Dynamics of Multibody SystemsFlexible multibody dynamics typically deals with non-linear structures whose segments undergo large rigid body motion superimposed by ﬂexible deformations

Dynamics of Multibody Systems/modelling concepts and applications/

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Zdravko TerzeUniversity of Zagreb

F. Mech. Eng. Naval Arch.

Albrecht EiberUniversity of Stuttgart

Institute B of Mechanics

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Text Box

(INTERNAL VERSION)

2

Foreword

Dynamics is a branch of mechanics that is concerned with the study of motionand the relation between forces and motion. The central focus of our study is dy-namics of systems of rigid bodies and its application to engineering problems. Fur-thermore, we are basically concerned with the computer aided dynamics of rigidbodies with the aim to give an insight into the contemporary classical dynamicsfrom the computational point of view. This should familiarise a reader with the ba-sic concepts of today’s computational dynamics whose modelling procedures andnumerical techniques are realized in various program packages.

The motivation for this approach stems from the fact that in the contemporaryengineering praxis a lot of dynamical problems arise but only very few of themcan be solved in the analytical form by following classical ’calculation by hand’approaches. For the majority of problems (large-scale problems, analytically non-solvable differential equations, non-linear tasks, coupled problems etc.) compu-tational methods have to be applied. This fact gives rise to many open questionsconcerning the optimal use of modelling procedures as well as the computationaltools available within the various program packages.

The experience shows that for an accurate and effective computation, mechani-cal and mathematical models of the given engineering problem have to be properlyestablished. The computational model should contain all necessary informationconsidering the mechanical phenomena under investigation. It also should be for-mulated properly to suit the computational method that is intended to be utilizedto obtain the final solution. On the other hand, many computational methods forthe various kinds of problems are at the user’s disposal today. Among them theappropriate ones for the problem at hand should be chosen and applied.

In this book our main goal is to provide the basic principles of the contemporarycomputational dynamics of rigid body systems as well as the necessary theoreticalbackground. The main issues of the multibody dynamics modelling concepts, ex-posed in the extent form the first introductional insights to the contemporary engi-neering applications, are in the focus of the study.

In course of the text, the more advanced concepts of multibody dynamics aregiven after selected basic topics of engineering mechanics are shortly introduced.Since these selected topics are necessary content of the most engineering curricula,a reader with an adequate educational background (higher semester undregraduatestudent of mechanical or aerospace engineering, graduate student starting researchin computational dynamics or engineer professionally interested in this domain)should have no problems in building up his expertise starting from the theoretical’common ground’.

A knowledge on the mathematical analysis, covered by the university courseson this topic, as well as the basic knowledge on engineering numerical analysis is

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assumed. Some degree of ’familiarity’ with the contemporary engineering compu-tational tools is not assumed but would be of help.

The starting point of the book is the question: What should be considered inestablishing a proper computational model that should be successfully solved ?

Authors

Chapter 1

Mechanical and MathematicalModelling

1.1 Introduction

1.1.1 Issues of applied dynamics

Dynamics can be classified into several sub-domains. Each of them has its ownmodelling assumptions and procedures. In most of the cases, the computationalmethods are also different. According to the characteristics of the problem andthe focus of the intended dynamical analysis, the sub-domain whose approach isbest suited to the problem at hand should be chosen. In sequel of the chapter, anoverview of the characteristics of sub-domains and problems of the contemporarydynamics is given.

Multibody dynamics

Multibody dynamics deals with the mechanical systems of interconnected rigid bod-ies that undergo large displacements and rotations [23], [21]. The bodies are inter-connected by kinematical constraint elements and coupling elements. Both visco-elastic and inertia properties of a real engineering system are discretised during theprocess of shaping of the system’s mechanical model. Mathematical modelling ofthe established discretised mechanical model leads to ordinary differential equations(ODE) (minimal form mathematical models) or to differential-algebraic equations(DAE) (mathematical models in descriptor form).

The concepts of multibody dynamics can be successfully utilized within theframework of the following engineering applications: vehicle systems, aircraft sub-systems, robotic systems, various kind of mechanisms, biomechanical systems,mechatronics [22].

1

CHAPTER 1. MECHANICAL AND MATHEMATICAL MODELLING 2

Structural dynamics

Structural dynamics deals with deformable mechanical structures whose segmentsgenerally do not undergo large displacements and rotations (not kinematical chains).The mass and visco-elastic properties of a system are distributed along the structure.

The basic mathematical modelling generally leads to partial differential equa-tions (PDE). The discretisation of a system that is usually performed in the sequelof mathematical modelling procedure yields a mathematical model in the form ofODE. By using finite element approach [2], very powerful computational proce-dures are available for tackling the problems of structural dynamics.

Typical structural dynamics applications are: plates, shells, aircraft structures,trusses, civil engineering structures.

Flexible multibody dynamics

In the framework of flexible multibody dynamics, segments of a system are consid-ered to be flexible.

Flexible multibody dynamics typically deals with non-linear structures whosesegments undergo large rigid body motion superimposed by flexible deformations[13]. Modelling and computational procedures of multibody dynamics and struc-tural dynamics are being combined in order to formulate efficient procedures forproblems of this kind. The methods of flexible multibody dynamics are subjects ofextensive ongoing research activities [24].

The applications of flexible multibody dynamics systems can be found in var-ious multibody systems with connected rigid and flexible segments. Some of theexamples are aircraft rotary wings, flexible robots, biomechanical systems and high-speed mechanisms.

Problems of dynamics

In dynamics various classes of problems can be distinguished.

• Inverse dynamics

Inverse dynamics deals with determination of applied and constraint forcesand torques for a mechanical system whose motion is prescribed [18], [4].

Besides ’full’ dynamic approach, in which all forces of a system are consid-ered in the computation, the quasi-static approach of inverse dynamics can beapplied. Within the framework of quasi-static approach, the inertia forces ofthe system are neglected.

In most of the cases, an inverse dynamics problem leads to a set of algebraicequations.


Dynamics

Dynamics of MBS

Dynamial behaviour(stability tests)

Inverse dynamics Forward dynamics Optimisation

Structural dynamics

Dynamical modelling

Deriving dynamical equations

Newton–Eulerapproach

Lagrange approach,Jourdain’s principle

etc.

Inversedynamics(Solvingof linearalgebraic

equations)

Minimal formformulation

Stability criteria

Linearization ofthe equations

Linear ODE(vibration analysis)

DAE system ODE system

Integration ofDAE

Integration ofODE

Descriptor formformulation

Reduction before

integration

Stabilityanalysis

Linearforwarddynamics

Reduction during

integration

Linear

analysis

Forwarddynamics

Forwarddynamics

Figure 1.1: Issues of applied dynamics

• Forward dynamics

Forward dynamics deals with determination of the motion of a system that issubjected to prescribed applied forces and torques [21], [17].

In the most engineering applications a forward dynamics problem leads tosolving of non-linear ordinary differential equations (the system bodies un-dergo large rotations, coupling elements of the system possess non-linear


characteristics etc). Depending on formulation of a mathematical model, theadditional algebraic equations may be imposed on the system.

• Vibrations

The solving of a vibrational problem in linear domain [15], [26] leads todetermination of system eigenvalues and modes in most of the cases. Thesystem stability problem can also be mentioned in this context.

