advances in nonlinear vibrationdownloads.hindawi.com/journals/specialissues/940489.pdf · advances...

Advances in Nonlinear Vibration

Journal of Applied Mathematics

Guest Editors: Livija Cveticanin, Kale Oyedeji, Vasile Marinca, and Nicolae Herisanu

Journal of Applied Mathematics


Guest Editors: Livija Cveticanin, Kale Oyedeji,Vasile Marinca, and Nicolae Herisanu

Copyright 2013 Hindawi Publishing Corporation. All rights reserved.

This is a special issue published in Journal of Applied Mathematics. All articles are open access articles distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the originalwork is properly cited.

Editorial Board

Saeid Abbasbandy, IranMina B. Abd-El-Malek, EgyptMohamed A. Abdou, EgyptSubhas Abel, IndiaMostafa Adimy, FranceCarlos J. S. Alves, PortugalMohamad Alwash, USAIgor Andrianov, GermanySabri Arik, TurkeyFrancis T.K. Au, Hong KongOlivier Bahn, CanadaRoberto Barrio, SpainAlfredo Bellen, ItalyJafar Biazar, IranHester Bijl, The NetherlandsAnjan Biswas, Saudi ArabiaStephane P.A. Bordas, USAJames Robert Buchanan, USAAlberto Cabada, SpainXiao Chuan Cai, USAJinde Cao, ChinaAlexandre Carvalho, BrazilSong Cen, ChinaQianshun S. Chang, ChinaTai-Ping Chang, TaiwanShih-sen Chang, ChinaRushan Chen, ChinaXinfu Chen, USAKe Chen, UKEric Cheng, Hong KongFrancisco Chiclana, UKJen-Tzung Chien, TaiwanC. S. Chien, TaiwanHan H. Choi, Republic of KoreaTin-Tai Chow, ChinaM. S. H. Chowdhury, MalaysiaCarlos Conca, ChileVitor Costa, PortugalLivija Cveticanin, SerbiaEric de Sturler, USAOrazio Descalzi, ChileKai Diethelm, GermanyVit Dolejsi, Czech RepublicBo-Qing Dong, ChinaMagdy A. Ezzat, Egypt

Meng Fan, ChinaYa Ping Fang, ChinaAntonio J. M. Ferreira, PortugalMichel Fliess, FranceM. A. Fontelos, SpainHuijun Gao, ChinaBernard J. Geurts, The NetherlandsJamshid Ghaboussi, USAPablo Gonzalez-Vera, SpainLaurent Gosse, ItalyK. S. Govinder, South AfricaJose L. Gracia, SpainYuantong Gu, AustraliaZhihong GUAN, ChinaNicola Guglielmi, ItalyFrederico G. Guimaraes, BrazilVijay Gupta, IndiaBo Han, ChinaMaoan Han, ChinaPierre Hansen, CanadaFerenc Hartung, HungaryXiaoqiao He, Hong KongLuis Javier Herrera, SpainJ. Hoenderkamp, The NetherlandsYing Hu, FranceNing Hu, JapanZhilong L. Huang, ChinaKazufumi Ito, USATakeshi Iwamoto, JapanGeorge Jaiani, GeorgiaZhongxiao Jia, ChinaTarun Kant, IndiaIdo Kanter, IsraelAbdul Hamid Kara, South AfricaHamid Reza Karimi, NorwayJae-Wook Kim, UKJong Hae Kim, Republic of KoreaKazutake Komori, JapanFanrong Kong, USAVadim . Krysko, RussiaJin L. Kuang, SingaporeMiroslaw Lachowicz, PolandHak-Keung Lam, UKTak-Wah Lam, Hong KongPGL Leach, Cyprus

Yongkun Li, ChinaWan-Tong Li, ChinaJ. Liang, ChinaChing-Jong Liao, TaiwanChong Lin, ChinaMingzhu Liu, ChinaChein-Shan Liu, TaiwanKang Liu, USAYansheng Liu, ChinaFawang Liu, AustraliaShutian Liu, ChinaZhijun Liu, ChinaJulian Lopez-Gomez, SpainShiping Lu, ChinaGert Lube, GermanyNazim Idrisoglu Mahmudov, TurkeyOluwole Daniel Makinde, South AfricaFrancisco J. Marcellan, SpainGuiomar Martn-Herran, SpainNicola Mastronardi, ItalyMichael McAleer, The NetherlandsStephane Metens, FranceMichael Meylan, AustraliaAlain Miranville, FranceRam N. Mohapatra, USAJaime E. Munoz Rivera, BrazilJavier Murillo, SpainRoberto Natalini, ItalySrinivasan Natesan, IndiaJiri Nedoma, Czech RepublicJianlei Niu, Hong KongRoger Ohayon, FranceJavier Oliver, SpainDonal ORegan, IrelandMartin Ostoja-Starzewski, USATurgut Ozis, TurkeyClaudio Padra, ArgentinaReinaldo Martinez Palhares, BrazilFrancesco Pellicano, ItalyJuan Manuel Pena, SpainRicardo Perera, SpainMalgorzata Peszynska, USAJames F. Peters, CanadaMark A. Petersen, South AfricaMiodrag Petkovic, Serbia

Vu Ngoc Phat, VietnamAndrew Pickering, SpainHector Pomares, SpainMaurizio Porfiri, USAMario Primicerio, ItalyMorteza Rafei, The NetherlandsRoberto Reno, ItalyJacek Rokicki, PolandDirk Roose, BelgiumCarla Roque, PortugalDebasish Roy, IndiaSamir H. Saker, EgyptMarcelo A. Savi, BrazilWolfgang Schmidt, GermanyEckart Schnack, GermanyMehmet Sezer, TurkeyNaseer Shahzad, Saudi ArabiaFatemeh Shakeri, IranJian Hua Shen, ChinaHui-Shen Shen, ChinaFernando Simoes, PortugalTheodore E. Simos, Greece

Abdel-Maksoud A. Soliman, EgyptXinyu Song, ChinaQiankun Song, ChinaYuri N. Sotskov, BelarusPeter J. C. Spreij, The NetherlandsNiclas Stromberg, SwedenRay KL Su, Hong KongJitao Sun, ChinaWenyu Sun, ChinaXianHua Tang, ChinaAlexander Timokha, NorwayMariano Torrisi, ItalyJung-Fa Tsai, TaiwanCh Tsitouras, GreeceKuppalapalle Vajravelu, USAAlvaro Valencia, ChileErik Van Vleck, USAEzio Venturino, ItalyJesus Vigo-Aguiar, SpainMichael N. Vrahatis, GreeceBaolin Wang, ChinaMingxin Wang, China

Qing-Wen Wang, ChinaGuangchen Wang, ChinaJunjie Wei, ChinaLi Weili, ChinaMartin Weiser, GermanyFrank Werner, GermanyShanhe Wu, ChinaDongmei Xiao, ChinaGongnan Xie, ChinaYuesheng Xu, USASuh-Yuh Yang, TaiwanBo Yu, ChinaJinyun Yuan, BrazilAlejandro Zarzo, SpainGuisheng Zhai, JapanJianming Zhan, ChinaZhihua Zhang, ChinaJingxin Zhang, AustraliaShan Zhao, USAChongbin Zhao, AustraliaRenat Zhdanov, USAHongping Zhu, China

Advances in Nonlinear Vibration, Livija Cveticanin, Kale Oyedeji, Vasile Marinca, and Nicolae HerisanuVolume 2013, Article ID 506419, 2 pages

Optimal Variational Method for Truly Nonlinear Oscillators, Vasile Marinca and Nicolae HerisanuVolume 2013, Article ID 620267, 6 pages

Nonlinear Dynamics of an Electrorheological Sandwich Beam with Rotary Oscillation, Kexiang Wei,Wenming Zhang, Ping Xia, and Yingchun LiuVolume 2012, Article ID 659872, 17 pages

Nonlinear Periodic Oscillation of a Cylindrical Microvoid Centered at an Isotropic IncompressibleOgden Cylinder, Wenzheng Zhang, Xuegang Yuan, and Hongwu ZhangVolume 2012, Article ID 872161, 9 pages

Some Properties of Motion Equations Describing the Nonlinear Dynamical Response of a MultibodySystem with Flexible Elements, Maria Luminita Scutaru and Sorin VlaseVolume 2012, Article ID 628503, 12 pages

Approximate Super- and Sub-harmonic Response of a Multi-DOFs System with Local CubicNonlinearities under Resonance, Yang CaiJinVolume 2012, Article ID 531480, 22 pages

Review on Mathematical and Mechanical Models of the Vocal Cord, L. CveticaninVolume 2012, Article ID 928591, 18 pages

An Optimal Approach to Study the Nonlinear Behaviour of a Rotating Electrical Machine,Nicolae Herisanu and Vasile MarincaVolume 2012, Article ID 465023, 10 pages

Analytical Approximate Solutions for the Cubic-Quintic Duffing Oscillator in Terms of ElementaryFunctions, A. Belendez, M. L. Alvarez, J. Frances, S. Bleda, T. Belendez, A. Najera, and E. ArribasVolume 2012, Article ID 286290, 16 pages

High-Order Energy Balance Method to Nonlinear Oscillators, Seher Durmaz and Metin Orhan KayaVolume 2012, Article ID 518684, 7 pages

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2013, Article ID 506419, 2 pageshttp://dx.doi.org/10.1155/2013/506419

EditorialAdvances in Nonlinear Vibration

Livija Cveticanin,1 Kale Oyedeji,2 Vasile Marinca,3 and Nicolae Herisanu4

1 Faculty of Technical Sciences, University of Novi Sad, Trg Dositeja Obradovica 6, 21000 Novi Sad, Serbia2Department of Physics, Morehouse College, Atlanta, GA, USA3 Center for Advanced and Fundamental Technical Research, Romanian Academy, Timisoara Branch, Bulevardul Mihai Viteazu,24, 300223 Timisoara, Romania

4 Politehnica University of Timisoara, Bulevardul Mihai Viteazu, 1, 300222 Timisoara, Romania

Correspondence should be addressed to Livija Cveticanin; [email protected]

Received 11 July 2013; Accepted 11 July 2013

Copyright 2013 Livija Cveticanin et al. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

1. Introduction

The main subject of the topic is nonlinear vibration and theoscillatory motion of the strong nonlinear system in all itsvaried aspects, including truly nonlinear oscillator, oscillatorwith time variable parameter, forced vibration, parametricalvibration, and analytical solving methods for these dynamicsystems.

