoclinic and heteroclinic orbits in a class ofperturbed Hamiltonian systems Until recently Zhang andYao [18] introduced the extended Melnikov method to theengineering field They came up with a simplification of theextended Melnikov method in the resonant case and utilizedit to analyze the Shilnikov type multipulse homoclinic bifur-cations and chaotic dynamics for the nonlinear nonplanaroscillations of the cantilever beam

The study on the second theory of the Shilnikov typemul-tipulse chaotic dynamics was stated by Haller and Wiggins[19] They presented the energy phase method to investigatethe existence of the multipulse jumping homoclinic andheteroclinic orbits in perturbed Hamiltonian systems Upto now few researchers have made use of the energy phasemethod to study the Shilnikov type multipulse homoclinicand heteroclinic bifurcations and chaotic dynamics of high-dimensional nonlinear systems in engineering applicationsMalhotra et al [20] used the energy-phase method to inves-tigate multipulse homoclinic orbits and chaotic dynamics forthe motion of flexible spinning discs Yu and Chen [21] madeuse of the energy-phase method to examine the Shilnikovtype multipulse homoclinic orbits of a harmonically excitedcircular plate

This paper focuses on the Shilnikov typemultipulse orbitsand chaotic dynamics for a simply supported laminatedcomposite piezoelectric rectangular plate under combinedparametric excitations and transverse load Based on thevon Karman type equations and Reddyrsquos third-order sheardeformation plate theory Hamiltonrsquos principle is employedto obtain the governing nonlinear equations of the lami-nated composite piezoelectric rectangular plate with com-bined parametric excitation and transverse load We applyGalerkinrsquos approach and the method of multiple scales to thepartial differential governing equations to obtain the four-dimensional averaged equation for the case of 1 3 internalresonance and primary parametric resonance From theaveraged equation the theory of normal form is used tofind the explicit formulas of normal form We study theheteroclinic bifurcations of the unperturbed system and thecharacteristic of the hyperbolic dynamics of the dissipativesystem respectively Finally we employ the extended Mel-nikov method to analyze the Shilnikov type multipulse orbitsand chaotic dynamics in the laminated composite piezoelec-tric plate In this paper the extended Melnikov function


x

a

y

b

qcosΩ3t

qxcosΩ1t

qycosΩ2t
















119906 (119909 119910 119911 119905) = 1199060(119909 119910 119905) + 119911120601

119909(119909 119910 119905)

minus 1199113

4

3ℎ2

(120601119909+

1205971199080

120597119909

)

(1a)

V (119909 119910 119911 119905) = V0(119909 119910 119905) + 119911120601

119910(119909 119910 119905)

minus 1199113

4

3ℎ2

(120601119910+

1205971199080

120597119910

)

(1b)

119908 (119909 119910 119911 119905) = 1199080(119909 119910 119905) (1c)



119909and 120601

119910represent the



120576119909119909

=

120597119906

120597119909

+

1

2

(

120597119908

120597119909

)

2

120576119909119911

=

1

2

(

120597119906

120597119911

+

120597119908

120597119909

)

120576119909119910

=

1

2

(

120597119906

120597119910

+

120597V120597119909

+

120597119908

120597119909

120597119908

120597119910

)

120576119910119910

=

120597V120597119910

+

1

2

(

120597119908

120597119910

)

2

120576119910119911

=

1

2

(

120597V120597119911

+

120597119908

120597119910

)

120576119911119911

=

120597119908

120597119911

(2)


120590119894119895= 120590119904

119894119895119896119897

120576119896119897minus 119890119894119895119896119864119896 (119894 119895 119896 119897 = 119909 119910 119911) (3)





119894119895is the





11986011

1205972

1199060

1205971199092

+ 11986066

1205972

1199060

1205971199102

+ (11986012

+ 11986066)

1205972V0

120597119909120597119910

+ 11986011

1205971199080

120597119909

1205972

1199080

1205971199092

+ 11986066

1205971199080

120597119909

1205972

1199080

1205971199102

+ (11986012

+ 11986066)

1205971199080

120597119910

1205972

1199080

120597119909120597119910

= 11986800+ 1198691

120601119909minus 11988811198683

1205970

120597119909

(4a)

11986066

1205972V0

1205971199092

+ 11986022

1205972V0

1205971199102

+ (11986021

+ 11986066)

1205972

1199060

120597119909120597119910

+ 11986066

1205971199080

120597119910

1205972

1199080

1205971199092

+ 11986022

1205971199080

120597119910

1205972

1199080

1205971199102

+ (11986021

+ 11986066)

1205971199080

120597119909

1205972

1199080

120597119909120597119910

= 1198680V0+ 1198691

120601119910minus 11988811198683

1205970

120597119910

(4b)

11986066

1205971199080

120597119909

1205972

1199060

1205971199102

minus 119867221198882

1

1205974

1199080

1205971199104

+ 1198881(211986566

+ 11986512

minus 2119867661198881minus 119867121198881)

1205973

120601119910

1205971199101205971199092

+ 1198881(11986522

minus 119867221198881)

1205973

120601119910

1205971199103

minus 119867111198882

1

1205974

1199080

1205971199094

+ 11986011

1205971199080

120597119909

1205972

1199060

1205971199092

+ (119865441198882

2

minus 2119863441198882+ 11986044)

120597120601119910

120597119910

+ 1198881(11986521

+ 211986566

minus 119867211198881minus 2119867661198881)

1205973

120601119909

1205971199102

120597119909

minus 1198882

1

(11986721

+ 411986766

+ 11986712)

1205974

1199080

1205971199102

1205971199092

+ (11986044

minus 119873119875

119910

cos (Ω4119905) + 119865

441198882

2

minus 2119863441198882)

1205972

1199080

1205971199102

minus

120597119873119875

119910

120597119910

cos (Ω2119905)

1205971199080

120597119910

+ (11986021

+ 411986066

+ 11986012)

times

1205971199080

120597119909

1205971199080

120597119910

1205972

1199080

120597119910120597119909

+ 1198881(11986511

minus 119867111198881)

1205973

120601119909

1205971199093

+ (11986021

+ 11986066)

1205971199080

120597119910

1205972

1199060

120597119910120597119909

+ 11986021

1205971199060

120597119909

1205972

1199080

1205971199102

times 11986066

1205971199080

120597119910

1205972V0

1205971199092

+ 11986022

1205971199080

120597119910

1205972V0

1205971199102


+

1

2

(11986012

+ 211986066) (

1205971199080

120597119910

)

2

1205972

1199080

1205971199092

+ 11986022

1205972

1199080

1205971199102

120597V0

120597119910

+ (11986012

+ 11986066)

1205971199080

120597119909

1205972V0

120597119910120597119909

+

1

2

(11986021

+ 211986066)

1205972

1199080

1205971199102

times (

1205971199080

120597119909

)

2

+

3

2

11986011(

1205971199080

120597119909

)

2

1205972

1199080

1205971199092

+ 11986011

1205972

1199080

1205971199092

1205971199060

120597119909

+ 11986012

1205972

1199080

1205971199092

120597V0

120597119910

+ 211986066

1205972

1199080

120597119910120597119909

120597V0

120597119909

+ 211986066

1205972

1199080

120597119910120597119909

1205971199060

120597119910

+

3

2

11986022(

1205971199080

120597119910

)

2

1205972

1199080

1205971199102

+ (11986055

+ 119902119909cos (Ω

1119905) minus 119873

119875

119909

cos (Ω4119905)

+ 119865551198882

2

minus 2119863551198882)

1205972

1199080

1205971199092

minus

120597119873119875

119909

120597119909

cos (Ω3119905)

120597119908

120597119909

+ (119865551198882

2

minus 2119863551198882+ 11986055)

120597120601119909

120597119909

minus 119902 cos (Ω3119905) + 119896119891

1205971199080

120597119905

= 11986800minus 1198882

1

1198686(

1205972

0

1205971199092

+

1205972

0

1205971199102

)

+ 11988811198683(

1205970

120597119909

+

120597V0

120597119910

) + 11988811198694(

120597

120601119909

120597119909

+

120597

120601119910

120597119910

)

(4c)

(11986311

minus 2119865111198881+ 119867111198882

1

)

1205972

120601119909

1205971199092

+ (11986366

minus 2119865661198881+ 119867661198882

1

)

1205972

120601119909

1205971199102

minus 1198881(11986511

minus 119867111198881)

1205973

1199080

1205971199093

minus (119865551198882

2

minus 2119863551198882+ 11986055)

1205971199080

120597119909

+ (11986312

+ 11986366

+ 119867661198882

1

minus 2119865661198881+ 119867121198882

1

minus 2119865121198881)

times

1205972

120601119910

120597119910120597119909

minus 1198881(211986566

+ 11986512

minus 2119867661198881minus 119867121198881)

times

1205973

1199080

1205971199102

120597119909

+ (2119863551198882minus 11986055

minus 119865551198882

2

) 120601119909

= 11986910+ 1198702

120601119909minus 11988811198694

1205970

120597119909

(4d)

(11986366

minus 2119865661198881+ 119867661198882

1

)

1205972

120601119910

1205971199092

minus 1198881(11986521

+ 211986566

minus 119867211198881minus 2119867661198881)

1205973

1199080

1205971199101205971199092

+ (119867211198882

1

+ 11986366

+ 11986321

minus 2119865211198881+ 119867661198882

1

minus 2119865661198881)

times

1205972

120601119909

120597119910120597119909

+ (119867221198882

1

+ 11986322

minus 2119865221198881)

1205972

120601119910

1205971199102

minus 1198881(11986522

minus 119867221198881)

1205973

1199080

1205971199103

minus (119865441198882

2

minus 2119863441198882+ 11986044)

times

1205971199080

120597119910

+ (2119863441198882minus 119865441198882

2

minus 11986044) 120601119910

= 1198691V0+ 1198702

120601119910minus 11988811198694

1205970

120597119910

(4e)


119909 = 0 119908 = 0 120601119910= 0 119873

119909119910= 0 119872

119909119909= 0

(5a)

119909 = 119886 119908 = 0 120601119910= 0 119873

119909119910= 0 119872

119909119909= 0

(5b)

119910 = 0 119908 = 0 120601119909= 0 119873

119909119910= 0 119872

119910119910= 0

(5c)

119910 = 119887 119908 = 0 120601119909= 0 119873

119909119910= 0 119872

119910119910= 0

(5d)

int

ℎ

0

119873119909119909

1003816100381610038161003816119909=0

119889119911 = minusint

ℎ

0

(1199020+ 119902119909cosΩ1119905) 119889119911 (5e)

int

ℎ

0

119873119910119910

10038161003816100381610038161003816119910=0

119889119911 = minusint

ℎ

0

(1199021+ 119902119910cosΩ2119905) 119889119911 (5f)


0 V0 1199080 120601119909 and 120601



0 V0 1199080 120601119909 and


1199060= 11990601(119905) cos 120587119909

2119886

cos120587119910

2119887

+ 11990602(119905) cos 3120587119909

2119886

cos120587119910

2119887

(6a)

V0= V1(119905) cos

120587119910

2119887

cos 1205871199092119886

+ V2(119905) cos

120587119910

2119887

cos 31205871199092119886

(6b)

1199080= 1199081(119905) sin 120587119909

119886

sin120587119910

119887

+ 1199082(119905) sin 3120587119909

119886

sin120587119910

119887

(6c)

120601119909= 1206011(119905) cos 120587119909

119886

sin120587119910

119887

+ 1206012(119905) cos 3120587119909

119886

sin120587119910

119887

(6d)

120601119910= 1206013(119905) cos

120587119910

119887

sin 120587119909

119886

+ 1206014(119905) cos

120587119910

119887

sin 3120587119909

119886

(6e)




01 11990602 V1 V2 1206011 1206012 1206013 and

1206014via 1199081and 119908

2as follows

11990601

= 11989611199082

1

+ 11989621199082

2

+ 119896311990811199082 (7a)

11990602

= 11989641199082

1

+ 11989651199082

2

+ 119896611990811199082 (7b)

V1= 11989671199082

1

+ 11989681199082

2

+ 119896911990811199082 (7c)

V2= 119896101199082

1

+ 119896111199082

2

+ 1198961211990811199082 (7d)

1206011= 119896191199081 120601

2= 119896201199082 (7e)

1206013= 119896211199081 120601

4= 119896221199082 (7f)



119906 =

1199060

119886

V =

V0

119887

119908 =

1199080

ℎ

120601119909

= 120601119909 120601

119910

= 120601119910 119909 =

119909

119886

119910 =

119910

119887

119902 =

1198872

119864ℎ3

119902 119902119909

=

1198872

119864ℎ3

119902119909 119902

119910

=

1198872

119864ℎ3

119902119910

119905 = 1205872

(

119864

119886119887120588

)

12

119905 Ω119894=

1

1205872

(

119886119887120588

119864

)

12

Ω119894

(119894 = 1 2)

119860119894119895=

(119886119887)12

119864ℎ2

119860119894119895 119861

119894119895=

(119886119887)12

119864ℎ3

119861119894119895

119863119894119895=

(119886119887)12

119864ℎ4

119863119894119895 119864

119894119895=

(119886119887)12

119864ℎ5

119864119894119895

119865119894119895=

(119886119887)12

119864ℎ6

119865119894119895 119867

119894119895=

(119886119887)12

119864ℎ8

119867119894119895

119868119894=

1

(119886119887)(119894+1)2

120588

119868119894

(8)


1+ 12058311+ 1205962

1

1199081


+ 11988651199082

1

1199082+ 11988661199082

2

1199081+ 11988671199083

1

+ 11988681199083

2

= 1198911cosΩ3119905

(9a)

2+ 12058322+ 1205962

2

1199082

+ (1198872cosΩ1119905 + 1198873cosΩ2119905 + 1198874cosΩ4119905) 1199082

+ 11988751199082

2

1199081+ 11988761199082

1

1199082+ 11988771199083

2

+ 11988781199083

1

= 1198912cosΩ3119905

(9b)



1205962

1

=

1205962

9

+ 1205761205901 120596

2

2

= 1205962

+ 1205761205902

Ω3= 120596 Ω

1= Ω2= Ω4=

2120596

3

1205962asymp 31205961

(10)

where 1205901and 120590



1199081(119905 120576) = 119909

10(1198790 1198791) + 120576119909

11(1198790 1198791) + sdot sdot sdot (11a)

1199082(119905 120576) = 119909

20(1198790 1198791) + 120576119909

21(1198790 1198791) + sdot sdot sdot (11b)

where 1198790= 119905 1198791= 120576119905


1= minus

1

2

12058311199091minus

1

2

12059011199092+

1

4

(1198862+ 1198863minus 1198864) 1199092

minus

3

2

11988671199092(1199092

1

+ 1199092

2

) minus

1

2

11988651199094(1199092

1

minus 1199092

2

)

minus 11988661199092(1199092

3

+ 1199092

4

) + 1198865119909111990921199093

(12a)

2= minus

1

2

12058311199092+

1

2

12059011199091+

1

4

(1198862+ 1198863minus 1198864) 1199091

+

3

2

11988671199091(1199092

1

+ 1199092

2

) +

1

2

11988651199093(1199092

1

minus 1199092

2

)

+ 11988661199091(1199092

3

+ 1199092

4

) + 1198865119909111990921199094

(12b)

3= minus

1

2

12058321199093minus

1

6

12059021199094minus

1

3

11988761199094(1199092

1

+ 1199092

2

)

minus

1

2

11988771199094(1199092

3

+ 1199092

4

) minus

1

6

11988781199092(31199092

1

minus 1199092

2

)

(12c)


4= minus

1

2

12058321199094+

1

6

12059021199093+

1

3

11988761199093(1199092

1

+ 1199092

2

)

+

1

2

11988771199093(1199092

3

+ 1199092

4

) +

1

6

11988781199091(1199092

1

minus 31199092

2

) minus

1

12

1198912

(12d)



2oplus 1198852and 119863

4





1= minus

1

2

12058311199091+ (1198910minus

1

2

1205901)1199092minus

3

2

11988671199092(1199092

1

+ 1199092

2

)

minus

1

2

11988651199094(1199092

1

minus 1199092

2

)

minus 11988661199092(1199092

3

+ 1199092

4

) + 1198865119909111990921199093

(13a)

2= minus

1

2

12058311199092+ (1198910+

1

2

1205901)1199091+

3

2

11988671199091(1199092

1

+ 1199092

2

)

+

1

2

11988651199093(1199092

1

minus 1199092

2

)

+ 11988661199091(1199092

3

+ 1199092

4

) + 1198865119909111990921199094

(13b)

3= minus

1

2

12058321199093minus

1

6

12059021199094minus

1

3

11988761199094(1199092

1

+ 1199092

2

)

minus

1

2

11988771199094(1199092

3

+ 1199092

4

) minus

1

6

11988781199092(31199092

1

minus 1199092

2

)

(13c)

4= minus

1

2

12058321199094+

1

6

12059021199093+

1

3

11988761199093(1199092

1

+ 1199092

2

)

+

1

2

11988771199093(1199092

3

+ 1199092

4

) +

1

6

11988781199091(1199092

1

minus 31199092

2

)

(13d)

where 1198910= (14)(119886

2+ 1198863minus 1198864)


1199091= 1199101minus

1

4

11988671199103

1

+

31198865(1205902minus 6)

1205902

2

1199102

1

1199103minus 119886611991011199102

3

minus 119886611991011199102

4

minus

31198865(1205903

2

minus 181205902

2

+ 2161205902minus 1296)

1205904

2

1199102

2

1199103

+

1198865(61205902

2

minus 721205902+ 432)

1205903

2

119910111991021199104

(14a)

1199092= 1199102+

3

2

11988671199103

2

+

3

4

11988671199102

1

1199102+

31198865

1205902

1199102

1

1199104

minus

31198865(1205902

2

minus 121205902+ 72)

1205903

2

1199102

2

1199104minus

61198865(1205902minus 6)

1205902

2

119910111991021199103

(14b)

1199093= 1199103minus

1198878

1205902

1199103

1

+

31198878(1205902

2

minus 121205902+ 72)

1205903

2

11991011199102

2

minus

1

3

1198876119910111991021199104

(14c)

1199094= 1199104+

1198878(1205903

2

minus 181205902

2

+ 2161205902minus 1296)

1205904

2

1199103

2

minus

31198878(1205902minus 6)

1205902

2

1199102

1

1199102+

1

3

1198876119910111991021199103

(14d)


1199101= minus1205831

1199101+ (1 minus 120590

1) 1199102 (15a)

1199102= 12059011199101minus 1205831

1199102+ 11988661199101(1199102

3

+ 1199102

4

) +

3

2

11988671199103

1

(15b)

1199103= minus1205832

1199103minus 12059021199104minus

1

3

11988761199102

1

1199104minus

1

2

11988771199104(1199102

3

+ 1199102

4

) (15c)

1199104= 12059021199103minus 1205832

1199104+

1

3

11988761199102

1

1199103+

1

2

11988771199103(1199102

3

+ 1199102

4

) minus 1198912

(15d)


= (12)1205831 1205832

= (12)1205832 1205902=

(16)1205902 and 119891

2


Further let

1199103= 119868 cos 120574 119910

4= 119868 sin 120574 (16)


1199101= minus1205831

1199101+ (1 minus 120590

1) 1199102 (17a)

1199102= 12059011199101minus 1205831

1199102+ 119886611991011198682

+

3

2

11988671199103

1

(17b)

119868 = minus1205832

119868 minus 1198912

sin 120574 (17c)

119868 120574 = 1205902119868 +

1

3

11988761199102

1

119868 +

1

2

11988771198683

minus 1198912

cos 120574 (17d)


[

1199101

1199102

] = radic3

radic10038161003816100381610038161198866

1003816100381610038161003816

radic10038161003816100381610038161198876

1003816100381610038161003816

[

1 minus 1205901

0

1205831

1] [

1199061

1199062

] (18)



1yield the


1= 1199062 (19a)

2= minus120583119906

1minus 12058331199062+ 12057811199063

1

+ 119886611990611198682

(19b)

119868 = minus1205832

119868 minus 1198912

sin 120574 (19c)

119868 120574 = 1205902119868 + 11988661199062

1

119868 + 12057221198683

minus 1198912

cos 120574 (19d)

where 120583 = 1205832

1

minus 1205901(1 minus 120590

1) 1205833= 21205831

1205781= 91198866119886721198876and

1205722= (12)119887

7


1205832

997888rarr 1205761205832

1205833997888rarr 120576120583

3 119891

2

997888rarr 1205761198912

1205781997888rarr 1205781 120572

2997888rarr 1205722 119886

6997888rarr 1198866

(20)


1=

120597119867

1205971199062

+ 1205761198921199061= 1199062 (21a)

2= minus

120597119867

1205971199061

+ 1205761198921199062= minus120583119906

1+ 12057811199063

1

+ 119886611990611198682

minus 12057612058331199062 (21b)

119868 =

120597119867

120597120574

+ 120576119892119868

= minus1205761205832

119868 minus 1205761198912

sin 120574 (21c)

119868 120574 = minus

120597119867

120597119868

+ 120576119892120574

= 1205902119868 + 12057221198683

+ 11988661198681199062

1

minus 1205761198912

cos 120574 (21d)


119867(1199061 1199062 119868 120574) =

1

2

1199062

2

+

1

2

1205831199062

1

minus

1

4

12057811199064

1

minus

1

2

11988661198682

1199062

1

minus

1

2

12059021198682

minus

1

4

12057221198684

(22)

and 1198921199061 1198921199062 119892119868 and 119892


1198921199061= 0 119892

1199062= minus12058331199062

119892119868

= minus1205832

119868 minus 1198912

sin 120574 119892120574

= minus1198912

cos 120574(23)



1 1199062) of (21a) (21b)


1= 1199062 (24a)

2= minus120583119906

1+ 12057811199063

1

+ 119886611990611198682

(24b)

Since 1205781


11988661198682




11986812

= plusmn[

1205832

1

minus 1205901(1 minus 120590

1)

1198866

]

12

(25)


0= (0 0)


119861 = plusmn

1

1205781

[1205832

1

minus 1205901(1 minus 120590

1) minus 11988661198682

]

12

(26)


0is the



1198681= 0 119868

2= [

1205832

1

minus 1205901(1 minus 120590

1)

1198866

]

12

(27)




(1198791 119868) that is lim


119906ℎ

plusmn

(1198791 119868) =


defined by

119872 = (119906 119868 120574) | 119906 = 119902plusmn(119868) 1198681lt 119868 lt 119868

2 0 le 120574 lt 2120587 (28)



119904

(119872) and 119882119906



119906

(119872)


Γ = (119906 119868 120574) | 119906 = 119906ℎ

plusmn

(1198791 119868) 1198681lt 119868 lt 119868

2

120574 = int

1198791

0

119863119868119867(119906ℎ

plusmn

(1198791 119868) 119868) 119889119904 + 120574

0

(29)


119868 = 0 (30a)

119868 120574 = 119863119868119867(119902plusmn(119868) 119868) 119868

1lt 119868 lt 119868

2 (30b)


where

119863119868119867(119902plusmn(119868) 119868) = minus

120597119867 (119902plusmn(119868) 119868)

120597119868

= 1205902119868 + 12057221198683

+ 11988661198681199062

1

(31)







119863119868119867(119902plusmn 119868) = 120590

2119868119903+ 12057221198683

119903

+ 11988661198681199031199062

1

= 0 (32)

Then we obtain

119868119903= plusmn

12059021205781+ 1198866[1205832

1

minus 1205901(1 minus 120590

1)]

1198862

6

minus 12057221205781

12

(33)



119903) (34)




1 1199062 119868 120574) The phase shift



61198682 and 120583


1= 1199062 (35a)

2= minus12057611199061+ 12057811199063

1

minus 12057612057621199062 (35b)


119867(1199061 1199062) =

1

2

1199062

2

+

1

2

12057611199062

1

minus

1

4

12057811199064

1

(36)






1199061(1198791) = plusmnradic

1205761

1205781

tanh(

radic21205761

2

1198791) (37a)

1199062(1198791) = plusmn

1205761

radic21205781


2

1198791) (37b)


120574 (1198791) = 1205961199031198791minus

1198866radic21205761

1205781

tanh(

radic21205761

2

1198791) + 1205740 (38)

where 120596119903= 1205902+ 12057221198682

+ 119886612057611205781

At 119868 = 119868119903 there is 120596


be expressed as

Δ120574 = [minus

21198866radic21205761

1205781

]

119868=119868119903

= minus

21198866

1205781

radic2 [1205832

1

minus 1205901(1 minus 120590

1) minus 11988661198682

119903

]

(39)



1 1199062) for the




plusmn(119868) in (24a)


M

I

120574

times

u2

u1

(a)

0 1205742120587

I = I1

I = Ir

I = I2

(b)




120576

Therefore we obtain

119872 = 119872120576= (119906 119868 120574) | 119906 = 119902

plusmn(119868) 1198681lt 119868 lt 119868

2 0 le 120574 lt 2120587

(40)


119868 = minus1205832

119868 minus 1198912

sin 120574 (41a)

120574 = 1205902+ 12057221198682

+ 11988661199062

1

minus

1198912

cos 120574119868

(41b)


1205832

997888rarr 1205761205832

119868 = 119868119903+ radic120576ℎ

1198912

997888rarr 1205761198912

1198791997888rarr

1198791

radic120576

(42)


ℎ = minus120583

2

119868119903minus 1198912

sin 120574 minus radic1205761205832

ℎ (43a)

120574 = minus

2120575

1205781

119868119903ℎ minus radic120576(

120575

1205781

ℎ2

+

1198912

119868119903

cos 120574) (43b)

where 120575 = 1198862

6

minus 12057221205781

When 120576 = 0 (43a) (43b) become

ℎ = minus120583

2

119868119903minus 1198912

sin 120574 (44a)

120574 = minus

2120575

1205781

119868119903ℎ (44b)


119863(ℎ 120574) = minus120583

2

119868119903120574 + 1198912

cos 120574 +

120575

1205781

119868119903ℎ2

(45)


1198750= (0 120574


1205832

119868119903

1198912

))

1198760= (0 120574

119904) = (0 120587 + arcsin(

1205832

119868119903

1198912

))

(46)


0and119876









120576 The phase




120574min minus

1198912

1205832

119868119903

cos 120574min = 120587 + arcsin1205832

119868119903

1198912

+

radic119891

2

2

minus 1205832

2

1198682

119903

1205832

119868119903

(47)


119903as

119860120576= (1199061 1199062 119868 120574) | 119906

1= 119861 119906

2= 0

1003816100381610038161003816119868 minus 119868119903

1003816100381610038161003816lt radic120576119862 120574 isin 119879

119871

(48)



120576 denoted as119882119904(119860

120576) and119882

119906

(119860120576) are the


119906






120576 leave and


h

120574

P0

120574min 120574c 120574s

Q0

(a)

h

120574Q120576

P120576

(b)








120574

|u|

h

120574c 120574c + 3Δ120574

p120576qc




119896(120576 119868 120574

0 1205832


0 and 120583

2




function 1198721(120576 119868119903 1205740 1205832


119903 1205740 1205832



(119872) and119882119906


119872(119868119903 1205740 1205832

1205781 1198866 1205761)

= int

+infin

minusinfin

⟨n (119901ℎ

(119905)) g (119901ℎ (119905) 1205832

0)⟩ 1198891198791


= int

+infin

minusinfin

(

120597119867

1205971199061

1198921199061+

120597119867

1205971199062

1198921199062+

120597119867

120597119868

119892119868

+

120597119867

120597120574

119892120574

)1198891198791

= minus

2radic21205833

31205781

12057632

1

minus 2radic211988661205832

1198682

119903

12057612

1

1205781

minus 1198912

119868119903[cos(120574

0minus 1198866

radic21205761

1205781

) minus cos(1205740+ 1198866

radic21205761

1205781

)]

(49)


) (119896 = 1 2 ) is defined as

119872119896(120576 119868119903 1205740 1205832

)

=

119896minus1

sum

119895=0

119872(119868119903 119895Δ120574 (119868

119903) + Γ119895(120576 119868119903 1205740 1205832

) + 1205740 1205832

)

(50)

where

Γ119895(120576 119868119903 1205740 1205832

) =

Ω (1199090(119868119903) 119868119903)

120582 (119868119903)

119895

sum

119903=1

log100381610038161003816100381610038161003816100381610038161003816

120589 (119868119903)

120576119872119903(120576 119868119903 1205740 1205832

)

100381610038161003816100381610038161003816100381610038161003816

(51)

for 119895 = 1 119896 minus 1 and Γ0(120576 119868119903 1205740 1205832


119895(120576 119868119903 1205740 1205832


119896(120576 119868119903 1205740 1205832


119903degenerates



119872119896(119868119903 1205740 1205832

1205781 1198866 1205761)

=

119896minus1

sum

119895=0

119872(119868119903 1205740+ 119895Δ120574 (119868

119903) 1205832

1205781 1198866 1205761)

= minus1198912

119868119903[cos(120574

0minus

1198866radic21205761

1205781

)

minus cos(1205740+

1198866radic21205761

1205781

)]

minus

2radic21205833

31205781

12057632

1

minus 2radic211988661205832

1198682

119903

12057612

1

1205781

minus 1198912

119868119903[cos(120574

0minus

1198866radic21205761

1205781

minus

21198866radic21205761

1205781

)

minus cos(1205740+

1198866radic21205761

1205781

minus

21198866radic21205761

1205781

)]

minus

2radic21205833

31205781

12057632

1

minus 2radic211988661205832

1198682

119903

12057612

1

1205781

+ sdot sdot sdot

minus 1198912

119868119903[cos(120574

0minus

1198866radic21205761

1205781

minus 2 (119896 minus 1) 1198866

radic21205761

1205781

)

minus cos(1205740+

1198866radic21205761

1205781


1198866radic21205761

1205781

)]

minus

2radic21205833

31205781

12057632

1

minus 2radic211988661205832

1198682

119903

12057612

1

1205781

= minus1198912

119868119903[cos(120574

0minus

1198866radic21205761

1205781


1198866radic21205761

1205781

)

minus cos(1205740+

1198866radic21205761

1205781

)]

minus

2radic21198961205833

31205781

12057632

1

minus 2radic211989611988661205832

1198682

119903

12057612

1

1205781

(52)


119896minus1= 1205740+ (119896 minus


119872119896(119868119903 1205740 1205832

1205781 1198866 1205761)

= 119872119896(119868119903 120574119896minus1


Δ120574

2

1205832

1205781 1198866 1205761)

= minus1198912

119868119903[cos(120574

119896minus1+

1

2

119896Δ120574)


minus

1

2

119896Δ120574)]

+

11989612058331205761

31198866

Δ120574 + 1205832

1198682

119903

(119896Δ120574)

= 21198912

119868119903sin 120574119896minus1

sin(

1

2

119896Δ120574)

+

12058331205761

31198866

(119896Δ120574) + 1205832

1198682

119903

(119896Δ120574)

(53)


10038161003816100381610038161003816100381610038161003816100381610038161003816

(12) 119896Δ120574

sin ((12) 119896Δ120574)

(12058331205761+ 311988661205832

1198682

119903

)

311988661198912

119868119903

10038161003816100381610038161003816100381610038161003816100381610038161003816

lt 1

1

2

119896Δ120574 = 119899120587 119899 = 0 plusmn1 plusmn2

(54)


119885119899

minus

= (119868119903 120574119896minus1

1205832

1205781 1198866 1205761) | 119872119896= 0119863

1205740

119872119896

= 0 (55)



119896minus1isin [0 120587]

120574119896minus11

= minus arcsin(12) 119896Δ120574

sin ((12) 119896Δ120574)

(12058331205761+ 311988661205832

1198682

119903

)

(311988661198912

119868119903)

120574119896minus12

= 120587 minus 120574119896minus11

(56)


3 1205761 1205832

1198866 and 119891

2


119896minus1= 120574119896minus11

and 120574119896minus1

= 120574119896minus12

=

120587 minus 120574119896minus11


119895(119868119903 1205740119894

1205832



119906

(119872120576) intersect


1205832120578111988661205761

plusmn120576(120574119896minus1119894


plusmn0

(120574119896minus1119894


119903 1205740

=

120574119896minus1119894


119895(119868119903 1205740119894

1205832

1205781 1198866 1205761)


0119894

= 120574119896minus1119894

minus (119896 minus 1)(Δ1205742) + 119895Δ120574


119895(119868119903 1205740119894

1205832



01



plusmn120576(120574119896minus11

)

and sum1205832120578111988661205761

plusmn120576(120574119896minus12


1205832120578111988661205761

plusmn0

(120574119896minus11

) and sum1205832120578111988661205761

plusmn0

(120574119896minus12


119906


n = ((minus1205831199061+ 12057811199063

1

+ 11988661198682

1199061) minus1199062 0 0) (57)


plusmn0

(120574119896minus11

)


plusmn0

(120574119896minus12





119894(120583) and 119888



the end points 119889119894(120583) to 119888


end points 119889119894(120583) and 119888



0119894

(119868119903 120583) minus Δ120574

minus



119894(120583) for 119894 =

2 119873 while the line 120574 = 1205740119894

(119868119903 120583) + Δ120574

+



(120583) at the end point 119889119894+1


119894(120583) and 119889

119894+1(120583) is zero

namely

ℎ (119888119894(120583)) minus ℎ (119889

119894+1(120583)) = 0 (58)



sum1205832120578111988661205761

0

(1205740119894


119894(120583) and 119889


1


(12057401



2(120583) on the


119873


0

(1205740119873


119873(120583) on the segments 119874


119889119873+1



sum1205832120578111988661205761

0

(1205740119894


1(120583) and 119874

2(120583) of (44a) (44b) It is known



(119868119903 120583) +

Δ120574+

(119868119903) and 120574 = 120574

0119894

(119868119903 120583) minus Δ120574

minus

(119868119903)


0

(120574119896minus11

) andsum1205832120578111988661205761

0

(120574119896minus12


01

minus


01

minusΔ120574+








119897and









120574

|u|h

Γ3

Γ2

Γ1

M

O2O3






1


6 where

1205781



6 1198867 and


1gt 0



2


2and 119886



4is the


1 1198862 1198864






3 respectively when




120574

|u|h

M

Γsum (12057401 12057402)

12057402 + Δ120574+ 12057401 + Δ120574+



12057402

) are depicted


2 1205831 1198862





2= 2 sim 200


183 1205902

= 197 1205831

= 02 1205832

= 02 1198862

= 230 1198863

=

120 1198864= 130 119886

5= minus101 119886

6= minus203 119886

7= minus205 119887

6=

407 1198877= minus308 119887

8= 109 119909

10= minus001 119909

20= minus005 119909

30=

minus001 11990940


1 1198912) and (119909

3 1198912) respec-






1 The figure demon-



1=


2=



50 100 150minus2

minus15

minus1

minus05

0

05

1

x1

f2

(a)

20 40 60 80 100 120 140 160 180

minus3

minus25

minus2

minus15

x3

f2

(b)



= minus001 11990920

= minus005 11990930

=

minus001 11990940


1198912


1198912

)

0 01 02 03 04 05 06 07minus25

minus2

minus15

minus1

minus05

0

05

1

15

x1

1205831

(a)

