collisions of multibody systems

Collisions of multibody systems Chau-Chin Chang,Ronald L.HustonAbstract This paper presents a computational procedure for studying collisions of multibody systems.It combines the procedures of impact analysis and the methods of modern multibody dynamics(including the use of Kane's equations,lower body arrays,generalized speeds,and differentiation algorithms).By assuming the duration of impact to be very short and that the con®gurations of the systems have only small changes during the colliding process,we can automatically generate the governing dy-namical equations.By using Newton's impact law,the partial velocities of the contact points determine impulse force components.Then by back substituting into the governing equations,the changes of velocities during the collision,the components of internal impulses,and the subsequent motions of the systems after collision may be determined.List of symbolsa k acceleration of G k in R k 1;...;Na`p generalized mass matrix coef®cients`;p 1;...;nB j body j of multibody systemc;d contact pointse coef®cient of restitutione ijk permutation coef®cients i;j;k 1;2;3F k equivalent applied(active)force acting onB k k 1;...;NFÃk equivalent inertia(passive)force acting onB k k 1;...;N^F k impulse due to F k k 1;...;NF`generalized applied(active)force acting on the system for y` ` 1;...;nFÃ`generalized inertia(passive)force acting onB k ` 1;...;nf`generalized force matrix coef®cients` 1;...;nG k mass center of Body B k k 1;...;N I k central inertia dyadic of B k for G k k 1;...;N I`generalized momentum of the multibody system for y` ` 1;...;nM k torque of couple of equivalent applied(active) force system acting on B k k 1;...;NMÃk torque of couple of equivalent inertia(passive) force system acting on B k k 1;...;N^M k angular impulse due to M k k 1;...;Nm k mass of B k k 1;...;NN number of bodies in a multibody systemn number of degrees of freedom of a multibody systemn jm mutually perpendicular unit vectors®xed inB j m 1;2;3n c k unit vector normal to B k at c k 1;...;Nn km mutually perpendicular unit vectors®xed inB k m 1;2;3n om mutually perpendicular unit vectors®xed in R m 1;2;3P`generalized momentum of the multibody system for y` ` 1;...;nR inertial reference framet timev k velocity of G k in R k 1;...;Nv k`m n om components of partial angular velocity of B k for y` k 1;...;N;` 1;...;n;m 1;2;3 X;Y;Z Cartesian coordinate axesx`generalized coordinates ` 1;...;ny`generalized speeds ` 1;...;na k angular acceleration of B k in R k 1;...;ND y`change in y` ` 1;...;nq c k position vector locating G k relative to Cx k angular velocity of B k in R k 1;...;Nx k`m n om components of partial angular velocity of B k for y` k 1;...;N;` 1;...;n;m 1;2;3IntroductionThe study of impact in multibody systems is an important branch of dynamics.Wittenburg(1977)used graph theory and Newton±Euler's method,Haug et al.(1986)used variational principles,Khulief and Shabana(1987)used continuous force model and effective mass,Lankarani and Nikravesh(1990)used continuous contact force model and Hertz contact law,while Batlle and Condomines (1991)used Lagrangian formulation and impulsive drivers to maintain the continuity of a set of generalized velocities during the impact time interval to study impact of multi-body systems.Computational Mechanics27(2001)436±444ÓSpringer-Verlag2001436Received24January2001Chau-Chin ChangAutomotive Division of Mechanical Engineering,Southern Taiwan University of Technology,Tainan,Taiwan710ROCRonald L.Huston(&)Department of Mechanical,Industrial and Nuclear Engineering,University of Cincinnati,Cincinnati,Ohio45221-0072USAColliding bodies are effectively rigid during collision if there is negligible deformation outside the contact region and if the contact region remains small in comparison with the size of the bodies.With the additional