Multibody System Dynamics: Roots and Perspectives · Multibody System Dynamics: Roots and...

Multibody System Dynamics 1: 149–188, 1997. 149c 1997 Kluwer Academic Publishers. Printed in the Netherlands.

Multibody System Dynamics:Roots and Perspectives

W. SCHIEHLENInstitute B of Mechanics, University of Stuttgart, D-70550 Stuttgart, Germany

(Received: 21 January 1997; accepted in revised form: 15 April 1997)

Abstract. The paper reviews the roots, the state-of-the-art and perspectives of multibody systemdynamics. Some historical remarks show that multibody system dynamics is based on classicalmechanics and its engineering applications ranging from mechanisms, gyroscopes, satellites androbots to biomechanics. The state-of-the-art in rigid multibody systems is presented with reference totextbooks and proceedings. Multibody system dynamics is characterized by algorithms or formalisms,respectively, ready for computer implementation. As a result simulation and animation are mostimportant. The state-of-the-art in flexible multibody systems is considered in a companion review byShabana.

Future research fields in multibody dynamics are identified as standardization of data, couplingwith CAD systems, parameter identification, real-time animation, contact and impact problems,extension to control and mechatronic systems, optimal system design, strength analysis and interac-tion with fluids. Further, there is a strong interest on multibody systems in analytical and numericalmathematics resulting in reduction methods for rigorous treatment of simple models and special inte-gration codes for ODE and DAE representations supporting the numerical efficiency. New softwareengineering tools with modular approaches promise improved efficiency still required for the moredemanding needs in biomechanics, robotics and vehicle dynamics.

Key words: dynamics of rigid bodies, multibody systems, computational methods, data models,parameter identification, optimal design, strength analysis, DAE integration codes.

1. Historical Remarks

The dynamics of multibody systems is based on classical mechanics. The mostsimple element of a multibody system is a free particle which can be treated byNewton’s equations published in 1686 in his “Philosophiae Naturalis PrincipiaMathematica” [111]. The principal element, the rigid body, was introduced in 1775by Euler in his contribution entitled “Nova methodus motum corporum rigidarumdeterminandi” [43]. For the modeling of constraints and joints, Euler already usedthe free body principle resulting in reaction forces. The equations obtained areknown in multibody dynamics as Newton–Euler equations.

A system of constrained rigid bodies was considered in 1743 by d’Alembert inhis “Traite de Dynamique” [32] where he distinguished between applied and reac-tion forces. D’Alembert called the reaction forces “lost forces” having the principleof virtual work in mind. A mathematical consistent formulation of d’Alembert’sprinciple is due to Lagrange [89] combining d’Alembert’s fundamental idea with

150 W. SCHIEHLEN

the principle of virtual work. As a result a minimal set of ordinary differentialequations (ODE) of second order is found.

A systematic analysis of constrained mechanical systems was established in1788 by Lagrange [89], too. The variational principle applied to the total kineticand potential energy of the system considering its kinematical constraints and thecorresponding generalized coordinates result in the Lagrangian equations of thefirst and the second kind. Lagrange’s equations of the first kind represent a set ofdifferential-algebraical equations (DAE) while the second kind leads to a minimalset of ordinary differential equations (ODE).

An extension of d’Alembert’s principle valid for holonomic systems only waspresented in 1913 by Jourdain [76]. For nonholonomic systems the variationswith respect to the translational and rotational velocities resulting in generalizedvelocities are required. Then, a minimal set of ordinary differential equations(ODE) of first order is obtained. The approach of generalized velocities, identifiedas partial velocities, was also introduced by Kane and Levinson [81]. The resultingKane’s equations represent a compact description of multibody systems. Moredetails on the history of classical mechanics including rigid body dynamics can befound in Pasler [120] and Szabo [181].

The first applications of the dynamics of rigid bodies are related to gyrodynam-ics, mechanism theory and biomechanics. Euler’s equations for the kinematics anddynamics of a single gyro date back to 1758. For more than a century, the researchon the solution of Euler’s equations attracted mathematicians and mechanicians.At the beginning of this century the engineering applications of the single gyro-scope got more important. Then, gyroscopic systems received also some attention.Grammel mentioned in 1920 in the first edition of his book “Der Kreisel – SeineTheorie und seine Anwendungen” [51] a two-gyro system but he did not discussits dynamics. Thirty years later in the second edition of the same book a smallsection was already devoted to gyroscopic systems. Magnus presented in 1971in his book “Kreisel” [99] a large section on gyrosystems including a rigorousstability theory. For example, a cardanic suspended gyro has to be modeled accu-rately as a three-body system (Figure 1). In 1977 Magnus [100] organized the firstIUTAM Symposium on Dynamics of Multibody Systems with quite a number ofcontributions to gyroscopic problems.

Mechanism theory deals also with the motion of constrained mechanical sys-tems. However, the application of the powerful graphical methods, developed, e.g.,in 1913 by Wittenbauer [194], was restricted to planar mechanisms. Later in 1955,matrix methods were introduced by Denavit and Hartenberg [33] for spatial kine-matics which formed the basis for the dynamical analysis of spatial linkages firstpublished by Uicker [187].

Early applications of rigid body dynamics are also found in biomechanics. Fis-cher [44] modeled in 1906 the walking motion of humans by rigid bodies. In thesecond half of this century biomechanics was strongly supported by research inathletic training and sports. For example, Chaffin [27] presented in 1969 a com-

MULTIBODY SYSTEM DYNAMICS: ROOTS AND PERSPECTIVES 151

outer body

rotor body

foundation

inner body

Figure 1. Three-body system.

puterized biomechanical model to study gross body motions. Kane and Scher [80]investigated in the same year the falling cat phenomena by rigid bodies. Vukobra-tovic et al. [189] discussed in 1970 the stability of the biped human locomotion,and Huston and Passerello [69] presented in 1971 a complete human body model.

Classical mechanics, rigid body systems and their applications have been char-acterized by strong restrictions on the model complexity until the 1960s. Thenonlinearity of large rotations and the highly nonlinear gyroscopic coupling in theequations of motion together with very inefficient numerical methods for solvingdifferential equations were insurmountable. However, the requirements for morecomplex models of satellites and spacecrafts, and the fast development of moreand more powerful computers led to a new branch of mechanics: multibody systemdynamics. The results of classical mechanics had to be reviewed and extendedas a basis of computer algorithms, the multibody formalisms. One of the first for-malisms is due to Hooker and Margulies [65] in 1965. This approach was developedfor satellites consisting of an arbitrary number of rigid bodies interconnected byspherical joints. Another formalism was published in 1967 by Roberson and Wit-tenburg [140]. In addition to these numerical formalisms, the progress in computerhardware and software allowed formula manipulation with the result of symboli-cal equations of motion, too. First contributions in 1977 are due to Levinson [92]and Schiehlen and Kreuzer [148]. In the 1980s complete software systems for themodeling, simulation and animation were offered on the market as described bySchwertassek and Roberson [165]. The state-of-the-art in 1990 was documented by

152 W. SCHIEHLEN

the Multibody System Handbook [152]. Reviews on multibody dynamics includ-ing modeling analysis methods and applications were presented by Kortum andSchiehlen [84] and Huston [71].

The scientific research in multibody system dynamics has been devoted toimprovements in modeling considering nonholonomic constraints, flexibility, fric-tion, contact, impact, and control. New methods evolved with respect to simulationby recursive formalisms, to closed kinematical loops, reaction forces and torques,and to pre- and postprocessing by data models, CAD coupling, signal analysis,animation and strength evaluation. The state-of-the-art will be discussed in thefollowing in more details.

2. State-of-the-Art in Rigid Multibody Systems

The modeling of rigid multibody systems will be presented and related to some ofthe algorithms widely used today. Then, the textbooks and some of the proceedingsvolumes will be reviewed.

2.1. MODELING AND FORMALISMS

The method of multibody systems utilizes a finite set of elements such as rigidbodies and/or particles, bearings, joints and supports, springs and dampers, activeforce and/or position actuators. For the unique mathematical description of theseelements a datamodel has been defined as a standardized basis for all kinds ofcomputer codes by Otter et al. [118]. First steps with respect to internationalstandardization have been achieved, Durr et al. [34].

The following assumptions were agreed upon:1. A multibody system consists of rigid bodies and ideal joints. A body may

degenerate to a particle or to a body without inertia. The ideal joints includethe rigid joint, the joint with completely given motion (rheonomic constraint)and the vanishing joint (free motion).

2. The topology of the multibody system is arbitrary. Chains, trees and closedloops are admitted.

3. Joints and actuators are summarized in open libraries of standard elements.4. Subsystems may be added to existing components of the multibody system.

A multibody system as defined is characterized on the basis of a datamodelby the class mbs consisting of an arbitrary number of objects of the classes partand interact (see Figure 2). The class part describes rigid bodies. Each part ischaracterized by at least one body-fixed frame. It may have a mass, a center ofmass and a tensor of inertia summarized in the class body. The class interactdescribes the interaction between a frame on part i and a frame on part (i +1). The interaction may be realized by a joint, by a force actuator or a sensorresulting in the classes joint, force or sensor, respectively. Thus, the class interactis characterized by two types of information: the frames to be connected and the


ÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉ

ÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉ

part ipart referenceframe i

part ( i +1 )

frames to be connectedpart referenceframe ( i + 1 )

inertial frame

interact

Figure 2. Multibody system to be represented by the datamodel.

connecting element itself. These classes form the basis of the class mbs representingthe assembled multibody system. The model assembly using the datamodel is theneasily executed. According to the definitions, the datamodel represents holonomic,rheonomic multibody systems.

The multibody system has to be described mathematically by equations ofmotions for the dynamical analysis. The general theory for holonomic and non-holonomic systems will be presented using a minimal number of generalized coor-dinates for a unique representation of the motion.

2.1.1. Kinematics of Multibody Systems

According to the free body diagram of a mechanical system at first all constraintsare omitted and a system of p bodies holds 6p degrees of freedom. The position ofthe system is given relative to the inertial frame by the 3� 1-translation vector

ri = [ri1 ri2 ri3]T ; i = 1(1)p; (1)

of the center of mass Ci and the 3� 3-rotation tensor

Si = Si(�i; �i; i); (2)

written down for each body. The rotation tensor Si depends on three angles �i, �i, i and corresponds with the direction cosine matrix relating the inertial frame Iand the body-fixed frame i to each other. The 3p translational coordinates and the3p rotational coordinates (angles) can be summarized in a 6p� 1-position vector

x = [r11 r12 r13 r21 � � � �p �p p]T : (3)

Equations (1) and (2) read now

ri = ri(x); Si = Si(x): (4)

Secondly, the q holonomic, rheonomic constraints are added to the mechanicalsystem given explicitly or implicitly by

x = x(y; t) or �(x; t) = 0; (5)

154 W. SCHIEHLEN

respectively, where the f � 1-position vector

y = [y1 y2 y3 � � � yf ]T (6)

is used for summarizing the f generalized coordinates of the system and � meansa q� 1-vector function. The number of generalized coordinates corresponds to thenumber of degrees of freedom, f = 6p� q, with respect to the system’s position.Then, translation and rotation of each body follow from (4) and (5) as

ri = ri(y; t); Si = Si(y; t); (7)

and the velocities are found by differentiation with respect to the inertial frame:

vi = _ri =@ri

@yT_y +

@ri

@t= JT i(y; t) _y + �vi(y; t); (8)

!i = _si =@si

@yT_y +

@si

@t= JRi(y; t) _y + �!i(y; t): (9)

The 3 � f -Jacobian matrices JT i and JRi defined by (8) and (9) characterizethe virtual translational and rotational displacement of the system, respectively.Later, they are also required for the application of d’Alembert’s principle. Theinfinitesimal 3 � 1-rotation vector si used in (9) follows analytically from thecorresponding infinitesimal skew-symmetrical 3� 3-rotation tensor. However, thematrix JRi can also be found by a geometrical analysis of the angular velocityvector !i with respect to the angles �i; �i; i, see, e.g., [149].

The accelerations are obtained by a second differentiation with respect to theinertial frame:

ai = JT i(y; t) �y +@vi

@yT_y +

@vi

@t; (10)

�i = JRi(y; t) �y +@!i

@yT_y +

@!i

@t: (11)

For scleronomic constraints the partial time-derivatives in (8), (9) and (10), (11)vanish.

Thirdly, the r nonholonomic, rheonomic constraints, e.g., due to rigid wheels,are introduced explicitly or implicitly by

_y = _y(y;z; t) or (y; _y; t) = 0; (12)

respectively, with the g � 1-velocity vector

z(t) = [z1 z2 z3 : : : zg]T (13)

summarizing the g generalized velocities of the system. Further, means a r� 1-vector function. The number of generalized velocities characterizes the number of


degrees of freedom, g = f � r, with respect to the system’s velocity. From (8), (9)and (12) the translational and rotational velocity of each body follow immediatelyas

vi = vi(y;z; t); !i = !i(y;z; t): (14)

The accelerations are found again by differentiation with respect to inertialframe:

ai =@vi

@zT_z +

@vi

@yT_y +

@vi

@t= LT i(y;z; t) _z + _��vi(y;z; t); (15)

�i =@!i

@zT_z +

@!i

@yT_y +

@!i

@t= LRi(y;z; t) _z + _��!i(y;z; t): (16)

Here, the 3�g-matricesLT i andLRi describe the virtual translational and rotationalvelocity of the system needed also for the application of Jourdain’s principle.Further, it has to be mentioned that the partial time-derivatives vanish in (15) and(16) for scleronomic systems.

