computational_dynamics_of_multibody_GetPDFServlet

a d s is ide tio opm an dy tic uss ar y s A systematic analysis of constrained mechanical systems was established in 1788 by Lagrange �4�, too. The variational principle applied to the total kinetic and potential energy of the system 1990 was documented in the Multibody System Handbook �16�. Reviews on multibody dynamics including analysis methods and Do considering its kinematical constraints and the corresponding gen- eralized coordinates results in the Lagrangian equations of the first and the second kind. Lagrange’s equations of the first kind repre- sent a set of differential-algebraical equations �DAE� while the second kind leads to a minimal set of ordinary differential equa- tions �ODE�. An extension of d’Alembert’s principle valid for holonomic systems only was presented in 1909 by Jourdain �5�. For nonholo- nomic systems the variations with respect to the translational and rotational velocities resulting in generalized velocities are re- quired. Then, a minimal set of ordinary differential equations �ODE� of first order is obtained. The approach of generalized velocities, identified as partial velocities, was also introduced by Kane and Levinson �6�. The resulting Kane’s equations represent a compact description of multibody systems. More details on the applications were presented by Kortüm and Schiehlen �17� and Huston �18�. Today, software packages for multibody dynamics analysis are widely used in academia and industry see, e.g., http:// real.uwaterloo.ca/~mbody/#Software. Recent research topics cover datamodels from CAD �standard- ization, coupling�, system parameter identification, real time simu- lation, contact and impact problems, extension to electronics and mechatronics, dynamic strength analysis, optimization of design and control devices, integration codes in particular for differential- algebraic systems, challenging applications in biomechanics, ro- botics and vehicle dynamics. In particular, elastic or flexible multibody systems, respectively, contact and impact problems, and actively controlled mechatronic systems represent key issues for researchers worldwide. The progress achieved in flexible multibody systems was documented by Shabana �19,20� who es- tablished the Absolute Nodal Coordinate Formulation �ANCF�. Bauchau �21� edited computational multibody dynamics ap- proaches including impact problems. Multibody dynamics as independent branch of mechanics was agreed to and established in 1977 at an IUTAM Symposium held in Munich, Germany and chaired by Magnus �22�. The computa- Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript receieved: May 3, 2005. Final manuscript received: May 24, 2005. Review conducted by: Subhash C. Sinha. Journal of Computational and Nonlinear Dynamics JANUARY 2006, Vol. 1 / 3 Peter Eberhard e-mail: eberhard@mechb.uni-stuttgart.de Werner Schiehlen e-mail: schiehlen@mechb.uni-stuttgart.de Institute B of Mechanics, University of Stuttgart, 70569 Stuttgart, Germany Comput Multibo Formali Multibody dynamics systems such as a w depends on computa theory. Recent devel respectively, contact mentals in multibody are presented. In par approaches are disc molecular dynamics Keywords: multibod neering applications 1 Historical Remarks Multibody system dynamics is related to classical and analyti- cal mechanics. The most simple element of a multibody system is a free particle introduced by Newton �1� 1686 in his ”Philosophiae Naturalis Principia Mathematica.” The essential element, the rigid body, was defined in 1776 by Euler �2� in his contribution entitled ”Nova methodus motum corporum rigidarum determinandi.” For the modeling of constraints and joints, Euler already used the free body principle resulting in reaction forces. The equations obtained are known in multibody dynamics as Newton-Euler equations. A system of constrained rigid bodies was considered in 1743 by d’Alembert �3� in his ”Traité de Dynamique” where he distin- guished between free and hinder components of the motion. D’ Alembert applied the equilibrium conditions to the applied forces and the reaction forces having the principle of virtual work in mind. A mathematical consistent formulation of d’Alembert’s principle is due to Lagrange �4� combining d’Alembert’s funda- mental idea with the principle of virtual work. As a result a mini- mal set of ordinary differential equations �ODE� of second order is found. Copyright © 20 wnloaded 20 Oct 2012 to 60.171.124.72. Redistribution subject to ASM tional Dynamics of y Systems: History, ms, and Applications based on analytical mechanics and is applied to engineering variety of machines and all kind of vehicles. Multibody dynamics nal dynamics and is closely related to control design and vibration ents in multibody dynamics focus on elastic or flexible systems, d impact problems, and actively controlled systems. Some funda- namics, recursive algorithms and methods for