In this issue, 1 review and 8 original research papersare published. The papers analyze the oscillatory motionand vibration of the strong nonlinear oscillatory systemswhich appear and are applied in all the fields of science andengineering. The topics of the papers are

(i) strong nonlinear oscillatory systems (2 papers);

(ii) asymptotic methods in truly nonlinear oscillators (2papers);

(iii) forced vibrations in the multiple-degree-of-freedomoscillatory system (1 paper);

(iv) application of the asymptotic methods in biomechan-ical (1 paper) solid structures (1 paper);

(v) nonlinear phenomena in stability and vibration of anelastic body (2 papers).

We are going to introduce the selected papers in thecategories of nonlinear vibrations of a cubic-quintic Duffingoscillator, a biomechanical system, a truly nonlinear oscillator

with its application to an electricalmachine,multiple-degree-of-freedom oscillator andmultibody system, and elastic body(sandwich beam and cylinder).

2. Analytical Solution Procedures for theCubic-Quintic Duffing Oscillator

In the paper entitled Analytical approximate solutions forthe cubic-quintic Duffing oscillator in terms of elementaryfunctions, A. Belendez et al. present a procedure for solvingthe cubic-quintic Duffing oscillator. The method is basedon the expansion of the restoring force into the Chebyshevpolynomials and transformation of the original nonlineardifferential equation into a cubic Duffing one. Approximatesolutions are expressed in the form of the complete ellipticintegral of the first kind and the cosine Jacobi ellipticfunction. Using the series expansion of the functions intoelementary functions and applying the harmonic balancemethod, the periodic solution of the original nonlinearoscillator is obtained.

S. Durmaz and M. O. Kaya in their paper entitled High-order energy balance method to nonlinear oscillators giveanother analytical procedure for solving the cubic-quinticDuffing oscillator. Namely, the energy balance method devel-oped for high-order nonlinear oscillators is adopted for solv-ing the cubic-quintic Duffing oscillator. The approximationis done up to the third order, and the maximal relative errorof the frequency which decreases to 0.008% is analyticallysolved by applying the energy balance method. There is a

2 Journal of Applied Mathematics

good agreement of the analytically calculated frequencies andperiodic solutions with the numerically calculated ones.

3. Application of the Nonlinear VibrationModel to a Biomedical System

In the paper Review onmathematical andmechanical modelsof the vocal cord by L. Cveticanin, a review on mathematicaland mechanical models of the vocal cords is given. The basicmodel of the vocal cords, that is, vocal folds, is a two-massnonlinear oscillator system which is assumed to be the basicone for mechanical description in voice production. Themodel is described with a system of two coupled second-order differential equations. The two-mass model is modi-fied into three, five, and more mass systems, systems withtime variable parameters, and three-dimensional systems butalso simplified into a one-mass system with coupled two-direction deflection function and with friction function. Thecorresponding mathematical models are the second-ordernonlinear differential equations. The regular vibrations andthe chaotic motion are investigated. Based on the obtainedresults, the pathology of vocal cords is determined.

4. Vibration of a Truly Nonlinear Oscillatorand Its Application in Electrical Machine

In the paper of V. Marinca and N. Herisanu entitled Optimalvariational method for truly nonlinear oscillator, a novelanalytical approximate solving technique is introduced. Themain advantage of the method is that it allows the adjust-ment of convergence regions during calculation. Besides,the approach does not depend upon any small or largeparameter. The suggested method is applied for solvingthe truly nonlinear oscillators. The analytically obtainedsolutions are compared with the numerical ones and showgood agreement.

The paper entitled An optimal approach to study thenonlinear behaviour of a rotating electrical machine byN. Herisanu and V. Marinca considers the application ofthe optimal variational method for solving the oscillatorwith cubic nonlinearity and time variable coefficients. Theprocedure involves the presence of arbitrary convergencecontrol parameters which have to satisfy the condition of theoptimal control. The method is applied for rotating electricalmachines.

5. Approximate Solution ofa Multiple-Degree-of-Freedom Oscillatorwith Cubic Nonlinearity

The paper entitled Approximate super- and sub-harmonicresponse of amulti-DOFs systemwith local cubic nonlinearitiesunder resonance by Y. CaJin deals with the problem ofmultiple-degree-of-freedom dynamical system with cubicnonlinearities and with superharmonic or subharmonic exci-tation. The single modal natural resonance theory is appliedfor linearization and simplification in the system. While thesystem is controlled by multiple modes, modal analysis isused for linearization and to obtain the dominant modes. An

example of a damped oscillator with ten degrees of freedom isconsidered. The approximate solutions are validated by com-paring them with the results of direct numerical integration.The difference between solutions may be neglected.

In the paper of M. L. Scutaru and S. Vlase entitledSome properties of motion equations describing the nonlin-ear dynamical response of a multibody system with flexibleelements, the motion of a multibody system with flexibleelements is considered. The system is discretized, and themotion is described with a system of strong nonlinearsecond-order differential equations with variable parameters.The linearization is done only for short time intervals. Thecoefficients are frozen and considered as constants. For suchassumptions, the exact solution of the system is obtained, andbased on the obtained result, the characteristics of the systemare discussed.

6. Vibration Properties of the NonlinearElastic Body

In the Nonlinear dynamics of an electrorheological sandwichbeam with rotary oscillation, by K. Wei et al. and P. Xia,the nonlinear properties and the parametric instability ofa rotating electrorheological sandwich beam with rotaryoscillation are numerically analyzed. The system is describedwith a system of coupled nonlinear differential equations.Thediscretization is introduced, and the multiple scale methodis applied for analyzing the parametric structural instability.The influence of the electrical field on the stability is inves-tigated. It is concluded that the electrorheological materialshave a significant influence on vibration characteristics andparametric instability. These materials can be used to adjustthe stability of the rotating flexible beams.

Nonlinear periodical vibrations of cylinders are con-sidered in Nonlinear periodic oscillation of a cylindricalmicrovoid centered at an isotropic incompressible Ogden cylin-der by W. Zhang et al. The vibration of an infinitely longcylinder composed of an isotropic incompressible Ogdenmaterial with a microvoid at its center, where the outersurface of the cylinder is subjected to a uniform radial tensileload, is analyzed. Using the incompressibility condition andthe boundary conditions, the motion of the microvoid isdescribedwith a second-order nonlinear ordinary differentialequation. The qualitative analysis of the equation shows thattwo types of solutions exist: a nonlinear periodic oscillatoryone and a blow-up solution. Namely, for certain parametervalue of the incompressible Ogden material, there is a jumpphenomenon in the amplitude-load diagram.

Acknowledgment

The guest editors would like to take this opportunity to thankall the contributions from the authors and reviewers.

Livija CveticaninKale Oyedeji

Vasile MarincaNicolae Herisanu

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2013, Article ID 620267, 6 pageshttp://dx.doi.org/10.1155/2013/620267

Research ArticleOptimal Variational Method for Truly Nonlinear Oscillators

Vasile Marinca1,2 and Nicolae Herisanu1,2

1 Faculty of Mechanical Engineering, Politehnica University of Timisoara, Bd. M. Viteazu, 1, 300222 Timisoara, Romania2 Center of Advanced Research in Engineering Sciences, Romanian Academy, Timisoara Branch,Bd. M. Viteazu, 24, 300223, Timisoara, Romania

Correspondence should be addressed to Vasile Marinca; [email protected]

Received 6 August 2012; Accepted 19 November 2012

Academic Editor: Kale Oyedeji

Copyright 2013 V. Marinca and N. Herisanu.This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

The Optimal Variational Method (OVM) is introduced and applied for calculating approximate periodic solutions of trulynonlinear oscillators.Themain advantage of this procedure consists in that it provides a convenient way to control the convergenceof approximate solutions in a very rigorous way and allows adjustment of convergence regions where necessary.This approach doesnot depend upon any small or large parameters. A very good agreement was found between approximate and numerical solution,which proves that OVM is very efficient and accurate.

1. Introduction

It is known that the study of nonlinear differential equationsis restricted to a variety of special classes of equations andthe method of solution usually involves one or more tech-niques to achieve analytical approximations to the solutions.Many researchers have recently paid much attention to findand develop approximate solutions. Perturbation methodshave been successfully employed to determine approximatesolutions to weakly nonlinear problems [13]. But the use ofperturbation theory in many problems is invalid or it simplybreaks down for parameters beyond a certain specified range.Therefore, new analytical techniques should be developed toovercome these shortcomings. Such a new technique shouldwork over a large range of parameters and yield accurateanalytical approximate solutions beyond the coverage andability of the classical perturbation methods. There has beena great need for effective algorithms to avoid the workrequired by traditional techniques, but it is difficult to obtainconvergent results in the cases of strong nonlinearity.

Recently, many new approaches have been proposed forthis purpose, such as various modified Lindstedt-Poincaremethods [4], some linearizationmethods [5, 6], the Adomiandecomposition method [7], the optimal homotopy asymp-totic method [8, 9], the optimal variational iteration method

[10], the energy balance method [11], and so on. A varia-tional principle for nonlinear oscillations by constructing theHamiltonian was studied in [11]. Variational principles havea great importance in physics and engineering since theyestablish connections between these disciplines and theirapplications are useful in devising various approximate tech-niques. Variational methods have been and continue to bepopular tools for nonlinear analysis. They combine physicalinsights into the nature of the solution of the problem andthe solutions obtained using possible trial functions are thebest. It is known that computing a Lagrangian for dynamicalsystems with more general Newtonian forces is nowadaysapplicable only to systems with force derivable from apotential function (basically, conservative systems). Strictlyspeaking, conservative dynamical systems do not exist inour Newtonian environment. As a result, the Lagrangianrepresentation of conservative Newtonian systems is in gen-eral only a crude approximation of physical reality. Theproblem of the existence of a Lagrangian, Hamiltonian, orRouthian can be studied today with a variety of modernand sophisticated mathematical tools which include the useof functional analysis, prolongation theory, and differentialgeometry, to cite only a few. This question is called InverseProblem in Newtonian Mechanics [12] and consists inthe identification of the methods for the construction of a


Lagrangian, Hamiltonian, or Routhian form given equationsof motion.