2.1.2. Newton–Euler Equations

For the application of Newton’s and Euler’s equations to multibody systems thefree body diagram has to be used again. Now the rigid bearings and supportsare replaced by adequate constraints forces and torques as discussed later in thissection.

Newton’s and Euler’s equations read for each body in the inertial frame

mi _vi = fei + f

ri ; i = 1(1)p; (17)

Ii _!i + e!iI i!i = lei + l

ri ; i = 1(1)p: (18)

The inertia is represented by the mass mi and the 3 � 3-inertia tensor Ii withrespect to the center of mass Ci of each body. The external forces and torques in(17) and (18) are composed by the 3�1-applied force vector fei and torque vector leidue to springs, dampers, actuators, weight, etc., and by the 3� 1-constraint forcevector f ri and torque vector lri . All torques are related to the center of mass Ci. Theapplied forces and torques, respectively, depend on the motion by appropriate lawsand they may be coupled to the constraint forces and torques in the case of friction.

The constraint forces and torques originate from the reactions in joints, bearings,supports or wheels. They can be reduced by distribution matrices to the generalizedconstraint forces. The number of the generalized constraint forces is equal to thetotal number of constraints (q+r) in the system. Introducing the (q+r)�1-vectorof generalized constraint forces

� = [�1 �2 �3 : : : �q+r]T (19)

156 W. SCHIEHLEN

and the 3 � (q + r)-distribution matrices

F i = F i(y;z; t); Li = Li(y;z; t) (20)

it turns out

f ri = F i �; lri = Li �; i = 1(1)p; (21)

for each body. The constraint forces or the distribution matrices, respectively, arederived by geometrical analysis or they can be found analytically, see Equations (33)to (35).

The ideal applied forces and torques depend only on the kinematical variablesof the system, they are independent of the constraint forces. Ideal applied forcesare due to the elements of multibody systems, and further actions on the system,e.g., gravity. The forces may be characterized by proportional, differential and/orintegral behavior.

The proportional forces are characterized by the system’s position and timefunctions

f ei = fei (x; t): (22)

E.g., conservative spring and weight forces as well as purely time-varying forcesare proportional forces.

The proportional-differential forces depend on the position and the velocity:

f ei = fei (x; _x; t): (23)

A parallel spring-damper configuration is a typical example for this kind of forces.The proportional-integral forces are a function of the position and integrals of theposition:

f ei = fei (x;w; t); _w = _w(x;w; t); (24)

where the p�1-vectorw describes the position integrals. E.g., serial spring-damperconfigurations and the eigendynamics of actuators result in proportional-integralforces. In vehicle systems proportional-integral forces appear, e.g., with modernengine mounts for simultaneous noise and vibration reduction. The same laws holdalso for ideal applied torques.

In the case of nonideal constraints with sliding friction or contact forces, respec-tively, the applied forces are coupled with the constraint forces.

The Newton–Euler equations of the complete system are summarized in matrixnotation by the following vectors and matrices. The inertia properties are writtenin the 6p� 6p-diagonal matrix

��M = diag fm1E m2E � � � I1 � � � Ipg ; (25)


where the 3 � 3-identity matrix E is used. The 6p � 1-force vectors �qc, �qe, �qr

representing the Coriolis forces, the ideal applied forces and the constraint forces,respectively, are given by the following scheme,

�q =hfT1 f

T2 � � � lT1 � � � lTp

iT: (26)

Further, the 6p � f -matrix �J and 6p � g-matrix �L as well as the 6p � (q + r)-distribution matrix �Q are introduced as global matrices, e.g.,

�J =hJTT1 J

TT2 � � � J

TR1 � � � J

TRp

iT: (27)

Now, the Newton–Euler equations can be represented in the inertial frame asfollows for holonomic systems

��M �J �y + �qc(y; _y; t) = �qe(y; _y; t) + �Q � (28)

and for nonholonomic systems

��M �L _z + �qc(y;z; t) = �qe(y;z; t) + �Q �; (29)

respectively. If the nonholonomic constraints disappear, e.g. z = _y, (29) reducesto (28), showing a close relation between both representations.

2.1.3. Equations of Motion

The Newton–Euler equations are combined algebraical and differential equationsand the question arises if they can be separated for solution into purely algebraicaland differential equations. There is a positive answer given by the dynamicalprinciples. In a first step, the system’s motion can be found by integration ofthe separated differential equations and in a second step the constraint forcesare calculated algebraically. For ideal applied forces both steps can be executedsuccessively while contact forces require simultaneous execution.

Holonomic systems with proportional or proportional-differential forces resultin ordinary multibody systems. The equations of motion follow from the Newton–Euler equations, applying d’Alembert’s principle.

The equations of motion of holonomic systems are found according to d’Alem-bert’s principle by premultiplication of (28) with �J

T as

M(y; t) �y + k(y; _y; t) = q(y; _y; t): (30)

Here the number of equations is reduced from 6p to f , the f � f -inertia matrixM(y; t) is completely symmetrized, M(y; t) = �J

T ��M �J > 0, and the constraintforces and torques are eliminated. The remaining f � 1-vector k describes thegeneralized Coriolis forces and the f �1-vector q includes the generalized appliedforces. Equation (30) may also be obtained from Lagrange’s equations of the second

158 W. SCHIEHLEN

kind, however, the procedure is computationally less efficient due to additionaldifferentiations of the kinetic energy required.

Nonholonomic systems with proportional-integral forces produce general multi-body systems. The equations of motion are obtained from Newton–Euler equa-tions (29) where the proportional-integral forces (24) and Jourdain’s principle hasto be regarded. However, the equations of motion are not sufficient, they have tobe completed by the explicit nonholonomic constraint equations (12). Thus, thecomplete equations read as

M (y;z; t) _z + k(y;z; t) = q(y;z;w; t);

_y = _y(y;z; t); _w = _w(y;z; t): (31)

Now the number of dynamical equations is reduced from 6p to g and the g � g-symmetric inertia matrix M(y; t) = �L

T ��M �L > 0 appears. Further, k and q areg � 1-vectors of generalized Coriolis and applied forces. Equations (31) are alsodenoted as Kane’s equations in literature.

In addition to the mechanical representation (31) of a multibody system, thereexists also the possibility to use the more general representation of dynamicalsystems in the state space, i.e.,

_x = f(x; t); (32)

where x means the n � 1-state vector composed of generalized coordinates andvelocities, and t the time, respectively.

Equation (30) is also true for unconstraint systems. Then, it yields y = x, andthe global Jacobian matrix �J is a quadratic 6p � 6p-matrix. Adding the implicitconstraints (5) again, Lagrange’s equations of the first kind are obtained as

M(x) �x+ k(x; _x; t) = q(x; _x; t)��Tx �; (33)

where � is the q � 1-vector of Lagrangian multipliers. By comparison with (28)it turns out that the Lagrangian multipliers may be interpreted as generalizedconstraint forces and the distribution matrix �Q can be obtained from the implicitformulation of the constraints. However, the 6p scalar equations (33) cannot besolved due to the 6p+ q unknowns in the vectors x, �. Therefore, the implicitconstraint equations (5) have to be called again

�(x; t) = 0: (34)

It remains a set of 6p+ q differential algebraical equations of index 3. One popularapproach to solve (33), (34) is to reduce the index by two differentiations withrespect to time"

M �Tx

�x 0

# "�x

�

#=

"q � k

��t �_�x _x

#: (35)


However, due to the derivative, Equations (35) are numerically unstable, see Sec-tion 3.9.

Furthermore, Equations (31) can be used for holonomic systems, too. Then, thef generalized coordinates and the f generalized velocities are independent fromeach other resulting often in a strong simplification of the dynamical equations.Such a separation of kinematics and kinetics has been successfully used by Eulerfor his kinematical and dynamical equations of a gyroscope.

Even if the constraint forces were completely omitted by the dynamical prin-ciples, they are also of engineering interest for the load in joints, bearings andsupports, and they are absolutely necessary for the computation of contact andfriction forces. From the 6p coordinates of the constraint force vector �qr there areonly (q + r) coordinates linear independent according to (21). Therefore, only the(q+ r)� 1-vector � of the generalized constraint forces is needed. The results aregiven for holonomic systems only, r = 0, but they can be transferred to nonholo-nomic systems without any problem.

According to the d’Alembert’s principle premultiplication of (28) by �QT ��M

�1

results immediately in the equations of reaction

N (y; t) �+ q(y; _y; t) = k(y; _y; t) (36)

where N (y; t) is the symmetrical q � q-reaction matrix and q and k are q � 1-vectors. The equations of reaction (36) are purely algebraical equations as knownfrom problems in statics.

2.1.4. Formalisms

The equations of motion presented may be automatically generated by formalismsas described in the Multibody Systems Handbook [152]. There are two differentkinds of formalisms, the numerical and the symbolical ones (Figure 3). The numer-ical equations of motion have to be generated for each timestep of the integrationcode and for each parameter variation. The symbolical equations were generatedonly once, they are especially helpful for real time applications and parameteroptimization. Symbolical equations may be obtained by formula manipulators likeMAPLE or with special formalisms, e.g., NEWEUL.

From a numerical point of view recursive algorithms are very efficient forsystems with a large number of joints what means more than 6 to 10 in a serialtopology. The main idea of the recursive procedure is to avoid the inversion of theinertia matrix ��M in Equation (30) which is required for numerical integration, see[153]. More recently, Stelzle et al. [180] made a comparative study of recursivemethods. For flexible multibody systems the recursive approach has proven to bealso very attractive, Amirouche and Xie [5], Ider [73], and Kim and Haug [82].

160 W. SCHIEHLEN

Model description

Data input

Formalism

Simulation

Local output

Global result

Pa

ram

ete

r va

ria

tion

Numerical equations

Ne

xt t

ime

ste

p

Mo

de

l va

ria

tion

Model description

Data input

Formalism

Simulation

Local output

Global result

Pa

ram

ete

r va

ria

tion

Symbolical equations

Ne

xt t

ime

ste

p

Mo

de

l va

ria

tion

Figure 3. Numerical and symbolical formalisms.

2.2. TEXTBOOKS AND PROCEEDINGS

The first international symposium on multibody dynamics was sponsored by theInternational Union of Theoretical and Applied Mechanics (IUTAM) and organized1977 by Magnus [100] in Munich, Germany. A NATO Advanced Study Instituteon computer-aided analysis and optimization held 1983 in Iowa City, U.S.A., wasalso devoted to multibody dynamics, Haug [59]. At the 8th symposium of theInternational Association of Vehicle System Dynamics (IAVSD) in 1985 a generallecture on multibody systems software was delivered by Kortum and Schiehlen [84].A second IUTAM Symposium on Dynamics of Multibody Systems took place 1985in Udine, Italy, Bianchi and Schiehlen [22]. Dynamical problems of rigid-elasticsystems and structures were considered 1990 at an IUTAM Symposium in Moscow,


U.S.S.R., Banichuk et al. [12]. The first Fields Institute Workshop entitled “TheFalling Cat and Related Problems” was held 1992 in Waterloo, Ontario, Canada,representing the interest of applied mathematicians in multibody dynamics, Enos[41]. Another NATO Advanced Study Institute was held 1993 in Lisbon, Portugalwith special emphasis on computational methods for multibody systems, Pereiraand Ambrosio [122]. Most recently, an IUTAM Symposium on optimization ofmultibody systems took place in Stuttgart, Germany, see Bestle and Schiehlen[21]. It turns out that multibody system dynamics is now a well established andvery lively branch of mechanics.

The first textbook on multibody dynamics was written in 1977 by Wittenburg[195]. Starting with rigid body kinematics and dynamics, classical problems ofone rigid body are presented as well as general multibody systems. A textbookpublished 1986 by Schiehlen [149] presents in unified manner multibody systems,finite element systems and continuous systems as equivalent models for mechanicalsystems. The computer-aided analysis of multibody systems was considered in 1988in a textbook by Nikravesh [113] for the first time. Roberson and Schwertassek[141] discuss the origin of multibody systems, they deal with one and several rigidbodies. Comments on linearized equations and computer simulation techniques areincluded. Basic methods of computer aided kinematics and dynamics of mechanicalsystems are shown in 1989 by Haug [60] for planar and spatial systems.