0 01 02 03 04 05 06 07minus32

minus3

minus28

minus26

minus24

minus22

minus2

minus18

x1

1205831

x3

1205831

x3

(b)




= minus001 11990920

= minus005 11990930

= minus001 11990940


1205831


3

1205831

)

the planes (1199091 1205831) and (119909









3 1198862)




4varies from 119886

4= 2 to 119886

4= 120


2is

chosen as 1198862


1 1198864) and (119909

3 1198864)







0 10 20 30 40 50 60

minus3

minus2

minus1

0

1

x1

a2

(a)

0 10 20 30 40 50 60minus34

minus32

minus3

minus28

minus26

minus24

minus22

minus2

minus18

a2

x3

(b)



= 02 and 1205832


2


= minus001 11990920

= minus005 11990930

= minus001 11990940


1198862


1198862

)

0 20 40 60 80 100 120minus25

minus2

minus15

minus1

minus05

0

05

1

x1

a4

(a)

0 20 40 60 80 100 120

minus35

minus3

minus25

minus2

a4

x3

(b)




= 02 and 1205832


= 23 and 1198863


= minus001 11990920

= minus005 11990930

= minus001 11990940

=


1198864


1198864

)


1are selected as











1= 1437 120590

2=

1142 1205831= 02 120583

2= 02 119886

2= 300 119886

3= 750 119886

4= 450

1198865

= minus1166 1198866

= 1227 1198912

= 1227 1198867

= minus268 1198876

=

minus22 1198877

= minus969 1198878

= minus2232 11990910

= minus108 11990920

= 0511990930

= minus001 11990940



minus1minus050051

minus05

0

05

minus25

minus2

minus15

minus1

x1

x2

x3

(a)


minus1

minus05

0

05

1

x1

x2

(b)


= 827 and 1205831


1199092

1199093


1199092

)

minus50

5

minus50

5minus6

minus4

minus2

0

2

4

6

x1x2

x3

(a)


minus10

minus5

0

5

10

x1

x2

(b)


= 1437 1205902

= 1142 1198862

= 300 1198863

= 750 1198864

= 450 1198865

= minus1166 1198866

=

1227 1198912

= 1227 1198867

= minus268 1198876

= minus22 1198877

= minus969 1198878

= minus2232 11990910

= minus108 11990920

= 05 11990930

= minus001 11990940


1

1199092

1199093


1199092

)





2= 313 119886

5=

minus1501 1198878

= 409 11990910

= minus001 11990920

= minus009 11990930

=

minus005 11990940


7 Conclusions




minus50

5

minus10

0

10

minus6

minus4

minus2

0

2

x1x2

x3

(a)

minus5 0 5 10

minus10

minus5

0

5

10

x1

x2

(b)


= 361 1205902

= 313 1198862

= 230 1198863

= 120 1198864

= 130 1198865

= minus1501 1198866

=

minus203 1198912

= 9238 1198867

= minus205 1198876

= 407 1198877

= minus308 1198878

= 409 11990910

= minus001 11990920

= minus009 11990930

= minus005 11990940


1

1199092

1199093


1199092

)



119896(120576 119868 120574

0 1205832




2 the in-plane


4 and the damping




1 respec-










Acknowledgments



References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










x

a

y

b

qcosΩ3t

qxcosΩ1t

qycosΩ2t
















119906 (119909 119910 119911 119905) = 1199060(119909 119910 119905) + 119911120601

119909(119909 119910 119905)

minus 1199113

4

3ℎ2

(120601119909+

1205971199080

120597119909

)

(1a)

V (119909 119910 119911 119905) = V0(119909 119910 119905) + 119911120601

119910(119909 119910 119905)

minus 1199113

4

3ℎ2

(120601119910+

1205971199080

120597119910

)

(1b)

119908 (119909 119910 119911 119905) = 1199080(119909 119910 119905) (1c)



119909and 120601

119910represent the



120576119909119909

=

120597119906

120597119909

+

1

2

(

120597119908

120597119909

)

2

120576119909119911

=

1

2

(

120597119906

120597119911

+

120597119908

120597119909

)

120576119909119910

=

1

2

(

120597119906

120597119910

+

120597V120597119909

+

120597119908

120597119909

120597119908

120597119910

)

120576119910119910

=

120597V120597119910

+

1

2

(

120597119908

120597119910

)

2

120576119910119911

=

1

2

(

120597V120597119911

+

120597119908

120597119910

)

120576119911119911

=

120597119908

120597119911

(2)


120590119894119895= 120590119904

119894119895119896119897

120576119896119897minus 119890119894119895119896119864119896 (119894 119895 119896 119897 = 119909 119910 119911) (3)





119894119895is the





11986011

1205972

1199060

1205971199092

+ 11986066

1205972

1199060

1205971199102

+ (11986012

+ 11986066)

1205972V0

120597119909120597119910

+ 11986011

1205971199080

120597119909

1205972

1199080

1205971199092

+ 11986066

1205971199080

120597119909

1205972

1199080

1205971199102

+ (11986012

+ 11986066)

1205971199080

120597119910

1205972

1199080

120597119909120597119910

= 11986800+ 1198691

120601119909minus 11988811198683

1205970

120597119909

(4a)

11986066

1205972V0

1205971199092

+ 11986022

1205972V0

1205971199102

+ (11986021

+ 11986066)

1205972

1199060

120597119909120597119910

+ 11986066

1205971199080

120597119910

1205972

1199080

1205971199092

+ 11986022

1205971199080

120597119910

1205972

1199080

1205971199102

+ (11986021

+ 11986066)

1205971199080

120597119909

1205972

1199080

120597119909120597119910

= 1198680V0+ 1198691

120601119910minus 11988811198683

1205970

120597119910

(4b)

11986066

1205971199080

120597119909

1205972

1199060

1205971199102

minus 119867221198882

1

1205974

1199080

1205971199104

+ 1198881(211986566

+ 11986512

minus 2119867661198881minus 119867121198881)

1205973

120601119910

1205971199101205971199092

+ 1198881(11986522

minus 119867221198881)

1205973

120601119910

1205971199103

minus 119867111198882

1

1205974

1199080

1205971199094

+ 11986011

1205971199080

120597119909

1205972

1199060

1205971199092

+ (119865441198882

2

minus 2119863441198882+ 11986044)

120597120601119910

120597119910

+ 1198881(11986521

+ 211986566

minus 119867211198881minus 2119867661198881)

1205973

120601119909

1205971199102

120597119909

minus 1198882

1

(11986721

+ 411986766

+ 11986712)

1205974

1199080

1205971199102

1205971199092

+ (11986044

minus 119873119875

119910

cos (Ω4119905) + 119865

441198882

2

minus 2119863441198882)

1205972

1199080

1205971199102

minus

120597119873119875

119910

120597119910

cos (Ω2119905)

1205971199080

120597119910

+ (11986021

+ 411986066

+ 11986012)

times

1205971199080

120597119909

1205971199080

120597119910

1205972

1199080

120597119910120597119909

+ 1198881(11986511

minus 119867111198881)

1205973

120601119909

1205971199093

+ (11986021

+ 11986066)

1205971199080

120597119910

1205972

1199060

120597119910120597119909

+ 11986021

1205971199060

120597119909

1205972

1199080

1205971199102

times 11986066

1205971199080

120597119910

1205972V0

1205971199092

+ 11986022

1205971199080

120597119910

1205972V0

1205971199102


+

1

2

(11986012

+ 211986066) (

1205971199080

120597119910

)

2

1205972

1199080

1205971199092

+ 11986022

1205972

1199080

1205971199102

120597V0

120597119910

+ (11986012

+ 11986066)

1205971199080

120597119909

1205972V0

120597119910120597119909

+

1

2

(11986021

+ 211986066)

1205972

1199080

1205971199102

times (

1205971199080

120597119909

)

2

+

3

2

11986011(

1205971199080

120597119909

)

2

1205972

1199080

1205971199092

+ 11986011

1205972

1199080

1205971199092

1205971199060

120597119909

+ 11986012

1205972

1199080

1205971199092

120597V0

120597119910

+ 211986066

1205972

1199080

120597119910120597119909

120597V0

120597119909

+ 211986066

1205972

1199080

120597119910120597119909

1205971199060

120597119910

+

3

2

11986022(

1205971199080

120597119910

)

2

1205972

1199080

1205971199102

+ (11986055

+ 119902119909cos (Ω

1119905) minus 119873

119875

119909

cos (Ω4119905)

+ 119865551198882

2

minus 2119863551198882)

1205972

1199080

1205971199092

minus

120597119873119875

119909

120597119909

cos (Ω3119905)

120597119908

120597119909

+ (119865551198882

2

minus 2119863551198882+ 11986055)

120597120601119909

120597119909

minus 119902 cos (Ω3119905) + 119896119891

1205971199080

120597119905

= 11986800minus 1198882

1

1198686(

1205972

0

1205971199092

+

1205972

0

1205971199102

)

+ 11988811198683(

1205970

120597119909

+

120597V0

120597119910

) + 11988811198694(

120597

120601119909

120597119909

+

120597

120601119910

120597119910

)

(4c)

(11986311

minus 2119865111198881+ 119867111198882

1

)

1205972

120601119909

1205971199092

+ (11986366

minus 2119865661198881+ 119867661198882

1

)

1205972

120601119909

1205971199102

minus 1198881(11986511

minus 119867111198881)

1205973

1199080

1205971199093

minus (119865551198882

2

minus 2119863551198882+ 11986055)

1205971199080

120597119909

+ (11986312

+ 11986366

+ 119867661198882

1

minus 2119865661198881+ 119867121198882

1

minus 2119865121198881)

times

1205972

120601119910

120597119910120597119909

minus 1198881(211986566

+ 11986512

minus 2119867661198881minus 119867121198881)

times

1205973

1199080

1205971199102

120597119909

+ (2119863551198882minus 11986055

minus 119865551198882

2

) 120601119909

= 11986910+ 1198702

120601119909minus 11988811198694

1205970

120597119909

(4d)

(11986366

minus 2119865661198881+ 119867661198882

1

)

1205972

120601119910

1205971199092

minus 1198881(11986521

+ 211986566

minus 119867211198881minus 2119867661198881)

1205973

1199080

1205971199101205971199092

+ (119867211198882

1

+ 11986366

+ 11986321

minus 2119865211198881+ 119867661198882

1

minus 2119865661198881)

times

1205972

120601119909

120597119910120597119909

+ (119867221198882

1

+ 11986322

minus 2119865221198881)

1205972

120601119910

1205971199102

minus 1198881(11986522

minus 119867221198881)

1205973

1199080

1205971199103

minus (119865441198882

2

minus 2119863441198882+ 11986044)

times

1205971199080

120597119910

+ (2119863441198882minus 119865441198882

2

minus 11986044) 120601119910

= 1198691V0+ 1198702

120601119910minus 11988811198694

1205970

120597119910

(4e)


119909 = 0 119908 = 0 120601119910= 0 119873

119909119910= 0 119872

119909119909= 0

(5a)

119909 = 119886 119908 = 0 120601119910= 0 119873

119909119910= 0 119872

119909119909= 0

(5b)

119910 = 0 119908 = 0 120601119909= 0 119873

119909119910= 0 119872

119910119910= 0

(5c)

119910 = 119887 119908 = 0 120601119909= 0 119873

119909119910= 0 119872

119910119910= 0

(5d)

int

ℎ

0

119873119909119909

1003816100381610038161003816119909=0

119889119911 = minusint

ℎ

0

(1199020+ 119902119909cosΩ1119905) 119889119911 (5e)

int

ℎ

0

119873119910119910

10038161003816100381610038161003816119910=0

119889119911 = minusint

ℎ

0

(1199021+ 119902119910cosΩ2119905) 119889119911 (5f)


0 V0 1199080 120601119909 and 120601



0 V0 1199080 120601119909 and


1199060= 11990601(119905) cos 120587119909

2119886

cos120587119910

2119887

+ 11990602(119905) cos 3120587119909

2119886

cos120587119910

2119887

(6a)

V0= V1(119905) cos

120587119910

2119887

cos 1205871199092119886

+ V2(119905) cos

120587119910

2119887

cos 31205871199092119886

(6b)

1199080= 1199081(119905) sin 120587119909

119886

sin120587119910

119887

+ 1199082(119905) sin 3120587119909

119886

sin120587119910

119887

(6c)

120601119909= 1206011(119905) cos 120587119909

119886

sin120587119910

119887

+ 1206012(119905) cos 3120587119909

119886

sin120587119910

119887

(6d)

120601119910= 1206013(119905) cos

120587119910

119887

sin 120587119909

119886

+ 1206014(119905) cos

120587119910

119887

sin 3120587119909

119886

(6e)




01 11990602 V1 V2 1206011 1206012 1206013 and

1206014via 1199081and 119908

2as follows

11990601

= 11989611199082

1

+ 11989621199082

2

+ 119896311990811199082 (7a)

11990602

= 11989641199082

1

+ 11989651199082

2

+ 119896611990811199082 (7b)

V1= 11989671199082

1

+ 11989681199082

2

+ 119896911990811199082 (7c)

V2= 119896101199082

1

+ 119896111199082

2

+ 1198961211990811199082 (7d)

1206011= 119896191199081 120601

2= 119896201199082 (7e)

1206013= 119896211199081 120601

4= 119896221199082 (7f)



119906 =

1199060

119886

V =

V0

119887

119908 =

1199080

ℎ

120601119909

= 120601119909 120601

119910

= 120601119910 119909 =

119909

119886

119910 =

119910

119887

119902 =

1198872

119864ℎ3

119902 119902119909

=

1198872

119864ℎ3

119902119909 119902

119910

=

1198872

119864ℎ3

119902119910

119905 = 1205872

(

119864

119886119887120588

)

12

119905 Ω119894=

1

1205872

(

119886119887120588

119864

)

12

Ω119894

(119894 = 1 2)

119860119894119895=

(119886119887)12

119864ℎ2

119860119894119895 119861

119894119895=

(119886119887)12

119864ℎ3

119861119894119895

119863119894119895=

(119886119887)12

119864ℎ4

119863119894119895 119864

119894119895=

(119886119887)12

119864ℎ5

119864119894119895

119865119894119895=

(119886119887)12

119864ℎ6

119865119894119895 119867

119894119895=

(119886119887)12

119864ℎ8

119867119894119895

119868119894=

1

(119886119887)(119894+1)2

120588

119868119894

(8)


1+ 12058311+ 1205962

1

1199081


+ 11988651199082

1

1199082+ 11988661199082

2

1199081+ 11988671199083

1

+ 11988681199083

2

= 1198911cosΩ3119905

(9a)

2+ 12058322+ 1205962

2

1199082

+ (1198872cosΩ1119905 + 1198873cosΩ2119905 + 1198874cosΩ4119905) 1199082

+ 11988751199082

2

1199081+ 11988761199082

1

1199082+ 11988771199083

2

+ 11988781199083

1

= 1198912cosΩ3119905

(9b)



1205962

1

=

1205962

9

+ 1205761205901 120596

2

2

= 1205962

+ 1205761205902

Ω3= 120596 Ω

1= Ω2= Ω4=

2120596

3

1205962asymp 31205961

(10)

where 1205901and 120590



1199081(119905 120576) = 119909

10(1198790 1198791) + 120576119909

11(1198790 1198791) + sdot sdot sdot (11a)

1199082(119905 120576) = 119909

20(1198790 1198791) + 120576119909

21(1198790 1198791) + sdot sdot sdot (11b)

where 1198790= 119905 1198791= 120576119905


1= minus

1

2

12058311199091minus

1

2

12059011199092+

1

4

(1198862+ 1198863minus 1198864) 1199092

minus

3

2

11988671199092(1199092

1

+ 1199092

2

) minus

1

2

11988651199094(1199092

1

minus 1199092

2

)

minus 11988661199092(1199092

3

+ 1199092

4

) + 1198865119909111990921199093

(12a)

2= minus

1

2

12058311199092+

1

2

12059011199091+

1

4

(1198862+ 1198863minus 1198864) 1199091

+

3

2

11988671199091(1199092

1

+ 1199092

2

) +

1

2

11988651199093(1199092

1

minus 1199092

2

)

+ 11988661199091(1199092

3

+ 1199092

4

) + 1198865119909111990921199094

(12b)

3= minus

1

2

12058321199093minus

1

6

12059021199094minus

1

3

11988761199094(1199092

1

+ 1199092

2

)

minus

1

2

11988771199094(1199092

3

+ 1199092

4

) minus

1

6

11988781199092(31199092

1

minus 1199092

2

)

(12c)


4= minus

1

2

12058321199094+

1

6

12059021199093+

1

3

11988761199093(1199092

1

+ 1199092

2

)

+

1

2

11988771199093(1199092

3

+ 1199092

4

) +

1

6

11988781199091(1199092

1

minus 31199092

2

) minus

1

12

1198912

(12d)



2oplus 1198852and 119863

4





1= minus

1

2

12058311199091+ (1198910minus

1

2

1205901)1199092minus

3

2

11988671199092(1199092

1

+ 1199092

2

)

minus

1

2

11988651199094(1199092

1

minus 1199092

2

)

minus 11988661199092(1199092

3

+ 1199092

4

) + 1198865119909111990921199093

(13a)

2= minus

1

2

12058311199092+ (1198910+

1

2

1205901)1199091+

3

2

11988671199091(1199092

1

+ 1199092

2

)

+

1

2

11988651199093(1199092

1

minus 1199092

2

)

+ 11988661199091(1199092

3

+ 1199092

4

) + 1198865119909111990921199094

(13b)

3= minus

1

2

12058321199093minus

1

6

12059021199094minus

1

3

11988761199094(1199092

1

+ 1199092

2

)

minus

1

2

11988771199094(1199092

3

+ 1199092

4

) minus

1

6

11988781199092(31199092

1

minus 1199092

2

)

(13c)

4= minus

1

2

12058321199094+

1

6

12059021199093+

1

3

11988761199093(1199092

1

+ 1199092

2

)

+

1

2

11988771199093(1199092

3

+ 1199092

4

) +

1

6

11988781199091(1199092

1

minus 31199092

2

)

(13d)

where 1198910= (14)(119886

2+ 1198863minus 1198864)


1199091= 1199101minus

1

4

11988671199103

1

+

31198865(1205902minus 6)

1205902

2

1199102

1

1199103minus 119886611991011199102

3

minus 119886611991011199102

4

minus

31198865(1205903

2

minus 181205902

2

+ 2161205902minus 1296)

1205904

2

1199102

2

1199103

+

1198865(61205902

2

minus 721205902+ 432)

1205903

2

119910111991021199104

(14a)

1199092= 1199102+

3

2

11988671199103

2

+

3

4

11988671199102

1

1199102+

31198865

1205902

1199102

1

1199104

minus

31198865(1205902

2

minus 121205902+ 72)

1205903

2

1199102

2

1199104minus

61198865(1205902minus 6)

1205902

2

119910111991021199103

(14b)

1199093= 1199103minus

1198878

1205902

1199103

1

+

31198878(1205902

2

minus 121205902+ 72)

1205903

2

11991011199102

2

minus

1

3

1198876119910111991021199104

(14c)

1199094= 1199104+

1198878(1205903

2

minus 181205902

2

+ 2161205902minus 1296)

1205904

2

1199103

2

minus

31198878(1205902minus 6)

1205902

2

1199102

1

1199102+

1

3

1198876119910111991021199103

(14d)


1199101= minus1205831

1199101+ (1 minus 120590

1) 1199102 (15a)

1199102= 12059011199101minus 1205831

1199102+ 11988661199101(1199102

3

+ 1199102

4

) +

3

2

11988671199103

1

(15b)

1199103= minus1205832

1199103minus 12059021199104minus

1

3

11988761199102

1

1199104minus

1

2

11988771199104(1199102

3

+ 1199102

4

) (15c)

1199104= 12059021199103minus 1205832

1199104+

1

3

11988761199102

1

1199103+

1

2

11988771199103(1199102

3

+ 1199102

4

) minus 1198912

(15d)


= (12)1205831 1205832

= (12)1205832 1205902=

(16)1205902 and 119891

2


Further let

1199103= 119868 cos 120574 119910

4= 119868 sin 120574 (16)


1199101= minus1205831

1199101+ (1 minus 120590

1) 1199102 (17a)

1199102= 12059011199101minus 1205831

1199102+ 119886611991011198682

+

3

2

11988671199103

1

(17b)

119868 = minus1205832

119868 minus 1198912

sin 120574 (17c)

119868 120574 = 1205902119868 +

1

3

11988761199102

1

119868 +

1

2

11988771198683

minus 1198912

cos 120574 (17d)


[

1199101

1199102

] = radic3

radic10038161003816100381610038161198866

1003816100381610038161003816

radic10038161003816100381610038161198876

1003816100381610038161003816

[

1 minus 1205901

0

1205831

1] [

1199061

1199062

] (18)



1yield the


1= 1199062 (19a)

2= minus120583119906

1minus 12058331199062+ 12057811199063

1

+ 119886611990611198682

(19b)

119868 = minus1205832

119868 minus 1198912

sin 120574 (19c)

119868 120574 = 1205902119868 + 11988661199062

1

119868 + 12057221198683

minus 1198912

cos 120574 (19d)

where 120583 = 1205832

1

minus 1205901(1 minus 120590

1) 1205833= 21205831

1205781= 91198866119886721198876and

1205722= (12)119887

7


1205832

997888rarr 1205761205832

1205833997888rarr 120576120583

3 119891

2

997888rarr 1205761198912

1205781997888rarr 1205781 120572

2997888rarr 1205722 119886

6997888rarr 1198866

(20)


1=

120597119867

1205971199062

+ 1205761198921199061= 1199062 (21a)

2= minus

120597119867

1205971199061

+ 1205761198921199062= minus120583119906

1+ 12057811199063

1

+ 119886611990611198682

minus 12057612058331199062 (21b)

119868 =

120597119867

120597120574

+ 120576119892119868

= minus1205761205832

119868 minus 1205761198912

sin 120574 (21c)

119868 120574 = minus

120597119867

120597119868

+ 120576119892120574

= 1205902119868 + 12057221198683

+ 11988661198681199062

1

minus 1205761198912

cos 120574 (21d)


119867(1199061 1199062 119868 120574) =

1

2

1199062

2

+

1

2

1205831199062

1

minus

1

4

12057811199064

1

minus

1

2

11988661198682

1199062

1

minus

1

2

12059021198682

minus

1

4

12057221198684

(22)

and 1198921199061 1198921199062 119892119868 and 119892


1198921199061= 0 119892

1199062= minus12058331199062

119892119868

= minus1205832

119868 minus 1198912

sin 120574 119892120574

= minus1198912

cos 120574(23)



1 1199062) of (21a) (21b)


1= 1199062 (24a)

2= minus120583119906

1+ 12057811199063

1

+ 119886611990611198682

(24b)

Since 1205781


11988661198682




11986812

= plusmn[

1205832

1

minus 1205901(1 minus 120590

1)

1198866

]

12

(25)


0= (0 0)


119861 = plusmn

1

1205781

[1205832

1

minus 1205901(1 minus 120590

1) minus 11988661198682

]

12

(26)


0is the



1198681= 0 119868

2= [

1205832

1

minus 1205901(1 minus 120590

1)

1198866

]

12

(27)




(1198791 119868) that is lim


119906ℎ

plusmn

(1198791 119868) =


defined by

119872 = (119906 119868 120574) | 119906 = 119902plusmn(119868) 1198681lt 119868 lt 119868

2 0 le 120574 lt 2120587 (28)



119904

(119872) and 119882119906



119906

(119872)


Γ = (119906 119868 120574) | 119906 = 119906ℎ

plusmn

(1198791 119868) 1198681lt 119868 lt 119868

2

120574 = int

1198791

0

119863119868119867(119906ℎ

plusmn

(1198791 119868) 119868) 119889119904 + 120574

0

(29)


119868 = 0 (30a)

119868 120574 = 119863119868119867(119902plusmn(119868) 119868) 119868

1lt 119868 lt 119868

2 (30b)


where

119863119868119867(119902plusmn(119868) 119868) = minus

120597119867 (119902plusmn(119868) 119868)

120597119868

= 1205902119868 + 12057221198683

+ 11988661198681199062

1

(31)







119863119868119867(119902plusmn 119868) = 120590

2119868119903+ 12057221198683

119903

+ 11988661198681199031199062

1

= 0 (32)

Then we obtain

119868119903= plusmn

12059021205781+ 1198866[1205832

1

minus 1205901(1 minus 120590

1)]

1198862

6

minus 12057221205781

12

(33)



119903) (34)




1 1199062 119868 120574) The phase shift



61198682 and 120583


1= 1199062 (35a)

2= minus12057611199061+ 12057811199063

1

minus 12057612057621199062 (35b)


119867(1199061 1199062) =

1

2

1199062

2

+

1

2

12057611199062

1

minus

1

4

12057811199064

1

(36)






1199061(1198791) = plusmnradic

1205761

1205781

tanh(

radic21205761

2

1198791) (37a)

1199062(1198791) = plusmn

1205761

radic21205781


2

1198791) (37b)


120574 (1198791) = 1205961199031198791minus

1198866radic21205761

1205781

tanh(

radic21205761

2

1198791) + 1205740 (38)

where 120596119903= 1205902+ 12057221198682

+ 119886612057611205781

At 119868 = 119868119903 there is 120596


be expressed as

Δ120574 = [minus

21198866radic21205761

1205781

]

119868=119868119903

= minus

21198866

1205781

radic2 [1205832

1

minus 1205901(1 minus 120590

1) minus 11988661198682

119903

]

(39)



1 1199062) for the




plusmn(119868) in (24a)


M

I

120574

times

u2

u1

(a)

0 1205742120587

I = I1

I = Ir

I = I2

(b)




120576

Therefore we obtain

119872 = 119872120576= (119906 119868 120574) | 119906 = 119902

plusmn(119868) 1198681lt 119868 lt 119868

2 0 le 120574 lt 2120587

(40)


119868 = minus1205832

119868 minus 1198912

sin 120574 (41a)

120574 = 1205902+ 12057221198682

+ 11988661199062

1

minus

1198912

cos 120574119868

(41b)


1205832

997888rarr 1205761205832

119868 = 119868119903+ radic120576ℎ

1198912

997888rarr 1205761198912

1198791997888rarr

1198791

radic120576

(42)


ℎ = minus120583

2

119868119903minus 1198912

sin 120574 minus radic1205761205832

ℎ (43a)

120574 = minus

2120575

1205781

119868119903ℎ minus radic120576(

120575

1205781

ℎ2

+

1198912

119868119903

cos 120574) (43b)

where 120575 = 1198862

6

minus 12057221205781

When 120576 = 0 (43a) (43b) become

ℎ = minus120583

2

119868119903minus 1198912

sin 120574 (44a)

120574 = minus

2120575

1205781

119868119903ℎ (44b)


119863(ℎ 120574) = minus120583

2

119868119903120574 + 1198912

cos 120574 +

120575

1205781

119868119903ℎ2

(45)


1198750= (0 120574


1205832

119868119903

1198912

))

1198760= (0 120574

119904) = (0 120587 + arcsin(

1205832

119868119903

1198912

))

(46)


0and119876









120576 The phase




120574min minus

1198912

1205832

119868119903

cos 120574min = 120587 + arcsin1205832

119868119903

1198912

+

radic119891

2

2

minus 1205832

2

1198682

119903

1205832

119868119903

(47)


119903as

119860120576= (1199061 1199062 119868 120574) | 119906

1= 119861 119906

2= 0

1003816100381610038161003816119868 minus 119868119903

1003816100381610038161003816lt radic120576119862 120574 isin 119879

119871

(48)



120576 denoted as119882119904(119860

120576) and119882

119906

(119860120576) are the


119906






120576 leave and


h

120574

P0

120574min 120574c 120574s

Q0

(a)

h

120574Q120576

P120576

(b)








120574

|u|

h

120574c 120574c + 3Δ120574

p120576qc




119896(120576 119868 120574

0 1205832


0 and 120583

2




function 1198721(120576 119868119903 1205740 1205832


119903 1205740 1205832



(119872) and119882119906


119872(119868119903 1205740 1205832

1205781 1198866 1205761)

= int

+infin

minusinfin

⟨n (119901ℎ

(119905)) g (119901ℎ (119905) 1205832

0)⟩ 1198891198791


= int

+infin

minusinfin

(

120597119867

1205971199061

1198921199061+

120597119867

1205971199062

1198921199062+

120597119867

120597119868

119892119868

+

120597119867

120597120574

119892120574

)1198891198791

= minus

2radic21205833

31205781

12057632

1

minus 2radic211988661205832

1198682

119903

12057612

1

1205781

minus 1198912

119868119903[cos(120574

0minus 1198866

radic21205761

1205781

) minus cos(1205740+ 1198866

radic21205761

1205781

)]

(49)


) (119896 = 1 2 ) is defined as

119872119896(120576 119868119903 1205740 1205832

)

=

119896minus1

sum

119895=0

119872(119868119903 119895Δ120574 (119868

119903) + Γ119895(120576 119868119903 1205740 1205832

) + 1205740 1205832

)

(50)

where

Γ119895(120576 119868119903 1205740 1205832

) =

Ω (1199090(119868119903) 119868119903)

120582 (119868119903)

119895

sum

119903=1

log100381610038161003816100381610038161003816100381610038161003816

120589 (119868119903)

120576119872119903(120576 119868119903 1205740 1205832

)

100381610038161003816100381610038161003816100381610038161003816

(51)

for 119895 = 1 119896 minus 1 and Γ0(120576 119868119903 1205740 1205832


119895(120576 119868119903 1205740 1205832


119896(120576 119868119903 1205740 1205832


119903degenerates



119872119896(119868119903 1205740 1205832

1205781 1198866 1205761)

=

119896minus1

sum

119895=0

119872(119868119903 1205740+ 119895Δ120574 (119868

119903) 1205832

1205781 1198866 1205761)

= minus1198912

119868119903[cos(120574

0minus

1198866radic21205761

1205781

)

minus cos(1205740+

1198866radic21205761

1205781

)]

minus

2radic21205833

31205781

12057632

1

minus 2radic211988661205832

1198682

119903

12057612

1

1205781

minus 1198912

119868119903[cos(120574

0minus

1198866radic21205761

1205781

minus

21198866radic21205761

1205781

)

minus cos(1205740+

1198866radic21205761

1205781

minus

21198866radic21205761

1205781

)]

minus

2radic21205833

31205781

12057632

1

minus 2radic211988661205832

1198682

119903

12057612

1

1205781

+ sdot sdot sdot

minus 1198912

119868119903[cos(120574

0minus

1198866radic21205761

1205781

minus 2 (119896 minus 1) 1198866

radic21205761

1205781

)

minus cos(1205740+

1198866radic21205761

1205781


1198866radic21205761

1205781

)]

minus

2radic21205833

31205781

12057632

1

minus 2radic211988661205832

1198682

119903

12057612

1

1205781

= minus1198912

119868119903[cos(120574

0minus

1198866radic21205761

1205781


1198866radic21205761

1205781

)

minus cos(1205740+

1198866radic21205761

1205781

)]

minus

2radic21198961205833

31205781

12057632

1

minus 2radic211989611988661205832

1198682

119903

12057612

1

1205781

(52)


119896minus1= 1205740+ (119896 minus


119872119896(119868119903 1205740 1205832

1205781 1198866 1205761)

= 119872119896(119868119903 120574119896minus1


Δ120574

2

1205832

1205781 1198866 1205761)

= minus1198912

119868119903[cos(120574

119896minus1+

1

2

119896Δ120574)


minus

1

2

119896Δ120574)]

+

11989612058331205761

31198866

Δ120574 + 1205832

1198682

119903

(119896Δ120574)

= 21198912

119868119903sin 120574119896minus1

sin(

1

2

119896Δ120574)

+

12058331205761

31198866

(119896Δ120574) + 1205832

1198682

119903

(119896Δ120574)

(53)


10038161003816100381610038161003816100381610038161003816100381610038161003816

(12) 119896Δ120574

sin ((12) 119896Δ120574)

(12058331205761+ 311988661205832

1198682

119903

)

311988661198912

119868119903

10038161003816100381610038161003816100381610038161003816100381610038161003816

lt 1

1

2

119896Δ120574 = 119899120587 119899 = 0 plusmn1 plusmn2

(54)


119885119899

minus

= (119868119903 120574119896minus1

1205832

1205781 1198866 1205761) | 119872119896= 0119863

1205740

119872119896

= 0 (55)



119896minus1isin [0 120587]

120574119896minus11

= minus arcsin(12) 119896Δ120574

sin ((12) 119896Δ120574)

(12058331205761+ 311988661205832

1198682

119903

)

(311988661198912

119868119903)

120574119896minus12

= 120587 minus 120574119896minus11

(56)


3 1205761 1205832

1198866 and 119891

2


119896minus1= 120574119896minus11

and 120574119896minus1

= 120574119896minus12

=

120587 minus 120574119896minus11


119895(119868119903 1205740119894

1205832



119906

(119872120576) intersect


1205832120578111988661205761

plusmn120576(120574119896minus1119894


plusmn0

(120574119896minus1119894


119903 1205740

=

120574119896minus1119894


119895(119868119903 1205740119894

1205832

1205781 1198866 1205761)


0119894

= 120574119896minus1119894

minus (119896 minus 1)(Δ1205742) + 119895Δ120574


119895(119868119903 1205740119894

1205832



01



plusmn120576(120574119896minus11

)

and sum1205832120578111988661205761

plusmn120576(120574119896minus12


1205832120578111988661205761

plusmn0

(120574119896minus11

) and sum1205832120578111988661205761

plusmn0

(120574119896minus12


119906


n = ((minus1205831199061+ 12057811199063

1

+ 11988661198682

1199061) minus1199062 0 0) (57)


plusmn0

(120574119896minus11

)


plusmn0

(120574119896minus12





119894(120583) and 119888



the end points 119889119894(120583) to 119888


end points 119889119894(120583) and 119888



0119894

(119868119903 120583) minus Δ120574

minus



119894(120583) for 119894 =

2 119873 while the line 120574 = 1205740119894

(119868119903 120583) + Δ120574

+



(120583) at the end point 119889119894+1


119894(120583) and 119889

119894+1(120583) is zero

namely

ℎ (119888119894(120583)) minus ℎ (119889

119894+1(120583)) = 0 (58)



sum1205832120578111988661205761

0

(1205740119894


119894(120583) and 119889


1


(12057401



2(120583) on the


119873


0

(1205740119873


119873(120583) on the segments 119874


119889119873+1



sum1205832120578111988661205761

0

(1205740119894


1(120583) and 119874

2(120583) of (44a) (44b) It is known



(119868119903 120583) +

Δ120574+

(119868119903) and 120574 = 120574

0119894

(119868119903 120583) minus Δ120574

minus

(119868119903)


0

(120574119896minus11

) andsum1205832120578111988661205761

0

(120574119896minus12


01

minus


01

minusΔ120574+








119897and









120574

|u|h

Γ3

Γ2

Γ1

M

O2O3






1


6 where

1205781



6 1198867 and


1gt 0



2


2and 119886



4is the


1 1198862 1198864






3 respectively when




120574

|u|h

M

Γsum (12057401 12057402)