In this paper we construct accurate approximations toperiodic solutions and frequencies of the so-called trulynonlinear oscillator (TNO). Following Mickens and Oyedeji[13, 14], the most general form of a TNO is given by thefollowing differential equation:

+ () = (, ) , (0) = , (0) = 0, (1)

where the dot denotes the derivative with respect to variable, is a positive arbitrary parameter, and the functions ()and (, ) have the properties:

() = () ,

(, ) = (, ) ,

(2)

and () does not have for small a dominant termproportional to .

In the present work, we consider () = 1/3 + 3,(, ) = 0 and therefore the truly nonlinear oscillator ismodeled by the following nonlinear differential equation:

+

1/3+

3= 0

(3)

subject to initial conditions

(0) = , (0) = 0, (4)

where , , and are known parameters.In (3), there exists no small or large parameter.

2. Basic Idea of Optimal VariationalMethod and Solution

In order to develop an application of the OVM, we considerthe following differential equation:

+ () = 0 (5)

with the initial conditions given by (4) and being anarbitrary odd function.

Introducing a new independent variable and a newunknown () as

= , () = () , (6)

where is the frequency of the system (5), then this becomes

2

+

1 () = 0,

(7)

where prime denotes derivative with respect to the newvariable .

The initial conditions (4) become

(0) = 1,

(0) = 0. (8)

The variational principle for (7) can be easily establishedif there exists a function

=

/2

0

(, ,

) ,

(9)

which admits as extremals the solutions of (7) and (8) where is the Lagrangian of the system (7):

(, ,

) =

1

2

2

2

+

1 () . (10)

The function () is given by the equation

()

= () .(11)

We assume that the approximate periodic solutions of (7)and (8) can be expressed as

() =

=1

cos (2 1) , (12)

where are arbitrary unknowns at this moment and is a

positive integer number. Choosing the solution (12) has beenmade in accordance to the properties (2).

Substituting (12) into (9) results in

(

1,

2, . . . ,

, ) =

/2

0

[

1

2

2

2+

1 ()] .

(13)

Applying the Ritz method [9], we require

1

=

2

= =

= 0. (14)

From (14) and from the initial condition (81) which

becomes

1+

2+ +

= 1 (15)

we can obtain optimally the parameters , = 1, 2, . . . , ,

and the frequency .We remark that the choice of the approximate solution

(12) is not unique. We can alternatively choose anotherexpression of the approximate periodic solution in the form

() =

=1

cos (4 1) (16)

and so on. With the parameters (called convergence-

control parameters) and the frequency known, the approx-imate periodic solutions is well determined.

The validity of the proposed approach is illustrated on theTNOgiven by (3). Using the transformations (6), equation (3)can be written in the form

2

+

2/3

1/3+

2

3= 0

(17)

and the Lagrangian becomes

(, ,

) =

1

2

2

2+

3

4

2/3

4/3+

1

4

2

4. (18)

If we consider = 3 into (12), then the approximateperiodic solutions become

() =

1cos +

2cos 3 +

3cos 5. (19)

Journal of Applied Mathematics 3

0.2 0.4 0.6 0.8 1 1.2 1.4

2

4

2

4

x

t

Figure 1: Comparison between the approximate solution (27) andnumerical solution of (3) in the case = = 1, = 5, red solidline: numerical integration results, blue dashed line: approximatesolution.

0.1 0.2 0.3 0.4 0.5 0.6 0.7

10

5

5

10

x

t


Now, substituting (19) into (18) and (13), we have succes-sively

/2

0

2 =

4

(

2

1+ 9

2

2+ 25

2

3) ,

/2

0

4 =

16

[3 (

4

1+

4

2+

4

3)

+ 12 (

2

1

2

2+

2

1

2

3+

2

2

2

3)

+4

3

1

2+ 12

2

1

2

3+ 12

1

2

2

3] .

(20)

To calculate the middle term 4/3 into (18) we use theseries expansion [15]:

cos1/3 = (cos 15

cos 3 + 110

cos 5

7

110

cos 7 + 122

cos 9

13

374

cos 11 + ) ,

(21)

where = 1.15959526696393. If we denote () = 4/3where is given by (19), then using the relation

(

0+ ) = (

0) +

1!

1(

0) +

2

2!

11(

0) +

(22)

with 0

=

1cos , =

2cos 3 +

3cos 5, we can write

successively

4/3= (

1cos )4/3 + 4

3

(

2cos 3 +

3cos 5)

(

1cos )1/3 +

=

4/3

1cos

(cos 15

cos 3 + 110

cos 5 7110

cos 7 + )

+

4

1/3

1

3

(

2cos 3 +

3cos 5)

(cos 15

cos 3 + 110

cos 5 7110

cos 7 + )

=

1/3

1[

1

2

1

2

15

2+

1

15

3

+ (

2

5

1

2

3

2

3

5

3) cos 2

+ (

1

20

1+

103

165

2+

126

165

3) cos 4

+ ] ,

(23)

/2

0

4/3 =

1/3

1

2

(

1

2

1

2

15

2+

1

15

3) .

(24)

Substituting (19) and (24) into (13) we obtain2

=

1

4

2(

2

1+ 9

2

2+ 25

2

3)

+

1/3

1

40

2/3(15

1 4

2+ 2

3)

+

2

32

[3 (

4

1+

4

2+

4

3)

+ 12 (

2

1

2

2+

2

1

2

3+

2

2

2

3)

+4

3

1

2+ 12

2

1

2

3+ 12

1

2

2

3] .

(25)

From (25), (14), and (15) we can obtain the unknowns 1,

2, 3, and .

We remark that with the approximation given by (19),there are four things to be calculated: the optimal valuesof the convergence-control parameters

1, 2, 3, and the

frequency . As it can be seen, the complexity of theequations is such that only numerical solutions can be foundfor particular values of the parameters , , and , which willbe further developed.


0.2 0.4 0.6 0.8 1

4

2

2

4

x

t


0.1 0.2 0.3 0.4 0.5

10

5

5

10

x

t


3. Numerical Examples

In order to show the validity and accuracy of the OVM, weconsider the following cases.

(1) In the first case we consider = = 1 and = 5 andwe obtain

1= 0.956942,

2= 0.041274,

3= 0.0017835, = 4.287825.

(26)

The approximate solution of (3) becomes

() = 4.78471 cos + 0.20637 cos 3

+ 0.0089175 cos 5.(27)

The value of obtained by numerical integration is

= 4.281323.

(2) In the second case, for = = 1 and = 10 we have

1= 0.955367,

2= 0.0427864,

3= 0.00184695, = 8.489069,

(28)

0.2 0.4 0.6 0.8

4

2

2

4

x

t


0.1 0.2 0.3 0.4

10

5

5

10

x

t

Figure 6: Comparison between the approximate solution andnumerical solution of (37) in the case = = 3, = 10, red solidline: numerical integration results, blue dashed line: approximatesolution.

and therefore the approximate solution becomes

() = 9.55367 cos

+ 0.427864 cos 3 + 0.0184695 cos 5.(29)

In this case, the value of obtained by numericalintegration is

= 8.485057.

(3) In the third case, we consider = = 2 and = 5and thus

1= 0.956797,

2= 0.0414033,

3= 0.00179965, = 6.063427.

(30)

The approximate solution of (3) can be written as

() = 4.783985 cos

+ 0.2070165 cos 3 + 0.00899825 cos 5.(31)

The result of numerical integration for the frequency is in this case

= 6.059008.


(4) In the case = = 2 and = 10 we get

1= 0.955313,

2= 0.0428326,

3= 0.00185462, = 12.00502,

(32)

and the approximate solution of (3) will be

() = 9.55313 cos

+ 0.428326 cos 3 + 0.0185462 cos 5.(33)

By numerical integration, we get in this case thefrequency

= 12.003983.

(5) For = = 3 and = 5, we obtain

1= 0.956908,

2= 0.0413049,

3= 0.00178722, = 7.426642,

(34)

and the approximate solution of (3) becomes

() = 4.78454 cos

+ 0.2065245 cos 3 + 0.00893611 cos 5.(35)

In this case the numerical integration result for thefrequency is

= 7.434185.

(6) In the last case, we consider = = 3 and = 10and therefore

1= 0.955363,

2= 0.0427894,

3= 0.0018479, = 14.703456,

(36)

and the approximate solution of (3) becomes

() = 9.55363 cos

+ 0.427894 cos 3 + 0.018479 cos 5.(37)

For comparison purposes, the frequency of the systemobtained directly by numerical integration is

=

14.705077.

Figures 1, 2, 3, 4, 5, and 6 present a comparison betweenthe present solutions (27)(37), respectively, and the numer-ical integration results for (3) and (4). Thus, it is easierto emphasize the accuracy of the obtained results, sincewithin these graphical representations, the analytical resultsobtained through OVM are nearly identical with numericalones. Also, the approximate frequencies obtained by OVMare in very good agreement with those obtained by numericalintegration, which also proves the validity of the approximateresults.

4. Conclusions

In this paper we introduce the Optimal Variational Methodto propose a new analytic approximate periodic solution toTNO. Our procedure is valid even if the nonlinear equationdoes not contain small or large parameters. OVM providesus with a simple and rigorous way to control and adjustthe convergence of a solution through several convergence-control parameters

whose values are optimally deter-

mined. This new method is very rapid and effective, and weprove it by comparing the approximate periodic solutionsand frequencies obtained through the proposedmethod withnumerical integration results. An excellent agreement hasbeen demonstrated between the analytical and numericalintegration results, and also for large values of the initialamplitudes, which validates the effectiveness of the proposedmethod. The proposed procedure can also be used to findanalytical approximate solutions to other classes of conser-vative oscillators.

The main advantage of the method consists in that itprovides us with a great freedom in choosing the approximateperiodic solution dependent on an arbitrary number ofinitially unknown parameters. It is interesting to remarkthat unlike other known approximate methods applicablefor such problems (such as the harmonic balance method,the multiple scales method, and so on), the frequency of the system is not obtained imposing some conditionsto avoid secular terms, which is a usual procedure withinother methods, but this frequency results directly from theconditions tominimize the residual functional along with theinitial conditions.