In his first textbook from 1989, Shabana [170] deals in particular with flexiblemultibody systems. This is a new and promising research area. Huston [70] presentskinematics, force and inertia concepts, multibody kinetics, numerical methods aswell as flexible multibody systems. Another textbook on flexible multibody systemsis due to Bremer and Pfeiffer [25], a broad variety of engineering examples is foundin that book. Computational methods for multibody dynamics are treated in 1992by Amirouche [4] with special emphasis on matrix methods. Garcia de Jalon andBayo [47] present efficient methods for the kinematic and dynamic simulation ofmultibody systems to meet the real time challenge. Shabana’s second book [171]from 1994 is devoted to computational dynamics of rigid multibody systems. Manydetailed examples show the execution of the computations required. Angeles andKecskemethy [8] summarize the contributions to a postgraduate course offered1995 at the International Center of Mechanical Sciences in Udine, Italy.

The textbook of Stejskal and Valasek [179] starts from the CAD design of spatialmechanics, discusses free bodies, describes the constraints by lower and higherkinematic pairs, it presents the dynamic analysis and reports on computational andnumerical matters.

In addition to the textbooks, the software for multibody systems is comparedand tested by benchmarks in the handbook by Schiehlen [152] and a collectionof codes was published by Kortum and Sharp [85] in 1993. The benchmarks ofthe handbook [152] are a seven-body mechanism and a robot, while in the codecollection [85] a road and rail vehicle are chosen as benchmarks.

162 W. SCHIEHLEN

NDF

A A

B B

C CD D

E E

Figure 4. Data exchange (a) bidirectional, (b) via a neutal data format NDF.

3. Perspectives

Multibody system dynamics is applied to a broad variety of engineering problemsfrom aerospace to civil engineering, from vehicle design to micromechanical analy-sis, from robotics to biomechanics. The fields of application are steadily increasing,in particular as multibody dynamics is considered as the basis of mechatronics, e.g.,controlled mechanical systems. These challenging applications require more fun-damental research on a number of topics which are presented in the following.

3.1. DATAMODELS FROM CAD

Within the multibody systems community many computer codes have been devel-oped, however, they differ widely in terms of model description, choice of basicprinciples of mechanics and topological structure so that a uniform descriptionof models does not exist. These distinct differences in the model description pre-clude the exchange of data between different formalisms. A most desirable dataexchange, however, would permit the alternate use of validated multibody systemmodels with different simulation systems, independent of the description formatselected, see Thomson et al. [182].

For the transfer of model data, e.g., between several multibody formalisms, oneapproach is to exchange the data directly between the formalisms. To exchange thedata between the different formalisms, data interfaces are necessary for adapting thedata representation. For this approach, bidirectional converters are required betweenall multibody formalisms within a simulation enviroment, in order to implement theexchange of models among each other. Each arrow in Figure 4a represents such aconverter. It is evident that the bidirectional approach is a costly solution comparedto the option of introducing a standardized, formalism-independent descriptionformat and providing conversion programs for this format only. In this more generalapproach, two data converters are required per formalism for the data exchange,so that changes with the interface of a formalism only have an effect on thesetwo converters. The incorporation of additional formalisms into the exchange-network does not affect the formalisms already included. One essential advantageof the second approach is that a neutral data format is provided (Figure 4b). The


ÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇ

ÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇ

ÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇ


ADAMS

Userinterface

SOLVER

Postprocessor

ADAMS

ADAMS

ADAMS

Modeldescription

ÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉ

ÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉ

ÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉ


DADS

Userinterface

SOLVER

Postprocessor

DADS

DADS

DADS

Modeldescription

ËËËËËËËËËËËËËËËËËË

ËËËËËËËËËËËËËËËËËËËËËËËË

ËËËËËËËËËËËËËËËËËËËËËËËË

ËËËËËËËËËËËËËËËËËË

Formalism 3

Preprocessor

SOLVER

Postprocessor

Formalism 3

Formalism 3

Formalism 3

Modeldescription

ÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊ

ÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊ

ÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊ

ÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊ

Formalism 4

Preprocessor

SOLVER

Postprocessor

Formalism 4

Formalism 4

Formalism 4

Modeldescription

Figure 5. Actual state of multibody system simulations.

responsibility for generating the converters for this description format lies with thesystem vendors.

The goal of such a standardization process is the unification and standardizationof the neutral data format, independent of any formalism. Only by means of sucha standardization it will be possible in future to achieve a situation in which asingle, unique model description is sufficient for describing a mechanical systemwith precision to make it accessible for analysis using any program package orformalism. The standardized model description is then used as the basis for theinput.

In a first step the input format required by the relevant formalism is generatedfrom the standardized model description and evaluated using a preprocessor orconverter. In a subsequent step, the formalism generates the mathematical modelequations.

On considering this process in greater detail for mechanical systems, the follow-ing comparison can be made between the actual status (Figure 5) and the desiredgoal (Figure 6). Until now a special model has to be created for each MBS formalismsuch as ADAMS or DADS, see [152]. This is read by the formalism with a specialpreprocessor/user interface. It is followed by the simulation and the evaluation ofthe results with postprocessors specific to the formalism. After standardization,there will be one neutral data model in which the mechanical system is stored instandardized form (Figure 6). The various MBS formalisms can access this modelusing their own preprocessors. After simulation, the results are stored again in aneutral data model from where they can be forwarded for data analysis, animation,etc., by postprocessors. The definition of a standardized result description may bepostponed in order to concentrate on the standardized input form.

The standardization of the multibody system data requires international coop-eration within STEP, see Durr et al. [34]. However, scientific support is necessary,too, as shown by Daberkow and Schiehlen [31]. In particular, a modular approach

164 W. SCHIEHLEN


ÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇÇ

Signal

Userinterface

SOLVER

Postprocessor

ADAMS

ADAMS


ÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉ

Animation

Userinterface

SOLVER

Postprocessor

DADS

DADS

ËËËËËËËËËËËËËËË

ËËËËËËËËËËËËËËËËËËËËËËËËËËËËËËËËËËËËËËËËËË

Strength

Preprocessor

SOLVER

Postprocessor

Formalism 3

Formalism 3

ÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊ

ÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊ

Preprocessor

SOLVER

Visualisation

Formalism 4

Formalism 4

Model description

Standardized result description

Neutral Data Format

analysis analysis

Figure 6. Desired goal for multibody system simulations.

for the modeling and simulation of multibody systems is most important as shownby Junker [77].

In addition to the standardization of input data, the postprocessing of the sim-ulation results by animation requires standardization, too. However, this aspectextends beyond the area of multibody dynamics.

3.2. PARAMETER IDENTIFICATION

In addition to the equations of motion, for multibody simulations the model datahave not only to be handled, they have to be determined first. For this purposeparameter identification is an essential part of multibody dynamics.

The equations of motion of mechanical systems undergoing large displace-ments are highly nonlinear, however they remain linear with respect to the systemparameters. This is of great advantage for the parameter identification.

All constant system parameters like distances, masses, moments of inertia,spring and damper coefficients and coefficients in nonlinear force laws, or combi-nations hereof, respectively, may be summarized in a � � 1-parameter vector p.Then, the characteristic vectors and matrices of the equations of motion (30) canbe rewritten as

M(y;p; t)�y + k(y; _y;p; t) = q(y; _y;u;p; t) (37)

where

M =

�Xj=1

pjM j(y; t); (38)


k =

�Xj=1

pjkj(y; _y; t); (39)

q =

�Xj=1

pjqj(y; _y;u; t) (40)

are linear with respect to the elements pj of the � � 1-parameter vector p, whichis composed of �k known parameters and �u = �� k unknown parameters to beidentified. Further, them�1-vector u represents the input variables of the system.The effort for parameter identification is generally increasing with the number �uof unknown parameters.

Therefore, information on parameters should be obtained first from all sourcesavailable. For example, the mass of a body can often be found simply by weightingand distances between bearings may be measured directly. The remaining unknownparameters have to be found experimentally using identification techniques.

From a more general point of view, the mathematical model (37) to (40) of thesystem may be reformulated as

R(x;u;p; t) _x = f(x;p; t); (41)

where x(t) now means the n � 1-state vector of the system and u(t) the m � 1-excitation vector. The n�n-weighting matrix of state derivativesR as well as then� 1-vector f on the right hand side are assumed to be linear with respect to theparameters,

R =

�Xj=1

pjRj(x;u; t); (42)

f =

�Xj=1

pjfj(x;u; t): (43)

In a typical experimental configuration for parameter identification the systemis driven by some broadband noise excitationu(t), while measurements of the statevectorx(t) are sampled and processed by a digital identification algorithm using themathematical model of the system, which consists of a set of ordinary differentialequations. As digital computers are based on algebraic operations only, the maindifficulty of identification of the time-continuous model (41) is the conversion intoan algebraic parameter identification problem.

If the state derivatives _x(t) are measured as well as the state x(t) and theexcitation u(t), the model (41) with the parametrization (42) and (43) results forevery sampled time instant tk = k�t,k = 1; : : : ; N , inn linear, algebraic equationsfor the unknown parameters. Then, the unknown parameters may be estimated usingN � �u=n samples by some identification technique for algebraic models like the

166 W. SCHIEHLEN

least-squares method, Isermann [74]. However, often direct measurements of statederivatives _x(t) are not available. Then, numerical techniques like state variablefiltering, Hsia [67], may be used to approximate the derivatives. This approach,however, shows some disadvantages in case of noise corrupted measurementsespecially if higher order derivatives are considered.

An alternative way for conversion of (41) into an algebraic problem is thenumerical integration of the model based on the sampled values x(tk);u(tk),k = 1; : : : ; N . The numerical effort and severe drift problems, however, make thismethod less suitable for practical applications.

Another approach is the transformation of the measured signals, for example bythe fast Fourier transform (FFT). Identification methods based on fast Fourier trans-form (FFT) algorithms are proven to be well suited for linear systems, Nathe [110].In the case of nonlinear systems, however, this method shows some restrictions.Thus, frequency domain methods seem to be less valuable for the identification ofnonlinear multibody systems.

Finally, the transformation of the mathematical model into difference formmay be looked as another solution of the problem. For linear multibody systems,such a method has been treated in detail by Schwarz [160]. Due to the nonlineartransformation of the model, however, the structure of the system is lost, thetransformed model no longer contains the physical parameters explicitly. As themethod is based on an explicit solution of the model equations, it is restricted tolinear systems and therefore not well suited for the identification of multibodysystems.

Regarding the difficulties mentioned above, covariance methods for identifica-tion of linear systems and nonlinear time-continuous systems were developed bySchiehlen and Kallenbach [79, 150]. The methods are based on stationary, ergodic,coloured noise excitations u(t). For identification, the measured signals u(t) andx(t) are processed by a linear, stable filter.

An extension of the covariance analysis to nonlinear functions of applied forcesis due to Krause and Schiehlen [87]. Nonlinear springs with cubic characteristicsand Coulomb’s friction may be identified for consideration in a multibody sys-tem simulation. Further, the sensitivity of parameter identification with respect tomeasurement noise has been investigated by Bestle and Krause [16].

The parameter identification of multibody system models has a high priority dueto the physically motivated discretization approach in multibody system dynamics.Using the right parameters, multibody system models are accurate and efficient.Therefore, parameter identification methods have to be adjusted and improvedcontinuously.

3.3. OPTIMAL DESIGN OF MULTIBODY SYSTEMS

Due to development of faster computing facilities the multibody system approach ischanging from a purely analyzing method to a more synthesizing tool. Optimization


��

� ��

p1

��

��

��

��

�1

��

� ��

p1

��

��

��

��

�1

��

� ��

p1

��

��

��

��

�1

��

� ��

optim

izat

ion

crite

ria �

(p)

p1

��

��

��

��

�1

desi

gn v

aria

bles

p multimodel performance evaluation

multicriteria optimization method

Figure 7. Multimodel analysis concept.

methods are applied to optimize multibody systems with respect to their dynamicbehavior, Grubel et al. [52], and Bestle [19]. The dynamics of multibody systemsis determined by parameters like the mass and moments of inertia of the materialbodies, geometrical data, stiffness and damping coefficients, or control parametersof actuators. Each of these parameters may serve as design variable for optimizingthe dynamic behavior.

Applications to technical problems clearly show that often several conflictingtechnical specifications and goals have to be taken into account. This situation canonly be treated by definition of several different performance criteria. Due to thepresence of more than one criterium the design problem has to be considered as amulticriteria optimization problem. The multicriteria optimization approach seemsto offer a promising way to handle the situation of conflicting system specificationsand requirements and to define optimal solutions.

Engineering applications also show that the analysis of different aspects of asystem has to be based on different models. For example, vertical vehicle dynam-ics can be studied with quarter-car or half-car models whereas studies of lateraldynamics require “bicycle models” or even spatial models. An optimization con-cept on the basis of simultaneously investigating several different models withshared parameters was demonstrated by Bestle and Eberhard [20].

According to Figure 7 the process of performance evaluation, i.e. computing theh� 1-performance vector from a given �� 1-design parameter vector p, has tobe split up into several submodel analyses. Each submodel is specially designed forevaluating the performance of the engineering system with respect to a subset ofdesign goals. For explaining the overall design concept, it may be just consideredas a black-box function between some input parameters �pi of the submodel i andthe output criterion values �

i= �

i(�pi).