In the framework of some very important industrial applications (non-linearvibrations within the vehicle sub-systems, acoustical problems etc.) non-linear vibrational problems have to be tackled.

• Optimisation

A problem of the optimisation of mechanical systems (weight, costs, struc-tural deformations and stresses, dynamical trajectories are some of the quan-tities that can be optimised) is very important in engineering and lies far outof the scope of this book [3], [7].

However, it can be stated that specialised methods and algorithms that al-low for optimisation of mechanical systems according to the specified criteriamay be applied. In some cases an improved design can be possibly achievedwithout utilization of the specialised optimisation algorithms: there are op-timisation problems where repeated dynamical simulations and variations ofdesign parameters can lead to the improved solutions.

In Figure 1.1 issues of applied dynamics as well as the problems of dynamicsand solving methods are depicted schematically.

1.1.2 Modelling of Mechanical Systems

Modelling of mechanical systems has two major steps that are illustrated in Figure1.2.

The first step is mapping of reality (an engineering object) into a set of thesimplified entities in order to establish a mechanical model [20]. Mechanical modelmust include the effects under consideration, but should not be too complex i.e. themodels should be as complex as necessary but as simple as possible (A. Einstein:’Everything should be made as simple as possible, but not simpler’). Mechanicalmodelling is not an unique but an iterative process. It needs a lot of engineeringexperience since proper analogies between reality and the model, dependent on thegoals of the analysis, have to be established.

Once the mechanical model is built, a mathematical model, i. e. a set of the gov-erning equations which describe the model’s dynamical behavior, has to be formu-lated in the second step [8]. Mathematical modelling is also not an unique process.


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Figure 1.2: Steps of modelling

It depends on the goals of analysis and the computational procedures and tools aswell as the computer hardware that are intended to be used.

Mechanical modelling

Mechanical modelling is a process that is affected by the character of the problemand focus of the intended analysis in the first place. Second, the characteristics ofa real objects are important, but only within the scope of a given task and intendedanalysis.

A real object can be modelled using different mechanical elements: an aircraftcan be considered as a sole rigid body within the scope of flight mechanics, but it hasto be modelled as a multibody system to analyse the landing dynamics phenomena.If its space trajectory is under investigation, a large space station can be modelledas a particle, but on the other hand, a tennis ball has to be considered to be an elasticbody in the case of its impact analysis.

The crucial modelling criteria is that mechanical model should be able to de-scribe (take into account) those mechanical properties of a real system that are underthe consideration with the desired accuracy.

Mathematical modelling

Mathematical modelling is a process of formulating a mathematical text (a set ofequations of motion, for example) referred to the established mechanical model byfollowing physical laws and principles (Newtonian classical mechanics, smooth ornon-smooth theory).

A good and effective mathematical model has to reflect the type and characterof the analysis that is to be performed (for example, linear or non-linear analysis),


but also has to be properly formulated to suit the computational procedures andalgorithms that are intended to be used for a manipulation and evaluation of thegenerated equations [8].

In some special cases, a solution of the established mathematical model maybe found analytically. If so, the obtained solution is ’exact’ under the assumptionsmade during mechanical and mathematical modelling. Nevertheless, in most of thecases computational procedures have to be utilized to find numerical solutions.

In the past three decades numerous computational techniques and algorithmshave been established to generate the governing equations for various classes ofproblems and specific kinds of the analysis (multibody systems, structural systems,systems with the unilateral or variable constraints etc.). These algorithms are thecore of various program packages that are offered in the market today [21].

Although very often the intended mechanical analysis can be carried out bystarting initially from different mathematical models, an appropriate mathematicalmodelling can influence a computational procedure itself to a great extent (reducingthe computation time or gaining more accurate results).

1.2 Mechanical Modelling

As it was mentioned in Chapter 1.1.2, mechanical modelling is a process of mappingof reality to a set of simplified elements. The established set of elements (mechan-ical model) has to be able to describe those mechanical properties of a real systemwhich influence dynamical phenomena under consideration.

Given the goals of analysis and characteristics of a real system whose dynamicalbehaviour has to be investigated, a first step toward establishing a proper mechanicalmodel is a decision whether the system is to be modelled as a multibody system orthe modelling principles of structural dynamics are to be applied.

Many engineering systems consist of large number of bodies interconnectedby the constraint elements such as joints, bearings, springs, dampers or actuators.These systems can be successfully modelled as multibody systems. It can be statedgenerally that if bodies in a system undergo large motion and small vibrations, avery powerful tool is the modelling using multibody system approach.

If multibody system concept is adopted for modelling purposes, a real systemwill be discretised by means of the elements that will be reviewed in the sequel ofthe chapter. The discussion will be confined to the modelling principles of classicalmultibody dynamics (the models are established as systems of interconnected rigidbodies) and flexibility of segments is not considered.

Since kinematical structure of a system determines its characteristics to a greatextent, the types and the character of kinematical constraints and the way they de-termine a behaviour of the system will be discussed in detail. The classification of


the forces that appear in multibody systems will be also overviewed.Another part of mechanical modelling is an idealised description of a real load.

It may be introduced in a model as concentrated forces and moments as well as theforces and moments distributed over line, surface or volume [2]. An appropriatemodelling of a load is also dependent on the particular task and the establishedmodel itself.

Once the mechanical model is established, a corresponding mathematical modelhas to be formulated.

1.2.1 Elements of Multibody Systems

As it is depicted in the Fig. 1.3, multibody systems consist of elements with inertiaand constraint elements and coupling elements without inertia [20]. Elements thatposses inertia are a rigid body or, as a special case, a particle. Therefore, systems ofparticles and lumped mass systems may be regarded as special cases of multibodysystems.

Within the coupling elements two types of actuators can be distinguished:

• Actuators that prescribe the particular applied forces as the functions of time(’force actuators’).

The motion of a system caused by this type of actuators is generally notknown. It is a subject of the forward dynamic analysis of a given task.

• Actuators that prescribe a motion of the system, i.e. prescribe the particulardisplacements or rotations of the system’s bodies as functions of time (’dis-placement actuators’).

The forces imposed by the actuators of this type are generally not known.These forces are subject of the inverse dynamic analysis of the system athand [18].

Since these actuators prescribe system’s motion (a system is constrained toevolve in time in the specific way), the actuators of this type can be consideredas kinematical constraints. Consequently, the forces imposed by them areclassified as constraint forces (see classification of forces and kinematicalconstraints in the sequel of the chapter). The actuators of this type are alsocalled ’kinematical drivers’.

1.2.2 System forces

Forces that appear in multibody systems can be classified into the categories asdiscussed in the sequel. Classification of system forces is important due to the fact


Passive elements

mass point

spring

damper

rod

support, bearings, joints

actuator (force / moment) actuator (displacement / rotation)

(kinematical driver)

Coupling elements Constraint elements

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Coupling elements Constraint elements

Active elements

rigid body withnodal points Pi andcenter of gravity C

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Figure 1.3: Elements of multibody system

that different types of forces ’play a different role’ in the process of establishing ofthe mathematical model of a system (see Chapter 1.3.2).

• External and internal forces

This classification is based on the individual choice of the system’s boundary.

The external forces act from the outside of the boundary.

The internal forces act inside the boundary of the system. They always appearin pairs.

