混合坐标法不同边界条件比较

JSV 081:0 issue "792813#*MS 0102 Journal of Sound and Vibration "0885# 081"0#\ 278�287 LETTER TO THE EDITOR RESONANCE CONDITIONS AND DEFORMABLE BODY CO!ORDINATE SYSTEMS A[ A[ SHABANA Department of Mechanical Engineering\ University of Illinois at Chicago\ 731 West Taylor Street\ Chicago\ IL 59596!6911\ U[S[A[ "Received 06 March 0884\ and in _nal form 00 August 0884# 0[ INTRODUCTION In rigid body dynamics\ the co!ordinates of three points on the rigid body can be used to completely de_ne the location and orientation of the rigid body co!ordinate system[ Similarly\ in ~exible body dynamics\ the body co!ordinate system and the component modes cannot be arbitrarily or independently selected[ The shape functions are de_ned in a co!ordinate system and\ therefore\ the expression of these functions de_ne the nature of the deformable body co!ordinate system[ This note demonstrates that resonance conditions are not absolute in the sense that di}erent resonance frequencies can be obtained for the same system if the deformation is de_ned in di}erent co!ordinate systems that can have arbitrary rigid body displacements[ As a consequence\ speaking of the resonance phenomenon must be associated with the selection of the co!ordinate system of the elastic body[ Only geometric conditions are required to de_ne the co!ordinate system and\ as such\ natural force conditons are of much less signi_cance in de_ning the resonance conditions[ The analysis presented in this note clearly explains why two di}erent sets of mode shapes and two di}erent sets of co!ordinate systems can be used to obtain approximately the same displacement solutions in multi!body simulations[ In view of the analysis presented in this note\ the fundamental relationship between the selection of the component modes and the deformable body co!ordinate system is discussed\ and it is demonstrated that a proper selection of the deformable body co!ordinate system not only leads to a consistent formulation\ but also leads to a proper de_nition of the resonance conditions[ In most existing rigid multi!body codes\ a centroidal body _xed co!ordinate system is used in order to simplify the inertia matrix and the centrifugal forces[ In ~exible multi!body dynamics\ body _xed or ~oating co!ordinate systems are often used to de_ne the deformation of the elastic bodies[ Conventional mode shapes as de_ned in standard vibration texts �09�01� are described in co!ordinate systems de_ned by the geometric boundary conditions[ For example\ cantilever mode shapes of a beam are de_ned in a body _xed co!ordinate system the origin of which is rigidly attached to one of the beam ends[ Free�free modes\ on the other hand\ de_ne a ~oating frame of reference the origin of which is not rigidly attached to a material point on the elastic beam[ The relationship between the selected modes of vibration and the co!ordinate system is fundamental in multi!body dynamics\ since improper selection of the co!ordinate system and the associated modes of deformation not only leads to inconsistent formulation and numerical problems\ but also leads to the wrong resonance conditions[ The mode shapes and the co!ordinate systems of deformable bodies that undergo large rigid body displacements cannot be selected arbitrarily or independently[ For instance\ the mode shapes obtained for a chassis of a vehicle using free�free boundary conditions do not de_ne the deformation in a body _xed co!ordinate system[ In principle\ however\ any 278 9911�359X:85:059278�09 ,07[99:9 7 0885 Academic Press Limited JSV 081:0 issue "792813#*MS 0102 LETTER TO THE EDITOR289 shape function or vector can be de_ned in any co!ordinate system[ Therefore\ if the mode shapes obtained using the free�free end conditions are to be used with a body _xed co!ordinate system\ these mode shapes must be modi_ed using a co!ordinate transformation that de_nes the relationship between the ~oating co!ordinate system and the body _xed co!ordinate system[ This transformation\ in addition to the fact that it is di.cult to accomplish in many applications\ may also lead to the loss of the orthogonality property of the original mode shapes[ For this reason\ in ~exible multi!body simulations\ it is more convenient to use body co!ordinate systems the origins of which are not rigidly attached to material points on the deformable bodies[ In many investigations on the dynamics of ~exible multi!body systems\ di}erent sets of modes\ such as attachment modes\ constraint modes and Ritz vectors �6�\ are combined in order to obtain an optimum set of functions that accurately describe the deformation of the constrained elastic bodies[ Di}erent geometric boundary conditions\ that de_ne di}erent eigenfunctions and di}erent natural frequencies\ de_ne di}erent co!ordinate systems[ The use of a combination of di}erent sets of modes obtained using di}erent geometric conditions must be carefully examined^ otherwise\ an inconsistent