168 W. SCHIEHLEN

op

timiz

atio

n c

rite

ria

y(p

)

de

sig

n v

aria

ble

s p

multicriteria optimization method

multimodel performance evaluation

equalityconstraints

inactivecriteria

sca

lar

op

timiz

atio

na

lgo

rith

mequalityconstraints

inequalityconstraints

singleobjective

mu

lticr

iteria

op

timiz

atio

nst

rate

gy

nonlinear programmingproblem

vector optimizationproblem

objectivefunctions

inequalityconstraints

Figure 8. Multicriteria optimization concept.

The input parameter vectors �pi of the submodels have to be linked to the globalset of design variables p, e.g., as a nonlinear vector function

�pi = �pi(p) (44)

to be defined by the designer. Using this relation, the criterion functions � i may

also be considered as functions of the global design variables p:

� i = �

i(�pi(p)) = � i(p): (45)

The total h� 1-vector criteria (p) is a union of the subsets of criteria � i(p); i =

1(1)m:

(p) = [ � 1T; �

2T; � � � ; �

mT

]T where n =mXi=1

�ni: (46)

In the problem formulation phase of the design process, the criteria should beconsidered to be just an instrument of performance evaluation. It is already part ofthe multicriteria optimization concept to classify them as objective functions fj(p),equality constraints gj(p) = 0 or inequality constraintshj(p) � 0 (Figure 8). Someof the performance criteria may even be neglected and considered as inactive ina first run in order to simplify the design process. Such a classification may bechanged several times within the design process to get a feeling for the engineeringsystem and the potentials of its optimization. Since several criteria may remain as


objective functions to be minimized, the problem has to be stated as a multicriteriaoptimization problem:

optp2P

f(p) where P := fp j g(p) = 0;h(p) � 0; pl � p � pug: (47)

The operator “opt” is used for simultaneous minimization of the individualobjective functions fj(p). In general, this is not possible due to conflicts arisingfrom different criteria. The multicriteria optimization approach, however, offers aconcept of defining optimal solutions also in the situation of conflicting objectivefunctions. Design points pP 2 P are called Edgeworth–Pareto (EP-) optimal, ifthere exists no feasible design point pwhere fj(p) � fj(p

P ) 8j^f(p) 6= f(pP ),Stadler [176]. In general, EP-optimal solutions are not unique and points withdifferent images are not comparable. The designer has to make, therefore, his finaldecision on an acceptable optimal solution out of the set of EP-optimal points.

EP-optimal solutions of the multicriteria optimization problem have to be founditeratively requiring several performance evaluations. Applied to dynamic systemdesign, performance evaluation involves a time-consuming numerical integrationof differential equations of motion. Therefore, not all multicriteria optimizationstrategies seem to be appropriate for dynamic system design. Strategies whichreduce the vector optimization problem to nonlinear programming problems haveproven to be very efficient. Several such strategies on the basis of the principles ofscalarization, hierarchization or a combination of it have been developed, Bestleand Eberhard [17]. The resulting nonlinear programming problems can be solvedvery efficiently with sequential quadratic programming (SQP) algorithms. Thedrawback, however, is the requirement of gradient information to be computedfrom the submodels. Taking into consideration the structure given in Figure 8, weobtain

d dp

=mXi=1

d� i

d�pid�pi

dp: (48)

The first term results from (45) provided by the specific submodel. The secondterm depends on the relation of the modelspecific parameters to the design vari-ables, see Equation (44). The major computational effort, however, results fromcomputing the sensitivity information d� i

=d�pi for each submodel.The design concept described above has been implemented in the program

system NEWOPT/AIMS by Bestle and Eberhard [18]. The submodels connectthe input variables �pi with the output variables �

i= �

i(�pi). The designer has

the possibilities to define these relations by any computer program, by analyticfunctions or a simulation program based on the multibody system approach.

The most important type of criteria with respect to dynamic system design arecriteria of the integral type for some time interval ti0 < t < ti1:

170 W. SCHIEHLEN

� ij = Gi1

j (ti1;yi1;zi1; �pi) +

ti1Zti0

F ij (t;y

i;zi; _zi; �pi) dt: (49)

These criteria do not only depend on the system parameters �pi, but also on thestate variables yi;zi describing the dynamic behavior of the submodel i. With themultibody system approach the state variables yi;zi are given implicitly by thedifferential equations of motion (31) and the initial corresponding conditions

yi0 : �i0(ti0;yi0; �pi) = 0; zi0 : _�i0(ti0;yi0;zi0; �pi) = 0: (50)

For each performance evaluation, an initial value problem has to be solved numer-ically. Simultaneously the performance functions (49) can be computed where thesecond term evaluates the dynamic behavior within the time interval consideredand the first term accounts for cases where special values for the final state yi1;zi1

or a minimal time ti1 must be achieved. The final time ti1 may be fixed or givenimplicitly by the final state condition Hi1(ti1;yi1;zi1; �pi) = 0.

The gradient for this type of criterion function can be computed most reliableand efficient using the adjoint variable approach. This approach results in a setof additional differential equations closely related to the linearized equations ofmotion. The finite differences approach which is usually used in a context of com-plicated relations �

i(�pi) has shown to be rather inefficient, inexact and unreliable,

see Bestle [19].In vehicle dynamics important, but contradicting criteria are riding comfort

and riding safety. In principle, one complex three-dimensional model including alleffects would be sufficient for investigating the problem. However, experience hasshown that three-dimensional models with detailed description of the suspensionsystems will require too much computational time for being included into an iter-ative, interactive design process. Therefore, the models used have to be simplifiedto provide just the interesting effects. For example, the spatial vehicle model withsimplified suspension systems in Figure 9 is sufficient to yield information oncomfort while driving over a rough road surface.

For achieving feasible EP-optimal design points a multicriteria optimizationstrategy has to be applied. Results for three different strategies will be shown inthe following.

As a first strategy, the weighted objectives method is proposed. All criteria areconsidered as objective functions yielding f = , and only some bounds on thedesign variables result in inequality constraints. A second strategy uses weightingfactors chosen differently, i.e. w = [1; 1; 1; 1; 1; 1; 100; 100; 1; 10]T . Minimizingthe utility function

u(p) =10Xj=1

wjfj(p) (51)


zs

zz

��

�

��

��

��

��xxyy

Figure 9. Spatial vehicle model.

results in optimal criterion values. A third strategy which is closely related tothe kind of engineering thinking is goal programming. The method allows topredefine goals which should be achieved. This may be combined with the ideaof hierarchization. Here three levels of importance are assigned to the objectives.E.g., the group of most important objectives consists of criteria 2 and 5, the secondimportant group of criteria 1, 3, 7 and the least important criteria are 4, 6 and 10.Criteria 8 and 9 are chosen as constraints. Not all goals need to be defined at thevery beginning of the optimization but can be successively introduced on the basisof the optimization results for the higher levels of importance.

Application of optimization methods to dynamic system design is somehow lag-ging behind the theoretical and algorithmic improvements in optimization theory.This is certainly due to the computational effort and the restriction to the classi-cal nonlinear programming problem. Recent advances in computer technology incombination with multicriteria and multimodel optimization ideas seem to open upnew ways for designing dynamic systems. The focus has to change from the solu-tion of the optimization problem to its flexible and convenient formulation. Then,the optimal solution is not a single design point any more, but a set of EP-optimalpoints from which the designer may chose according to his preferences. All aspectsof system requirements can be taken into account simultaneously. For more details,see Eberhard [35].

The optimal design of multibody systems may be considered as an efficient toolfor the synthesis of engineering systems. Hansen and Tortorelli [58] used such anapproach for the synthesis of planar mechanisms with up to 6 bodies and 15 designvariables.

172 W. SCHIEHLEN

3.4. DYNAMIC STRENGTH ANALYSIS FOR MULTIBODY SYSTEMS

By definition rigid multibody systems are qualified for the motion analysis butthey do not answer the question on the strength of the bodies subject to dynamicload. Therefore, the strength analysis requires the concept of flexible multibodysystems which is reviewed in detail by Shabana [172]. Then, the results obtainedin research on strength analysis of material bodies can be applied and combinedwith the multibody system approach, see Melzer [102].

A multibody system may consist of p rigid and ne elastic bodies subject to qholonomic constraints. Then, the system holds

f = 6p� q +neXi=1

nqi (52)

degrees of freedom. The system is uniquely described by 6p � q generalizedcoordinates describing the rigid body motion of all bodies and the sum of the nqigeneralized elastic coordinates describing the elastic deformations of thene flexiblebodies. The f � 1-vector of the generalized coordinates may be now defined as

yq(t) =

264y(t)qi(t)

...

375 ; i = 1(1)ne; (53)

where the 6p � q � 1-vector y(t) summarizes the rigid motion coordinates andthe nqi � 1-vectors qi(t) characterize the elastic coordinates of ne flexible bodies.Then, following Melzer [101] the equations of motion read in extension of (30) as

M(yq; t) �yq(t) + kc(yq; _yq; t) + ki(yq; _yq) = q(yq; _yq; t); (54)

where the symmetric inertia matrixM , the vector kc of the generalized gyroscopicand Coriolis force, the vector ki of the internal elastic forces and the vector q ofthe generalized applied forces are used. These equations of motion may also becomputed semi-symbolically by finite element preprocessing.

For a pure rigid body system, the vector of the internal elastic forces is complete-ly vanishing, ki � 0, and the well known Equations (30) of an ordinary multibodysystem are achieved. In the case of a vanishing rigid body motion, y � 0, from(54) follow the equations of motion well known in structural dynamics

ME�q(t) +DE _q(t) +KEq(t) = f(t); (55)

where the ne �nqi�1-vector q summarizes all the elastic coordinates andME;DE

andKE are the inertia, damping and stiffness matrix, respectively, of the structuralsystem.

The displacement field in a flexible body is given by elastic coordinates forsmall quantities as

u(c; t) = �(c)q(t); (56)


where c is the position vector of a material point in the reference position and�(c)is the space-dependent shape function. The 6 � 1-strain vector

� = [�11 �22 �33 2�12 2�23 2�31]T (57)

is obtained by partial differentiation of the displacement field resulting in

� = Lq + �L(q)q; (58)

where L and �L(q) are matrices depending linearly and quadraticly on the shapefunction �(c), respectively. Finally, the stress vector

� = [�11 �22 �33 �12 �23 �31]T (59)

is found by Hooke’s law represented by matrix H for linear elastic, isotropic andhomogeneous material as

� =H�+ �n; (60)

where �n represents the stress distribution in the reference position of the flexiblebody.

As a result from the dynamic analysis, the time histories�(c; t) at each materialpoint of a flexible body are available. As a consequence, for the strength analysisthe fatigue life prediction is most important. Even if reliable life predictions areonly possible by experiments, the computational life prediction is an emerging toolin engineering design.

According to Buxbaum [26] the amplitude, the frequency and the sequence ofthe loads are most important to the life of a part subject to vibratory fatigue. Thisinformation may be obtained by multibody system simulation.

Life predictions are mainly related to one dimensional loads. Only a few papersare devoted to more dimensional loads see [53, 64, 144, 184]. Therefore, a restric-tion to one dimensional loads is reasonable.

In the literature [26, 54], three concepts for the computational life predictionsare found:1. nominal stress concept,2. local concept,3. damage mechanics concept.

The damage mechanics concept is related to the growth of cracks in a partwhich is not acceptable in mechanical engineering. The local concept is veryexpensive due to an elastic-plastic approach. Therefore, the nominal stress conceptis recommended using the stresses in a part with smooth surface.

The Wohler-diagram represents the experimental results from fatigue experi-ments. The nominal stress amplitude is related to the cycle frequency which hasto be found by cycle counting methods like the rainflow counting, see Watson andDabell [192]. However, in practice the oscillations of the stresses do not featureconstant amplitude and constant mean. The cumulative damage in fatigue can be

174 W. SCHIEHLEN

estimated using the hypothesis of Palmgren [119] and Miner [103] which havebeen modified more recently by Zenner and Liu [197]. A first application of thedynamic stress analysis was published by Melzer [102] for a rotating beam and atwo-link robot. There is no doubt that much research work is needed in this fieldof multibody dynamics.

3.5. CONTACT AND IMPACT PROBLEMS

Rigid and/or flexible bodies moving in space are subject to collisions what mechan-ically means impact and contact. Therefore, the fundamental laws of the impactdue to Newton and Poisson may be applied in multibody dynamics, too. Contactproblems usually include friction phenomena which may be modeled by Coulom-b’s law. Then, the reaction forces exercise also an influence on the motion even ifthey do not contribute to the virtual work of the system.