assumptions of collision occurring over a short time interval and no changes in con®gurations during the collision process,we can predict the velocity changes during collision and post collision ing these assumptions we present a procedure for making ef®cient computational analyses of collisions of multibody systems.In developing the procedure we employ an impulse form of Kane's equations(1985)obtained by integrating the equations over the impact time interval to obtain a balance of generalized momenta and generalized impulses. These equations in turn lead to a set of linear algebraic equations for velocity changes.The solution of these equations then forms the basis for studying the movement of the system during the post-impact period.In the analysis,the collisions may occur between dif-ferent multibody systems or within the same multibody system.During the collision process,there is an impulse generated at each contact point.By neglecting friction at colliding points,the unknown impulse generated by the collision may be obtained by using Newton's impact law. Substituting the impulse back into the algebraic equations, the changes of velocities during colliding process,and thereby,the initial conditions for the analysis of the sub-sequent motions of the system are determined.The pro-cedure also provides for the computation of the components of internal impulses at the joints of the sys-tems during the collision process.The balance of the paper is divided into®ve parts with the®rst of these providing a summary of multibody dy-namics analysis useful in the sequel.The next part presents the impact analysis itself.The third part studies the col-lision of multibody systems.The®nal two parts contain an example and concluding remarks.Preliminary considerations1.KinematicsConsider a system with N bodies having n degrees of freedom,described by n generalized coordinates x`` 1;...;n .These coordinates will in general include both translation and rotation variables.Let n variables y` ` 1;...;n ,called`generalized speeds'be introduced such that the y`are independent linear combinations of the derivatives of the generalized coordinates.For exam-ple,the individual y`could be set equal to the time de-rivatives of individual generalized coordinates, respectively.It is often convenient however to let the y` associated with rotational degrees of freedom be compo-nents of the relative angular velocities of the bodies.It has been shown that the angular velocity x k and the mass center velocity v k of a typical body B k of the system may then be expressed as(Huston,1990):x k x k`m y`n om ` 1;...;n;m 1;2;3 1 andv k v k`m y`n om ` 1;...;n;m 1;2;3 2 where the n om m 1;2;3 are mutually perpendicularunit vectors®xed in an inertia frame R.The coef®cientsx k`m and v k`m ` 1;...;n called`partial angular veloc-ity components'and`partial velocity components'(Hu-ston,1990),form arrays which depend only upon the geometrical parameters of the systems.[Regarding nota-tion,repeated indices,such as`and m in Eqs.(1)and(2), indicate a sum over the range of the index.]By differentiating Eqs.(1)and(2),the angular accel-eration of B k and the acceleration of the mass center G kmay be expressed asa k x k`m_y` _x k`m y` n omk 1;...;N;` 1;...;n;m 1;2;3 3anda k v k`m_y` _v k`m y` n omk 1;...;N;` 1;...;n;m 1;2;3 4 Observe that the x k`m and v k`m are the coef®cients of y`.That is,x k`m n omo x ko y`o a ko_y`x k` 5andv k`m n omo v ko y`o a ko_y`v k` 6where x k`and v k`are called`partial angular velocity'(Kaneet al.,1985)of body B k and`partial velocity'(Kane et al., 1985)of G k with respect to generalized speeds y`,respec-tively.2.KineticsLet the multibody system be subjected to a force®eldwhich may be represented on a typical body B k by a single force F k passing through mass center G k together with a couple with torque M k.This force®eld may arise from externally applied forces such as gravity or contact forcesor from constraint forces due to either holonomic or non-holonomic constraints.Then the contribution of F k andM k to the generalized active force F`associated with y`is (Kane et al.,1985)F`o v ko y`ÁF ko x ko y`ÁM k v k`ÁF k x k`ÁM k7By using Eqs.