Acknowledgment

This work was supported by CNCS-UEFISCDI, Project no.PN-II-ID-PCE-2012-4-0358.

References

[1] A.H.Nayfeh andD.T.Mook,NonlinearOscillations, JohnWileyand Sons, New York, NY, USA, 1979.

[2] P. Hagedorn, Nonlinear Oscillations, vol. 10, Clarendon Press,Oxford, UK, 1988.

[3] R. E. Mickens,Oscillations in Planar Dynamical Systems, vol. 37,World Scientific, Singapore, 1996.

[4] J.-H. He, Modified Lindstedt-Poincare methods for somestrongly nonlinear oscillators. Part.I. expansion of a constant,International Journal of Non-Linear Mechanics, vol. 37, no. 2, pp.309314, 2002.

[5] A. Belendez, C. Pascual, C. Neipp, T. Belendez, and A.Hernandez, An equivalent linearization method for conser-vative nonlinear oscillators, International Journal of NonlinearSciences and Numerical Simulation, vol. 9, no. 1, pp. 917, 2008.

[6] J. I. Ramos, Linearized Galerkin and artificial parameter tech-niques for the determination of periodic solutions of nonlinearoscillators,AppliedMathematics and Computation, vol. 196, no.2, pp. 483493, 2008.

[7] G. Adomian, A review of the decomposition method inapplied mathematics, Journal of Mathematical Analysis andApplications, vol. 135, no. 2, pp. 501544, 1988.


[8] V. Marinca and N. Herisanu, Determination of periodicsolutions for the motion of a particle on a rotating parabola bymeans of the optimal homotopy asymptotic method, Journal ofSound and Vibration, vol. 329, no. 9, pp. 14501459, 2010.

[9] V. Marinca and N. Herisanu, Nonlinear Dynamical Systems inEngineering. Some Approximate Approaches, Springer, Berlin,Germany, 2011.

[10] N. Herisanu and V. Marinca, A modified variational iterationmethod for strongly nonlinear problems, Nonlinear ScienceLetters A, vol. 1, pp. 183192, 2010.

[11] J. H. He, Preliminary report on the energy balance for nonlin-ear oscillations, Mechanics Research Communications, vol. 29,pp. 107111, 2003.

[12] V. Obadeanu and V. Marinca, The Inverse Problem in AnalyticMechanics, University of Timisoara, Timisoara, Romania, 1992.

[13] R. E. Mickens and K. Oyedeji, Construction of approximateanalytical solutions to a new class of nonlinear oscillatorequation, Journal of Sound and Vibration, vol. 162, no. 4, pp.579582, 1985.

[14] R. E.Mickens, A generalized iteration procedure for calculatingapproximations to periodic solutions of truly nonlinear oscilla-tors, Journal of Sound and Vibration, vol. 287, no. 4-5, pp. 10451051, 2005.

[15] R. E. Mickens, Iteration method solutions for conservativeand limit-cycle X1/3 force oscillators, Journal of Sound andVibration, vol. 292, no. 35, pp. 964968, 2006.

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2012, Article ID 659872, 17 pagesdoi:10.1155/2012/659872

Research ArticleNonlinear Dynamics of an ElectrorheologicalSandwich Beam with Rotary Oscillation

Kexiang Wei,1 Wenming Zhang,2 Ping Xia,1 and Yingchun Liu1

1 Department of Mechanical Engineering, Hunan Institute of Engineering, Xiangtan 411101, China2 State Key Laboratory of Mechanical System and Vibration, Shanghai Jiao Tong University,Shanghai 200240, China

Correspondence should be addressed to Kexiang Wei, [email protected]

Received 29 August 2012; Accepted 28 November 2012

Academic Editor: Vasile Marinca

Copyright q 2012 Kexiang Wei et al. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.

The dynamic characteristics and parametric instability of a rotating electrorheological ER sand-wich beam with rotary oscillation are numerically analyzed. Assuming that the angular velocityof an ER sandwich beam varies harmonically, the dynamic equation of the rotating beam is firstderived based on Hamiltons principle. Then the coupling and nonlinear equation is discretizedand solved by the finite element method. The multiple scales method is employed to determine theparametric instability of the structures. The effects of electric field on the natural frequencies, lossfactor, and regions of parametric instability are presented. The results obtained indicate that theER material layer has a significant effect on the vibration characteristics and parametric instabilityregions, and the ER material can be used to adjust the dynamic characteristics and stability of therotating flexible beams.

1. Introduction

The dynamics of rotating flexible beams have been the subject of extensive research dueto a number of important applications in engineering such as manipulators, helicopters,turbine blades, and so forth. Much research about the dynamic modeling and vibrationcharacteristics of fixed-shaft rotating beams has been published in recent decades. Chungand Yoo 1 investigated the dynamic characteristics of rotating beams using finite elementmethod FEM and obtained the time responses and distribution of the deformations andstresses at a given rotating speed. The nonlinear dynamics of a rotating beam with flexibleroot attached to a rotating hub with elastic foundation has been analyzed by Al-Qaisia2. He discussed the effect of root flexibility, hub stiffness, torque type, torque period and


excitation frequency and amplitude on the dynamic behavior of the rotating beam-hub.Lee et al. 3 investigated divergence instability and vibration of a rotating Timoshenkobeam with precone and pitch angles. The nonlinear modal analysis of a rotating beamhas been studied by Arvin and Bakhtiari-Nejad 4. The stability and some dynamiccharacteristics of the nonlinear normal modes such as the phase portrait, Poincare section,and power spectrum diagrams have been inspected. But most research to date has examinedonly the effects of steady velocity on the vibration characteristics of the flexible beam, withoutconsidering the dynamic characteristics of speed variation of beams. Rotating flexible beamswith variable speeds, such as manipulators, demonstrate complex dynamic characteristicsbecause of changes in angular velocity. The beam can suffer from dynamic instabilityunder certain movement parameters. Therefore, the vibration stability of flexible beamswith variable angular velocity has attracted increasing attention in recent years. Abbas 5studied the dynamics of rotating flexible beams fixed on a flexible foundation, and usingFEM analyzed the effects of rotation speed and flexible foundation on the static buckling loadand region of vibration instability. Young and Lin 6 investigated the parametrically excitedvibration of beams with random rotary speed. Sinha et al. 7 analyzed the dynamic stabilityand control of rotating flexible beams with different damping coefficients and boundaryconditions. Chung et al. 8 studied the dynamic stability of a fixed-shaft cantilever beamwith periodically harmonic swing under different swing frequencies and speeds. Turhanand Bulut 9 studied the vibration characteristics of a rotating flexible beam with a centralrigid body under periodically variable speeds, and simulated the dynamic stability of thesystem under different movement parameters. Nonlinear vibration of a variable speedrotating beam has been analyzed by Younesian and Esmailzadeh 10. They investigatedthe parameter sensitivity and the effect of different parameters including the hub radius,structural damping, acceleration, and the deceleration rates on the vibration amplitude.

Electrorheological ER materials are a kind of smart material whose physicalproperties can be instantaneously and reversibly controlled with the application of an electricfield. These unusual properties enable ER materials to be employed in numerous potentialengineering applications, such as shock absorbers, clutch/brake systems, valves andadaptive structures. One of the most commonly studied ER structures is the ER sandwichbeam, in which an ER material layer is sandwiched between two containing surface layers11. These sandwich structures have the adaptive control capability of varying the dampingand stiffness of the beam by changing the strength of the applied electric field. Since Gandhiet al. 12 first proposed the application of ER fluids to adaptive structures, much has beenachieved in the vibration control of beams 1315. More recently, the dynamic stabilityproblems of ER sandwich beams have attracted some attention. Yeh et al. 16 studied thedynamic stability problem of an ER sandwich beam subjected to an axial dynamic force.They found that the ER core had a significant effect on the dynamic stability regions. Yehand Shih 17 investigated the critical load, parametric instability, and dynamic response ofa simply supported ER adaptive beam subjected to periodic axial force. However, researchinto the application of ER materials to vibration control of rotating motion beams is rare.In our previous work 18, the feasibility of applying ER fluids to the vibration control ofrotating flexible beams was discussed. Results demonstrated that the vibration of the beamcaused by the rotating motion at different rotation speeds and acceleration could be quicklysuppressed by applying electric fields to the ER material layer. When the angular velocity ofthe rotating ER sandwich beam is variable, the rotating beam would suffer from parametricinstability at some critical movement parameters. In order to successfully apply ER materialsto the vibration control of rotating beams and optimize the control effects, it is needed to


investigate the nonlinear dynamic characteristics and vibration stabilities of the rotating ERsandwich beam.

In this paper, the dynamic characteristics and parametric instability of a rotating ERsandwich beam with rotary oscillation is investigated. Assuming the ER sandwich beamto rotate around a fixed axis with time-varying harmonic periodic motion, the rotating ERsandwich beam is regarded as a parametrically excited system. Based on Hamiltons principleand finite element method FEM, the governing equations of the rotating beam are obtained.The multiple scales method is employed to determine the regions of instability for simpleand combination resonances. The effects of electric field on the natural frequency, loss factor,and regions of parametric instability are investigated. The results of the stability analysis areverified by investigating the time responses of the ER sandwich beam.

2. Properties of ER Fluids

ER fluids behave as Newtonian fluids in the absence of an electric field. On applicationof an electric field, their physical appearance changes to resemble a solid gel. However,their rheological response changes before and after the yield point. Due to this differencein rheological behavior before and after the yield point, the rheology of ER fluids isapproximately modeled in pre-yield and post-yield regimes Figure 1. The pre-yield regimecan be modeled by a linear viscoelastic model, and the post-yield regime be modeled by theBingham plastic model.