Glocker and Pfeiffer [49] discussed the impact problem as unilateral contactproblem using a complementary approach, many details of which are publishedin a book by the same authors [126]. In another paper, Glocker and Pfeiffer [50]considered planar friction, too. In addition to analysis and simulation, Han andGilmore [57] added an experimental validation of the multibody impact motionwith friction. Canonical impulse-momentum equations are used by Lankarani andNikravesh [90]. Another approach applies finite element modeling of the contactconditions in multibody system dynamics, Amirouche et al. [6]. The impact analysisof an impulsive motion in nonholonomic deformable multibody systems is treatedby Shabana and Rismantab-Sany [169]. Various applications are also found ingears and transmission systems, Pennestri and Freudenstein [121]. It turns out thata number of results are available for impact problems in multibody dynamics butmore work is needed to understand the micromechanical phenomena influencingthe macromechanical multibody motion with contact.

3.6. INTERACTION WITH FLUIDS

The interaction of fluids with multibody systems has been discussed in satellitedynamics and vehicle dynamics. Beginning with the space age, Abramson [1]considered the dynamic behavior of liquids in moving containers. Further, thestabilization problem of spinning satellites filled with liquid was investigated byPfeiffer [124]. More recently, Chen and Pletcher [28] have studied numerically andexperimentally three-dimensional liquid sloshing flows. A finite element solutionfor the numerical simulation of rotating flows was presented by Codina and Soto[29]. The sloshing phenomena actuated by gravity with slow motion of a spacecraftwere modeled by Hung and Pan [68].

The results from satellite dynamics were also applied to road vehicles as shownby Bauer [14]. The stationary dynamics of an articulated tank vehicle was inves-tigated by Slibar and Troger [175]. Simulation models for such kind of vehicles


were presented by Schlieschke [155]. The roll motion of articulated vehicles wasstudied by Rakheja et al. [132]. The same authors [134] analyzed the steady turningstability of partially filled tank vehicles with arbitrary tank geometry. Further, in[135] the cited authors dealt with the dynamic response of articulated tank vehiclesdue to liquid load shift. Popov et al. [127, 128] presented the dynamic responsesof tank vehicles with rectangular and cylindrical containers. Simulations on thebasis of discrete models of tank vehicles were persecuted by Rauh and Rill [137].Field testing and validation of the directional dynamics model of a tank vehiclewere performed by Rakheja et al. [133]. More recently the strongly instationaryphenomena due to breaking of tank vehicles was considered by Ranganathan andYang [136].

All the models for sloshing fluids in rigid bodies like satellites and vehiclesare characterized by strong simplifications of the fluid motion. It is expected thatwith the growing power of computer hardware and computational fluid dynamicsmore realistic models of the sloshing fluid with a free surface may be found.Multibody system dynamics makes the problem even more complex due to thehighly nonlinear motion of the containers which may be still considered as rigid. Amajor field of application will be vehicle engineering and tank trucks in particular.

Another kind of interaction between fluids and multibody systems is foundin aerodynamics which may be used to investigate phenomena of aeroelasticity.O’Heron et al. [114] presented a detailed study of the aerodynamics of a tilt wingplane.

3.7. EXTENSION TO CONTROL AND MECHATRONICS

The applied forces and torques acting on multibody systems may be subject tocontrol. Then, the multibody system is considered as the plant for which a con-troller has to be designed. This point of view is found in early space application,e.g., the attitude control of satellites [10, 147]. Today, mechatronics is understoodas an interdisciplinary approach to controlled mechanical systems usally modeledas multibody systems. In particular robot dynamics and vehicle dynamics are sub-ject to control, e.g., [130, 167, 177, 190]. A more general view is presented bySchweitzer and Mansour [161]. Special emphasis is taken to the control of flexiblemultibody systems by Pfeiffer and Gebler [125] with respect to robots, while Modiand Suleman [105] present the application of control to orbiting flexible struc-tures. Some problems connected with the design of controllers using a multimodelapproach are shown by Haug [62]. It is useful to distinguish between a simple mod-el for the control design and a complex model for the validation of the control law.The mechatronic aspects of multibody dynamics were pointed out by Hiller [63]:modeling, control design and simulation have to be performed simultaneously. Itis expected that mechatronics, multibody dynamics and control engineering willfertilize each other in the years to come.

176 W. SCHIEHLEN

3.8. NONHOLONOMIC SYSTEMS

Nonholonomic systems are known for a long time in mechanics. Most of the text-books deal with nonholonomic systems, e.g. Hamel [56]. The classical examplesare the ice-skate and an axle with two rigid wheels subject to lateral friction. There-fore, nonholonomic systems are of engineering interest in vehicle dynamics andfor mobile robots. More recently, nonholonomic systems attracted mathematicians,too, looking for analytical solutions in a reduced minimal state space.

Bates and Sniatycki [13] reported on the nonholonomic reduction from a mathe-matical point of view. Essen [42] discussed nonholonomic dynamics as a geometricproblem. Kalaba and Udwadia [78] found the equations of nonholonomic dynam-ical systems using Gauss’s principle. Li-Fu Liang [93] considered the dynamicsof nonholonomic systems as a variational problem. Noether’s theorem was citedin papers by Luo [97] and Hui-dan Yu et al. [196] showing the strong intereston theoretical approaches. Yao-huang Luo and Yong-da Zhao [98] used Routh’sequations for nonholonomic systems with variable mass. Mladenova [104] con-sidered nonholonomic coordinates in rigid body dynamics. Rismantab-Sany andShabana [138, 139] dealt with nonholonomic deformable systems and their numeri-cal solution by differential-algebraical equations. Control aspects of nonholonomicsystems were treated in [154]. A number of control problems including the fallingcat is presented in the proceedings volume edited by Enos [41].

3.9. DAE INTEGRATION CODES

Differential-algebraic equations (DAE) may appear during the modeling processof a multibody system as shown, e.g., by Equations (33) to (35). In particular, theimplicit formulation of the constraint equation (5) is more convenient for multibodysystems with closed kinematical loops. But actively controlled multibody systemswith mechatronical components may also require a representation by differential-algebraical equations.

Bae and Haug [11] considered closed-loop multibody systems with a recursiveformalism. A numerical solution of differential-algebraic equations of motion ofdeformable mechanical systems with nonholonomic constraints was presented byRismantab-Sany and Shabana [139]. Fuhrer and Schwertasek [46] reported onnew developments in the generation and solution of multibody system equationsfeaturing the DAE approach. Another contribution to the numerical simulation ofmechanical systems using methods for DAEs is due to Anantharaman and Hiller[7]. The index of differential-algebraic equations of constrained multibody systemswas outlined by Blajer [23].

The problem of the numerical integration of differential-algebraical equationsis the inherent instability due to the index 3 found for mechanical systems. An earlyapproach to overcome the instability is the Baumgarte stabilization [15]. The ideabehind is an artificial feedback in the constraint equations (5) as follows:


��

��

��

��

��

��

��

��

��

Figure 10. Stabilization and projection approach.

��+ 2� _�+ �2� = 0; (61)

where �; � have to be chosen properly. The Baumgarte stabilization does notsolve the original problem since Equation (61) changes its dynamics. However,in many practical applications the stabilization approach works well. Fundamentalcontributions to the development of methods for the numerical integration of DAEsare due to Petzold [123], Gear [48] and Ascher and Petzold [9].

The early integration codes for differential-algebraical equations like DASSLdo not consider the special properties of mechanical systems, and they provedto be very inefficient, see Leister [91]. Then, the mathematicians got interestedin the numerics of DAE closely related to stiff ordinary differential equations(ODE), Hairer and Wanner [55]. The principal idea was to replace the stabilizationcondition (61) by a projection on the manifold of the constraints for each step ofthe integration. This means that the dynamics of the mechanical problem remainsunchanged. A visualization of both approaches is presented in Figure 10 designedby Schirle [156]. Projecting methods have been analyzed and implemented byFuhrer [45]. One of the first projecting multistep methods was published by Eich[39] which proved to be very successfull in multibody dynamics.

An extrapolation integrator for constrained multibody systems was developedby Lubich [94], the code is called MEXX, see Lubich et al. [95]. On the basis ofRunge–Kutta-methods, Simeon [174] developed the multibody simulation package

178 W. SCHIEHLEN

MBSPACK while von Schwerin and Winckler [166] used multistep methods in thesimulation code MBSSIM.

The application of differential-algebraical equations of mechanical systems incontrol engineering is also addressed as control analysis and synthesis of linearmechanical descriptor systems, see Schupphaus and Muller [159]. It turned out thatthe state space representation of control systems by ordinary differential equationscan be extended to the descriptor systems representing differential-algebraicalequations. A complete theory for linear mechanical systems is today availablefrom Muller et al. [108]. A survey of differential-algebraical equations in vehiclesystem dynamics has been published by Simeon et al. [173]. The available DAEintegration codes have recently been tested and compared by Schirle [156] for vehi-cle dynamics applications. Considerable progress on the efficiency of DAE codescould be reported. A modular modeling of the lateral dynamics of an autonomouslycontrolled vehicle including hydraulic power steering is shown by Rukgauer andSchiehlen [142]. Modeling and simulation of mechatronic systems is supported bythe program package NEWMOS recently published by Rukgauer [143].

The modeling process for mechatronic systems and mechanical systems withclosed kinematical loops is more efficient on the basis of implicit constraint equa-tions for coupling components and closing the loop. The great progress achievedwith DAE integration codes offers an opportunity to improve the simulation tools.More research in multibody dynamics is needed to evaluate joint ODE/DAE sim-ulation environments.

3.10. REAL-TIME SIMULATION AND ANIMATION

Efficient and fast simulation is always desirable in computational dynamics but itis really necessary for hardware-in-the-loop and operator-in-the-loop applications.There are two approaches to achieve real-time simulation: high-speed hardwareand efficient software. Multibody system dynamics is called to contribute to theefficiency of the software by recursive and/or symbolic formalisms and fast integra-tion codes. Tsai and Haug [185, 186] considered recursive formalisms, topologicalrequirements, parallel algorithms and numerical results for real-time multibodysystem simulation. Eichberger [40] showed the benefits of parallel multibody sim-ulation. Schaller et al. [146] presented a parallel extrapolation method for multibodysystem simulation. Hardware-in-the loop simulations for vehicle system dynamicswere presented by Schiehlen and Schafer [151] and Schafer [145]. Operator-in-the-loop applications are found in driving simulators as summarized by Haug [61].

The animation of motion is considered as a typical postprocessing procedureof multibody dynamical simualtions. The principles used are known from com-puter graphics, e.g. Watt [193]. Another approach is the application of geometricinformation from CAD modeling as it was pointed out by Daberkow [30]. Asan example, the animation of a spatial closed-loop torus mechanism is shown inFigure 11, see also Schirm [157].


Figure 11. Animation of torus motion.

A general problem of real-time simulation is the increasing complexity ofthe models under consideration. The efficiency of computer hardware and thecomplexity of the models required are growing simultaneously with the result thatthe efficiency of the multibody systems methods has to be improved continuously,too. Further requirements are due to the realism of the graphics used in the display.

3.11. CHALLENGING APPLICATIONS

Multibody system dynamics has a broad variety of applications, some of whichwill be mentioned here.

In biomechanics the walking motion is an important topic for some time. Acombination of motion and force control allows walking without impact as Blajerand Schiehlen [24] have shown. However, the models of the planar motion haveto be extended to spatial motion to include the small rotations with respect to thevertical axis, too. Another challenging problem is the dynamics of the middle ear[36, 37]. The mechanism of the middle ear has been considered as an acousticalproblem by Onchi [115] and from a medical point of view, e.g., by Huttenbrink[72]. The multibody modeling is related to the parameters to be identified. There,the laser vibrometry is used to compare measurements and computer simulations[38, 178]. In addition to the passive dynamics of the middle ear piezoactuators maybe analyzed as well, see, e.g., Kim and Jones [83].

However, there are much more problems in biomechanics which can be modeledand solved by multibody dynamics. The applications are ranging from vehicleoccupants to sport sciences, see, e.g., Morecki [106] or Nigg and Herzog [112].

180 W. SCHIEHLEN

Multibody dynamics is also a solid basis for nonlinear dynamics. The inher-ent nonlinearity and the possible small number of degrees of freedom allow theapplication of numerical methods of nonlinear dynamics to multibody systems, see[88, 168, 183]. In particular, impact and friction induced vibrations show chaoticbehavior as reported by Popp and Stelter [129], and Ostreich et al. [116]. The noisegeneration in railway wheels due to rail-wheel contact forces can be also consid-ered as a highly nonlinear phenomenon [158]. More recent development includealso the control of chaos, see Ott et al. [117].

The control aspects in multibody dynamics are getting more and more important.At last three IUTAM symposia were devoted to the control of mechanical systems,i.e. [12, 161, 188]. The problems of stability, controllability and observability wereconsidered in detail by Muller [107]. Closely related are descriptor systems whichmay be a new way to model mechatronic systems [109]. Using observers descriptorsystems allow a fault diagnosis [66].