(5)and(6),F`may be written asF` v k`m F km x k`m M km` 1;...;n;m 1;2;3;no sum on k 8 where F km and M km are the n om components of F k and M k. Regarding the index notation,if there is a sum over k,Eq.(8)represents the contribution from all the bodies. Hence,the generalized active force F`associated with generalized speed y`of the whole system may be written asF` v k`ÁF k x k`ÁM k k 1;...;N;` 1;...;n9437orF` v k`m F km x k`m M kmk 1;...;N;` 1;...;n;m 1;2;3 10 Similarly,let the inertia forces on B k be represented by asingle force FÃk passing through G k together with a couple with torque MÃk.Then FÃk and MÃk may be expressed as (Kane et al.,1985)FÃk Àm k a k no sum on k 11 andMÃk ÀI kÁa kÀx k I kÁx k no sum on k 12 where m k is the mass of B k and I k is its central inertia dyadic.The generalized inertia force FÃ`associated with generalized speed y`of the system can then be written as (Kane et al.,1985)FÃ` v k`ÁFÃk x k`ÁMÃk k 1;...;N;` 1;...;n13 orFÃ` v k`m FÃkm x k`m MÃkmk 1;...;N;` 1;...;n;m 1;2;3 14 where FÃkm and MÃkm are the n om components of FÃk and MÃk.3.DynamicsKane's equations state that the equations of motion may be expressed asF` FÃ` 0 ` 1;...;n 15 By using Eqs.(1)±(14),the dynamic equation(15)may be written asa`p_y p f` `;p 1;...;n 16 wherea`p m k v k`m v kpm I kmz x k`m x kpzk 1;...;N;`;p 1;...;n;m;z 1;2;317 andf` F`À m k v k`m_v kpm y p I kmz x k`m_x kpz y pe rms I ksz x k`m x kqr x kpz y q y pk 1;...;N;`;p;q 1;...;n;m;r;s;z 1;2;3 18 where I kmz are n om and n oz components of central dyadic I k and where e rms is the standard permutation symbol,and where there is now a sum over the triply repeated index k. The a`p are sometimes referred to as`generalized mass elements'.Impact analysis1.Generalized impulsesAn`impulsive force'is de®ned as a large force acting over a short time interval.The term`large'means that an im-pulsive force is signi®cantly larger than other forces(for example,gravity)exerted on a given body or system during the impact time interval.An`impulse'is de®ned as the time integral of the impulsive force over the impact time interval.Consider a multibody system with N bodies pos-sessing n degrees of freedom,characterized by n generalized coordinates x` ` 1;...;n .Let the gene-ralized speeds y`be de®ned as relative angular velocity components and relative displacement velocity com-ponents between adjoining bodies.Let the impulsive forces and moments on a typical body B k be equivalent to an impulsive force F k passing through mass center G k together with an impulsive couple with torque M k.Then the generalized impulse I`of the system associated with the generalized speed y`isde®ned asI` v k`ÁZ t2t1F k d t x k`ÁZ t2t1M k d t@1Av k`Á^F k x k`Á^M k k 1;...;N;` 1;...;n19 orI` v k`m^F km x k`m^M kmk 1;...;N;` 1;...;n;m 1;2;3 20 where t1;t2 is the impact time interval,^F k is the impulse of F k,^M k is the angular impulse of M k.The quantities^Fkm and^Mkm are n om components of^F k and^M k, respectively.2.Generalized momentumThe generalized momentum P`of the system associated with generalized speed y`is de®ned asP` v k`Ám k v k x k`ÁI kÁx kk 1;...;N;` 1;...;n 21Substituting from Eqs.(1),(2),(5)and(6)into(21),the generalized momentum may be expressed asP` m k v k`m v kpm I kmz x k`m x kpz y pk 1;...;N;`;p 1;...;n;m;z 1;2;3a`p y p `;p 1;...;n 22where the a`p are the same as the generalized mass ele-ments of Eq.(17).3.Dynamical analysisIntegrating Eq.(15)through the time interval t1;t2 yields Z t2t1F`d tZ t2t1FÃ`d t 0 ` 1;...;n 23By using Eq.(9),the®rst term of Eq.