research into 2d dynamics and control of small...

Research ArticleResearch into 2D Dynamics and Control of Small Oscillations ofa Cross-Beam during Transportation by Two Overhead Cranes

Alexander V Perig1 Alexander N Stadnik2

Alexander A Kostikov3 and Sergey V Podlesny2

1Manufacturing Processes and Automation Engineering Department Engineering Automation FacultyDonbass State Engineering Academy Shkadinova 72 Kramatorsk Donetsk Region 84313 Ukraine2Department of Technical Mechanics Engineering Automation Faculty Donbass State Engineering AcademyShkadinova 72 Kramatorsk Donetsk Region 84313 Ukraine3Informatics and Engineering Graphics Department Engineering Automation Faculty Donbass State Engineering AcademyShkadinova 72 Kramatorsk Donetsk Region 84313 Ukraine

Correspondence should be addressed to Alexander V Perig olexanderperiggmailcom

Received 13 December 2016 Accepted 5 January 2017 Published 15 February 2017

Academic Editor Tai Thai

Copyright copy 2017 Alexander V Perig 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

A new mathematical model of a 3DOF 2D mechanical system ldquotransported cross-beam two moving bridge cranesrdquo has beenproposed Small system oscillations have been derived through the introduction of Lagrange equations The numerical estimationof 3DOF systemmotion has been carried out with equation-basedModelica languageThe present article uses the Lagrangemethodand numerical and optimization methods realized with JModelicaorg and Optimica freeware The absolute swaying of the cross-beam with respect to the displacement of the two moving bridge cranes was estimated The phase portraits of the 3DOF systemfor linear and angular coordinates were presented An open loop optimal control problem was posed for the motion of the bridgecranes A ldquobang-bangrdquo control strategy was implemented for the derivation of an optimal control solution which enables the travelof two bridge cranes at a prescribed distance for minimum time and minimum swaying of a heavy cross-beamThe derived resultsof the numerical simulation can be easily practically realized by crane operators with good agreement with simple engineeringestimations The proposed control strategy enables synchronous motion of two bridge cranes with a cross-beam that practicallysolves the posed problem of unwanted excessive oscillations of a heavy cross-beam during transportation

1 Background and Introduction

Current civil engineering technologies often assume a mod-ular concept for the assembly of buildings and other largestructures Individual modules are manufactured at special-ized plants and plant departments with further transporta-tion of modular sections to a storage warehouse Furthertransportation routes of individual modules lead from thewarehouse to the erecting yard at the construction siteIndividual modules and modular sections are many timesvery heavy and bulky objects Construction plant opera-tion requires simultaneous usage of several cranes workingtogether at the various stages of transportation of these heavymodules [1 2]

An example of modular unit construction would be theerection of the tower of a wind-powered generator withseveral modular sections utilizing two boom cranes workingtogether This practical case of the two-crane assembly of awind machine tower requires 3D turning and tumbling ofcylindrical modular sections from a horizontal to a verticalposition In contrast to this example the present article(Figures 1 2 3 and 6) is focused on the solution of a problemof horizontal transportation of a finished heavy steel beamfrom a fabrication shop to a storage warehouse

A shipbuilding example of collaborative working of twoor more cranes would be the problem of transportation of aheavy bulk marine engine weighing in the range of 3ndash3000tons to the installation site

HindawiShock and VibrationVolume 2017 Article ID 9605657 21 pageshttpsdoiorg10115520179605657

2 Shock and Vibration

Figure 1 Typical plant photos of heavy cross-beam transportation with two overhead cranes

z

D

E

xOb

x1

x2

C1

C2

A

B

m1g

m3g

m2g

l1

l2

120572120574

120573

C3

(120574 = 1205740 + 120575)

a2

a2

a cos (1205740)

zC3

xC3

Figure 2 The computational scheme for transportation of heavy cross-beam DE by two bridge cranes 119860 and 119861Other examples in civil engineering include the lifting-

and-handling problems of modular building elements ofconstruction bays of a building ceiling and floor slab panelsand spandrel walls Construction problems of roofs floorsand domes of roofed stadiums and shopping centers requirepreliminary manufacturing of the heavy bulk trussed framemodules with further transportation to building sites

It is important to note that attempts to transport heavybulk stores with a single crane are very undesirable because ofthe high probability of torsional oscillations of a heavy loadMoreover usage of two cranes working together for heavy

load transportation is a truly natural way for handlingdisplacements in civil and mechanical engineering as natureitself has supplied humans with two hands and two legs formore efficient performance of lifting-and-handling opera-tions

Overhead crane dynamics has been studied in theresearch efforts of such scientists as Abdel-Rahman et al(2003) [1] Deen Ali et al (2005) [2] Arena et al (2015) [3]Cartmell et al (1998) [4] Castelli et al (2014) [5] Cha etal (2010) [6] Goodwin (1997) [7] Huang et al (2015) [8]Lahouar et al (2009) [9] Pigani and Gallina (2014) [10]

Shock and Vibration 3

z

B

b

O

D

E

C1 A

C2

x1

x2

120573

120572

m1g

m2g

l1

l2

C3

m3g

1205740 Δz0

Δz0 = l1 minus l2 + b

Δz0a

a2

a2

DE = a

AE = l1 BD = l2

sin (1205740) =

sin (1205740) =

(l1 minus l2 + b)

a

a cos (1205740)

Figure 3 The computational scheme of the initial position of mechanical system during transportation of heavy cross-beam DE by twobridge cranes 119860 and 119861Sawodny et al (2002) [11] Smoczek (2014) [12] Zi et al (2008)[13]

Abdel-Rahman et al (2003) have discussed mechanicaldesign schemes and mathematical models of gantry cranesand bidirectional gantry cranes with translational displace-ments of payloads [1] However payload transportation bytwo overhead cranes with cross-beam usage is not addressedin [1]

Arena et al (2015) have proposed several 3D kinematicand dynamic models of container cranes with constantdistances between fixed points of supporting rigid and elasticcables [3] However some disadvantage of Arenarsquos model isassociated with a prescribed imposition of geometric con-straints on the distances between fixed points for supportingcables A 2D kinematic model of the present article generali-zes Arenarsquos model in the vertical plane [3]

Castelli et al (2014) have studied a kinetostatic model ofa Cartesian cable-suspended robot [5] A kinematic model ofCastelli et al (2014) is focused on the transportation of a pay-load using double cables with variable lengths which providetranslational linear motion of the payload without obstaclesThere is an analogy between a swaying Cartesian cable-suspended robot (Castelli 2014) and the swaying cross-beamED in our problem However during transportation the pay-load may have additional extraneous transverse oscillationsMoreover successful payload transportation requires a syn-chronized change of lengths of both supporting cables

Cha et al (2010) have studied extra-heavy cargo lifting bytwo floating cranes with a barge ship connected between thetwo cranes [6]

Huang et al (2015) have proposed a new double-pendulum mechanical model describing the transportationof a large payload with distributed mass which is attached toone trolley by one suspension cable and two rigging cables [8]Huang et al (2015) have suppressed payload vibration duringtransportation by control of trolley acceleration [8] Mechan-ical disadvantage of Huangrsquos model is a usage of only one sus-pension cable However there are possible strong engineering

benefits of using two suspension cables without riggingcables which is beyond the scope of Huangrsquos research [8]

Kostikov et al (2016) and Perig et al (2014) have studiedcargo transportation by crane with one slewing pivot point([14ndash17])

Analysis of references [1ndash18] clearly shows that thedynamics of transportation of the cross-beam by two over-head cranes is not fully addressed in available works [1ndash18]This confirms the actuality relevance and industrial signifi-cance (Figure 1) of the present research

2 Research Actuality andPrime Novelty of Research

The aim of the present research is the dynamic descriptionof the transportation of a heavy cross-beam by two overheadcranes (Figures 1 2 3 and 6)

The objective of this research is a 3DOF 2D guidedmechanical system ldquocross-beam ED cables AE and BDmoving bridge crane A moving bridge crane Brdquo

The subject of this research is the study of small oscilla-tions of a heavy cross-beam induced by the guided motionof two overhead cranes

The prime novelty of the present research is the introduc-tion of a new mathematical model of a 3DOF 2Dmechanicalsystem ldquocross-beam two moving bridge cranesrdquo with furtherdynamic and optimization analysis of small system oscilla-tions and numerical evaluation of guided system motion inthe first approach

The main contribution of the present original researcharticle to the field of engineering transportation is as followsThe 2D problem of horizontal transportation of a heavysteel structure by two bridge cranes operating at differenthorizontal levels with different length hoisting cables hasbeen addressed

The theoretical novelty of the research is the formulationof differential equations for the motion of a 2D mechanical


system ldquocross-beam two moving bridge cranesrdquo determina-tion of natural frequencies and periods of oscillations and thenumerical derivation of laws of system motion for coherentinitial conditions An open loop optimal control problemwas formulated and solved with JModelicaorg and Optimicafreeware The posed optimal control problem numericalsolution provides minimization of transportation time forbodies of the mechanical system together with minimalswaying of the heavy bulk load

The practical sphere of engineering applications of thepresent transportation problem is in lifting-and-handlingmachinery and civil engineering

The plant photo in Figure 1 shows the two bridge craneswith crane operatorsrsquo cabins which move along the differenthorizontal levelsThe different lengths of supporting cables 1198971and 1198972 for holding and carriage of heavy cross-beam by twobridge cranes are clearly observable in Figures 1 2 3 and 6

Figure 1 contains an experimental plant photo of a cross-beam being transported with two independently movingcranes and provides a schematic illustration of the cross-beam inclination angles The straight inclined line within thecross-beam which passes through the fixing points of con-junction of the carrying cablesrsquo roping to the cross-beamforms a slope angle 1205740 to the horizon As amatter of engineer-ing convenience the present article introduces three inter-connected angles 120574 1205740 and 120575 which uniquely determine theangle of rotation of the cross-beamThe angle 120574 in Figure 2 isthe current angle of rotation of a cross-beam EDwith respectto the horizontal 119909 where cross-beam ED displacementoccurs along the horizontal 119909 direction Factually the angle120574 in Figure 2 is a kind of incidence angle which characterizesthe current angle of a cross-beamEDwith the direction of thevelocity vector of the horizontal displacements of the cranesIt will be shown further that the law of the change of angle 120574will be a harmonic one with respect to a certain position ofstatic equilibrium of cross-beam ED which is characterizedwith angle 1205740 (Figures 1 and 3) The new angle 120575 in Figure 2will be introduced as an angle of rotation of a cross-beamED with respect to the angle 1205740 (Figures 1 and 3) accordingto the formula 120575 = 120574 minus 1205740 Therefore the introduction ofangle 120575 provides a way to select the harmonic component120575 of angle 120574 of cross-beam ED rotation It is necessary tonote that angular velocity of cross-beam ED rotation whichis necessary for calculation of cross-beam kinetic energy willbe determined in the sameway using 120575 and 120574 angles as120596ED =119889120575119889119905 = 119889120574119889119905 It will be shown later that both 120575 and 120574 anglesare second-order infinitesimal quantities with respect to thesmall angle 120572 of the deviation of cable 1 from the vertical inFigure 2 Therefore it is possible to consider that the cross-beam ED makes a translational movement with 120596ED =119889120575119889119905 = 119889120574119889119905 asymp 03 Kinematics of the Mechanical System

Geometric Constraints Coordinatesand Velocities

The mechanical model of the present problem (Figure 1) isshown in Figures 2 3 and 6 Dynamic equations for the

motion of the mechanical system will be derived using theLagrange equations [14 16ndash18] for generalized coordinates11990911199092 and 120572 So it is necessary to determine kinetic and potentialenergies of the mechanical system as functions of our chosengeneralized coordinates For this purposewe have to calculatethe coordinates of bodies 119860 119861 and material point 1198623 of thecross-beam body DE

The Cartesian coordinates of point 119861 which coincideswith the point 1198622 are as follows119909119861 = 119886 cos (1205740) + 1199092119911119861 = 119887119909119861 = 1199091 + 1198971 sdot sin (120572) + 119886 sdot cos (120574) minus 1198972 sdot sin (120573) 119911119861 = minus1198971 sdot cos (120572) + 119886 sdot sin (120574) + 1198972 sdot cos (120573)

(1)

System (1) of transcendental equations which determinethe Cartesian coordinates of point 119861 contains variables 11990911199092 and 120572 It is necessary to solve system (1) with respect to theunknown angles 120573 and 120574 that is to derive 120573 = 119891120573(1199091 1199092 120572)and 120574 = 119891120574(1199091 1199092 120572) for 120574 = 1205740 + 120575 where the current angle120575 is counted off from the initial inclination angle 1205740 in thedirection of cross-beam ED motion in Figures 2 and 3 Thesimplest way to solve system (1) with respect to the 120573 120574 and120575 is usage of a small angle assumption (Appendix AndashD)

In order to calculate the initial inclination angle 1205740(Figures 1 and 3) of cross-beam ED with a horizon 119909 itis possible to introduce an additional geometric parameterΔ1199110 = 119886 sdot sin(1205740) in Figure 3 by substitution of the initial timemoment 1199050 = 0 (s) into the second lower equations of system(1) which yields the following expression 119911119861 = 119887 = (minus1) sdot 1198971 sdot1+119886sdotsin(1205740)+1198972 sdot1 andΔ1199110 = 119886sdotsin(1205740) = 119887+1198971minus1198972 or the sinusof 1205740 angle is sin(1205740) = (119887+1198971minus1198972)119886 = Δ1199110119886 (Figure 3)Thesereasons confirm alternative results in formulae (A2)ndash(A4)

Equation (A12) establishes a simplified linear depen-dence between swing angles 120572 and 120573 So one gets from (A12)that the small angle120573 in Figure 2 can be estimated on the basisof (A12) in the following way120573 asymp ( 11198972) sdot (1199091 minus 1199092 + 1198971120572) (2)

Equation (2) determines the swing angle 120573 as the sim-plified linear function of the variables 1199091 1199092 120572 that is 120573 =119891120573(1199091 1199092 120572)

The rate of change of the swing angle 120573 that is the smallangular velocity 120596120573 = (119889120573119889119905) can be estimated on the basisof (2) in the following way(119889120573119889119905 ) asymp ( 11198972) sdot ((1198891199091119889119905 ) minus (1198891199092119889119905 ) + 1198971 (119889120572119889119905 )) (3)

Equation (B9) determines the small inclination angle 120575 tobe a simplified linear function of the variables 1199091 1199092 120572 thatis 120575 = 119891120575(1199091 1199092 120572) Algebraic expression (B9) accuratelyproves the correctness of statement (A9) concerning the sec-ond order of infinitely small 120575 (infinitesimality of 120575 = 119874(1205722))for a small inclination angle 120575 which determines the current


inclination of the cross-beam ED in Figure 2 Equation (B9)allows the development of an alternative approximate expres-sion for the small angle 120574 for cross-beam ED inclination inFigure 2120574 = 1205740 + 120575 asymp 1205740+ ( 1(2 sdot 119886 sdot 1198972 sdot cos (1205740)) ((1199091 minus 1199092 + 1198971120572)2 minus 1198971 sdot 1198972sdot 1205722)) (4)

The derived alternative equations (B9) and (4) providea way for the estimation of the value of the angular velocity120596120575 = (119889120575119889119905) = 120596120574 = (119889120574119889119905) by the following expression(119889120574119889119905 ) = (119889120575119889119905 ) asymp 1(119886 sdot 1198972 sdot cos (1205740)) ((1199091 minus 1199092 + 1198971120572)sdot ((1198891199091119889119905 ) minus (1198891199092119889119905 ) + 1198971 (119889120572119889119905 )) minus 1198971 sdot 1198972 sdot 120572sdot (119889120572119889119905 ))

(5)

Algebraic expression (5) also proves the correctness ofstatement (A9) concerning the second order of infinitelysmall 120596120575 = 120596120574 (infinitesimality of 120596120575 = (119889120575119889119905) = 120596120574 =(119889120574119889119905) = 119874(1205722)) for the angular velocity of the cross-beamED in Figure 2

After simplifications (C4) yields that1199111198623 asymp ((12) sdot 1198971) sdot 1205722+ 1(4 sdot 1198972) ((1199091 minus 1199092 + 1198971120572)2 minus 1198971 sdot 1198972 sdot 1205722)+ ((12) sdot (119887 minus 1198971 minus 1198972)) (6)

4 Dynamics of the Mechanical SystemPotential and Kinetic Energies GeneralizedForces and Lagrange Equations

The potential energy for the mechanical system in Figure 2can be calculated as Π = (1198983119892) sdot 1199111198623 (7)

Substitution of approximated expression (6) into (7)yields Π asymp (1198983119892) sdot (((12) sdot 1198971) sdot 1205722+ 1(4 sdot 1198972) ((1199091 minus 1199092 + 1198971120572)2 minus 1198971 sdot 1198972 sdot 1205722)+ ((12) sdot (119887 minus 1198971 minus 1198972)))

(8)

Estimation of potential energy enables calculation of thegeneralized forces 1198761199091 1198761199092 and 119876120572 which can be calculatedas partial derivatives of potential energyΠwith respect to thegeneral coordinates 1199091 1199092 and 120572 (Appendix E) Substitutionof (E2) (E4) and (E9) into a one-column matrix yields thefollowing vector of generalized forces