Robotics and mechatronics are closely related to each other. The basics, objec-tives and examples of mechatronics have been shown by Schweitzer [163]. How-ever, mechatronics may also be used for the design of human oriented machinesas outlined by Schweitzer [164]. The interaction of robotics and society is still achallenging topic for interdisciplinary research with some relation to multibodydynamics. Another very successful application with great potential for industry aremagnetic bearings, Schweitzer et al. [162].

Vehicle dynamics is also subject to more control. An excellent example is theresearch project “Integration of distributed systems of mechatronics with specialemphasis to real time simulations”. This project was devoted to automated wreckingof an automobile to show the interdisciplinary integration from multibody dynamicsand control engineering to information processing. A detailed report is due toLuckel [96]. The fundamentals of vehicle dynamics and control are summarized byKortum and Lugner [86]. Recent applications have been presented at the AdvancedVehicle Control conference the proceedings of which are published by Wallentowitz[191]. There is no doubt that vehicle control problems fit perfectly to multibodysystem dynamics.

Another area of new and challenging applications of multibody systems isthe structural and occupant crashworthiness. There, the nonlinear structural issues,vehicle modeling and occupant modeling are combined in a unique manner. Recentadvances in the area are well described by Ambrosio et al. [2, 3], Jager [75] andPrasad and Chou [131].

4. Conclusions

The state-of-the-art of multibody system dynamics is presented based upon somehistorical remarks. The modeling procedure and the simulation tools are reviewed.The textbooks and conference proceedings are mentioned. The perspectives aremainly devoted to data models, parameter identification and optimal design. The


topics of real-time animation, contact and impact problems, mechatronics andcontrol, strength analysis, interaction with fluids, nonholonomic systems and DAEintegration codes are outlined. Challenging applications include biomechanics,chaos and nonlinearity, robotics and society and vehicle control. An extensive listof references is presented. As a matter of fact, multibody system dynamics turn outto be a very lively and promissing research subject.

References

1. Abramson, H.N., ‘The dynamic behaviour of liquids in moving containers’, NASA-SP-106,Washington, 1966.

2. Ambrosio, J., Pereira, M. and Dias, J., ‘Distributed and discrete nonlinear deformations onmultibody dynamics’, Nonlinear Dynamics 10, 1996, 359–379.

3. Ambrosio, J., Pereira, M. and Silva, F., Crashworthiness of Transportation Systems: StructuralImpact and Occupant Protection, NATO-ASI Series E, Vol. 332, Kluwer Academic Publishers,Dordrecht, 1997.

4. Amirouche, F.M.L., Computational Methods for Multibody Dynamics, Prentice-Hall, Engle-wood Cliffs, NJ, 1992.

5. Amirouche, F.M.L. and Xie, M., ‘Explicit matrix formulation of the dynamical equations forflexible multibody systems: Recursive approach’, Computers & Structures 46, 1993, 311–321.

6. Amirouche, F.M.L., Xie, M. and Shareef, N.H., ‘FE modeling of contact conditions in multibodysystem dynamics’, Nonlinear Dynamics 4, 1993, 83–102.

7. Anantharaman, M. and Hiller, M., ‘Numerical simulation of mechanical systems using methodsfor differential-algebraic equations’, International Journal for Numerical Methods in Engineer-ing 32, 1991, 1531–1542.

8. Angeles, J. and Kecskemethy, A., Kinematics and Dynamics of Multibody Systems, CISMCourses and Lectures, Vol. 360, Springer-Verlag, Wien, 1995.

9. Ascher, U. and Petzold, L., ‘Projected implicit Runge–Kutta methods for differential-algebraicequations’, SIAM Journal on Numerical Analysis 28, 1991, 1097–1120.

10. Aseltine, J.A. (ed.), Peaceful Uses of Automation in Outer Space, Plenum Press, New York,1966.

11. Bae, Dae-Sung and Haug, E.J., ‘Recursive formulation for constraint mechanical system dynam-ics. Part II: Closed loop systems,’ Mechanics of Structures and Machines 15, 1988, 481–506.

12. Banichuk, N.V., Klimov, D.M. and Schiehlen, W. (eds), Dynamical Problems of Rigid-ElasticSystems and Structures, Springer-Verlag, Berlin, 1991.

13. Bates, L. and Sniatycki, J., ‘Nonholonomic reduction’, Reports on Mathematical Physics 32,1993, 99–115.

14. Bauer, H.F., ‘Dynamic behaviour of an elastic separating wall in vehicle containers. Part I’,International Journal of Vehicle Design 2, 1975, 44–77.

15. Baumgarte, J., ‘Stabilization of constraints and integrals of motion in dynamical systems’,Computer Methods in Applied Mechanics and Engineering 1, 1972, 1–16.

16. Bestle, D. and Krause, R., ‘Sensitivity of parameter identification on measurement noise’,Problems in Engineering and Mechanics 7, 1992, 37–42.

17. Bestle, D. and Eberhard, P., ‘Automated approach for optimizing dynamic systems’, in Compu-tational Optimal Control, R. Bulirsch and D. Kraft (eds), Birkhauser, Basel, 1994, 225–235.

18. Bestle, D. and Eberhard, P., NEWOPT/AIMS2.2 – Ein Programmpaket zur Analyse und Opti-mierung von mechanischen Systemen, User’s Manual, Institute B of Mechanics, Stuttgart, 1994.

19. Bestle, D., Analyse und Optimierung von Mehrkorpersystemen, Springer-Verlag, Berlin, 1994.20. Bestle, D. and Eberhard, P., ‘Multicriteria multimodel design optimization’, in IUTAM Sym-

posium on Optimization of Mechanical Systems, D. Bestle and W. Schiehlen (eds), KluwerAcademic Publishers, Dordrecht, 1996, 33–40.

21. Bestle, D. and Schiehlen, W. (eds), IUTAM Symposium on Optimisation of Mechanical Systems,Kluwer Academic Publishers, Dordrecht, 1996.

182 W. SCHIEHLEN

22. Bianchi, G. and Schiehlen, W. (eds), Dynamics of Multibody Systems, Springer-Verlag, Berlin,1986.

23. Blajer, W., ‘Index of differential-algebrahic equations governing the dynamics of constrainedmechanical systems’, Applied Mathematical Modelling 16, 1992, 70–77.

24. Blajer, W. and Schiehlen, W., ‘Walking without impacts as a motion/force control problem’,Journal of Dynamic Systems, Measurement, and Control. Trans. ASME 114, 1992, 660–665.

25. Bremer, H. and Pfeiffer, F., Elastische Mehrkorpersysteme, Teubner, Stuttgart, 1992.26. Buxbaum, O., Betriebsfestigkeit. Sichere und wirtschaftliche Bemessung schwing-

bruchgefahrdeter Bauteile, Verlag Stahleisen, Dusseldorf, 1986.27. Chaffin, D.B., ‘A computerized biomechanical model-development for the use in studying gross

body actions’, Journal of Biomechanics 2, 1969, 429–441.28. Chen, K.-H. and Pletcher, R.H., ‘Primitive variable, strongly implicit calculation procedure for

viscous flows at all speeds’, AIAA Journal 29, 1993, 1241–1249.29. Codina, R. and Soto, O., ‘Finite element solution of the Stokes problem with Coriolis force’,

in Computational Fluid Dynamics ’94. Proceedings of the Second European Fluid DynamicsConference, S. Wagner, E.H. Hirschel, J. Periaux and R. Piva (eds), Wiley, New York, 1994,113–120.

30. Daberkow, A., Zur CAD-gestutzten Modellierung von Mehrkorpersystemen, Fortschritt-Berichte, Reihe 20, Nr. 80, VDI-Verlag, Dusseldorf, 1993.

31. Daberkow, A. and Schiehlen, W., ‘Concept, development and implementation of DAMOS-C: The object oriented approach to multibody systems’, in Computers in Engineering 1991,Proceedings ASME International Computers in Engineering Conference, K. Ishii (ed.), ASME,New York, 1994, 937–951.

32. D’ Alembert, J., Traite de Dynamique, Paris, 1743.33. Denavit, J. and Hartenberg, R.S., ‘A kinematic motion for lower pair mechanisms based on

matrices’, Journal of Applied Mechanics 22, 1955, 215–221.34. Durr, R., Neerpasch, U., Schiehlen, W. and Witte, L., ‘Mechatronik und STEP, Standardisierung

eines neutralen Datenformats in STEP fur die Simulation mechatronischer Systeme’, Produkt-daten Journal 2, 1995, 12–19.

35. Eberhard, P., Zur Mehrkriterienoptimierung von Mehrkorpersystemen, Fortschritt-Berichte,Reihe 11, Nr. 227, VDI-Verlag, Dusseldorf, 1996.

36. Eiber, A. and Kauf, A., ‘Berechnete Verschiebungen der Mittelohrknochen unter statischerBelastung’, HNO 42, 1994, 754–759.

37. Eiber, A., Kauf, A. and Schiehlen, W., ‘Biomechanics of the Middle Ear’, in Proceedings of the9th World Congress on TMM, A. Rovetta (ed.), Edizioni Unicopli, Milano, 1995, 2415–2419.

38. Eiber, A., Kauf, A., Maassen, M.M., Burkhardt, C., Rodriguez, J. and Zenner,H.P., ‘Vergleich von Laservibrometriemessungen und Computersimulationen der Gehor-knochelchenbewegungen’, HNO 45, 1997, to appear.

39. Eich, E., Projizierende Mehrschrittverfahren zur numerischen Losung von Bewegungsgleichun-gen technischer Systeme mit Zwangsbedingungen und Unstetigkeiten, Fortschritt-Berichte, Rei-he 18, Nr. 109, VDI-Verlag, Dusseldorf, 1992.

40. Eichberger, A., ‘Benefits of parallel multibody simulation’, International Journal Methods Eng.37, 1994, 1557–1572.

41. Enos, M.J. (ed.), Dynamics and Control of Mechanical Systems, American Mathematical Soci-ety, Providence, 1993.

42. Essen, H., ‘Geometry of nonholonomic dynamics’, Journal of Applied Mechanics 61, 1994,689–694.

43. Euler, L., ‘Nova methods motum corporum rigidarum determinandi’, Novi Commentarii Acad-emiae Scientiarum Petropolitanae 20, 1776, 208–238. See also Euler, L., Opera Omnia, SeriesII, Vol. 9, 99–125.

44. Fischer, O., Einfuhrung in die Mechanik lebender Mechanismen, Leipzig, 1906.45. Fuhrer, C., Differential-algebraische Gleichungssysteme in mechanischen Mehrkorper-

systemen, Dissertation, Technische Universitat Munchen, Munchen, 1988.46. Fuhrer, C. and Schwertassek, R., ‘Generation and solution of multibody system equations’,

International Journal of Non-Linear Mechanics 25, 1990, 127–141.


47. Garcia de Jalon, J. and Bayo, E., Kinematic and Dynamic Simulation of Multibody Systems,Springer-Verlag, New York, 1994.

48. Gear, C.W., ‘Differential-algebraic equation index transformations’, SIAM Journal of Scientificand Statistical Computing 9, 1988, 39–47.

49. Glocker, C. and Pfeiffer, F., ‘Dynamical systems with unilateral contacts’, Nonlinear Dynamics3, 1992, 245–259.

50. Glocker, C. and Pfeiffer, F., ‘Complementarity problems in multibody systems with planarfriction’, Archive of Applied Mechanics 63, 1993, 452–463.

51. Grammel, R., Der Kreisel – Seine Theorie und seine Anwendungen, First ed., Vieweg, Braun-schweig, 1920. Second ed., Springer-Verlag, Berlin, 1950.

52. Grubel, G., Finsterwalder, R., Gramlich, G., Joos, H.-D. and Lewald, A., ‘ANDECS-A compu-tation environment for control applications of optimization’, Computational Optimal Control,R. Bulirsch and D. Kraft (eds), Birkhauser, Basel, 1994, 237–254.

53. Hafele, P. and Dietmann, H., ‘Weiterentwicklung der Modifizierten Oktaederschubspannnungs-hypothese (MOSH)’, Konstruktion 46, 1994, 51–58.

54. Haibach, E., Betriebsfestigkeit. Verfahren und Daten zur Bauteilberechnung, VDI-Verlag,Dusseldorf, 1989.

55. Hairer, E. and Wanner, G., Solving Ordinary Differential Equations II. Stiff and Differential-Algebraic Problems, Springer-Verlag, Berlin, 1991.

56. Hamel, G., Theoretische Mechanik, Springer-Verlag, Berlin, 1949.57. Han, I. and Gilmore, B.J., ‘Multi-body impact motion with friction: Analysis, simulation and

experimental validation’, Journal of Mechanical Design 115, 1993, 412–422.58. Hansen, J.M. and Tortorelli, D.A., ‘An efficient method for synthesis of mechanisms using an

optimization method’, in Optimization of Mechanical System, D. Bestle and W. Schiehlen (eds),Kluwer Academic Publishers, Dordrecht, 1996, 129–138.

59. Haug, E.J. (ed.), Computer-Aided Analysis and Optimization of Mechnical System Dynamics,Springer-Verlag, Berlin, 1984.