(23)may be written as438Z t2t1F`d tZ t2t1v k`ÁF k x k`ÁM k d tZ t2t1v k`ÁF k d tZ t2t1x k`ÁM k d tk 1;...;N;` 1;...;n 24In general,the partial velocity vectors v k`and partial angular velocity vectors x k`are functions of time and the con®guration of the system.Our assumption is that the impact duration is short and thus during the impact the con®guration of the system remains essen-tially the same.This means that the partial velocity and partial angular velocity vectors are nearly constant during the impact time interval.Hence,Eq.(24)can be written asZ t2t1F`d t%v k`ÁZ t2t1F k d t x k`ÁZ t2t1M k d t@1A I`k 1;...;N;` 1;...;n 25By using Eqs.(11)±(13),the second term of Eq.(23)may be written asZ t2t1FÃ`d tZ t2t1v k`Á Àm k a kx k`Á ÀI kÁa kÀx kÂI kÁx k d tk 1;...;N;` 1;...;n 26The®rst term in the parenthesis on the right hand side of Eq.(26)may be expressed asZ t2t1v k`Á Àm k a k d t Àm k v k`ÁZ t2t1a k d tÀm k v k`Á v k t2 Àv k t1k 1;...;N;` 1;...;n27where v k t1 and v k t2 are the mass center velocities of body B k at beginning and end of the impact.Similarly,the second term in the parenthesis can be expressed asZ t2t1x k`Á ÀI kÁa k d t Àx k`ÁI kÁZ t2t1a k d tÀx k`ÁI kÁ x k t2 Àx k t1k 1;...;N;` 1;...;n28where x k t1 and x k t2 are the angular velocities of body B k at the beginning and end of the impact.By using Eqs.(1)and(5),the third term in the paren-thesis on the right side of Eq.(26)may be written as Z t2t1x k`Á Àx kÂI kÁx k d tÀZ t2t1e rms I ksz x k`m x kpr x kqz y p y q d t 29Although the generalized speeds y p and y q in Eq.(29)are not constant during the impact time interval,theyare nevertheless bounded.By using the mean value theorem of integration,the value of this term is equalto the mean value of the integrand multiplied by the impact time interval.Since the impact duration is short,this term can be neglected,when compared with the®rst two terms of Eq.(26).Hence,Eq.(26)may bewritten asZ t2t1FÃ`d t Àm k v k`Á v k t2 Àv k t1Àx k`ÁI kÁ x k t2 Àx k t1À v k`Ám k v k t2 x k`ÁI kÁx k t2v k`Ám k v k t1 x k`ÁI kÁx k t1ÀP` t2 P` t1k 1;...;N;` 1;...;n 30 where P` t1 and P` t2 are the generalized momentum ofthe system with respect to the generalized speeds y`at the beginning and end of the impact.erning equationsSubstituting Eqs.(25)and(30)into Eq.(23)yieldsP` t2 ÀP` t1 I` ` 1;...;n 31This expression shows that the changes of generalized momentum of the system during the impact interval isequal to the generalized impulse acting on the system.Using Eq.(22),we®nd from Eq.(31)that the governing dynamical equations may be written asa`p y p t2 Àa`p y p t1 I` `;p 1;...;n 32ora`p D y p I` `;p 1;...;n 33 resulting inD y p aÀ1`p I` 34 where the D y p are the changes of the generalized speeds during the impact interval,and where aÀ1`p are the elementsof the inverse of the generalized mass matrix.Equation(33)[or(34)]forms a system of linear algebraic equations for the D y p.These equations form the basis for the analysis of the effects of impulses on a multibody system.That is,by knowing the D y p,we can determine velocity and angular velocity changes experi-enced by the system.We can then use these values to set initial conditions for the subsequent motion of the system.4395.Internal impulses at the jointsA feature and bene®t of Eq.(33)is that there are exactly the same number of equations as there are degrees-of-freedom of the system.Hence,the unknown generalized speeds of the constrained system are uniquely determined. Sometimes,however,it is of interest to determine in-ternal impulsive forces and moments at the joints of the system where the movement is restrained.We may de-termine these forces and moments as follows:let the constraints at the joint be temporarily released.Then let the internal impulsive forces and moments be considered as though they were externally applied.The system will then have