Q asymp (1198983119892) sdot 1(2 sdot 1198972)sdot ( minus1199091 + 1199092 minus 11989711205721199091 minus 1199092 + 1198971120572minus (1198971 sdot 1198972 + 11989721) sdot 120572 minus 1199091 sdot 1198971 + 1199092 sdot 1198971) (9)

Simplification of (F7) results in the following approxi-mated expression

119879 asymp (12) sdot (1198981 + 1198983) sdot (1198891199091119889119905 )2 + (12) sdot 1198982sdot (1198891199092119889119905 )2 + (12) sdot 1198983 sdot 11989721 sdot (119889120572119889119905 )2 + 1198983 sdot 1198971sdot (1198891199091119889119905 ) sdot (119889120572119889119905 )

(10)

The matrix form of the left-hand sides of Lagrangeequations (G1) (G7) and (G13) or (G5) (G11) and (G17)yields the following matrix expression

119889119889119905 (120597119879120597q) minus 120597119879120597q asymp ((1198981 + 1198983) 0 (1198983 sdot 1198971)0 1198982 0(1198983 sdot 1198971) 0 (1198983 sdot 11989721))

sdot(((((

(119889211990911198891199052 )(119889211990921198891199052 )(11988921205721198891199052 ))))))

minus0(11)

The vector form of Lagrange equations is as follows119889119889119905 (120597119879120597q) minus 120597119879120597q = Q (12)

where the left-hand side of (12) is determined by matrixproduct (11) and the right-hand side of (12) is determined bythe vector-column (9)


0 2 4 6 8 10

Time (s)

minus01

00

01

02

03

04

05

06

Coo

rdin

ates

of c

arts

1 an

d 2

Transported cross-beam with two moving bridge cranes

x1

x2

(a)

0 2 4 6 8 10

Time (s)

minus020

minus015

minus010

minus005

000

005

010

015

020

x2minusx1

(b)

Figure 4 JModelicaorg-derived numerical plots for linear coordinates 1199091 = 1199091(119905) 1199092 = 1199092(119905) (a) and 1199092 minus 1199091 = 1199092(119905) minus 1199091(119905) (b) computedfor system (13) with nonzero initial velocities 119881119861(0) = 05 (ms) and 120596120573(0) asymp minus0083 (rads)

Substitution of (11) and (9) into (12) results in thefollowing matrix form of Lagrange equations for the motionof the system in Figures 2 and 3

((1198981 + 1198983) 0 (1198983 sdot 1198971)0 1198982 0(1198983 sdot 1198971) 0 (1198983 sdot 11989721)) sdot((((((

(119889211990911198891199052 )(119889211990921198891199052 )(11988921205721198891199052 )))))))asymp (1198983119892) sdot 1(2 sdot 1198972)

sdot ( minus1199091 + 1199092 minus 11989711205721199091 minus 1199092 + 1198971120572minus (1198971 sdot 1198972 + 11989721) sdot 120572 minus 1199091 sdot 1198971 + 1199092 sdot 1198971)(13)

Derived equation (G6) (G12) (G18) or (13) allows thedetermination of natural periods of system oscillations nat-ural frequencies of mechanical system and deriving motionequations for each of the three bodies of studied mechanicalsystem for different initial conditions in Figures 1ndash3 Thenumerical solution of Lagrange equation (G6) (G12) (G18)or (13) enables the determination of motion patterns and for-mulating applied engineering recommendations concerningmotion of the system

5 First Computational Example ofNumerical Integration of Derived LagrangeEquations for the Studied MechanicalSystem with JModelicaorg Freeware

Both Figure 3 and formulae (2)-(3) yield that at the initialtime moment 1199050 = 0 (s) (2) results in 1205730 asymp ((1199091)0 minus (1199092)0 + 1198971 sdot(1205720))1198972 = 0 The time derivative of this expression providesan analogical algebraic equation for initial velocities 120596120573(0) =119889(1205730)119889119905 asymp ((1198811)0minus(1198812)0+1198971 sdot((119889(1205720))119889119905))1198972 of system in Fig-ures 1ndash3 which results in the following equations 1198972 sdot 120596120573(0) =(1198811)0minus(1198812)0+1198971 sdot120596120572(0) and120596120572(0) = 119889(1205720)119889119905= ((1198812)0minus(1198811)0+1198972 sdot 120596120573(0))1198971 = ((119881119861)0 minus (119881119860)0 + 1198972 sdot 120596120573(0))1198971 The numericalJModelicaorg-enhanced solution of the posed problem inFigures 4 and 5 of the present article was derived in the typicalengineering case (1198812)0 = (119881119861)0 = 0 of right-hand sidemotionof the crane 119861 along 119909 direction when the system motionstarted from the system equilibrium position with zero-initial-values of three generalized initial coordinates (1199091)0 =(1199092)0 = (1205720) = 0 Assumption of (1198811)0 = (119881119860)0 = 0 (ms)and (1198812)0 = (119881119861)0 = 0 yields 120596120573(0) = 119889(1205730)119889119905 asymp (1198971 sdot 120596120572(0) minus(1198812)0)1198972 Additional assumption120596120572(0) = 0 results in120596120573(0) =119889(1205730)119889119905 asymp ((minus1) sdot (1198812)0)1198972 = ((minus1) sdot (119881119861)0)1198972 The derivationof numerical solutions of inverse dynamic problems formechanical system in Figures 1ndash3 requires the assignment ofdifferent initial values of system coordinates and velocitieswhich satisfy the abovementioned constraint equations forinitial coordinates1205730 asymp ((1199091)0minus(1199092)0+1198971 sdot(1205720))1198972 = 0 and ini-tial velocities120596120572(0) = 119889(1205720)119889119905= ((1198812)0minus(1198811)0+1198972sdot120596120573(0))1198971 =((119881119861)0 minus (119881119860)0 + 1198972 sdot 120596120573(0))1198971 These questions concerning theinfluence of initial conditions were properly addressed by the


0 2 4 6 8 10

Time (s)

minus003

minus002

minus001

000

001

002

003

120572

(a)

0 2 4 6 8 10

Time (s)

minus003

minus002

minus001

000

001

002

003

minus004

120573

(b)

0 2 4 6 8 10

Time (s)

minus00004

minus00002

00000

00002

00004

00006

00008

00010

120575

(c)

0 2 4 6 8 10

Time (s)

08478

08480

08482

08484

08486

08488

08490

120574

(d)

0 2 4 6 8 10

Time (s)

minus0003

minus0002

minus0001

0000

0001

0002

0003

d(120574

)dt

(e)

08478 08480 08482 08484 08486 08488 08490minus0003

minus0002

minus0001

0000

0001

0002

0003

120574

d(120574

)dt

(f)

Figure 5 JModelicaorg-derived numerical plots for angular coordinates 120572 = 120572(119905) (a) 120573 = 120573(119905) (b) 120575 = 120575(119905) (c) 120574 = 120574(119905) (d) 120596120574 = 120596120574(119905) =(119889(120574(119905))119889119905) (e) and 120596120574 = 120596120574(120574) that is (119889(120574(119905))119889119905) = 119891(120574) (f) computed for system (13) with nonzero initial velocities 119881119861(0) = 05 (ms)and 120596120573(0) asymp minus0083 (rads)


z

D

E

xOb

x1

x2

C1

C2

A

B 120575x2

120575x1

F2 = u2(t)

F1 = u1(t)

120575120572

120572

l1

l2m1g

m2g

m3g

a2

a2 120574

120573

(120574 = 1205740 + 120575)

C3

a cos (1205740)

zC3

xC3

Figure 6 The computational scheme for transportation of heavy cross-beam DE by two guided bridge cranes 119860 and 119861 with applied controlforces to carts 119860 and 119861 with forcesrsquo magnitudes 1198651 = 1199061(119905) and 1198652 = 1199062(119905)authors in the present research and one of the typical cases isshown in numerical plots in Figures 4 and 5

It is useful to get the numerical solution of system (13) ofLagrange equations (G6) (G12) and (G18) with freewarecode JModelicaorg [14 15 19] for the following numericalvalues of parameters of the mechanical system in Figure 2gravity acceleration is119892=981 (ms2)masses ofmovable carts119860 and119861 are1198981 =1198982 = 1000 (kg)mass of the cross-beamED is1198983 = 10000 (kg) length of the left cableAE is 1198971 =6 (m) lengthof the right cable BD is 1198972 = 5 (m) vertical distance betweenthe carts is 119887= 2 (m) length of the cross-beamED is 119886= 4 (m)

The initial conditions for system (13) of Lagrange equa-tions (G6) (G12) and (G18) are as follows it is possibleto assume that at the initial time moment 119905 = 0 (s) the rightupper cart 119861 in Figure 2 moves rectilinearly to the right thatis initial coordinate of the left cart 119860 is 1199091(0) = 0 (m) initialvelocity of the left cart 119860 is 119881119860(0) = (119889(1199091(0))119889119905) = 0 (ms)initial coordinate of the right cart 119861 is 1199092(0) = 0 (m) initialvelocity of the right cart 119861 which moves to the right is(1198812)0 = 119881119861(0) = (119889(1199092(0))119889119905) = 05 (ms) = 0 initial swingangle of the left rope AE is 120572(0) = 0 (rad) initial angularvelocity of the left cable AE is 120596120572(0) = (119889(120572(0))119889119905) =0 (rads)The abovementioned constraint equations yield that120596120573(0) = 119889(1205730)119889119905 asymp ((minus1) sdot (1198812)0)1198972 = ((minus1) sdot (119881119861)0)1198972= (minus1) sdot 05 (ms)6 (m) asymp ndash0083 (rads) The correspondingcomputational results are shown in Figures 4 and 5

Computational Figures 4 and 5 show the increase of cartsrsquocoordinates with respect tomotion time 119905The computationalcurves in Figures 4 and 5 also show the additional harmonicoscillations Computational dependencies of the angularcoordinates in Figure 5 are harmonic curves The computa-tional amplitudes in Figure 5 confirm the correctness of smallangle assumption in the present article

Computational Figure 5(f) shows an unstable focus at thephase plane (120574(119905) 119889(120574(119905))119889119905) that means a certain increase in

amplitude during cross-beam oscillations However it is pos-sible to neglect some increase of 120574 amplitude by taking intoaccount the fact that both 120574(119905) and (119889(120574(119905))119889119905) are second-order infinitesimal quantities So from an engineering stand-point some small increase in 120574(119905) during a cross-beam trans-portation is negotiable and not essentialMoreover Figure 5(f)shows the necessity of application of additional dampingdevices for suppression of a 120574(119905) oscillations6 Dynamics of the Guided Mechanical System

The formulation of the optimal control problem for guidedmotion of bridge cranes 119860 and 119861 requires the application oftwo independent external control forces to moving carts 119860and 119861 with magnitudes 1198651 = 1199061(119905) [N] and 1198652 = 1199062(119905) [N] ofvariable forces in Figure 6

Substitution of (I9) (I14) and (I19) into the one-columnmatrix yields the following vector of generalized forces for theguided mechanical systemQ120575 asymp (1198983119892) sdot 1(2 sdot 1198972)

sdot ( minus1199091 + 1199092 minus 11989711205721199091 minus 1199092 + 1198971120572minus (1198971 sdot 1198972 + 11989721) sdot 120572 minus 1199091 sdot 1198971 + 1199092 sdot 1198971)+(119865111986520 )

(14)

Substitution of (11) and (14) into (12) results in thefollowing matrix form of the Lagrange equations for themotion of the guided mechanical system in Figure 6


((1198981 + 1198983) 0 (1198983 sdot 1198971)0 1198982 0(1198983 sdot 1198971) 0 (1198983 sdot 11989721)) sdot(((((

(119889211990911198891199052 )(119889211990921198891199052 )(11988921205721198891199052 ))))))asymp (1198983119892) sdot 1(2 sdot 1198972)


(15)

where 1198651 = 1199061(119905) [N] and 1198652 = 1199062(119905) [N] in Figure 6Matrix equation (15) shows that the addition of control

into system (15) can be done through the introduction ofcontrol forces 1198651 = 1199061(119905) [N] and 1198652 = 1199062(119905) [N] in the formof the additional vector components (1199061(119905) 1199062(119905) 0)119879 addedto the right-hand side of equation (15)

Formulation of the governing system of ordinary differ-ential equations (J4) (J5) and (J6) or (15) for motion of theguided mechanical system in Figure 6 allows us to pose thefollowing optimal control problem

It is necessary to find the control functions 1199061(119905) and1199062(119905) for control forces 1198651 = 1199061(119905) [N] and 1198652 = 1199062(119905) [N]in Figure 6 which provide minimization of the objectivefunction 119869 = 119905119891 + int119905119891

0(120572 (119905))2 119889119905 (16)

where 120572(119905) [rad] is the variable angle of the cable AE with thevertical axis 119911 and 119905119891 [s] is the time of travel of the bridgecranes 119860 and 119861 of the prescribed desired distance 119878

Minimization of the objective function 119869 (16) has to bedone for the following constraints119889119889119905 (1199091 (119905)) = 1199091119901 (119905) (17)119889119889119905 (1199092 (119905)) = 1199092119901 (119905) (18)119889119889119905 (120572 (119905)) = 120572119901 (119905) (19)

(1198981 + 1198983) sdot ( 119889119889119905 (1199091119901 (119905))) + 1198983 sdot 1198971 sdot ( 119889119889119905 (120572119901 (119905)))asymp 1199061 (119905) minus (1198983119892) sdot 1(2 sdot 1198972)sdot (1199091 (119905) minus 1199092 (119905) + 1198971 sdot 120572 (119905)) (20)

1198982 sdot ( 119889119889119905 (1199092119901 (119905)))asymp 1199062 (119905) + (1198983119892) sdot 1(2 sdot 1198972)sdot (1199091 (119905) minus 1199092 (119905) + 1198971 sdot 120572 (119905)) (21)

1198983 sdot 1198971 sdot ( 119889119889119905 (1199091119901 (119905))) + 1198983 sdot 11989721 sdot ( 119889119889119905 (120572119901 (119905)))asymp (1198983119892) sdot 1(2 sdot 1198972)sdot (minus (1198971 sdot 1198972 + 11989721) sdot 120572 (119905) minus 1199091 (119905) sdot 1198971 + 1199092 (119905) sdot 1198971) (22)

1199091 (0) = 01199091119901 (0) = 01199092 (0) = 01199092119901 (0) = 0120572 (0) = 0120572119901 (0) = 0(23)

1199091 (119905119891) = 1198781199091119901 (119905119891) = 01199092 (119905119891) = 1198781199092119901 (119905119891) = 0120572 (119905119891) = 0120572119901 (119905119891) = 0(24)

10038161003816100381610038161199061 (119905)1003816100381610038161003816 le 1199061max10038161003816100381610038161199062 (119905)1003816100381610038161003816 le 1199062max (25)

Introduction of control (1199061(119905) 1199062(119905) 0)119879 into equations(J4) (J5) (J6) and (15) provides possibility of deriving anoptimal control in dependence from the form of aminimizedfunctional 119869 (16) with equal weight coefficients

7 Second Computational Example ofa Numerical Solution of Derived GoverningEquations for the Studied GuidedMechanical System with JModelicaorg andOptimica Freeware

It is possible to achieve a numerical solution of system (16)ndash(25) forminimized functional (16) governing equations (17)ndash(22) initial conditions (23) terminal conditions (24) andlimitations on the control forces (25) with module Optimicaof freeware code JModelicaorg [14 15 19] for the followingnumerical values of parameters of the mechanical system in


Figure 6 gravity acceleration is 119892 = 981 (ms2) masses ofmovable carts 119860 and 119861 are 1198981 = 1198982 = 1000 (kg) mass of thecross-beam ED is1198983 = 10000 (kg) length of the left rope AEis 1198971 = 6 (m) length of the right rope BD is 1198972 = 5 (m) verticaldistance between the carts is 119887 = 2 (m) length of the cross-beam ED is 119886 = 4 (m) prescribed desired transportationdistance for bridge cranes 119860 and 119861 is 119878 = 40 (m) themaximum value of the first control force is 1199061max = 100 (N)and themaximumvalue of the second control force is 1199062max =100 (N) Derived plots of motion of guided system with zeroinitial conditions (23) are shown in Figure 7

8 Discussion of Derived Results

The present problem is a new and mathematically complexresearch problem Complexity of the problem is connectedwith the imposition of essentially nonlinear geometric con-straints that cause a lot of issues and problems with thederivation of differential equations of motion

Development of the governing equations was based onthe application of a small parameter linearization methodThenumerical solution of the problem confirmed the correct-ness of applying linearization techniques Derived simulationresults were in good agreement with the basic conservationlaws of classical mechanics