• Applied and constraint forces

The applied forces are forces imposed on the system by the coupling ele-ments as well as the forces which can be described by physical laws [14].They influence the way how the system evolves in time (as well as the system


constraint forces). Some examples of the applied forces are: gravity force,force actuators, springs, dampers, forces due to the magnetic field.

The constraint forces are imposed to the system by the kinematical constraintelements (joints, bearings, actuators that prescribe motion of the system). Inthe case of ideal constraints, these forces are collinear to direction of therestricted motion (see discussion on the kinematical constraints below). ’Theyinfluence the possible motion of the system’.

In Figure 1.5 system forces that appear in double pendulum are analysed and clas-sified by deriving the free-body diagram of the system.

1.2.3 Kinematical constraints

Kinematical constraints are mechanical entities that are imposed by joints, bear-ings and the system prescribed motions (kinematical drivers). They restrict systemmotion, reduce its degrees of freedom and are represented by the equations thatdescribe the kinematical restrictions imposed on a system.

Kinematical constraints are independent if these equations are linearly indepen-dent [18], [27] (number of independent kinematical constraints is equal to numberof the linearly independent equations between the constraint forces, rank of the ma-trix in the equation (1.83)).

Kinematical constraints can be independent of time (scleronomic constraints) orcan prescribe the motion of the system as a function of time (rheonomic constraints).

If kinematical constraints are represented by the equations comprising only dis-placements and rotations i.e. the constraints are at the position level since thereare no velocities or accelerations present in the equations, the constraints are calledholonomic constraints. If constraint equations are at the velocity level (containingtime derivatives of position coordinates) but can be directly transformed by integra-tion into the position level, they are also holonomic.

If constraint equations are at the velocity level and they can not be directlytransformed into the position level (unless system is integrated as a whole), they arecalled non-holonomic constraints [20], [30].

In the case of ideal kinematical constraints (the joints and bearings as well asthe kinematical drivers are assumed to be rigid and frictionless), the direction ofconstraint force that is imposed by the particular kinematical constraint is directedalong the direction of the constraint itself.

The considerations in this chapter are restricted to ideal and holonomic con-straints.


System coordinates and degrees of freedom

In the case of a totally unconstrained ’free’ system of p rigid bodies, a degree offreedom (DOF) of a system is 6p. It stems from the fact that 6p independent coordi-nates are necessary to describe kinematical configuration (position and orientationof the system’s bodies) uniquely. Consenquently, if motion of mechanical system ofp unconstrained bodies is restricted to a plane (a planar system), it posses 3 p DOF.

If q holonomic constraints are added to the system, its degree of freedom is reduced.

• If all q constraints are independent, a degree of freedom of the system isf = 6p− q.

• If only r of the q constraints are independent, a degree of freedom of thesystem is f = 6p − r. The number r of independent constraints is equal tothe rank of matrix Q in equation (1.83).

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Figure 1.4: Degree of freedom of mechanical system: double pendulum

If system possesses f DOF, there are f independent coordinates necessary todescribe the configuration of the system uniquely. This means that position of eachbody, expressed most often via Cartesian position of body center of mass, and itsorientation can be expressed as mathematical function of adopted independent co-ordinates and system geometrical characteristics (length of linkages, for example).These coordinates are called generalised coordinates and can be chosen in differentways appropriate to the particular problem. A choice of a set of generalised coordi-nates may strongly influence the process of mathematical modelling as well as theprocess of solving the equations (see Chapter 1.3).


In Figure 1.4 the concept of generalised coordinates and system degree of free-dom is explained using example of simple mechanical (multibody) system i.e. dou-ble pendulum. Since it is a planar system of two bodies constrained by four con-straint equations imposed by joints A and B (holonomic constraints), planar doublependulum posess f= 2 DOF. Consenquently, two generalized coordinates may beintroduced, in order to describe the configuration of the system.

So, if parameters α1 and α2 are adopted as a set of generalized coordinates, thesystem evolution in time is completely defined by keeping track of the coordinatesα1(t) and α2(t). By knowing time functions α1(t),α2(t) a kinematical configurationof the system (the coordinates of the bodies center of mass x1,z1,x2,z2 and bodiesabsolute orientation ϕ1 and ϕ2) can be determined by the equations:

x1(t) = 0.5l1 sinα1(t), z1(t) = 0.5l1 cosα1(t),

x2(t) = 0.5l2 sinα2(t), z2(t) = 0.5l2 cosα2(t),

ϕ1(t) = α1(t), ϕ2(t) = α2(t).

Velocities and accelerations of a system can be expressed via generalized velocitiesα1, α2 and generalized accelerations α1, α2 by differentiation of the configurationequtions given above. This is typical for holonomic systems.

If other set of independent coordinates is introduced, for example parametersα1 and α2, the configuration equations would have a different shape. Although achoice of any set of coordinates that can serve for unique description of systemkinematical configuration is principally a valid one, this choice may influence anumerical efficency as well as accuracy of overall calculation.

This can be illustrated by following. Let’s assume that body coordinates z1 andz2 are chosen as system generalized cordinates. If system is restricted to move thatapplies x ≥ 0, kinematical configuration of the body 1 is determined by:

x1 =√

l21 − z21 , z1 = z1, α1 = arccos

z1l1.

At the velocity level, it can be shown that x1 is expressed by x1 = − z1

x1

z1, whichmeans that for the position of a system that is characterized by x1 = 0 (body 1 alignwith z axis, ϕ1 = 0) equation fails to give required dependency. Furthermore, inall configurations of the system around position ϕ1 = 0 numerical accuracy of thecalculation will be low (why?).

The similar situation will occur in the context of body 2 around positionϕ2 = 0.Without going into details it can be said that this happens because the chosen gen-eralized coordinates fail to follow the motion of the system in those particular kine-matical configurations (coordinates z1 and z2 are exactly ortogonal to the directionof the system motion, i.e. system velocities, in those positions).


If such situation is likely to occur, the choice of set of generalized coordinatesis not an optimal one and, at least in the vicinity of critical system configurations,one should switch to an another set of coordinates. A reader, interested in researchthat addresses more advanced issues of configuration space geometrical propertiesas well as the optimal choice of the generalized coordinates, can find more detailsin [28], [30].

Generally, mathematical modelling of holonomic system dynamics using a setof f generalized coordinates is called mathematical modelling in minimal form(Chapter 1.3). If number of coordinates exceedsf (number of system DOF), themodel is in descriptor form. The extreme case is full descriptor form in the con-text of which the maximal number of 6p coordinates is used to describe systemconfiguration (Chapter 1.3).

Types of mechanical systems

Mechanical systems can be classified in terms of the number of its DOF and howthe imposed kinematical constraints are arranged (Figure 1.6 and Figure 1.7).

• Statical determination

If all q constraints are independent, a system is statically determined. On theother hand, if only r of the q constraints are independent, than n=q − r con-straints are superflous or redundant. The system is statically n times overde-termined. In this case constraint forces can not be calculated without intro-ducing further modelling assumptions (elastic properties, for example). InFigure 1.6 and Figure 1.7, the systems 1.6 d) and 1.7 d) are statically overde-termined.

• Kinematical determination

A system is kinematically determined, f = 0, if displacements and rotationsof all its members are completely determined by the constraints.

Within this type two cases can be distinguished. If all kinematical constraintsdo not depend on time, the system is a statical one. In Figure 1.7, the systemsc) and d) are of this type.