12057402 + Δ120574+ 12057401 + Δ120574+



12057402

) are depicted


2 1205831 1198862





2= 2 sim 200


183 1205902

= 197 1205831

= 02 1205832

= 02 1198862

= 230 1198863

=

120 1198864= 130 119886

5= minus101 119886

6= minus203 119886

7= minus205 119887

6=

407 1198877= minus308 119887

8= 109 119909

10= minus001 119909

20= minus005 119909

30=

minus001 11990940


1 1198912) and (119909

3 1198912) respec-






1 The figure demon-



1=


2=



50 100 150minus2

minus15

minus1

minus05

0

05

1

x1

f2

(a)

20 40 60 80 100 120 140 160 180

minus3

minus25

minus2

minus15

x3

f2

(b)



= minus001 11990920

= minus005 11990930

=

minus001 11990940


1198912


1198912

)

0 01 02 03 04 05 06 07minus25

minus2

minus15

minus1

minus05

0

05

1

15

x1

1205831

(a)

0 01 02 03 04 05 06 07minus32

minus3

minus28

minus26

minus24

minus22

minus2

minus18

x1

1205831

x3

1205831

x3

(b)




= minus001 11990920

= minus005 11990930

= minus001 11990940


1205831


3

1205831

)

the planes (1199091 1205831) and (119909









3 1198862)




4varies from 119886

4= 2 to 119886

4= 120


2is

chosen as 1198862


1 1198864) and (119909

3 1198864)







0 10 20 30 40 50 60

minus3

minus2

minus1

0

1

x1

a2

(a)

0 10 20 30 40 50 60minus34

minus32

minus3

minus28

minus26

minus24

minus22

minus2

minus18

a2

x3

(b)



= 02 and 1205832


2


= minus001 11990920

= minus005 11990930

= minus001 11990940


1198862


1198862

)

0 20 40 60 80 100 120minus25

minus2

minus15

minus1

minus05

0

05

1

x1

a4

(a)

0 20 40 60 80 100 120

minus35

minus3

minus25

minus2

a4

x3

(b)




= 02 and 1205832


= 23 and 1198863


= minus001 11990920

= minus005 11990930

= minus001 11990940

=


1198864


1198864

)


1are selected as











1= 1437 120590

2=

1142 1205831= 02 120583

2= 02 119886

2= 300 119886

3= 750 119886

4= 450

1198865

= minus1166 1198866

= 1227 1198912

= 1227 1198867

= minus268 1198876

=

minus22 1198877

= minus969 1198878

= minus2232 11990910

= minus108 11990920

= 0511990930

= minus001 11990940



minus1minus050051

minus05

0

05

minus25

minus2

minus15

minus1

x1

x2

x3

(a)


minus1

minus05

0

05

1

x1

x2

(b)


= 827 and 1205831


1199092

1199093


1199092

)

minus50

5

minus50

5minus6

minus4

minus2

0

2

4

6

x1x2

x3

(a)


minus10

minus5

0

5

10

x1

x2

(b)


= 1437 1205902

= 1142 1198862

= 300 1198863

= 750 1198864

= 450 1198865

= minus1166 1198866

=

1227 1198912

= 1227 1198867

= minus268 1198876

= minus22 1198877

= minus969 1198878

= minus2232 11990910

= minus108 11990920

= 05 11990930

= minus001 11990940


1

1199092

1199093


1199092

)





2= 313 119886

5=

minus1501 1198878

= 409 11990910

= minus001 11990920

= minus009 11990930

=

minus005 11990940


7 Conclusions




minus50

5

minus10

0

10

minus6

minus4

minus2

0

2

x1x2

x3

(a)

minus5 0 5 10

minus10

minus5

0

5

10

x1

x2

(b)


= 361 1205902

= 313 1198862

= 230 1198863

= 120 1198864

= 130 1198865

= minus1501 1198866

=

minus203 1198912

= 9238 1198867

= minus205 1198876

= 407 1198877

= minus308 1198878

= 409 11990910

= minus001 11990920

= minus009 11990930

= minus005 11990940


1

1199092

1199093


1199092

)



119896(120576 119868 120574

0 1205832




2 the in-plane


4 and the damping




1 respec-










Acknowledgments



References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















119906 (119909 119910 119911 119905) = 1199060(119909 119910 119905) + 119911120601

119909(119909 119910 119905)

minus 1199113

4

3ℎ2

(120601119909+

1205971199080

120597119909

)

(1a)

V (119909 119910 119911 119905) = V0(119909 119910 119905) + 119911120601

119910(119909 119910 119905)

minus 1199113

4

3ℎ2

(120601119910+

1205971199080

120597119910

)

(1b)

119908 (119909 119910 119911 119905) = 1199080(119909 119910 119905) (1c)



119909and 120601

119910represent the



120576119909119909

=

120597119906

120597119909

+

1

2

(

120597119908

120597119909

)

2

120576119909119911

=

1

2

(

120597119906

120597119911

+

120597119908

120597119909

)

120576119909119910

=

1

2

(

120597119906

120597119910

+

120597V120597119909

+

120597119908

120597119909

120597119908

120597119910

)

120576119910119910

=

120597V120597119910

+

1

2

(

120597119908

120597119910

)

2

120576119910119911

=

1

2

(

120597V120597119911

+

120597119908

120597119910

)

120576119911119911

=

120597119908

120597119911

(2)


120590119894119895= 120590119904

119894119895119896119897

120576119896119897minus 119890119894119895119896119864119896 (119894 119895 119896 119897 = 119909 119910 119911) (3)





119894119895is the





11986011

1205972

1199060

1205971199092

+ 11986066

1205972

1199060

1205971199102

+ (11986012

+ 11986066)

1205972V0

120597119909120597119910

+ 11986011

1205971199080

120597119909

1205972

1199080

1205971199092

+ 11986066

1205971199080

120597119909

1205972

1199080

1205971199102

+ (11986012

+ 11986066)

1205971199080

120597119910

1205972

1199080

120597119909120597119910

= 11986800+ 1198691

120601119909minus 11988811198683

1205970

120597119909

(4a)

11986066

1205972V0

1205971199092

+ 11986022

1205972V0

1205971199102

+ (11986021

+ 11986066)

1205972

1199060

120597119909120597119910

+ 11986066

1205971199080

120597119910

1205972

1199080

1205971199092

+ 11986022

1205971199080

120597119910

1205972

1199080

1205971199102

+ (11986021

+ 11986066)

1205971199080

120597119909

1205972

1199080

120597119909120597119910

= 1198680V0+ 1198691

120601119910minus 11988811198683

1205970

120597119910

(4b)

11986066

1205971199080

120597119909

1205972

1199060

1205971199102

minus 119867221198882

1

1205974

1199080

1205971199104

+ 1198881(211986566

+ 11986512

minus 2119867661198881minus 119867121198881)

1205973

120601119910

1205971199101205971199092

+ 1198881(11986522

minus 119867221198881)

1205973

120601119910

1205971199103

minus 119867111198882

1

1205974

1199080

1205971199094

+ 11986011

1205971199080

120597119909

1205972

1199060

1205971199092

+ (119865441198882

2

minus 2119863441198882+ 11986044)

120597120601119910

120597119910

+ 1198881(11986521

+ 211986566

minus 119867211198881minus 2119867661198881)

1205973

120601119909

1205971199102

120597119909

minus 1198882

1

(11986721

+ 411986766

+ 11986712)

1205974

1199080

1205971199102

1205971199092

+ (11986044

minus 119873119875

119910

cos (Ω4119905) + 119865

441198882

2

minus 2119863441198882)

1205972

1199080

1205971199102

minus

120597119873119875

119910

120597119910

cos (Ω2119905)

1205971199080

120597119910

+ (11986021

+ 411986066

+ 11986012)

times

1205971199080

120597119909

1205971199080

120597119910

1205972

1199080

120597119910120597119909

+ 1198881(11986511

minus 119867111198881)

1205973

120601119909

1205971199093

+ (11986021

+ 11986066)

1205971199080

120597119910

1205972

1199060

120597119910120597119909

+ 11986021

1205971199060

120597119909

1205972

1199080

1205971199102

times 11986066

1205971199080

120597119910

1205972V0

1205971199092

+ 11986022

1205971199080

120597119910

1205972V0

1205971199102


+

1

2

(11986012

+ 211986066) (

1205971199080

120597119910

)

2

1205972

1199080

1205971199092

+ 11986022

1205972

1199080

1205971199102

120597V0

120597119910

+ (11986012

+ 11986066)

1205971199080

120597119909

1205972V0

120597119910120597119909

+

1

2

(11986021

+ 211986066)

1205972

1199080

1205971199102

times (

1205971199080

120597119909

)

2

+

3

2

11986011(

1205971199080

120597119909

)

2

1205972

1199080

1205971199092

+ 11986011

1205972

1199080

1205971199092

1205971199060

120597119909

+ 11986012

1205972

1199080

1205971199092

120597V0

120597119910

+ 211986066

1205972

1199080

120597119910120597119909

120597V0

120597119909

+ 211986066

1205972

1199080

120597119910120597119909

1205971199060

120597119910

+

3

2

11986022(

1205971199080

120597119910

)

2

1205972

1199080

1205971199102

+ (11986055

+ 119902119909cos (Ω

1119905) minus 119873

119875

119909

cos (Ω4119905)

+ 119865551198882

2

minus 2119863551198882)

1205972

1199080

1205971199092

minus

120597119873119875

119909

120597119909

cos (Ω3119905)

120597119908

120597119909

+ (119865551198882

2

minus 2119863551198882+ 11986055)

120597120601119909

120597119909

minus 119902 cos (Ω3119905) + 119896119891

1205971199080

120597119905

= 11986800minus 1198882

1

1198686(

1205972

0

1205971199092

+

1205972

0

1205971199102

)

+ 11988811198683(

1205970

120597119909

+

120597V0

120597119910

) + 11988811198694(

120597

120601119909

120597119909

+

120597

120601119910

120597119910

)

(4c)

(11986311

minus 2119865111198881+ 119867111198882

1

)

1205972

120601119909

1205971199092

+ (11986366

minus 2119865661198881+ 119867661198882

1

)

1205972

120601119909

1205971199102

minus 1198881(11986511

minus 119867111198881)

1205973

1199080

1205971199093

minus (119865551198882

2

minus 2119863551198882+ 11986055)

1205971199080

120597119909

+ (11986312

+ 11986366

+ 119867661198882

1

minus 2119865661198881+ 119867121198882

1

minus 2119865121198881)

times

1205972

120601119910

120597119910120597119909

minus 1198881(211986566

+ 11986512

minus 2119867661198881minus 119867121198881)

times

1205973

1199080

1205971199102

120597119909

+ (2119863551198882minus 11986055

minus 119865551198882

2

) 120601119909

= 11986910+ 1198702

120601119909minus 11988811198694

1205970

120597119909

(4d)

(11986366

minus 2119865661198881+ 119867661198882

1

)

1205972

120601119910

1205971199092

minus 1198881(11986521

+ 211986566

minus 119867211198881minus 2119867661198881)

1205973

1199080

1205971199101205971199092

+ (119867211198882

1

+ 11986366

+ 11986321

minus 2119865211198881+ 119867661198882

1

minus 2119865661198881)

times

1205972

120601119909

120597119910120597119909

+ (119867221198882

1

+ 11986322

minus 2119865221198881)

1205972

120601119910

1205971199102

minus 1198881(11986522

minus 119867221198881)

1205973

1199080

1205971199103

minus (119865441198882

2

minus 2119863441198882+ 11986044)

times

1205971199080

120597119910

+ (2119863441198882minus 119865441198882

2

minus 11986044) 120601119910

= 1198691V0+ 1198702

120601119910minus 11988811198694

1205970

120597119910

(4e)


119909 = 0 119908 = 0 120601119910= 0 119873

119909119910= 0 119872

119909119909= 0

(5a)

119909 = 119886 119908 = 0 120601119910= 0 119873

119909119910= 0 119872

119909119909= 0

(5b)

119910 = 0 119908 = 0 120601119909= 0 119873

119909119910= 0 119872

119910119910= 0

(5c)

119910 = 119887 119908 = 0 120601119909= 0 119873

119909119910= 0 119872

119910119910= 0

(5d)

int

ℎ

0

119873119909119909

1003816100381610038161003816119909=0

119889119911 = minusint

ℎ

0

(1199020+ 119902119909cosΩ1119905) 119889119911 (5e)

int

ℎ

0

119873119910119910

10038161003816100381610038161003816119910=0

119889119911 = minusint

ℎ

0

(1199021+ 119902119910cosΩ2119905) 119889119911 (5f)


0 V0 1199080 120601119909 and 120601



0 V0 1199080 120601119909 and


1199060= 11990601(119905) cos 120587119909

2119886

cos120587119910

2119887

+ 11990602(119905) cos 3120587119909

2119886

cos120587119910

2119887

(6a)

V0= V1(119905) cos

120587119910

2119887

cos 1205871199092119886

+ V2(119905) cos

120587119910

2119887

cos 31205871199092119886

(6b)

1199080= 1199081(119905) sin 120587119909

119886

sin120587119910

119887

+ 1199082(119905) sin 3120587119909

119886

sin120587119910

119887

(6c)

120601119909= 1206011(119905) cos 120587119909

119886

sin120587119910

119887

+ 1206012(119905) cos 3120587119909

119886

sin120587119910

119887

(6d)

120601119910= 1206013(119905) cos

120587119910

119887

sin 120587119909

119886

+ 1206014(119905) cos

120587119910

119887

sin 3120587119909

119886

(6e)




01 11990602 V1 V2 1206011 1206012 1206013 and

1206014via 1199081and 119908

2as follows

11990601

= 11989611199082

1

+ 11989621199082

2

+ 119896311990811199082 (7a)

11990602

= 11989641199082

1

+ 11989651199082

2

+ 119896611990811199082 (7b)

V1= 11989671199082

1

+ 11989681199082

2

+ 119896911990811199082 (7c)

V2= 119896101199082

1

+ 119896111199082

2

+ 1198961211990811199082 (7d)

1206011= 119896191199081 120601

2= 119896201199082 (7e)

1206013= 119896211199081 120601

4= 119896221199082 (7f)



119906 =

1199060

119886

V =

V0

119887

119908 =

1199080

ℎ

120601119909

= 120601119909 120601

119910

= 120601119910 119909 =

119909

119886

119910 =

119910

119887

119902 =

1198872

119864ℎ3

119902 119902119909

=

1198872

119864ℎ3

119902119909 119902

119910

=

1198872

119864ℎ3

119902119910

119905 = 1205872

(

119864

119886119887120588

)

12

119905 Ω119894=

1

1205872

(

119886119887120588

119864

)

12

Ω119894

(119894 = 1 2)

119860119894119895=

(119886119887)12

119864ℎ2

119860119894119895 119861

119894119895=

(119886119887)12

119864ℎ3

119861119894119895

119863119894119895=

(119886119887)12

119864ℎ4

119863119894119895 119864

119894119895=

(119886119887)12

119864ℎ5

119864119894119895

119865119894119895=

(119886119887)12

119864ℎ6

119865119894119895 119867

119894119895=

(119886119887)12

119864ℎ8

119867119894119895

119868119894=

1

(119886119887)(119894+1)2

120588

119868119894

(8)


1+ 12058311+ 1205962

1

1199081


+ 11988651199082

1

1199082+ 11988661199082

2

1199081+ 11988671199083

1

+ 11988681199083

2

= 1198911cosΩ3119905

(9a)

2+ 12058322+ 1205962

2

1199082

+ (1198872cosΩ1119905 + 1198873cosΩ2119905 + 1198874cosΩ4119905) 1199082

+ 11988751199082

2

1199081+ 11988761199082

1

1199082+ 11988771199083

2

+ 11988781199083

1

= 1198912cosΩ3119905

(9b)



1205962

1

=

1205962

9

+ 1205761205901 120596

2

2

= 1205962

+ 1205761205902

Ω3= 120596 Ω

1= Ω2= Ω4=

2120596

3

1205962asymp 31205961

(10)

where 1205901and 120590



1199081(119905 120576) = 119909

10(1198790 1198791) + 120576119909

11(1198790 1198791) + sdot sdot sdot (11a)

1199082(119905 120576) = 119909

20(1198790 1198791) + 120576119909

21(1198790 1198791) + sdot sdot sdot (11b)

where 1198790= 119905 1198791= 120576119905


1= minus

1

2

12058311199091minus

1

2

12059011199092+

1

4

(1198862+ 1198863minus 1198864) 1199092

minus

3

2

11988671199092(1199092

1

+ 1199092

2

) minus

1

2

11988651199094(1199092

1

minus 1199092

2

)

minus 11988661199092(1199092

3

+ 1199092

4

) + 1198865119909111990921199093

(12a)

2= minus

1

2

12058311199092+

1

2

12059011199091+

1

4

(1198862+ 1198863minus 1198864) 1199091

+

3

2

11988671199091(1199092

1

+ 1199092

2

) +

1

2

11988651199093(1199092

1

minus 1199092

2

)

+ 11988661199091(1199092

3

+ 1199092

4

) + 1198865119909111990921199094

(12b)

3= minus

1

2

12058321199093minus

1

6

12059021199094minus

1

3

11988761199094(1199092

1

+ 1199092

2

)

minus

1

2

11988771199094(1199092

3

+ 1199092

4

) minus

1

6

11988781199092(31199092

1

minus 1199092

2

)

(12c)


4= minus

1

2

12058321199094+

1

6

12059021199093+

1

3

11988761199093(1199092

1

+ 1199092

2

)

+

1

2

11988771199093(1199092

3

+ 1199092

4

) +

1

6

11988781199091(1199092

1

minus 31199092

2

) minus

1

12

1198912

(12d)



2oplus 1198852and 119863

4





1= minus

1

2

12058311199091+ (1198910minus

1

2

1205901)1199092minus

3

2

11988671199092(1199092

1

+ 1199092

2

)

minus

1

2

11988651199094(1199092

1

minus 1199092

2

)

minus 11988661199092(1199092

3

+ 1199092

4

) + 1198865119909111990921199093

(13a)

2= minus

1

2

12058311199092+ (1198910+

1

2

1205901)1199091+

3

2

11988671199091(1199092

1

+ 1199092

2

)

+

1

2

11988651199093(1199092

1

minus 1199092

2

)

+ 11988661199091(1199092

3

+ 1199092

4

) + 1198865119909111990921199094

(13b)

3= minus

1

2

12058321199093minus

1

6

12059021199094minus

1

3

11988761199094(1199092

1

+ 1199092

2

)

minus

1

2

11988771199094(1199092

3

+ 1199092

4

) minus

1

6

11988781199092(31199092

1

minus 1199092

2

)

(13c)

4= minus

1

2

12058321199094+

1

6

12059021199093+

1

3

11988761199093(1199092

1

+ 1199092

2

)

+

1

2

11988771199093(1199092

3

+ 1199092

4

) +

1

6

11988781199091(1199092

1

minus 31199092

2

)

(13d)

where 1198910= (14)(119886

2+ 1198863minus 1198864)


1199091= 1199101minus

1

4

11988671199103

1

+

31198865(1205902minus 6)

1205902

2

1199102

1

1199103minus 119886611991011199102

3

minus 119886611991011199102

4

minus

31198865(1205903

2

minus 181205902

2

+ 2161205902minus 1296)

1205904

2

1199102

2

1199103

+

1198865(61205902

2

minus 721205902+ 432)

1205903

2

119910111991021199104

(14a)

1199092= 1199102+

3

2

11988671199103

2

+

3

4

11988671199102

1

1199102+

31198865

1205902

1199102

1

1199104

minus

31198865(1205902

2

minus 121205902+ 72)

1205903

2

1199102

2

1199104minus

61198865(1205902minus 6)

1205902

2

119910111991021199103

(14b)

1199093= 1199103minus

1198878

1205902

1199103

1

+

31198878(1205902

2

minus 121205902+ 72)

1205903

2

11991011199102

2

minus

1

3

1198876119910111991021199104

(14c)

1199094= 1199104+

1198878(1205903

2

minus 181205902

2

+ 2161205902minus 1296)

1205904

2

1199103

2

minus

31198878(1205902minus 6)

1205902

2

1199102

1

1199102+

1

3

1198876119910111991021199103

(14d)


1199101= minus1205831

1199101+ (1 minus 120590

1) 1199102 (15a)

1199102= 12059011199101minus 1205831

1199102+ 11988661199101(1199102

3

+ 1199102

4

) +

3

2

11988671199103

1

(15b)

1199103= minus1205832

1199103minus 12059021199104minus

1

3

11988761199102

1

1199104minus

1

2

11988771199104(1199102

3

+ 1199102

4

) (15c)

1199104= 12059021199103minus 1205832

1199104+

1

3

11988761199102

1

1199103+

1

2

11988771199103(1199102

3

+ 1199102

4

) minus 1198912

(15d)


= (12)1205831 1205832

= (12)1205832 1205902=

(16)1205902 and 119891

2


Further let

1199103= 119868 cos 120574 119910

4= 119868 sin 120574 (16)


1199101= minus1205831

1199101+ (1 minus 120590

1) 1199102 (17a)

1199102= 12059011199101minus 1205831

1199102+ 119886611991011198682

+

3

2

11988671199103

1

(17b)

119868 = minus1205832

119868 minus 1198912

sin 120574 (17c)

119868 120574 = 1205902119868 +

1

3

11988761199102

1

119868 +

1

2

11988771198683

minus 1198912

cos 120574 (17d)


[

1199101

1199102

] = radic3

radic10038161003816100381610038161198866

1003816100381610038161003816

radic10038161003816100381610038161198876

1003816100381610038161003816

[

1 minus 1205901

0

1205831

1] [

1199061

1199062

] (18)



1yield the


1= 1199062 (19a)

2= minus120583119906

1minus 12058331199062+ 12057811199063

1

+ 119886611990611198682

(19b)

119868 = minus1205832

119868 minus 1198912

sin 120574 (19c)

119868 120574 = 1205902119868 + 11988661199062

1

119868 + 12057221198683

minus 1198912

cos 120574 (19d)

where 120583 = 1205832

1

minus 1205901(1 minus 120590

1) 1205833= 21205831

1205781= 91198866119886721198876and

1205722= (12)119887

7


1205832

997888rarr 1205761205832

1205833997888rarr 120576120583

3 119891

2

997888rarr 1205761198912

1205781997888rarr 1205781 120572

2997888rarr 1205722 119886

6997888rarr 1198866

(20)


1=

120597119867

1205971199062

+ 1205761198921199061= 1199062 (21a)

2= minus

120597119867

1205971199061

+ 1205761198921199062= minus120583119906

1+ 12057811199063

1

+ 119886611990611198682

minus 12057612058331199062 (21b)

119868 =

120597119867

120597120574

+ 120576119892119868

= minus1205761205832

119868 minus 1205761198912

sin 120574 (21c)

119868 120574 = minus

120597119867

120597119868

+ 120576119892120574

= 1205902119868 + 12057221198683

+ 11988661198681199062

1

minus 1205761198912

cos 120574 (21d)


119867(1199061 1199062 119868 120574) =

1

2

1199062

2

+

1

2

1205831199062

1

minus

1

4

12057811199064

1

minus

1

2

11988661198682

1199062

1

minus

1

2

12059021198682

minus

1

4

12057221198684

(22)

and 1198921199061 1198921199062 119892119868 and 119892


1198921199061= 0 119892

1199062= minus12058331199062

119892119868

= minus1205832

119868 minus 1198912

sin 120574 119892120574

= minus1198912

cos 120574(23)



1 1199062) of (21a) (21b)


1= 1199062 (24a)

2= minus120583119906

1+ 12057811199063

1

+ 119886611990611198682

(24b)

Since 1205781


11988661198682




11986812

= plusmn[

1205832

1

minus 1205901(1 minus 120590

1)

1198866

]

12

(25)


0= (0 0)


119861 = plusmn

1

1205781

[1205832

1

minus 1205901(1 minus 120590

1) minus 11988661198682

]

12

(26)


0is the



1198681= 0 119868

2= [

1205832

1

minus 1205901(1 minus 120590

1)

1198866

]

12

(27)




(1198791 119868) that is lim


119906ℎ

plusmn

(1198791 119868) =


defined by

119872 = (119906 119868 120574) | 119906 = 119902plusmn(119868) 1198681lt 119868 lt 119868

2 0 le 120574 lt 2120587 (28)



119904

(119872) and 119882119906



119906

(119872)


Γ = (119906 119868 120574) | 119906 = 119906ℎ

plusmn

(1198791 119868) 1198681lt 119868 lt 119868

2

120574 = int

1198791

0

119863119868119867(119906ℎ

plusmn

(1198791 119868) 119868) 119889119904 + 120574

0

(29)


119868 = 0 (30a)

119868 120574 = 119863119868119867(119902plusmn(119868) 119868) 119868

1lt 119868 lt 119868

2 (30b)


where

119863119868119867(119902plusmn(119868) 119868) = minus

120597119867 (119902plusmn(119868) 119868)

120597119868

= 1205902119868 + 12057221198683

+ 11988661198681199062

1

(31)







119863119868119867(119902plusmn 119868) = 120590

2119868119903+ 12057221198683

119903

+ 11988661198681199031199062

1

= 0 (32)

Then we obtain

119868119903= plusmn

12059021205781+ 1198866[1205832

1

minus 1205901(1 minus 120590

1)]

1198862

6

minus 12057221205781

12

(33)



119903) (34)




1 1199062 119868 120574) The phase shift



61198682 and 120583


1= 1199062 (35a)

2= minus12057611199061+ 12057811199063

1

minus 12057612057621199062 (35b)


119867(1199061 1199062) =

1

2

1199062

2

+

1

2

12057611199062

1

minus

1

4

12057811199064

1

(36)






1199061(1198791) = plusmnradic

1205761

1205781

tanh(

radic21205761

2

1198791) (37a)

1199062(1198791) = plusmn

1205761

radic21205781


2

1198791) (37b)


120574 (1198791) = 1205961199031198791minus

1198866radic21205761

1205781

tanh(

radic21205761

2

1198791) + 1205740 (38)

where 120596119903= 1205902+ 12057221198682

+ 119886612057611205781

At 119868 = 119868119903 there is 120596


be expressed as

Δ120574 = [minus

21198866radic21205761

1205781

]

119868=119868119903

= minus

21198866

1205781

radic2 [1205832

1

minus 1205901(1 minus 120590

1) minus 11988661198682

119903

]

(39)



1 1199062) for the




plusmn(119868) in (24a)


M

I

120574

times

u2

u1

(a)

0 1205742120587

I = I1

I = Ir

I = I2

(b)




120576

Therefore we obtain

119872 = 119872120576= (119906 119868 120574) | 119906 = 119902

plusmn(119868) 1198681lt 119868 lt 119868

2 0 le 120574 lt 2120587

(40)


119868 = minus1205832

119868 minus 1198912

sin 120574 (41a)

120574 = 1205902+ 12057221198682

+ 11988661199062

1

minus

1198912

cos 120574119868

(41b)


1205832

997888rarr 1205761205832

119868 = 119868119903+ radic120576ℎ

1198912

997888rarr 1205761198912

1198791997888rarr

1198791

radic120576

(42)


ℎ = minus120583

2

119868119903minus 1198912

sin 120574 minus radic1205761205832

ℎ (43a)

120574 = minus

2120575

1205781

119868119903ℎ minus radic120576(

120575

1205781

ℎ2

+

1198912

119868119903

cos 120574) (43b)

where 120575 = 1198862

6

minus 12057221205781

When 120576 = 0 (43a) (43b) become

ℎ = minus120583

2

119868119903minus 1198912

sin 120574 (44a)

120574 = minus

2120575

1205781

119868119903ℎ (44b)


119863(ℎ 120574) = minus120583

2

119868119903120574 + 1198912

cos 120574 +

120575

1205781

119868119903ℎ2

(45)


1198750= (0 120574


1205832

119868119903

1198912

))

1198760= (0 120574

119904) = (0 120587 + arcsin(

1205832

119868119903

1198912

))

(46)


0and119876









120576 The phase




120574min minus

1198912

1205832

119868119903

cos 120574min = 120587 + arcsin1205832

119868119903

1198912

+

radic119891

2

2

minus 1205832

2

1198682

119903

1205832

119868119903

(47)


119903as

119860120576= (1199061 1199062 119868 120574) | 119906

1= 119861 119906

2= 0

1003816100381610038161003816119868 minus 119868119903

1003816100381610038161003816lt radic120576119862 120574 isin 119879

119871

(48)



120576 denoted as119882119904(119860

120576) and119882

119906

(119860120576) are the


119906






120576 leave and


h

120574

P0

120574min 120574c 120574s

Q0

(a)

h

120574Q120576

P120576

(b)








120574

|u|

h

120574c 120574c + 3Δ120574

p120576qc




119896(120576 119868 120574

0 1205832


0 and 120583

2




function 1198721(120576 119868119903 1205740 1205832


119903 1205740 1205832



(119872) and119882119906


119872(119868119903 1205740 1205832

1205781 1198866 1205761)

= int

+infin

minusinfin

⟨n (119901ℎ

(119905)) g (119901ℎ (119905) 1205832

0)⟩ 1198891198791


= int

+infin

minusinfin

(

120597119867

1205971199061

1198921199061+

120597119867

1205971199062

1198921199062+

120597119867

120597119868

119892119868

+

120597119867

120597120574

119892120574

)1198891198791

= minus

2radic21205833

31205781

12057632

1

minus 2radic211988661205832

1198682

119903

12057612

1

1205781

minus 1198912

119868119903[cos(120574

0minus 1198866

radic21205761

1205781

) minus cos(1205740+ 1198866

radic21205761

1205781

)]

(49)


) (119896 = 1 2 ) is defined as

119872119896(120576 119868119903 1205740 1205832

)

=

119896minus1

sum

119895=0

119872(119868119903 119895Δ120574 (119868

119903) + Γ119895(120576 119868119903 1205740 1205832

) + 1205740 1205832

)

(50)

where

Γ119895(120576 119868119903 1205740 1205832

) =

Ω (1199090(119868119903) 119868119903)

120582 (119868119903)

119895

sum

119903=1

log100381610038161003816100381610038161003816100381610038161003816

120589 (119868119903)

120576119872119903(120576 119868119903 1205740 1205832

)

100381610038161003816100381610038161003816100381610038161003816

(51)

for 119895 = 1 119896 minus 1 and Γ0(120576 119868119903 1205740 1205832


119895(120576 119868119903 1205740 1205832


119896(120576 119868119903 1205740 1205832


119903degenerates



119872119896(119868119903 1205740 1205832

1205781 1198866 1205761)

=

119896minus1

sum

119895=0

119872(119868119903 1205740+ 119895Δ120574 (119868

119903) 1205832

1205781 1198866 1205761)

= minus1198912

119868119903[cos(120574

0minus

1198866radic21205761

1205781

)

minus cos(1205740+

1198866radic21205761

1205781

)]

minus

2radic21205833

31205781

12057632

1

minus 2radic211988661205832

1198682

119903

12057612

1

1205781

minus 1198912

119868119903[cos(120574

0minus

1198866radic21205761

1205781

minus

21198866radic21205761

1205781

)

minus cos(1205740+

1198866radic21205761

1205781

minus

21198866radic21205761

1205781

)]

minus

2radic21205833

31205781

12057632

1

minus 2radic211988661205832

1198682

119903

12057612

1

1205781

+ sdot sdot sdot

minus 1198912

119868119903[cos(120574

0minus

1198866radic21205761

1205781

minus 2 (119896 minus 1) 1198866

radic21205761

1205781

)

minus cos(1205740+

1198866radic21205761

1205781


1198866radic21205761

1205781

)]

minus

2radic21205833

31205781

12057632

1

minus 2radic211988661205832

1198682

119903

12057612

1

1205781

= minus1198912

119868119903[cos(120574

0minus

1198866radic21205761

1205781


1198866radic21205761

1205781

)

minus cos(1205740+

1198866radic21205761

1205781

)]

minus

2radic21198961205833

31205781

12057632

1

minus 2radic211989611988661205832

1198682

119903

12057612

1

1205781

(52)


119896minus1= 1205740+ (119896 minus


119872119896(119868119903 1205740 1205832

1205781 1198866 1205761)

= 119872119896(119868119903 120574119896minus1


Δ120574

2

1205832

1205781 1198866 1205761)

= minus1198912

119868119903[cos(120574

119896minus1+

1

2

119896Δ120574)


minus

1

2

119896Δ120574)]

+

11989612058331205761

31198866

Δ120574 + 1205832

1198682

119903

(119896Δ120574)

= 21198912

119868119903sin 120574119896minus1

sin(

1

2

119896Δ120574)

+

12058331205761

31198866

(119896Δ120574) + 1205832

1198682

119903

(119896Δ120574)

(53)


10038161003816100381610038161003816100381610038161003816100381610038161003816

(12) 119896Δ120574

sin ((12) 119896Δ120574)

(12058331205761+ 311988661205832

1198682

119903

)

311988661198912

119868119903

10038161003816100381610038161003816100381610038161003816100381610038161003816

lt 1

1

2

119896Δ120574 = 119899120587 119899 = 0 plusmn1 plusmn2

(54)


119885119899

minus

= (119868119903 120574119896minus1

1205832

1205781 1198866 1205761) | 119872119896= 0119863

1205740

119872119896

= 0 (55)



119896minus1isin [0 120587]

120574119896minus11

= minus arcsin(12) 119896Δ120574

sin ((12) 119896Δ120574)

(12058331205761+ 311988661205832

1198682

119903

)

(311988661198912

119868119903)

120574119896minus12

= 120587 minus 120574119896minus11

(56)


3 1205761 1205832

1198866 and 119891

2


119896minus1= 120574119896minus11

and 120574119896minus1

= 120574119896minus12

=

120587 minus 120574119896minus11


119895(119868119903 1205740119894

1205832



119906

(119872120576) intersect


1205832120578111988661205761

plusmn120576(120574119896minus1119894


plusmn0

(120574119896minus1119894


119903 1205740

=

120574119896minus1119894


119895(119868119903 1205740119894

1205832

1205781 1198866 1205761)


0119894

= 120574119896minus1119894

minus (119896 minus 1)(Δ1205742) + 119895Δ120574


119895(119868119903 1205740119894

1205832



01



plusmn120576(120574119896minus11

)

and sum1205832120578111988661205761

plusmn120576(120574119896minus12


1205832120578111988661205761

plusmn0

(120574119896minus11

) and sum1205832120578111988661205761

plusmn0

(120574119896minus12


119906


n = ((minus1205831199061+ 12057811199063

1

+ 11988661198682

1199061) minus1199062 0 0) (57)


plusmn0

(120574119896minus11

)


plusmn0

(120574119896minus12





119894(120583) and 119888



the end points 119889119894(120583) to 119888


end points 119889119894(120583) and 119888



0119894

(119868119903 120583) minus Δ120574

minus



119894(120583) for 119894 =

2 119873 while the line 120574 = 1205740119894

(119868119903 120583) + Δ120574

+



(120583) at the end point 119889119894+1


119894(120583) and 119889

119894+1(120583) is zero

namely

ℎ (119888119894(120583)) minus ℎ (119889

119894+1(120583)) = 0 (58)



sum1205832120578111988661205761

0

(1205740119894


119894(120583) and 119889


1


(12057401



2(120583) on the


119873


0

(1205740119873


119873(120583) on the segments 119874


119889119873+1



sum1205832120578111988661205761

0

(1205740119894


1(120583) and 119874

2(120583) of (44a) (44b) It is known



(119868119903 120583) +

Δ120574+

(119868119903) and 120574 = 120574

0119894

(119868119903 120583) minus Δ120574

minus

(119868119903)