dynamic formulation is obtained[ It has been demonstrated �0� that di}erent sets of modes which correspond to di}erent sets of natural frequencies can lead to the same solution in multi!body simulations if the co!ordinate system and the mode shapes are properly de_ned[ A slider�crank mechanism with a ~exible connecting rod was used as an example to demonstrate that the natural frequency of the linear problem is not a signi_cant factor in the solution of the non!linear dynamic equations of ~exible multi!body systems[ Simply supported end conditions\ two!cantilever end conditions\ and free�free end conditions were used to model the ~exible connecting rod and obtain the same displacement solution[ Gofron and Shabana �8� demonstrated the equivalance of the elastic forces associated with two di}erent sets of modes which yield the same displacement solution[ It was shown that the elastic forces associated with simply supported end conditions are equipollent to the elastic forces associated with the two!cantilever end conditions and di}er only by a co!ordinate transformation[ While resonance as a physical phenomenon must be independent of the imaginary co!ordinate systems\ the frequency of excitation can be generated from di}erent frames of reference[ In this note we demonstrate that the resonance conditions depend on the selection of the body co!ordinate system[ If the eigenfunction obtained using a set of boundary conditions de_ned in a di}erent co!ordinate system\ one obtains di}erent resonance conditions and\ as a consequence\ the natural frequencies of the linear problem do not represent absolute quantities in ~exible multi!body dynamics[ Di}erent mode shapes that correspond to signi_cantly di}erent sets of natural frequencies can be used to obtain the same resonance conditions by using simple co!ordinate transformations[ This study is organized as follows[ In section 1 it is demonstrated\ using a simple slider�crank mechanism example\ that two di}erent sets of modes that correspond to two di}erent sets of natural frequencies can be used to obtain approximately the same displacement solution if the co!ordinate system is properly selected[ The background material presented in this section\ which is obtained from previously published work\ provides the motivation for understanding the analysis presented in the remainder of this work[ In section 2\ two di}erent sets of boundary conditions are presented and the relationship between the boundary conditions and the co!ordinate systems is discussed[ In section 3\ it is demonstrated that a change in the co!ordinate system can lead to a change in the de_nition of the mode shapes and the associated natural frequencies[ The results presented in this section clearly demonstrate that the absolute quantities are the JSV 081:0 issue "792813#*MS 0102 LETTER TO THE EDITOR 280 Figure 0[ The slider!crank mechanism[ dimensions and material properties of the defomable bodies[ In section 4\ the equations of vibration of a prismatic beam are obtained using two di}erent mode shapes which are obtained using two di}erent sets of boundary conditions[ It is demonstrated analytically that approximately the same displacement solutions can be obtained using the two di}erent shape functions[ In section 5\ the relationship between the formulation presented in this note and multi!body formulations is discused\ and section 6 is devoted to a summary and conclusions drawn from the work presented herein[ 1[ BACKGROUND The slider�crank mechansism shown in Figure 0 was used in many investigations to test the accuracy of ~exible multi!body formulations[ The dynamics of this mechanism with a ~exible connecting rod was examined using the _nite element and component modes techniques[ In the majority of previously published work\ the connecting rod was treated as a simply supported beam �4�[ The simply supported end conditions de_ne a connecting rod co!ordinate system the origin of which is located at\ but not rigidly attached to\ point A since the slope at point A can be di}erent from zero[ As a consequence\ the co!ordinate system resulting from the use of the simply supported end conditions is not a body _xed co!ordinate system[ It was demonstrated in reference �0� that di}erent end conditions that de_ne di}erent body co!ordinate systems can be used to obtain approximately the same displacement solution[ Three di}erent models were used] in the _rst model\ simply supported end conditions were used^ in the second model\ the connecting rod was modelled as two cantilever beams joined at the center of the rod^ and in the third model\ the free�free beam that satis_ed the mean axis conditions was used[ These three models have signi_cantly di}erent sets of natural frequencies as reported in reference �0�[ The three models used by Agrawal and Shabana �0� correspond to three di}erent co!ordinate systems for the connecting rod[ The simply supported and free�free end conditions de_ne