Otherwise, if f = 0, but at least one constraint is dependent on time, thesystem does not have a fixed configuration but evolves in time and can beconsidered as a kinematical or dynamical system.

Generally, the kinematical constraints that do not depend on time are called’scleronomic’ constraints and those constraints that depend on time are called’rheonomic’ ones.

CHAPTER 1. MECHANICAL AND MATHEMATICAL MODELLING 13��

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constraintexternal internal applied

Force

Figure 1.5: Forces in mechanical system: a double pendulum

If kinematical configuration of a system is not fully constrained by the kine-matical constraints, i.e. f =6p− r>0, the system has f degrees of freedomand it is kinematically undetermined. All examples presented in Fig. 1.6 aswell as the examples a) and b) in Fig. 1.7, are kinematically undetermined.Note: mechanisms are kinematically undetermined, f > 0, as long as theirmotion is not prescribed. For example, the well known four-bar linkage pos-sesses 1 DOF if there is no rheonomic constraint which determines its motion


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(Pendulum)

a)

b)

c)

d)

e)

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Figure 1.6: Systems with various constraints and DOF: rigid beam with varioussupports

(if so, it is kinematically determined, f= 0).

Types of statical and dynamical analysis

Depending on the kinematical structure of a mechanical system (Figure 1.6 andFigure 1.7), different kinds of analysis can be performed.

• Kinematically determined system

In the case of the kinematically determined system, a static analysis [25] or aninverse dynamic analysis can be performed (it depends on whether the system


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statically and kinematicallydetermined support

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statically overdeterminedsupport

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a)

b)

c)

d)

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Figure 1.7: Systems with various constraints and DOF: structures and mechanisms

is a statical one or its structure evolves in time).

In both cases a geometrical configuration of a system is not dependent onthe applied forces that are imposed on the system: its motion is completelydefined by kinematical constraints.

• Kinematically undetermined system

If a system is not fully kinematically constrained but possesses f degree offreedom, a time evolution of the system’s kinematical configuration is not


fully determined by kinematical constraints and it is dependent on system’sapplied forces. To determine the system’s motion, forward dynamical analy-sis must be performed.

If the system’s constraint forces are of interest, they can also be calculatedduring forward dynamic analysis or subsequently after a motion of the sys-tem is determined (depending on formulation of the system’s governing equa-tions).

1.3 Mathematical modelling

1.3.1 Basic laws and equations

Before formulating governing equations of multibody systems, we survey dynamicsof particles and rigid bodies based on the laws of classical mechanics. The vectorialentities like displacements, velocities, forces and torques, possessing a magnitudeand direction, are represented by vectors and designated using bold italic font like f

or r. Magnitudes of vectorial components along the axis of an adopted coordinatesystem represent coordinates of vectors in the chosen coordinate system.

For the metter of computation, vector entities are represented by arranging thecoordinates in one-dimensional arrays (one-column matrices). In computationalmechanics and control literature sometimes all types of one-column matrices aregenerally called ’vectors’ (one-column n-component arrays that, for example, mayrepresent the components of all forces acting on a multibody system or multiple val-ues of system control input) but they should not be confused with the vectors in thesense of classical mechanics. Similarly, tensors are arranged in multi-dimensionalarrays.

The aim of adopting matrix representations is to perform required vector/tensoroperations by using operations with matrices that can be easily utilised in computerapplications. Starting from the classical vector representation, matrix equations ofthe basic expressions needed for formulation of mathematical model of multibodysystems will be derived.

Dynamics of particles

By applying Newton’s second law, the equation of motion of a particle depicted inFigure 1.8 can be written [11],[25] as

f = mr = mv = ma =d

dt(mv) , (1.1)

where mv is linear momentum of a particle.


x

f

rvO

y

z �

Figure 1.8: Motion of particle

Vector r denotes position vector of a particle in inertal (non-accelerated) co-ordinate system. Velocity of a particle is given by v = r while acceleration isdetermined as a = r.

As known from basic textbooks of classical mechanics, in order to apply New-ton’s second law correctly, the derivatives in (1.1 ) must be derived with respect tothe inertial coordinate system.

The angular momentum of a particle with respect to O is given by

hO = r ×mv . (1.2)

EQUATIONS OF MOTION OF SYSTEM OF PARTICLES

rp

f i12

f1e

f2e

fpe

f i21

f i1p f i

p1

f i2p

f ip2

r2

r1

x

Oy

z

Figure 1.9: System of p particles

If system of p particles shown in Figure 1.9 is considered, Newton’s law for thei-th particle yields

f i = f ei + f i

i = miai (1.3)


where f ie denotes the resultant of external forces acting on the i-th particle. The

resultant of internal forces f ii is given by the equation

p∑

j=1

f ij = f ii . (1.4)

In the equation (1.4), f ij represents system internal forces which act between thebodies i and j. According to Newton’s third law [14], it is valid

f ij = −f ji . (1.5)

When summing up over the entire system of p particles, it can be written

p∑

i=1

f ei +

p∑

i=1

f ii =

p∑

i=1

miai ,

p∑

i=1

f ii = 0 , (1.6)

and finallyp

∑

i=1

f ei =

p∑

i=1

miai . (1.7)

The equation (1.7) can be further elaborated by using relations as follows.The position rC of the system mass centre C is defined by

mrC =

p∑

i=1

miri , (1.8)

where m =∑p

i=1mi is total mass of the system. Differentiation of (1.8) withrespect to time leads to linear momentum of the system of particles

mvC =

p∑

i=1

mivi . (1.9)

The second differentiation of (1.8) yields

maC =

p∑

i=1

miai , (1.10)

which can be introduced into (1.7)

p∑

i=1

f ei = maC . (1.11)

According to (1.11), the mass centre of system of particles moves as if entiremass of the system were concentrated at that point and all the external forces were


firAi

vi

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vA

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i–th particle,

x

Oy

z

Figure 1.10: Angular momentum of i-th particle

applied there [25].

ANGULAR MOMENTUM OF SYSTEM OF PARTICLES

Angular momentum of the i-th particle about an arbitrary moving point A (Fig-ure 1.10) reads as

hAi = rAi ×mivi (1.12)

and differentiation leads to

hAi = rAi ×mivi + rAi×mivi . (1.13)

If A coincides with the fixed point O (rAi× vi = vi × vi = 0), then it can be

writtenhAi = rAi ×mivi = rAi × f i , (1.14)

orhAi = lAi , (1.15)

where lAi is the resultant torque with respect to A (Figure 1.10).The angular momentum of system of particles about the moving point A is

hA =

p∑

i=1

rAi ×mivi . (1.16)

After differentiation of (1.16) with respect to time and some algebraic operations,the equation (1.13) for a system of particles has the form

hA = rAC ×mvC +

p∑

i=1

rAi × f ei , (1.17)


where rAC is the position vector of mass centre of a system of particles with respectto point A.

When point A coincides with the fixed origin O (rAC = rC = vC , vC×mvC =0 ) or point A coincides with the centre of mass C, equation (1.17) reduces to

hA =

p∑

i=1

rAi × f ei . (1.18)

So, if A coincides with C, equation (1.17) can also be written as

hC =

p∑

i=1

rCi × f ie , (1.19)

orhC = lC , (1.20)

where

lC =

p∑

i=1

rCi × f ie (1.21)

is the resultant moment of all external forces acting on the system of particles aboutthe mass centre C. It should be mentioned that dynamics of particle is completelydescribed by the Newton’s equation (1.3). An introduction of angular momentum,Eq. (1.12), results in redundant information. It has been introduced here as a pre-stage to dynamics of rigid body, where consideration of angular momentum leadsto the essential Euler’s equation.