0

(120574119896minus11

) andsum1205832120578111988661205761

0

(120574119896minus12


01

minus


01

minusΔ120574+








119897and









120574

|u|h

Γ3

Γ2

Γ1

M

O2O3






1


6 where

1205781



6 1198867 and


1gt 0



2


2and 119886



4is the


1 1198862 1198864






3 respectively when




120574

|u|h

M

Γsum (12057401 12057402)

12057402 + Δ120574+ 12057401 + Δ120574+



12057402

) are depicted


2 1205831 1198862





2= 2 sim 200


183 1205902

= 197 1205831

= 02 1205832

= 02 1198862

= 230 1198863

=

120 1198864= 130 119886

5= minus101 119886

6= minus203 119886

7= minus205 119887

6=

407 1198877= minus308 119887

8= 109 119909

10= minus001 119909

20= minus005 119909

30=

minus001 11990940


1 1198912) and (119909

3 1198912) respec-






1 The figure demon-



1=


2=



50 100 150minus2

minus15

minus1

minus05

0

05

1

x1

f2

(a)

20 40 60 80 100 120 140 160 180

minus3

minus25

minus2

minus15

x3

f2

(b)



= minus001 11990920

= minus005 11990930

=

minus001 11990940


1198912


1198912

)

0 01 02 03 04 05 06 07minus25

minus2

minus15

minus1

minus05

0

05

1

15

x1

1205831

(a)

0 01 02 03 04 05 06 07minus32

minus3

minus28

minus26

minus24

minus22

minus2

minus18

x1

1205831

x3

1205831

x3

(b)




= minus001 11990920

= minus005 11990930

= minus001 11990940


1205831


3

1205831

)

the planes (1199091 1205831) and (119909









3 1198862)




4varies from 119886

4= 2 to 119886

4= 120


2is

chosen as 1198862


1 1198864) and (119909

3 1198864)







0 10 20 30 40 50 60

minus3

minus2

minus1

0

1

x1

a2

(a)

0 10 20 30 40 50 60minus34

minus32

minus3

minus28

minus26

minus24

minus22

minus2

minus18

a2

x3

(b)



= 02 and 1205832


2


= minus001 11990920

= minus005 11990930

= minus001 11990940


1198862


1198862

)

0 20 40 60 80 100 120minus25

minus2

minus15

minus1

minus05

0

05

1

x1

a4

(a)

0 20 40 60 80 100 120

minus35

minus3

minus25

minus2

a4

x3

(b)




= 02 and 1205832


= 23 and 1198863


= minus001 11990920

= minus005 11990930

= minus001 11990940

=


1198864


1198864

)


1are selected as











1= 1437 120590

2=

1142 1205831= 02 120583

2= 02 119886

2= 300 119886

3= 750 119886

4= 450

1198865

= minus1166 1198866

= 1227 1198912

= 1227 1198867

= minus268 1198876

=

minus22 1198877

= minus969 1198878

= minus2232 11990910

= minus108 11990920

= 0511990930

= minus001 11990940



minus1minus050051

minus05

0

05

minus25

minus2

minus15

minus1

x1

x2

x3

(a)


minus1

minus05

0

05

1

x1

x2

(b)


= 827 and 1205831


1199092

1199093


1199092

)

minus50

5

minus50

5minus6

minus4

minus2

0

2

4

6

x1x2

x3

(a)


minus10

minus5

0

5

10

x1

x2

(b)


= 1437 1205902

= 1142 1198862

= 300 1198863

= 750 1198864

= 450 1198865

= minus1166 1198866

=

1227 1198912

= 1227 1198867

= minus268 1198876

= minus22 1198877

= minus969 1198878

= minus2232 11990910

= minus108 11990920

= 05 11990930

= minus001 11990940


1

1199092

1199093


1199092

)





2= 313 119886

5=

minus1501 1198878

= 409 11990910

= minus001 11990920

= minus009 11990930

=

minus005 11990940


7 Conclusions




minus50

5

minus10

0

10

minus6

minus4

minus2

0

2

x1x2

x3

(a)

minus5 0 5 10

minus10

minus5

0

5

10

x1

x2

(b)


= 361 1205902

= 313 1198862

= 230 1198863

= 120 1198864

= 130 1198865

= minus1501 1198866

=

minus203 1198912

= 9238 1198867

= minus205 1198876

= 407 1198877

= minus308 1198878

= 409 11990910

= minus001 11990920

= minus009 11990930

= minus005 11990940


1

1199092

1199093


1199092

)



119896(120576 119868 120574

0 1205832




2 the in-plane


4 and the damping




1 respec-










Acknowledgments



References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










+

1

2

(11986012

+ 211986066) (

1205971199080

120597119910

)

2

1205972

1199080

1205971199092

+ 11986022

1205972

1199080

1205971199102

120597V0

120597119910

+ (11986012

+ 11986066)

1205971199080

120597119909

1205972V0

120597119910120597119909

+

1

2

(11986021

+ 211986066)

1205972

1199080

1205971199102

times (

1205971199080

120597119909

)

2

+

3

2

11986011(

1205971199080

120597119909

)

2

1205972

1199080

1205971199092

+ 11986011

1205972

1199080

1205971199092

1205971199060

120597119909

+ 11986012

1205972

1199080

1205971199092

120597V0

120597119910

+ 211986066

1205972

1199080

120597119910120597119909

120597V0

120597119909

+ 211986066

1205972

1199080

120597119910120597119909

1205971199060

120597119910

+

3

2

11986022(

1205971199080

120597119910

)

2

1205972

1199080

1205971199102

+ (11986055

+ 119902119909cos (Ω

1119905) minus 119873

119875

119909

cos (Ω4119905)

+ 119865551198882

2

minus 2119863551198882)

1205972

1199080

1205971199092

minus

120597119873119875

119909

120597119909

cos (Ω3119905)

120597119908

120597119909

+ (119865551198882

2

minus 2119863551198882+ 11986055)

120597120601119909

120597119909

minus 119902 cos (Ω3119905) + 119896119891

1205971199080

120597119905

= 11986800minus 1198882

1

1198686(

1205972

0

1205971199092

+

1205972

0

1205971199102

)

+ 11988811198683(

1205970

120597119909

+

120597V0

120597119910

) + 11988811198694(

120597

120601119909

120597119909

+

120597

120601119910

120597119910

)

(4c)

(11986311

minus 2119865111198881+ 119867111198882

1

)

1205972

120601119909

1205971199092

+ (11986366

minus 2119865661198881+ 119867661198882

1

)

1205972

120601119909

1205971199102

minus 1198881(11986511

minus 119867111198881)

1205973

1199080

1205971199093

minus (119865551198882

2

minus 2119863551198882+ 11986055)

1205971199080

120597119909

+ (11986312

+ 11986366

+ 119867661198882

1

minus 2119865661198881+ 119867121198882

1

minus 2119865121198881)

times

1205972

120601119910

120597119910120597119909

minus 1198881(211986566

+ 11986512

minus 2119867661198881minus 119867121198881)

times

1205973

1199080

1205971199102

120597119909

+ (2119863551198882minus 11986055

minus 119865551198882

2

) 120601119909

= 11986910+ 1198702

120601119909minus 11988811198694

1205970

120597119909

(4d)

(11986366

minus 2119865661198881+ 119867661198882

1

)

1205972

120601119910

1205971199092

minus 1198881(11986521

+ 211986566

minus 119867211198881minus 2119867661198881)

1205973

1199080

1205971199101205971199092

+ (119867211198882

1

+ 11986366

+ 11986321

minus 2119865211198881+ 119867661198882

1

minus 2119865661198881)

times

1205972

120601119909

120597119910120597119909

+ (119867221198882

1

+ 11986322

minus 2119865221198881)

1205972

120601119910

1205971199102

minus 1198881(11986522

minus 119867221198881)

1205973

1199080

1205971199103

minus (119865441198882

2

minus 2119863441198882+ 11986044)

times

1205971199080

120597119910

+ (2119863441198882minus 119865441198882

2

minus 11986044) 120601119910

= 1198691V0+ 1198702

120601119910minus 11988811198694

1205970

120597119910

(4e)


119909 = 0 119908 = 0 120601119910= 0 119873

119909119910= 0 119872

119909119909= 0

(5a)

119909 = 119886 119908 = 0 120601119910= 0 119873

119909119910= 0 119872

119909119909= 0

(5b)

119910 = 0 119908 = 0 120601119909= 0 119873

119909119910= 0 119872

119910119910= 0

(5c)

119910 = 119887 119908 = 0 120601119909= 0 119873

119909119910= 0 119872

119910119910= 0

(5d)

int

ℎ

0

119873119909119909

1003816100381610038161003816119909=0

119889119911 = minusint

ℎ

0

(1199020+ 119902119909cosΩ1119905) 119889119911 (5e)

int

ℎ

0

119873119910119910

10038161003816100381610038161003816119910=0

119889119911 = minusint

ℎ

0

(1199021+ 119902119910cosΩ2119905) 119889119911 (5f)


0 V0 1199080 120601119909 and 120601



0 V0 1199080 120601119909 and


1199060= 11990601(119905) cos 120587119909

2119886

cos120587119910

2119887

+ 11990602(119905) cos 3120587119909

2119886

cos120587119910

2119887

(6a)

V0= V1(119905) cos

120587119910

2119887

cos 1205871199092119886

+ V2(119905) cos

120587119910

2119887

cos 31205871199092119886

(6b)

1199080= 1199081(119905) sin 120587119909

119886

sin120587119910

119887

+ 1199082(119905) sin 3120587119909

119886

sin120587119910

119887

(6c)

120601119909= 1206011(119905) cos 120587119909

119886

sin120587119910

119887

+ 1206012(119905) cos 3120587119909

119886

sin120587119910

119887

(6d)

120601119910= 1206013(119905) cos

120587119910

119887

sin 120587119909

119886

+ 1206014(119905) cos

120587119910

119887

sin 3120587119909

119886

(6e)




01 11990602 V1 V2 1206011 1206012 1206013 and

1206014via 1199081and 119908

2as follows

11990601

= 11989611199082

1

+ 11989621199082

2

+ 119896311990811199082 (7a)

11990602

= 11989641199082

1

+ 11989651199082

2

+ 119896611990811199082 (7b)

V1= 11989671199082

1

+ 11989681199082

2

+ 119896911990811199082 (7c)

V2= 119896101199082

1

+ 119896111199082

2

+ 1198961211990811199082 (7d)

1206011= 119896191199081 120601

2= 119896201199082 (7e)

1206013= 119896211199081 120601

4= 119896221199082 (7f)



119906 =

1199060

119886

V =

V0

119887

119908 =

1199080

ℎ

120601119909

= 120601119909 120601

119910

= 120601119910 119909 =

119909

119886

119910 =

119910

119887

119902 =

1198872

119864ℎ3

119902 119902119909

=

1198872

119864ℎ3

119902119909 119902

119910

=

1198872

119864ℎ3

119902119910

119905 = 1205872

(

119864

119886119887120588

)

12

119905 Ω119894=

1

1205872

(

119886119887120588

119864

)

12

Ω119894

(119894 = 1 2)

119860119894119895=

(119886119887)12

119864ℎ2

119860119894119895 119861

119894119895=

(119886119887)12

119864ℎ3

119861119894119895

119863119894119895=

(119886119887)12

119864ℎ4

119863119894119895 119864

119894119895=

(119886119887)12

119864ℎ5

119864119894119895

119865119894119895=

(119886119887)12

119864ℎ6

119865119894119895 119867

119894119895=

(119886119887)12

119864ℎ8

119867119894119895

119868119894=

1

(119886119887)(119894+1)2

120588

119868119894

(8)


1+ 12058311+ 1205962

1

1199081


+ 11988651199082

1

1199082+ 11988661199082

2

1199081+ 11988671199083

1

+ 11988681199083

2

= 1198911cosΩ3119905

(9a)

2+ 12058322+ 1205962

2

1199082

+ (1198872cosΩ1119905 + 1198873cosΩ2119905 + 1198874cosΩ4119905) 1199082

+ 11988751199082

2

1199081+ 11988761199082

1

1199082+ 11988771199083

2

+ 11988781199083

1

= 1198912cosΩ3119905

(9b)



1205962

1

=

1205962

9

+ 1205761205901 120596

2

2

= 1205962

+ 1205761205902

Ω3= 120596 Ω

1= Ω2= Ω4=

2120596

3

1205962asymp 31205961

(10)

where 1205901and 120590



1199081(119905 120576) = 119909

10(1198790 1198791) + 120576119909

11(1198790 1198791) + sdot sdot sdot (11a)

1199082(119905 120576) = 119909

20(1198790 1198791) + 120576119909

21(1198790 1198791) + sdot sdot sdot (11b)

where 1198790= 119905 1198791= 120576119905


1= minus

1

2

12058311199091minus

1

2

12059011199092+

1

4

(1198862+ 1198863minus 1198864) 1199092

minus

3

2

11988671199092(1199092

1

+ 1199092

2

) minus

1

2

11988651199094(1199092

1

minus 1199092

2

)

minus 11988661199092(1199092

3

+ 1199092

4

) + 1198865119909111990921199093

(12a)

2= minus

1

2

12058311199092+

1

2

12059011199091+

1

4

(1198862+ 1198863minus 1198864) 1199091

+

3

2

11988671199091(1199092

1

+ 1199092

2

) +

1

2

11988651199093(1199092

1

minus 1199092

2

)

+ 11988661199091(1199092

3

+ 1199092

4

) + 1198865119909111990921199094

(12b)

3= minus

1

2

12058321199093minus

1

6

12059021199094minus

1

3

11988761199094(1199092

1

+ 1199092

2

)

minus

1

2

11988771199094(1199092

3

+ 1199092

4

) minus

1

6

11988781199092(31199092

1

minus 1199092

2

)

(12c)


4= minus

1

2

12058321199094+

1

6

12059021199093+

1

3

11988761199093(1199092

1

+ 1199092

2

)

+

1

2

11988771199093(1199092

3

+ 1199092

4

) +

1

6

11988781199091(1199092

1

minus 31199092

2

) minus

1

12

1198912

(12d)



2oplus 1198852and 119863

4





1= minus

1

2

12058311199091+ (1198910minus

1

2

1205901)1199092minus

3

2

11988671199092(1199092

1

+ 1199092

2

)

minus

1

2

11988651199094(1199092

1

minus 1199092

2

)

minus 11988661199092(1199092

3

+ 1199092

4

) + 1198865119909111990921199093

(13a)

2= minus

1

2

12058311199092+ (1198910+

1

2

1205901)1199091+

3

2

11988671199091(1199092

1

+ 1199092

2

)

+

1

2

11988651199093(1199092

1

minus 1199092

2

)

+ 11988661199091(1199092

3

+ 1199092

4

) + 1198865119909111990921199094

(13b)

3= minus

1

2

12058321199093minus

1

6

12059021199094minus

1

3

11988761199094(1199092

1

+ 1199092

2

)

minus

1

2

11988771199094(1199092

3

+ 1199092

4

) minus

1

6

11988781199092(31199092

1

minus 1199092

2

)

(13c)

4= minus

1

2

12058321199094+

1

6

12059021199093+

1

3

11988761199093(1199092

1

+ 1199092

2

)

+

1

2

11988771199093(1199092

3

+ 1199092

4

) +

1

6

11988781199091(1199092

1

minus 31199092

2

)

(13d)

where 1198910= (14)(119886

2+ 1198863minus 1198864)


1199091= 1199101minus

1

4

11988671199103

1

+

31198865(1205902minus 6)

1205902

2

1199102

1

1199103minus 119886611991011199102

3

minus 119886611991011199102

4

minus

31198865(1205903

2

minus 181205902

2

+ 2161205902minus 1296)

1205904

2

1199102

2

1199103

+

1198865(61205902

2

minus 721205902+ 432)

1205903

2

119910111991021199104

(14a)

1199092= 1199102+

3

2

11988671199103

2

+

3

4

11988671199102

1

1199102+

31198865

1205902

1199102

1

1199104

minus

31198865(1205902

2

minus 121205902+ 72)

1205903

2

1199102

2

1199104minus

61198865(1205902minus 6)

1205902

2

119910111991021199103

(14b)

1199093= 1199103minus

1198878

1205902

1199103

1

+

31198878(1205902

2

minus 121205902+ 72)

1205903

2

11991011199102

2

minus

1

3

1198876119910111991021199104

(14c)

1199094= 1199104+

1198878(1205903

2

minus 181205902

2

+ 2161205902minus 1296)

1205904

2

1199103

2

minus

31198878(1205902minus 6)

1205902

2

1199102

1

1199102+

1

3

1198876119910111991021199103

(14d)


1199101= minus1205831

1199101+ (1 minus 120590

1) 1199102 (15a)

1199102= 12059011199101minus 1205831

1199102+ 11988661199101(1199102

3

+ 1199102

4

) +

3

2

11988671199103

1

(15b)

1199103= minus1205832

1199103minus 12059021199104minus

1

3

11988761199102

1

1199104minus

1

2

11988771199104(1199102

3

+ 1199102

4

) (15c)

1199104= 12059021199103minus 1205832

1199104+

1

3

11988761199102

1

1199103+

1

2

11988771199103(1199102

3

+ 1199102

4

) minus 1198912

(15d)


= (12)1205831 1205832

= (12)1205832 1205902=

(16)1205902 and 119891

2


Further let

1199103= 119868 cos 120574 119910

4= 119868 sin 120574 (16)


1199101= minus1205831

1199101+ (1 minus 120590

1) 1199102 (17a)

1199102= 12059011199101minus 1205831

1199102+ 119886611991011198682

+

3

2

11988671199103

1

(17b)

119868 = minus1205832

119868 minus 1198912

sin 120574 (17c)

119868 120574 = 1205902119868 +

1

3

11988761199102

1

119868 +

1

2

11988771198683

minus 1198912

cos 120574 (17d)


[

1199101

1199102

] = radic3

radic10038161003816100381610038161198866

1003816100381610038161003816

radic10038161003816100381610038161198876

1003816100381610038161003816

[

1 minus 1205901

0

1205831

1] [

1199061

1199062

] (18)



1yield the


1= 1199062 (19a)

2= minus120583119906

1minus 12058331199062+ 12057811199063

1

+ 119886611990611198682

(19b)

119868 = minus1205832

119868 minus 1198912

sin 120574 (19c)

119868 120574 = 1205902119868 + 11988661199062

1

119868 + 12057221198683

minus 1198912

cos 120574 (19d)

where 120583 = 1205832

1

minus 1205901(1 minus 120590

1) 1205833= 21205831

1205781= 91198866119886721198876and

1205722= (12)119887

7


1205832

997888rarr 1205761205832

1205833997888rarr 120576120583

3 119891

2

997888rarr 1205761198912

1205781997888rarr 1205781 120572

2997888rarr 1205722 119886

6997888rarr 1198866

(20)


1=

120597119867

1205971199062

+ 1205761198921199061= 1199062 (21a)

2= minus

120597119867

1205971199061

+ 1205761198921199062= minus120583119906

1+ 12057811199063

1

+ 119886611990611198682

minus 12057612058331199062 (21b)

119868 =

120597119867

120597120574

+ 120576119892119868

= minus1205761205832

119868 minus 1205761198912

sin 120574 (21c)

119868 120574 = minus

120597119867

120597119868

+ 120576119892120574

= 1205902119868 + 12057221198683

+ 11988661198681199062

1

minus 1205761198912

cos 120574 (21d)


119867(1199061 1199062 119868 120574) =

1

2

1199062

2

+

1

2

1205831199062

1

minus

1

4

12057811199064

1

minus

1

2

11988661198682

1199062

1

minus

1

2

12059021198682

minus

1

4

12057221198684

(22)

and 1198921199061 1198921199062 119892119868 and 119892


1198921199061= 0 119892

1199062= minus12058331199062

119892119868

= minus1205832

119868 minus 1198912

sin 120574 119892120574

= minus1198912

cos 120574(23)



1 1199062) of (21a) (21b)


1= 1199062 (24a)

2= minus120583119906

1+ 12057811199063

1

+ 119886611990611198682

(24b)

Since 1205781


11988661198682




11986812

= plusmn[

1205832

1

minus 1205901(1 minus 120590

1)

1198866

]

12

(25)


0= (0 0)


119861 = plusmn

1

1205781

[1205832

1

minus 1205901(1 minus 120590

1) minus 11988661198682

]

12

(26)


0is the



1198681= 0 119868

2= [

1205832

1

minus 1205901(1 minus 120590

1)

1198866

]

12

(27)




(1198791 119868) that is lim


119906ℎ

plusmn

(1198791 119868) =


defined by

119872 = (119906 119868 120574) | 119906 = 119902plusmn(119868) 1198681lt 119868 lt 119868

2 0 le 120574 lt 2120587 (28)



119904

(119872) and 119882119906



119906

(119872)


Γ = (119906 119868 120574) | 119906 = 119906ℎ

plusmn

(1198791 119868) 1198681lt 119868 lt 119868

2

120574 = int

1198791

0

119863119868119867(119906ℎ

plusmn

(1198791 119868) 119868) 119889119904 + 120574

0

(29)


119868 = 0 (30a)

119868 120574 = 119863119868119867(119902plusmn(119868) 119868) 119868

1lt 119868 lt 119868

2 (30b)


where

119863119868119867(119902plusmn(119868) 119868) = minus

120597119867 (119902plusmn(119868) 119868)

120597119868

= 1205902119868 + 12057221198683

+ 11988661198681199062

1

(31)







119863119868119867(119902plusmn 119868) = 120590

2119868119903+ 12057221198683

119903

+ 11988661198681199031199062

1

= 0 (32)

Then we obtain

119868119903= plusmn

12059021205781+ 1198866[1205832

1

minus 1205901(1 minus 120590

1)]

1198862

6

minus 12057221205781

12

(33)



119903) (34)




1 1199062 119868 120574) The phase shift



61198682 and 120583


1= 1199062 (35a)

2= minus12057611199061+ 12057811199063

1

minus 12057612057621199062 (35b)


119867(1199061 1199062) =

1

2

1199062

2

+

1

2

12057611199062

1

minus

1

4

12057811199064

1

(36)






1199061(1198791) = plusmnradic

1205761

1205781

tanh(

radic21205761

2

1198791) (37a)

1199062(1198791) = plusmn

1205761

radic21205781


2

1198791) (37b)


120574 (1198791) = 1205961199031198791minus

1198866radic21205761

1205781

tanh(

radic21205761

2

1198791) + 1205740 (38)

where 120596119903= 1205902+ 12057221198682

+ 119886612057611205781

At 119868 = 119868119903 there is 120596


be expressed as

Δ120574 = [minus

21198866radic21205761

1205781

]

119868=119868119903

= minus

21198866

1205781

radic2 [1205832

1

minus 1205901(1 minus 120590

1) minus 11988661198682

119903

]

(39)



1 1199062) for the




plusmn(119868) in (24a)


M

I

120574

times

u2

u1

(a)

0 1205742120587

I = I1

I = Ir

I = I2

(b)




120576

Therefore we obtain

119872 = 119872120576= (119906 119868 120574) | 119906 = 119902

plusmn(119868) 1198681lt 119868 lt 119868

2 0 le 120574 lt 2120587

(40)


119868 = minus1205832

119868 minus 1198912

sin 120574 (41a)

120574 = 1205902+ 12057221198682

+ 11988661199062

1

minus

1198912

cos 120574119868

(41b)


1205832

997888rarr 1205761205832

119868 = 119868119903+ radic120576ℎ

1198912

997888rarr 1205761198912

1198791997888rarr

1198791

radic120576

(42)


ℎ = minus120583

2

119868119903minus 1198912

sin 120574 minus radic1205761205832

ℎ (43a)

120574 = minus

2120575

1205781

119868119903ℎ minus radic120576(

120575

1205781

ℎ2

+

1198912

119868119903

cos 120574) (43b)

where 120575 = 1198862

6

minus 12057221205781

When 120576 = 0 (43a) (43b) become

ℎ = minus120583

2

119868119903minus 1198912

sin 120574 (44a)

120574 = minus

2120575

1205781

119868119903ℎ (44b)


119863(ℎ 120574) = minus120583

2

119868119903120574 + 1198912

cos 120574 +

120575

1205781

119868119903ℎ2

(45)


1198750= (0 120574


1205832

119868119903

1198912

))

1198760= (0 120574

119904) = (0 120587 + arcsin(

1205832

119868119903

1198912

))

(46)


0and119876









120576 The phase




120574min minus

1198912

1205832

119868119903

cos 120574min = 120587 + arcsin1205832

119868119903

1198912

+

radic119891

2

2

minus 1205832

2

1198682

119903

1205832

119868119903

(47)


119903as

119860120576= (1199061 1199062 119868 120574) | 119906

1= 119861 119906

2= 0

1003816100381610038161003816119868 minus 119868119903

1003816100381610038161003816lt radic120576119862 120574 isin 119879

119871

(48)



120576 denoted as119882119904(119860

120576) and119882

119906

(119860120576) are the


119906






120576 leave and


h

120574

P0

120574min 120574c 120574s

Q0

(a)

h

120574Q120576

P120576

(b)








120574

|u|

h

120574c 120574c + 3Δ120574

p120576qc




119896(120576 119868 120574

0 1205832


0 and 120583

2




function 1198721(120576 119868119903 1205740 1205832


119903 1205740 1205832



(119872) and119882119906


119872(119868119903 1205740 1205832

1205781 1198866 1205761)

= int

+infin

minusinfin

⟨n (119901ℎ

(119905)) g (119901ℎ (119905) 1205832

0)⟩ 1198891198791


= int

+infin

minusinfin

(

120597119867

1205971199061

1198921199061+

120597119867

1205971199062

1198921199062+

120597119867

120597119868

119892119868

+

120597119867

120597120574

119892120574

)1198891198791

= minus

2radic21205833

31205781

12057632

1

minus 2radic211988661205832

1198682

119903

12057612

1

1205781

minus 1198912

119868119903[cos(120574

0minus 1198866

radic21205761

1205781

) minus cos(1205740+ 1198866

radic21205761

1205781

)]

(49)


) (119896 = 1 2 ) is defined as

119872119896(120576 119868119903 1205740 1205832

)

=

119896minus1

sum

119895=0

119872(119868119903 119895Δ120574 (119868

119903) + Γ119895(120576 119868119903 1205740 1205832

) + 1205740 1205832

)

(50)

where

Γ119895(120576 119868119903 1205740 1205832

) =

Ω (1199090(119868119903) 119868119903)

120582 (119868119903)

119895

sum

119903=1

log100381610038161003816100381610038161003816100381610038161003816

120589 (119868119903)

120576119872119903(120576 119868119903 1205740 1205832

)

100381610038161003816100381610038161003816100381610038161003816

(51)

for 119895 = 1 119896 minus 1 and Γ0(120576 119868119903 1205740 1205832


119895(120576 119868119903 1205740 1205832


119896(120576 119868119903 1205740 1205832


119903degenerates



119872119896(119868119903 1205740 1205832

1205781 1198866 1205761)

=

119896minus1

sum

119895=0

119872(119868119903 1205740+ 119895Δ120574 (119868

119903) 1205832

1205781 1198866 1205761)

= minus1198912

119868119903[cos(120574

0minus

1198866radic21205761

1205781

)

minus cos(1205740+

1198866radic21205761

1205781

)]

minus

2radic21205833

31205781

12057632

1

minus 2radic211988661205832

1198682

119903

12057612

1

1205781

minus 1198912

119868119903[cos(120574

0minus

1198866radic21205761

1205781

minus

21198866radic21205761

1205781

)

minus cos(1205740+

1198866radic21205761

1205781

minus

21198866radic21205761

1205781

)]

minus

2radic21205833

31205781

12057632

1

minus 2radic211988661205832

1198682

119903

12057612

1

1205781

+ sdot sdot sdot

minus 1198912

119868119903[cos(120574

0minus

1198866radic21205761

1205781

minus 2 (119896 minus 1) 1198866

radic21205761

1205781

)

minus cos(1205740+

1198866radic21205761

1205781


1198866radic21205761

1205781

)]

minus

2radic21205833

31205781

12057632

1

minus 2radic211988661205832

1198682

119903

12057612

1

1205781

= minus1198912

119868119903[cos(120574

0minus

1198866radic21205761

1205781


1198866radic21205761

1205781

)

minus cos(1205740+

1198866radic21205761

1205781

)]

minus

2radic21198961205833

31205781

12057632

1

minus 2radic211989611988661205832

1198682

119903

12057612

1

1205781

(52)


119896minus1= 1205740+ (119896 minus


119872119896(119868119903 1205740 1205832

1205781 1198866 1205761)

= 119872119896(119868119903 120574119896minus1


Δ120574

2

1205832

1205781 1198866 1205761)

= minus1198912

119868119903[cos(120574

119896minus1+

1

2

119896Δ120574)


minus

1

2

119896Δ120574)]

+

11989612058331205761

31198866

Δ120574 + 1205832

1198682

119903

(119896Δ120574)

= 21198912

119868119903sin 120574119896minus1

sin(

1

2

119896Δ120574)

+

12058331205761

31198866

(119896Δ120574) + 1205832

1198682

119903

(119896Δ120574)

(53)


10038161003816100381610038161003816100381610038161003816100381610038161003816

(12) 119896Δ120574

sin ((12) 119896Δ120574)

(12058331205761+ 311988661205832

1198682

119903

)

311988661198912

119868119903

10038161003816100381610038161003816100381610038161003816100381610038161003816

lt 1

1

2

119896Δ120574 = 119899120587 119899 = 0 plusmn1 plusmn2

(54)


119885119899

minus

= (119868119903 120574119896minus1

1205832

1205781 1198866 1205761) | 119872119896= 0119863

1205740

119872119896

= 0 (55)



119896minus1isin [0 120587]

120574119896minus11

= minus arcsin(12) 119896Δ120574

sin ((12) 119896Δ120574)

(12058331205761+ 311988661205832

1198682

119903

)

(311988661198912

119868119903)

120574119896minus12

= 120587 minus 120574119896minus11

(56)


3 1205761 1205832

1198866 and 119891

2


119896minus1= 120574119896minus11

and 120574119896minus1

= 120574119896minus12

=

120587 minus 120574119896minus11


119895(119868119903 1205740119894

1205832



119906

(119872120576) intersect


1205832120578111988661205761

plusmn120576(120574119896minus1119894


plusmn0

(120574119896minus1119894


119903 1205740

=

120574119896minus1119894


119895(119868119903 1205740119894

1205832

1205781 1198866 1205761)


0119894

= 120574119896minus1119894

minus (119896 minus 1)(Δ1205742) + 119895Δ120574


119895(119868119903 1205740119894

1205832



01



plusmn120576(120574119896minus11

)

and sum1205832120578111988661205761

plusmn120576(120574119896minus12


1205832120578111988661205761

plusmn0

(120574119896minus11

) and sum1205832120578111988661205761

plusmn0

(120574119896minus12


119906


n = ((minus1205831199061+ 12057811199063

1

+ 11988661198682

1199061) minus1199062 0 0) (57)


plusmn0

(120574119896minus11

)


plusmn0

(120574119896minus12





119894(120583) and 119888



the end points 119889119894(120583) to 119888


end points 119889119894(120583) and 119888



0119894

(119868119903 120583) minus Δ120574

minus



119894(120583) for 119894 =

2 119873 while the line 120574 = 1205740119894

(119868119903 120583) + Δ120574

+



(120583) at the end point 119889119894+1


119894(120583) and 119889

119894+1(120583) is zero

namely

ℎ (119888119894(120583)) minus ℎ (119889

119894+1(120583)) = 0 (58)



sum1205832120578111988661205761

0

(1205740119894


119894(120583) and 119889


1


(12057401



2(120583) on the


119873


0

(1205740119873


119873(120583) on the segments 119874


119889119873+1



sum1205832120578111988661205761

0

(1205740119894


1(120583) and 119874

2(120583) of (44a) (44b) It is known



(119868119903 120583) +

Δ120574+

(119868119903) and 120574 = 120574

0119894

(119868119903 120583) minus Δ120574

minus

(119868119903)


0

(120574119896minus11

) andsum1205832120578111988661205761

0

(120574119896minus12


01

minus


01

minusΔ120574+








119897and









120574

|u|h

Γ3

Γ2

Γ1

M

O2O3






1


6 where

1205781



6 1198867 and


1gt 0



2


2and 119886



4is the


1 1198862 1198864






3 respectively when




120574

|u|h

M

Γsum (12057401 12057402)

12057402 + Δ120574+ 12057401 + Δ120574+



12057402

) are depicted


2 1205831 1198862





2= 2 sim 200


183 1205902

= 197 1205831

= 02 1205832

= 02 1198862

= 230 1198863

=

120 1198864= 130 119886

5= minus101 119886

6= minus203 119886

7= minus205 119887

6=

407 1198877= minus308 119887

8= 109 119909

10= minus001 119909

20= minus005 119909

30=

minus001 11990940


1 1198912) and (119909

3 1198912) respec-






1 The figure demon-



1=


2=



50 100 150minus2

minus15

minus1

minus05

0

05

1

x1

f2

(a)

20 40 60 80 100 120 140 160 180

minus3

minus25

minus2

minus15

x3

f2

(b)



= minus001 11990920

= minus005 11990930

=

minus001 11990940


1198912


1198912

)

0 01 02 03 04 05 06 07minus25

minus2

minus15

minus1

minus05

0

05

1

15

x1

1205831

(a)

0 01 02 03 04 05 06 07minus32

minus3

minus28

minus26

minus24

minus22

minus2

minus18

x1

1205831

x3

1205831

x3

(b)




= minus001 11990920

= minus005 11990930

= minus001 11990940


1205831


3

1205831

)

the planes (1199091 1205831) and (119909









3 1198862)




4varies from 119886

4= 2 to 119886

4= 120


2is

chosen as 1198862


1 1198864) and (119909

3 1198864)







0 10 20 30 40 50 60

minus3

minus2

minus1

0

1

x1

a2

(a)

0 10 20 30 40 50 60minus34

minus32

minus3

minus28

minus26

minus24

minus22

minus2

minus18

a2

x3

(b)



= 02 and 1205832


2


= minus001 11990920

= minus005 11990930

= minus001 11990940


1198862


1198862

)

0 20 40 60 80 100 120minus25

minus2

minus15

minus1

minus05

0

05

1

x1

a4

(a)

0 20 40 60 80 100 120

minus35

minus3

minus25

minus2

a4

x3

(b)




= 02 and 1205832


= 23 and 1198863


= minus001 11990920

= minus005 11990930

= minus001 11990940

=


1198864


1198864

)


1are selected as











1= 1437 120590

2=

1142 1205831= 02 120583

2= 02 119886

2= 300 119886

3= 750 119886

4= 450

1198865

= minus1166 1198866

= 1227 1198912

= 1227 1198867

= minus268 1198876

=

minus22 1198877

= minus969 1198878

= minus2232 11990910

= minus108 11990920

= 0511990930

= minus001 11990940



minus1minus050051

minus05

0

05

minus25

minus2

minus15

minus1

x1

x2

x3

(a)


minus1

minus05

0

05

1

x1

x2

(b)