The linear momentum of the system at the initial time119905 = 0 (s) is determined by the linear momentum of cart 119861and can be calculated as 1198982 sdot 119881119861(0) = 1000 (kg) sdot 05 (ms)= 500 (kgsdotms) The total mass of the whole system is (1198981 +1198982 + 1198983) = 12000 (kg) It is possible to determine the linearvelocity of the center of gravity of the mechanical systemat an arbitrary time with the introduction of a theoremdefining center-of-gravity motion conservation as 119881119862 =(1198982 sdot 119881119861(0))(1198981 + 1198982 + 1198983) = 500 (kgsdotms)12000 (kg) asymp0042 (ms) The derived value of 119881119862 determines the averagevelocity for the entire system in Figure 2 So it is possible toestimate the traversed path of the carts119860 and 119861 in Figure 4(a)for the moment of time 120591 = 10 (s) with the formula 119878 asymp 119881119862 sdot120591 asymp 0042 (ms) sdot 10 (s) asymp 042 (m) There is good agreementof this estimated value 119878 asymp 042 (m) with the numerical value1199091120591 asymp 05 (m) in Figure 4(a)

It is necessary to note that in addition to the motion dueto the velocity of the center of gravity there are additionalsystem motions with natural frequencies 1198961 asymp 1021 (1s)and 1198962 asymp 3794 (1s) (Appendix H) and the correspondingnatural periods 1205911 = 2 sdot 1205871198961 asymp 6155 (s) and 1205912 = 2 sdot1205871198962 asymp 1656 (s) These two natural periods and amplitudesare clearly observable in Figures 4 and 5

The numerical solution of an optimal control problem(16)ndash(25) allows determination of the control variables 1199061(119905)and 1199062(119905) for control forces 1198651 = 1199061(119905) [N] and 1198652 = 1199062(119905) [N]in Figure 7(b) which enable transportation of a cross-beamfrom the initial to the final position with minimum time andminimum swaying in Figures 7(c) and 7(d) These controlvariables 1199061(119905) and 1199062(119905) are the control forces 1198651 = 1199061(119905) and1198652 = 1199062(119905) applied to the first and second bridge cranes 119860and 119861 Optimica-derived Figure 7(b) shows that it is possibleto provide an optimal motion in the abovementioned senseby the following strategy when at the first half of motion

time the control forces 1199061(119905) and 1199062(119905) should be constant andpositive and at the second half of motion time the controlforces 1199061(119905) and 1199062(119905) should be negative As a result in thebeginning acting forces provide maximum acceleration ofthe bridge cranes and then decelerationThis control strategyis a ldquobang-bangrdquo control method with the switching point inthe time moment 119905lowast = 1199051198912 where 119905119891 is the time of travelof bridge cranes from one position to another It is shownin Figure 7(b) that during the last 2-3 seconds the controlforces have negative values because it is necessary to provideslowing-down (deceleration) of the system motion In thiscase the values of control forces are 025 sdot max1199061(119905) 1199062(119905)This kind of a control strategy enables travel to the finalposition with near zero-values of velocities1198811199091 asymp 1198811199092 asymp 1198811199093 asymp0 (ms) and near zero-values of swaying angles 120572 asymp 120573 asymp 0 Aneasy practical realization of these recommendations by bridgecranes operators is the real advantage of the found controlstrategy Really in the first half of the way crane operatorshave to accelerate bridge cranes and in the second half ofthe movement crane operators have to decelerate bridgecranesThis control strategy is relatively easy for realization inmanual mode using available crane control mechanismsTheproposed control strategy reduces transportation time andpartially decreases intensity of the labor of crane operators

Analysis of Figure 7(a) indicates that the proposed con-trol strategy enables synchronous motion of bridge craneswith quite small and negligible oscillations of a heavy cross-beam Analysis of Figure 7(c) shows that deviation of a cross-beam from equilibrium position in the horizontal directiondoes not exceed 0002 (rad) and in the vertical direction thecross-beam deviation is much smaller and does not exceed4sdot10minus6 (rad)

It is easy to confirm the correctness of the proposedoptimal control with engineering estimation Accordingto the law of variation of momentum along the horizontaldirection the impulse of control forces for 1198651 = 1199061(119905) [N]and 1198652 = 1199062(119905) [N] is (1199061 +1199062) sdot (1199051198912) asymp (100+ 100) sdot 49 (Nsdots)asymp 9800 (Nsdots) Moreover it follows from Figure 7(e) that thelinear momentum of the system to this time moment 49 s is(1198981 + 1198982 + 1198983) sdot 119881119862 asymp (1000 + 1000 + 10000) sdot 08 (kgsdotms)asymp 9600 (Nsdots) The results are approximately the same in thedeceleration region These simple engineering estimationsconfirm the correctness of the derived optimal controlstrategy

Further research directions are associated with dynamicanalysis of internal forces reactions of constraints intro-duction of additional elastic-damping devices developmentand analysis of other modes of optimal control introductionof additional elastic-damping devices and search for othernatural frequencies in 3D problem

9 Final Conclusions

The present problem is a truly new problem of lifting-and-handling machinery

This problem has arisen from applied engineering assign-ments associated with civil engineering and transportationproblems in plant departments and in field environmentconditions


0 20 40 60 80 100

Time (s)

minus10

0

10

20

30

40

50

x1

x2

x3


Coo

rdin

ates

of c

arts

1 2

and

the c

ente

r of

mas

s of a

cros

s-be

am

(a)

0 20 40 60 80 100

Time (s)u1

u2

minus100

minus50

0

50

100

u1

andu2

Con

trols

(b)

120572

120573

0 20 40 60 80 100

Time (s)

minus0003

minus0002

minus0001

0000

0001

0002

0003

120572an

d120573

(c)

0 20 40 60 80 100

Time (s)

minus0000003

minus0000002

minus0000001

0000000

0000001

0000002

0000003

0000004

0000005120575

(d)

0 20 40 60 80 100

Time (s)

00

01

02

03

04

05

06

07

08

09

12

1

2an

d c

3

c3

(e)

minus0000015

minus0000010

minus0000005

0000000

0000005

0000010

0000015

000

0001

000

0002

000

0003

000

0004

000

0005

000

0006

000

0007

000

0008

000

0009

848058e minus 1

120574

+

d(120574

)dt

(f)

Figure 7 JModelicaorg- and Optimica-derived numerical plots for linear coordinates 1199091 = 1199091(119905) 1199092 = 1199092(119905) and 1199093 = 1199093(119905) (a) controlforces 1198651 = 1199061(119905) [N] and 1198652 = 1199062(119905) [N] (b) angular coordinates 120572 = 120572(119905) and 120573 = 120573(119905) (c) angular coordinate 120575 = 120575(119905) (d) synchronizedvelocities 1198811199091 (119905) = (119889(1199091(119905))119889119905) 1198811199101 (119905) = (119889(1199101(119905))119889119905) and 1198811199091198623 (119905) = (119889(1199091198623 (119905))119889119905) of points 1198621 1198622 1198623 during guided system motion (e)and 120596120574 = 120596120574(120574) that is (119889(120574(119905))119889119905) = 119891(120574) obtained for synchronous motion of the mechanical system with an open loop control


There are previously known studies of vertical liftingspace canting and 3D rotation of the load up to 90 degwith two boom cranes working together But there is a lackof relevant and specific information concerning horizontaltransportation of the load with two bridge cranes

The present original research article is focused on thedevelopment of a mathematical model describing the hori-zontal transportation of the load (cross-beam) in the verticalplane

Transportation dynamics of heavy cross-beam moved bytwo overhead cranes was addressed with the introduction ofthe Lagrange equations using small angle assumptions

A 3DOF mechanical system ldquoheavy cross-beam twomoving cartsrdquo was studied

A system of Lagrange equations for a 3DOF mechanicalsystem was developed and numerically solved

A number of general trends and features of the heavycross-beam motion were found using mathematical andnumerical simulation techniques Amongmotion features arethe phenomena of insignificant horizontal swaying of a cross-beam in motion direction as well as infinitesimally smallangles of cross-beam oscillations around the center of mass

One of the methodologies of the search of optimalcontrol was proposed An optimal control strategy withminimization of transportation time and swaying angles 120572and 120573 was developed with an application of JModelicaorgand Optimica freeware

The proposedmathematical model for an optimal controlsystem allows development of the corresponding graphs andnomographic charts for formulation of applied engineeringrecommendations

The posed transport engineering problem will find fur-ther development and application with the introduction ofelastic-damping devices changing hoisting cables lengthsand switching to a 3D simulation

Appendix

A Linearization Techniques

It is possible to make a linearization of expressions (1)assuming that

sin (120572) asymp 120572cos (120572) asymp 1 minus (12) sdot 1205722sin (120573) asymp 120573cos (120573) asymp 1 minus (12) sdot 1205732

(A1)

However linearization of 120574 requires the usage of a moresophisticated approach At the initial moment of time 119905 = 0both cables AE and BD had the vertical positions with initialvalues of angles 1205720 = 1205730 = 0 in Figure 3 At 119905 = 0 the initialinclination angle of the cross-beam ED (119897ED = 119886) to the

horizontal is 1205740 (Figure 3) It is obvious that at the initial timemoment initial position of oscillating system yields1198971 + 119887 = 1198972 + Δ1199110 (A2)

The algebraic equation (A2) yieldsΔ1199110 = 1198971 minus 1198972 + 119887 (A3)

Equations (A2) and (A3) enable calculation of the initialangle 1205740

sin (1205740) = Δ1199110119886 = (1198971 minus 1198972 + 119887)119886 cos (1205740) = radic1 minus sin2 (1205740) (A4)

Further linearization of (1) requires the derivation ofsimplified trigonometric expressions for the current angle 120574using the formulae for sine and cosine of angular sum where120574 = 1205740 + 120575 (Figure 2)

cos (120574) = cos (1205740 + 120575)= cos (1205740) cos (120575) minus sin (1205740) sin (120575) sin (120574) = sin (1205740 + 120575)= sin (1205740) cos (120575) + cos (1205740) sin (120575)

(A5)

It is possible to make a linearization of (A5) for the smallangle 120575 assuming that

sin (120575) asymp 120575cos (120575) asymp 1 (A6)

After substitution of (A6) into (A5) one obtains

cos (120574) = cos (1205740 + 120575) asymp cos (1205740) minus 120575 sdot sin (1205740) (A7)

sin (120574) = sin (1205740 + 120575) asymp sin (1205740) + 120575 sdot cos (1205740) (A8)

As will subsequently be shown in further equations (4)(B9) and (5) the angle 120575 and the angular velocity of thecross-beam ED are the second-order infinitesimal quantitieswith respect to 120572 that is120575 = 119874 (1205722) (119889120575119889119905 ) = (119889120574119889119905 ) = 119874 (1205722) (A9)

Substitution of (A9) into (A7) approximately results in

cos (120574) asymp cos (1205740) (A10)

Combined usage of (A1) and (A10) provides a way tolinearize (1) for the abscissa119909119861 of point119861 Substitution of (A1)and (A10) into the upper lines of (1) yields that119909119861 = 119886 cos (1205740) + 1199092asymp 1199091 + 1198971 sdot 120572 + 119886 cos (1205740) minus 1198972 sdot 120573 (A11)

Approximate equation (A11) results in1199092 minus 1199091 asymp 1198971 sdot 120572 minus 1198972 sdot 120573 (A12)


B Estimation of Small Inclination Angles

Now it is necessary to find a simplified expression for 120574 =119891120574(1199091 1199092 120572) where 120574 = 1205740+120575 For this purpose it is necessaryto address system (1) for the applicate 119911119861 of point 119861 Substitu-tion of the second equations of system (A1) and (A8) intothe lower lines of (1) yields119911119861 = 119887asymp minus1198971 sdot (1 minus (12) sdot 1205722) + 119886 sdot sin (1205740) + 119886 sdot 120575sdot cos (1205740) + 1198972 sdot (1 minus (12) sdot 1205732)

(B1)

where the second term on the right-hand side of (B1) isdetermined by expression (A4) Therefore the substitutionof (A4) into (B1) produces119887 asymp minus1198971 sdot (1 minus (12) sdot 1205722) + (1198971 minus 1198972 + 119887) + 119886 sdot 120575sdot cos (1205740) + 1198972 sdot (1 minus (12) sdot 1205732) (B2)

119887 asymp minus1198971 + (12) sdot 1198971 sdot 1205722 + 1198971 minus 1198972 + 119887 + 119886 sdot 120575 sdot cos (1205740)+ 1198972 minus (12) sdot 1198972 sdot 1205732 (B3)

It is obvious that components 119887 1198971 and 1198972 with oppositesigns in (B3) result in zero-sum terms of (B3) So (B3) yields119886 sdot 120575 sdot cos (1205740) asymp (12) sdot 1198972 sdot 1205732 minus (12) sdot 1198971 sdot 1205722 (B4)

After transformation (B4) yields the small angle 120575 expres-sion 120575 asymp (1198972 sdot 1205732 minus 1198971 sdot 1205722)(2 sdot 119886 sdot cos (1205740)) (B5)

Equation (B5) determines the small inclination angle 120575as the simplified linear function of the variables 120572 and 120573 thatis 120575 = 119891120575(120572 120573) Equation (B5) provides the possibility of cal-culating an approximate expression for the small angle 120574 forcross-beam ED inclination in Figure 2

120574 = 1205740 + 120575 asymp 1205740 + ((1198972 sdot 1205732 minus 1198971 sdot 1205722)(2 sdot 119886 sdot cos (1205740)) ) (B6)

The derived equations (B5) and (B6) provide a way toestimate the value of the angular velocity 120596120575 = (119889120575119889119905) =120596120574 = (119889120574119889119905) by the following expression(119889120574119889119905 ) = (119889120575119889119905 )asymp 1(119886 sdot cos (1205740)) (1198972 sdot 120573 sdot (119889120573119889119905 ) minus 1198971 sdot 120572 sdot (119889120572119889119905 )) (B7)

It is possible tomake further transformations of (B5) andto derive the small inclination angle 120575 as the simplified linearfunction of the variables 1199091 1199092 120572 that is to get an expressionfor 120575 = 119891120575(1199091 1199092 120572) by substitution of (2) into (B5)120575 asymp 1(2 sdot 119886 sdot cos (1205740)) (1198972 sdot ( 11198972)2 sdot (1199091 minus 1199092 + 1198971120572)2minus 1198971 sdot 1205722) (B8)

Reducing expression (B8) to a common denominatoryields120575 asymp 1(2 sdot 119886 sdot 1198972 sdot cos (1205740)) ((1199091 minus 1199092 + 1198971120572)2 minus 1198971 sdot 1198972sdot 1205722) (B9)

C Linearized Coordinates ofCross-Beam Center

Now it is necessary to address the coordinates and displace-ments of the point 1198623 which is located at the center of thecross-beamEDThe coordinates of the point1198623 are necessaryfor determination of kinetic (Appendix F) and potentialenergies of the cross-beamED as well as for calculation of thegeneralized forces (Appendix E) in the Lagrange equations(Appendix G) The horizontal and vertical Cartesian coordi-nates of point 1198623 in Figure 2 are determined by the followingformulae 1199091198623 = 1199091 + 1198971 sdot sin (120572) + (1198862) sdot cos (120574) 1199111198623 = minus1198971 sdot cos (120572) + (1198862) sdot sin (120574) (C1)

Further linearization of (C1) is possible through theapplication of the previous expressions (A1) and (A7)-(A8)Substitution of (A1) and (A7)-(A8) yields1199091198623 asymp 1199091 + 1198971 sdot 120572 + (1198862) sdot (cos (1205740) minus 120575 sdot sin (1205740)) 1199111198623 asymp minus1198971 sdot (1 minus (12) sdot 1205722) + (1198862)sdot (sin (1205740) + 120575 sdot cos (1205740))

(C2)

It is possible to expand the brackets in (C2) to get thefollowing simplified expressions1199091198623 asymp 1199091 + 1198971 sdot 120572 minus ((1198862) sdot sin (1205740)) sdot 120575+ ((1198862) sdot cos (1205740)) 1199111198623 asymp ((12) sdot 1198971) sdot 1205722 + ((1198862) sdot cos (1205740)) sdot 120575+ ((1198862) sdot sin (1205740) minus 1198971)

(C3)


Substitution of (A4) and (B9) into the second equationof system (C3) results in the following expression for thevertical location applicate of the point 11986231199111198623 asymp ((12) sdot 1198971) sdot 1205722 + ((1198862) sdot cos (1205740))sdot 1(2 sdot 119886 sdot 1198972 sdot cos (1205740)) ((1199091 minus 1199092 + 1198971120572)2 minus 1198971 sdot 1198972sdot 1205722) + ((12) sdot (1198971 minus 1198972 + 119887) minus 1198971)

(C4)

D Linearized Velocities of Cross-Beam Center

The further calculation of kinetic energy of the cross-beamED requires an estimation of velocity of the point 1198623 Thefirst time derivatives of system (C1) give to one the systemof equations for V1198623 projections119889 (1199091198623)119889119905 = (1198891199091119889119905 ) + (1198971 sdot cos (120572)) sdot (119889120572119889119905 )minus ((1198862) sdot sin (120574)) sdot (119889120574119889119905 ) 119889 (1199111198623)119889119905 = (1198971 sdot sin (120572)) sdot (119889120572119889119905 ) + ((1198862) sdot cos (120574))sdot (119889120574119889119905 )