Dynamics of rigid body

Prior to deriving governing equation of rigid body dynamics, in the next sectionsome basic kinematical relations will be recapitulated.

BASIC KINEMATICAL RELATIONS

In Figure 1.11 the following notation is used:

ω angular velocity of a body,rA position vector of the point A (body-fixed reference point),(x, y, z) inertial coordinate system k,(x′, y′, z′) coordinate system fixed to the body k′.

Since a rigid body is considered (Figure 1.11), r ′ is a body-fixed position vectorwhich does not change its magnitude but only its orientation due to rotation of thebody. Its time derivative with respect to the inertial system k can be expressed as

r ′ = ω × r ′ . (1.22)


rA

�

vA

z�y�

x�

rC

�

�rC�

r

r �

�

��

x

Oy

z

Figure 1.11: Rigid body

With the previously introduced notation, r can be written as

r = rA + r ′ (1.23)

and the velocity is obtained as

v = vA + r ′ = vA + ω × r ′ . (1.24)

An orientation of a body in the inertial coordinate system can be determinedvia Euler angles ϕ, ϑ, ψ [12] that specify an orientation of a fixed body system k ′

with respect to the inertial system k. Other possibilities to describe orientation of abody-fixed ccordinate system include Bryant angles, Euler parameters, Rodriguezparameters etc. [5], [18].

A relation between ω and the derivatives of the Euler angles xR = [ϕ ϑ ψ]T canbe expressed in matrix form as

ω = HRxR . (1.25)

LINEAR MOMENTUM

Mass of a body is given by

m =

∫

m

dm , (1.26)

and the position vector of body centre of mass C in coordinate system k is

rC =1

m

∫

r dm . (1.27)


The position of C with respect to a body-fixed point A is given by

rAC = rC′ =

1

m

∫

r ′ dm . (1.28)

From (1.24), linear momentum of a body can be written in the form∫

m

(vA + ω × r ′) dm = vA

∫

m

dm+ ω ×

∫

m

r ′ dm , (1.29)

or∫

m

(vA + ω × r ′) dm = vAm+ ω ×mrC′

= m(vA + ω × rC′)

= mvC

= mrC . (1.30)

ANGULAR MOMENTUM

Absolute angular momentum of a body with respect to O (origin of the inertialcoordinate system) is determined by [5]

hO =

∫

m

r × r dm (1.31)

or after introducingr = rA + r ′ , (1.32)

hO =

∫

m

(rA + r ′) × (vA + ω × r ′) dm

= rA × (vA + ω × rC′)m+ rC

′ × vAm+

∫

m

r ′ × (ω × r ′) dm . (1.33)

The term∫

m

r ′ × (ω × r ′) dm can be written in the form

∫

m

r ′ × (ω × r ′) dm =

∫

m

(r′ 2E − r′r′) dm · ω

= IA · ω , (1.34)


where IA is the inertia tensor of the body with respect to A

IA =

∫

m

(r′ 2E − r′r′) dm , (1.35)

E is the unit vector and r′r′ denotes dyadic product.Finally, if body centre of mass C is chosen as the reference point A (rC

′ = 0 ;vA → vC ; rA → rC) and equation (1.34) is taken into account, (1.33) becomes

hO = rC × vCm+ IC · ω . (1.36)

EQUATIONS OF MOTION OF RIGID BODY

Newton’s equation determines dynamics of a body’s translational motion [14]

d

dt(mvC) = f , (1.37)

where:mvC is linear momentum of a rigid body,f is resultant of all forces acting on a body.

If mass of a body is constant (dm/dt = 0), the equation (1.37) becomes

maC = f . (1.38)

Euler’s equation determines dynamics of a body’s rotational motion

hO = lO , (1.39)

where:hO is absolute angular momentum with respect to the

fixed reference point O in the inertial space,lO is resultant torque with respect to O.

A result of the derivative of equation (1.36) with respect to time is

hO = vC × vCm+ rC ×maC + IC · ω + ω × IC · ω , (1.40)

and since vC × vCm = 0 ,

hO = rC ×maC + IC · ω + ω × IC · ω. (1.41)

By substituting equation (1.41) into equation (1.39) and by considering equation(1.38), Euler’s equation can be written in form

IC · ω + ω × IC · ω = lO − rC × f = lC , (1.42)


or simplyIC · ω + ω × IC · ω = lC. (1.43)

NEWTON-EULER EQUATIONS IN MATRIX FORM

By following the rules of matrix algebra, vector-valued equations (1.38) and (1.43)can be written in the matrix form [18]

maC = f , (1.44)

ICα + ωICω = lC . (1.45)

The matrix α stands for the body’s angular acceleration

α = ω

and vector product is performed using the skew-symetric matrix ω.Matrix equation (1.44) is derived from a non-coordinate expression (invariant

form) (1.38) by using an inertial coordinate system k. On the other hand, matrixequation (1.45) is derived from invariant form (1.43) by using a body-fixed coordi-nate system k′. By using a body-fixed coordinate system the components of inertiatensor of a body with respect to C remain constant. This is very convenient fromthe computational point of view.

1.3.2 Mathematical models and procedures

Mathematical models

As a result of mathematical modelling via different methods for formulation of thegoverning equations, two basic forms of the mathematical models can be distin-guished: descriptor form and minimal form.

Each of these forms possesses specific characteristics, being more or less ap-propriate for a particular dynamic analysis. Once the model is established, thesecharacteristics determine to a great extent the computational procedures that are tobe used in the subsequent computational process [1], [6].

• Descriptor form characteristics

– number of coordinates and differential equations are larger than thenumber of DOF

– type of differential equations: DAE

– lower degree of non-linearity of the differential equations

• Minimal form characteristics


– number of coordinates and differential equations is equal to the numberof DOF

– type of differential equations: ODE

– highly non-linear differential equations

Full descriptor form– 6p dynamical equations of

the free–body diagram, 6p“cartesian” coordinates x

– q kinematical constraintsequations

��

Minimal form– f equations of motion,

f generalised coordinates y

– q kinematical constraintequations

Governing equation(holonomic system)

p...bodies, q...constraints, f...DOF

Figure 1.12: Forms of mathematical model

Approaches to computational procedures

Independent on the form of the established mathematical model, two forms of ob-taining a solution of the governing equation can be distinguished: a closed formsolution and numerical (approximate) solution. If methods for obtaining numericalsolution have to be applied (this is a case for the most engineering applications),this can be done by using either symbolic or numerical approach to computationalprocedures [21].

• Closed form solution

Searching for a closed form solution ’pays off’ if there are indications that asolution of the established mathematical model can be found by using pureanalytical methods. In this case the result is expressed in the form of func-tions. The solution is ’exact’ under the assumptions that have been madeduring mechanical and mathematical modelling of a system (the obtained so-lution would be free of numerical errors of any kind).


Unfortunately, in most of the cases (except for some linear models and sim-pler tasks of small dimensionality), it is not possible to find a closed form so-lution and a numerical procedure has to be applied to obtain a solution of themodel (a numerical procedure may be launched immediately after mathemat-ical model is established, or some symbolic manipulations and simplificationscan be performed prior to the numerical calculations).