additional degrees of freedom corresponding to the removed constraints leading to additional dynamical equations containing the desired components of internal impulses.It also happens that,if relative angular velocity components and relative velocity components between the bodies are used as the generalized speeds,then these im-pulsive force and moment components appear singly(that is,uncoupled)in these additional equations.Finally,by specifying the incremental values of the generalized speeds corresponding to the removed constraints,we can deter-mine the values of these force and moment components. (The incremental values of these generalized speeds are often zero.)Collisions of multirigid body systems1.Collisions of two multibody systemsConsider a multibody system S1having N1bodies with n1 degrees of freedom colliding with a second multibody system S2having N2bodies with n2degrees of freedom as depicted in Fig.1.Consider speci®cally the two colliding bodies,B k in S1and B j in S2.Let q c k and q d j be position vectors from G k and G j,mass centers of B k and B j,re-spectively,to the contact points c and d,of B k and B j and let n c k be a unit vector along the normal of the colliding surfaces at c and d as shown in Fig.1.Then q c k and q d j may be expressed asq c k q c ki n ki SOK mi q c ki n omi;m 1;2;3;k 1;...;N1;no sum on k35 andq d j q d ji n ji SOJ mi q d ji n omi;m 1;2;3;j 1;...;N2;no sum on j 36 where q c ki(i 1;2;3)are the components of q c k relative tobody®xed unit vectors,SOK mi and SOJ mi are the trans-formation matrices between the body®xed and inertial reference frames.The unit vector along normal to the colliding surfaces of B k and B j may be expressed asn c k u c om n om m 1;2;3;no sum on o 37 where u c om are the components of n c k in the n om directions. Newton's impact law may then be written asc v k t1 D c v k Àd v j t1 D d v j Án c kÀe c v k t1 Àd v j t1 Án c k 38 where c v k t1 and d v j t1 are the velocity of contact points c and d at the beginning of the collision,D c v k and D d v j are the velocity changes of contact points c and d during the collision process,and e is the coef®cient of restitution between the colliding bodies.By using Eq.(1),(2)and(35),the velocity of c may be written asc v k v k x kÂq ckv kpm y p n om x kpn y p n onÂSOK hi q c ki n ohv kpm e nhm x kpn SOK hi q c ki y p n omc v kpm y p n om h;i;m;n 1;2;3;p 1;...;n1;k 1;...;N1;no sum on k39 where v k and x k are the mass center velocity and the angular velocity of body B k in R,and wherec v kpm v kpm e nhm x kpn SOK hi q ckih;i;m;n 1;2;3;p 1;...;n1;k 1;...;N1;no sum on k 40 The c v kpm are the partial velocity components of c with respect to the generalized speeds y p.During theshort 440collision time interval,the c v kpm which depend only on the generalized coordinates are essentially constant.There-fore,the velocity changes at point c during the collision time interval may be expressed asD c v k c v k t2 Àc v k t1c v kpm y p t2 Ày p t1 n om c v kpm D y p n omk 1;...;N1;p 1;...;n1;m 1;2;3 41 Similarly,the velocity at contact point d of body B j and the velocity change may be written asd v j v j x jÂq djv jqm e nhm x jqn SOJ hi q d ji y q n om d v jqm y q n om j 1;...;N2;h;i;m;n 1;2;3;q 1;...;n2;no sum on j 42 andD d v j d v j t2 Àd v j t1d v jqm y q t2 n omÀd v jqm y q t1 n omd v jqm D y q n om j 1;...;N2;q 1;...;n2;m 1;2;3 43 Neglecting friction,the impulse on B k may be represented by an impulse force^F c k acting at contact point c normal to the surface of B k at c.Similarly,the impulse on B j may be represented by an impulse force^F d j having equal magni-tude but opposite direction.