(D1)

It is more useful to linearize system (D1) through differ-entiation of system (C3) with respect to the time 119905119889 (1199091198623)119889119905 asymp (1198891199091119889119905 ) + 1198971 sdot (119889120572119889119905 ) minus ((1198862) sdot sin (1205740))sdot (119889120575119889119905 ) 119889 (1199111198623)119889119905 asymp ((12) sdot 1198971) sdot (2 sdot 120572) sdot (119889120572119889119905 )+ ((1198862) sdot cos (1205740)) sdot (119889120575119889119905 )

(D2)

The system of expressions (D2) can be further simplifiedby taking into account equation (5) that 1205963 = 120596120575 = (119889120575119889119905) =120596120574 = (119889120574119889119905) = 119874(1205722) asymp 0119889 (1199091198623)119889119905 asymp (1198891199091119889119905 ) + 1198971 sdot (119889120572119889119905 ) 119889 (1199111198623)119889119905 asymp (1198971 sdot 120572) sdot (119889120572119889119905 ) (D3)

It is possible to further simplify (D3) by taking intoaccount the fact that the product 120572 sdot (119889120572119889119905) is the second-order infinitesimal quantity that is120572 sdot (119889120572119889119905 ) = 119874 (1205722) (D4)

Substitution of (D4) into (D3) yields119889 (1199091198623)119889119905 asymp (1198891199091119889119905 ) + 1198971 sdot (119889120572119889119905 ) 119889 (1199111198623)119889119905 asymp 0 (D5)

E Generalized Forces

The first generalized force 1198761199091 which corresponds to thegeneralized linear coordinate 1199091 is as follows1198761199091 = minus 120597Π1205971199091asymp (minus1) sdot (1198983119892) sdot 1(4 sdot 1198972) sdot 2 sdot (1199091 minus 1199092 + 1198971120572) (E1)

After simplifications the previous equation (E1) for thefirst generalized force 1198761199091 takes the form1198761199091 asymp (minus1) sdot (1198983119892) sdot 1(2 sdot 1198972) sdot (1199091 minus 1199092 + 1198971120572) (E2)

The second generalized force 1198761199092 which corresponds tothe generalized linear coordinate 1199092 can be calculated as1198761199092 = minus 120597Π1205971199092asymp (minus1) sdot (1198983119892) sdot 1(4 sdot 1198972) sdot 2 sdot (1199091 minus 1199092 + 1198971120572)sdot (minus1)

(E3)

After simplifications (E3) for the second generalizedforce 1198761199092 yields1198761199092 asymp (+1) sdot (1198983119892) sdot 1(2 sdot 1198972) sdot (1199091 minus 1199092 + 1198971120572) (E4)

It is possible to calculate the third generalized force 119876120572which corresponds to the generalized angular coordinate 120572by the following expression

119876120572 = minus120597Π120597120572 asymp (minus1) sdot (1198983119892) sdot (((12) sdot 1198971) sdot (2 sdot 120572)+ 1(4 sdot 1198972) (2 sdot (1199091 minus 1199092 + 1198971120572) sdot 1198971 minus 1198971 sdot 1198972 sdot 2 sdot 120572)) (E5)


After simplifications (E5) produces

119876120572 asymp (minus1) sdot (1198983119892) sdot ((1198971 sdot 120572)+ 1(2 sdot 1198972) ((1199091 minus 1199092 + 1198971120572) sdot 1198971 minus 1198971 sdot 1198972 sdot 120572)) (E6)

Reducing a right-hand side fraction of (E6) to a commondenominator yields119876120572 asymp (1198983119892) sdot 1(2 sdot 1198972)sdot (minus1198971 sdot 1198972 sdot 120572 minus (1199091 minus 1199092 + 1198971120572) sdot 1198971) (E7)

Expanding the brackets in (E7) leads to the followingexpression119876120572 asymp (1198983119892) sdot 1(2 sdot 1198972)sdot (minus1198971 sdot 1198972 sdot 120572 minus 1199091 sdot 1198971 + 1199092 sdot 1198971 minus 11989721 sdot 120572) (E8)

Equation (E8) finally produces119876120572 asymp (1198983119892) sdot 1(2 sdot 1198972)sdot (minus (1198971 sdot 1198972 + 11989721) sdot 120572 minus 1199091 sdot 1198971 + 1199092 sdot 1198971) (E9)

F System Kinetic Energy

Calculation of kinetic energy of the mechanical system inFigure 2 requires some additional comments

It is obvious that the first cart 119860 (point 1198621) is involved inthe translational motion along the horizontal axis119874119909 Hencethe kinetic energy of a 2Dmodel of the bridge crane119860 can becalculated by the following expression1198791 = (12) sdot 119898119860 sdot (V1198621)2 = (12) sdot 119898119860 sdot (1198811198621)2= (12) sdot 1198981 sdot (1198891199091119889119905 )2 (F1)

It is clearly shown in Figure 2 that the second cart119861 (point1198622) is involved in the translational motion along the upperhorizontal axis which is parallel to axis 119874119909 and is locatedfrom 119874119909 at the distance 119887 Therefore the kinetic energy of a2Dmodel of the bridge crane 119861 can be analogously calculatedby the same formula for kinetic energy of translationalmotion as1198792 = (12) sdot 119898119861 sdot (V1198622)2 = (12) sdot 119898119861 sdot (1198811198622)2= (12) sdot 1198982 sdot (1198891199092119889119905 )2 (F2)

It is shown in Figures 1 and 2 that the heavy cross-beamED can be considered as a solid rod which is involved inparallel-plane motion It is well known from the course ofmechanics that the kinetic energy of the cross-beam ED inparallel-planemotion can be calculated with Konigrsquos theoremas 1198793 = (12) sdot 119898ED sdot (V1198623)2 + (12) sdot 1198691198623 sdot (ED)2 (F3)

where 119898ED = 1198983 is the mass of the cross-beam ED V1198623 =V1198623119909 + V1198623119911 is the vector of the linear velocity of the point1198623 (see formulae (D1)ndash(D5) of Appendix D) where point1198623is the center of mass of the cross-beam ED 1198691198623 = ((112) sdot1198983 sdot 1198862) is the moment of inertia of the cross-beam ED withrespect to the cross-beam central point 1198623 and 120596ED = 1205963 isthe vector of an angular velocity of the cross-beam ED whichis determined by formulae (B7) and (5)

Substitution of scalar quantities into Konigrsquos theoremresults in the following equation

1198793 = (12) sdot 1198983 sdot ((119889 (1199091198623)119889119905 )2 + (119889 (1199111198623)119889119905 )2)+ (12) sdot ( 112 (1198983 sdot 1198862)) sdot (119889120574119889119905 )2 (F4)

where approximate values of time derivatives (119889(1199091198623)119889119905)(119889(1199111198623)119889119905) and (119889120574119889119905) are determined by (5) and (D5)

It was shown above that the first time derivatives(119889(1199111198623)119889119905) and (119889120574119889119905) are second-order infinitesimal quan-tities Therefore substitution of (5) and (D5) into (F4)results in zero-values of the following two terms of (F4)(119889(1199111198623)119889119905)2 asymp 0 and (119889120574119889119905)2 asymp 0 Substitution of (5) and(D5) into (F4) proves the possibility of reducing the parallel-plane motion of the heavy cross-beam ED to planar motion

Simplified formula (F4) after linearization with (5) and(D5) yields that

1198793 asymp (12) sdot 1198983 sdot (119889 (1199091198623)119889119905 )2 (F5)

Substitution of (D5) into (F5) results in1198793 asymp (12) sdot 1198983 sdot ((1198891199091119889119905 ) + 1198971 sdot (119889120572119889119905 ))2 (F6)

It is possible to calculate the kinetic energy of mechanicalsystem by substitution of (F1) (F2) and (F6) into algebraicexpression for the total kinetic energy119879 = 1198791 + 1198792 + 1198793asymp (12) sdot 1198981 sdot (1198891199091119889119905 )2 + (12) sdot 1198982 sdot (1198891199092119889119905 )2+ (12) sdot 1198983 sdot (1198891199091119889119905 )2 + (12) sdot 1198983 sdot 11989721 sdot (119889120572119889119905 )2+ (12) sdot 1198983 sdot 2 sdot (1198891199091119889119905 ) sdot 1198971 sdot (119889120572119889119905 )

(F7)


G System Lagrange Equations

The first Lagrange equation for the generalized coordinate 1199091is as follows 119889119889119905 ( 1205971198791205971) minus 1205971198791205971199091 = 1198761199091 (G1)

where 119879 and 1198761199091 are approximately determined by (10) and(E2)

The partial derivative of the kinetic energy with respect tothe first generalized velocity (1198891199091119889119905) yields1205971198791205971 asymp (12) sdot (1198981 + 1198983) sdot 2 sdot (1198891199091119889119905 ) + 1198983 sdot 1198971sdot (119889120572119889119905 ) (G2)

The full derivative of expression (G2)with respect to time119905 is as follows119889119889119905 ( 1205971198791205971) asymp (1198981 + 1198983) sdot (119889211990911198891199052 ) + 1198983 sdot 1198971sdot (11988921205721198891199052 ) (G3)

It follows from (10) that the partial derivative of thekinetic energy with respect to the first generalized coordinate1199091 has a zero-value 1205971198791205971199091 asymp 0 (G4)

Substitution of (G3) and (G4) into (G1) results in thefollowing approximate expression of the left-hand side of thefirst Lagrange equation119889119889119905 ( 1205971198791205971) minus 1205971198791205971199091 asymp (1198981 + 1198983) sdot (119889211990911198891199052 ) + 1198983 sdot 1198971sdot (11988921205721198891199052 ) (G5)

Substitution of (G5) and (E2) into (G1) gives the firstsimplified Lagrange equation

(1198981 + 1198983) sdot (119889211990911198891199052 ) + 1198983 sdot 1198971 sdot (11988921205721198891199052 )asymp (minus1) sdot (1198983119892) sdot 1(2 sdot 1198972) sdot (1199091 minus 1199092 + 1198971120572) (G6)

The second Lagrange equation for the generalized coor-dinate 1199092 is as follows119889119889119905 ( 1205971198791205972) minus 1205971198791205971199092 = 1198761199092 (G7)


The partial derivative of the kinetic energy with respect tothe second generalized velocity (1198891199092119889119905) yields1205971198791205972 asymp (12) sdot 1198982 sdot 2 sdot (1198891199092119889119905 ) (G8)

The full derivative of expression (G8)with respect to time119905 is as follows 119889119889119905 ( 1205971198791205972) asymp 1198982 sdot (119889211990921198891199052 ) (G9)

It follows from (10) that the partial derivative of thekinetic energy with respect to the second generalized coor-dinate 1199092 has a zero-value 1205971198791205971199092 asymp 0 (G10)

Substitution of (G9) and (G10) into (G7) results in thefollowing approximate expression of the left-hand side of thesecond Lagrange equation119889119889119905 ( 1205971198791205972) minus 1205971198791205971199092 asymp 1198982 sdot (119889211990921198891199052 ) (G11)

Substitution of (G11) and (E4) into (G7) gives the secondsimplified Lagrange equation

1198982 sdot (119889211990921198891199052 ) asymp (+1) sdot (1198983119892) sdot 1(2 sdot 1198972)sdot (1199091 minus 1199092 + 1198971120572) (G12)

The third Lagrange equation for the generalized coordi-nate 120572 is as follows 119889119889119905 (120597119879120597 ) minus 120597119879120597120572 = 119876120572 (G13)


The partial derivative of the kinetic energy with respect tothe third generalized velocity (119889120572119889119905) yields120597119879120597 asymp (12) sdot 1198983 sdot 11989721 sdot 2 sdot (119889120572119889119905 ) + 1198983 sdot 1198971 sdot (1198891199091119889119905 ) (G14)

The full derivative of expression (G14) with respect totime 119905 is as follows119889119889119905 (120597119879120597 ) asymp 1198983 sdot 11989721 sdot (11988921205721198891199052 ) + 1198983 sdot 1198971 sdot (119889211990911198891199052 ) (G15)


It follows from (10) that the partial derivative of thekinetic energy with respect to the third generalized coordi-nate 120572 has a zero-value 120597119879120597120572 asymp 0 (G16)

Substitution of (G15) and (G16) into (G13) results in thefollowing approximate expression of the left-hand side of thethird Lagrange equation119889119889119905 (120597119879120597) minus 120597119879120597120572 asymp 1198983 sdot 11989721 sdot (11988921205721198891199052 ) + 1198983 sdot 1198971sdot (119889211990911198891199052 ) (G17)

Substitution of (G17) and (E9) into (G13) gives the thirdsimplified Lagrange equation

1198983 sdot 11989721 sdot (11988921205721198891199052 ) + 1198983 sdot 1198971 sdot (119889211990911198891199052 )asymp (1198983119892) sdot 1(2 sdot 1198972)sdot (minus (1198971 sdot 1198972 + 11989721) sdot 120572 minus 1199091 sdot 1198971 + 1199092 sdot 1198971) (G18)

H Numerical Estimation of NaturalFrequencies for the Mechanical System

It is possible to estimate the natural frequencies of themechanical system with the system (13) of Lagrange equa-tions (G6) (G12) and (G18) After generation of thedeterminant of the natural frequencies matrix for system (13)it is possible to write the following characteristic biquadraticequation for system (13) of Lagrange equations (G6) (G12)and (G18) in the form

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816((1198981 + 1198983)1198983 sdot 1205822 + 1198922 sdot 1198972) (minus 1198922 sdot 1198972) (1205822 + 1198922 sdot 1198972)(minus 1198922 sdot 1198972) (11989821198983 sdot 1205822 + 1198922 sdot 1198972) (minus119892 sdot 11989712 sdot 1198972 )(1205822 + 1198922 sdot 1198972) (minus119892 sdot 11989712 sdot 1198972 ) (1198971 sdot 1205822 + 1198922 sdot 1198972 sdot (1198971 + 1198972))

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816= 0 (H1)

The determinant (H1) yields the following characteristicsecular equation of sixth order for the above determinednumerical values in Section 5

0560 sdot 1205826 + 7073 sdot 1205824 minus 15879 sdot 1205822 minus 23602 = 0 (H2)

Thenumerical solution of the secular equation (H2) givesthe following set of roots

120582 = plusmn1676 plusmn1021 sdot 119894 plusmn3794 sdot 119894 (H3)

I Generalized Forces of the GuidedMechanical System

Calculation of the generalized forces 119876120575(1199091) 119876120575(1199092) and119876120575(120572) in (J1)ndash(J3) requires the estimation of isochronousvariation 120575(1199111198623) of the vertical coordinate 1199111198623 on the basis ofapproximate formula (D2)The second equation of formulae(D2) yields the following approximate expression for totalordinary differential of the vertical coordinate 1199111198623 119889 (1199111198623) asymp (1198971 sdot 120572) sdot 119889 (120572) + ((1198862) sdot cos (1205740)) sdot 119889 (120575) (I1)

It is possible to get an approximate expression for totalordinary differential119889(120575) in (I1) on the basis of the previouslyderived formula (5) which produces the following equation119889 (120575) asymp 1(119886 sdot 1198972 sdot cos (1205740)) ((1199091 minus 1199092 + 1198971120572)sdot (119889 (1199091) minus 119889 (1199092) + 1198971 sdot 119889 (120572)) minus 1198971 sdot 1198972 sdot 120572 sdot 119889 (120572)) (I2)

Substitution of (I2) into (I1) results in the followingexpression119889 (1199111198623) asymp (1198971 sdot 120572) sdot 119889 (120572) + ((1198862) sdot cos (1205740))(119886 sdot 1198972 sdot cos (1205740))sdot ((1199091 minus 1199092 + 1198971120572) sdot (119889 (1199091) minus 119889 (1199092) + 1198971 sdot 119889 (120572))minus 1198971 sdot 1198972 sdot 120572 sdot 119889 (120572)) (I3)

The formal change of differential symbol 119889 to the symbolof virtual displacement 120575 in (I3) corresponds to imaginaryfreezing of time variable 119905 and gives the following expressionfor the isochronous variation 120575(1199111198623) of the vertical coordinate1199111198623 120575 (1199111198623) asymp (1198971 sdot 120572) sdot 120575 (120572) + 1(2 sdot 1198972) sdot ((1199091 minus 1199092 + 1198971120572)sdot (120575 (1199091) minus 120575 (1199092) + 1198971 sdot 120575 (120572)) minus 1198971 sdot 1198972 sdot 120572 sdot 120575 (120572)) (I4)


where 120575(1199091) 120575(1199092) and 120575(120572) are the three independentisochronous variations of the three generalized coordinates11990911199092 and120572 which determine the current position of a 3DOFmechanical system in Figures 2 and 6

Calculation of the first generalized force 119876120575(1199091) requiresthe imposition of the first independent virtual displacement120575(1199091) = 0 to the mechanical system in Figure 6 withsimultaneous ldquofreezingrdquo or ldquosticking downrdquo of cart 119861 to theupper rails and ldquofreezingrdquo or ldquosticking to the leftrdquo of cable AEto the vertical axis 119911 factually assuming zero-values of anothertwo virtual displacements 120575(1199092) = 0 and 120575(120572) = 0119876120575(1199091) = ( 1120575 (1199091)) sdot ((sum119896 120575119860119886119896)120575(1199091))10038161003816100381610038161003816100381610038161003816100381610038161003816120575(1199092)=0 120575(120572)=0