60. Haug, E.J., Computer-Aided Kinematics and Dynamics of Mechanical Systems, Allyn andBacon, Boston, 1989.

61. Haug, E.J., ‘High speed multibody dynamic simulation and its impact on man-machine systems’,in Advanced Multibody System Dynamics, W. Schiehlen (ed.), Kluwer Academic Publishers,Dordrecht, 1993, 1–18.

62. Haug, J., Zur Modellierung aktiv geregelter elastischer Mehrkorpersysteme, Fortschritt-Berichte, Reihe 11, Nr. 230, VDI-Verlag, Dusseldorf, 1996.

63. Hiller, M., ‘Modeling and simualtion of complex multibody systems in mechatronic systems’,Eurotech Direct ’91 – Machine Systems, Proceedings Eur. Eng. Res. and Tech. TransferCongress, Birmingham, 1991, 159–163.

64. Hoffmann, M. and Seeger, T., ‘Mehrachsige Kerbbeanspruchung bei proportionaler Belastung’,Konstruktion 38, 1986, 63–70.

65. Hooker, W.W. and Margulies, G., ‘The dynamical attitude equations for n-body satellite’,Journal on Astronomical Science 12, 1965, 123–128.

66. Hou, M., Descriptor Systems: Observers and Fault Diagnosis, Fortschritt-Berichte, Reihe 8,Nr. 482, VDI-Verlag, Dusseldorf, 1995.

67. Hsia, T.C., System Identification. Least Squares Methods, Lexingon Books, Lexington, 1977.68. Hung, R. J. and Pan, H. L., ‘Modelling of sloshing modulated angular momentum fluctuations

actuated by gravity gradient associated with spacecraft slew motion’, Applied MathematicalModelling 20, 1996, 399–409.

69. Huston, R.L. and Passerello, C.E., ‘On the dynamics of a human body model’, Journal onBiomechanics 4, 1971, 369–378.

70. Huston, R.L., Multibody Dynamics, Butterworth-Heinemann, Boston, 1990.71. Huston, R.L., ‘Multibody dynamics – modeling and analysis methods’, Applied Mechanics

Review 44, 1991, 109–117 and Applied Mechanics Review 49, 1996, 35–40.72. Huttenbrink, K.B., ‘Die funktionelle Bedeutung der Aufhangebander der Gehor-

knochelchenkette’, Laryngorhinotologie 68, 1989, 146–151.

184 W. SCHIEHLEN

73. Ider, S.K., ‘FE based recursive formulation for real time dynamic simulation of flexible multi-body systems’, Computers & Structures 40, 1991, 939–945.

74. Isermann, R., Prozeßidentifikation, Springer-Verlag, Berlin, 1992.75. Jager, M. de, ‘Mathematical head-neck models for acceleration impacts’, Ph.D. Thesis, Univer-

sity of Technology, Eindhoven, 1996.76. Jourdain, P.E.B., ‘Note on an analogue at Gauss’ principle of least constraints’, Quarterly

Journal on Pure Applied Mathematics 40, 1909, 153–197.77. Junker, F., ‘Modular-hierarchisch strukturierte Modellbildung mechanischer Systeme’, Archive

of Applied Mechanics 65, 1995, 227–245.78. Kalaba, R.E. and Udwadia, F.E., ‘Equations of motion for nonholonomic, constrained dynamical

systems via Gauss’s principle’, Journal of Applied Mechanics 60, 1993, 662–668.79. Kallenbach, R., Kovarianzmethoden zur Parameteridentifikation zeitkontinuierlicher Systeme,

Fortschritt-Berichte, Reihe 11, Nr. 92, VDI-Verlag, Dusseldorf, 1987.80. Kane, T.R. and Scher, M.P., ‘A dynamical explanation of the falling cat phenomena’, Interna-

tional Journal on Solids Structures 5, 1969, 663–670.81. Kane, T.R. and Levinson, D.A., Dynamics: Theory and Applications, McGraw-Hill, New York,

1985.82. Kim, S.-S. and Haug, E.J., ‘Recursive formulation for flexible multibody dynamics. Part II.

Closed loop systems’, Computer Methods in Applied Mechanics and Engineering 74, 1989,251–269.

83. Kim, S.J. and Jones, J.D., ‘Optimal design of piezoactuators for active noise and vibrationcontrol’, AIAA Journal 29, 1991, 2047–2053.

84. Kortum, W. and Schiehlen, W., ‘General purpose vehicle system dynamics software based onmultibody formalisms’, Vehicle System Dynamics 14, 1985, 229–263.

85. Kortum, W. and Sharp, R.S., Multibody Computer Codes in Vehicle System Dynamics, Swets& Zeitlinger, Amsterdam, 1993.

86. Kortum, W. and Lugner, P., Systemdynamik und Regelung von Fahrzeugen, Springer-Verlag,Berlin, 1996.

87. Krause, R. and Schiehlen, W., ‘Parametric modeling of nonlinearities by covariance analy-sis’, in EUROMECH 280, Proceedings International Symposium on Identification of NonlinearMechanical Systems by Dynamic Tests, L. Jezequel and C.-H. Lamarque (eds), Balkema, Rot-terdam, 1992, 157–158.

88. Kreuzer, E., Numerische Untersuchung nichtlinearer dynamischer Systeme, Springer-Verlag,Berlin, 1987.

89. Lagrange, J.-L., Mecanique Analytique, L’Academie Royal des Sciences, Paris, 1788.90. Lankarani, H.M. and Nikravesh, P.E., ‘Canonical impulse-momentum equations for impact

analysis of multibody systems’, Journal of Mechanical Design 114, 1992, 180–186.91. Leister, G., Beschreibung und Simulation von Mehrkorpersystemen mit geschlossenen kinema-

tischen Schleifen, Fortschritt-Berichte, Reihe 11, Nr. 167, VDI-Verlag, Dusseldorf, 1992.92. Levinson, D.A., ‘Equations of motion for multi-rigid-body systems via symbolic manipulation’,

Journal of Spacecraft and Rockets 14, 1977, 479–487.93. Liang, Li-Fu, ‘Variational problem of general displacement in nonholonomic systems’, Acta

Mechanica Solida Sinica 6, 1993, 357–364.94. Lubich, Ch., ‘Extrapolation integrators for constrained multibody systems’, IMPACT Comp.

Sci. Eng. 3, 1991, 213–234.95. Lubich, Ch., Nowak, U., Pohle, U. and Engstler, Ch., MEXX – Numerical Software for the

Integration of Constrained Mechanical Multibody Systems, Konrad-Zuse-Zentrum fur Informa-tionstechnik, Berlin, 1992, Preprint SC 92-12.

96. Luckel, J., Integration verteilter Systeme der Mechatronik mit besonderer Berucksichtigungdes Echtzeitverhaltens, Arbeitsberichte 1993/94 und 1994/95, Unversitat GesamthochschulePaderborn, Paderborn, 1994 and 1995.

97. Luo, S., ‘Generalized Noether’s theorem of nonholonomic nonpotential system in noninertialreference frames’, Applied Mathematics and Mechanics 12, 1991, 927–934.

98. Luo, Yao-huang and Zhao, Yong-da, ‘Routh’s equations for general nonholonomic mechanicalsystems of variable mass’, Applied Mathematics and Mechanics 14, 1993, 285–298.


99. Magnus, K., Kreisel-Theorie und Anwendungen, Springer-Verlag, Berlin, 1971.100. Magnus, K. (ed.), Dynamics of Multibody Systems, Springer-Verlag, Berlin, 1978.101. Melzer, F., Symbolisch-numerische Modellierung elastischer Mehrkorpersysteme mit Anwen-

dung auf rechnerische Lebensdauervorhersage, Fortschritt-Berichte, Reihe 20, Nr. 139, VDI-Verlag, Dusseldorf, 1994.

102. Melzer, F., ‘Symbolic computations in flexible multibody system’, Nonlinear Dynamics 9, 1996,147–163.

103. Miner, M.A., ‘Cumulative damage in fatigue’, Journal of Applied Mechanics 12, 1945, 159–169.104. Mladenova, C.D., ‘Kinematics and dynamics of a rigid body in nonholonomic coordinates’,

Systems & Control Letters 22, 1994, 257–265.105. Modi, V.J. and Suleman, A.C., ‘Approach to dynamics and control of orbiting flexible structures’,

International Journal for Numerical Methods in Engineering 32, 1991, 1727–1748.106. Morecki, A., Biomechanics of Engineering. Modelling, Simulation, Control, Springer-Verlag,

Wien, 1987.107. Muller, P.C., Stabilitat und Matrizen, Springer-Verlag, Berlin, 1977.108. Muller, P.C., Rentrop, P., Kortum, W. and Fuhrer, C., ‘Constrained mechanical systems in

descriptor form: Identification, simulation and control’, Advanced Multibody System Dynamics,W. Schiehlen (ed.), Kluwer Academic Publishers, Dordrecht, 1993, 451–456.

109. Muller, P.C., ‘Descriptor Systems: A New Way to Model Mechatronic Systems’, in Proceedings3rd European Control Conference, Vol. 3, Part 2, 1995, 2725–2729.

110. Natke, H.G., Einfuhrung in Theorie und Praxis der Zeitreihen- und Modalanalyse, Vieweg,Braunschweig, 1983.

111. Newton, I., Philosophiae Naturalis Principia Mathematica, Royal Society, London, 1687.112. Nigg, B.M. and Herzog, W., Biomechanics of the Musculo-Skeletal System, John Wiley & Sons,

Toronto, 1994.113. Nikravesh, P.E., Computer-Aided Analysis of Mechanical Systems, Prentice-Hall, Engelwood

Cliffs, NJ, 1988.114. O’Heron, P., Nikravesh, P. and Arabyan, A., ‘A multibody model simulating tilt wing conver-

sion’, Technical Report CAEL-91-5, Computer Aided Engineering Laboratory, University ofArizona, 1991.

115. Onchi, Y., ‘Mechanism of the middle ear’, Journal of the Acoustical Society of America 33,1961, 794–805.

116. Ostreich, M., Hinrichs, N. and Popp, K., ‘Bifurcation and stability analysis of a non-smoothfriction oscillator’, Archive of Applied Mechanics 66, 1996, 301–314.

117. Ott, E., Grebogi, C. and Yorke, J.A., ‘Controlling chaos’, Physical Review Letters, 1990, 1196–1199.

118. Otter, M., Hocke, M., Daberkow, A. and Leister,G., ‘An object oriented data model for multibodysystems’, in Advanced Multibody System Dynamics, W. Schiehlen (ed.), Kluwer AcademicPublishers, Dordrecht, 1993, 19–48.

119. Palmgren, A., ‘Die Lebensdauer von Kugellagern’, VDI-Zeitschrift 69, 1924, 339–341.120. Pasler, M., Prinzipe der Mechanik, De Gruyter, Berlin, 1968.121. Pennestri, E. and Freudenstein, F., ‘Systematic approach to powerflow and static-force analysis

in epicyclic spur-gear trains’, Journal of Mechanical Design 115, 1993, 639–644.122. Pereira, M.F.O.S. and Ambrosio, J.A.C. (eds), Computer-Aided Analysis of Rigid and Flexible

Mechanical Systems, Kluwer Academic Publishers, Dordrecht, 1994.123. Petzold, L.R., ‘An efficient numerical method for solving stiff and nonstiff systems of ordinary

differential equations’, SIAM Journal on Numerical Analysis 18, 1981, 455–479.124. Pfeiffer, F., ‘Linearisierte Eigenschwingungen von Treibstoff in beliebigen Behaltern’,

Zeitschrift fur Flugwissenschaft 16, 1968, 188–194.125. Pfeiffer, F. and Gebler, B., ‘A multistage-approach to the dynamics and control of elastic robots’,

in 1988 IEEE International Conference on Robotics and Automation, Vol. 1, IEEE Comput.Soc. Press, Washington, 1988, 2–8.

126. Pfeiffer, F. and Glocker, C., Multibody Dynamics with Unilateral Contacts, Wiley, New York,1996.

186 W. SCHIEHLEN

127. Popov, G., Sankar, S., Sankar, T.S. and Vatistas, G.H., ‘Liquid sloshing in rectangular roadcontainers’, Computers & Fluids 21, 1992, 551–559.

128. Popov, G., Sankar, S., Sankar, T.S. and Vatistas, G.H., ‘Dynamics of liquid sloshing in horizontalcylindrical road containers’, Proceedings of the Institute of Mechanical Engineers Part C 207,1993, 399–406.

129. Popp, K. and Stelter, P., ‘Stick-slip vibrations and chaos’, Philosophical Transactions of theRoyal Society of London A336, 1990, 89–105.

130. Popp, K. and Schiehlen, W., Fahrzeugdynamik, Teubner, Stuttgart, 1993.131. Prasad, P. and Chou, C.C., ‘A review of mathematical occupant simulation models’, in Crash-

worthiness and Occupant Protection in Transportation Systems, ASME AMD Vol. 106/BEDVol. 13, T. Khalil and A. King (eds), 1989, 36–112.