• Symbolic approach

Symbolic mathematical operations consist of the manipulations with mathe-matical entities without assigning their numerical values.

If a computational tool gives possibilities for symbolic calculations, some-times a more efficient computational procedure can be achieved by simplify-ing the established mathematical model before an iterative numerical proce-dure is launched. Once mathematical model is established by using symbolicformalisms, it can be used for repeated numerical calculations e.g. in numer-ical integration schemes.

However, the extent to which an efficiency may be improved using the sym-bolic tools is dependent on the task (mathematical model) at hand. Symbolicmanipulations are often computationally more costly than the numerical pro-cedures. It is specially true for some types of problems in the context of whichsome very efficient numerical procedures can be utilized e.g. the sparse ma-trices techniques.

Although symbolic procedures are much in use in today’s computation, for-mulation and implementation of symbolic algorithms are the topics of theextensive ongoing research activities.

• Numerical approach

By using this approach, numerical values are assigned to symbolic items assoon as mathematical model is established and the whole computational pro-cess deals with numerical values.

The majority of the computational packages on today’s market are numeri-cally oriented, especially packages and tools that are designed for a generaluse [21].

1.3.3 Formulation of governing equations

When performing dynamic analysis of a given mechanical system, a formulationof governing equations is the main part of mathematical modelling. It is the firststage of mathematical modelling independent of the dynamical task at hand (inversedynamical problem, forward dynamics, optimization problems, etc.).


Derived mathematical model serves as a basic set of equations by means ofwhich the system’s motion and constraint forces can be determined. In most ofthe cases, the basic set of equations will have to be manipulated further to suit theintended analysis and computational procedure.

As it was already explained, an ’output’ of different formalisms consists ofmathematical models shaped in different forms which require different numericalprocedures and algorithms in order to obtain the final solution.

In sequel of the chapter the methods for formulation of the governing equationsof multibody systems are elaborated. These methods are among some of the mostcommonly used methods in the computational dynamics today. The main charac-teristics of each method as well as the application properties are provided briefly.Some illustrative examples are given in the Chapter 2.

Multibody systems of ’free’ bodies

Prior to investigation of a constrained mechanical system, a system of free bodies isconsidered to prescribe nature of the underlying dynamics. A multibody system of’free’ bodies, shown in Figure 1.12, is a mechanical system of rigid bodies whosemotion is not constrained by kinematical constraints of any kind. Therefore, ifsystem consists of p bodies, it possesses 6p degrees of freedom (DOF).

��

��

��

l1

f i2p

f ip2

f1e

��

�1

��

f2e

l2

�2

fpe

��

lp

f i1p

f ip1

�p

f i12

f i21x

O

yz v1

vp

v2

Figure 1.13: Free-body diagram of multibody system of ’free’ bodies

A determination of absolute position and orientation of the i-th body of a systemis given by the vector of the body mass centre expressed in the inertial coordinate


system (Cartesian system is used but the other coordinate systems can be chosen aswell)

xTi = [xi yi zi]T , (1.46)

and Euler angles of body’s absolute orientation

xRi = [ϕi ϑi ψi]T . (1.47)

(Note: In Eq. (1.46), (1.47) and subsequent text the index ’C’ is omitted since cen-tre of mass will always be referred to describe a position of the body.) By groupingequations (1.46) and (1.47) together, a body absolute position vector can be intro-duced in the form

xi = [xTTi xT

Ri]T . (1.48)

Newton-Euler equations of ’i-th’ bodyNewton-Euler equations are basic equations of rigid body dynamics, see Chap-

ter 1.1. Newton’s equation determines dynamics of a body’s translational motion,while a body’s rotational motion is determined by Euler’s equation.

Newton’s equation is given by

miai = f i , (1.49)

or in the matrix formmixTi = fi . (1.50)

Euler’s equation is expressed by

I iωi + ωi × I iωi = li , (1.51)

or, following the rules of the matrix algebra,

Iiαi + ωiIiωi = li , (1.52)

where a body’s angular acceleration is given by the equation

αi = ωi . (1.53)

A relation between body’s angular velocity ωi and time derivatives of Euler anglesxRi = [ϕi ϑi ψi]

T , by means of which absolute orientation of a body in the inertialcoordinate system is specified, can be given in the form [5]

ωi = HRixRi , (1.54)

and the differentiation with respect to time using the chain rule yields

αi = HRixRi + αi . (1.55)


In equation (1.55), all terms in which second derivative appear linearly are ex-pressed in the product HRixRi and all others are grouped in αi. By taking intoaccount equations (1.55), equation (1.52) can be written in the form

IiHRixRi + Iiαi + ωiIiωi = li . (1.56)

Furthermore, the equations (1.50) and (1.56) can be grouped together to formNewton-Euler equations of the i-th body in the matrix form:

[miE 0

0 Ii

] [E 0

0 HRi

] [xTi

xRi

]

+

[0

Iiαi + ωiIiωi

]

=

[fili

]

, (1.57)

or in shortMiHixi + qv

i = qai . (1.58)

The dimensions of the matrices in equation (1.58) are

dim[Mi] = 6 × 6, dim[Hi] = 6 × 6, dim[xi] = 6 × 1 ,

dim[qvi ] = 6 × 1, dim[qa

i ] = 6 × 1 .

Newton-Euler equations of ’p’ bodiesBy formulating equation (1.58) for each body in the system (i = 1...p), Newton-

Euler equations of the multibody system of ’free’ bodies can be obtained in theform:

MHx + qv = qa , (1.59)

where the matrices are specified as follows

x = [xT1 xT

2 ... xTp ]T , dim[x] = 6p× 1 , (1.60)

M =

m1E 0 0 0 · 0 0

0 I1 0 0 · 0 0

0 0 m2E 0 · 0 0

0 0 0 I2 · 0 0

· · · · · · ·

0 0 0 0 · mpE 0

0 0 0 0 · 0 Ip

, dim[M] = 6p× 6p , (1.61)

H =

E 0 0 0 · 0 0

0 HR1 0 0 · 0 0

0 0 E 0 · 0 0

0 0 0 HR2 · 0 0

· · · · · · ·

0 0 0 0 · E 0

0 0 0 0 · 0 HRp

, dim[H] = 6p× 6p , (1.62)


qv =

0

I1α1 + ω1I1ω1

0

I2α2 + ω2I2ω2

· ·

0

Ipαp + ωpIpωp

, dim[qv] = 6p× 1 , (1.63)

qa =

f1l1f2l2· ·

fplp

, dim[qa] = 6p× 1 . (1.64)

In the case of multibody system of free bodies, Newton-Euler equations (1.59)are the equations of motion of a system.

Equation (1.59) represents 6p dimensional ODE system. It can be integrated intime for specified initial conditions x0 , x0 to determine the system’s motion (vari-ables x , x , x).

Some computational issuesInertia matrix MH in Eq.(1.59) has non-symetrical properties which may decresesignificantly an efficiency of computation.

In the framework of integration of governing equations, a non-symetric inertiamatrix prevents use of very efficient numeric procedures which require its symetri-cal properties (e.g. Cholesky method).

Therefore, to improve an efficiency of the procedure, it may be helpful to sym-metrise the matrix MH. This can be done by premultiplying Eq.(1.59) by HT ,which reads

HTMHx + HTqv = HTqa . (1.65)

In many applications an integration of (1.65) requires a less computer-power thanintegration of (1.59).