dynamical analysis ular, methods from linear system analysis and nonlinear dynamics ed. Also, applications from vehicles, manufacturing science and e shown. �DOI: 10.1115/1.1961875� ystem dynamics, recursive algorithms, vibration analysis, engi- history of classical mechanics including rigid body dynamics can be found in Dugas �7�, Päsler �8�, and Szabó �9�. The first applications of the dynamics of rigid bodies are related to gyrodynamics, mechanism theory and biomechanics as re- viewed by Schiehlen �10�. However, the requirements for more complex models of satellites and spacecrafts, and the fast devel- opment of more and more powerful computers led to a new branch of mechanics: Multibody system dynamics. The results of classical mechanics had to be extended to computer algorithms, the multibody formalisms. One of the first formalisms is due to Hooker and Margulies �11� in 1965. This approach was developed for satellites consisting of an arbitrary number of rigid bodies interconnected by spherical joints. Another formalism was pub- lished in 1967 by Roberson and Wittenburg �12�. In addition to these numerical formalisms, the progress in computer hardware and software allowed formula manipulation with the result of symbolical equations of motion, too. First contributions in 1977 are due to Levinson �13� and Schiehlen and Kreuzer �14�. In the 1980s complete software systems for the modeling, simulation and animation were offered on the market as described by Schwertassek and Roberson �15�. The state-of-the-art achieved in 06 by ASME E license or copyright; see http://www.asme.org/terms/Terms_Use.cfm tional aspects of multibody dynamics were highlighted in 1990 at the Second World Congress on Computational Mechanics ri = �ri1 ri2 ri3�T, Si��i,�i,�i�, i = 1�1�p , �1� see, e.g., Schiehlen �10�. Thus, the position vector x of the free Do �WCCM II� held in Stuttgart, Germany and chaired by Argyris �23�. Seven years later Multibody System Dynamics was estab- lished as the first scientific journal fully devoted to multibody dynamics. Today, in addition to the world congresses on mechan- ics, many series of specialized scientific meetings regularly take place like IUTAM Symposia and EUROMECH Colloquia on multibody dynamics, the ASME International Conferences on Multibody Systems, Nonlinear Dynamics and Control �MSNDC�, the International Symposia on Multibody Dynamics: Monitoring and Simulation Techniques held in UK, the ECCOMAS Thematic Conferences on Multibody Dynamics, the Asian Conferences on Multibody Dynamics, and the International Symposia on Multi- body Systems and Mechatronics held in South America. Thus, multibody system dynamics is now a well established and very lively branch of mechanics. The first textbook on multibody dynamics was written in 1977 by Wittenburg �24�. Starting with rigid body kinematics and dy- namics, classical problems of one rigid body are presented as well as general multibody systems. The famous textbook on Kane’s equations appeared in 1985 written by Kane and Levinson �6�. A textbook published 1986 by Schiehlen �25� presents in a unified manner multibody systems, finite element systems and continuous systems as equivalent models for mechanical systems. The computer-aided analysis of multibody systems was considered in 1988 in a textbook by Nikravesh �26� for the first time. Roberson and Schwertassek �27� discussed the origin of multibody systems; they deal with one and several rigid bodies. Comments on linear- ized equations and computer simulation techniques are also in- cluded. Basic methods of computer aided kinematics and dynam- ics of mechanical systems are shown in 1989 by Haug �28� for planar and spatial systems. In his first textbook from 1989, Sha- bana �19� deals in particular with flexible multibody systems. Huston �29� presents kinematics, force and inertia concepts, multi- body kinetics, numerical methods as well as flexible multibody systems. Garcia de Jalon and Bayo �30� present efficient methods for the kinematic and dynamic simulation of multibody systems to meet the real time challenge. Shabana’s second book �31� from 1994 is devoted to computational dynamics of rigid multibody systems. Many detailed examples show the execution of the com- putations required. Angeles and Kecskemethy �32� summarize im- portant contributions in a postgraduate course offered 1995 at the International Center of Mechanical Sciences �CISM� in Udine, Italy. Contact problems in multibody dynamics were highlighted in 1996 by Pfeiffer and Glocker �33�. The symbolic modeling approach of multibody systems was described in 2003 in the book of Samin and Fisette �34�, while the application of commercial software to vehicle dynamics was treated 2004 in a textbook by Blundell and Harty �35�. Many of the early textbooks are now available in a second edition as indicated in the reference list. 2 Analytical Dynamics In this section the essential steps for generation of the equations of motion in multibody dynamics will be summarized based on the fundamental principles. 