^F c k and^F d j may be expressed as ^F ck^F c k n c k ^F c k u c om n om k 1;...;N1;m 1;2;3;no sum on k 44 and^F djÀ^F d j n c k À^F j d u c om n omj 1;...;N1;m 1;2;3 45^F c k is the only external impulse acting on S1.The general-ized impulse for S1is thus projection of^F c k along the partial velocities of c.Hence,the governing dynamical equations for S1during the collision may be written as a`p D y p I` ^F c kÁc v k` c v k`m u c om^F c kk 1;...;N1;m 1;2;3;`;p 1;...;n1;no sum on k 46 where c v k`is the partial velocity of point c with respect to the generalized speed y`.Similarly,the governing dynamical equations for S2 during the collision may be expressed asa sq D y q I s ^F d jÁd v j s Àd v jsm u c om^F d jm 1;2;3;s;q 1;...;n2;j 1;...;N2;no sum on jÀd v jsm u c om^F c k 47 where d v j s is the partial velocity of point d with respect tothe generalized speed y s.By using Eqs.(46)and(47),the changes of generalized speeds of S1and S2may be written asD y p aÀ1`p c v k`m u c om^F c kk 1;...;N1;`;p 1;...;n1;m 1;2;3;no sum on k 48andD y q ÀaÀ1sq d v jsm u c om^F c k s;q 1;...;n2;m 1;2;349From Eq.(41)we obtainD c v k c v kpm aÀ1`p c v k`m u c om^F c k n om 50andD d v j Àd v jqm aÀ1sq d v jsm u c om^F c k n om 51 Substituting from Eqs.(37),(39),(42),(50),and(51)intoEq.(38),we obtainc v kpm y pÀd v jqm y q aÀ1`p c v k`m c v kpm u c omaÀ1sq d v jsm d v jqm u c om ^F c k n omÁu c on n onÀe c v kpm y pÀd v jqm y q n omÁu c on n onno sum on j;k 52From this expression,the magnitude of the impulse generated by the collision may be written as^F ckÀ 1 e c v kpm y pÀd v jqm y q u c omaÀ1`p c v k`m c v kpm u c om aÀ1sq d v jsm d v jqm u c om u c om53where y p y p t1 and y q y q t1 are the generalizedspeeds of system S1and S2at the beginning of the colli-sion,respectively.Equation(53)may be expressed in matrix form as:^F À 1 e f Y T v À Yà T và g u =f u T v T A À1 v À u T và T Aà À1 và g u 54 where ^F is a1Â1matrix with element^F c k, Y is an n1Â1 matrix with elements y p, Yà is an n2Â1matrix with elements y q,[v]is an n1Â3matrix with elements c v k`m,và is an n2Â3matrix with elements d v jsm,[u]is a3Â1 matrix with elements u c om,[A]is an n1Ân1matrix with elements a`p,and Aà is an n2Ân2matrix with elementsa sq.Note that there is no sum on the k of c v k`m nor on the jof d v jsm of Eq.(52).Therefore,the v and và in Eq.(54)are two dimensional matrices.If system S2is®xed,such as a ground or wall,then D d v j,d v j and thus d v jqm are zero,and Eq.(53)reduces to^F ckÀ 1 e c v kpm y p u c omaÀ1`p c v k`m c v kpm u c om u c om55or in matrix form as^F À 1 e f Y T v u g=f u T v T A À1 v u g 56441By substituting from Eq.(53)into Eqs.(46)and(47),the changes of generalized speeds can be determined.Then by using the generalized speeds after collision as initial con-ditions,the subsequent motions of the systems may be determined.2.Collision within the same multibody system Consider collisions within a single multibody system S1. Speci®cally,suppose a collision occurs at point c of body B k with point d of body B j of the same system.Then an impulse^F c k at point c and an impulse^F d j at point d occur. Again,both impulses have the same magnitude but op-posite directions.The governing equations for the system may then be expressed asa`p D y p I`^F c kÁc v k` ^F d jÁd v j`^F c k u c om n omÁc v k`n n onÀ^F d j u c om n omÁd v j`n n on^F c k u c om n omÁc v k`n n onÀ^F c k u c om n omÁd v j`n n onc v k`mÀd v j`m u c om^F c kj;k 1;...;N1;` 1;...;n1;m 1;2;3;j k;no sum on k 57 where as before c v k`and d v j`are the partial velocities of points c and d with respect to the generalized speed y`,and u c om are components of the normal unit vector at points c and d.From Eq.(57),the changes of generalized speeds may then be found asD y p aÀ1`p c v k`mÀd v j`m u c om^F c k 58 Substituting Eq.(58)into Eqs.(41)and(42),and replacing the index q with p,we obtain D c v k c v kpm aÀ1`p c v k`mÀd v j`m u c om^F c k n omk 1;...N1;`;p 1;...;n1;m 1;2;3;no sum on k 59 andD d v j d v jpm aÀ1`p c v k`mÀd v j`m u c om^F c k n omj;k 1;...N1;`;p 1;...;n1;m 1;2;3;no sum on j;k 60 From Eq.