= 1198651 sdot 120575 (1199091) + (1205751198601 (m3g))120575(1199091)120575 (1199091) (I5)

119876120575(1199091) = 1198651 sdot 120575 (1199091) + (minus1) sdot (1198983119892) sdot (120575 (1199111198623))120575(1199091)120575 (1199091) (I6)

where the virtual work of the force 1198652 in (I5)-(I6) at thevirtual displacement 120575(1199091) = 0 has a zero-value because inour imaginationwemade ldquofreezingrdquo or ldquosticking downrdquo of thecart 119861 to the upper rails with 120575(1199092) = 0

The second term (1205751198601(m3g))120575(1199091) in the numerator of(I5) is the virtual work of the gravity forcem3g of the cross-beam ED in the case of 120575(1199091) = 0 with 120575(1199092) = 0 and120575(120572) = 0 The minus sign in the numerator of (I6) showsthat gravity forcem3g has negative work due to lifting up thepoint 1198623 in this case in Figure 6 It is possible to calculate thealgebraic expression of the isochronous variation (120575(1199111198623))120575(1199091)in the present case through the substitution of 120575(1199092) = 0 and120575(120572) = 0 into (I4)

(120575 (1199111198623))120575(1199091) = (120575 (1199111198623))10038161003816100381610038161003816120575(1199092)=0 120575(120572)=0asymp 1(2 sdot 1198972)sdot ((1199091 minus 1199092 + 1198971120572) sdot (120575 (1199091))) (I7)

Substitution of (I7) into (I6) results in the followingexpression

119876120575(1199091)asymp 1198651 sdot 120575 (1199091) + (minus1) sdot (1198983119892) sdot 1 (2 sdot 1198972) sdot (1199091 minus 1199092 + 1198971120572) sdot 120575 (1199091)120575 (1199091) (I8)

After simplifications (I8) produces that

119876120575(1199091) asymp 1198651 minus (1198983119892) sdot 1(2 sdot 1198972) sdot (1199091 minus 1199092 + 1198971120572) (I9)

Derived expression (I9) generalizes previous formula(E2) in the case of 1198651 = 0 Formula (I9) has been inde-pendently derived through virtual displacements Similarityof expressions (E2) and (I9) proves the correctness of theproposed mathematical model Substitution of 1198651 = 0into (I9) again yields (E2) that proves and confirms thecorrectness of (E2)

Calculation of the second generalized force 119876120575(1199092)requires the imposition of the second independent virtualdisplacement 120575(1199092) = 0 to the mechanical system in Figure 6with simultaneous ldquofreezingrdquo or ldquosticking downrdquo of cart 119860 tothe lower rails and ldquofreezingrdquo or ldquosticking to the leftrdquo of cableAE to the vertical axis 119911 factually assuming zero-values ofanother two virtual displacements 120575(1199091) = 0 and 120575(120572) = 0

119876120575(1199092) = ( 1120575 (1199092))sdot ((sum

119896

120575119860119886119896)120575(1199092)

)10038161003816100381610038161003816100381610038161003816100381610038161003816120575(1199091)=0 120575(120572)=0= 1198652 sdot 120575 (1199092) + (1205751198602 (m3g))120575(1199092)120575 (1199092)

(I10)


where the virtual work of the force 1198651 in (I10)-(I11) at thevirtual displacement 120575(1199092) = 0 has a zero-value because inour imagination we made ldquofreezingrdquo or ldquosticking downrdquo ofthe cart 119860 to the lower rails with 120575(1199091) = 0

The second term (1205751198602(m3g))120575(1199092) in the numerator of(I10) is the virtual work of the gravity forcem3g of the cross-beam ED in the case of 120575(1199092) = 0 with 120575(1199091) = 0 and120575(120572) = 0 The minus sign in the numerator of (I11) showsthat gravity forcem3g has negative work due to lifting up thepoint 1198623 in this case in Figure 6 It is possible to calculate thealgebraic expression of the isochronous variation (120575(1199111198623))120575(1199092)in the present case through the substitution of 120575(1199091) = 0 and120575(120572) = 0 into (I4)(120575 (1199111198623))120575(1199092) = (120575 (1199111198623))10038161003816100381610038161003816120575(1199091)=0 120575(120572)=0asymp 1(2 sdot 1198972)sdot ((1199091 minus 1199092 + 1198971120572) sdot (minus120575 (1199092)))

(I12)



119876120575(1199092) asymp 1198652 sdot 120575 (1199092) + (minus1) sdot (1198983119892) sdot 1 (2 sdot 1198972) sdot (1199091 minus 1199092 + 1198971120572) sdot (minus1) sdot 120575 (1199092)120575 (1199092) (I13)


119876120575(1199092) asymp 1198652 + (1198983119892) sdot 1(2 sdot 1198972) sdot (1199091 minus 1199092 + 1198971120572) (I14)

Derived expression (I14) generalizes previous formula(E4) in the case of 1198652 = 0 Formula (I14) has beenindependently derived through virtual displacements Sim-ilarity of expressions (E4) and (I14) proves the correctnessof proposed mathematical model Substitution of 1198652 = 0into (I14) yields again (E4) that proves and confirms thecorrectness of (E4)

Calculation of the third generalized force 119876120575(120572) requiresthe imposition of the third independent virtual displacement120575(120572) = 0 to the mechanical system in Figure 6 withsimultaneous ldquofreezingrdquo or ldquosticking downrdquo of cart 119860 to thelower rails and ldquofreezingrdquo or ldquosticking downrdquo of cart 119861 tothe upper rails factually assuming zero-values of another twovirtual displacements 120575(1199091) = 0 and 120575(1199092) = 0119876120575(120572) = ( 1120575 (120572)) sdot ((sum

119896

120575119860119886119896)120575(120572)

)10038161003816100381610038161003816100381610038161003816100381610038161003816120575(1199091)=0 120575(1199092)=0= (1205751198603 (m3g))120575(120572)120575 (120572) (I15)

119876120575(120572) = (minus1) sdot (1198983119892) sdot (120575 (1199111198623))120575(120572)120575 (120572) (I16)

where the virtual works of the forces 1198651 and 1198652 in (I15)-(I16)at the virtual displacement 120575(120572) = 0 have zero-values becausein our imagination we made ldquofreezingrdquo or ldquosticking downrdquo ofcart 119860 to the lower rails and ldquofreezingrdquo or ldquosticking downrdquo ofcart 119861 to the upper rails with 120575(1199091) = 0 and 120575(1199092) = 0

The term (1205751198603(m3g))120575(120572) in the numerator of (I15) is thevirtual work of the gravity forcem3g of the cross-beam ED inthe case of 120575(120572) = 0 with 120575(1199091) = 0 and 120575(1199092) = 0 The minussign in the numerator of (I16) shows that gravity force m3ghas negative work due to lifting up the point 1198623 in this casein Figure 6 It is possible to calculate the algebraic expressionof the isochronous variation (120575(1199111198623))120575(120572) in the present casethrough the substitution of 120575(1199091) = 0 and 120575(1199092) = 0 into(I4)(120575 (1199111198623))120575(120572) = (120575 (1199111198623))10038161003816100381610038161003816120575(1199091)=0 120575(1199092)=0asymp (1198971 sdot 120572) sdot 120575 (120572) + 1(2 sdot 1198972)sdot ((1199091 minus 1199092 + 1198971120572) sdot (1198971 sdot 120575 (120572)) minus 1198971 sdot 1198972 sdot 120572 sdot 120575 (120572))

(I17)


119876120575(120572) asymp (minus1) sdot (1198983119892) sdot ((1198971 sdot 120572) sdot 120575 (120572) + 1 (2 sdot 1198972) sdot ((1199091 minus 1199092 + 1198971120572) sdot (1198971 sdot 120575 (120572)) minus 1198971 sdot 1198972 sdot 120572 sdot 120575 (120572)))120575 (120572) (I18)


119876120575(120572) asymp (minus1) sdot (1198983119892) sdot ((1198971 sdot 120572) + 1(2 sdot 1198972)sdot (1198971 sdot (1199091 minus 1199092 + 1198971120572) minus 1198971 sdot 1198972 sdot 120572)) (I19)

The derived expression (I19) is completely identical toprevious formula (E6) Formula (I19) has been indepen-dently derived through virtual displacements Coincidenceof expressions (E6) and (I19) proves the correctness of theproposed mathematical model

J Lagrange Equations of the GuidedMechanical System

Application of independent control forces 1198651 = 1199061(119905) [N] and1198652 = 1199062(119905) [N] requires the estimation of new values of threegeneralized forces1198761205751199091 1198761205751199092 and119876120575120572 in the right-hand sidesof the following Lagrange equations for Figure 6

119889119889119905 ( 1205971198791205971) minus 1205971198791205971199091 = 119876120575(1199091) (J1)119889119889119905 ( 1205971198791205972) minus 1205971198791205971199092 = 119876120575(1199092) (J2)119889119889119905 (120597119879120597 ) minus 120597119879120597120572 = 119876120575(120572) (J3)


where magnitudes of generalized forces 119876120575(1199091) 119876120575(1199092) and119876120575(120572) are determined by the action of three independentexternal forces 1198651 = 1199061(119905) [N] 1198652 = 1199062(119905) [N] and 1198983119892 [N](Figure 6) The left-hand sides of Lagrange equations (J1)ndash(J3) are determined by (G5) (G11) and (G17)

Substitution of (G5) and (I9) into (J1) results in thefollowing first Lagrange equation for the guided mechanicalsystem (1198981 + 1198983) sdot (119889211990911198891199052 ) + 1198983 sdot 1198971 sdot (11988921205721198891199052 )asymp 1198651 minus (1198983119892) sdot 1(2 sdot 1198972) sdot (1199091 minus 1199092 + 1198971120572) (J4)

Substitution of (G11) and (I14) into (J2) results intothe following second Lagrange equation for the guidedmechanical system1198982 sdot (119889211990921198891199052 ) asymp 1198652 + (1198983119892) sdot 1(2 sdot 1198972)sdot (1199091 minus 1199092 + 1198971120572) (J5)

Substitution of (G17) and (I19) into (J3) results into thefollowing third Lagrange equation for the guided mechanicalsystem1198983 sdot 11989721 sdot (11988921205721198891199052 ) + 1198983 sdot 1198971 sdot (119889211990911198891199052 ) asymp (minus1) sdot (1198983119892)sdot ((1198971 sdot 120572) + 1(2 sdot 1198972)sdot (1198971 sdot (1199091 minus 1199092 + 1198971120572) minus 1198971 sdot 1198972 sdot 120572))

(J6)

Nomenclature

DOF Degree of freedomODE Ordinary differential equationpoint 119860 Point of bridge crane 1 positionpoint 119861 Point of bridge crane 2 positionDE Cross-beam 3point 1198623 Point of gravity center of cross-beam 3119892 Scalar value of gravitational acceleration

(ms2)119905 Current time (s)1198971 Length of the left cable AE that is119897AE = 1198971 (m)1198972 Length of the right cable BD that is119897BD = 1198972 (m)119886 Length of cross-beam DE (m)119887 Vertical distance between the carts 119860 and119861 (m)1199091 The horizontal coordinate of bridge cranep 119860 (m)1198891199091119889119905 The horizontal projection of bridge cranep 119860 velocity (ms)

119889211990911198891199052 The horizontal projection of bridge crane p119860 acceleration (ms2)1199092 The horizontal coordinate of bridge crane p119861 (m)1198891199092119889119905 The horizontal projection of bridge crane p119861 velocity (ms)119889211990921198891199052 The horizontal projection of bridge crane p119861 acceleration (ms2)1199091198623 The horizontal coordinate of p 1198623 (m)119889(1199091198623)119889119905 The horizontal projection of p 1198623 velocity(ms)1198892(1199091198623)1198891199052 The horizontal projection of p 1198623 accelera-tion (ms2)1199111198623 The vertical coordinate of p 1198623 (m)119889(1199111198623)119889119905 The vertical projection of p1198623 velocity (ms)1198892(1199111198623)1198891199052 The vertical projection of p 1198623 acceleration(ms2)120572 The angular distance between cable AE andvertical line (rad)119889120572119889119905 The angular velocity of cable AE around p119860(rads)11988921205721198891199052 The angular acceleration of cable AE aroundp 119860 (rads2)120573 The angular distance between cable BD andvertical line (rad)119889120573119889119905 The angular velocity of cable BD around p 119861(rads)11988921205731198891199052 The angular acceleration of cable BD aroundp 119861 (rads2)120574 The inclination angle of cross-beam DE withhorizontal line (rad)119889120574119889119905 The angular velocity of cross-beam DEaround horizontal line or around p 119864 (rads)11988921205741198891199052 The angular acceleration of cross-beam DEaround horizontal line or around p 119864(rads2)1198981 Mass of bridge crane p 119860 (kg)1198982 Mass of bridge crane p 119861 (kg)1198983 Mass of cross-beam DE (kg)119879 Kinetic energy of the system ldquoheavy cross-beam two moving cartsrdquo (J = N-m)Π Potential energy of the system ldquoheavy cross-beam two moving cartsrdquo (J = N-m)1198761199091 1198761199092 119876120572 Generalized forces (N)120582119894 Roots of the secular equation1198961 1198962 First and second natural frequencies of sys-tem oscillations (1s)1205911 1205912 First and second natural periods of systemoscillations (s)

Additional Points

Highlights A new 2D model of a 3DOF system ldquotwo movingbridge cranes transported cross-beamrdquo was proposed Anew open loop optimal control problem was formulatedand solved with JModelicaorg freeware Simple engineeringmethodology for optimal control realization was proposed


Disclosure

The submission of the authorsrsquo paper implies that it has notbeen previously published that it is not under considerationfor publication elsewhere and that it will not be publishedelsewhere in the same form without the written permissionof the editors

Competing Interests

The authors Alexander V Perig Alexander N StadnikAlexander A Kostikov and Sergey V Podlesny declare thatthere is no conflict of interests regarding the publication ofthis paper

Authorsrsquo Contributions

All authors participated in the design of this work andperformed equally All authors read and approved the finalmanuscript

References

[1] E M Abdel-Rahman A H Nayfeh and Z N MasoudldquoDynamics and control of cranes a reviewrdquo Journal of Vibrationand Control vol 9 no 7 pp 863ndash908 2003

[2] M S A Deen Ali N R Babu and K Varghese ldquoCollision freepath planning of cooperative crane manipulators using geneticalgorithmrdquo Journal of Computing in Civil Engineering vol 19no 2 pp 182ndash193 2005

[3] A Arena A Casalotti W Lacarbonara and M P CartmellldquoDynamics of container cranes three-dimensional modelingfull-scale experiments and identificationrdquo International Journalof Mechanical Sciences vol 93 pp 8ndash21 2015

[4] M P Cartmell L Morrish and A J Taylor ldquoDynamics ofspreadermotion in a gantry cranerdquo Proceedings of the InstitutionofMechanical Engineers Part C Journal ofMechanical Engineer-ing Science vol 212 no 2 pp 85ndash105 1998

[5] G Castelli E Ottaviano and P Rea ldquoA Cartesian Cable-SuspendedRobot for improving end-usersrsquomobility in an urbanenvironmentrdquo Robotics and Computer-Integrated Manufactur-ing vol 30 no 3 pp 335ndash343 2014

[6] J-H Cha S-H Ham K-Y Lee and M-I Roh ldquoApplicationof a topological modelling approach of multi-body systemdynamics to simulation of multi-floating cranes in shipyardsrdquoProceedings of the Institution of Mechanical Engineers Part KJournal of Multi-Body Dynamics vol 224 no 4 pp 365ndash3732010

[7] K Goodwin ldquoRoboCrane construction of bridgesrdquo Transporta-tion Research Record no 1575 pp 42ndash46 1997

[8] J Huang Z Liang and Q Zang ldquoDynamics and swingcontrol of double-pendulum bridge cranes with distributed-mass beamsrdquoMechanical Systems and Signal Processing vol 54-55 pp 357ndash366 2015

[9] S Lahouar E Ottaviano S Zeghoul L Romdhane andM Ceccarelli ldquoCollision free path-planning for cable-drivenparallel robotsrdquo Robotics and Autonomous Systems vol 57 no11 pp 1083ndash1093 2009

[10] L Pigani and P Gallina ldquoCable-direct-driven-robot (CDDR)with a 3-link passive serial supportrdquo Robotics and Computer-Integrated Manufacturing vol 30 no 3 pp 265ndash276 2014

[11] O Sawodny H Aschemann and S Lahres ldquoAn automatedgantry crane as a large workspace robotrdquo Control EngineeringPractice vol 10 no 12 pp 1323ndash1338 2002

[12] J Smoczek ldquoFuzzy crane control with sensorless payload deflec-tion feedback for vibration reductionrdquoMechanical Systems andSignal Processing vol 46 no 1 pp 70ndash81 2014