Existing studies 16, 17, 19, 20 demonstrate that the ER materials behave as linearvisco-elastic properties when they are filled in a sandwich beam configuration. So the shearstress is related to the shear strain by the complex shear modulus G,

G. 2.1

The complex shear modulus G is a function of the electric field strength applied onthe ER fluids, and can be written in the form

G G1 G2i G1(1 i

), 2.2

where G1 is the storage modulus, G2 is the loss modulus, G2/G1 is the loss factor, andi

1. So sandwich beams filled with ER fluids behave like visco-elastic damping beamswith controllable shear modulus.

3. Finite Element Modeling of Rotating ER Sandwich Beams

Because ER materials exhibit linear shear behavior at small strain levels similar to manyvisco-elastic damping materials, it is found that the models developed for the viscoelasticallydamped structures were potentially applicable to ER materials beams 20. So in the presentstudy, the finite element model for a rotating beam with a constrained damping layer 21, 22is adopted to model the rotating ER sandwich beam.

3.1. Basic Kinematic Relationships of the Rotating Beam

The structure of an ER sandwich beam is shown in Figure 2. The ER material layer is sand-wiched between two elastic surface layers. The beam with a length L and width b rotates in ahorizontal plane at an angular velocity about the axis Y .


Pre-yield Post-yield

Increasing electric field

Shear strain

Shea

r st

ress

G

Figure 1: The shear stress-shear strain relationship of ER fluids.

Elastic layer

Elastic layer

ER layer

Z

X

z

r

O

t

L

A

A

x

z y A-A

b

h1

h2

h3

Figure 2: Rotating sandwich beam filled with ER fluid core.

It is assumed that no slipping occurs at the interface between the elastic layer andthe ER fluid layer, and the transverse displacement w in a section does not vary along thebeams thickness. From the geometry of the deflected beam Figure 3, the shear strain andlongitudinal deflection u2 of the ER fluid layer can be expressed as 11

u1 u3

h2

h

h2w,x,

u2 u1 u3

2h1 h3

4w,x,

3.1

with

h h2 h1 h3

2, 3.2

where uk k 1, 2, 3 are the longitudinal displacements of the mid-plane of the kth layer; wis the transverse displacements of the beam, and subscript , x denotes partial differentiation


Elastic layer

Elastic layer

ER layer

z, w

u1

u2

u3

w

x, u

w/x

Figure 3: Kinematic relationships of deflected beam.

with respect to coordinate x; hk k 1, 2, 3 is the thickness of the kth layer; and k 1, 2, 3denote the upper face layer, the ER core layer, and the lower face layer, respectively.

3.2. Governing Equations

The kinetic energy for the rotating ER sandwich beam can be expressed as

T 12

L

0

3

k1

kAku2kdx

12

L

0

3

k1

kAkw2dx, 3.3

where k and Ak k 1, 2, 3 are the density and cross-section area of the kth layer; L is thelength of beam.

Assuming the shear strains in the elastic surface layers as well as the longitudinal andtransverse stresses in the ER fluid layer are negligible, the strain energy of the system can beexpressed as

U1 12

L

0

(E1A1u

21,x

)dx

12

L

0

(E3A3u

23,x

)dx

12

L

0E1I1 E3I3w2,xxdx

12

L

0GA22dx,

3.4

where Ek k 1, 3 is the Youngs modulus of the upper and lower surface layers, respec-tively; Ik k 1, 3 is the moment of inertia of the upper and lower surface layers, respectively;G G1 G2i is the complex shear modulus of the ER fluid; is the shear strain of the ERmaterial layer.

The potential energies attributable to centrifugal forces are written as 21, 22

U2 12

L

0Px, tw2,xdx 3.5

with

Px, t Att2[rL x 1

2

(L2 x2

)], 3.6


where is the rotating speed, At is the cross-section of the sandwich beam, t is the density ofthe system, x is the distance from the fixed end of the beam to any section on which centrifugalforces are acting, r is the hub radius.

The work done by external forces is exerted by the rotational torque and the externaldistributed force acting on the beam. In this study, only the transverse load q is considered.The total work by the external forces can be expressed as

W L

0qbwdx. 3.7

The governing equations of the rotating ER sandwich beam are obtained by applyingHamiltons principle

t2

t1

T U Wdt 0. 3.8

3.3. Finite Element Discretization

The finite element method FEM is used to discretize the rotating ER sandwich beam in thisstudy. The elemental model presented here consists of two nodes, each of which has fourdegrees of freedom. Nodal displacements are given by

qi {u1j u3j wj wj,x u1k u3k wk wk,x

}T, 3.9

where j and k are elemental node numbers, and u1, u3, w, w,x denote the longitudinal dis-placement of upper layer and lower layer, the transverse displacement, and the rotationalangle, respectively.

The deflection vector {u1 u2 u3 w w,x} can be expressed in terms of the nodal deflec-tion vector qi and finite element shape functions

{u1 u2 u3 w w,x

}{N1 N2 N3 N4 N4,x

}Tqi, 3.10

where N1, N2, N3, and N4 are the finite element shape functions and are given by

N1 [1 0 0 0 0 0 0],

N2 12

(N1 N3

h1 h32

N4,x),

N3 [0 1 0 0 0 0 0],

N4 [0 0 1 32 23 ( 22 3)Li 0 0 32 23

(2 3)Li],

3.11

with x/Li and Li is the length of the element.Substituting 3.10 into 3.33.7 and Hamiltons principle 3.8, the element equa-

tions of the rotating sandwich beam can be obtained as follows

Meqe 2Ceqe [Ke1

2(Ke2 Me) Ce

]qe Fe, 3.12


where Me, Ce, Ke1, and Ke2 are the element mass, the element gyroscopic, the element stiffness,

and the element motion-induced stiffness matrices of the rotating beam, respectively; Fe is theelement load vector. These element matrices and vector may be expressed as

Me Li

0

3

k1

[kAk

(NTkNk N

T4N4

)]dx,

Ce Li

0

[3

k1

kAk(NTkN4 NkNT4

)]

dx,

Ke1 Li

0

3

k1

(EkAkNTk,xNk,x EkIkN

T4,xxN4,xx

)dx

Li

0

GA2h22

N1 N3 hN4,xT N1 N3 hN4,xdx,

Ke2 12

Li

0

(3

k1

kAk

)[L2 xi x2

]NT4,xN4,xdx

rLi

0

(3

k1

kAk

)

L xi xNT4,xN4,xdx,

Fe Li

0

{3

k1

[kAk

2r xi xNTk]

3

k1

[kAkr xi xNT4

]}

dx n

i1

Fqi.

3.13

Assembling each element, the global equation of the rotating ER sandwich beam is

Mq 2Cq [K1 2K2 M C

]q F, 3.14

where M is the global mass matrix; C is the global gyroscopic matrices; K1 is the globalstiffness matrices, which is complex due to the complex shear modulus G of the ER material;K2 is the global motion-induced stiffness matrices, and F is the global load vector.

Since the first longitudinal natural frequency of a beam is far separated from thefirst transverse natural frequency, the gyroscopic coupling terms in 3.14 could be assumednegligible and ignored 23. With this assumption, 3.14 can be simplified as

Mq [K1 2K2 M

]q F. 3.15

It is assumed that the ER sandwich beam rotates around a fixed axis for a sinusoidalperiodic swing and the speed is

0 sin t, 3.16

where 0 is the maximum angular speed of the rotating beam and is the frequency of theswing. Substituting 3.16 into 3.15, the dynamic equation for the rotating ER sandwichbeam without applied external forces can be obtained as

Mq K1q (0 sin t

)2K2 Mq 0. 3.17


Assume is the normalized modal matrix of M1K1, 3.17 can be transformed to thefollowing N coupled Mathieu equations ifa linear transformation q is introduced andonly the homogeneous part of the equation:

i 2i i 20 sin

2tN

k1

hikk 0, 3.18

where 2i are the eigenvalues of M1K1 and hik are the elements of the complex matrix H

1M1K2 M. i and hik are written asi i,R ii,I , hik hik,R ihik,I , i

1. 3.19

4. Stability Analysis

Equation 3.18 represents a typical parametrically excited system because the last term onits left-hand side is a periodic function of time. When the system parameters reach specialresonance conditions, the rotating beam will suffer divergence instability 24. The deter-mination problem of these conditions is called dynamic stability analysis. In this section,stability of the solutions of 3.18 will be studied by multiscale method.

It is assumed that the dimensional maximum angular speed of the ER rotating beamcan be expressed as a function of a small value < 1:

20 4. 4.1

Based on the multi-scale method, the solution for 3.18 can be written as

it, i0T0, T1, . . . i1T0, T1, . . . , i 1 n, 4.2

where i0 and i1 represent the displacement function of fast and slow scales, respectively;T0 t is the fast time scale; and T1 t is the slow time scale.

Substituting 4.1 and 4.2 into 3.18, and comparing the same-order exponent, weobtain

0 : D20i0 2i i0 0, 4.3

1 : D20i1 2i i1 2D0D1i0

(e2iT0 e2iT0 2

) n

k1

hikk0, 4.4

where Dn /Tn n 0, 1, hik is the uniterm at row i and column k in matrix H. It should benoted that the effective excitation frequency is 2 in 4.4, which is originated from sin2tof 3.18. This is different from the equation of motion for an axially oscillating cantileverbeam, in which has sin t instead of sin2t 9.

Using the first order approximation, the general solution of 4.3 can be expressed inthe form

i0 AiT1, T2eiiT0 AiT1, T2eiiT0 , 4.5

where AiT1, T2 is the complex function of slow time scale, and AiT1, T2 denotes the com-plex conjugate of AiT1, T2.


The solution of 4.5 is substituted into 4.4 to obtain

D20i1 2i i1 2iiD1AieiiT0

n

k1

hikAk[eik2T0 eik2T0 2eikT0

] cc,

4.6

where cc represents the complex conjugate of all previous items. The complex functions Aishould be chosen to satisfy the conditions that i1 is bounded. If the terms on the right-hand side of 4.6 have the excitation frequency i, resonance occurs because the excitationfrequency coincides with the natural frequency. These terms, called the secular terms, shouldbe eliminated from 4.6. So the frequency of perturbation 2 needs to be checked for itsnearness to the individual natural frequency as well as their combinations. To this order ofapproximation, there are three main categories of simple and combination resonances 25.Their respective dynamic stability behaviours will be analyzed below.

a Combination Resonance of Sum Type

If the variation frequency 2 approaches the sum of any two natural frequencies of the sys-tem, summation parametric resonance may occur. The nearness of 2 to p,R q,R can beexpressed by introducing a detuning parameters defined by

12(p,R q,R

)

12, 4.7

where p,R and q,R are, respectively, the pth and qth natural frequency of the ER rotatingbeam.