Constrained multibody systems

MATHEMATICAL MODEL IN FULL DESCRIPTOR FORM

Constrained multibody system (Figure 1.14) is a mechanical system of rigid bodieswhose motion is constrained by kinematical constraints. If a system consists of p


bodies whose motion is constrained by q kinematical constraints, the system pos-sesses f = 6p− q DOF .

f1e1

f1ek

��

��

��

��

l2e1

l2eh

Figure 1.14: Constrained multibody system (mechanical model)

The following notation is used:f

ej

i . . . j-th (j = 1...k) applied external force that acts on the i-th body (i = 1...p),l

ej

i . . . j-th (j = 1...h) applied external torque that acts on the i-th body (i = 1...p).In Figure 1.13 this is illustrated at the body i = 1.

Forces in a constrained multibody systemApplied forces

In Figure 1.15, a free-body diagram of constrained multibody system is derived.Resultant applied force that acts on the i-th body is

f i = f ei + f i

i , (1.66)

where resultant force of the external applied forces, reduced to the centre of massCi, can be written as

f ei =

k∑

j=1

fej

i (1.67)

and resultant force of the internal applied forces (internal springs, dampers, etc.),reduced to the centre of mass, is given by the sum of the internal applied forcesbetween the bodies i and j

f ii =

p∑

j=1

f iij . (1.68)


f c12

f1e

l1c �

�

��

��

l1

f2e

l2c

l2

fpe

lpc

lp

f c21

f i1p

f ip1

f i2p

f ip2

f cp0

Figure 1.15: Free-body diagram of constrained multibody system

Reduction of forces with different application points to the specific point impliesthe formulation of an equivalent couple of force and torque acting at this point.

Resultant torque about the centre of mass Ci of the applied forces and torquesthat act on the i-th body (Figure 1.15) is

li =h∑

j=1

lej

i +

p∑

j=1

l iij + lredi

, (1.69)

where l iij is an internal applied torque (internal torsional spring, for example)

that acts between bodies i and j and lrediis the torque due to the reduction of the

forces f ei and f i

i to the centre of mass Ci.

Constraint forcesResultant constraint force that acts on the i-th body, reduced to the centre of

mass Ci, is given by the equation

f ci =

p∑

j=0

f cij , (1.70)

where f cij is a constraint force (Figure 1.15) that acts between bodies i and j (i, j =

0...p, the index 0 stands for the ’body’ of ’external world’).If an index i or j is equal to zero, a constraint force is of the external type (a

force due to a kinematical constraint with the ’external world’). If none of indices


i, j is zero, a constraint force is of the internal type (due to a kinematical constraintthat restricts relative motion of the bodies).

Resultant torque about the centre of mass Ci of the constraint forces and torquesthat act on the i-th body can be given in the form

lci =

p∑

j=0

l cij + l c

redi, (1.71)

where l cij is a constraint torque that acts between bodies i and j or a body and the

’external world’ (i, j = 0...p) and l credi

is a reduction torque of the forces f cij .

Newton-Euler equations of ’i-th’ bodyAs it was the case with a system of ’free’ bodies, we start derivation of gov-

erning equations of constrained system by considering Newton-Euler equations ofa single body. When a body is kinematically constrained, constraint forces andtorques influence a motion of a body and have to be considered in the framework ofNewton-Euler equations along with applied forces and torques.

Newton’s equation is given by

miai = f i + f ci , (1.72)

or in the matrix formmixTi = fi + f c

i . (1.73)

Euler’s equation is expressed by

I iωi + ωi × I iωi = li + l ci , (1.74)

or following the rules of matrix algebra

Iiαi + ωiIiωi = li + lci . (1.75)

By considering (1.55), equation (1.74) can be expressed in the form

IiHRixRi + Iiαi + ωiIiωi = li + lci . (1.76)

After introduction of a body absolute position vector (1.60), equations (1.73) and(1.76) can be grouped together to form Newton-Euler equations of the i-th body

[miE 0

0 Ii

] [E 0

0 HRi

] [xTi

xRi

]

+

[0

Iiαi + ωiIiωi

]

=

[fili

]

+

[f ci

lci

]

,

(1.77)or in short

MiHixi + qvi = qa

i + qci . (1.78)


Dimensions of the matrices in equation (1.78) are

dim[Mi] = 6 × 6, dim[Hi] = 6 × 6, dim[xi] = 6 × 1 ,

dim[qvi ] = 6 × 1, dim[qa

i ] = 6 × 1, dim[qci ] = 6 × 1 .

Newton-Euler equations of constrained system of ’p’ bodiesBy formulating equation (1.78) for each body in a system (i = 1...p), Newton-

Euler equations of a constrained multibody system can be arranged in the form

MHx + qv = qa + qc . (1.79)

As it will be described in the sequel, system constraint forces qc in equation (1.79)can be expressed via kinematical constraint equations and additional parameters.

Governing equations of constrained multibody systemsKinematical constraint equations

Newton-Euler equations (1.79) are part of the governing equations of constrainedmultibody system. Since motion of system bodies is kinematically constrained,components of system position vector x are not independent but satisfy a set of qkinematical constraint equations, which can be put in the form [18]

g(x, t) = 0 , dim[g] = q . (1.80)

By differentiation of (1.80) with respect to time, the equation that expresses therelation between the system velocities is obtained as

∂g

∂xx +

∂g

∂t= 0 , (1.81)

or in the short form

Qx = −∂g

∂t, (1.82)

where the matrix Q is defined as

Q(x, t) =∂g

∂x, dim[Q] = q × 6p . (1.83)

If (1.80) is differentiated twice, the equation that expresses dependency betweensystem accelerations can be formulated. After application of chain rule of differen-tiation, kinematical constraint equations at the level of acceleration can be writtenin the short form

Qx = c . (1.84)


Constraint forces via kinematical constraintsIt can be shown [20] that system constraint forces qc which are caused by ideal

kinematical constraints (a friction is not considered, constraint forces qc are orthog-onal to ’directions’ of the imposed kinematical constraints) can be expressed viamatrix Q and q unknowns λi, (i = 1...q) that are usually called ’Lagrange multipli-ers’.

If vector λ of Lagrange multipliers is introduced in the form

λ = [λ1 λ2 ...λq]T , dim[λ] = q × 1 , (1.85)

system constraint forces can be expressed by equation

qc = QT λ . (1.86)

In the context of the equation (1.86) it can be stated, without going to the details,that directions of system constraint forces qc are expressed by the columns of trans-posed matrix QT while the magnitudes of constraint forces are given by Lagrangemultipliers vector λ.

Matrix Q, defined in (1.83), by means of which system constrained forces qc areexpressed in (1.86), is usually called ’system constraint matrix’. System constraintmatrix is a one of the most important matrices in domain of dynamics of constrainedmultibody systems. By checking its rank it can be examined if a system is properlyconstrained i.e. if all constraints imposed on a system are independent or some ofthem are superfluous. Number of independent constraints is equal to the rank of Q(see Chapter 1.2.3).

System governing equationsAfter insertion of (1.86) into (1.79), Newton-Euler equations of a constrained

multibody system can be written in the form

MHx + qv = qa + QT λ . (1.87)

Unlike Newton-Euler equations of a ’free’ multibody system (1.59), a set of theequations (1.87) can not be solved and integrated in time directly, since it containsq additional algebraic unknowns λ.