= 827 and 1205831


1199092

1199093


1199092

)

minus50

5

minus50

5minus6

minus4

minus2

0

2

4

6

x1x2

x3

(a)


minus10

minus5

0

5

10

x1

x2

(b)


= 1437 1205902

= 1142 1198862

= 300 1198863

= 750 1198864

= 450 1198865

= minus1166 1198866

=

1227 1198912

= 1227 1198867

= minus268 1198876

= minus22 1198877

= minus969 1198878

= minus2232 11990910

= minus108 11990920

= 05 11990930

= minus001 11990940


1

1199092

1199093


1199092

)





2= 313 119886

5=

minus1501 1198878

= 409 11990910

= minus001 11990920

= minus009 11990930

=

minus005 11990940


7 Conclusions




minus50

5

minus10

0

10

minus6

minus4

minus2

0

2

x1x2

x3

(a)

minus5 0 5 10

minus10

minus5

0

5

10

x1

x2

(b)


= 361 1205902

= 313 1198862

= 230 1198863

= 120 1198864

= 130 1198865

= minus1501 1198866

=

minus203 1198912

= 9238 1198867

= minus205 1198876

= 407 1198877

= minus308 1198878

= 409 11990910

= minus001 11990920

= minus009 11990930

= minus005 11990940


1

1199092

1199093


1199092

)



119896(120576 119868 120574

0 1205832




2 the in-plane


4 and the damping




1 respec-










Acknowledgments



References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











01 11990602 V1 V2 1206011 1206012 1206013 and

1206014via 1199081and 119908

2as follows

11990601

= 11989611199082

1

+ 11989621199082

2

+ 119896311990811199082 (7a)

11990602

= 11989641199082

1

+ 11989651199082

2

+ 119896611990811199082 (7b)

V1= 11989671199082

1

+ 11989681199082

2

+ 119896911990811199082 (7c)

V2= 119896101199082

1

+ 119896111199082

2

+ 1198961211990811199082 (7d)

1206011= 119896191199081 120601

2= 119896201199082 (7e)

1206013= 119896211199081 120601

4= 119896221199082 (7f)



119906 =

1199060

119886

V =

V0

119887

119908 =

1199080

ℎ

120601119909

= 120601119909 120601

119910

= 120601119910 119909 =

119909

119886

119910 =

119910

119887

119902 =

1198872

119864ℎ3

119902 119902119909

=

1198872

119864ℎ3

119902119909 119902

119910

=

1198872

119864ℎ3

119902119910

119905 = 1205872

(

119864

119886119887120588

)

12

119905 Ω119894=

1

1205872

(

119886119887120588

119864

)

12

Ω119894

(119894 = 1 2)

119860119894119895=

(119886119887)12

119864ℎ2

119860119894119895 119861

119894119895=

(119886119887)12

119864ℎ3

119861119894119895

119863119894119895=

(119886119887)12

119864ℎ4

119863119894119895 119864

119894119895=

(119886119887)12

119864ℎ5

119864119894119895

119865119894119895=

(119886119887)12

119864ℎ6

119865119894119895 119867

119894119895=

(119886119887)12

119864ℎ8

119867119894119895

119868119894=

1

(119886119887)(119894+1)2

120588

119868119894

(8)


1+ 12058311+ 1205962

1

1199081


+ 11988651199082

1

1199082+ 11988661199082

2

1199081+ 11988671199083

1

+ 11988681199083

2

= 1198911cosΩ3119905

(9a)

2+ 12058322+ 1205962

2

1199082

+ (1198872cosΩ1119905 + 1198873cosΩ2119905 + 1198874cosΩ4119905) 1199082

+ 11988751199082

2

1199081+ 11988761199082

1

1199082+ 11988771199083

2

+ 11988781199083

1

= 1198912cosΩ3119905

(9b)



1205962

1

=

1205962

9

+ 1205761205901 120596

2

2

= 1205962

+ 1205761205902

Ω3= 120596 Ω

1= Ω2= Ω4=

2120596

3

1205962asymp 31205961

(10)

where 1205901and 120590



1199081(119905 120576) = 119909

10(1198790 1198791) + 120576119909

11(1198790 1198791) + sdot sdot sdot (11a)

1199082(119905 120576) = 119909

20(1198790 1198791) + 120576119909

21(1198790 1198791) + sdot sdot sdot (11b)

where 1198790= 119905 1198791= 120576119905


1= minus

1

2

12058311199091minus

1

2

12059011199092+

1

4

(1198862+ 1198863minus 1198864) 1199092

minus

3

2

11988671199092(1199092

1

+ 1199092

2

) minus

1

2

11988651199094(1199092

1

minus 1199092

2

)

minus 11988661199092(1199092

3

+ 1199092

4

) + 1198865119909111990921199093

(12a)

2= minus

1

2

12058311199092+

1

2

12059011199091+

1

4

(1198862+ 1198863minus 1198864) 1199091

+

3

2

11988671199091(1199092

1

+ 1199092

2

) +

1

2

11988651199093(1199092

1

minus 1199092

2

)

+ 11988661199091(1199092

3

+ 1199092

4

) + 1198865119909111990921199094

(12b)

3= minus

1

2

12058321199093minus

1

6

12059021199094minus

1

3

11988761199094(1199092

1

+ 1199092

2

)

minus

1

2

11988771199094(1199092

3

+ 1199092

4

) minus

1

6

11988781199092(31199092

1

minus 1199092

2

)

(12c)


4= minus

1

2

12058321199094+

1

6

12059021199093+

1

3

11988761199093(1199092

1

+ 1199092

2

)

+

1

2

11988771199093(1199092

3

+ 1199092

4

) +

1

6

11988781199091(1199092

1

minus 31199092

2

) minus

1

12

1198912

(12d)



2oplus 1198852and 119863

4





1= minus

1

2

12058311199091+ (1198910minus

1

2

1205901)1199092minus

3

2

11988671199092(1199092

1

+ 1199092

2

)

minus

1

2

11988651199094(1199092

1

minus 1199092

2

)

minus 11988661199092(1199092

3

+ 1199092

4

) + 1198865119909111990921199093

(13a)

2= minus

1

2

12058311199092+ (1198910+

1

2

1205901)1199091+

3

2

11988671199091(1199092

1

+ 1199092

2

)

+

1

2

11988651199093(1199092

1

minus 1199092

2

)

+ 11988661199091(1199092

3

+ 1199092

4

) + 1198865119909111990921199094

(13b)

3= minus

1

2

12058321199093minus

1

6

12059021199094minus

1

3

11988761199094(1199092

1

+ 1199092

2

)

minus

1

2

11988771199094(1199092

3

+ 1199092

4

) minus

1

6

11988781199092(31199092

1

minus 1199092

2

)

(13c)

4= minus

1

2

12058321199094+

1

6

12059021199093+

1

3

11988761199093(1199092

1

+ 1199092

2

)

+

1

2

11988771199093(1199092

3

+ 1199092

4

) +

1

6

11988781199091(1199092

1

minus 31199092

2

)

(13d)

where 1198910= (14)(119886

2+ 1198863minus 1198864)


1199091= 1199101minus

1

4

11988671199103

1

+

31198865(1205902minus 6)

1205902

2

1199102

1

1199103minus 119886611991011199102

3

minus 119886611991011199102

4

minus

31198865(1205903

2

minus 181205902

2

+ 2161205902minus 1296)

1205904

2

1199102

2

1199103

+

1198865(61205902

2

minus 721205902+ 432)

1205903

2

119910111991021199104

(14a)

1199092= 1199102+

3

2

11988671199103

2

+

3

4

11988671199102

1

1199102+

31198865

1205902

1199102

1

1199104

minus

31198865(1205902

2

minus 121205902+ 72)

1205903

2

1199102

2

1199104minus

61198865(1205902minus 6)

1205902

2

119910111991021199103

(14b)

1199093= 1199103minus

1198878

1205902

1199103

1

+

31198878(1205902

2

minus 121205902+ 72)

1205903

2

11991011199102

2

minus

1

3

1198876119910111991021199104

(14c)

1199094= 1199104+

1198878(1205903

2

minus 181205902

2

+ 2161205902minus 1296)

1205904

2

1199103

2

minus

31198878(1205902minus 6)

1205902

2

1199102

1

1199102+

1

3

1198876119910111991021199103

(14d)


1199101= minus1205831

1199101+ (1 minus 120590

1) 1199102 (15a)

1199102= 12059011199101minus 1205831

1199102+ 11988661199101(1199102

3

+ 1199102

4

) +

3

2

11988671199103

1

(15b)

1199103= minus1205832

1199103minus 12059021199104minus

1

3

11988761199102

1

1199104minus

1

2

11988771199104(1199102

3

+ 1199102

4

) (15c)

1199104= 12059021199103minus 1205832

1199104+

1

3

11988761199102

1

1199103+

1

2

11988771199103(1199102

3

+ 1199102

4

) minus 1198912

(15d)


= (12)1205831 1205832

= (12)1205832 1205902=

(16)1205902 and 119891

2


Further let

1199103= 119868 cos 120574 119910

4= 119868 sin 120574 (16)


1199101= minus1205831

1199101+ (1 minus 120590

1) 1199102 (17a)

1199102= 12059011199101minus 1205831

1199102+ 119886611991011198682

+

3

2

11988671199103

1

(17b)

119868 = minus1205832

119868 minus 1198912

sin 120574 (17c)

119868 120574 = 1205902119868 +

1

3

11988761199102

1

119868 +

1

2

11988771198683

minus 1198912

cos 120574 (17d)


[

1199101

1199102

] = radic3

radic10038161003816100381610038161198866

1003816100381610038161003816

radic10038161003816100381610038161198876

1003816100381610038161003816

[

1 minus 1205901

0

1205831

1] [

1199061

1199062

] (18)



1yield the


1= 1199062 (19a)

2= minus120583119906

1minus 12058331199062+ 12057811199063

1

+ 119886611990611198682

(19b)

119868 = minus1205832

119868 minus 1198912

sin 120574 (19c)

119868 120574 = 1205902119868 + 11988661199062

1

119868 + 12057221198683

minus 1198912

cos 120574 (19d)

where 120583 = 1205832

1

minus 1205901(1 minus 120590

1) 1205833= 21205831

1205781= 91198866119886721198876and

1205722= (12)119887

7


1205832

997888rarr 1205761205832

1205833997888rarr 120576120583

3 119891

2

997888rarr 1205761198912

1205781997888rarr 1205781 120572

2997888rarr 1205722 119886

6997888rarr 1198866

(20)


1=

120597119867

1205971199062

+ 1205761198921199061= 1199062 (21a)

2= minus

120597119867

1205971199061

+ 1205761198921199062= minus120583119906

1+ 12057811199063

1

+ 119886611990611198682

minus 12057612058331199062 (21b)

119868 =

120597119867

120597120574

+ 120576119892119868

= minus1205761205832

119868 minus 1205761198912

sin 120574 (21c)

119868 120574 = minus

120597119867

120597119868

+ 120576119892120574

= 1205902119868 + 12057221198683

+ 11988661198681199062

1

minus 1205761198912

cos 120574 (21d)


119867(1199061 1199062 119868 120574) =

1

2

1199062

2

+

1

2

1205831199062

1

minus

1

4

12057811199064

1

minus

1

2

11988661198682

1199062

1

minus

1

2

12059021198682

minus

1

4

12057221198684

(22)

and 1198921199061 1198921199062 119892119868 and 119892


1198921199061= 0 119892

1199062= minus12058331199062

119892119868

= minus1205832

119868 minus 1198912

sin 120574 119892120574

= minus1198912

cos 120574(23)



1 1199062) of (21a) (21b)


1= 1199062 (24a)

2= minus120583119906

1+ 12057811199063

1

+ 119886611990611198682

(24b)

Since 1205781


11988661198682




11986812

= plusmn[

1205832

1

minus 1205901(1 minus 120590

1)

1198866

]

12

(25)


0= (0 0)


119861 = plusmn

1

1205781

[1205832

1

minus 1205901(1 minus 120590

1) minus 11988661198682

]

12

(26)


0is the



1198681= 0 119868

2= [

1205832

1

minus 1205901(1 minus 120590

1)

1198866

]

12

(27)




(1198791 119868) that is lim


119906ℎ

plusmn

(1198791 119868) =


defined by

119872 = (119906 119868 120574) | 119906 = 119902plusmn(119868) 1198681lt 119868 lt 119868

2 0 le 120574 lt 2120587 (28)



119904

(119872) and 119882119906



119906

(119872)


Γ = (119906 119868 120574) | 119906 = 119906ℎ

plusmn

(1198791 119868) 1198681lt 119868 lt 119868

2

120574 = int

1198791

0

119863119868119867(119906ℎ

plusmn

(1198791 119868) 119868) 119889119904 + 120574

0

(29)


119868 = 0 (30a)

119868 120574 = 119863119868119867(119902plusmn(119868) 119868) 119868

1lt 119868 lt 119868

2 (30b)


where

119863119868119867(119902plusmn(119868) 119868) = minus

120597119867 (119902plusmn(119868) 119868)

120597119868

= 1205902119868 + 12057221198683

+ 11988661198681199062

1

(31)







119863119868119867(119902plusmn 119868) = 120590

2119868119903+ 12057221198683

119903

+ 11988661198681199031199062

1

= 0 (32)

Then we obtain

119868119903= plusmn

12059021205781+ 1198866[1205832

1

minus 1205901(1 minus 120590

1)]

1198862

6

minus 12057221205781

12

(33)



119903) (34)




1 1199062 119868 120574) The phase shift



61198682 and 120583


1= 1199062 (35a)

2= minus12057611199061+ 12057811199063

1

minus 12057612057621199062 (35b)


119867(1199061 1199062) =

1

2

1199062

2

+

1

2

12057611199062

1

minus

1

4

12057811199064

1

(36)






1199061(1198791) = plusmnradic

1205761

1205781

tanh(

radic21205761

2

1198791) (37a)

1199062(1198791) = plusmn

1205761

radic21205781


2

1198791) (37b)


120574 (1198791) = 1205961199031198791minus

1198866radic21205761

1205781

tanh(

radic21205761

2

1198791) + 1205740 (38)

where 120596119903= 1205902+ 12057221198682

+ 119886612057611205781

At 119868 = 119868119903 there is 120596


be expressed as

Δ120574 = [minus

21198866radic21205761

1205781

]

119868=119868119903

= minus

21198866

1205781

radic2 [1205832

1

minus 1205901(1 minus 120590

1) minus 11988661198682

119903

]

(39)



1 1199062) for the




plusmn(119868) in (24a)


M

I

120574

times

u2

u1

(a)

0 1205742120587

I = I1

I = Ir

I = I2

(b)




120576

Therefore we obtain

119872 = 119872120576= (119906 119868 120574) | 119906 = 119902

plusmn(119868) 1198681lt 119868 lt 119868

2 0 le 120574 lt 2120587

(40)


119868 = minus1205832

119868 minus 1198912

sin 120574 (41a)

120574 = 1205902+ 12057221198682

+ 11988661199062

1

minus

1198912

cos 120574119868

(41b)


1205832

997888rarr 1205761205832

119868 = 119868119903+ radic120576ℎ

1198912

997888rarr 1205761198912

1198791997888rarr

1198791

radic120576

(42)


ℎ = minus120583

2

119868119903minus 1198912

sin 120574 minus radic1205761205832

ℎ (43a)

120574 = minus

2120575

1205781

119868119903ℎ minus radic120576(

120575

1205781

ℎ2

+

1198912

119868119903

cos 120574) (43b)

where 120575 = 1198862

6

minus 12057221205781

When 120576 = 0 (43a) (43b) become

ℎ = minus120583

2

119868119903minus 1198912

sin 120574 (44a)

120574 = minus

2120575

1205781

119868119903ℎ (44b)


119863(ℎ 120574) = minus120583

2

119868119903120574 + 1198912

cos 120574 +

120575

1205781

119868119903ℎ2

(45)


1198750= (0 120574


1205832

119868119903

1198912

))

1198760= (0 120574

119904) = (0 120587 + arcsin(

1205832

119868119903

1198912

))

(46)


0and119876









120576 The phase




120574min minus

1198912

1205832

119868119903

cos 120574min = 120587 + arcsin1205832

119868119903

1198912

+

radic119891

2

2

minus 1205832

2

1198682

119903

1205832

119868119903

(47)


119903as

119860120576= (1199061 1199062 119868 120574) | 119906

1= 119861 119906

2= 0

1003816100381610038161003816119868 minus 119868119903

1003816100381610038161003816lt radic120576119862 120574 isin 119879

119871

(48)



120576 denoted as119882119904(119860

120576) and119882

119906

(119860120576) are the


119906






120576 leave and


h

120574

P0

120574min 120574c 120574s

Q0

(a)

h

120574Q120576

P120576

(b)








120574

|u|

h

120574c 120574c + 3Δ120574

p120576qc




119896(120576 119868 120574

0 1205832


0 and 120583

2




function 1198721(120576 119868119903 1205740 1205832


119903 1205740 1205832



(119872) and119882119906


119872(119868119903 1205740 1205832

1205781 1198866 1205761)

= int

+infin

minusinfin

⟨n (119901ℎ

(119905)) g (119901ℎ (119905) 1205832

0)⟩ 1198891198791


= int

+infin

minusinfin

(

120597119867

1205971199061

1198921199061+

120597119867

1205971199062

1198921199062+

120597119867

120597119868

119892119868

+

120597119867

120597120574

119892120574

)1198891198791

= minus

2radic21205833

31205781

12057632

1

minus 2radic211988661205832

1198682

119903

12057612

1

1205781

minus 1198912

119868119903[cos(120574

0minus 1198866

radic21205761

1205781

) minus cos(1205740+ 1198866

radic21205761

1205781

)]

(49)


) (119896 = 1 2 ) is defined as

119872119896(120576 119868119903 1205740 1205832

)

=

119896minus1

sum

119895=0

119872(119868119903 119895Δ120574 (119868

119903) + Γ119895(120576 119868119903 1205740 1205832

) + 1205740 1205832

)

(50)

where

Γ119895(120576 119868119903 1205740 1205832

) =

Ω (1199090(119868119903) 119868119903)

120582 (119868119903)

119895

sum

119903=1

log100381610038161003816100381610038161003816100381610038161003816

120589 (119868119903)

120576119872119903(120576 119868119903 1205740 1205832

)

100381610038161003816100381610038161003816100381610038161003816

(51)

for 119895 = 1 119896 minus 1 and Γ0(120576 119868119903 1205740 1205832


119895(120576 119868119903 1205740 1205832


119896(120576 119868119903 1205740 1205832


119903degenerates



119872119896(119868119903 1205740 1205832

1205781 1198866 1205761)

=

119896minus1

sum

119895=0

119872(119868119903 1205740+ 119895Δ120574 (119868

119903) 1205832

1205781 1198866 1205761)

= minus1198912

119868119903[cos(120574

0minus

1198866radic21205761

1205781

)

minus cos(1205740+

1198866radic21205761

1205781

)]

minus

2radic21205833

31205781

12057632

1

minus 2radic211988661205832

1198682

119903

12057612

1

1205781

minus 1198912

119868119903[cos(120574

0minus

1198866radic21205761

1205781

minus

21198866radic21205761

1205781

)

minus cos(1205740+

1198866radic21205761

1205781

minus

21198866radic21205761

1205781

)]

minus

2radic21205833

31205781

12057632

1

minus 2radic211988661205832

1198682

119903

12057612

1

1205781

+ sdot sdot sdot

minus 1198912

119868119903[cos(120574

0minus

1198866radic21205761

1205781

minus 2 (119896 minus 1) 1198866

radic21205761

1205781

)

minus cos(1205740+

1198866radic21205761

1205781


1198866radic21205761

1205781

)]

minus

2radic21205833

31205781

12057632

1

minus 2radic211988661205832

1198682

119903

12057612

1

1205781

= minus1198912

119868119903[cos(120574

0minus

1198866radic21205761

1205781


1198866radic21205761

1205781

)

minus cos(1205740+

1198866radic21205761

1205781

)]

minus

2radic21198961205833

31205781

12057632

1

minus 2radic211989611988661205832

1198682

119903

12057612

1

1205781

(52)


119896minus1= 1205740+ (119896 minus


119872119896(119868119903 1205740 1205832

1205781 1198866 1205761)

= 119872119896(119868119903 120574119896minus1


Δ120574

2

1205832

1205781 1198866 1205761)

= minus1198912

119868119903[cos(120574

119896minus1+

1

2

119896Δ120574)


minus

1

2

119896Δ120574)]

+

11989612058331205761

31198866

Δ120574 + 1205832

1198682

119903

(119896Δ120574)

= 21198912

119868119903sin 120574119896minus1

sin(

1

2

119896Δ120574)

+

12058331205761

31198866

(119896Δ120574) + 1205832

1198682

119903

(119896Δ120574)

(53)


10038161003816100381610038161003816100381610038161003816100381610038161003816

(12) 119896Δ120574

sin ((12) 119896Δ120574)

(12058331205761+ 311988661205832

1198682

119903

)

311988661198912

119868119903

10038161003816100381610038161003816100381610038161003816100381610038161003816

lt 1

1

2

119896Δ120574 = 119899120587 119899 = 0 plusmn1 plusmn2

(54)


119885119899

minus

= (119868119903 120574119896minus1

1205832

1205781 1198866 1205761) | 119872119896= 0119863

1205740

119872119896

= 0 (55)



119896minus1isin [0 120587]

120574119896minus11

= minus arcsin(12) 119896Δ120574

sin ((12) 119896Δ120574)

(12058331205761+ 311988661205832

1198682

119903

)

(311988661198912

119868119903)

120574119896minus12

= 120587 minus 120574119896minus11

(56)


3 1205761 1205832

1198866 and 119891

2


119896minus1= 120574119896minus11

and 120574119896minus1

= 120574119896minus12

=

120587 minus 120574119896minus11


119895(119868119903 1205740119894

1205832



119906

(119872120576) intersect


1205832120578111988661205761

plusmn120576(120574119896minus1119894


plusmn0

(120574119896minus1119894


119903 1205740

=

120574119896minus1119894


119895(119868119903 1205740119894

1205832

1205781 1198866 1205761)


0119894

= 120574119896minus1119894

minus (119896 minus 1)(Δ1205742) + 119895Δ120574


119895(119868119903 1205740119894

1205832



01



plusmn120576(120574119896minus11

)

and sum1205832120578111988661205761

plusmn120576(120574119896minus12


1205832120578111988661205761

plusmn0

(120574119896minus11

) and sum1205832120578111988661205761

plusmn0

(120574119896minus12


119906


n = ((minus1205831199061+ 12057811199063

1

+ 11988661198682

1199061) minus1199062 0 0) (57)


plusmn0

(120574119896minus11

)


plusmn0

(120574119896minus12





119894(120583) and 119888



the end points 119889119894(120583) to 119888


end points 119889119894(120583) and 119888



0119894

(119868119903 120583) minus Δ120574

minus



119894(120583) for 119894 =

2 119873 while the line 120574 = 1205740119894

(119868119903 120583) + Δ120574

+



(120583) at the end point 119889119894+1


119894(120583) and 119889

119894+1(120583) is zero

namely

ℎ (119888119894(120583)) minus ℎ (119889

119894+1(120583)) = 0 (58)



sum1205832120578111988661205761

0

(1205740119894


119894(120583) and 119889


1


(12057401



2(120583) on the


119873


0

(1205740119873


119873(120583) on the segments 119874


119889119873+1



sum1205832120578111988661205761

0

(1205740119894


1(120583) and 119874

2(120583) of (44a) (44b) It is known



(119868119903 120583) +

Δ120574+

(119868119903) and 120574 = 120574

0119894

(119868119903 120583) minus Δ120574

minus

(119868119903)


0

(120574119896minus11

) andsum1205832120578111988661205761

0

(120574119896minus12


01

minus


01

minusΔ120574+








119897and









120574

|u|h

Γ3

Γ2

Γ1

M

O2O3






1


6 where

1205781



6 1198867 and


1gt 0



2


2and 119886



4is the


1 1198862 1198864






3 respectively when




120574

|u|h

M

Γsum (12057401 12057402)

12057402 + Δ120574+ 12057401 + Δ120574+



12057402

) are depicted


2 1205831 1198862





2= 2 sim 200


183 1205902

= 197 1205831

= 02 1205832

= 02 1198862

= 230 1198863

=

120 1198864= 130 119886

5= minus101 119886

6= minus203 119886

7= minus205 119887

6=

407 1198877= minus308 119887

8= 109 119909

10= minus001 119909

20= minus005 119909

30=

minus001 11990940


1 1198912) and (119909

3 1198912) respec-






1 The figure demon-



1=


2=



50 100 150minus2

minus15

minus1

minus05

0

05

1

x1

f2

(a)

20 40 60 80 100 120 140 160 180

minus3

minus25

minus2

minus15

x3

f2

(b)



= minus001 11990920

= minus005 11990930

=

minus001 11990940


1198912


1198912

)

0 01 02 03 04 05 06 07minus25

minus2

minus15

minus1

minus05

0

05

1

15

x1

1205831

(a)

0 01 02 03 04 05 06 07minus32

minus3

minus28

minus26

minus24

minus22

minus2

minus18

x1

1205831

x3

1205831

x3

(b)




= minus001 11990920

= minus005 11990930

= minus001 11990940


1205831


3

1205831

)

the planes (1199091 1205831) and (119909









3 1198862)




4varies from 119886

4= 2 to 119886

4= 120


2is

chosen as 1198862


1 1198864) and (119909

3 1198864)







0 10 20 30 40 50 60

minus3

minus2

minus1

0

1

x1

a2

(a)

0 10 20 30 40 50 60minus34

minus32

minus3

minus28

minus26

minus24

minus22

minus2

minus18

a2

x3

(b)



= 02 and 1205832


2


= minus001 11990920

= minus005 11990930

= minus001 11990940


1198862


1198862

)

0 20 40 60 80 100 120minus25

minus2

minus15

minus1

minus05

0

05

1

x1

a4

(a)

0 20 40 60 80 100 120

minus35

minus3

minus25

minus2

a4

x3

(b)




= 02 and 1205832


= 23 and 1198863


= minus001 11990920

= minus005 11990930

= minus001 11990940

=


1198864


1198864

)


1are selected as











1= 1437 120590

2=

1142 1205831= 02 120583

2= 02 119886

2= 300 119886

3= 750 119886

4= 450

1198865

= minus1166 1198866

= 1227 1198912

= 1227 1198867

= minus268 1198876

=

minus22 1198877

= minus969 1198878

= minus2232 11990910

= minus108 11990920

= 0511990930

= minus001 11990940



minus1minus050051

minus05

0

05

minus25

minus2

minus15

minus1

x1

x2

x3

(a)


minus1

minus05

0

05

1

x1

x2

(b)


= 827 and 1205831


1199092

1199093


1199092

)

minus50

5

minus50

5minus6

minus4

minus2

0

2

4

6

x1x2

x3

(a)


minus10

minus5

0

5

10

x1

x2

(b)


= 1437 1205902

= 1142 1198862

= 300 1198863

= 750 1198864

= 450 1198865

= minus1166 1198866

=

1227 1198912

= 1227 1198867

= minus268 1198876

= minus22 1198877

= minus969 1198878

= minus2232 11990910

= minus108 11990920

= 05 11990930

= minus001 11990940


1

1199092

1199093


1199092

)





2= 313 119886

5=

minus1501 1198878

= 409 11990910

= minus001 11990920

= minus009 11990930

=

minus005 11990940


7 Conclusions




minus50

5

minus10

0

10

minus6

minus4

minus2

0

2

x1x2

x3

(a)

minus5 0 5 10

minus10

minus5

0

5

10

x1

x2

(b)


= 361 1205902

= 313 1198862

= 230 1198863

= 120 1198864

= 130 1198865

= minus1501 1198866

=

minus203 1198912

= 9238 1198867

= minus205 1198876

= 407 1198877

= minus308 1198878

= 409 11990910

= minus001 11990920

= minus009 11990930

= minus005 11990940


1

1199092

1199093


1199092

)



119896(120576 119868 120574

0 1205832




2 the in-plane


4 and the damping




1 respec-










Acknowledgments



References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










4= minus

1

2

12058321199094+

1

6

12059021199093+

1

3

11988761199093(1199092

1

+ 1199092

2

)

+

1

2

11988771199093(1199092

3

+ 1199092

4

) +

1

6

11988781199091(1199092

1

minus 31199092

2

) minus

1

12

1198912

(12d)



2oplus 1198852and 119863

4





1= minus

1

2

12058311199091+ (1198910minus

1

2

1205901)1199092minus

3

2

11988671199092(1199092

1

+ 1199092

2

)

minus

1

2

11988651199094(1199092

1

minus 1199092

2

)

minus 11988661199092(1199092

3

+ 1199092

4

) + 1198865119909111990921199093

(13a)

2= minus

1

2

12058311199092+ (1198910+

1

2

1205901)1199091+

3

2

11988671199091(1199092

1

+ 1199092

2

)

+

1

2

11988651199093(1199092

1

minus 1199092

2

)

+ 11988661199091(1199092

3

+ 1199092

4

) + 1198865119909111990921199094

(13b)

3= minus

1

2

12058321199093minus

1

6

12059021199094minus

1

3

11988761199094(1199092

1

+ 1199092

2

)

minus

1

2

11988771199094(1199092

3

+ 1199092

4

) minus

1

6

11988781199092(31199092

1

minus 1199092

2

)

(13c)

4= minus

1

2

12058321199094+

1

6

12059021199093+

1

3

11988761199093(1199092

1

+ 1199092

2

)

+

1

2

11988771199093(1199092

3

+ 1199092

4

) +

1

6

11988781199091(1199092

1

minus 31199092

2

)

(13d)

where 1198910= (14)(119886

2+ 1198863minus 1198864)


1199091= 1199101minus

1

4

11988671199103

1

+

31198865(1205902minus 6)

1205902

2

1199102

1

1199103minus 119886611991011199102

3

minus 119886611991011199102

4

minus

31198865(1205903

2

minus 181205902

2

+ 2161205902minus 1296)

1205904

2

1199102

2

1199103

+

1198865(61205902

2

minus 721205902+ 432)

1205903

2

119910111991021199104

(14a)

1199092= 1199102+

3

2

11988671199103

2

+

3

4

11988671199102

1

1199102+

31198865

1205902

1199102

1

1199104

minus

31198865(1205902

2

minus 121205902+ 72)

1205903

2

1199102

2

1199104minus

61198865(1205902minus 6)

1205902

2

119910111991021199103

(14b)

1199093= 1199103minus

1198878

1205902

1199103

1

+

31198878(1205902

2

minus 121205902+ 72)

1205903

2

11991011199102

2

minus

1

3

1198876119910111991021199104

(14c)

1199094= 1199104+

1198878(1205903

2

minus 181205902

2

+ 2161205902minus 1296)

1205904

2

1199103

2

minus

31198878(1205902minus 6)

1205902

2

1199102

1

1199102+

1

3

1198876119910111991021199103

(14d)


1199101= minus1205831

1199101+ (1 minus 120590

1) 1199102 (15a)

1199102= 12059011199101minus 1205831

1199102+ 11988661199101(1199102

3

+ 1199102

4

) +

3

2

11988671199103

1

(15b)

1199103= minus1205832

1199103minus 12059021199104minus

1

3

11988761199102

1

1199104minus

1

2

11988771199104(1199102

3

+ 1199102

4

) (15c)

1199104= 12059021199103minus 1205832

1199104+

1

3

11988761199102

1

1199103+

1

2

11988771199103(1199102

3

+ 1199102

4

) minus 1198912

(15d)


= (12)1205831 1205832

= (12)1205832 1205902=

(16)1205902 and 119891

2


Further let

1199103= 119868 cos 120574 119910

4= 119868 sin 120574 (16)


1199101= minus1205831

1199101+ (1 minus 120590

1) 1199102 (17a)

1199102= 12059011199101minus 1205831

1199102+ 119886611991011198682

+

3

2

11988671199103

1

(17b)

119868 = minus1205832

119868 minus 1198912

sin 120574 (17c)

119868 120574 = 1205902119868 +

1

3

11988761199102

1

119868 +

1

2

11988771198683

minus 1198912

cos 120574 (17d)


[

1199101

1199102

] = radic3

radic10038161003816100381610038161198866

1003816100381610038161003816

radic10038161003816100381610038161198876

1003816100381610038161003816

[

1 minus 1205901

0

1205831

1] [

1199061

1199062

] (18)



1yield the


1= 1199062 (19a)

2= minus120583119906

1minus 12058331199062+ 12057811199063

1

+ 119886611990611198682

(19b)

119868 = minus1205832

119868 minus 1198912

sin 120574 (19c)

119868 120574 = 1205902119868 + 11988661199062

1

119868 + 12057221198683

minus 1198912

cos 120574 (19d)

where 120583 = 1205832

1

minus 1205901(1 minus 120590

1) 1205833= 21205831

1205781= 91198866119886721198876and

1205722= (12)119887

7


1205832

997888rarr 1205761205832

1205833997888rarr 120576120583

3 119891

2

997888rarr 1205761198912

1205781997888rarr 1205781 120572

2997888rarr 1205722 119886

6997888rarr 1198866

(20)


1=

120597119867

1205971199062

+ 1205761198921199061= 1199062 (21a)

2= minus

120597119867

1205971199061

+ 1205761198921199062= minus120583119906

1+ 12057811199063

1

+ 119886611990611198682

minus 12057612058331199062 (21b)

119868 =

120597119867

120597120574

+ 120576119892119868

= minus1205761205832

119868 minus 1205761198912

sin 120574 (21c)

119868 120574 = minus

120597119867

120597119868

+ 120576119892120574

= 1205902119868 + 12057221198683

+ 11988661198681199062

1

minus 1205761198912

cos 120574 (21d)


119867(1199061 1199062 119868 120574) =

1

2

1199062

2

+

1

2

1205831199062

1

minus

1

4

12057811199064

1

minus

1

2

11988661198682

1199062

1

minus

1

2

12059021198682

minus

1

4

12057221198684

(22)

and 1198921199061 1198921199062 119892119868 and 119892


1198921199061= 0 119892

1199062= minus12058331199062

119892119868

= minus1205832

119868 minus 1198912

sin 120574 119892120574

= minus1198912

cos 120574(23)



1 1199062) of (21a) (21b)


1= 1199062 (24a)

2= minus120583119906

1+ 12057811199063

1

+ 119886611990611198682

(24b)

Since 1205781


11988661198682




11986812

= plusmn[

1205832

1

minus 1205901(1 minus 120590

1)

1198866

]

12

(25)


0= (0 0)


119861 = plusmn

1

1205781

[1205832

1

minus 1205901(1 minus 120590

1) minus 11988661198682

]

12

(26)


0is the



1198681= 0 119868

2= [

1205832

1

minus 1205901(1 minus 120590

1)

1198866

]

12

(27)




(1198791 119868) that is lim


119906ℎ

plusmn

(1198791 119868) =


defined by

119872 = (119906 119868 120574) | 119906 = 119902plusmn(119868) 1198681lt 119868 lt 119868