[13] B Zi B Y Duan J L Du and H Bao ldquoDynamic modeling andactive control of a cable-suspended parallel robotrdquoMechatron-ics vol 18 no 1 pp 1ndash12 2008

[14] A A Kostikov A V Perig D Y Mikhieienko and R R LozunldquoNumerical JModelicaorg-based approach to a simulation ofCoriolis effects on guided boom-driven payload swaying duringnon-uniform rotary crane boom slewingrdquo Journal of the Brazil-ian Society of Mechanical Sciences and Engineering vol 39 no3 pp 737ndash756 2017

[15] A A Kostikov A V Perig and R R Lozun ldquoSimulation-assisted teaching of graduate students in transport a case studyof the application of acausal freeware JModelicaorg to solutionof Sakawarsquos open-loop optimal control problem for payloadmotion during crane boom rotationrdquo International Journal ofMechanical Engineering Education vol 45 no 1 pp 3ndash27 2017

[16] A V Perig A N Stadnik and A I Deriglazov ldquoSphericalpendulum small oscillations for slewing crane motionrdquo TheScientific World Journal vol 2014 Article ID 451804 10 pages2014

[17] A V Perig A N Stadnik A I Deriglazov and S V Podlesnyldquo3DOF spherical pendulumoscillationswith a uniform slewingpivot center and a small angle assumptionrdquo Shock andVibrationvol 2014 Article ID 203709 32 pages 2014

[18] J G Papastavridis Analytical Mechanics A ComprehensiveTreatise on the Dynamics of Constrained Systems for EngineersPhysicists Oxford University Press New York NY USA 2001

[19] A B Modelon ldquoJModelicaorg User Guide Version 117rdquo 2015httpwwwjmodelicaorgapi-docsusersguideJModelicaUs-ersGuide-1170pdf

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpswwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



Figure 1 Typical plant photos of heavy cross-beam transportation with two overhead cranes

z

D

E

xOb

x1

x2

C1

C2

A

B

m1g

m3g

m2g

l1

l2

120572120574

120573

C3

(120574 = 1205740 + 120575)

a2

a2

a cos (1205740)

zC3

xC3

Figure 2 The computational scheme for transportation of heavy cross-beam DE by two bridge cranes 119860 and 119861Other examples in civil engineering include the lifting-

and-handling problems of modular building elements ofconstruction bays of a building ceiling and floor slab panelsand spandrel walls Construction problems of roofs floorsand domes of roofed stadiums and shopping centers requirepreliminary manufacturing of the heavy bulk trussed framemodules with further transportation to building sites

It is important to note that attempts to transport heavybulk stores with a single crane are very undesirable because ofthe high probability of torsional oscillations of a heavy loadMoreover usage of two cranes working together for heavy

load transportation is a truly natural way for handlingdisplacements in civil and mechanical engineering as natureitself has supplied humans with two hands and two legs formore efficient performance of lifting-and-handling opera-tions

Overhead crane dynamics has been studied in theresearch efforts of such scientists as Abdel-Rahman et al(2003) [1] Deen Ali et al (2005) [2] Arena et al (2015) [3]Cartmell et al (1998) [4] Castelli et al (2014) [5] Cha etal (2010) [6] Goodwin (1997) [7] Huang et al (2015) [8]Lahouar et al (2009) [9] Pigani and Gallina (2014) [10]


z

B

b

O

D

E

C1 A

C2

x1

x2

120573

120572

m1g

m2g

l1

l2

C3

m3g

1205740 Δz0


Δz0a

a2

a2

DE = a

AE = l1 BD = l2

sin (1205740) =

sin (1205740) =

(l1 minus l2 + b)

a

a cos (1205740)


























(1)










(5)






(8)





(10)



sdot(((((

(119889211990911198891199052 )(119889211990921198891199052 )(11988921205721198891199052 ))))))

minus0(11)




0 2 4 6 8 10

Time (s)

minus01

00

01

02

03

04

05

06

Coo

rdin

ates

of c

arts

1 an

d 2


x1

x2

(a)

0 2 4 6 8 10

Time (s)

minus020

minus015

minus010

minus005

000

005

010

015

020

x2minusx1

(b)




(119889211990911198891199052 )(119889211990921198891199052 )(11988921205721198891199052 )))))))asymp (1198983119892) sdot 1(2 sdot 1198972)






0 2 4 6 8 10

Time (s)

minus003

minus002

minus001

000

001

002

003

120572

(a)

0 2 4 6 8 10

Time (s)

minus003

minus002

minus001

000

001

002

003

minus004

120573

(b)

0 2 4 6 8 10

Time (s)

minus00004

minus00002

00000

00002

00004

00006

00008

00010

120575

(c)

0 2 4 6 8 10

Time (s)

08478

08480

08482

08484

08486

08488

08490

120574

(d)

0 2 4 6 8 10

Time (s)

minus0003

minus0002

minus0001

0000

0001

0002

0003

d(120574

)dt

(e)

08478 08480 08482 08484 08486 08488 08490minus0003

minus0002

minus0001

0000

0001

0002

0003

120574

d(120574

)dt

(f)



z

D

E

xOb

x1

x2

C1

C2

A

B 120575x2

120575x1

F2 = u2(t)

F1 = u1(t)

120575120572

120572

l1

l2m1g

m2g

m3g

a2

a2 120574

120573

(120574 = 1205740 + 120575)

C3

a cos (1205740)

zC3

xC3










(14)




(119889211990911198891199052 )(119889211990921198891199052 )(11988921205721198891199052 ))))))asymp (1198983119892) sdot 1(2 sdot 1198972)


(15)





0(120572 (119905))2 119889119905 (16)






1199091 (0) = 01199091119901 (0) = 01199092 (0) = 01199092119901 (0) = 0120572 (0) = 0120572119901 (0) = 0(23)

1199091 (119905119891) = 1198781199091119901 (119905119891) = 01199092 (119905119891) = 1198781199092119901 (119905119891) = 0120572 (119905119891) = 0120572119901 (119905119891) = 0(24)

10038161003816100381610038161199061 (119905)1003816100381610038161003816 le 1199061max10038161003816100381610038161199062 (119905)1003816100381610038161003816 le 1199062max (25)
















9 Final Conclusions




0 20 40 60 80 100

Time (s)

minus10

0

10

20

30

40

50

x1

x2

x3


Coo

rdin

ates

of c

arts

1 2

and

the c

ente

r of

mas

s of a

cros

s-be

am

(a)

0 20 40 60 80 100

Time (s)u1

u2

minus100

minus50

0

50

100

u1

andu2

Con

trols

(b)

120572

120573

0 20 40 60 80 100

Time (s)

minus0003

minus0002

minus0001

0000

0001

0002

0003

120572an

d120573

(c)

0 20 40 60 80 100

Time (s)

minus0000003

minus0000002

minus0000001

0000000

0000001

0000002

0000003

0000004

0000005120575

(d)

0 20 40 60 80 100

Time (s)

00

01

02

03

04

05

06

07

08

09

12

1

2an

d c

3

c3

(e)

minus0000015

minus0000010

minus0000005

0000000

0000005

0000010

0000015

000

0001

000

0002

000

0003

000

0004

000

0005

000

0006

000

0007

000

0008

000

0009

848058e minus 1

120574

+

d(120574

)dt

(f)












Appendix




(A1)








(A5)








cos (120574) asymp cos (1205740) (A10)






(B1)













(C2)


(C3)



(C4)



(D1)


(D2)








(E3)





















1198793 asymp (12) sdot 1198983 sdot (119889 (1199091198623)119889119905 )2 (F5)



(F7)































1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816= 0 (H1)













= 1198651 sdot 120575 (1199091) + (1205751198601 (m3g))120575(1199091)120575 (1199091) (I5)











119876120575(1199092) = ( 1120575 (1199092))sdot ((sum

119896

120575119860119886119896)120575(1199092)

)10038161003816100381610038161003816100381610038161003816100381610038161003816120575(1199091)=0 120575(120572)=0= 1198652 sdot 120575 (1199092) + (1205751198602 (m3g))120575(1199092)120575 (1199092)

(I10)




(I12)








119896

120575119860119886119896)120575(120572)

)10038161003816100381610038161003816100381610038161003816100381610038161003816120575(1199091)=0 120575(1199092)=0= (1205751198603 (m3g))120575(120572)120575 (120572) (I15)

119876120575(120572) = (minus1) sdot (1198983119892) sdot (120575 (1199111198623))120575(120572)120575 (120572) (I16)



(I17)














(J6)

Nomenclature




Additional Points



Disclosure


Competing Interests




References






















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










z

B

b

O

D

E

C1 A

C2

x1

x2

120573

120572

m1g

m2g

l1

l2

C3

m3g

1205740 Δz0


Δz0a

a2

a2

DE = a

AE = l1 BD = l2

sin (1205740) =

sin (1205740) =

(l1 minus l2 + b)

a

a cos (1205740)


























(1)










(5)






(8)





(10)



sdot(((((

(119889211990911198891199052 )(119889211990921198891199052 )(11988921205721198891199052 ))))))

minus0(11)




0 2 4 6 8 10

Time (s)

minus01

00

01

02

03

04

05

06

Coo

rdin

ates

of c

arts

1 an

d 2


x1

x2

(a)

0 2 4 6 8 10

Time (s)

minus020

minus015

minus010

minus005

000

005

010

015

020

x2minusx1

(b)




(119889211990911198891199052 )(119889211990921198891199052 )(11988921205721198891199052 )))))))asymp (1198983119892) sdot 1(2 sdot 1198972)






0 2 4 6 8 10

Time (s)

minus003

minus002

minus001

000

001

002

003

120572

(a)

0 2 4 6 8 10

Time (s)

minus003

minus002

minus001

000

001

002

003

minus004

120573

(b)

0 2 4 6 8 10

Time (s)

minus00004

minus00002

00000

00002

00004

00006

00008

00010

120575

(c)

0 2 4 6 8 10

Time (s)

08478

08480

08482

08484

08486

08488

08490

120574

(d)

0 2 4 6 8 10

Time (s)

minus0003

minus0002

minus0001

0000

0001

0002

0003

d(120574

)dt

(e)

08478 08480 08482 08484 08486 08488 08490minus0003

minus0002

minus0001

0000

0001

0002

0003

120574

d(120574

)dt

(f)



z

D

E

xOb

x1

x2

C1

C2

A

B 120575x2

120575x1

F2 = u2(t)

F1 = u1(t)

120575120572

120572

l1

l2m1g

m2g

m3g

a2

a2 120574

120573

(120574 = 1205740 + 120575)

C3

a cos (1205740)

zC3

xC3










(14)




(119889211990911198891199052 )(119889211990921198891199052 )(11988921205721198891199052 ))))))asymp (1198983119892) sdot 1(2 sdot 1198972)


(15)





0(120572 (119905))2 119889119905 (16)






1199091 (0) = 01199091119901 (0) = 01199092 (0) = 01199092119901 (0) = 0120572 (0) = 0120572119901 (0) = 0(23)

1199091 (119905119891) = 1198781199091119901 (119905119891) = 01199092 (119905119891) = 1198781199092119901 (119905119891) = 0120572 (119905119891) = 0120572119901 (119905119891) = 0(24)

10038161003816100381610038161199061 (119905)1003816100381610038161003816 le 1199061max10038161003816100381610038161199062 (119905)1003816100381610038161003816 le 1199062max (25)
















9 Final Conclusions




0 20 40 60 80 100

Time (s)

minus10

0

10

20

30

40

50

x1

x2

x3


Coo

rdin

ates

of c

arts

1 2

and

the c

ente

r of

mas

s of a

cros

s-be

am

(a)

0 20 40 60 80 100

Time (s)u1

u2

minus100

minus50

0

50

100

u1

andu2

Con

trols

(b)

120572

120573

0 20 40 60 80 100

Time (s)

minus0003

minus0002

minus0001

0000

0001

0002

0003

120572an

d120573

(c)

0 20 40 60 80 100

Time (s)

minus0000003

minus0000002

minus0000001

0000000

0000001

0000002

0000003

0000004

0000005120575

(d)

0 20 40 60 80 100

Time (s)

00

01

02

03

04

05

06

07

08

09

12

1

2an

d c

3

c3

(e)

minus0000015

minus0000010

minus0000005

0000000

0000005

0000010

0000015

000

0001

000

0002

000

0003

000

0004

000

0005

000

0006

000

0007

000

0008

000

0009

848058e minus 1

120574

+

d(120574

)dt

(f)












Appendix




(A1)








(A5)








cos (120574) asymp cos (1205740) (A10)






(B1)













(C2)


(C3)



(C4)



(D1)


(D2)








(E3)





















1198793 asymp (12) sdot 1198983 sdot (119889 (1199091198623)119889119905 )2 (F5)



(F7)































1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816= 0 (H1)













= 1198651 sdot 120575 (1199091) + (1205751198601 (m3g))120575(1199091)120575 (1199091) (I5)











119876120575(1199092) = ( 1120575 (1199092))sdot ((sum

119896

120575119860119886119896)120575(1199092)

)10038161003816100381610038161003816100381610038161003816100381610038161003816120575(1199091)=0 120575(120572)=0= 1198652 sdot 120575 (1199092) + (1205751198602 (m3g))120575(1199092)120575 (1199092)

(I10)




(I12)








119896

120575119860119886119896)120575(120572)

)10038161003816100381610038161003816100381610038161003816100381610038161003816120575(1199091)=0 120575(1199092)=0= (1205751198603 (m3g))120575(120572)120575 (120572) (I15)

119876120575(120572) = (minus1) sdot (1198983119892) sdot (120575 (1199111198623))120575(120572)120575 (120572) (I16)



(I17)














(J6)

Nomenclature




Additional Points



Disclosure


Competing Interests




References






















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation


















(1)










(5)






(8)





(10)



sdot(((((

(119889211990911198891199052 )(119889211990921198891199052 )(11988921205721198891199052 ))))))

minus0(11)




0 2 4 6 8 10

Time (s)

minus01

00

01

02

03

04

05

06

Coo

rdin

ates

of c

arts

1 an

d 2


x1

x2

(a)

0 2 4 6 8 10

Time (s)

minus020

minus015

minus010

minus005

000

005

010

015

020

x2minusx1

(b)




(119889211990911198891199052 )(119889211990921198891199052 )(11988921205721198891199052 )))))))asymp (1198983119892) sdot 1(2 sdot 1198972)






0 2 4 6 8 10

Time (s)

minus003

minus002

minus001

000

001

002

003

120572

(a)

0 2 4 6 8 10

Time (s)

minus003

minus002

minus001

000

001

002

003

minus004

120573

(b)

0 2 4 6 8 10

Time (s)

minus00004

minus00002

00000

00002

00004

00006

00008

00010

120575

(c)

0 2 4 6 8 10

Time (s)

08478

08480

08482

08484

08486

08488

08490

120574

(d)

0 2 4 6 8 10

Time (s)

minus0003

minus0002

minus0001

0000

0001

0002

0003

d(120574

)dt

(e)

08478 08480 08482 08484 08486 08488 08490minus0003

minus0002

minus0001

0000

0001

0002

0003

120574

d(120574

)dt

(f)



z

D

E

xOb

x1

x2

C1

C2

A

B 120575x2

120575x1

F2 = u2(t)

F1 = u1(t)

120575120572

120572

l1

l2m1g

m2g

m3g

a2

a2 120574

120573

(120574 = 1205740 + 120575)

C3

a cos (1205740)

zC3

xC3










(14)




(119889211990911198891199052 )(119889211990921198891199052 )(11988921205721198891199052 ))))))asymp (1198983119892) sdot 1(2 sdot 1198972)


(15)





0(120572 (119905))2 119889119905 (16)






1199091 (0) = 01199091119901 (0) = 01199092 (0) = 01199092119901 (0) = 0120572 (0) = 0120572119901 (0) = 0(23)

1199091 (119905119891) = 1198781199091119901 (119905119891) = 01199092 (119905119891) = 1198781199092119901 (119905119891) = 0120572 (119905119891) = 0120572119901 (119905119891) = 0(24)

10038161003816100381610038161199061 (119905)1003816100381610038161003816 le 1199061max10038161003816100381610038161199062 (119905)1003816100381610038161003816 le 1199062max (25)
















9 Final Conclusions




0 20 40 60 80 100

Time (s)

minus10

0

10

20

30

40

50

x1

x2

x3


Coo

rdin

ates

of c

arts

1 2

and

the c

ente

r of

mas

s of a

cros

s-be

am

(a)

0 20 40 60 80 100

Time (s)u1

u2

minus100

minus50

0

50

100

u1

andu2

Con

trols

(b)

120572

120573

0 20 40 60 80 100

Time (s)

minus0003

minus0002

minus0001

0000

0001

0002

0003

120572an

d120573

(c)

0 20 40 60 80 100

Time (s)

minus0000003

minus0000002

minus0000001

0000000

0000001

0000002

0000003

0000004

0000005120575

(d)

0 20 40 60 80 100

Time (s)