132. Rakheja, S., Sankar, S. and Ranganathan, R., ‘Roll plane analysis of articulated tank vehiclesduring steady turning’, Vehicle System Dynamics 17, 1988, 81–104.

133. Rakheja, S., Ranganathan, R. and Sankar, S., ‘Field testing and validation of directional dynam-ics model of a tank truck’, International Journal of Vehicle Design 13, 1992, 251–275.

134. Ranganathan, R., Rakheja, S. and Sankar, S., ‘Steady turning stability of partially filled tankvehicles with arbitrary tank geometry’, Journal of Dynamic Systems, Measurement, and Control111, 1989, 481–489.

135. Ranganathan, R., Rakheja, S. and Sankar, S., ‘Influence of liquid load shift on the dynamicresponse of articulated tank vehicles’, Vehicle System Dynamics 19, 1990, 177–200.

136. Ranganathan, R. and Yang, Y.S., ‘Impact of liquid load shift on the braking characteristics ofpartially filled tank vehicles’, Vehicle System Dynamics 26, 1996, 223–240.

137. Rauh, J. and Rill, G., Simulation von Tankfahrzeugen, VDI-Berichte 1007, VDI-Verlag,Dusseldorf, 1992, 681–697.

138. Rismantab-Sany, J. and Shabana, A.A., ‘Impulsive motion of nonholonomic deformable multi-body systems. Part I, Kinematic and dynamic equations’, Journal of Sound and Vibration 127,1988, 193–204.

139. Rismantab-Sany, J. and Shabana A.A., ‘Numerical solution of differential-algebraic equationsof motion of deformable mechanical systems with nonholonomic constraints’, Computers &Structures 33, 1989, 1017–1029.

140. Roberson, R.E. and Wittenburg, J., ‘A dynamical formalism for an arbitrary number of inter-connected rigid bodies, with reference to the problem of satellite attitude control’, Proceedings3rd Congr. Int. Fed. Autom. Control, Butterworth, Vol. 1, Book 3, Paper 46 D, London, 1967.

141. Roberson, R.E. and Schwertassek, R., Dynamics of Multibody Systems, Springer-Verlag, Berlin,1988.

142. Rukgauer, A. and Schiehlen, W., ‘Efficient simulation of the behaviour of articulated vehicles,in Proceedings AVEC’96, H. Wallentowitz (ed.), Aachen University of Technology, Aachen,1996, 1057–1069.

143. Rukgauer, A., ‘Modulare Simulation mechatronischer Systeme mit Anwendung in der Fahrzeug-dynamik’, Dissertation, Universitat Stuttgart, Stuttgart, 1997.

144. Sanetra, C., ‘Untersuchung zum Festigkeitsverhalten bei mehrachsiger Randombeanspruchungunter Biegung und Torsion’, Dissertation, Technische Hochschule Clausthal, Clausthal, 1991.

145. Schafer, P., ‘Echtzeitsimulation aktiver Mehrkorpersysteme auf Transputernetzen’, Fortschritt-Berichte, Reihe 11, Nr. 202, VDI-Verlag, Dusseldorf, 1994.

146. Schaller, C., Fuhrer, C. and Simeon, B., ‘Simulation of multi-body systems by a parallelextrapolation method’, Mechanics of Structures and Machines 22, 1994, 473–486.

147. Schiehlen, W., Dynamics of Satellites, Springer-Verlag, Wien, 1970.148. Schiehlen, W. and Kreuzer, E., ‘Rechnergestutzes Aufstellen der Bewegungsgleichungen

gewohnlicher Mehrkorpersysteme’, Ingenieur-Archiv 46, 1977, 185–194.149. Schiehlen, W., Technische Dynamik, Teubner, Stuttgart, 1986.150. Schiehlen, W. and Kallenbach, R.G., ‘Identification of robot system parameters’, in Theory

of Robots. Proceedings IFAC/IFIP/IMACS Symposium, P. Kopacek, I. Troch, and K. Desoyer(eds), Pergamon Press, Oxford, 1988, 119–124.


151. Schiehlen, W. and Schafer, P., ‘Modeling of vehicles with controlled components’, in TheDynamics of Vehicles on Roads and on Tracks, Proceedings 11th IAVSD Symposium, R.J.Anderson (ed.), Swets & Zeitlinger, Amsterdam, 1989, 488–501.

152. Schiehlen, W. (ed.), Multibody Systems Handbook, Springer-Verlag, Berlin, 1990.153. Schiehlen, W., ‘Computational aspects in multibody system dynamics’, Computer Methods in

Applied Mechanics and Engineering 90, 1991, 569–582.154. Schiehlen, W., ‘Dynamics and control of nonholonomic mobile robot systems’, in Preprints

of Symposium on Robot Control 1994, Vol. 1, L. Sciavicco, C. Bonivento and F. Nicolo (eds),Napoli, 1994, 329–334.

155. Schieschke, R.-G., Konzeption und Realisierung komplexer fahrdynamischer Simulationsmod-elle am Beispiel eines Sattelzuges mit beweglicher Ladung, Fortschritt-Berichte, Reihe 12,Nr. 106, VDI-Verlag, Dusseldorf, 1988.

156. Schirle, Th., ‘Kopplung von Mehrkorpersystemen uber einen nichtminimalen Ansatz’, Studi-enarbeit STUD-134, Institut B fur Mechanik, Universitat Stuttgart, Stuttgart, 1996.

157. Schirm, W., Symbolisch-numerische Behandlung von kinematischen Schleifen inMehrkorpersystemen, Fortschritt-Berichte, Reihe 11, Nr. 198, VDI-Verlag, Dusseldorf, 1993.

158. Schneider, E., Popp, K. and Irretier, H., ‘Noise Generation in Railway Wheels due to Rail-WheelContact Forces’, J. Sound Vibr. 120, 1989, 227–244.

159. Schupphaus, R. and Muller, P.C., ‘Control analysis and synthesis of linear mechanical descrip-tor systems’, Advanced Multibody System Dynamics, W. Schiehlen (ed.), Kluwer AcademicPublishers, Dordrecht, 1993, 463–468.

160. Schwarz, R.G., Identifikation mechanischer Mehrkorpersysteme, Fortschritt-Berichte, Reihe 8,Nr. 30, VDI-Verlag, Dusseldorf, 1980.

161. Schweitzer, G. and Mansour, M. (eds), Dynamics of Controlled Mechanical Systems, Springer-Verlag, Berlin, 1989.

162. Schweitzer, G., Traxler, A. and Bleuler, H., Magnetlager: Grundlagen, Eigenschaften undAnwendungen beruhrungsfreier elektromagnetischer Lager, Springer-Verlag, Berlin, 1993.

163. Schweitzer, G, ‘Mechatronics – basics, objectives, examples’, Journal of Systems and ControlEngineering 210, Proceedings IMechE, 1996, 1–11.

164. Schweitzer, G., ‘Mechatronics for the design of human oriented machines’, IEEE/ASME Trans.on Mechatronics 2, 1997, to appear.

165. Schwertassek, R. and Roberson, R.E., ‘A perspective on computer-oriented multibody dynami-cal formalisms and their implementations’, in Dynamics of Multibody Systems, G. Bianchi andW. Schiehlen (eds), Springer-Verlag, Berlin, 1986, 263–273.

166. Schwerin, R.v. and Winckler, M., A Guide to the Integrator Library MBSSIM Version 1.00,Interdisziplinares Zentrum fur Wissenschaftliches Rechnen (IWR), Preprint 94–75, Heidelberg,1994.

167. Segel, L. (ed.), The Dynamics of Vehicles on Roads and Tracks, Swets & Zeitlinger, Lisse, 1996.168. Seydel, R., From Equilibrium to Chaos, Practical Bifurcation and Stability Analysis, Elsevier,

New York, 1988.169. Shabana, A.A. and Rismantab-Sany, J., ‘Impulsive motion of nonholonomic deformable multi-

body systems. Part II: Impact analysis’, Journal Sound Vib. 127, 1988, 205–219.170. Shabana, A.A., Dynamics of Multibody Systems, Wiley, New York, 1989.171. Shabana, A.A., Computational Dynamics, Wiley, New York, 1994.172. Shabana, A.A., ‘Flexible Multibody Dynamics: Review of Past and Recent Developments’,

Multibody System Dynamics 1, 1997, this issue.173. Simeon, B., Fuhrer, C. and Rentrop, P., ‘Differential-algebraic equations in vehicle system

dynamics’, Survey on Math. in Industry 1, 1991, 1–37.174. Simeon, B., Numerische Integration mechanischer Mehrkorpersysteme: Projizierende Deskrip-

torformen, Algorithmen und Rechenprogramme, Fortschritt-Berichte, Reihe 20, Nr. 130, VDI-Verlag, Dusseldorf, 1994.

175. Slibar, A. and Troger, H., ‘Das stationare Fahrverhalten des Tank-Sattelaufliegers’, in Proceed-ings IUTAM-Symposium, H.B. Pacejka (ed.), Swets & Zeitlinger, Amsterdam, 1976, 152–172.

176. Stadler, W. (ed.), Multicriteria Optimization in Engineering and in the Sciences, Plenum Press,New York, 1988.

188 W. SCHIEHLEN

177. Stadler, W., Analytical Robotics and Mechatronics, McGraw-Hill, London, 1995.178. Stasche, N., Foth, H.J. and Hormann, K., ‘Laser-Doppler-Vibrometrie des Trommelfells’, HNO

41, 1993, 1–6.179. Stejskal, V. and Valasek, M., Kinematics and Dynamimcs of Machinery, Marcel Dekker, New

York, 1996.180. Stelzle, W., Kecskemethy, A. and Hiller, M., ‘A comperative study of recursive methods’,

Archive of Applied Mechanics 66, 1995, 9–19.181. Szabo, I., Geschichte der mechanischen Prinzipien, Birkhauser, Basel, 1977.182. Thomson, B., Neerpasch, U. and Schiehlen, W., DAMOS-C ein neutrales Datenformat zur

standardisierten Anbindung von MKS an Simulationssysteme mit offener Systemarchitektur,VDI-Berichte 1153, VDI-Verlag, Dusseldorf, 1994, 183–205.

183. Troger, H. and Steindl, A., Nonlinear Stability and Bifurcation Theory, Springer-Verlag, Wien,1991.

184. Troost, A., Akin, O. and Klubberg, F., ‘Dauerfestigkeitsverhalten metallischer Werkstoffebei zweiachsiger Beanspruchung durch drei phasenverschoben schwingende Lastspannungen’,Konstruktion 39, 1987, 479–488.

185. Tsai, F.F. and Haug, E.J., ‘Real-time multibody system dynamic simulation. Part I: A parallelalgorithm and numerical results’, Mechanics of Structures and Machines 19, 1991, 99–127.

186. Tsai, F.F. and Haug, E.J., ‘Real-time multibody system dynamic simulation. Part II: A modifiedrecursive formulation and topological analysis’, Mechanics of Structures and Machines 19,1991, 129–162.

187. Uicker, J.J. Jr., ‘On the dynamic analysis of spatial linkages using 4 by 4 matrices’, Ph.D.Thesis, Northwestern University, Evanston, 1965.

188. Van Campen, D. (ed.), Interaction between Dynamics and Control in Advanced MechanicalSystems, Kluwer Academic Publishers, Dordrecht, 1997.

189. Vukobratovic, M., Frank, A.A. and Juricic, D., ‘On the stability of biped locomotion’, IEEETransactions on Biomedical Engineering BME-17, 1970, 25–36.

190. Vukobratovic, M. and Stokic, D., Control of Manipulation Robots, Springer-Verlag, Berlin,1982.

191. Wallentowitz, H. (ed.), AVEC’96, Proceedings International Symposium on Advanced VehicleControl, Aachen University of Technology, Aachen, 1996.

192. Watson, P. and Dabell, B.J., ‘Cycle counting and fatigue damage’, J. Soc. Env. Eng. 9, 1976,3–8.

193. Watt, A., Fundamentals of Three-Dimensional Computer Graphics, Addison-Wesley, Reading,MA, 1989.

194. Wittenbauer, F., Graphische Dynamik, Springer-Verlag, Berlin, 1923.195. Wittenburg, J., Dynamics of Systems of Rigid Bodies, Teubner, Stuttgart, 1977.196. Yu, Hui-dan, Zhang, Jie-fang and Xu, You-shen, ‘Noether’s theory for nonholonomic systems

relative to non-inertial reference frame’, Applied Mathematics and Mechanics 14, 1993, 527–536.

197. Zenner, H. and Liu, J., ‘Vorschlag zur Verbesserung der Lebensdauerabschatzung nach demNennspannungskonzept’, Konstruktion 44, 1992, 9–17.

Multibody System Dynamics: Roots and Perspectives · Multibody System Dynamics: Roots and...

Documents

Transcript of Multibody System Dynamics: Roots and Perspectives · Multibody System Dynamics: Roots and...