Substituting 4.7 into 4.6, the condition required to eliminate secular terms in 4.6can be obtained as

2ipD1Ap 2hppAp hpqAqeiT1p,Iq,IT0 0,2iqD1Aq 2hqqAq hqpAqeiT1p,Iq,IT0 0,

4.8

where Aq and Aq are the complex conjugates of Ap and Aq, respectively. It should beremarked that the i and hik i p, q in 4.8 are complex due to the complex shear modulusG of the ER materials layer, which are shown in 3.19.

From the condition that nontrivial solutions of 4.8 should be bounded, the bound-aries of the unstable regions in this case are given by 9, 22

12(p,R q,R

) (p,I q,I

)

4(p,Iq,I

)1/2

012

(hpq,Rh

qp,R h

pq,Ih

qp,I

)

p,Rq,R 16p,Iq,I

1/2

,

4.9

where, p,R and q,I are, respectively, the real and imaginary components of the systemscomplex eigenvalues; and hij,R and h

ij,I respectively represent the real and imaginary compo-

nents of hij .


When p q, 4.9 can be simplified into the critical condition for instability of order nharmonic resonance,

2p,R 12

012[(

hpp,R)2

(hpp,I

)2]

(p,R

)2 16(p,I

)2

1/2

. 4.10

b Combination Resonance of Difference Type

When the excitation frequency 2 varies around the difference between the natural frequen-cies at orders p and q, this phenomenon is called the combination resonance of differencetype. Its boundary condition of instability can be obtained by changing the sign of i in thesituation above. The boundary curve of the corresponding stability and instability curves is

p,R q,R (p,I q,I

)

4(p,Iq,I

)1/2

012

(hpq,Rh

qp,R hpq,Ihqp,I

)

p,Rq,R 16p,Iq,I

1/2

.

4.11

c No-Resonance Case

Consider the case that the excitation frequency 2 is far away from p,Rq,R for all possiblepositive integer values of p and q. In this case, the condition required to eliminate the secularterms in 4.6 is

D1Ai 0, i 1 n. 4.12

So the particular solution of 4.6 is

i1 0n

k1

hikAk

[eik2T0

k 22 2i

eik2T0

k 22 2i

]

20N

k1, k / i

hikAkeikT0

2k2i

cc.

4.13

Because there does not exist the case where 2 is simultaneously near p,R q,Rand p,Rq,R, there is no unstable solution for 4.7. Hence the system is said to be alwaysstable when 2 is away from p,R q,R.

5. Numerical Simulation and Discussion

To validate the reliability of the calculation methods in this paper, we first assumed that theangular speed of the rotating ER sandwich beam 0 and regarded it as a static cantileverbeam. The structural and material parameters of the beam in 19 were used to calculatethe natural frequencies and modal loss factors for the first five orders when the electric fieldintensity E 3.5 kV/mm. The results are shown in Table 1. We can see from the table thatalthough the natural frequencies at each order obtained through the method in this paper


Table 1: Comparison of natural frequencies and loss factors obtained herein with those of 19 L 381 mm, b 25.4 mm, h1 h3 0.79 mm, h2 0.5 mm, E 3.5 kV/mm, G2 6125001 0.011i.

Mode Natural frequency f Hz Loss factor Present Ref. Present Ref.

1 10.011 10.005 0.00393 0.003952 40.091 40.051 0.00507 0.005123 89.125 89.028 0.00459 0.004614 152.926 152.702 0.00336 0.003395 236.396 235.761 0.00244 0.00250

0 0.5 1 1.5 2 2.5 3

Electric fields (kV/mm)

Com

plex

she

ar m

odul

us (P

a)

The shear storage modulus

The loss modulus

102

103

104

105

106

Figure 4: Complex shear modulus of ER fluids at different electric fields.

are slightly higher than that obtained from the Mead-Markus modeling method in 19, thedifference is minimal. The loss factors obtained through the two methods are basically thesame. We also used the geometric and material parameters of rotating beams at the activerestraint damping layer in 26 to calculate the natural frequencies and modal damping ratiofor the first two orders under different rotation speeds Table 2 and found that the resultobtained from the method in this paper is almost the same as that obtained from 26.

The effects of an electric field on the dynamic characteristics and parametric instabilityof the rotating ER sandwich beam were studied. The sandwich beam was constructed withan ER material core and two elastic faces made of aluminum. The material properties andgeometrical parameters are shown in Table 3. The ER materials used in this study are sameas those described by Don 27. Its density is 1200 kg/m2, and complex modulus can beexpressed as

G 50000E2 2600E 1700i, 5.1

where E is the electric fields in kV/mm. The shear storage modulus G1 and the loss modulusG2 are shown in Figure 4.

The dynamic characteristics of the rotating sandwich beam with an ER core wereinvestigated first. Let the angular velocity of the rotating beam 0. Then the naturalfrequencies and damping loss factors can be obtained by the eigenvalue equations

{[K1 2K2 M

] 2M

}{} 0, 5.2


Table 2: Comparison of a rotating beam for natural frequencies and modal damping ratio obtained hereinwith those of 26 L 300 mm, b 12.7 mm, h1 0.762 mm, h3 2.286 mm, h2 0.25 mm, G2 26150010.38i.

Angular velocity r.p.mNatural frequency Modal damping ratio

f1 Hz f2 Hz 1 2

0 Ref. 20.15 104.0 0.0382 0.0235Present 20.14 103.9 0.0384 0.0233

600 Ref. 20.58 106.8 0.0365 0.0220Present 20.53 106.6 0.0366 0.0222

1000 Ref. 21.20 111.2 0.0340 0.0201Present 21.17 111.1 0.0340 0.0204

Table 3: Parameters of ER sandwich beam.

Parameters of beam geometry L 300 mm, b 20 mm, h1 h3 0.5 mm, h2 2 mmElastic layer properties Al 1 3 2700 kg/m3, E1 E3 70 GpaER fluid properties 27 2 1200 kg/m3, G 50000E2 2600E 1700i

where is the complex frequency rad/s and {} is the corresponding eigenvector. Thecomplex eigenvalue {}2 is expressed as

2 2(1 i

), 5.3

where is the damping loss factor and is the natural frequency.Comparisons of the natural frequencies and loss factors of ER sandwich beams with

different rotating speed are shown in Figures 5 and 6, respectively. Figure 5 shows the effectsof electric field strength on the first three natural frequencies. It is observed that the incrementof the electric field strength increases the natural frequencies of the ER sandwich beam atdifferent rotation speeds. Thus, the stiffness of the rotating beam increases with the strengthof the applied electric field. Figure 6 illustrates the effect of electric field strength on theloss factors. At all rotation speeds, the loss factor first increases as the electric field strengthincreases. But the loss factor declines with the strength of the electric field when the electricfield strength exceeds 0.5 kV/mm. This trend is very obvious in lower modes and less evidentin higher modes. Figures 5 and 6 also demonstrate that the natural frequency increases andthe loss factor decreases with an increase in rotating speed. That is because the stiffness of therotating ER beam increases with rotating speed, whereas its damping decreases with rotatingspeed. Thus the natural frequencies and loss factors of the rotating ER beam can be alteredby varying the strength of the applied electric field.

The multiple scale method was used to obtain the parametric instability region of therotating ER sandwich beam with periodically variable angular velocity. The effects of electricfield strength on the region of parametric instability are shown in Figure 7. Figures 7aand 7b illustrate the instability regions for the first and second order parametricallyexcited resonance, respectively, and Figures 7c and 7d are the instability regions forparametrically excited combination resonance of sum and difference types. It is noted thatincreasing the electric field strength will increase the excitation frequency so that the unstableregions shift to the right. The critical maximal rotating speed i.e., the maximal rotating speedwhen parametric instability occurs increases and the width of unstable region decreases withan increase in the strength of the electric field. Thus increasing the strength of the applied


0 0.5 1 1.5 2 2.5 30

20

40

60

80

100

120

Nat

ural

freq

uenc

y

(Hz)

Electric field strength E (kV/mm)

123

a Rotating speed 0 r.p.m.

0 0.5 1 1.5 2 2.5 30

20

40

60

80

100

120

Nat

ural

freq

uenc

y

(Hz)


123

b Rotating speed 200 r.p.m.

0 0.5 1 1.5 2 2.5 30

20

40

60

80

100

120

Nat

ural

freq

uenc

y

(Hz)


123

c Rotating speed 400 r.p.m.

0 0.5 1 1.5 2 2.5 30

20

40

60

80

100

120N

atur

al fr

eque

ncy

(Hz)


123

d Rotating speed 600 r.p.m.

Figure 5: Effect of strength of electric field on the first three natural frequencies at different rotation speeds.

electric field not only moves the region of instability to a higher frequency, but also reducesthe width of the region. That is, increasing the electric field strength will increase the stabilityof the beam.

The results of the stability analysis can be verified by investigating the time responsesfor points A and B in Figure 7a. The time responses for the transverse displacement arecomputed at the free end of the ER sandwich beam by 3.18 using the fourth-order Runge-Kutta method. The co-ordinates of points A and B in Figure 7a are 0.8, 3 and 1, 3. Asshown in Figure 7a, point A is in the stable region and point B is in the unstable regionwithout an applied electric field, whereas points A and B are both in the stable region whenthe electric field strength E 0.5 kV/mm.

Comparisons of the time responses of points A and B without electric field are shownin Figure 8. The time response for point A, as shown in Figure 8a, is bounded by a limitedvalue. However, for point B, which is within the unstable region, the amplitude of the time


0 0.5 1 1.5 2 2.5 30

0.02

0.04

0.06

0.08

Los

s fa

ctor


123

a Rotating speed 0 r.p.m.

0 0.5 1 1.5 2 2.5 30

0.02

0.04

0.06

0.08

Los

s fa

ctor


123

b Rotating speed 200 r.p.m.

0 0.5 1 1.5 2 2.5 30

0.02

0.04

0.06

0.08

Los

s fa

ctor


123

c Rotating speed 400 r.p.m.

0 0.5 1 1.5 2 2.5 30

0.02

0.04

0.06

0.08L

oss

fact

or


123

d Rotating speed 600 r.p.m.