00

01

02

03

04

05

06

07

08

09

12

1

2an

d c

3

c3

(e)

minus0000015

minus0000010

minus0000005

0000000

0000005

0000010

0000015

000

0001

000

0002

000

0003

000

0004

000

0005

000

0006

000

0007

000

0008

000

0009

848058e minus 1

120574

+

d(120574

)dt

(f)












Appendix




(A1)








(A5)








cos (120574) asymp cos (1205740) (A10)






(B1)













(C2)


(C3)



(C4)



(D1)


(D2)








(E3)





















1198793 asymp (12) sdot 1198983 sdot (119889 (1199091198623)119889119905 )2 (F5)



(F7)































1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816= 0 (H1)













= 1198651 sdot 120575 (1199091) + (1205751198601 (m3g))120575(1199091)120575 (1199091) (I5)











119876120575(1199092) = ( 1120575 (1199092))sdot ((sum

119896

120575119860119886119896)120575(1199092)

)10038161003816100381610038161003816100381610038161003816100381610038161003816120575(1199091)=0 120575(120572)=0= 1198652 sdot 120575 (1199092) + (1205751198602 (m3g))120575(1199092)120575 (1199092)

(I10)




(I12)








119896

120575119860119886119896)120575(120572)

)10038161003816100381610038161003816100381610038161003816100381610038161003816120575(1199091)=0 120575(1199092)=0= (1205751198603 (m3g))120575(120572)120575 (120572) (I15)

119876120575(120572) = (minus1) sdot (1198983119892) sdot (120575 (1199111198623))120575(120572)120575 (120572) (I16)



(I17)














(J6)

Nomenclature




Additional Points



Disclosure


Competing Interests




References






















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












(5)






(8)





(10)



sdot(((((

(119889211990911198891199052 )(119889211990921198891199052 )(11988921205721198891199052 ))))))

minus0(11)




0 2 4 6 8 10

Time (s)

minus01

00

01

02

03

04

05

06

Coo

rdin

ates

of c

arts

1 an

d 2


x1

x2

(a)

0 2 4 6 8 10

Time (s)

minus020

minus015

minus010

minus005

000

005

010

015

020

x2minusx1

(b)




(119889211990911198891199052 )(119889211990921198891199052 )(11988921205721198891199052 )))))))asymp (1198983119892) sdot 1(2 sdot 1198972)






0 2 4 6 8 10

Time (s)

minus003

minus002

minus001

000

001

002

003

120572

(a)

0 2 4 6 8 10

Time (s)

minus003

minus002

minus001

000

001

002

003

minus004

120573

(b)

0 2 4 6 8 10

Time (s)

minus00004

minus00002

00000

00002

00004

00006

00008

00010

120575

(c)

0 2 4 6 8 10

Time (s)

08478

08480

08482

08484

08486

08488

08490

120574

(d)

0 2 4 6 8 10

Time (s)

minus0003

minus0002

minus0001

0000

0001

0002

0003

d(120574

)dt

(e)

08478 08480 08482 08484 08486 08488 08490minus0003

minus0002

minus0001

0000

0001

0002

0003

120574

d(120574

)dt

(f)



z

D

E

xOb

x1

x2

C1

C2

A

B 120575x2

120575x1

F2 = u2(t)

F1 = u1(t)

120575120572

120572

l1

l2m1g

m2g

m3g

a2

a2 120574

120573

(120574 = 1205740 + 120575)

C3

a cos (1205740)

zC3

xC3










(14)




(119889211990911198891199052 )(119889211990921198891199052 )(11988921205721198891199052 ))))))asymp (1198983119892) sdot 1(2 sdot 1198972)


(15)





0(120572 (119905))2 119889119905 (16)






1199091 (0) = 01199091119901 (0) = 01199092 (0) = 01199092119901 (0) = 0120572 (0) = 0120572119901 (0) = 0(23)

1199091 (119905119891) = 1198781199091119901 (119905119891) = 01199092 (119905119891) = 1198781199092119901 (119905119891) = 0120572 (119905119891) = 0120572119901 (119905119891) = 0(24)

10038161003816100381610038161199061 (119905)1003816100381610038161003816 le 1199061max10038161003816100381610038161199062 (119905)1003816100381610038161003816 le 1199062max (25)
















9 Final Conclusions




0 20 40 60 80 100

Time (s)

minus10

0

10

20

30

40

50

x1

x2

x3


Coo

rdin

ates

of c

arts

1 2

and

the c

ente

r of

mas

s of a

cros

s-be

am

(a)

0 20 40 60 80 100

Time (s)u1

u2

minus100

minus50

0

50

100

u1

andu2

Con

trols

(b)

120572

120573

0 20 40 60 80 100

Time (s)

minus0003

minus0002

minus0001

0000

0001

0002

0003

120572an

d120573

(c)

0 20 40 60 80 100

Time (s)

minus0000003

minus0000002

minus0000001

0000000

0000001

0000002

0000003

0000004

0000005120575

(d)

0 20 40 60 80 100

Time (s)

00

01

02

03

04

05

06

07

08

09

12

1

2an

d c

3

c3

(e)

minus0000015

minus0000010

minus0000005

0000000

0000005

0000010

0000015

000

0001

000

0002

000

0003

000

0004

000

0005

000

0006

000

0007

000

0008

000

0009

848058e minus 1

120574

+

d(120574

)dt

(f)












Appendix




(A1)








(A5)








cos (120574) asymp cos (1205740) (A10)






(B1)













(C2)


(C3)



(C4)



(D1)


(D2)








(E3)





















1198793 asymp (12) sdot 1198983 sdot (119889 (1199091198623)119889119905 )2 (F5)



(F7)































1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816= 0 (H1)













= 1198651 sdot 120575 (1199091) + (1205751198601 (m3g))120575(1199091)120575 (1199091) (I5)











119876120575(1199092) = ( 1120575 (1199092))sdot ((sum

119896

120575119860119886119896)120575(1199092)

)10038161003816100381610038161003816100381610038161003816100381610038161003816120575(1199091)=0 120575(120572)=0= 1198652 sdot 120575 (1199092) + (1205751198602 (m3g))120575(1199092)120575 (1199092)

(I10)




(I12)








119896

120575119860119886119896)120575(120572)

)10038161003816100381610038161003816100381610038161003816100381610038161003816120575(1199091)=0 120575(1199092)=0= (1205751198603 (m3g))120575(120572)120575 (120572) (I15)

119876120575(120572) = (minus1) sdot (1198983119892) sdot (120575 (1199111198623))120575(120572)120575 (120572) (I16)



(I17)














(J6)

Nomenclature




Additional Points



Disclosure


Competing Interests




References






















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 2 4 6 8 10

Time (s)

minus01

00

01

02

03

04

05

06

Coo

rdin

ates

of c

arts

1 an

d 2


x1

x2

(a)

0 2 4 6 8 10

Time (s)

minus020

minus015

minus010

minus005

000

005

010

015

020

x2minusx1

(b)




(119889211990911198891199052 )(119889211990921198891199052 )(11988921205721198891199052 )))))))asymp (1198983119892) sdot 1(2 sdot 1198972)






0 2 4 6 8 10

Time (s)

minus003

minus002

minus001

000

001

002

003

120572

(a)

0 2 4 6 8 10

Time (s)

minus003

minus002

minus001

000

001

002

003

minus004

120573

(b)

0 2 4 6 8 10

Time (s)

minus00004

minus00002

00000

00002

00004

00006

00008

00010

120575

(c)

0 2 4 6 8 10

Time (s)

08478

08480

08482

08484

08486

08488

08490

120574

(d)

0 2 4 6 8 10

Time (s)

minus0003

minus0002

minus0001

0000

0001

0002

0003

d(120574

)dt

(e)

08478 08480 08482 08484 08486 08488 08490minus0003

minus0002

minus0001

0000

0001

0002

0003

120574

d(120574

)dt

(f)



z

D

E

xOb

x1

x2

C1

C2

A

B 120575x2

120575x1

F2 = u2(t)

F1 = u1(t)

120575120572

120572

l1

l2m1g

m2g

m3g

a2

a2 120574

120573

(120574 = 1205740 + 120575)

C3

a cos (1205740)

zC3

xC3










(14)




(119889211990911198891199052 )(119889211990921198891199052 )(11988921205721198891199052 ))))))asymp (1198983119892) sdot 1(2 sdot 1198972)


(15)





0(120572 (119905))2 119889119905 (16)






1199091 (0) = 01199091119901 (0) = 01199092 (0) = 01199092119901 (0) = 0120572 (0) = 0120572119901 (0) = 0(23)

1199091 (119905119891) = 1198781199091119901 (119905119891) = 01199092 (119905119891) = 1198781199092119901 (119905119891) = 0120572 (119905119891) = 0120572119901 (119905119891) = 0(24)

10038161003816100381610038161199061 (119905)1003816100381610038161003816 le 1199061max10038161003816100381610038161199062 (119905)1003816100381610038161003816 le 1199062max (25)
















9 Final Conclusions




0 20 40 60 80 100

Time (s)

minus10

0

10

20

30

40

50

x1

x2

x3


Coo

rdin

ates

of c

arts

1 2

and

the c

ente

r of

mas

s of a

cros

s-be

am

(a)

0 20 40 60 80 100

Time (s)u1

u2

minus100

minus50

0

50

100

u1

andu2

Con

trols

(b)

120572

120573

0 20 40 60 80 100

Time (s)

minus0003

minus0002

minus0001

0000

0001

0002

0003

120572an

d120573

(c)

0 20 40 60 80 100

Time (s)

minus0000003

minus0000002

minus0000001

0000000

0000001

0000002

0000003

0000004

0000005120575

(d)

0 20 40 60 80 100

Time (s)

00

01

02

03

04

05

06

07

08

09

12

1

2an

d c

3

c3

(e)

minus0000015

minus0000010

minus0000005

0000000

0000005

0000010

0000015

000

0001

000

0002

000

0003

000

0004

000

0005

000

0006

000

0007

000

0008

000

0009

848058e minus 1

120574

+

d(120574

)dt

(f)












Appendix




(A1)








(A5)








cos (120574) asymp cos (1205740) (A10)






(B1)













(C2)


(C3)



(C4)



(D1)


(D2)








(E3)





















1198793 asymp (12) sdot 1198983 sdot (119889 (1199091198623)119889119905 )2 (F5)



(F7)































1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816= 0 (H1)













= 1198651 sdot 120575 (1199091) + (1205751198601 (m3g))120575(1199091)120575 (1199091) (I5)











119876120575(1199092) = ( 1120575 (1199092))sdot ((sum

119896

120575119860119886119896)120575(1199092)

)10038161003816100381610038161003816100381610038161003816100381610038161003816120575(1199091)=0 120575(120572)=0= 1198652 sdot 120575 (1199092) + (1205751198602 (m3g))120575(1199092)120575 (1199092)

(I10)




(I12)








119896

120575119860119886119896)120575(120572)

)10038161003816100381610038161003816100381610038161003816100381610038161003816120575(1199091)=0 120575(1199092)=0= (1205751198603 (m3g))120575(120572)120575 (120572) (I15)

119876120575(120572) = (minus1) sdot (1198983119892) sdot (120575 (1199111198623))120575(120572)120575 (120572) (I16)



(I17)














(J6)

Nomenclature




Additional Points



Disclosure


Competing Interests




References






















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 2 4 6 8 10

Time (s)

minus003

minus002

minus001

000

001

002

003

120572

(a)

0 2 4 6 8 10

Time (s)

minus003

minus002

minus001

000

001

002

003

minus004

120573

(b)

0 2 4 6 8 10

Time (s)

minus00004

minus00002

00000

00002

00004

00006

00008

00010

120575

(c)

0 2 4 6 8 10

Time (s)

08478

08480

08482

08484

08486

08488

08490

120574

(d)

0 2 4 6 8 10

Time (s)

minus0003

minus0002

minus0001

0000

0001

0002

0003

d(120574

)dt

(e)

08478 08480 08482 08484 08486 08488 08490minus0003

minus0002

minus0001

0000

0001

0002

0003

120574

d(120574

)dt

(f)



z

D

E

xOb

x1

x2

C1

C2

A

B 120575x2

120575x1

F2 = u2(t)

F1 = u1(t)

120575120572

120572

l1

l2m1g

m2g

m3g

a2

a2 120574

120573

(120574 = 1205740 + 120575)

C3

a cos (1205740)

zC3

xC3










(14)




(119889211990911198891199052 )(119889211990921198891199052 )(11988921205721198891199052 ))))))asymp (1198983119892) sdot 1(2 sdot 1198972)


(15)





0(120572 (119905))2 119889119905 (16)






1199091 (0) = 01199091119901 (0) = 01199092 (0) = 01199092119901 (0) = 0120572 (0) = 0120572119901 (0) = 0(23)

1199091 (119905119891) = 1198781199091119901 (119905119891) = 01199092 (119905119891) = 1198781199092119901 (119905119891) = 0120572 (119905119891) = 0120572119901 (119905119891) = 0(24)

10038161003816100381610038161199061 (119905)1003816100381610038161003816 le 1199061max10038161003816100381610038161199062 (119905)1003816100381610038161003816 le 1199062max (25)
















9 Final Conclusions




0 20 40 60 80 100

Time (s)

minus10

0

10

20

30

40

50

x1

x2

x3


Coo

rdin

ates

of c

arts

1 2

and

the c

ente

r of

mas

s of a

cros

s-be

am

(a)

0 20 40 60 80 100

Time (s)u1

u2

minus100

minus50

0

50

100

u1

andu2

Con

trols

(b)

120572

120573

0 20 40 60 80 100

Time (s)

minus0003

minus0002

minus0001

0000

0001

0002

0003

120572an

d120573

(c)

0 20 40 60 80 100

Time (s)

minus0000003

minus0000002

minus0000001

0000000

0000001

0000002

0000003

0000004

0000005120575

(d)

0 20 40 60 80 100

Time (s)

00

01

02

03

04

05

06

07

08

09

12

1

2an

d c

3

c3

(e)

minus0000015

minus0000010

minus0000005

0000000

0000005

0000010

0000015

000

0001

000

0002

000

0003

000

0004

000

0005

000

0006

000

0007

000

0008

000

0009

848058e minus 1

120574

+

d(120574

)dt

(f)












Appendix




(A1)








(A5)








cos (120574) asymp cos (1205740) (A10)






(B1)













(C2)


(C3)



(C4)



(D1)


(D2)








(E3)





















1198793 asymp (12) sdot 1198983 sdot (119889 (1199091198623)119889119905 )2 (F5)



(F7)































1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816= 0 (H1)













= 1198651 sdot 120575 (1199091) + (1205751198601 (m3g))120575(1199091)120575 (1199091) (I5)











119876120575(1199092) = ( 1120575 (1199092))sdot ((sum

119896

120575119860119886119896)120575(1199092)

)10038161003816100381610038161003816100381610038161003816100381610038161003816120575(1199091)=0 120575(120572)=0= 1198652 sdot 120575 (1199092) + (1205751198602 (m3g))120575(1199092)120575 (1199092)

(I10)




(I12)








119896

120575119860119886119896)120575(120572)

)10038161003816100381610038161003816100381610038161003816100381610038161003816120575(1199091)=0 120575(1199092)=0= (1205751198603 (m3g))120575(120572)120575 (120572) (I15)

119876120575(120572) = (minus1) sdot (1198983119892) sdot (120575 (1199111198623))120575(120572)120575 (120572) (I16)



(I17)














(J6)

Nomenclature




Additional Points



Disclosure


Competing Interests




References






















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










z

D

E

xOb

x1

x2

C1

C2

A

B 120575x2

120575x1

F2 = u2(t)

F1 = u1(t)

120575120572

120572

l1

l2m1g

m2g

m3g

a2

a2 120574

120573

(120574 = 1205740 + 120575)

C3

a cos (1205740)

zC3

xC3










(14)




(119889211990911198891199052 )(119889211990921198891199052 )(11988921205721198891199052 ))))))asymp (1198983119892) sdot 1(2 sdot 1198972)


(15)





0(120572 (119905))2 119889119905 (16)






1199091 (0) = 01199091119901 (0) = 01199092 (0) = 01199092119901 (0) = 0120572 (0) = 0120572119901 (0) = 0(23)

1199091 (119905119891) = 1198781199091119901 (119905119891) = 01199092 (119905119891) = 1198781199092119901 (119905119891) = 0120572 (119905119891) = 0120572119901 (119905119891) = 0(24)

10038161003816100381610038161199061 (119905)1003816100381610038161003816 le 1199061max10038161003816100381610038161199062 (119905)1003816100381610038161003816 le 1199062max (25)
















9 Final Conclusions




0 20 40 60 80 100

Time (s)

minus10

0

10

20

30

40

50

x1

x2

x3


Coo

rdin

ates

of c

arts

1 2

and

the c

ente

r of

mas

s of a

cros

s-be

am

(a)

0 20 40 60 80 100

Time (s)u1

u2

minus100

minus50

0

50

100

u1

andu2

Con

trols

(b)

120572

120573

0 20 40 60 80 100

Time (s)

minus0003

minus0002

minus0001

0000

0001

0002

0003

120572an

d120573

(c)

0 20 40 60 80 100

Time (s)

minus0000003

minus0000002

minus0000001

0000000

0000001

0000002

0000003

0000004

0000005120575

(d)