2.1 Mechanical Modeling. First of all the engineering system has to be replaced by the elements of the multibody system ap- proach: Rigid and/or flexible bodies, joints, gravity, springs, dampers and position and/or force actuators. The system con- strained by bearings and joints is disassembled as free body sys- tem using an appropriate number of inertial, moving reference and body fixed frames for the mathematical description. 2.2 Kinematics. A system of p free rigid bodies holds f =6p degrees of freedom characterized by translation vectors and rota- tion tensors with respect to the inertial frame as 4 / Vol. 1, JANUARY 2006 wnloaded 20 Oct 2012 to 60.171.124.72. Redistribution subject to ASM system can be written as x = �r11 r12 r13 r21 … �p �p �p�T. �2� Then, the free system’s translation and rotation remain as ri = ri�x�, Si = Si�x� . �3� Assembling the system by q holonomic, rheonomic constraints reduces the number of degrees of freedom to f =6p−q. The cor- responding constraint equations may be written in explicit or im- plicit form, respectively, as x = x�y,t� or ��x,t� = 0 �4� where the position vector y summarizes the f generalized coordi- nates of the holonomic system y�t� = �y1 y2 y3 ¯ yf�T. �5� Then, for the holonomic system’s translation and rotation it re- mains ri = ri�y,t�, Si = Si�y,t� . �6� By differentiation the absolute translational and rotational velocity vectors are found �i = r˙i = � ri � yT y˙ + � ri � t = JT i�y,t�y˙ + �¯i�y,t� , �7� �i = s˙i = � si � yT y˙ + � si � t = JR i�y,t�y˙ + �¯i�y,t� �8� where si means a vector of infinitesimal rotations following from the corresponding rotation tensor, see, e.g., Schiehlen �10�, and �¯i, �¯i are the local velocities. Further, the Jacobian matrices JT i and JR i for translation and rotation are defined by Eqs. �7� and �8�. The system may be subject to additional r nonholonomic con- straints which do not affect the f =6p−q positional degrees of freedom. But they reduce the velocity dependent degrees of free- dom to g= f −r=6p−q−r. The corresponding constraint equations can be written explicitly or implicitly, too, y˙ = y˙�y,z,t� or ��y, y˙,t� = 0 �9� where the g generalized velocities are summarized by the vector z�t� = �z1 z2 z3… zg�T. �10� For the system’s translational and rotational velocities it follows from Eqs. �7�–�9� �i = �i�y,z,t� and �i =�i�y,z,t� . �11� By differentiation the acceleration vectors are obtained, e.g., the translational acceleration as ai = � �i � zT z˙ + � �i � yT + � �i � t = LT i�y,z,t�z˙ + �˙�i�y,z,t� �12� where �˙�i denotes the so-called local accelerations. A similar equation yields the rotational acceleration. The Jaco- bian matrices LT i and LR i, respectively, are related to the gener- alized velocities, for translations as well as for rotations. 2.3 Newton-Euler Equations. Newton’s equations and Eul- er’s equations are based on the velocities and accelerations from Sec. 2.2 as well as on the applied forces and torques, and the constraint forces and torques acting on all bodies. The reactions or constraint forces and torques, respectively, can be reduced to a minimal number of generalized constraint forces also known as Lagrange multipliers. In matrix notation the following equations are obtained, see also Schiehlen �10�. Free body system kinematics and holonomic constraint forces: Transactions of the ASME E license or copyright; see http://www.asme.org/terms/Terms_Use.cfm M�x¨ + q¯c�x, x˙,t� = q¯e�x, x˙,t� + Q¯ g, Q¯ = −�xT. �13� Do Holonomic system kinematics and constraints: M�J¯ y¨ + q¯c�y, y˙,t� = q¯e�y, y˙,t� + Q¯ g . �14� Nonholonomic system kinematics and constraints: M�L¯ z˙ + q¯c�y,z,t� = q¯e�y,z,t� + Q¯ g . �15� On the left hand side of Eqs. �13�–�15� the inertia forces appear characterized by the inertia matrix M�, the global Jacobian matri- ces J¯ , L¯ and the vector q¯c of the Coriolis forces. On the right hand side the vector q¯e of the applied forces, which include control forces, and the constraint forces composed by a global distribution matrix Q¯ and the vector of the generalized constraint forces g are found. Each of the Eqs. �13�–�15� represents 6p scalar equations. However, the number of unknowns is different. In Eq. �13� there are 6p+q unknowns resulting from the vectors x and g. In Eq. �14� the number of unknowns is exactly 6p= f +q represented by the vectors y and g, while in Eq. �15� the number of unknowns is 12p−q due to the additional velocity vector z and an extended constraint vector g. Obviously, the Newton-Euler equations have to be supplemented for the simulation of motion. 