(38),we then obtain the magnitude of theimpulse asor in matrix form aswhere và is n1Â3matrix,not n2Â3matrix. ExampleTo illustrate the procedure consider the collision of two identical open-chain systems consisting of nine identical pin-connected links where each link has a mass of1kg and a length of1m.The systems move on a smooth horizontal surface.Initially,let the links of one of the systems be con®g-ured as a straight line and let the entire system be rotating at an angular rate of0.2rad/s about one of its ends as though it were a single rigid body.Let the links of the other system be initially con®gured into a semicircular arc moving in translation at a speed of1m/s in the Y-direc-tion,as though it were also a single rigid body.Figure2 depicts the two systems.Let the two systems,called S1and S2,move such that their ends d and c will collide,as depicted in Fig.3.Let the coef®cient of restitution between the colliding links be0.8. The objective is then to determine the motions of S1and S2 after collision.Let the links be numbered or labeled as shown in Figs.2 and3.Then S1has9degrees of freedom and S2has11 degrees of freedom which can be described bygeneralized ^FÀ 1 e f Y T v À Y T và g uf u T v T A À1f v À và gÀ u T và T A À1f v À và gg u62^F ckÀ 1 e c v kpm y pÀd v jpm y p u c omaÀ1`p c v kpm c v k`mÀd v j`m u c omÀaÀ1`p d v jpm c v k`mÀd v j`m u c om u c om61 442coordinates x 1p p 1;...;9 and x 2q q 1;...;11 de®ned as follows:x 11is the angle between link 1of S 1and the negative X axis;the x 1p p 2;...;9 are the relative angles between the successive adjoining links of S 1;x 21is the angle between link 1of S 2and the X axis;the x 2q q 2;...;9 are the relative angles between the successive adjoining links of S 2;and x 2;10and x 2;11are the X and Y coordinates of the open end of link 1of S 2.The generalized speeds y 1p p 1;...;9 andy 2q q 1;...;11 are de®ned simply as time derivatives of the generalized coordinates:y 1p _x1p p 1;...;9 and y 2q _x2q q 1;...;11 63The values of the generalized coordinates and the gener-alized speeds just before impact are:x 11 x 12 x 13 ÁÁÁ x 19 0 64 y 11 0:2rad/s ;y 12 y 13 ÁÁÁ y 19 065andx 21 À4p =10rad ;x 22 x 23 ÁÁÁ x 29 p =10radx 2;10 x 0and x 2;11 y 066y 21 y 22 y 23 ÁÁÁ y 2;10 0andy 2;11 1m/s67where x 0and y 0are constants.By using the foregoing procedures we may readily ob-tain the components of the partial velocities of the contact points c and d as well as the elements of the generalized mass matrices.Then by using Eq.(53)the generated impulses at the contact points are^F c k 1:2714n o 2N Ás ;^F d j À1:2714n o 2N Ás68where n o 2is a unit vector in the Y direction.By using Eqs.(46)and (47)the changes in the generalized speeds during the impact are found to be:D y 11 À0:0001rad/s ;D y 12 0:0007rad/s ;D y 13 À0:0026rad/s ;D y 14 0:0098rad/s ;D y 15 À0:0365rad/s ;D y 16 0:1362rad/s D y 17 À0:5084rad/s ;D y 18 1:8973rad/s ;D y 19 À7:0807rad/s69and D y 21 À0:0366rad/s ;D y 22 0:0040rad/s ;D y 23 À0:0085rad/s ;D y 24 À0:0081rad/s ;D y 25 À0:0210rad/s ;D y 26 0:0003rad/s D y 27 À0:1105rad/s ;D y 28 0:2602rad/s ;D y 29 À1:1798rad/s ;D y 2;10 À0:0036m/s ;D y 2;11 0:0506m/s70The components of internal impulse force at each joint inthe inertia frame due to the collisions of systems S 1and S 2are shown in Table 1.Finally,using the generalized speeds after impact as initial conditions for Eq.(16)the subsequent motionofFig.3.Collision of linear chain systemsTable ponents of internal impulse force at each joint in the inertia frameSystem S 1System S 2XY X Y Joint 100.000020Joint 20)0.000030.0210)0.0450Joint 300.000170.0726)0.0747Joint 40)0.000430.1495)0.0782Joint 500.001970.2461)0.0417Joint 60)0.006330.35030.0534Joint 700.024670.44370.2168Joint 80)0.091330.47320.4865Joint 900.340670.48200.8057Fig.4.Con®gurations of the multibody systems after impact443。