2 0 le 120574 lt 2120587 (28)



119904

(119872) and 119882119906



119906

(119872)


Γ = (119906 119868 120574) | 119906 = 119906ℎ

plusmn

(1198791 119868) 1198681lt 119868 lt 119868

2

120574 = int

1198791

0

119863119868119867(119906ℎ

plusmn

(1198791 119868) 119868) 119889119904 + 120574

0

(29)


119868 = 0 (30a)

119868 120574 = 119863119868119867(119902plusmn(119868) 119868) 119868

1lt 119868 lt 119868

2 (30b)


where

119863119868119867(119902plusmn(119868) 119868) = minus

120597119867 (119902plusmn(119868) 119868)

120597119868

= 1205902119868 + 12057221198683

+ 11988661198681199062

1

(31)







119863119868119867(119902plusmn 119868) = 120590

2119868119903+ 12057221198683

119903

+ 11988661198681199031199062

1

= 0 (32)

Then we obtain

119868119903= plusmn

12059021205781+ 1198866[1205832

1

minus 1205901(1 minus 120590

1)]

1198862

6

minus 12057221205781

12

(33)



119903) (34)




1 1199062 119868 120574) The phase shift



61198682 and 120583


1= 1199062 (35a)

2= minus12057611199061+ 12057811199063

1

minus 12057612057621199062 (35b)


119867(1199061 1199062) =

1

2

1199062

2

+

1

2

12057611199062

1

minus

1

4

12057811199064

1

(36)






1199061(1198791) = plusmnradic

1205761

1205781

tanh(

radic21205761

2

1198791) (37a)

1199062(1198791) = plusmn

1205761

radic21205781


2

1198791) (37b)


120574 (1198791) = 1205961199031198791minus

1198866radic21205761

1205781

tanh(

radic21205761

2

1198791) + 1205740 (38)

where 120596119903= 1205902+ 12057221198682

+ 119886612057611205781

At 119868 = 119868119903 there is 120596


be expressed as

Δ120574 = [minus

21198866radic21205761

1205781

]

119868=119868119903

= minus

21198866

1205781

radic2 [1205832

1

minus 1205901(1 minus 120590

1) minus 11988661198682

119903

]

(39)



1 1199062) for the




plusmn(119868) in (24a)


M

I

120574

times

u2

u1

(a)

0 1205742120587

I = I1

I = Ir

I = I2

(b)




120576

Therefore we obtain

119872 = 119872120576= (119906 119868 120574) | 119906 = 119902

plusmn(119868) 1198681lt 119868 lt 119868

2 0 le 120574 lt 2120587

(40)


119868 = minus1205832

119868 minus 1198912

sin 120574 (41a)

120574 = 1205902+ 12057221198682

+ 11988661199062

1

minus

1198912

cos 120574119868

(41b)


1205832

997888rarr 1205761205832

119868 = 119868119903+ radic120576ℎ

1198912

997888rarr 1205761198912

1198791997888rarr

1198791

radic120576

(42)


ℎ = minus120583

2

119868119903minus 1198912

sin 120574 minus radic1205761205832

ℎ (43a)

120574 = minus

2120575

1205781

119868119903ℎ minus radic120576(

120575

1205781

ℎ2

+

1198912

119868119903

cos 120574) (43b)

where 120575 = 1198862

6

minus 12057221205781

When 120576 = 0 (43a) (43b) become

ℎ = minus120583

2

119868119903minus 1198912

sin 120574 (44a)

120574 = minus

2120575

1205781

119868119903ℎ (44b)


119863(ℎ 120574) = minus120583

2

119868119903120574 + 1198912

cos 120574 +

120575

1205781

119868119903ℎ2

(45)


1198750= (0 120574


1205832

119868119903

1198912

))

1198760= (0 120574

119904) = (0 120587 + arcsin(

1205832

119868119903

1198912

))

(46)


0and119876









120576 The phase




120574min minus

1198912

1205832

119868119903

cos 120574min = 120587 + arcsin1205832

119868119903

1198912

+

radic119891

2

2

minus 1205832

2

1198682

119903

1205832

119868119903

(47)


119903as

119860120576= (1199061 1199062 119868 120574) | 119906

1= 119861 119906

2= 0

1003816100381610038161003816119868 minus 119868119903

1003816100381610038161003816lt radic120576119862 120574 isin 119879

119871

(48)



120576 denoted as119882119904(119860

120576) and119882

119906

(119860120576) are the


119906






120576 leave and


h

120574

P0

120574min 120574c 120574s

Q0

(a)

h

120574Q120576

P120576

(b)








120574

|u|

h

120574c 120574c + 3Δ120574

p120576qc




119896(120576 119868 120574

0 1205832


0 and 120583

2




function 1198721(120576 119868119903 1205740 1205832


119903 1205740 1205832



(119872) and119882119906


119872(119868119903 1205740 1205832

1205781 1198866 1205761)

= int

+infin

minusinfin

⟨n (119901ℎ

(119905)) g (119901ℎ (119905) 1205832

0)⟩ 1198891198791


= int

+infin

minusinfin

(

120597119867

1205971199061

1198921199061+

120597119867

1205971199062

1198921199062+

120597119867

120597119868

119892119868

+

120597119867

120597120574

119892120574

)1198891198791

= minus

2radic21205833

31205781

12057632

1

minus 2radic211988661205832

1198682

119903

12057612

1

1205781

minus 1198912

119868119903[cos(120574

0minus 1198866

radic21205761

1205781

) minus cos(1205740+ 1198866

radic21205761

1205781

)]

(49)


) (119896 = 1 2 ) is defined as

119872119896(120576 119868119903 1205740 1205832

)

=

119896minus1

sum

119895=0

119872(119868119903 119895Δ120574 (119868

119903) + Γ119895(120576 119868119903 1205740 1205832

) + 1205740 1205832

)

(50)

where

Γ119895(120576 119868119903 1205740 1205832

) =

Ω (1199090(119868119903) 119868119903)

120582 (119868119903)

119895

sum

119903=1

log100381610038161003816100381610038161003816100381610038161003816

120589 (119868119903)

120576119872119903(120576 119868119903 1205740 1205832

)

100381610038161003816100381610038161003816100381610038161003816

(51)

for 119895 = 1 119896 minus 1 and Γ0(120576 119868119903 1205740 1205832


119895(120576 119868119903 1205740 1205832


119896(120576 119868119903 1205740 1205832


119903degenerates



119872119896(119868119903 1205740 1205832

1205781 1198866 1205761)

=

119896minus1

sum

119895=0

119872(119868119903 1205740+ 119895Δ120574 (119868

119903) 1205832

1205781 1198866 1205761)

= minus1198912

119868119903[cos(120574

0minus

1198866radic21205761

1205781

)

minus cos(1205740+

1198866radic21205761

1205781

)]

minus

2radic21205833

31205781

12057632

1

minus 2radic211988661205832

1198682

119903

12057612

1

1205781

minus 1198912

119868119903[cos(120574

0minus

1198866radic21205761

1205781

minus

21198866radic21205761

1205781

)

minus cos(1205740+

1198866radic21205761

1205781

minus

21198866radic21205761

1205781

)]

minus

2radic21205833

31205781

12057632

1

minus 2radic211988661205832

1198682

119903

12057612

1

1205781

+ sdot sdot sdot

minus 1198912

119868119903[cos(120574

0minus

1198866radic21205761

1205781

minus 2 (119896 minus 1) 1198866

radic21205761

1205781

)

minus cos(1205740+

1198866radic21205761

1205781


1198866radic21205761

1205781

)]

minus

2radic21205833

31205781

12057632

1

minus 2radic211988661205832

1198682

119903

12057612

1

1205781

= minus1198912

119868119903[cos(120574

0minus

1198866radic21205761

1205781


1198866radic21205761

1205781

)

minus cos(1205740+

1198866radic21205761

1205781

)]

minus

2radic21198961205833

31205781

12057632

1

minus 2radic211989611988661205832

1198682

119903

12057612

1

1205781

(52)


119896minus1= 1205740+ (119896 minus


119872119896(119868119903 1205740 1205832

1205781 1198866 1205761)

= 119872119896(119868119903 120574119896minus1


Δ120574

2

1205832

1205781 1198866 1205761)

= minus1198912

119868119903[cos(120574

119896minus1+

1

2

119896Δ120574)


minus

1

2

119896Δ120574)]

+

11989612058331205761

31198866

Δ120574 + 1205832

1198682

119903

(119896Δ120574)

= 21198912

119868119903sin 120574119896minus1

sin(

1

2

119896Δ120574)

+

12058331205761

31198866

(119896Δ120574) + 1205832

1198682

119903

(119896Δ120574)

(53)


10038161003816100381610038161003816100381610038161003816100381610038161003816

(12) 119896Δ120574

sin ((12) 119896Δ120574)

(12058331205761+ 311988661205832

1198682

119903

)

311988661198912

119868119903

10038161003816100381610038161003816100381610038161003816100381610038161003816

lt 1

1

2

119896Δ120574 = 119899120587 119899 = 0 plusmn1 plusmn2

(54)


119885119899

minus

= (119868119903 120574119896minus1

1205832

1205781 1198866 1205761) | 119872119896= 0119863

1205740

119872119896

= 0 (55)



119896minus1isin [0 120587]

120574119896minus11

= minus arcsin(12) 119896Δ120574

sin ((12) 119896Δ120574)

(12058331205761+ 311988661205832

1198682

119903

)

(311988661198912

119868119903)

120574119896minus12

= 120587 minus 120574119896minus11

(56)


3 1205761 1205832

1198866 and 119891

2


119896minus1= 120574119896minus11

and 120574119896minus1

= 120574119896minus12

=

120587 minus 120574119896minus11


119895(119868119903 1205740119894

1205832



119906

(119872120576) intersect


1205832120578111988661205761

plusmn120576(120574119896minus1119894


plusmn0

(120574119896minus1119894


119903 1205740

=

120574119896minus1119894


119895(119868119903 1205740119894

1205832

1205781 1198866 1205761)


0119894

= 120574119896minus1119894

minus (119896 minus 1)(Δ1205742) + 119895Δ120574


119895(119868119903 1205740119894

1205832



01



plusmn120576(120574119896minus11

)

and sum1205832120578111988661205761

plusmn120576(120574119896minus12


1205832120578111988661205761

plusmn0

(120574119896minus11

) and sum1205832120578111988661205761

plusmn0

(120574119896minus12


119906


n = ((minus1205831199061+ 12057811199063

1

+ 11988661198682

1199061) minus1199062 0 0) (57)


plusmn0

(120574119896minus11

)


plusmn0

(120574119896minus12





119894(120583) and 119888



the end points 119889119894(120583) to 119888


end points 119889119894(120583) and 119888



0119894

(119868119903 120583) minus Δ120574

minus



119894(120583) for 119894 =

2 119873 while the line 120574 = 1205740119894

(119868119903 120583) + Δ120574

+



(120583) at the end point 119889119894+1


119894(120583) and 119889

119894+1(120583) is zero

namely

ℎ (119888119894(120583)) minus ℎ (119889

119894+1(120583)) = 0 (58)



sum1205832120578111988661205761

0

(1205740119894


119894(120583) and 119889


1


(12057401



2(120583) on the


119873


0

(1205740119873


119873(120583) on the segments 119874


119889119873+1



sum1205832120578111988661205761

0

(1205740119894


1(120583) and 119874

2(120583) of (44a) (44b) It is known



(119868119903 120583) +

Δ120574+

(119868119903) and 120574 = 120574

0119894

(119868119903 120583) minus Δ120574

minus

(119868119903)


0

(120574119896minus11

) andsum1205832120578111988661205761

0

(120574119896minus12


01

minus


01

minusΔ120574+








119897and









120574

|u|h

Γ3

Γ2

Γ1

M

O2O3






1


6 where

1205781



6 1198867 and


1gt 0



2


2and 119886



4is the


1 1198862 1198864






3 respectively when




120574

|u|h

M

Γsum (12057401 12057402)

12057402 + Δ120574+ 12057401 + Δ120574+



12057402

) are depicted


2 1205831 1198862





2= 2 sim 200


183 1205902

= 197 1205831

= 02 1205832

= 02 1198862

= 230 1198863

=

120 1198864= 130 119886

5= minus101 119886

6= minus203 119886

7= minus205 119887

6=

407 1198877= minus308 119887

8= 109 119909

10= minus001 119909

20= minus005 119909

30=

minus001 11990940


1 1198912) and (119909

3 1198912) respec-






1 The figure demon-



1=


2=



50 100 150minus2

minus15

minus1

minus05

0

05

1

x1

f2

(a)

20 40 60 80 100 120 140 160 180

minus3

minus25

minus2

minus15

x3

f2

(b)



= minus001 11990920

= minus005 11990930

=

minus001 11990940


1198912


1198912

)

0 01 02 03 04 05 06 07minus25

minus2

minus15

minus1

minus05

0

05

1

15

x1

1205831

(a)

0 01 02 03 04 05 06 07minus32

minus3

minus28

minus26

minus24

minus22

minus2

minus18

x1

1205831

x3

1205831

x3

(b)




= minus001 11990920

= minus005 11990930

= minus001 11990940


1205831


3

1205831

)

the planes (1199091 1205831) and (119909









3 1198862)




4varies from 119886

4= 2 to 119886

4= 120


2is

chosen as 1198862


1 1198864) and (119909

3 1198864)







0 10 20 30 40 50 60

minus3

minus2

minus1

0

1

x1

a2

(a)

0 10 20 30 40 50 60minus34

minus32

minus3

minus28

minus26

minus24

minus22

minus2

minus18

a2

x3

(b)



= 02 and 1205832


2


= minus001 11990920

= minus005 11990930

= minus001 11990940


1198862


1198862

)

0 20 40 60 80 100 120minus25

minus2

minus15

minus1

minus05

0

05

1

x1

a4

(a)

0 20 40 60 80 100 120

minus35

minus3

minus25

minus2

a4

x3

(b)




= 02 and 1205832


= 23 and 1198863


= minus001 11990920

= minus005 11990930

= minus001 11990940

=


1198864


1198864

)


1are selected as











1= 1437 120590

2=

1142 1205831= 02 120583

2= 02 119886

2= 300 119886

3= 750 119886

4= 450

1198865

= minus1166 1198866

= 1227 1198912

= 1227 1198867

= minus268 1198876

=

minus22 1198877

= minus969 1198878

= minus2232 11990910

= minus108 11990920

= 0511990930

= minus001 11990940



minus1minus050051

minus05

0

05

minus25

minus2

minus15

minus1

x1

x2

x3

(a)


minus1

minus05

0

05

1

x1

x2

(b)


= 827 and 1205831


1199092

1199093


1199092

)

minus50

5

minus50

5minus6

minus4

minus2

0

2

4

6

x1x2

x3

(a)


minus10

minus5

0

5

10

x1

x2

(b)


= 1437 1205902

= 1142 1198862

= 300 1198863

= 750 1198864

= 450 1198865

= minus1166 1198866

=

1227 1198912

= 1227 1198867

= minus268 1198876

= minus22 1198877

= minus969 1198878

= minus2232 11990910

= minus108 11990920

= 05 11990930

= minus001 11990940


1

1199092

1199093


1199092

)





2= 313 119886

5=

minus1501 1198878

= 409 11990910

= minus001 11990920

= minus009 11990930

=

minus005 11990940


7 Conclusions




minus50

5

minus10

0

10

minus6

minus4

minus2

0

2

x1x2

x3

(a)

minus5 0 5 10

minus10

minus5

0

5

10

x1

x2

(b)


= 361 1205902

= 313 1198862

= 230 1198863

= 120 1198864

= 130 1198865

= minus1501 1198866

=

minus203 1198912

= 9238 1198867

= minus205 1198876

= 407 1198877

= minus308 1198878

= 409 11990910

= minus001 11990920

= minus009 11990930

= minus005 11990940


1

1199092

1199093


1199092

)



119896(120576 119868 120574

0 1205832




2 the in-plane


4 and the damping




1 respec-










Acknowledgments



References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1yield the


1= 1199062 (19a)

2= minus120583119906

1minus 12058331199062+ 12057811199063

1

+ 119886611990611198682

(19b)

119868 = minus1205832

119868 minus 1198912

sin 120574 (19c)

119868 120574 = 1205902119868 + 11988661199062

1

119868 + 12057221198683

minus 1198912

cos 120574 (19d)

where 120583 = 1205832

1

minus 1205901(1 minus 120590

1) 1205833= 21205831

1205781= 91198866119886721198876and

1205722= (12)119887

7


1205832

997888rarr 1205761205832

1205833997888rarr 120576120583

3 119891

2

997888rarr 1205761198912

1205781997888rarr 1205781 120572

2997888rarr 1205722 119886

6997888rarr 1198866

(20)


1=

120597119867

1205971199062

+ 1205761198921199061= 1199062 (21a)

2= minus

120597119867

1205971199061

+ 1205761198921199062= minus120583119906

1+ 12057811199063

1

+ 119886611990611198682

minus 12057612058331199062 (21b)

119868 =

120597119867

120597120574

+ 120576119892119868

= minus1205761205832

119868 minus 1205761198912

sin 120574 (21c)

119868 120574 = minus

120597119867

120597119868

+ 120576119892120574

= 1205902119868 + 12057221198683

+ 11988661198681199062

1

minus 1205761198912

cos 120574 (21d)


119867(1199061 1199062 119868 120574) =

1

2

1199062

2

+

1

2

1205831199062

1

minus

1

4

12057811199064

1

minus

1

2

11988661198682

1199062

1

minus

1

2

12059021198682

minus

1

4

12057221198684

(22)

and 1198921199061 1198921199062 119892119868 and 119892


1198921199061= 0 119892

1199062= minus12058331199062

119892119868

= minus1205832

119868 minus 1198912

sin 120574 119892120574

= minus1198912

cos 120574(23)



1 1199062) of (21a) (21b)


1= 1199062 (24a)

2= minus120583119906

1+ 12057811199063

1

+ 119886611990611198682

(24b)

Since 1205781


11988661198682




11986812

= plusmn[

1205832

1

minus 1205901(1 minus 120590

1)

1198866

]

12

(25)


0= (0 0)


119861 = plusmn

1

1205781

[1205832

1

minus 1205901(1 minus 120590

1) minus 11988661198682

]

12

(26)


0is the



1198681= 0 119868

2= [

1205832

1

minus 1205901(1 minus 120590

1)

1198866

]

12

(27)




(1198791 119868) that is lim


119906ℎ

plusmn

(1198791 119868) =


defined by

119872 = (119906 119868 120574) | 119906 = 119902plusmn(119868) 1198681lt 119868 lt 119868

2 0 le 120574 lt 2120587 (28)



119904

(119872) and 119882119906



119906

(119872)


Γ = (119906 119868 120574) | 119906 = 119906ℎ

plusmn

(1198791 119868) 1198681lt 119868 lt 119868

2

120574 = int

1198791

0

119863119868119867(119906ℎ

plusmn

(1198791 119868) 119868) 119889119904 + 120574

0

(29)


119868 = 0 (30a)

119868 120574 = 119863119868119867(119902plusmn(119868) 119868) 119868

1lt 119868 lt 119868

2 (30b)


where

119863119868119867(119902plusmn(119868) 119868) = minus

120597119867 (119902plusmn(119868) 119868)

120597119868

= 1205902119868 + 12057221198683

+ 11988661198681199062

1

(31)







119863119868119867(119902plusmn 119868) = 120590

2119868119903+ 12057221198683

119903

+ 11988661198681199031199062

1

= 0 (32)

Then we obtain

119868119903= plusmn

12059021205781+ 1198866[1205832

1

minus 1205901(1 minus 120590

1)]

1198862

6

minus 12057221205781

12

(33)



119903) (34)




1 1199062 119868 120574) The phase shift



61198682 and 120583


1= 1199062 (35a)

2= minus12057611199061+ 12057811199063

1

minus 12057612057621199062 (35b)


119867(1199061 1199062) =

1

2

1199062

2

+

1

2

12057611199062

1

minus

1

4

12057811199064

1

(36)






1199061(1198791) = plusmnradic

1205761

1205781

tanh(

radic21205761

2

1198791) (37a)

1199062(1198791) = plusmn

1205761

radic21205781


2

1198791) (37b)


120574 (1198791) = 1205961199031198791minus

1198866radic21205761

1205781

tanh(

radic21205761

2

1198791) + 1205740 (38)

where 120596119903= 1205902+ 12057221198682

+ 119886612057611205781

At 119868 = 119868119903 there is 120596


be expressed as

Δ120574 = [minus

21198866radic21205761

1205781

]

119868=119868119903

= minus

21198866

1205781

radic2 [1205832

1

minus 1205901(1 minus 120590

1) minus 11988661198682

119903

]

(39)



1 1199062) for the




plusmn(119868) in (24a)


M

I

120574

times

u2

u1

(a)

0 1205742120587

I = I1

I = Ir

I = I2

(b)




120576

Therefore we obtain

119872 = 119872120576= (119906 119868 120574) | 119906 = 119902

plusmn(119868) 1198681lt 119868 lt 119868

2 0 le 120574 lt 2120587

(40)


119868 = minus1205832

119868 minus 1198912

sin 120574 (41a)

120574 = 1205902+ 12057221198682

+ 11988661199062

1

minus

1198912

cos 120574119868

(41b)


1205832

997888rarr 1205761205832

119868 = 119868119903+ radic120576ℎ

1198912

997888rarr 1205761198912

1198791997888rarr

1198791

radic120576

(42)


ℎ = minus120583

2

119868119903minus 1198912

sin 120574 minus radic1205761205832

ℎ (43a)

120574 = minus

2120575

1205781

119868119903ℎ minus radic120576(

120575

1205781

ℎ2

+

1198912

119868119903

cos 120574) (43b)

where 120575 = 1198862

6

minus 12057221205781

When 120576 = 0 (43a) (43b) become

ℎ = minus120583

2

119868119903minus 1198912

sin 120574 (44a)

120574 = minus

2120575

1205781

119868119903ℎ (44b)


119863(ℎ 120574) = minus120583

2

119868119903120574 + 1198912

cos 120574 +

120575

1205781

119868119903ℎ2

(45)


1198750= (0 120574


1205832

119868119903

1198912

))

1198760= (0 120574

119904) = (0 120587 + arcsin(

1205832

119868119903

1198912

))

(46)


0and119876









120576 The phase




120574min minus

1198912

1205832

119868119903

cos 120574min = 120587 + arcsin1205832

119868119903

1198912

+

radic119891

2

2

minus 1205832

2

1198682

119903

1205832

119868119903

(47)


119903as

119860120576= (1199061 1199062 119868 120574) | 119906

1= 119861 119906

2= 0

1003816100381610038161003816119868 minus 119868119903

1003816100381610038161003816lt radic120576119862 120574 isin 119879

119871

(48)



120576 denoted as119882119904(119860

120576) and119882

119906

(119860120576) are the


119906






120576 leave and


h

120574

P0

120574min 120574c 120574s

Q0

(a)

h

120574Q120576

P120576

(b)








120574

|u|

h

120574c 120574c + 3Δ120574

p120576qc




119896(120576 119868 120574

0 1205832


0 and 120583

2




function 1198721(120576 119868119903 1205740 1205832


119903 1205740 1205832



(119872) and119882119906


119872(119868119903 1205740 1205832

1205781 1198866 1205761)

= int

+infin

minusinfin

⟨n (119901ℎ

(119905)) g (119901ℎ (119905) 1205832

0)⟩ 1198891198791


= int

+infin

minusinfin

(

120597119867

1205971199061

1198921199061+

120597119867

1205971199062

1198921199062+

120597119867

120597119868

119892119868

+

120597119867

120597120574

119892120574

)1198891198791

= minus

2radic21205833

31205781

12057632

1

minus 2radic211988661205832

1198682

119903

12057612

1

1205781

minus 1198912

119868119903[cos(120574

0minus 1198866

radic21205761

1205781

) minus cos(1205740+ 1198866

radic21205761

1205781

)]

(49)


) (119896 = 1 2 ) is defined as

119872119896(120576 119868119903 1205740 1205832

)

=

119896minus1

sum

119895=0

119872(119868119903 119895Δ120574 (119868

119903) + Γ119895(120576 119868119903 1205740 1205832

) + 1205740 1205832

)

(50)

where

Γ119895(120576 119868119903 1205740 1205832

) =

Ω (1199090(119868119903) 119868119903)

120582 (119868119903)

119895

sum

119903=1

log100381610038161003816100381610038161003816100381610038161003816

120589 (119868119903)

120576119872119903(120576 119868119903 1205740 1205832

)

100381610038161003816100381610038161003816100381610038161003816

(51)

for 119895 = 1 119896 minus 1 and Γ0(120576 119868119903 1205740 1205832


119895(120576 119868119903 1205740 1205832


119896(120576 119868119903 1205740 1205832


119903degenerates



119872119896(119868119903 1205740 1205832

1205781 1198866 1205761)

=

119896minus1

sum

119895=0

119872(119868119903 1205740+ 119895Δ120574 (119868

119903) 1205832

1205781 1198866 1205761)

= minus1198912

119868119903[cos(120574

0minus

1198866radic21205761

1205781

)

minus cos(1205740+

1198866radic21205761

1205781

)]

minus

2radic21205833

31205781

12057632

1

minus 2radic211988661205832

1198682

119903

12057612

1

1205781

minus 1198912

119868119903[cos(120574

0minus

1198866radic21205761

1205781

minus

21198866radic21205761

1205781

)

minus cos(1205740+

1198866radic21205761

1205781

minus

21198866radic21205761

1205781

)]

minus

2radic21205833

31205781

12057632

1

minus 2radic211988661205832

1198682

119903

12057612

1

1205781

+ sdot sdot sdot

minus 1198912

119868119903[cos(120574

0minus

1198866radic21205761

1205781

minus 2 (119896 minus 1) 1198866

radic21205761

1205781

)

minus cos(1205740+

1198866radic21205761

1205781


1198866radic21205761

1205781

)]

minus

2radic21205833

31205781

12057632

1

minus 2radic211988661205832

1198682

119903

12057612

1

1205781

= minus1198912

119868119903[cos(120574

0minus

1198866radic21205761

1205781


1198866radic21205761

1205781

)

minus cos(1205740+

1198866radic21205761

1205781

)]

minus

2radic21198961205833

31205781

12057632

1

minus 2radic211989611988661205832

1198682

119903

12057612

1

1205781

(52)


119896minus1= 1205740+ (119896 minus


119872119896(119868119903 1205740 1205832

1205781 1198866 1205761)

= 119872119896(119868119903 120574119896minus1


Δ120574

2

1205832

1205781 1198866 1205761)

= minus1198912

119868119903[cos(120574

119896minus1+

1

2

119896Δ120574)


minus

1

2

119896Δ120574)]

+

11989612058331205761

31198866

Δ120574 + 1205832

1198682

119903

(119896Δ120574)

= 21198912

119868119903sin 120574119896minus1

sin(

1

2

119896Δ120574)

+

12058331205761

31198866

(119896Δ120574) + 1205832

1198682

119903

(119896Δ120574)

(53)


10038161003816100381610038161003816100381610038161003816100381610038161003816

(12) 119896Δ120574

sin ((12) 119896Δ120574)

(12058331205761+ 311988661205832

1198682

119903

)

311988661198912

119868119903

10038161003816100381610038161003816100381610038161003816100381610038161003816

lt 1

1

2

119896Δ120574 = 119899120587 119899 = 0 plusmn1 plusmn2

(54)


119885119899

minus

= (119868119903 120574119896minus1

1205832

1205781 1198866 1205761) | 119872119896= 0119863

1205740

119872119896

= 0 (55)



119896minus1isin [0 120587]

120574119896minus11

= minus arcsin(12) 119896Δ120574

sin ((12) 119896Δ120574)

(12058331205761+ 311988661205832

1198682

119903

)

(311988661198912

119868119903)

120574119896minus12

= 120587 minus 120574119896minus11

(56)


3 1205761 1205832

1198866 and 119891

2


119896minus1= 120574119896minus11

and 120574119896minus1

= 120574119896minus12

=

120587 minus 120574119896minus11


119895(119868119903 1205740119894

1205832



119906

(119872120576) intersect


1205832120578111988661205761

plusmn120576(120574119896minus1119894


plusmn0

(120574119896minus1119894


119903 1205740

=

120574119896minus1119894


119895(119868119903 1205740119894

1205832

1205781 1198866 1205761)


0119894

= 120574119896minus1119894

minus (119896 minus 1)(Δ1205742) + 119895Δ120574


119895(119868119903 1205740119894

1205832



01



plusmn120576(120574119896minus11

)

and sum1205832120578111988661205761

plusmn120576(120574119896minus12


1205832120578111988661205761

plusmn0

(120574119896minus11

) and sum1205832120578111988661205761

plusmn0

(120574119896minus12


119906


n = ((minus1205831199061+ 12057811199063

1

+ 11988661198682

1199061) minus1199062 0 0) (57)


plusmn0

(120574119896minus11

)


plusmn0

(120574119896minus12





119894(120583) and 119888



the end points 119889119894(120583) to 119888


end points 119889119894(120583) and 119888



0119894

(119868119903 120583) minus Δ120574

minus



119894(120583) for 119894 =

2 119873 while the line 120574 = 1205740119894

(119868119903 120583) + Δ120574

+



(120583) at the end point 119889119894+1


119894(120583) and 119889

119894+1(120583) is zero

namely

ℎ (119888119894(120583)) minus ℎ (119889

119894+1(120583)) = 0 (58)



sum1205832120578111988661205761

0

(1205740119894


119894(120583) and 119889


1


(12057401



2(120583) on the


119873


0

(1205740119873


119873(120583) on the segments 119874


119889119873+1



sum1205832120578111988661205761

0

(1205740119894


1(120583) and 119874

2(120583) of (44a) (44b) It is known



(119868119903 120583) +

Δ120574+

(119868119903) and 120574 = 120574

0119894

(119868119903 120583) minus Δ120574

minus

(119868119903)


0

(120574119896minus11

) andsum1205832120578111988661205761

0

(120574119896minus12


01

minus


01

minusΔ120574+








119897and









120574

|u|h

Γ3

Γ2

Γ1

M

O2O3






1


6 where

1205781



6 1198867 and


1gt 0



2


2and 119886



4is the


1 1198862 1198864






3 respectively when




120574

|u|h

M

Γsum (12057401 12057402)

12057402 + Δ120574+ 12057401 + Δ120574+



12057402

) are depicted


2 1205831 1198862





2= 2 sim 200


183 1205902

= 197 1205831

= 02 1205832

= 02 1198862

= 230 1198863

=

120 1198864= 130 119886

5= minus101 119886

6= minus203 119886

7= minus205 119887

6=

407 1198877= minus308 119887

8= 109 119909

10= minus001 119909

20= minus005 119909

30=

minus001 11990940


1 1198912) and (119909

3 1198912) respec-






1 The figure demon-



1=


2=



50 100 150minus2

minus15

minus1

minus05

0

05

1

x1

f2

(a)

20 40 60 80 100 120 140 160 180

minus3

minus25

minus2

minus15

x3

f2

(b)



= minus001 11990920

= minus005 11990930

=

minus001 11990940


1198912


1198912

)

0 01 02 03 04 05 06 07minus25

minus2

minus15

minus1

minus05

0

05

1

15

x1

1205831

(a)

0 01 02 03 04 05 06 07minus32

minus3

minus28

minus26

minus24

minus22

minus2

minus18

x1

1205831

x3

1205831

x3

(b)




= minus001 11990920

= minus005 11990930

= minus001 11990940


1205831


3

1205831

)

the planes (1199091 1205831) and (119909









3 1198862)




4varies from 119886

4= 2 to 119886

4= 120


2is

chosen as 1198862


1 1198864) and (119909

3 1198864)







0 10 20 30 40 50 60

minus3

minus2

minus1

0

1

x1

a2

(a)

0 10 20 30 40 50 60minus34

minus32

minus3

minus28

minus26

minus24

minus22

minus2

minus18

a2

x3

(b)



= 02 and 1205832


2


= minus001 11990920

= minus005 11990930

= minus001 11990940


1198862


1198862

)

0 20 40 60 80 100 120minus25

minus2

minus15

minus1

minus05

0

05

1

x1

a4

(a)

0 20 40 60 80 100 120

minus35

minus3

minus25

minus2

a4

x3

(b)




= 02 and 1205832


= 23 and 1198863


= minus001 11990920

= minus005 11990930

= minus001 11990940

=


1198864


1198864

)


1are selected as











1= 1437 120590

2=

1142 1205831= 02 120583

2= 02 119886

2= 300 119886

3= 750 119886

4= 450

1198865

= minus1166 1198866

= 1227 1198912

= 1227 1198867

= minus268 1198876

=

minus22 1198877

= minus969 1198878

= minus2232 11990910

= minus108 11990920

= 0511990930

= minus001 11990940



minus1minus050051

minus05

0

05

minus25

minus2

minus15

minus1

x1

x2

x3

(a)


minus1

minus05

0

05

1

x1

x2

(b)


= 827 and 1205831


1199092

1199093


1199092

)

minus50

5

minus50

5minus6

minus4

minus2

0

2

4

6

x1x2

x3

(a)


minus10

minus5

0

5

10

x1

x2

(b)


= 1437 1205902

= 1142 1198862

= 300 1198863

= 750 1198864

= 450 1198865

= minus1166 1198866

=

1227 1198912

= 1227 1198867

= minus268 1198876

= minus22 1198877

= minus969 1198878

= minus2232 11990910

= minus108 11990920

= 05 11990930

= minus001 11990940


1

1199092

1199093


1199092

)





2= 313 119886

5=

minus1501 1198878

= 409 11990910

= minus001 11990920

= minus009 11990930

=

minus005 11990940


7 Conclusions




minus50

5

minus10

0

10

minus6

minus4

minus2

0

2

x1x2

x3

(a)

minus5 0 5 10

minus10

minus5

0

5

10

x1

x2

(b)


= 361 1205902

= 313 1198862

= 230 1198863

= 120 1198864

= 130 1198865

= minus1501 1198866

=

minus203 1198912

= 9238 1198867

= minus205 1198876

= 407 1198877

= minus308 1198878

= 409 11990910

= minus001 11990920

= minus009 11990930

= minus005 11990940


1

1199092

1199093


1199092

)



119896(120576 119868 120574

0 1205832




2 the in-plane


4 and the damping




1 respec-










Acknowledgments



References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










where

119863119868119867(119902plusmn(119868) 119868) = minus

120597119867 (119902plusmn(119868) 119868)

120597119868

= 1205902119868 + 12057221198683

+ 11988661198681199062

1

(31)







119863119868119867(119902plusmn 119868) = 120590

2119868119903+ 12057221198683

119903

+ 11988661198681199031199062

1

= 0 (32)

Then we obtain

119868119903= plusmn

12059021205781+ 1198866[1205832

1

minus 1205901(1 minus 120590

1)]

1198862

6

minus 12057221205781

12

(33)



119903) (34)




1 1199062 119868 120574) The phase shift



61198682 and 120583


1= 1199062 (35a)

2= minus12057611199061+ 12057811199063

1

minus 12057612057621199062 (35b)