2.4 Equations of Motion of Rigid Body Systems. The equa- tions of motion are complete sets of equations to be solved by vibration analysis and/or numerical integration. There are two ap- proaches used resulting in differential-algebraic equations �DAE� or ordinary differential equations �ODE�, respectively. For the DAE approach the implicit constraint equations �4� are differentiated twice and added to the Newton-Euler equations �13� resulting in �M� �xT �x 0 ��x¨g � = �q¯e − q¯c −�˙ t −� ˙ x x˙ � . �16� Equation �16� is numerically unstable due to a double zero ei- genvalue originating from the differentiation of the constraints. During the last decade great progress was achieved in the stabili- zation of the solutions of Eq. �16�. This is, e.g., well documented in Eich-Soellner and Führer �36�. The ODE approach is based on the elimination of the constraint forces using the orthogonality of generalized motions and con- straints, J¯TQ¯ =0, also known as d’Alembert’s principle �3� for ho- lonomic systems. Then, it remains a minimal number of equations M�y,t�y¨ + k�y, y˙,t� = q�y, y˙,t� . �17� The orthogonality may also be used for nonholonomic systems, L¯TQ¯ =0, corresponding to Jourdain’s principle �5� and Kane’s equations �6�. However, the explicit form of the nonholonomic constraints �9� has to be added, y˙ = y˙�y,z,t�, M�y,z,t�z˙ + k�y,z,t� = q�y,z,t� . �18� Equations �17� and �18� can now be solved by any standard time integration code. 2.5 Equations of Motion of Flexible Systems. The equations presented can also be extended to flexible bodies, Fig. 1. For the analysis of small structural vibrations the relative nodal coordinate formulation �RNCF� with a floating frame of reference is used while for large deformations the absolute nodal coordinate formu- lation �ANCF� turned out to be very efficient, see, e.g., Melzer �37� and Shabana �19,20�. Within the RNCF the small f f relative coordinates describing the elastic deformations are added to the large fr rigid body coor- dinates of the reference frame moving with translation r�t� and rotation S�t� resulting in an extended position vector Journal of Computational and Nonlinear Dynamics wnloaded 20 Oct 2012 to 60.171.124.72. Redistribution subject to ASM y�t� = �yr T y f T�T �19� where the subvectors yr, y f summarize the corresponding coordi- nates. Then, the extended equations of motion read as M�y,t�y¨ + k�y, y˙,t� + ki�y, y˙� = q�y, y˙,t� . �20� In comparison to Eq. �17� the additional term ki�y, y˙� = �0 00 K �y + �0 00 D �y˙ �21� depends only on the stiffness and damping matrices K and D of the flexible bodies. Moreover, the inertia matrix shows the inertia coupling due to the relative coordinates M = �Mrr MrfMrfT M f f � . �22� Within the ANCF for highly flexible bodies fa absolute coordi- nates are summarized in a vector ya characterizing the material points of the bodies by an appropriate shape function. Then, the equations of motion read as M y¨a + Ka�ya�ya = q�ya,t� �23� where M is a constant mass matrix and the vector k of the gener- alized Coriolis forces is vanishing due to the absolute coordinates. This is true for standard finite elements like Euler beams or bricks. However, for Timoshenko beams with rotary inertia Eq. �20� may be found again as pointed out by von Dombrowski �38�. In any case, the stiffness matrix Ka is highly nonlinear and requires spe- cial evaluation as shown by Shabana �19�. 3 Recursive Formalisms For time integration of holonomic systems the mass matrix in Eq. �17� has to be inverted what is numerically costly for systems with many degrees of freedom, y¨�t� = M−1�y,t��q�y,y˙,t� − k�y,y˙,t�� . �24� Recursive algorithms avoid this matrix inversion. The funda- mental requirement, however, is a chain or tree topology of the multibody system as shown in Fig. 2. Loop topologies are not included. Contributions on recursive algorithms are due, e.g., to Hollerbach �39�, Featherstone �40�, Bae and Haug �41�, Brandl, Johanni, and Otter �42�, and Schiehlen �43�. 3.1 Kinematics. Recursive kinematics use the relative motion between two neighboring bodies and the related constraints as shown in Fig. 3. The absolute translational and rotational velocity Fig. 1 Reference systems for flexible multibody systems JANUARY 2006, Vol. 1 / 5 E license or copyright; see http://www.asme.org/terms/Terms_Use.cfm ˜ Do vector wi of body i is related to the absolute velocity vector wi−1 of body i−1 and the generalized coordinates yi of the joint i be- tween these two bodies. It yields �25� Here, the tilde notation for the cross product is used, i.e., it holds ab=a�b, for two arbitrary vectors a, b see, e.g., Roberson and Schwertassek �27�. Using the fundamentals of relative motion of rigid bodies, it remains for the absolute acceleration bi = Ci bi−1 + Ji y¨i + �i �y˙i,wi−1� �26� where the vector bi summarizes the translational and rotational accelerations of body i. For the total system one gets for the absolute acceleration in matrix notation b = C b + J y¨ + � �27� where the geometry matrix C is a lower block-subdiagonal matrix and the Jacobian matrix J is a block-diagonal matrix as follows C = � 0 0 0 ¯ 0 C2 0 0 ¯ 0 0 C3 0 ¯ 0 ] ] � � 0 0 0 0 Cp 0 �, J = � J1 0 0 ¯ 0 0