a ~oating co!ordinate system\ the origin of which is not rigidly attached to a material point of the beam[ The two cantilever end conditions\ on the other hand\ de_ne a body _xed co!ordinate system\ the origin of which is rigidly attached to the geometric center of the beam in the underformed state[ The mid!point de~ection of the connecting rod is shown as a function of time in Figure 1[ In this _gure\ the results presented are obtained using two di}erent beam models[ In the _rst model\ the connecting rod is modelled as a simply supported beam\ while in the second model\ the connecting rod is modelled as two cantilever beams joined at the rod center[ In both cases\ the results are obtained using six modes of vibration and all the driving conditions\ and dimensions and material properties are the same as reported by Gofron and Shabana �7�[ It is clear from the results presented in Figure 1 that while the two sets of boundary conditions lead to two di}erent sets of natural frequencies\ the same displacement solution can be obtained[ In the remainder of this note\ it will be explained how the resonance conditions in multi!body simulatins are not dependent on the natural frequencies of the JSV 081:0 issue "792813#*MS 0102 LETTER TO THE EDITOR281 Figure 1[ The mid!point de~ection of the connecting rod[ **\ Simply supported^ � � �\ two!cantilever[ linear problem\ and it will be demonstrated that di}erent co!ordinate systems lead to di}erent de_nitions of the modal mass\ sti}ness and frequency coe.cients[ 2[ BOUNDARY CONDITIONS AND CO!ORDINATE SYSTEMS In this section\ we consider two sets of boundary conditions that de_ne two di}erent co!ordinate systems[ In the _rst case\ we consider a beam with simply supported end conditions\ while in the second case we consider a beam with free�free boundary conditions[ 2[0[ Simply supported beam Consider a simply supported beam which has length l\ cross!sectional area a\ mass density r\ modulus of elasticity E\ second moment of area Iz and volume V[ The partial di}erential equation that governs the linear transverse vibration of the beam is 11v 1t1 � EIz ra 13v 1x3 �9\ "0# where v is the transverse displacement\ t is time\ and x is the spatial co!ordinate[ The simply supported end conditions are v"9\ t#�9\ 11v"9\ t#:1x1 �9\ v"l\ t#�9\ 11v"l\ t#:1x1 �9[ "1# Substituting these end conditions into the partial di}erential equation\ one obtains the natural frequencies and eigenfunctions of the simply supported beam[ The _rst eigenfunction of the beam is f0 � sin"px:l# "2# and the corresponding natural frequency is v0 � "p1:l1#zEIz :ra [ "3# JSV 081:0 issue "792813#*MS 0102 LETTER TO THE EDITOR 282 Figure 2[ Simply supported end conditions[ If the beam is subjected to a harmonic force acting at its center\ the equation of motion associated with the _rst mode of vibration is m0 q� � k0 q�F9 sin vf t\ "4# where m0 and k0 are the modal mass and sti}ness coe.cients\ q is the modal co!ordinate\ F9 is the amplitude of the forcing function\ and vf is the frequency of excitation[ Clearly\ the resonance in this case occurs when vf �v0[ Note that because of the end conditions\ the eigenfunction of equation "2# must be de_ned in the co!ordinate system shown in Figure 2[ If another co!ordinate system is used\ the eigenfunction must be rede_ned accordingly[ For this reason\ in multi!body dynamics\ the boundary conditions are called the reference conditions[ 2[1[ Free�free end conditions Free�free modes are used in many aerospace and machine design applications to de_ne the shape of deformation of ~exible structures[ It can be shown that the free�free modes satisfy the mean!axis conditions which are obtained by minimizing the kinetic energy with respect to an observer stationed on the ~exible body �1\ 3\ 5�[ If free�free end conditions are used\ the eigenfunction associated with the _rst deformation mode of vibration of the beam is �2� f�cosh lx l �cos lx l � s0sinh lxl �sin lxl 1\ "5# where l�3=62993963\ s�9=871491104[ "6# The natural frequency that corresponds to the eigenfucntion of equation "5# is v0 � "3=629#1 l1 XEIzra \ "7# which is signi_cantly di}erent from the natural frequency associated with the _rst eigenfunction of the simply supported beam[ 3[ CHANGE IN THE CO!ORDINATE SYSTEM The co!ordinate system in which the eigenfunction of equation "5# is de_ned is shown in Figure 3[ It is clear from this _gure that the deformation at the two ends of the beam are not equal to zero as de_ned in this co!ordinate system[ In this section\ we attempt to de_ne the eigenfunction of equation "5# in another co!ordinate system[ This co!ordinate JSV 081:0 issue "792813#*MS 0102 LETTER TO THE EDITOR283 Figure 3[ Free�free end conditions[ system is shown in Figure 4\ and it is clear that in this co!ordinate system the displacements of the end points are equal to zero[ The eigenfunctions of equation "5# is de_ned in the new co!ordinate system as fm �f�f"9#\ "8# which\ upon