119867(1199061 1199062) =

1

2

1199062

2

+

1

2

12057611199062

1

minus

1

4

12057811199064

1

(36)






1199061(1198791) = plusmnradic

1205761

1205781

tanh(

radic21205761

2

1198791) (37a)

1199062(1198791) = plusmn

1205761

radic21205781


2

1198791) (37b)


120574 (1198791) = 1205961199031198791minus

1198866radic21205761

1205781

tanh(

radic21205761

2

1198791) + 1205740 (38)

where 120596119903= 1205902+ 12057221198682

+ 119886612057611205781

At 119868 = 119868119903 there is 120596


be expressed as

Δ120574 = [minus

21198866radic21205761

1205781

]

119868=119868119903

= minus

21198866

1205781

radic2 [1205832

1

minus 1205901(1 minus 120590

1) minus 11988661198682

119903

]

(39)



1 1199062) for the




plusmn(119868) in (24a)


M

I

120574

times

u2

u1

(a)

0 1205742120587

I = I1

I = Ir

I = I2

(b)




120576

Therefore we obtain

119872 = 119872120576= (119906 119868 120574) | 119906 = 119902

plusmn(119868) 1198681lt 119868 lt 119868

2 0 le 120574 lt 2120587

(40)


119868 = minus1205832

119868 minus 1198912

sin 120574 (41a)

120574 = 1205902+ 12057221198682

+ 11988661199062

1

minus

1198912

cos 120574119868

(41b)


1205832

997888rarr 1205761205832

119868 = 119868119903+ radic120576ℎ

1198912

997888rarr 1205761198912

1198791997888rarr

1198791

radic120576

(42)


ℎ = minus120583

2

119868119903minus 1198912

sin 120574 minus radic1205761205832

ℎ (43a)

120574 = minus

2120575

1205781

119868119903ℎ minus radic120576(

120575

1205781

ℎ2

+

1198912

119868119903

cos 120574) (43b)

where 120575 = 1198862

6

minus 12057221205781

When 120576 = 0 (43a) (43b) become

ℎ = minus120583

2

119868119903minus 1198912

sin 120574 (44a)

120574 = minus

2120575

1205781

119868119903ℎ (44b)


119863(ℎ 120574) = minus120583

2

119868119903120574 + 1198912

cos 120574 +

120575

1205781

119868119903ℎ2

(45)


1198750= (0 120574


1205832

119868119903

1198912

))

1198760= (0 120574

119904) = (0 120587 + arcsin(

1205832

119868119903

1198912

))

(46)


0and119876









120576 The phase




120574min minus

1198912

1205832

119868119903

cos 120574min = 120587 + arcsin1205832

119868119903

1198912

+

radic119891

2

2

minus 1205832

2

1198682

119903

1205832

119868119903

(47)


119903as

119860120576= (1199061 1199062 119868 120574) | 119906

1= 119861 119906

2= 0

1003816100381610038161003816119868 minus 119868119903

1003816100381610038161003816lt radic120576119862 120574 isin 119879

119871

(48)



120576 denoted as119882119904(119860

120576) and119882

119906

(119860120576) are the


119906






120576 leave and


h

120574

P0

120574min 120574c 120574s

Q0

(a)

h

120574Q120576

P120576

(b)








120574

|u|

h

120574c 120574c + 3Δ120574

p120576qc




119896(120576 119868 120574

0 1205832


0 and 120583

2




function 1198721(120576 119868119903 1205740 1205832


119903 1205740 1205832



(119872) and119882119906


119872(119868119903 1205740 1205832

1205781 1198866 1205761)

= int

+infin

minusinfin

⟨n (119901ℎ

(119905)) g (119901ℎ (119905) 1205832

0)⟩ 1198891198791


= int

+infin

minusinfin

(

120597119867

1205971199061

1198921199061+

120597119867

1205971199062

1198921199062+

120597119867

120597119868

119892119868

+

120597119867

120597120574

119892120574

)1198891198791

= minus

2radic21205833

31205781

12057632

1

minus 2radic211988661205832

1198682

119903

12057612

1

1205781

minus 1198912

119868119903[cos(120574

0minus 1198866

radic21205761

1205781

) minus cos(1205740+ 1198866

radic21205761

1205781

)]

(49)


) (119896 = 1 2 ) is defined as

119872119896(120576 119868119903 1205740 1205832

)

=

119896minus1

sum

119895=0

119872(119868119903 119895Δ120574 (119868

119903) + Γ119895(120576 119868119903 1205740 1205832

) + 1205740 1205832

)

(50)

where

Γ119895(120576 119868119903 1205740 1205832

) =

Ω (1199090(119868119903) 119868119903)

120582 (119868119903)

119895

sum

119903=1

log100381610038161003816100381610038161003816100381610038161003816

120589 (119868119903)

120576119872119903(120576 119868119903 1205740 1205832

)

100381610038161003816100381610038161003816100381610038161003816

(51)

for 119895 = 1 119896 minus 1 and Γ0(120576 119868119903 1205740 1205832


119895(120576 119868119903 1205740 1205832


119896(120576 119868119903 1205740 1205832


119903degenerates



119872119896(119868119903 1205740 1205832

1205781 1198866 1205761)

=

119896minus1

sum

119895=0

119872(119868119903 1205740+ 119895Δ120574 (119868

119903) 1205832

1205781 1198866 1205761)

= minus1198912

119868119903[cos(120574

0minus

1198866radic21205761

1205781

)

minus cos(1205740+

1198866radic21205761

1205781

)]

minus

2radic21205833

31205781

12057632

1

minus 2radic211988661205832

1198682

119903

12057612

1

1205781

minus 1198912

119868119903[cos(120574

0minus

1198866radic21205761

1205781

minus

21198866radic21205761

1205781

)

minus cos(1205740+

1198866radic21205761

1205781

minus

21198866radic21205761

1205781

)]

minus

2radic21205833

31205781

12057632

1

minus 2radic211988661205832

1198682

119903

12057612

1

1205781

+ sdot sdot sdot

minus 1198912

119868119903[cos(120574

0minus

1198866radic21205761

1205781

minus 2 (119896 minus 1) 1198866

radic21205761

1205781

)

minus cos(1205740+

1198866radic21205761

1205781


1198866radic21205761

1205781

)]

minus

2radic21205833

31205781

12057632

1

minus 2radic211988661205832

1198682

119903

12057612

1

1205781

= minus1198912

119868119903[cos(120574

0minus

1198866radic21205761

1205781


1198866radic21205761

1205781

)

minus cos(1205740+

1198866radic21205761

1205781

)]

minus

2radic21198961205833

31205781

12057632

1

minus 2radic211989611988661205832

1198682

119903

12057612

1

1205781

(52)


119896minus1= 1205740+ (119896 minus


119872119896(119868119903 1205740 1205832

1205781 1198866 1205761)

= 119872119896(119868119903 120574119896minus1


Δ120574

2

1205832

1205781 1198866 1205761)

= minus1198912

119868119903[cos(120574

119896minus1+

1

2

119896Δ120574)


minus

1

2

119896Δ120574)]

+

11989612058331205761

31198866

Δ120574 + 1205832

1198682

119903

(119896Δ120574)

= 21198912

119868119903sin 120574119896minus1

sin(

1

2

119896Δ120574)

+

12058331205761

31198866

(119896Δ120574) + 1205832

1198682

119903

(119896Δ120574)

(53)


10038161003816100381610038161003816100381610038161003816100381610038161003816

(12) 119896Δ120574

sin ((12) 119896Δ120574)

(12058331205761+ 311988661205832

1198682

119903

)

311988661198912

119868119903

10038161003816100381610038161003816100381610038161003816100381610038161003816

lt 1

1

2

119896Δ120574 = 119899120587 119899 = 0 plusmn1 plusmn2

(54)


119885119899

minus

= (119868119903 120574119896minus1

1205832

1205781 1198866 1205761) | 119872119896= 0119863

1205740

119872119896

= 0 (55)



119896minus1isin [0 120587]

120574119896minus11

= minus arcsin(12) 119896Δ120574

sin ((12) 119896Δ120574)

(12058331205761+ 311988661205832

1198682

119903

)

(311988661198912

119868119903)

120574119896minus12

= 120587 minus 120574119896minus11

(56)


3 1205761 1205832

1198866 and 119891

2


119896minus1= 120574119896minus11

and 120574119896minus1

= 120574119896minus12

=

120587 minus 120574119896minus11


119895(119868119903 1205740119894

1205832



119906

(119872120576) intersect


1205832120578111988661205761

plusmn120576(120574119896minus1119894


plusmn0

(120574119896minus1119894


119903 1205740

=

120574119896minus1119894


119895(119868119903 1205740119894

1205832

1205781 1198866 1205761)


0119894

= 120574119896minus1119894

minus (119896 minus 1)(Δ1205742) + 119895Δ120574


119895(119868119903 1205740119894

1205832



01



plusmn120576(120574119896minus11

)

and sum1205832120578111988661205761

plusmn120576(120574119896minus12


1205832120578111988661205761

plusmn0

(120574119896minus11

) and sum1205832120578111988661205761

plusmn0

(120574119896minus12


119906


n = ((minus1205831199061+ 12057811199063

1

+ 11988661198682

1199061) minus1199062 0 0) (57)


plusmn0

(120574119896minus11

)


plusmn0

(120574119896minus12





119894(120583) and 119888



the end points 119889119894(120583) to 119888


end points 119889119894(120583) and 119888



0119894

(119868119903 120583) minus Δ120574

minus



119894(120583) for 119894 =

2 119873 while the line 120574 = 1205740119894

(119868119903 120583) + Δ120574

+



(120583) at the end point 119889119894+1


119894(120583) and 119889

119894+1(120583) is zero

namely

ℎ (119888119894(120583)) minus ℎ (119889

119894+1(120583)) = 0 (58)



sum1205832120578111988661205761

0

(1205740119894


119894(120583) and 119889


1


(12057401



2(120583) on the


119873


0

(1205740119873


119873(120583) on the segments 119874


119889119873+1



sum1205832120578111988661205761

0

(1205740119894


1(120583) and 119874

2(120583) of (44a) (44b) It is known



(119868119903 120583) +

Δ120574+

(119868119903) and 120574 = 120574

0119894

(119868119903 120583) minus Δ120574

minus

(119868119903)


0

(120574119896minus11

) andsum1205832120578111988661205761

0

(120574119896minus12


01

minus


01

minusΔ120574+








119897and









120574

|u|h

Γ3

Γ2

Γ1

M

O2O3






1


6 where

1205781



6 1198867 and


1gt 0



2


2and 119886



4is the


1 1198862 1198864






3 respectively when




120574

|u|h

M

Γsum (12057401 12057402)

12057402 + Δ120574+ 12057401 + Δ120574+



12057402

) are depicted


2 1205831 1198862





2= 2 sim 200


183 1205902

= 197 1205831

= 02 1205832

= 02 1198862

= 230 1198863

=

120 1198864= 130 119886

5= minus101 119886

6= minus203 119886

7= minus205 119887

6=

407 1198877= minus308 119887

8= 109 119909

10= minus001 119909

20= minus005 119909

30=

minus001 11990940


1 1198912) and (119909

3 1198912) respec-






1 The figure demon-



1=


2=



50 100 150minus2

minus15

minus1

minus05

0

05

1

x1

f2

(a)

20 40 60 80 100 120 140 160 180

minus3

minus25

minus2

minus15

x3

f2

(b)



= minus001 11990920

= minus005 11990930

=

minus001 11990940


1198912


1198912

)

0 01 02 03 04 05 06 07minus25

minus2

minus15

minus1

minus05

0

05

1

15

x1

1205831

(a)

0 01 02 03 04 05 06 07minus32

minus3

minus28

minus26

minus24

minus22

minus2

minus18

x1

1205831

x3

1205831

x3

(b)




= minus001 11990920

= minus005 11990930

= minus001 11990940


1205831


3

1205831

)

the planes (1199091 1205831) and (119909









3 1198862)




4varies from 119886

4= 2 to 119886

4= 120


2is

chosen as 1198862


1 1198864) and (119909

3 1198864)







0 10 20 30 40 50 60

minus3

minus2

minus1

0

1

x1

a2

(a)

0 10 20 30 40 50 60minus34

minus32

minus3

minus28

minus26

minus24

minus22

minus2

minus18

a2

x3

(b)



= 02 and 1205832


2


= minus001 11990920

= minus005 11990930

= minus001 11990940


1198862


1198862

)

0 20 40 60 80 100 120minus25

minus2

minus15

minus1

minus05

0

05

1

x1

a4

(a)

0 20 40 60 80 100 120

minus35

minus3

minus25

minus2

a4

x3

(b)




= 02 and 1205832


= 23 and 1198863


= minus001 11990920

= minus005 11990930

= minus001 11990940

=


1198864


1198864

)


1are selected as











1= 1437 120590

2=

1142 1205831= 02 120583

2= 02 119886

2= 300 119886

3= 750 119886

4= 450

1198865

= minus1166 1198866

= 1227 1198912

= 1227 1198867

= minus268 1198876

=

minus22 1198877

= minus969 1198878

= minus2232 11990910

= minus108 11990920

= 0511990930

= minus001 11990940



minus1minus050051

minus05

0

05

minus25

minus2

minus15

minus1

x1

x2

x3

(a)


minus1

minus05

0

05

1

x1

x2

(b)


= 827 and 1205831


1199092

1199093


1199092

)

minus50

5

minus50

5minus6

minus4

minus2

0

2

4

6

x1x2

x3

(a)


minus10

minus5

0

5

10

x1

x2

(b)


= 1437 1205902

= 1142 1198862

= 300 1198863

= 750 1198864

= 450 1198865

= minus1166 1198866

=

1227 1198912

= 1227 1198867

= minus268 1198876

= minus22 1198877

= minus969 1198878

= minus2232 11990910

= minus108 11990920

= 05 11990930

= minus001 11990940


1

1199092

1199093


1199092

)





2= 313 119886

5=

minus1501 1198878

= 409 11990910

= minus001 11990920

= minus009 11990930

=

minus005 11990940


7 Conclusions




minus50

5

minus10

0

10

minus6

minus4

minus2

0

2

x1x2

x3

(a)

minus5 0 5 10

minus10

minus5

0

5

10

x1

x2

(b)


= 361 1205902

= 313 1198862

= 230 1198863

= 120 1198864

= 130 1198865

= minus1501 1198866

=

minus203 1198912

= 9238 1198867

= minus205 1198876

= 407 1198877

= minus308 1198878

= 409 11990910

= minus001 11990920

= minus009 11990930

= minus005 11990940


1

1199092

1199093


1199092

)



119896(120576 119868 120574

0 1205832




2 the in-plane


4 and the damping




1 respec-










Acknowledgments



References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










M

I

120574

times

u2

u1

(a)

0 1205742120587

I = I1

I = Ir

I = I2

(b)




120576

Therefore we obtain

119872 = 119872120576= (119906 119868 120574) | 119906 = 119902

plusmn(119868) 1198681lt 119868 lt 119868

2 0 le 120574 lt 2120587

(40)


119868 = minus1205832

119868 minus 1198912

sin 120574 (41a)

120574 = 1205902+ 12057221198682

+ 11988661199062

1

minus

1198912

cos 120574119868

(41b)


1205832

997888rarr 1205761205832

119868 = 119868119903+ radic120576ℎ

1198912

997888rarr 1205761198912

1198791997888rarr

1198791

radic120576

(42)


ℎ = minus120583

2

119868119903minus 1198912

sin 120574 minus radic1205761205832

ℎ (43a)

120574 = minus

2120575

1205781

119868119903ℎ minus radic120576(

120575

1205781

ℎ2

+

1198912

119868119903

cos 120574) (43b)

where 120575 = 1198862

6

minus 12057221205781

When 120576 = 0 (43a) (43b) become

ℎ = minus120583

2

119868119903minus 1198912

sin 120574 (44a)

120574 = minus

2120575

1205781

119868119903ℎ (44b)


119863(ℎ 120574) = minus120583

2

119868119903120574 + 1198912

cos 120574 +

120575

1205781

119868119903ℎ2

(45)


1198750= (0 120574


1205832

119868119903

1198912

))

1198760= (0 120574

119904) = (0 120587 + arcsin(

1205832

119868119903

1198912

))

(46)


0and119876









120576 The phase




120574min minus

1198912

1205832

119868119903

cos 120574min = 120587 + arcsin1205832

119868119903

1198912

+

radic119891

2

2

minus 1205832

2

1198682

119903

1205832

119868119903

(47)


119903as

119860120576= (1199061 1199062 119868 120574) | 119906

1= 119861 119906

2= 0

1003816100381610038161003816119868 minus 119868119903

1003816100381610038161003816lt radic120576119862 120574 isin 119879

119871

(48)



120576 denoted as119882119904(119860

120576) and119882

119906

(119860120576) are the


119906






120576 leave and


h

120574

P0

120574min 120574c 120574s

Q0

(a)

h

120574Q120576

P120576

(b)








120574

|u|

h

120574c 120574c + 3Δ120574

p120576qc




119896(120576 119868 120574

0 1205832


0 and 120583

2




function 1198721(120576 119868119903 1205740 1205832


119903 1205740 1205832



(119872) and119882119906


119872(119868119903 1205740 1205832

1205781 1198866 1205761)

= int

+infin

minusinfin

⟨n (119901ℎ

(119905)) g (119901ℎ (119905) 1205832

0)⟩ 1198891198791


= int

+infin

minusinfin

(

120597119867

1205971199061

1198921199061+

120597119867

1205971199062

1198921199062+

120597119867

120597119868

119892119868

+

120597119867

120597120574

119892120574

)1198891198791

= minus

2radic21205833

31205781

12057632

1

minus 2radic211988661205832

1198682

119903

12057612

1

1205781

minus 1198912

119868119903[cos(120574

0minus 1198866

radic21205761

1205781

) minus cos(1205740+ 1198866

radic21205761

1205781

)]

(49)


) (119896 = 1 2 ) is defined as

119872119896(120576 119868119903 1205740 1205832

)

=

119896minus1

sum

119895=0

119872(119868119903 119895Δ120574 (119868

119903) + Γ119895(120576 119868119903 1205740 1205832

) + 1205740 1205832

)

(50)

where

Γ119895(120576 119868119903 1205740 1205832

) =

Ω (1199090(119868119903) 119868119903)

120582 (119868119903)

119895

sum

119903=1

log100381610038161003816100381610038161003816100381610038161003816

120589 (119868119903)

120576119872119903(120576 119868119903 1205740 1205832

)

100381610038161003816100381610038161003816100381610038161003816

(51)

for 119895 = 1 119896 minus 1 and Γ0(120576 119868119903 1205740 1205832


119895(120576 119868119903 1205740 1205832


119896(120576 119868119903 1205740 1205832


119903degenerates



119872119896(119868119903 1205740 1205832

1205781 1198866 1205761)

=

119896minus1

sum

119895=0

119872(119868119903 1205740+ 119895Δ120574 (119868

119903) 1205832

1205781 1198866 1205761)

= minus1198912

119868119903[cos(120574

0minus

1198866radic21205761

1205781

)

minus cos(1205740+

1198866radic21205761

1205781

)]

minus

2radic21205833

31205781

12057632

1

minus 2radic211988661205832

1198682

119903

12057612

1

1205781

minus 1198912

119868119903[cos(120574

0minus

1198866radic21205761

1205781

minus

21198866radic21205761

1205781

)

minus cos(1205740+

1198866radic21205761

1205781

minus

21198866radic21205761

1205781

)]

minus

2radic21205833

31205781

12057632

1

minus 2radic211988661205832

1198682

119903

12057612

1

1205781

+ sdot sdot sdot

minus 1198912

119868119903[cos(120574

0minus

1198866radic21205761

1205781

minus 2 (119896 minus 1) 1198866

radic21205761

1205781

)

minus cos(1205740+

1198866radic21205761

1205781


1198866radic21205761

1205781

)]

minus

2radic21205833

31205781

12057632

1

minus 2radic211988661205832

1198682

119903

12057612

1

1205781

= minus1198912

119868119903[cos(120574

0minus

1198866radic21205761

1205781


1198866radic21205761

1205781

)

minus cos(1205740+

1198866radic21205761

1205781

)]

minus

2radic21198961205833

31205781

12057632

1

minus 2radic211989611988661205832

1198682

119903

12057612

1

1205781

(52)


119896minus1= 1205740+ (119896 minus


119872119896(119868119903 1205740 1205832

1205781 1198866 1205761)

= 119872119896(119868119903 120574119896minus1


Δ120574

2

1205832

1205781 1198866 1205761)

= minus1198912

119868119903[cos(120574

119896minus1+

1

2

119896Δ120574)


minus

1

2

119896Δ120574)]

+

11989612058331205761

31198866

Δ120574 + 1205832

1198682

119903

(119896Δ120574)

= 21198912

119868119903sin 120574119896minus1

sin(

1

2

119896Δ120574)

+

12058331205761

31198866

(119896Δ120574) + 1205832

1198682

119903

(119896Δ120574)

(53)


10038161003816100381610038161003816100381610038161003816100381610038161003816

(12) 119896Δ120574

sin ((12) 119896Δ120574)

(12058331205761+ 311988661205832

1198682

119903

)

311988661198912

119868119903

10038161003816100381610038161003816100381610038161003816100381610038161003816

lt 1

1

2

119896Δ120574 = 119899120587 119899 = 0 plusmn1 plusmn2

(54)


119885119899

minus

= (119868119903 120574119896minus1

1205832

1205781 1198866 1205761) | 119872119896= 0119863

1205740

119872119896

= 0 (55)



119896minus1isin [0 120587]

120574119896minus11

= minus arcsin(12) 119896Δ120574

sin ((12) 119896Δ120574)

(12058331205761+ 311988661205832

1198682

119903

)

(311988661198912

119868119903)

120574119896minus12

= 120587 minus 120574119896minus11

(56)


3 1205761 1205832

1198866 and 119891

2


119896minus1= 120574119896minus11

and 120574119896minus1

= 120574119896minus12

=

120587 minus 120574119896minus11


119895(119868119903 1205740119894

1205832



119906

(119872120576) intersect


1205832120578111988661205761

plusmn120576(120574119896minus1119894


plusmn0

(120574119896minus1119894


119903 1205740

=

120574119896minus1119894


119895(119868119903 1205740119894

1205832

1205781 1198866 1205761)


0119894

= 120574119896minus1119894

minus (119896 minus 1)(Δ1205742) + 119895Δ120574


119895(119868119903 1205740119894

1205832



01



plusmn120576(120574119896minus11

)

and sum1205832120578111988661205761

plusmn120576(120574119896minus12


1205832120578111988661205761

plusmn0

(120574119896minus11

) and sum1205832120578111988661205761

plusmn0

(120574119896minus12


119906


n = ((minus1205831199061+ 12057811199063

1

+ 11988661198682

1199061) minus1199062 0 0) (57)


plusmn0

(120574119896minus11

)


plusmn0

(120574119896minus12





119894(120583) and 119888



the end points 119889119894(120583) to 119888


end points 119889119894(120583) and 119888



0119894

(119868119903 120583) minus Δ120574

minus



119894(120583) for 119894 =

2 119873 while the line 120574 = 1205740119894

(119868119903 120583) + Δ120574

+



(120583) at the end point 119889119894+1


119894(120583) and 119889

119894+1(120583) is zero

namely

ℎ (119888119894(120583)) minus ℎ (119889

119894+1(120583)) = 0 (58)



sum1205832120578111988661205761

0

(1205740119894


119894(120583) and 119889


1


(12057401



2(120583) on the


119873


0

(1205740119873


119873(120583) on the segments 119874


119889119873+1



sum1205832120578111988661205761

0

(1205740119894


1(120583) and 119874

2(120583) of (44a) (44b) It is known



(119868119903 120583) +

Δ120574+

(119868119903) and 120574 = 120574

0119894

(119868119903 120583) minus Δ120574

minus

(119868119903)


0

(120574119896minus11

) andsum1205832120578111988661205761

0

(120574119896minus12


01

minus


01

minusΔ120574+








119897and









120574

|u|h

Γ3

Γ2

Γ1

M

O2O3






1


6 where

1205781



6 1198867 and


1gt 0



2


2and 119886



4is the


1 1198862 1198864






3 respectively when




120574

|u|h

M

Γsum (12057401 12057402)

12057402 + Δ120574+ 12057401 + Δ120574+



12057402

) are depicted


2 1205831 1198862





2= 2 sim 200


183 1205902

= 197 1205831

= 02 1205832

= 02 1198862

= 230 1198863

=

120 1198864= 130 119886

5= minus101 119886

6= minus203 119886

7= minus205 119887

6=

407 1198877= minus308 119887

8= 109 119909

10= minus001 119909

20= minus005 119909

30=

minus001 11990940


1 1198912) and (119909

3 1198912) respec-






1 The figure demon-



1=


2=



50 100 150minus2

minus15

minus1

minus05

0

05

1

x1

f2

(a)

20 40 60 80 100 120 140 160 180

minus3

minus25

minus2

minus15

x3

f2

(b)



= minus001 11990920

= minus005 11990930

=

minus001 11990940


1198912


1198912

)

0 01 02 03 04 05 06 07minus25

minus2

minus15

minus1

minus05

0

05

1

15

x1

1205831

(a)

0 01 02 03 04 05 06 07minus32

minus3

minus28

minus26

minus24

minus22

minus2

minus18

x1

1205831

x3

1205831

x3

(b)




= minus001 11990920

= minus005 11990930

= minus001 11990940


1205831


3

1205831

)

the planes (1199091 1205831) and (119909









3 1198862)




4varies from 119886

4= 2 to 119886

4= 120


2is

chosen as 1198862


1 1198864) and (119909

3 1198864)







0 10 20 30 40 50 60

minus3

minus2

minus1

0

1

x1

a2

(a)

0 10 20 30 40 50 60minus34

minus32

minus3

minus28

minus26

minus24

minus22

minus2

minus18

a2

x3

(b)



= 02 and 1205832


2


= minus001 11990920

= minus005 11990930

= minus001 11990940


1198862


1198862

)

0 20 40 60 80 100 120minus25

minus2

minus15

minus1

minus05

0

05

1

x1

a4

(a)

0 20 40 60 80 100 120

minus35

minus3

minus25

minus2

a4

x3

(b)




= 02 and 1205832


= 23 and 1198863


= minus001 11990920

= minus005 11990930

= minus001 11990940

=


1198864


1198864

)


1are selected as











1= 1437 120590

2=

1142 1205831= 02 120583

2= 02 119886

2= 300 119886

3= 750 119886

4= 450

1198865

= minus1166 1198866

= 1227 1198912

= 1227 1198867

= minus268 1198876

=

minus22 1198877

= minus969 1198878

= minus2232 11990910

= minus108 11990920

= 0511990930

= minus001 11990940



minus1minus050051

minus05

0

05

minus25

minus2

minus15

minus1

x1

x2

x3

(a)


minus1

minus05

0

05

1

x1

x2

(b)


= 827 and 1205831


1199092

1199093


1199092

)

minus50

5

minus50

5minus6

minus4

minus2

0

2

4

6

x1x2

x3

(a)


minus10

minus5

0

5

10

x1

x2

(b)


= 1437 1205902

= 1142 1198862

= 300 1198863

= 750 1198864

= 450 1198865

= minus1166 1198866

=

1227 1198912

= 1227 1198867

= minus268 1198876

= minus22 1198877

= minus969 1198878

= minus2232 11990910

= minus108 11990920

= 05 11990930

= minus001 11990940


1

1199092

1199093


1199092

)





2= 313 119886

5=

minus1501 1198878

= 409 11990910

= minus001 11990920

= minus009 11990930

=

minus005 11990940


7 Conclusions




minus50

5

minus10

0

10

minus6

minus4

minus2

0

2

x1x2

x3

(a)

minus5 0 5 10

minus10

minus5

0

5

10

x1

x2

(b)


= 361 1205902

= 313 1198862

= 230 1198863

= 120 1198864

= 130 1198865

= minus1501 1198866

=

minus203 1198912

= 9238 1198867

= minus205 1198876

= 407 1198877

= minus308 1198878

= 409 11990910

= minus001 11990920

= minus009 11990930

= minus005 11990940


1

1199092

1199093


1199092

)



119896(120576 119868 120574

0 1205832




2 the in-plane


4 and the damping




1 respec-










Acknowledgments



References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










h

120574

P0

120574min 120574c 120574s

Q0

(a)

h

120574Q120576

P120576

(b)








120574

|u|

h

120574c 120574c + 3Δ120574

p120576qc




119896(120576 119868 120574

0 1205832


0 and 120583

2




function 1198721(120576 119868119903 1205740 1205832


119903 1205740 1205832



(119872) and119882119906


119872(119868119903 1205740 1205832

1205781 1198866 1205761)

= int

+infin

minusinfin

⟨n (119901ℎ

(119905)) g (119901ℎ (119905) 1205832

0)⟩ 1198891198791


= int

+infin

minusinfin

(

120597119867

1205971199061

1198921199061+

120597119867

1205971199062

1198921199062+

120597119867

120597119868

119892119868

+

120597119867

120597120574

119892120574

)1198891198791

= minus

2radic21205833

31205781

12057632

1

minus 2radic211988661205832

1198682

119903

12057612

1

1205781

minus 1198912

119868119903[cos(120574

0minus 1198866

radic21205761

1205781

) minus cos(1205740+ 1198866

radic21205761

1205781

)]

(49)


) (119896 = 1 2 ) is defined as

119872119896(120576 119868119903 1205740 1205832

)

=

119896minus1

sum

119895=0

119872(119868119903 119895Δ120574 (119868

119903) + Γ119895(120576 119868119903 1205740 1205832

) + 1205740 1205832

)

(50)

where

Γ119895(120576 119868119903 1205740 1205832

) =

Ω (1199090(119868119903) 119868119903)

120582 (119868119903)

119895

sum

119903=1

log100381610038161003816100381610038161003816100381610038161003816

120589 (119868119903)

120576119872119903(120576 119868119903 1205740 1205832

)

100381610038161003816100381610038161003816100381610038161003816

(51)

for 119895 = 1 119896 minus 1 and Γ0(120576 119868119903 1205740 1205832


119895(120576 119868119903 1205740 1205832


119896(120576 119868119903 1205740 1205832


119903degenerates



119872119896(119868119903 1205740 1205832

1205781 1198866 1205761)

=

119896minus1

sum

119895=0

119872(119868119903 1205740+ 119895Δ120574 (119868

119903) 1205832

1205781 1198866 1205761)

= minus1198912

119868119903[cos(120574

0minus

1198866radic21205761

1205781

)

minus cos(1205740+

1198866radic21205761

1205781

)]

minus

2radic21205833

31205781

12057632

1

minus 2radic211988661205832

1198682

119903

12057612

1

1205781

minus 1198912

119868119903[cos(120574

0minus

1198866radic21205761

1205781

minus

21198866radic21205761

1205781

)

minus cos(1205740+

1198866radic21205761

1205781

minus

21198866radic21205761

1205781

)]

minus

2radic21205833

31205781

12057632

1

minus 2radic211988661205832

1198682

119903

12057612

1

1205781

+ sdot sdot sdot

minus 1198912

119868119903[cos(120574

0minus

1198866radic21205761

1205781

minus 2 (119896 minus 1) 1198866

radic21205761

1205781

)

minus cos(1205740+

1198866radic21205761

1205781


1198866radic21205761

1205781

)]

minus

2radic21205833

31205781

12057632

1

minus 2radic211988661205832

1198682

119903

12057612

1

1205781

= minus1198912

119868119903[cos(120574

0minus

1198866radic21205761

1205781


1198866radic21205761

1205781

)

minus cos(1205740+

1198866radic21205761

1205781

)]

minus

2radic21198961205833

31205781

12057632

1

minus 2radic211989611988661205832

1198682

119903

12057612

1

1205781

(52)


119896minus1= 1205740+ (119896 minus


119872119896(119868119903 1205740 1205832

1205781 1198866 1205761)

= 119872119896(119868119903 120574119896minus1


Δ120574

2

1205832

1205781 1198866 1205761)

= minus1198912

119868119903[cos(120574

119896minus1+

1

2

119896Δ120574)


minus

1

2

119896Δ120574)]

+

11989612058331205761

31198866

Δ120574 + 1205832

1198682

119903

(119896Δ120574)

= 21198912

119868119903sin 120574119896minus1

sin(

1

2

119896Δ120574)

+

12058331205761

31198866

(119896Δ120574) + 1205832

1198682

119903

(119896Δ120574)

(53)


10038161003816100381610038161003816100381610038161003816100381610038161003816

(12) 119896Δ120574

sin ((12) 119896Δ120574)

(12058331205761+ 311988661205832

1198682

119903

)

311988661198912

119868119903

10038161003816100381610038161003816100381610038161003816100381610038161003816

lt 1

1

2

119896Δ120574 = 119899120587 119899 = 0 plusmn1 plusmn2

(54)


119885119899

minus

= (119868119903 120574119896minus1

1205832

1205781 1198866 1205761) | 119872119896= 0119863

1205740

119872119896

= 0 (55)



119896minus1isin [0 120587]

120574119896minus11

= minus arcsin(12) 119896Δ120574

sin ((12) 119896Δ120574)

(12058331205761+ 311988661205832

1198682

119903

)

(311988661198912

119868119903)

120574119896minus12

= 120587 minus 120574119896minus11

(56)


3 1205761 1205832

1198866 and 119891

2


119896minus1= 120574119896minus11

and 120574119896minus1

= 120574119896minus12

=

120587 minus 120574119896minus11


119895(119868119903 1205740119894

1205832



119906

(119872120576) intersect


1205832120578111988661205761

plusmn120576(120574119896minus1119894


plusmn0

(120574119896minus1119894


119903 1205740

=

120574119896minus1119894


119895(119868119903 1205740119894

1205832

1205781 1198866 1205761)


0119894

= 120574119896minus1119894

minus (119896 minus 1)(Δ1205742) + 119895Δ120574


119895(119868119903 1205740119894

1205832



01



plusmn120576(120574119896minus11

)

and sum1205832120578111988661205761

plusmn120576(120574119896minus12


1205832120578111988661205761

plusmn0

(120574119896minus11

) and sum1205832120578111988661205761

plusmn0

(120574119896minus12


119906


n = ((minus1205831199061+ 12057811199063

1

+ 11988661198682

1199061) minus1199062 0 0) (57)


plusmn0

(120574119896minus11

)


plusmn0

(120574119896minus12





119894(120583) and 119888



the end points 119889119894(120583) to 119888


end points 119889119894(120583) and 119888



0119894

(119868119903 120583) minus Δ120574

minus



119894(120583) for 119894 =

2 119873 while the line 120574 = 1205740119894

(119868119903 120583) + Δ120574

+



(120583) at the end point 119889119894+1


119894(120583) and 119889

119894+1(120583) is zero

namely

ℎ (119888119894(120583)) minus ℎ (119889

119894+1(120583)) = 0 (58)



sum1205832120578111988661205761

0

(1205740119894


119894(120583) and 119889


1


(12057401



2(120583) on the


119873


0

(1205740119873


119873(120583) on the segments 119874


119889119873+1



sum1205832120578111988661205761

0

(1205740119894


1(120583) and 119874

2(120583) of (44a) (44b) It is known



(119868119903 120583) +

Δ120574+

(119868119903) and 120574 = 120574

0119894

(119868119903 120583) minus Δ120574

minus

(119868119903)