Figure 6: Effect of strength of electric field on the first three loss factors at different rotation speeds.

response increases with time, as illustrated in Figure 8b. Figure 9 shows the time responsesfor points A and B when the electric field strength E 0.5 kV/mm. It is demonstratedthat points A and B are both stable because the time responses are bounded. Therefore, itis verified that the stability results of Figure 7a agree well with the behavior of the timeresponses in Figures 8 and 9.

6. Conclusion

The dynamic characteristics and parametric instability of rotating ER sandwich beams witha periodically variable angular velocity were studied using FEM and a multi-scale method.The effects of electric field on the natural frequency, loss factor, and regions of parametricinstability were investigated. When the strength of the electric field is increased, the stiffnessof the ER sandwich beam increases at different rotation speeds and the instability region


0 0.5 1 1.5 2 2.5 3 3.50

2

4

6

8

10

ab

c

de

A B

/

2 0

/(2w0)

a First-order excited resonance

/

2 0

4 6 8 10 120

2

4

6

8

10

d ea bc

/(2w0)

b Second-order excited resonance

/

2 0

1.5 2 2.5 3 3.50

2

4

6

8

10

a

b

c

/(2w0)

c Combination resonance of difference types

/

2 0

2.5 3 3.5 4 4.5 5 5.50

2

4

6

8

10

ab

c

/(2w0)

d Combination resonance of sum types

Figure 7: Instability boundaries for various applied electric fields: curve a, E 0 kV/mm; curve b, E 0.5 kV/mm; curve c, E 1.0 kV/mm; curve d, E 1.5 kV/mm; curve e, E 2 kV/mm.

0 2 4 6 820

10

0

10

20

Time (s)

Res

pons

e am

plit

ude(m

m)

a Time responses of point A in Figure 7a

0 2 4 6 820

10

0

10

20

Time (s)

Res

pons

e am

plit

ude(m

m)

b Time responses of point B in Figure 7a

Figure 8: Time responses of the transverse displacement at electric field E 0 kV/mm.


0 2 4 6 820

10

0

10

20

Time (s)

Res

pons

e am

plit

ude(m

m)

a Time responses of point A in Figure 7a

0 2 4 6 820

10

0

10

20

Time (s)

Res

pons

e am

plit

ude(m

m)

b Time responses of point B in Figure 7a

Figure 9: Time responses of the transverse displacement at electric field E 0.5 kV/mm.

of the rotating beam moves toward the high-frequency section. The unstable regions narrowwith an increase in the strength of the electric field, while the maximum critical angular speedrequired for the beam to have parametric instability increases as electric field increases. Hencethe vibration characteristics and dynamic stability of rotating ER sandwich beams can beadjusted when they are subjected to an electric field. It was demonstrated that the ER materiallayer can be used to improve the parametric instability of rotating flexible beams.

Acknowledgments

This work was supported by the National Natural Science Foundation of China 11172100and 51075138, and the Scientific Research Fund of Hunan Provincial Education Departmentof China 10A021.

References

1 J. Chung and H. H. Yoo, Dynamic analysis of a rotating cantilever beam by using the finite elementmethod, Journal of Sound and Vibration, vol. 249, no. 1, pp. 147164, 2002.

2 A. A. Al-Qaisia, Dynamics of a rotating beam with flexible root and flexible hub, Structural Engi-neering and Mechanics, vol. 30, no. 4, pp. 427444, 2008.

3 S. Y. Lee, S. M. Lin, and Y. S. Lin, Instability and vibration of a rotating Timoshenko beam withprecone, International Journal of Mechanical Sciences, vol. 51, no. 2, pp. 114121, 2009.

4 H. Arvin and F. Bakhtiari-Nejad, Non-linear modal analysis of a rotating beam, International Journalof Non-Linear Mechanics, vol. 46, no. 6, pp. 877897, 2011.

5 B. A. H. Abbas, Dynamic stability of a rotating Timoshenko beam with a flexible root, Journal ofSound and Vibration, vol. 108, no. 1, pp. 2532, 1986.

6 T. H. Young and T. M. Lin, Stability of rotating pretwisted, tapered beams with randomly varyingspeeds, Journal of Vibration and Acoustics, Transactions of the ASME, vol. 120, no. 3, pp. 784790, 1998.

7 S. C. Sinha, D. B. Marghitu, and D. Boghiu, Stability and control of a parametrically excited rotatingbeam, Journal of Dynamic Systems, Measurement and Control, Transactions of the ASME, vol. 120, no. 4,pp. 462469, 1998.

8 J. Chung, D. Jung, and H. H. Yoo, Stability analysis for the flapwise motion of a cantilever beam withrotary oscillation, Journal of Sound and Vibration, vol. 273, no. 4-5, pp. 10471062, 2004.

9 O. Turhan and G. Bulut, Dynamic stability of rotating blades beams eccentrically clamped to ashaft with fluctuating speed, Journal of Sound and Vibration, vol. 280, no. 35, pp. 945964, 2005.


10 D. Younesian and E. Esmailzadeh, Non-linear vibration of variable speed rotating viscoelasticbeams, Nonlinear Dynamics, vol. 60, no. 1-2, pp. 193205, 2010.

11 K. Wei, G. Meng, W. Zhang, and S. Zhou, Vibration characteristics of rotating sandwich beams filledwith electrorheological fluids, Journal of Intelligent Material Systems and Structures, vol. 18, no. 11, pp.11651173, 2007.

12 M. V. Gandhi, B. S. Thompson, and S. B. Choi, A new generation of innovative ultra-advancedintelligent composite materials featuring electro-rheological fluids: an experimental investigation,Journal of Composite Materials, vol. 23, no. 12, pp. 12321255, 1989.

13 C. Y. Lee and C. C. Cheng, Dynamic characteristics of sandwich beam with embedded electro-rheological fluid, Journal of Intelligent Material Systems and Structures, vol. 9, no. 1, pp. 6068, 1998.

14 T. Fukuda, T. Takawa, and K. Nakashima, Optimum vibration control of CFRP sandwich beam usingelectro-rheological fluids and piezoceramic actuators, Smart Materials and Structures, vol. 9, no. 1, pp.121125, 2000.

15 S. B. Choi, Electric field-dependent vibration characteristics of a plate featuring an electrorheologicalfluid, Journal of Sound and Vibration, vol. 234, no. 4, pp. 705712, 2000.

16 J. Y. Yeh, L. W. Chen, and C. C. Wang, Dynamic stability of a sandwich beam with a constrainedlayer and electrorheological fluid core, Composite Structures, vol. 64, no. 1, pp. 4754, 2004.

17 Z. F. Yeh and Y. S. Shih, Critical load, dynamic characteristics and parametric instability ofelectrorheological material-based adaptive beams, Computers and Structures, vol. 83, no. 25-26, pp.21622174, 2005.

18 K. X. Wei, G. Meng, S. Zhou, and J. Liu, Vibration control of variable speed/acceleration rotatingbeams using smart materials, Journal of Sound and Vibration, vol. 298, no. 4-5, pp. 11501158, 2006.

19 M. Yalcintas and J. P. Coulter, Electrorheological material based adaptive beams subjected to variousboundary conditions, Journal of Intelligent Material Systems and Structures, vol. 6, no. 5, pp. 700717,1995.

20 M. Yalcintas and J. P. Coulter, Electrorheological material based non-homogeneous adaptive beams,Smart Materials and Structures, vol. 7, no. 1, pp. 128143, 1998.

21 E. H. K. Fung and D. T. W. Yau, Vibration characteristics of a rotating flexible arm with ACLDtreatment, Journal of Sound and Vibration, vol. 269, no. 1-2, pp. 165182, 2004.

22 H. Saito and K. Otomi, Parametric response of viscoelastically supported beams, Journal of Soundand Vibration, vol. 63, no. 2, pp. 169178, 1979.

23 H. H. Yoo and S. H. Shin, Vibration analysis of rotating cantilever beams, Journal of Sound andVibration, vol. 212, no. 5, pp. 807808, 1998.

24 H. Y. Hu, Applied Nonlinear Dynamics, Press of Aeronautical Industries, Beijing, China, 2000.25 T. H. Tan, H. P. Lee, and G. S. B. Leng, Parametric instability of spinning pretwisted beams subjected

to spin speed perturbation, Computer Methods in Applied Mechanics and Engineering, vol. 148, no. 1-2,pp. 139163, 1997.

26 C. Y. Lin and L. W. Chen, Dynamic stability of a rotating beam with a constrained damping layer,Journal of Sound and Vibration, vol. 267, no. 2, pp. 209225, 2003.

27 D. L. Don, An investigation of electrorheological material adaptive structure [M.S. thesis], Lehigh University,Bethlehem, Pa, USA, 1993.

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2012, Article ID 872161, 9 pagesdoi:10.1155/2012/872161

Research ArticleNonlinear Periodic Oscillation of a CylindricalMicrovoid Centered at an Isotropic IncompressibleOgden Cylinder

Wenzheng Zhang,1 Xuegang Yuan,1, 2 and Hongwu Zhang1

1 State Key Laboratory of Structural Analysis for Industrial Equipment,Department of Engineering Mechanics, Faculty of Vehicle Engineering and Mechanics,Dalian University of Technology, Dalian 116024, China

2 School of Science, Dalian Nationalities University, Dalian 116600, China

Correspondence should be addressed to Xuegang Yuan, [email protected]

Received 31 August 2012; Accepted 5 December 2012

Academic Editor: Kale Oyedeji

Copyright q 2012 Wenzheng Zhang et al. This is an open access article distributed under theCreative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.

We study the dynamic mathematical model for an infinitely long cylinder composed of an isotropicincompressible Ogden material with a microvoid at its center, where the outer surface of thecylinder is subjected to a uniform radial tensile load. Using the incompressibility condition andthe boundary conditions, we obtain a second-order nonlinear ordinary differential equation thatdescribes the motion of the microvoid with time. Qualitatively, we find that this equation hastwo types of solutions. One is a classical nonlinear periodic solution which describes that themotion of the microvoid is a nonlinear periodic oscillation; the other is a blow-up solution.Significantly, for the isotropic incompressible Ogden material, there exist some special values ofmaterial parameters, the phase diagrams of the motion equation have homoclinic orbits, whichmeans that the amplitude of a nonlinear periodic oscillation increases discontinuously with theincreasing load.