0 20 40 60 80 100

Time (s)

00

01

02

03

04

05

06

07

08

09

12

1

2an

d c

3

c3

(e)

minus0000015

minus0000010

minus0000005

0000000

0000005

0000010

0000015

000

0001

000

0002

000

0003

000

0004

000

0005

000

0006

000

0007

000

0008

000

0009

848058e minus 1

120574

+

d(120574

)dt

(f)












Appendix




(A1)








(A5)








cos (120574) asymp cos (1205740) (A10)






(B1)













(C2)


(C3)



(C4)



(D1)


(D2)








(E3)





















1198793 asymp (12) sdot 1198983 sdot (119889 (1199091198623)119889119905 )2 (F5)



(F7)































1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816= 0 (H1)













= 1198651 sdot 120575 (1199091) + (1205751198601 (m3g))120575(1199091)120575 (1199091) (I5)











119876120575(1199092) = ( 1120575 (1199092))sdot ((sum

119896

120575119860119886119896)120575(1199092)

)10038161003816100381610038161003816100381610038161003816100381610038161003816120575(1199091)=0 120575(120572)=0= 1198652 sdot 120575 (1199092) + (1205751198602 (m3g))120575(1199092)120575 (1199092)

(I10)




(I12)








119896

120575119860119886119896)120575(120572)

)10038161003816100381610038161003816100381610038161003816100381610038161003816120575(1199091)=0 120575(1199092)=0= (1205751198603 (m3g))120575(120572)120575 (120572) (I15)

119876120575(120572) = (minus1) sdot (1198983119892) sdot (120575 (1199111198623))120575(120572)120575 (120572) (I16)



(I17)














(J6)

Nomenclature




Additional Points



Disclosure


Competing Interests




References






















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











(119889211990911198891199052 )(119889211990921198891199052 )(11988921205721198891199052 ))))))asymp (1198983119892) sdot 1(2 sdot 1198972)


(15)





0(120572 (119905))2 119889119905 (16)






1199091 (0) = 01199091119901 (0) = 01199092 (0) = 01199092119901 (0) = 0120572 (0) = 0120572119901 (0) = 0(23)

1199091 (119905119891) = 1198781199091119901 (119905119891) = 01199092 (119905119891) = 1198781199092119901 (119905119891) = 0120572 (119905119891) = 0120572119901 (119905119891) = 0(24)

10038161003816100381610038161199061 (119905)1003816100381610038161003816 le 1199061max10038161003816100381610038161199062 (119905)1003816100381610038161003816 le 1199062max (25)
















9 Final Conclusions




0 20 40 60 80 100

Time (s)

minus10

0

10

20

30

40

50

x1

x2

x3


Coo

rdin

ates

of c

arts

1 2

and

the c

ente

r of

mas

s of a

cros

s-be

am

(a)

0 20 40 60 80 100

Time (s)u1

u2

minus100

minus50

0

50

100

u1

andu2

Con

trols

(b)

120572

120573

0 20 40 60 80 100

Time (s)

minus0003

minus0002

minus0001

0000

0001

0002

0003

120572an

d120573

(c)

0 20 40 60 80 100

Time (s)

minus0000003

minus0000002

minus0000001

0000000

0000001

0000002

0000003

0000004

0000005120575

(d)

0 20 40 60 80 100

Time (s)

00

01

02

03

04

05

06

07

08

09

12

1

2an

d c

3

c3

(e)

minus0000015

minus0000010

minus0000005

0000000

0000005

0000010

0000015

000

0001

000

0002

000

0003

000

0004

000

0005

000

0006

000

0007

000

0008

000

0009

848058e minus 1

120574

+

d(120574

)dt

(f)












Appendix




(A1)








(A5)








cos (120574) asymp cos (1205740) (A10)






(B1)













(C2)


(C3)



(C4)



(D1)


(D2)








(E3)





















1198793 asymp (12) sdot 1198983 sdot (119889 (1199091198623)119889119905 )2 (F5)



(F7)































1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816= 0 (H1)













= 1198651 sdot 120575 (1199091) + (1205751198601 (m3g))120575(1199091)120575 (1199091) (I5)











119876120575(1199092) = ( 1120575 (1199092))sdot ((sum

119896

120575119860119886119896)120575(1199092)

)10038161003816100381610038161003816100381610038161003816100381610038161003816120575(1199091)=0 120575(120572)=0= 1198652 sdot 120575 (1199092) + (1205751198602 (m3g))120575(1199092)120575 (1199092)

(I10)




(I12)








119896

120575119860119886119896)120575(120572)

)10038161003816100381610038161003816100381610038161003816100381610038161003816120575(1199091)=0 120575(1199092)=0= (1205751198603 (m3g))120575(120572)120575 (120572) (I15)

119876120575(120572) = (minus1) sdot (1198983119892) sdot (120575 (1199111198623))120575(120572)120575 (120572) (I16)



(I17)














(J6)

Nomenclature




Additional Points



Disclosure


Competing Interests




References






















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation





















9 Final Conclusions




0 20 40 60 80 100

Time (s)

minus10

0

10

20

30

40

50

x1

x2

x3


Coo

rdin

ates

of c

arts

1 2

and

the c

ente

r of

mas

s of a

cros

s-be

am

(a)

0 20 40 60 80 100

Time (s)u1

u2

minus100

minus50

0

50

100

u1

andu2

Con

trols

(b)

120572

120573

0 20 40 60 80 100

Time (s)

minus0003

minus0002

minus0001

0000

0001

0002

0003

120572an

d120573

(c)

0 20 40 60 80 100

Time (s)

minus0000003

minus0000002

minus0000001

0000000

0000001

0000002

0000003

0000004

0000005120575

(d)

0 20 40 60 80 100

Time (s)

00

01

02

03

04

05

06

07

08

09

12

1

2an

d c

3

c3

(e)

minus0000015

minus0000010

minus0000005

0000000

0000005

0000010

0000015

000

0001

000

0002

000

0003

000

0004

000

0005

000

0006

000

0007

000

0008

000

0009

848058e minus 1

120574

+

d(120574

)dt

(f)












Appendix




(A1)








(A5)








cos (120574) asymp cos (1205740) (A10)






(B1)













(C2)


(C3)



(C4)



(D1)


(D2)








(E3)





















1198793 asymp (12) sdot 1198983 sdot (119889 (1199091198623)119889119905 )2 (F5)



(F7)































1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816= 0 (H1)













= 1198651 sdot 120575 (1199091) + (1205751198601 (m3g))120575(1199091)120575 (1199091) (I5)











119876120575(1199092) = ( 1120575 (1199092))sdot ((sum

119896

120575119860119886119896)120575(1199092)

)10038161003816100381610038161003816100381610038161003816100381610038161003816120575(1199091)=0 120575(120572)=0= 1198652 sdot 120575 (1199092) + (1205751198602 (m3g))120575(1199092)120575 (1199092)

(I10)




(I12)








119896

120575119860119886119896)120575(120572)

)10038161003816100381610038161003816100381610038161003816100381610038161003816120575(1199091)=0 120575(1199092)=0= (1205751198603 (m3g))120575(120572)120575 (120572) (I15)

119876120575(120572) = (minus1) sdot (1198983119892) sdot (120575 (1199111198623))120575(120572)120575 (120572) (I16)



(I17)














(J6)

Nomenclature




Additional Points



Disclosure


Competing Interests




References






















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 20 40 60 80 100

Time (s)

minus10

0

10

20

30

40

50

x1

x2

x3


Coo

rdin

ates

of c

arts

1 2

and

the c

ente

r of

mas

s of a

cros

s-be

am

(a)

0 20 40 60 80 100

Time (s)u1

u2

minus100

minus50

0

50

100

u1

andu2

Con

trols

(b)

120572

120573

0 20 40 60 80 100

Time (s)

minus0003

minus0002

minus0001

0000

0001

0002

0003

120572an

d120573

(c)

0 20 40 60 80 100

Time (s)

minus0000003

minus0000002

minus0000001

0000000

0000001

0000002

0000003

0000004

0000005120575

(d)

0 20 40 60 80 100

Time (s)

00

01

02

03

04

05

06

07

08

09

12

1

2an

d c

3

c3

(e)

minus0000015

minus0000010

minus0000005

0000000

0000005

0000010

0000015

000

0001

000

0002

000

0003

000

0004

000

0005

000

0006

000

0007

000

0008

000

0009

848058e minus 1

120574

+

d(120574

)dt

(f)












Appendix




(A1)








(A5)








cos (120574) asymp cos (1205740) (A10)






(B1)













(C2)


(C3)



(C4)



(D1)


(D2)








(E3)





















1198793 asymp (12) sdot 1198983 sdot (119889 (1199091198623)119889119905 )2 (F5)



(F7)































1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816= 0 (H1)













= 1198651 sdot 120575 (1199091) + (1205751198601 (m3g))120575(1199091)120575 (1199091) (I5)











119876120575(1199092) = ( 1120575 (1199092))sdot ((sum

119896

120575119860119886119896)120575(1199092)

)10038161003816100381610038161003816100381610038161003816100381610038161003816120575(1199091)=0 120575(120572)=0= 1198652 sdot 120575 (1199092) + (1205751198602 (m3g))120575(1199092)120575 (1199092)

(I10)




(I12)








119896

120575119860119886119896)120575(120572)

)10038161003816100381610038161003816100381610038161003816100381610038161003816120575(1199091)=0 120575(1199092)=0= (1205751198603 (m3g))120575(120572)120575 (120572) (I15)

119876120575(120572) = (minus1) sdot (1198983119892) sdot (120575 (1199111198623))120575(120572)120575 (120572) (I16)



(I17)














(J6)

Nomenclature




Additional Points



Disclosure


Competing Interests




References






















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation



















Appendix




(A1)








(A5)








cos (120574) asymp cos (1205740) (A10)






(B1)













(C2)


(C3)



(C4)



(D1)


(D2)








(E3)





















1198793 asymp (12) sdot 1198983 sdot (119889 (1199091198623)119889119905 )2 (F5)



(F7)































1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816= 0 (H1)













= 1198651 sdot 120575 (1199091) + (1205751198601 (m3g))120575(1199091)120575 (1199091) (I5)











119876120575(1199092) = ( 1120575 (1199092))sdot ((sum

119896

120575119860119886119896)120575(1199092)

)10038161003816100381610038161003816100381610038161003816100381610038161003816120575(1199091)=0 120575(120572)=0= 1198652 sdot 120575 (1199092) + (1205751198602 (m3g))120575(1199092)120575 (1199092)

(I10)




(I12)








119896

120575119860119886119896)120575(120572)

)10038161003816100381610038161003816100381610038161003816100381610038161003816120575(1199091)=0 120575(1199092)=0= (1205751198603 (m3g))120575(120572)120575 (120572) (I15)

119876120575(120572) = (minus1) sdot (1198983119892) sdot (120575 (1199111198623))120575(120572)120575 (120572) (I16)



(I17)














(J6)

Nomenclature




Additional Points



Disclosure


Competing Interests




References






















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












(B1)













(C2)


(C3)



(C4)



(D1)


(D2)








(E3)





















1198793 asymp (12) sdot 1198983 sdot (119889 (1199091198623)119889119905 )2 (F5)



(F7)































1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816= 0 (H1)













= 1198651 sdot 120575 (1199091) + (1205751198601 (m3g))120575(1199091)120575 (1199091) (I5)











119876120575(1199092) = ( 1120575 (1199092))sdot ((sum

119896

120575119860119886119896)120575(1199092)

)10038161003816100381610038161003816100381610038161003816100381610038161003816120575(1199091)=0 120575(120572)=0= 1198652 sdot 120575 (1199092) + (1205751198602 (m3g))120575(1199092)120575 (1199092)

(I10)




(I12)








119896

120575119860119886119896)120575(120572)

)10038161003816100381610038161003816100381610038161003816100381610038161003816120575(1199091)=0 120575(1199092)=0= (1205751198603 (m3g))120575(120572)120575 (120572) (I15)

119876120575(120572) = (minus1) sdot (1198983119892) sdot (120575 (1199111198623))120575(120572)120575 (120572) (I16)



(I17)














(J6)

Nomenclature




Additional Points



Disclosure


Competing Interests




References






















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











(C4)



(D1)


(D2)








(E3)





















1198793 asymp (12) sdot 1198983 sdot (119889 (1199091198623)119889119905 )2 (F5)



(F7)































1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816= 0 (H1)













= 1198651 sdot 120575 (1199091) + (1205751198601 (m3g))120575(1199091)120575 (1199091) (I5)











119876120575(1199092) = ( 1120575 (1199092))sdot ((sum

119896

120575119860119886119896)120575(1199092)

)10038161003816100381610038161003816100381610038161003816100381610038161003816120575(1199091)=0 120575(120572)=0= 1198652 sdot 120575 (1199092) + (1205751198602 (m3g))120575(1199092)120575 (1199092)

(I10)




(I12)








119896

120575119860119886119896)120575(120572)

)10038161003816100381610038161003816100381610038161003816100381610038161003816120575(1199091)=0 120575(1199092)=0= (1205751198603 (m3g))120575(120572)120575 (120572) (I15)

119876120575(120572) = (minus1) sdot (1198983119892) sdot (120575 (1199111198623))120575(120572)120575 (120572) (I16)



(I17)














(J6)

Nomenclature




Additional Points



Disclosure


Competing Interests




References






















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation


























1198793 asymp (12) sdot 1198983 sdot (119889 (1199091198623)119889119905 )2 (F5)



(F7)































1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816= 0 (H1)













= 1198651 sdot 120575 (1199091) + (1205751198601 (m3g))120575(1199091)120575 (1199091) (I5)











119876120575(1199092) = ( 1120575 (1199092))sdot ((sum

119896

120575119860119886119896)120575(1199092)

)10038161003816100381610038161003816100381610038161003816100381610038161003816120575(1199091)=0 120575(120572)=0= 1198652 sdot 120575 (1199092) + (1205751198602 (m3g))120575(1199092)120575 (1199092)

(I10)




(I12)








119896

120575119860119886119896)120575(120572)

)10038161003816100381610038161003816100381610038161003816100381610038161003816120575(1199091)=0 120575(1199092)=0= (1205751198603 (m3g))120575(120572)120575 (120572) (I15)

119876120575(120572) = (minus1) sdot (1198983119892) sdot (120575 (1199111198623))120575(120572)120575 (120572) (I16)



(I17)














(J6)

Nomenclature




Additional Points



Disclosure


Competing Interests




References






















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation







































1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816= 0 (H1)













= 1198651 sdot 120575 (1199091) + (1205751198601 (m3g))120575(1199091)120575 (1199091) (I5)











119876120575(1199092) = ( 1120575 (1199092))sdot ((sum

119896

120575119860119886119896)120575(1199092)

)10038161003816100381610038161003816100381610038161003816100381610038161003816120575(1199091)=0 120575(120572)=0= 1198652 sdot 120575 (1199092) + (1205751198602 (m3g))120575(1199092)120575 (1199092)

(I10)




(I12)








119896

120575119860119886119896)120575(120572)

)10038161003816100381610038161003816100381610038161003816100381610038161003816120575(1199091)=0 120575(1199092)=0= (1205751198603 (m3g))120575(120572)120575 (120572) (I15)

119876120575(120572) = (minus1) sdot (1198983119892) sdot (120575 (1199111198623))120575(120572)120575 (120572) (I16)



(I17)














(J6)

Nomenclature




Additional Points



Disclosure


Competing Interests




References






















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












= 1198651 sdot 120575 (1199091) + (1205751198601 (m3g))120575(1199091)120575 (1199091) (I5)











119876120575(1199092) = ( 1120575 (1199092))sdot ((sum

119896

120575119860119886119896)120575(1199092)

)10038161003816100381610038161003816100381610038161003816100381610038161003816120575(1199091)=0 120575(120572)=0= 1198652 sdot 120575 (1199092) + (1205751198602 (m3g))120575(1199092)120575 (1199092)

(I10)




(I12)








119896

120575119860119886119896)120575(120572)

)10038161003816100381610038161003816100381610038161003816100381610038161003816120575(1199091)=0 120575(1199092)=0= (1205751198603 (m3g))120575(120572)120575 (120572) (I15)

119876120575(120572) = (minus1) sdot (1198983119892) sdot (120575 (1199111198623))120575(120572)120575 (120572) (I16)



(I17)














(J6)

Nomenclature




Additional Points



Disclosure


Competing Interests




References






















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation
















119896

120575119860119886119896)120575(120572)

)10038161003816100381610038161003816100381610038161003816100381610038161003816120575(1199091)=0 120575(1199092)=0= (1205751198603 (m3g))120575(120572)120575 (120572) (I15)

119876120575(120572) = (minus1) sdot (1198983119892) sdot (120575 (1199111198623))120575(120572)120575 (120572) (I16)



(I17)














(J6)

Nomenclature




Additional Points



Disclosure


Competing Interests




References






















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation














(J6)

Nomenclature




Additional Points



Disclosure


Competing Interests




References






















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Disclosure


Competing Interests




References






















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









research into 2d dynamics and control of small...

Documents