0

(120574119896minus11

) andsum1205832120578111988661205761

0

(120574119896minus12


01

minus


01

minusΔ120574+








119897and









120574

|u|h

Γ3

Γ2

Γ1

M

O2O3






1


6 where

1205781



6 1198867 and


1gt 0



2


2and 119886



4is the


1 1198862 1198864






3 respectively when




120574

|u|h

M

Γsum (12057401 12057402)

12057402 + Δ120574+ 12057401 + Δ120574+



12057402

) are depicted


2 1205831 1198862





2= 2 sim 200


183 1205902

= 197 1205831

= 02 1205832

= 02 1198862

= 230 1198863

=

120 1198864= 130 119886

5= minus101 119886

6= minus203 119886

7= minus205 119887

6=

407 1198877= minus308 119887

8= 109 119909

10= minus001 119909

20= minus005 119909

30=

minus001 11990940


1 1198912) and (119909

3 1198912) respec-






1 The figure demon-



1=


2=



50 100 150minus2

minus15

minus1

minus05

0

05

1

x1

f2

(a)

20 40 60 80 100 120 140 160 180

minus3

minus25

minus2

minus15

x3

f2

(b)



= minus001 11990920

= minus005 11990930

=

minus001 11990940


1198912


1198912

)

0 01 02 03 04 05 06 07minus25

minus2

minus15

minus1

minus05

0

05

1

15

x1

1205831

(a)

0 01 02 03 04 05 06 07minus32

minus3

minus28

minus26

minus24

minus22

minus2

minus18

x1

1205831

x3

1205831

x3

(b)




= minus001 11990920

= minus005 11990930

= minus001 11990940


1205831


3

1205831

)

the planes (1199091 1205831) and (119909









3 1198862)




4varies from 119886

4= 2 to 119886

4= 120


2is

chosen as 1198862


1 1198864) and (119909

3 1198864)







0 10 20 30 40 50 60

minus3

minus2

minus1

0

1

x1

a2

(a)

0 10 20 30 40 50 60minus34

minus32

minus3

minus28

minus26

minus24

minus22

minus2

minus18

a2

x3

(b)



= 02 and 1205832


2


= minus001 11990920

= minus005 11990930

= minus001 11990940


1198862


1198862

)

0 20 40 60 80 100 120minus25

minus2

minus15

minus1

minus05

0

05

1

x1

a4

(a)

0 20 40 60 80 100 120

minus35

minus3

minus25

minus2

a4

x3

(b)




= 02 and 1205832


= 23 and 1198863


= minus001 11990920

= minus005 11990930

= minus001 11990940

=


1198864


1198864

)


1are selected as











1= 1437 120590

2=

1142 1205831= 02 120583

2= 02 119886

2= 300 119886

3= 750 119886

4= 450

1198865

= minus1166 1198866

= 1227 1198912

= 1227 1198867

= minus268 1198876

=

minus22 1198877

= minus969 1198878

= minus2232 11990910

= minus108 11990920

= 0511990930

= minus001 11990940



minus1minus050051

minus05

0

05

minus25

minus2

minus15

minus1

x1

x2

x3

(a)


minus1

minus05

0

05

1

x1

x2

(b)


= 827 and 1205831


1199092

1199093


1199092

)

minus50

5

minus50

5minus6

minus4

minus2

0

2

4

6

x1x2

x3

(a)


minus10

minus5

0

5

10

x1

x2

(b)


= 1437 1205902

= 1142 1198862

= 300 1198863

= 750 1198864

= 450 1198865

= minus1166 1198866

=

1227 1198912

= 1227 1198867

= minus268 1198876

= minus22 1198877

= minus969 1198878

= minus2232 11990910

= minus108 11990920

= 05 11990930

= minus001 11990940


1

1199092

1199093


1199092

)





2= 313 119886

5=

minus1501 1198878

= 409 11990910

= minus001 11990920

= minus009 11990930

=

minus005 11990940


7 Conclusions




minus50

5

minus10

0

10

minus6

minus4

minus2

0

2

x1x2

x3

(a)

minus5 0 5 10

minus10

minus5

0

5

10

x1

x2

(b)


= 361 1205902

= 313 1198862

= 230 1198863

= 120 1198864

= 130 1198865

= minus1501 1198866

=

minus203 1198912

= 9238 1198867

= minus205 1198876

= 407 1198877

= minus308 1198878

= 409 11990910

= minus001 11990920

= minus009 11990930

= minus005 11990940


1

1199092

1199093


1199092

)



119896(120576 119868 120574

0 1205832




2 the in-plane


4 and the damping




1 respec-










Acknowledgments



References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










= int

+infin

minusinfin

(

120597119867

1205971199061

1198921199061+

120597119867

1205971199062

1198921199062+

120597119867

120597119868

119892119868

+

120597119867

120597120574

119892120574

)1198891198791

= minus

2radic21205833

31205781

12057632

1

minus 2radic211988661205832

1198682

119903

12057612

1

1205781

minus 1198912

119868119903[cos(120574

0minus 1198866

radic21205761

1205781

) minus cos(1205740+ 1198866

radic21205761

1205781

)]

(49)


) (119896 = 1 2 ) is defined as

119872119896(120576 119868119903 1205740 1205832

)

=

119896minus1

sum

119895=0

119872(119868119903 119895Δ120574 (119868

119903) + Γ119895(120576 119868119903 1205740 1205832

) + 1205740 1205832

)

(50)

where

Γ119895(120576 119868119903 1205740 1205832

) =

Ω (1199090(119868119903) 119868119903)

120582 (119868119903)

119895

sum

119903=1

log100381610038161003816100381610038161003816100381610038161003816

120589 (119868119903)

120576119872119903(120576 119868119903 1205740 1205832

)

100381610038161003816100381610038161003816100381610038161003816

(51)

for 119895 = 1 119896 minus 1 and Γ0(120576 119868119903 1205740 1205832


119895(120576 119868119903 1205740 1205832


119896(120576 119868119903 1205740 1205832


119903degenerates



119872119896(119868119903 1205740 1205832

1205781 1198866 1205761)

=

119896minus1

sum

119895=0

119872(119868119903 1205740+ 119895Δ120574 (119868

119903) 1205832

1205781 1198866 1205761)

= minus1198912

119868119903[cos(120574

0minus

1198866radic21205761

1205781

)

minus cos(1205740+

1198866radic21205761

1205781

)]

minus

2radic21205833

31205781

12057632

1

minus 2radic211988661205832

1198682

119903

12057612

1

1205781

minus 1198912

119868119903[cos(120574

0minus

1198866radic21205761

1205781

minus

21198866radic21205761

1205781

)

minus cos(1205740+

1198866radic21205761

1205781

minus

21198866radic21205761

1205781

)]

minus

2radic21205833

31205781

12057632

1

minus 2radic211988661205832

1198682

119903

12057612

1

1205781

+ sdot sdot sdot

minus 1198912

119868119903[cos(120574

0minus

1198866radic21205761

1205781

minus 2 (119896 minus 1) 1198866

radic21205761

1205781

)

minus cos(1205740+

1198866radic21205761

1205781


1198866radic21205761

1205781

)]

minus

2radic21205833

31205781

12057632

1

minus 2radic211988661205832

1198682

119903

12057612

1

1205781

= minus1198912

119868119903[cos(120574

0minus

1198866radic21205761

1205781


1198866radic21205761

1205781

)

minus cos(1205740+

1198866radic21205761

1205781

)]

minus

2radic21198961205833

31205781

12057632

1

minus 2radic211989611988661205832

1198682

119903

12057612

1

1205781

(52)


119896minus1= 1205740+ (119896 minus


119872119896(119868119903 1205740 1205832

1205781 1198866 1205761)

= 119872119896(119868119903 120574119896minus1


Δ120574

2

1205832

1205781 1198866 1205761)

= minus1198912

119868119903[cos(120574

119896minus1+

1

2

119896Δ120574)


minus

1

2

119896Δ120574)]

+

11989612058331205761

31198866

Δ120574 + 1205832

1198682

119903

(119896Δ120574)

= 21198912

119868119903sin 120574119896minus1

sin(

1

2

119896Δ120574)

+

12058331205761

31198866

(119896Δ120574) + 1205832

1198682

119903

(119896Δ120574)

(53)


10038161003816100381610038161003816100381610038161003816100381610038161003816

(12) 119896Δ120574

sin ((12) 119896Δ120574)

(12058331205761+ 311988661205832

1198682

119903

)

311988661198912

119868119903

10038161003816100381610038161003816100381610038161003816100381610038161003816

lt 1

1

2

119896Δ120574 = 119899120587 119899 = 0 plusmn1 plusmn2

(54)


119885119899

minus

= (119868119903 120574119896minus1

1205832

1205781 1198866 1205761) | 119872119896= 0119863

1205740

119872119896

= 0 (55)



119896minus1isin [0 120587]

120574119896minus11

= minus arcsin(12) 119896Δ120574

sin ((12) 119896Δ120574)

(12058331205761+ 311988661205832

1198682

119903

)

(311988661198912

119868119903)

120574119896minus12

= 120587 minus 120574119896minus11

(56)


3 1205761 1205832

1198866 and 119891

2


119896minus1= 120574119896minus11

and 120574119896minus1

= 120574119896minus12

=

120587 minus 120574119896minus11


119895(119868119903 1205740119894

1205832



119906

(119872120576) intersect


1205832120578111988661205761

plusmn120576(120574119896minus1119894


plusmn0

(120574119896minus1119894


119903 1205740

=

120574119896minus1119894


119895(119868119903 1205740119894

1205832

1205781 1198866 1205761)


0119894

= 120574119896minus1119894

minus (119896 minus 1)(Δ1205742) + 119895Δ120574


119895(119868119903 1205740119894

1205832



01



plusmn120576(120574119896minus11

)

and sum1205832120578111988661205761

plusmn120576(120574119896minus12


1205832120578111988661205761

plusmn0

(120574119896minus11

) and sum1205832120578111988661205761

plusmn0

(120574119896minus12


119906


n = ((minus1205831199061+ 12057811199063

1

+ 11988661198682

1199061) minus1199062 0 0) (57)


plusmn0

(120574119896minus11

)


plusmn0

(120574119896minus12





119894(120583) and 119888



the end points 119889119894(120583) to 119888


end points 119889119894(120583) and 119888



0119894

(119868119903 120583) minus Δ120574

minus



119894(120583) for 119894 =

2 119873 while the line 120574 = 1205740119894

(119868119903 120583) + Δ120574

+



(120583) at the end point 119889119894+1


119894(120583) and 119889

119894+1(120583) is zero

namely

ℎ (119888119894(120583)) minus ℎ (119889

119894+1(120583)) = 0 (58)



sum1205832120578111988661205761

0

(1205740119894


119894(120583) and 119889


1


(12057401



2(120583) on the


119873


0

(1205740119873


119873(120583) on the segments 119874


119889119873+1



sum1205832120578111988661205761

0

(1205740119894


1(120583) and 119874

2(120583) of (44a) (44b) It is known



(119868119903 120583) +

Δ120574+

(119868119903) and 120574 = 120574

0119894

(119868119903 120583) minus Δ120574

minus

(119868119903)


0

(120574119896minus11

) andsum1205832120578111988661205761

0

(120574119896minus12


01

minus


01

minusΔ120574+








119897and









120574

|u|h

Γ3

Γ2

Γ1

M

O2O3






1


6 where

1205781



6 1198867 and


1gt 0



2


2and 119886



4is the


1 1198862 1198864






3 respectively when




120574

|u|h

M

Γsum (12057401 12057402)

12057402 + Δ120574+ 12057401 + Δ120574+



12057402

) are depicted


2 1205831 1198862





2= 2 sim 200


183 1205902

= 197 1205831

= 02 1205832

= 02 1198862

= 230 1198863

=

120 1198864= 130 119886

5= minus101 119886

6= minus203 119886

7= minus205 119887

6=

407 1198877= minus308 119887

8= 109 119909

10= minus001 119909

20= minus005 119909

30=

minus001 11990940


1 1198912) and (119909

3 1198912) respec-






1 The figure demon-



1=


2=



50 100 150minus2

minus15

minus1

minus05

0

05

1

x1

f2

(a)

20 40 60 80 100 120 140 160 180

minus3

minus25

minus2

minus15

x3

f2

(b)



= minus001 11990920

= minus005 11990930

=

minus001 11990940


1198912


1198912

)

0 01 02 03 04 05 06 07minus25

minus2

minus15

minus1

minus05

0

05

1

15

x1

1205831

(a)

0 01 02 03 04 05 06 07minus32

minus3

minus28

minus26

minus24

minus22

minus2

minus18

x1

1205831

x3

1205831

x3

(b)




= minus001 11990920

= minus005 11990930

= minus001 11990940


1205831


3

1205831

)

the planes (1199091 1205831) and (119909









3 1198862)




4varies from 119886

4= 2 to 119886

4= 120


2is

chosen as 1198862


1 1198864) and (119909

3 1198864)







0 10 20 30 40 50 60

minus3

minus2

minus1

0

1

x1

a2

(a)

0 10 20 30 40 50 60minus34

minus32

minus3

minus28

minus26

minus24

minus22

minus2

minus18

a2

x3

(b)



= 02 and 1205832


2


= minus001 11990920

= minus005 11990930

= minus001 11990940


1198862


1198862

)

0 20 40 60 80 100 120minus25

minus2

minus15

minus1

minus05

0

05

1

x1

a4

(a)

0 20 40 60 80 100 120

minus35

minus3

minus25

minus2

a4

x3

(b)




= 02 and 1205832


= 23 and 1198863


= minus001 11990920

= minus005 11990930

= minus001 11990940

=


1198864


1198864

)


1are selected as











1= 1437 120590

2=

1142 1205831= 02 120583

2= 02 119886

2= 300 119886

3= 750 119886

4= 450

1198865

= minus1166 1198866

= 1227 1198912

= 1227 1198867

= minus268 1198876

=

minus22 1198877

= minus969 1198878

= minus2232 11990910

= minus108 11990920

= 0511990930

= minus001 11990940



minus1minus050051

minus05

0

05

minus25

minus2

minus15

minus1

x1

x2

x3

(a)


minus1

minus05

0

05

1

x1

x2

(b)


= 827 and 1205831


1199092

1199093


1199092

)

minus50

5

minus50

5minus6

minus4

minus2

0

2

4

6

x1x2

x3

(a)


minus10

minus5

0

5

10

x1

x2

(b)


= 1437 1205902

= 1142 1198862

= 300 1198863

= 750 1198864

= 450 1198865

= minus1166 1198866

=

1227 1198912

= 1227 1198867

= minus268 1198876

= minus22 1198877

= minus969 1198878

= minus2232 11990910

= minus108 11990920

= 05 11990930

= minus001 11990940


1

1199092

1199093


1199092

)





2= 313 119886

5=

minus1501 1198878

= 409 11990910

= minus001 11990920

= minus009 11990930

=

minus005 11990940


7 Conclusions




minus50

5

minus10

0

10

minus6

minus4

minus2

0

2

x1x2

x3

(a)

minus5 0 5 10

minus10

minus5

0

5

10

x1

x2

(b)


= 361 1205902

= 313 1198862

= 230 1198863

= 120 1198864

= 130 1198865

= minus1501 1198866

=

minus203 1198912

= 9238 1198867

= minus205 1198876

= 407 1198877

= minus308 1198878

= 409 11990910

= minus001 11990920

= minus009 11990930

= minus005 11990940


1

1199092

1199093


1199092

)



119896(120576 119868 120574

0 1205832




2 the in-plane


4 and the damping




1 respec-










Acknowledgments



References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119896minus1isin [0 120587]

120574119896minus11

= minus arcsin(12) 119896Δ120574

sin ((12) 119896Δ120574)

(12058331205761+ 311988661205832

1198682

119903

)

(311988661198912

119868119903)

120574119896minus12

= 120587 minus 120574119896minus11

(56)


3 1205761 1205832

1198866 and 119891

2


119896minus1= 120574119896minus11

and 120574119896minus1

= 120574119896minus12

=

120587 minus 120574119896minus11


119895(119868119903 1205740119894

1205832



119906

(119872120576) intersect


1205832120578111988661205761

plusmn120576(120574119896minus1119894


plusmn0

(120574119896minus1119894


119903 1205740

=

120574119896minus1119894


119895(119868119903 1205740119894

1205832

1205781 1198866 1205761)


0119894

= 120574119896minus1119894

minus (119896 minus 1)(Δ1205742) + 119895Δ120574


119895(119868119903 1205740119894

1205832



01



plusmn120576(120574119896minus11

)

and sum1205832120578111988661205761

plusmn120576(120574119896minus12


1205832120578111988661205761

plusmn0

(120574119896minus11

) and sum1205832120578111988661205761

plusmn0

(120574119896minus12


119906


n = ((minus1205831199061+ 12057811199063

1

+ 11988661198682

1199061) minus1199062 0 0) (57)


plusmn0

(120574119896minus11

)


plusmn0

(120574119896minus12





119894(120583) and 119888



the end points 119889119894(120583) to 119888


end points 119889119894(120583) and 119888



0119894

(119868119903 120583) minus Δ120574

minus



119894(120583) for 119894 =

2 119873 while the line 120574 = 1205740119894

(119868119903 120583) + Δ120574

+



(120583) at the end point 119889119894+1


119894(120583) and 119889

119894+1(120583) is zero

namely

ℎ (119888119894(120583)) minus ℎ (119889

119894+1(120583)) = 0 (58)



sum1205832120578111988661205761

0

(1205740119894


119894(120583) and 119889


1


(12057401



2(120583) on the


119873


0

(1205740119873


119873(120583) on the segments 119874


119889119873+1



sum1205832120578111988661205761

0

(1205740119894


1(120583) and 119874

2(120583) of (44a) (44b) It is known



(119868119903 120583) +

Δ120574+

(119868119903) and 120574 = 120574

0119894

(119868119903 120583) minus Δ120574

minus

(119868119903)


0

(120574119896minus11

) andsum1205832120578111988661205761

0

(120574119896minus12


01

minus


01

minusΔ120574+








119897and









120574

|u|h

Γ3

Γ2

Γ1

M

O2O3






1


6 where

1205781



6 1198867 and


1gt 0



2


2and 119886



4is the


1 1198862 1198864






3 respectively when




120574

|u|h

M

Γsum (12057401 12057402)

12057402 + Δ120574+ 12057401 + Δ120574+



12057402

) are depicted


2 1205831 1198862





2= 2 sim 200


183 1205902

= 197 1205831

= 02 1205832

= 02 1198862

= 230 1198863

=

120 1198864= 130 119886

5= minus101 119886

6= minus203 119886

7= minus205 119887

6=

407 1198877= minus308 119887

8= 109 119909

10= minus001 119909

20= minus005 119909

30=

minus001 11990940


1 1198912) and (119909

3 1198912) respec-






1 The figure demon-



1=


2=



50 100 150minus2

minus15

minus1

minus05

0

05

1

x1

f2

(a)

20 40 60 80 100 120 140 160 180

minus3

minus25

minus2

minus15

x3

f2

(b)



= minus001 11990920

= minus005 11990930

=

minus001 11990940


1198912


1198912

)

0 01 02 03 04 05 06 07minus25

minus2

minus15

minus1

minus05

0

05

1

15

x1

1205831

(a)

0 01 02 03 04 05 06 07minus32

minus3

minus28

minus26

minus24

minus22

minus2

minus18

x1

1205831

x3

1205831

x3

(b)




= minus001 11990920

= minus005 11990930

= minus001 11990940


1205831


3

1205831

)

the planes (1199091 1205831) and (119909









3 1198862)




4varies from 119886

4= 2 to 119886

4= 120


2is

chosen as 1198862


1 1198864) and (119909

3 1198864)







0 10 20 30 40 50 60

minus3

minus2

minus1

0

1

x1

a2

(a)

0 10 20 30 40 50 60minus34

minus32

minus3

minus28

minus26

minus24

minus22

minus2

minus18

a2

x3

(b)



= 02 and 1205832


2


= minus001 11990920

= minus005 11990930

= minus001 11990940


1198862


1198862

)

0 20 40 60 80 100 120minus25

minus2

minus15

minus1

minus05

0

05

1

x1

a4

(a)

0 20 40 60 80 100 120

minus35

minus3

minus25

minus2

a4

x3

(b)




= 02 and 1205832


= 23 and 1198863


= minus001 11990920

= minus005 11990930

= minus001 11990940

=


1198864


1198864

)


1are selected as











1= 1437 120590

2=

1142 1205831= 02 120583

2= 02 119886

2= 300 119886

3= 750 119886

4= 450

1198865

= minus1166 1198866

= 1227 1198912

= 1227 1198867

= minus268 1198876

=

minus22 1198877

= minus969 1198878

= minus2232 11990910

= minus108 11990920

= 0511990930

= minus001 11990940



minus1minus050051

minus05

0

05

minus25

minus2

minus15

minus1

x1

x2

x3

(a)


minus1

minus05

0

05

1

x1

x2

(b)


= 827 and 1205831


1199092

1199093


1199092

)

minus50

5

minus50

5minus6

minus4

minus2

0

2

4

6

x1x2

x3

(a)


minus10

minus5

0

5

10

x1

x2

(b)


= 1437 1205902

= 1142 1198862

= 300 1198863

= 750 1198864

= 450 1198865

= minus1166 1198866

=

1227 1198912

= 1227 1198867

= minus268 1198876

= minus22 1198877

= minus969 1198878

= minus2232 11990910

= minus108 11990920

= 05 11990930

= minus001 11990940


1

1199092

1199093


1199092

)





2= 313 119886

5=

minus1501 1198878

= 409 11990910

= minus001 11990920

= minus009 11990930

=

minus005 11990940


7 Conclusions




minus50

5

minus10

0

10

minus6

minus4

minus2

0

2

x1x2

x3

(a)

minus5 0 5 10

minus10

minus5

0

5

10

x1

x2

(b)


= 361 1205902

= 313 1198862

= 230 1198863

= 120 1198864

= 130 1198865

= minus1501 1198866

=

minus203 1198912

= 9238 1198867

= minus205 1198876

= 407 1198877

= minus308 1198878

= 409 11990910

= minus001 11990920

= minus009 11990930

= minus005 11990940


1

1199092

1199093


1199092

)



119896(120576 119868 120574

0 1205832




2 the in-plane


4 and the damping




1 respec-










Acknowledgments



References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










120574

|u|h

Γ3

Γ2

Γ1

M

O2O3






1


6 where

1205781



6 1198867 and


1gt 0



2


2and 119886



4is the


1 1198862 1198864






3 respectively when




120574

|u|h

M

Γsum (12057401 12057402)

12057402 + Δ120574+ 12057401 + Δ120574+



12057402

) are depicted


2 1205831 1198862





2= 2 sim 200


183 1205902

= 197 1205831

= 02 1205832

= 02 1198862

= 230 1198863

=

120 1198864= 130 119886

5= minus101 119886

6= minus203 119886

7= minus205 119887

6=

407 1198877= minus308 119887

8= 109 119909

10= minus001 119909

20= minus005 119909

30=

minus001 11990940


1 1198912) and (119909

3 1198912) respec-






1 The figure demon-



1=


2=



50 100 150minus2

minus15

minus1

minus05

0

05

1

x1

f2

(a)

20 40 60 80 100 120 140 160 180

minus3

minus25

minus2

minus15

x3

f2

(b)



= minus001 11990920

= minus005 11990930

=

minus001 11990940


1198912


1198912

)

0 01 02 03 04 05 06 07minus25

minus2

minus15

minus1

minus05

0

05

1

15

x1

1205831

(a)

0 01 02 03 04 05 06 07minus32

minus3

minus28

minus26

minus24

minus22

minus2

minus18

x1

1205831

x3

1205831

x3

(b)




= minus001 11990920

= minus005 11990930

= minus001 11990940


1205831


3

1205831

)

the planes (1199091 1205831) and (119909









3 1198862)




4varies from 119886

4= 2 to 119886

4= 120


2is

chosen as 1198862


1 1198864) and (119909

3 1198864)







0 10 20 30 40 50 60

minus3

minus2

minus1

0

1

x1

a2

(a)

0 10 20 30 40 50 60minus34

minus32

minus3

minus28

minus26

minus24

minus22

minus2

minus18

a2

x3

(b)



= 02 and 1205832


2


= minus001 11990920

= minus005 11990930

= minus001 11990940


1198862


1198862

)

0 20 40 60 80 100 120minus25

minus2

minus15

minus1

minus05

0

05

1

x1

a4

(a)

0 20 40 60 80 100 120

minus35

minus3

minus25

minus2

a4

x3

(b)




= 02 and 1205832


= 23 and 1198863


= minus001 11990920

= minus005 11990930

= minus001 11990940

=


1198864


1198864

)


1are selected as











1= 1437 120590

2=

1142 1205831= 02 120583

2= 02 119886

2= 300 119886

3= 750 119886

4= 450

1198865

= minus1166 1198866

= 1227 1198912

= 1227 1198867

= minus268 1198876

=

minus22 1198877

= minus969 1198878

= minus2232 11990910

= minus108 11990920

= 0511990930

= minus001 11990940



minus1minus050051

minus05

0

05

minus25

minus2

minus15

minus1

x1

x2

x3

(a)


minus1

minus05

0

05

1

x1

x2

(b)


= 827 and 1205831


1199092

1199093


1199092

)

minus50

5

minus50

5minus6

minus4

minus2

0

2

4

6

x1x2

x3

(a)


minus10

minus5

0

5

10

x1

x2

(b)


= 1437 1205902

= 1142 1198862

= 300 1198863

= 750 1198864

= 450 1198865

= minus1166 1198866

=

1227 1198912

= 1227 1198867

= minus268 1198876

= minus22 1198877

= minus969 1198878

= minus2232 11990910

= minus108 11990920

= 05 11990930

= minus001 11990940


1

1199092

1199093


1199092

)





2= 313 119886

5=

minus1501 1198878

= 409 11990910

= minus001 11990920

= minus009 11990930

=

minus005 11990940


7 Conclusions




minus50

5

minus10

0

10

minus6

minus4

minus2

0

2

x1x2

x3

(a)

minus5 0 5 10

minus10

minus5

0

5

10

x1

x2

(b)


= 361 1205902

= 313 1198862

= 230 1198863

= 120 1198864

= 130 1198865

= minus1501 1198866

=

minus203 1198912

= 9238 1198867

= minus205 1198876

= 407 1198877

= minus308 1198878

= 409 11990910

= minus001 11990920

= minus009 11990930

= minus005 11990940


1

1199092

1199093


1199092

)



119896(120576 119868 120574

0 1205832




2 the in-plane


4 and the damping




1 respec-










Acknowledgments



References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










50 100 150minus2

minus15

minus1

minus05

0

05

1

x1

f2

(a)

20 40 60 80 100 120 140 160 180

minus3

minus25

minus2

minus15

x3

f2

(b)



= minus001 11990920

= minus005 11990930

=

minus001 11990940


1198912


1198912

)

0 01 02 03 04 05 06 07minus25

minus2

minus15

minus1

minus05

0

05

1

15

x1

1205831

(a)

0 01 02 03 04 05 06 07minus32

minus3

minus28

minus26

minus24

minus22

minus2

minus18

x1

1205831

x3

1205831

x3

(b)




= minus001 11990920

= minus005 11990930

= minus001 11990940


1205831


3

1205831

)

the planes (1199091 1205831) and (119909









3 1198862)




4varies from 119886

4= 2 to 119886

4= 120


2is

chosen as 1198862


1 1198864) and (119909

3 1198864)







0 10 20 30 40 50 60

minus3

minus2

minus1

0

1

x1

a2

(a)

0 10 20 30 40 50 60minus34

minus32

minus3

minus28

minus26

minus24

minus22

minus2

minus18

a2

x3

(b)



= 02 and 1205832


2


= minus001 11990920

= minus005 11990930

= minus001 11990940


1198862


1198862

)

0 20 40 60 80 100 120minus25

minus2

minus15

minus1

minus05

0

05

1

x1

a4

(a)

0 20 40 60 80 100 120

minus35

minus3

minus25

minus2

a4

x3

(b)




= 02 and 1205832


= 23 and 1198863


= minus001 11990920

= minus005 11990930

= minus001 11990940

=


1198864


1198864

)


1are selected as











1= 1437 120590

2=

1142 1205831= 02 120583

2= 02 119886

2= 300 119886

3= 750 119886

4= 450

1198865

= minus1166 1198866

= 1227 1198912

= 1227 1198867

= minus268 1198876

=

minus22 1198877

= minus969 1198878

= minus2232 11990910

= minus108 11990920

= 0511990930

= minus001 11990940



minus1minus050051

minus05

0

05

minus25

minus2

minus15

minus1

x1

x2

x3

(a)


minus1

minus05

0

05

1

x1

x2

(b)


= 827 and 1205831


1199092

1199093


1199092

)

minus50

5

minus50

5minus6

minus4

minus2

0

2

4

6

x1x2

x3

(a)


minus10

minus5

0

5

10

x1

x2

(b)


= 1437 1205902

= 1142 1198862

= 300 1198863

= 750 1198864

= 450 1198865

= minus1166 1198866

=

1227 1198912

= 1227 1198867

= minus268 1198876

= minus22 1198877

= minus969 1198878

= minus2232 11990910

= minus108 11990920

= 05 11990930

= minus001 11990940


1

1199092

1199093


1199092

)





2= 313 119886

5=

minus1501 1198878

= 409 11990910

= minus001 11990920

= minus009 11990930

=

minus005 11990940


7 Conclusions




minus50

5

minus10

0

10

minus6

minus4

minus2

0

2

x1x2

x3

(a)

minus5 0 5 10

minus10

minus5

0

5

10

x1

x2

(b)


= 361 1205902

= 313 1198862

= 230 1198863

= 120 1198864

= 130 1198865

= minus1501 1198866

=

minus203 1198912

= 9238 1198867

= minus205 1198876

= 407 1198877

= minus308 1198878

= 409 11990910

= minus001 11990920

= minus009 11990930

= minus005 11990940


1

1199092

1199093


1199092

)



119896(120576 119868 120574

0 1205832




2 the in-plane


4 and the damping




1 respec-










Acknowledgments



References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 10 20 30 40 50 60

minus3

minus2

minus1

0

1

x1

a2

(a)

0 10 20 30 40 50 60minus34

minus32

minus3

minus28

minus26

minus24

minus22

minus2

minus18

a2

x3

(b)



= 02 and 1205832


2


= minus001 11990920

= minus005 11990930

= minus001 11990940


1198862


1198862

)

0 20 40 60 80 100 120minus25

minus2

minus15

minus1

minus05

0

05

1

x1

a4

(a)

0 20 40 60 80 100 120

minus35

minus3

minus25

minus2

a4

x3

(b)




= 02 and 1205832


= 23 and 1198863


= minus001 11990920

= minus005 11990930

= minus001 11990940

=


1198864


1198864

)


1are selected as











1= 1437 120590

2=

1142 1205831= 02 120583

2= 02 119886

2= 300 119886

3= 750 119886

4= 450

1198865

= minus1166 1198866

= 1227 1198912

= 1227 1198867

= minus268 1198876

=

minus22 1198877

= minus969 1198878

= minus2232 11990910

= minus108 11990920

= 0511990930

= minus001 11990940



minus1minus050051

minus05

0

05

minus25

minus2

minus15

minus1

x1

x2

x3

(a)


minus1

minus05

0

05

1

x1

x2

(b)


= 827 and 1205831


1199092

1199093


1199092

)

minus50

5

minus50

5minus6

minus4

minus2

0

2

4

6

x1x2

x3

(a)


minus10

minus5

0

5

10

x1

x2

(b)


= 1437 1205902

= 1142 1198862

= 300 1198863

= 750 1198864

= 450 1198865

= minus1166 1198866

=

1227 1198912

= 1227 1198867

= minus268 1198876

= minus22 1198877

= minus969 1198878

= minus2232 11990910

= minus108 11990920

= 05 11990930

= minus001 11990940


1

1199092

1199093


1199092

)





2= 313 119886

5=

minus1501 1198878

= 409 11990910

= minus001 11990920

= minus009 11990930

=

minus005 11990940


7 Conclusions




minus50

5

minus10

0

10

minus6

minus4

minus2

0

2

x1x2

x3

(a)

minus5 0 5 10

minus10

minus5

0

5

10

x1

x2

(b)


= 361 1205902

= 313 1198862

= 230 1198863

= 120 1198864

= 130 1198865

= minus1501 1198866

=

minus203 1198912

= 9238 1198867

= minus205 1198876

= 407 1198877

= minus308 1198878

= 409 11990910

= minus001 11990920

= minus009 11990930

= minus005 11990940


1

1199092

1199093


1199092

)



119896(120576 119868 120574

0 1205832




2 the in-plane


4 and the damping




1 respec-










Acknowledgments



References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










minus1minus050051

minus05

0

05

minus25

minus2

minus15

minus1

x1

x2

x3

(a)


minus1

minus05

0

05

1

x1

x2

(b)


= 827 and 1205831


1199092

1199093


1199092

)

minus50

5

minus50

5minus6

minus4

minus2

0

2

4

6

x1x2

x3

(a)


minus10

minus5

0

5

10

x1

x2

(b)


= 1437 1205902

= 1142 1198862

= 300 1198863

= 750 1198864

= 450 1198865

= minus1166 1198866

=

1227 1198912

= 1227 1198867

= minus268 1198876

= minus22 1198877

= minus969 1198878

= minus2232 11990910

= minus108 11990920

= 05 11990930

= minus001 11990940


1

1199092

1199093


1199092

)





2= 313 119886

5=

minus1501 1198878

= 409 11990910

= minus001 11990920

= minus009 11990930

=

minus005 11990940


7 Conclusions




minus50

5

minus10

0

10

minus6

minus4

minus2

0

2

x1x2

x3

(a)

minus5 0 5 10

minus10

minus5

0

5

10

x1

x2

(b)


= 361 1205902

= 313 1198862

= 230 1198863

= 120 1198864

= 130 1198865

= minus1501 1198866

=

minus203 1198912

= 9238 1198867

= minus205 1198876

= 407 1198877

= minus308 1198878

= 409 11990910

= minus001 11990920

= minus009 11990930

= minus005 11990940


1

1199092

1199093


1199092

)



119896(120576 119868 120574

0 1205832




2 the in-plane


4 and the damping




1 respec-










Acknowledgments



References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










minus50

5

minus10

0

10

minus6

minus4

minus2

0

2

x1x2

x3

(a)

minus5 0 5 10

minus10

minus5

0

5

10

x1

x2

(b)


= 361 1205902

= 313 1198862

= 230 1198863

= 120 1198864

= 130 1198865

= minus1501 1198866

=

minus203 1198912

= 9238 1198867

= minus205 1198876

= 407 1198877

= minus308 1198878

= 409 11990910

= minus001 11990920

= minus009 11990930

= minus005 11990940


1

1199092

1199093


1199092

)



119896(120576 119868 120574

0 1205832




2 the in-plane


4 and the damping




1 respec-










Acknowledgments



References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article modeling and chaotic dynamics of the

Documents