1. Introduction

Cylindrical structures are very common used in social productions and human lives. Theresearches on the dynamic oscillation problems of such structures composed of hyperelasticmaterials are of important significance. As is well known, such problems can be formulatedas initial boundary value problems of nonlinear evolution equations. Knowles 1 firstlystudied the free radial oscillation of an incompressible cylindrical tube composed of anisotropic Mooney-Rivlin material; in the limiting case of a thin walled cylindrical tube,the equation reduces to the Ermakov-Pinney equation. Then, Shahinpoor and Nowinski


2 and Rogers and Baker 3 used the nonlinear superposition principle for the Ermakov-Pinney equation to derive solutions. The works appeared in this area have been reviewedby Rogers and Ames 4. In 2007, Mason and Maluleke 5 introduced the Lie pointsymmetry into this area and investigated the nonlinear radial oscillations of a transverselyisotropic incompressible cylindrical tube subjected to time dependent net applied surfacepressures; moreover, they proved that for radial and tangential transversely isotropic tubesthe differential equations may be reduced to the Abel equations of the second kind. Inaddition, with the development of the mathematical theory, Yuan et al. 6 investigatedthe dynamic inflation problems for infinitely long cylindrical tubes composed of a class oftransversely isotropic incompressible Ogden materials from the equation itself and discussedthe influences of material parameters, structure parameter and applied pressures on thedynamic behaviors of the tubes in detail. Ren 7 studied the dynamical responses, suchas motion and destruction of hyperelastic cylindrical shells subjected to dynamic loads onthe inner surface. Other references on the dynamic responses for hyperelastic cylindricalstructures may be found in Dai and Kong 8, Yuan et al. 9, and so on.

The purpose of this paper is to investigate the nonlinear periodic oscillation of acylindrical microvoid centered at an infinitely long cylinder, where the cylinder is composedof an isotropic incompressible Ogden material 10 and its outer surface is subjected toa uniform radial tensile load. In Section 2, the basic governing equations, the boundaryconditions and the initial conditions are presented. In Section 3, a second order nonlinearordinary differential equation describing the motion of the microvoid is obtained. Then,in Section 4, some nonlinear dynamic analyses of the equation are performed in detail.Meanwhile, some numerical examples are given.

2. Mathematical Model

The mathematical model examined in this paper is listed as follows.

(a) Basic Governing Equations

In the absence of body force, the equilibrium differential equation, the incompressibilitycondition and the strain-energy function associated with the known Ogden material are givenby

rr, tr

1rrr, t r, t 0

2rR, tt2

, 2.1

r 1, 2.2

W 11

[1r

1 2

]22

[2r

2 2

]. 2.3

(b) Boundary Conditions and Initial Conditions

The boundary conditions are given by

rrA, t 0, rrB, t p0[

B

rB

]. 2.4


The initial conditions are as follows:

rR, 0 R, rR, 0 0. 2.5

In 2.12.5, ir, t iW/i pr, t, i r, are the principal Cauchy stresses,pr, t is the hydrostatic pressure related to the incompressibility condition, r r/R, r/R are the radial and the circumference stretches, respectively, r rR, t, 0 < A R B isthe radial deformation function with time to be determined, and A and B are the radius of themicrovoid and the outer radius of the cylinder in the undeformed configuration, respectively.0 is the constant mass density of the material. i, i i 1, 2 are material parameters. Theboundary conditions in 2.4 mean that the surface of the microvoid is traction free and theouter surface of the cylinder is subjected to a uniform radial tensile load, denoted by p0. Theinitial conditions in 2.5 mean that the cylinder is in an undeformed state at time t 0.

3. Solutions

From the incompressibility condition 2.2 we find that

rR, t (R2 A2 r21t

)1/2, 3.1

where r1t is an undetermined radial motion function of the radius of the microvoid. From3.1, it is easy to know that the radial motion of the cylinder can be completely described byr1t.

From 3.1, it is not difficult to show that

2rR, tt2

r

(

r1r1 ln r r12 ln r 12r12

(r1r

)2)

. 3.2

Substituting 3.2 into 2.1, integrating it with respect to r from rA, t to rB, t andusing the traction boundary conditions in 2.4, we obtain

120r1r1 ln

(B2 A2 r21

r21

)

120r

21 ln

(B2 A2 r21

r21

)

120r

21

(A2 B2

B2 A2 r21

)

p0

B(B2 A2 r21

)1/2

rB,t

rA,t

[r

W

r W

]dr

r 0,

3.3

where rA, t A2 A2 r21t1/2 r1t, rB, t B2 A2 r21t

1/2.Obviously, 3.3 is a second order nonlinear ordinary differential equation with respect

to r1t. Next we study the qualitative properties of solutions of 3.3.


For convenience, we introduce the following dimensionless notations:

k (R

r

)1

(

1 r21 A2r2

)1/2, xt

r1tB

,

xt r1tB

,

(B2 A2)1/2

B.

3.4

Thus, the initial conditions in 2.5 reduce to

x0 (

1 2)1/2

, x0 0, 3.5

and 3.3 may be rewritten as

120B

2xx ln

(2 x2

x2

)

120B

2x2 ln

(2 x2

x2

)

120B

2x2(

2

2 x2

)

p0(

12 x2

)1/2 Fx, 0,

3.6

where

F(x, , 1, 2, 1, 2

) 2x21/2

x/121/21k

1 2k2 1k1 2k2kk2 1 dk. 3.7

4. Nonlinear Dynamic Analyses

Multiplying 3.6 by xx, we obtain the following first integral:

Ux, x2 V(x, p0,

) 0, 4.1

where

Ux, 140B

2x2 ln

(2 x2

x2

)

,

V(x, p0,

)x

121/2

(

Fs, p0(

12 s2

)1/2)

s ds.

4.2


0 1 2 3 4 50

0.5

1

1.5

2

2.5

31 = 1, 2 = 1

1 = 1, 2 = 0.5

1 = 0.5, 2 = 0.5

x

P

Pm

Pha/ = 2.95757 = 2, = 0.9999

Figure 1: Curves of P x for 0 < 1, 2 1.

Attentively, Ux, > 0 is valid for any x > 1 21/2, which means that 4.1 has realsolutions only when V x, p0, < 0. However, V x, p0, < 0 is equivalent to V x, p0, min xm.

ii For the case that 1 2 1, we have limxGx, , 1, 1, 111 21/2,which means 4.3 has a horizontal asymptote, written as Pha 1 1 1 21/2. For the given values of and , curves of P versus x are shown inFigure 1 for 0 < 1, 2 1.


0 1 2 30

10

20

30

50

40

x

P

= 2, = 0.9999

1 = 2.1, 2 = 2.5

1 = 2, 2 = 2

Figure 2: Curves of P x for 1, 2 > 1.

0

1

2

3

4

5

6

7

0 1 2 3 4 5 6 7

x

P

= 12 = 11.5

c = 10.863

= 4

= 0 = 0.9999, 1 = 1.5, 2 = 0.5

= 8

a

5

5.5

6

6.5

7

7.5

P

= 12

= 11.5

c = 10.863

= 8

= 0.9999, 1 = 1.5, 2 = 0.5

P1P2

0 1 2 3 4 5 6 7

x

b

Figure 3: Curves of P x for 1 > 1, 0 < 2 < 1.

iii If 1, 2 > 1, we have limxGx, , 1, 2, . P increases strictly with theincreasing x. Particularly, if 1 2 2, it is easy to prove that there is anotherasymptote, written as Pax 1/2 ln12x. For the given values of and, Figure 2 shows the relationships of P versus x for 1, 2 > 1.

iv For the case that 1 > 1, 0 < 2 < 1 or 0 < 1 < 1, 2 > 1 here, we only discuss thecase that 1 > 1, 0 < 2 < 1, it can be proved that there exists a critical value of ,written as c, such that P increases monotonically if 0 < c and has a local max-imum and a local minimum if > c, written as P1 and P2, respectively. Curves ofP versus x are given in Figure 3 for 1 > 1, 0 < 2 < 1 and for the given values of .


0 1 2 3 4 5 6 7

x

P

6

6.2

6.4

6.6

6.8

7

7.2

7.4

7.6

7.8

= 0.999

= 0.9999

= 0.99999

= 12, 1 = 1.5, 2 = 0.5

Figure 4: Curves of P x for different values of .

4.1.2. Influence of Structure Parameter

Once the values of 1, 2, and are given, the influence of on the relationships of P versusx is shown in Figure 4.

4.2. Number of Equilibrium Points

1 For the case that 0 < 1, 2 < 1, it can be seen from Figure 1 that there are twodifferent roots of 4.3 as 0 < P < Pm, written as x1 and x2 x1 < x2. It means that3.6 has two equilibrium points x1, 0 and x2, 0; moreover, x1, 0 is a center andx2, 0 is a saddle point.

2 If 1 2 1, 3.6 has a unique equilibrium point as 0 P < Pha, written as x3, 0;moreover, x3, 0 is a center. While P > Pha, 3.6 has no equilibrium point.

3 For the case that 1, 2 > 1, 3.6 has only one equilibrium point as shown inFigure 2, written as x4, 0, and it is also a center.

4 If 1 > 1, 0 < 2 < 1, from the above analyses we know that P has a local maximumP1 and a local minimum P2 as > c, where c is a critical value of . Equation3.6 has a unique equilibrium point as P > P1 or P < P2, written as x5, 0, andit is a center. Equation 3.6 has exactly three equilibrium points as P1 > P > P2,written as x6, 0, x7, 0, and x8, 0, respectively, where x6 < x7 < x8; moreover,x6, 0 and x8, 0 are centers and x7, 0 is a saddle point. For the given values of, 1, 2, and , the phase diagrams of 4.1 are shown in Figure 5. It is foundthat the radial oscillation of the microvoid presents a nonlinear periodic oscillation;moreover, the amplitude of the oscillation increases gradually as P increases from 0to Pcr. However, the increase of the amplitude of the nonlinear periodic oscillationis discontinuous as P passes through Pcr. Another interesting phen

advances in nonlinear vibrationdownloads.hindawi.com/journals/specialissues/940489.pdf · advances...

Documents