the use of equation "5#\ leads to fm �cosh lx l �cos lx l � s0sinh lxl �sin lxl 1�1[ "09# The modal mass and sti}ness coe.cients are de_ned in this case as m�g l 9 raf1m dx\ k�g l 9 EI1"11fm:1x1#1 dx[ "00# Note that these modal mass and sti}ness coe.cients are de_ned using the kinematic relationship of equation "8# without enforcing any natural boundary conditions[ Using the modal mass and sti}ness coe.cients of equation "00#\ one can show that the natural frequency of the system obtained using the modi_ed shape function of equation "09# is v0 �Xkm�"2=052#1l1 XEIzra [ "01# Figure 4[ Change in the co!ordinate system[ JSV 081:0 issue "792813#*MS 0102 LETTER TO THE EDITOR 284 This natural frequency is very close to the natural frequency of equation "3#\ which was obtained using the simply supported end conditions[ Despite the fact that two di}erent mode shapes are used\ the error in the natural frequency is approximately 0)[ 4[ EQUATIONS OF MOTION In the case of the simply supported beam\ the shape function of equation "2# can be used to de_ne the modal mass and sti}ness coe.cients[ It can be shown\ in the case of the simply supported beam discussed in section 2\ that the vibration equation of the beam under the e}ect of the concentrated force that acts at the beam center can be written as "equation "4## "m:1#q� �"p3EIz :1l2#q�F9 sin vf t[ "02# If the shape function of equation "09# is used\ the virtual work can be used to de_ne the generalized force associated with this mode of deformation[ This leads to the equation of motion of the beam obtained using the modi_ed shape function of equation "09# and the beam model described in the preceding section[ This equation is "4m#q� �""3=629#3EIz :l2#q��2=1045F9 sin vf t[ "03# Dividing by the corresponding modal mass coe.cients\ equations "02# and "03# can be written as q� �"2=031:l#3"EIz :ra#q�"1F9 :m# sin vf t "04# and q� �"2=052:l#3"EIz :ra#q�"�9=53201F9 :m# sin vf t[ "05# The solution of equation "04#\ which was obtained using the simply supported end conditions\ is q� 1F9 :"2=031#3"EIz :l2# 0� v1f "2=031:l#3"EIz :ra# sin vf t\ "06# while the solution of equation "05#\ which was obtained using the modi_ed shape function of equation "09#\ is q� �9=53201F9 :"2=052#3"EIz :l2# 0� v1f "2=052:l#3"EIz :ra# sin vf t[ "07# The modal co!ordinates obtained using the preceding two equations are signi_cantly di}erent because the shape function of equation "09# is not normalized such that fm"l:1#�0[ The physical displacements obtained using the modal co!ordinates of the preceding two equations\ however\ are in a good agreement[ For example\ the displacement at the mid point of the simply supported beam can be obtained using the shape function of equation "2# and the time dependent modal co!ordinate of equation "06# as v�f0 l11q"t#� 1F9 :"2=031#3"EIz :l2#0� v1f "2=031:l#3"EIz :ra# sin vf t\ "08# JSV 081:0 issue "792813#*MS 0102 LETTER TO THE EDITOR285 Figure 5[ The multi!body formulation[ while the mid!point displacement obtained using the modi_ed shape function of equation "09# and the modal co!ordinate of equation "07# is v�fm 0 l11q"t#� 1=95F9 :"2=052#3"EIz :l2#0� v1f "2=052:l#3"EIz :ra# sin vf t\ "19# which shows that two physical displacements of the mid!point obtained using two di}erent sets of modes are in good agreement[ 5[ RELATIONSHIP WITH MULTI!BODY FORMULATIONS The same results as obtained in the preceding section can be obtained using a multi!body methodology without the need to change the beam co!ordinate system or rede_ne the eigenfunction[ In this case\ we select a _xed co!ordinate system Xf Yf which is attached to the end point of the beam at A\ as shown in Figure 5[ The location of the beam co!ordinate system XY with respect to the _xed co!ordinate system is de_ned by the reference co!ordinate Ry [ The location of an arbitrary point on the beam with respect to the _xed co!ordinate system can be written in terms of the reference co!ordinate as v�Ry �f"x#q"t#\ "10# where q is the modal co!ordinate and f is the shape function de_ned in equation "5#[ This shape function de_nes the deformation of the beam with respect to its co!ordinate system XY[ If point A is a _xed point\ we have the following algebraic kinematic constraint equation] v"9\ t#�9�Ry �f"9#q"t#\ "11# from which Ry ��f"9#q"t#[ "12# Substituting equation "12# into equation "10# yields v��f"x#�f"9#�q"t#�fm q"t#\ "13# where fm is the same shape function de_ned by equation "09#[ Note that in this case the beam co!ordinate system XY is a ~oating frame\ and it moves so as to preserve the shape of deformation de_ned by equation "5#[ JSV 081:0 issue "792813#*MS 0102 LETTER TO THE EDITOR 286 6[ SUMMARY AND CONCLUSIONS In multi!body simulations\ there is a subtle and fundamental relationship between the selection of the mode shapes and the co!ordinate systems of constrained deformable bodies[ In many investigations on robotics and machine dynamics\ di}erent sets of modes are used