To make a set of the equations (1.87) complete and solvable, kinematical con-straint equations have to be added to mathematical model and considered simulta-neously with Newton-Euler equations.

With this aim in view, Newton-Euler equations (1.87) and q kinematical con-straint equation (1.80) are grouped together, forming the governing equations of aconstrained multibody systems

MHx + qv = qa + QT λ


g(x, t) = 0 . (1.88)

Equation (1.88) is 6p+q dimensional DAE system (DAE of index 3) that can besolved and integrated in time to obtain motion of a system (variables x , x , x) andsystem constraint forces qc = QT λ. For numerical time integration, the specificsolving procedures for DAE systems have to be used [1].

To utilize more convenient numerical procedure for integration, the governingequations of constrained multibody systems are very often formulated by usingkinematical constraint equations at the acceleration level (equation (1.84)), insteadof those formulated at the position level (Eq. (1.80)).

In this way, the governing equations can be formulated in the form (6p + qdimensional DAE of index 1)

MHx + qv = qa + QT λ

Qx = c . (1.89)

A set of the equations (1.88) as well as the set (1.89) represent the governingequations of a constrained multibody system expressed in the full descriptor form.

As it was already explained, by integrating (1.88) or (1.89) for a specified initialconditions x0 , x0 a system’s motion as well as system’s constraint forces can bedetermined.

Because of inherent numerical instability of DAE system presented by (1.88)and (1.89), a numerical time integration of these equations is a challenging taskwhich has to be treated very carefully [9].

Similarly as it was the case with Eq. (1.59), inertia matrix MH in Eq. (1.87) isa non-symetric one which makes time integration of (1.88) and (1.89) less efficient.As it was explained, premultiplication of (1.87) by HT symmetrise inertia matrixMH and brings (1.87) in the form

HTMHx + HTqv = HTqa + HTQT λ , (1.90)

which can be integrated more efficiently.

Characteristics of mathematical model in full descriptor form

• Basic equations: Newton-Euler equations, constraint forces are included

• Absolute coordinates, 6 coordinates per body: Cartesian coordinates of thebody mass centers and body’s Euler angles (or other parameters)

• Easy-to-obtain mathematical model, straightforward universal approach [18]


• Once mathematical model of the multibody system at hand is established,the model can be easily re-formulated if kinematical structure of a system ischanged.

Since kinematical structure of a system is reflected within the governing equa-tions only through the kinematical constraints equations g(x, t) = 0 andsystem constraint matrix Q (equation (1.83)), just these terms have to bechanged/redefined if a new kinematical configuration of the system is intro-duced (see also Fig. 1.12).

• Appropriate for forward and inverse dynamical problem

– Inverse dynamics

All constraint and applied forces are included in the mathematical modeland can be obtained by using standard procedures.

– Forward dynamics

Model allows for determination of system motion and constraint forcessimultaneously.

Since model is based on the six coordinates per body, a position andorientation of each body are automatically calculated in the course ofsimulation.

• Appropriate for computer algorithms

Model is easy to establish by using standard matrix algebra operations.

It is suitable for implementation in general purpose multibody algorithms[21].

• Characteristics of the equations

Mathematical model in the full descriptor is expressed by DAE equations.

DAE equations are generally more difficult to solve than ODE systems. Con-straint violation stabilisation procedures are generally needed [4], [29], [30].

If it is required, governing equations expressed in full descriptor form can be re-duced to mathematical model in minimal form (that represents equations of motionof a constrained multibody system) which will be described in the sequel.


MATHEMATICAL MODEL IN MINIMAL FORM

Reduction of model from full descriptor to minimal formIn order to shape mathematical model in minimal form i.e. to establish equa-

tions of motion of a constrained multibody system, a minimal set of coordinates y,dim[y] = f (f= number of DOF), have to be chosen. By means of y, kinematicalconfiguration of a system is uniquely described.

A relation between system full descriptor absolute coordinates

x = [xT1 xT

2 ... xTp ]T , dim[x] = 6p× 1

and a minimal form coordinates y, dim[y] = f , depends on system kinematicalconstraints equations and can be expressed explicitly by equation

x = f(y, t) , (1.91)

(an implicit formulation ϕ(x, x, t) = 0 can be formulated as well). The corre-sponding equation at the velocity lavel takes form

x =∂f

∂yy +

∂f

∂t= Jy +

∂f

∂t, (1.92)

and equation at the acceleration level reads as

x = Jy + a , (1.93)

where matrix J is given in the form

J =∂f

∂y, dim[J] = 6p× f . (1.94)

Matrix J is usually called ’Jacobian matrix’. It is not unique but depends on achosen set of coordinates by means of which a system kinematical configuration isdescribed.

It can be shown [20] that Jacobian matrix J has a property of being orthogonal-complementary matrix to system constraint matrix Q. A relation between these twomatrices can be expressed by

QJ = JTQT = 0 . (1.95)

The orthogonality between matrices J and Q stems from the fact that these ma-trices span different ’subspaces’ which are mutually orthogonal. Namely, columnsof transposed constraint matrix QT as well as Jacobian matrix J are vectors whichform the basis of two vectorial subspaces: q-dimensional subspace of system con-straint forces that is spanned by Q and f -dimensional subspace of system velocities,spanned by J.


The orthogonality given by (1.95) holds only for ideal mechanical systems andcan be briefly explained by the fact that system velocities are always orthogonal tosystem constraints. In analytical mechanics this is related to Jourdain’s principlewhich is based on virtual power. Other principles are d’Alembert’s principle, for-mulated by Lagrange on the basis of virtual work, and Gauss’s principle based onminimal constraints. The details of the introduced vectorial subspaces as well asstrict mathematical proof are not given here and an interested reader is referred tothe literature [4], [28].

By introducing equations (1.91), (1.92) and (1.93) into equation (1.87), Newton-Euler equations of constrained systems can be expressed via minimal set of coordi-nates y

MH′(Jy + a) + qv′ = qa′ + Q′T λ (1.96)

where matrices that are expressed by a new set of coordinates are denoted by ’′’.Furthermore, if equation (1.96) is multiplied from the left side by transposed

Jacobian matrix JT , two important effects will be achieved:

• elimination of constraint forces by means of orthogonality relation (1.95)(principle of virtual work),

• reduction of dimension of the equation from 6p to f .

In this way, it can be written

JTMH′Jy + JT (MH′a + qv′) = JTqa′ , (1.97)

or in the short formMgeny + qv

gen = qagen . (1.98)

In equation (1.96), generalised mass matrix is defined as

Mgen = JTMH′J , dim[Mgen] = f × f , (1.99)

vector of centrifugal, Coriolis and gyroscopic terms reads as

qvgen = JT (MH′a + qv′) , dim[qv

gen] = f × 1 (1.100)

and vector of generalized applied forces is

qagen = JTqa′ , dim[qa

gen] = f × 1 . (1.101)

A set of the equations (1.98) represents equations of motion of a constrainedmultibody system (mathematical model in minimal form, f -dimensional ODE sys-tem). It can be integrated in time for given initial conditions y0, y0 to obtain sys-tem’s motion.


Since term of constraint forces QT λ vanishes from the governing equations dueto the left multiplication by JT , it is obvious that this term has not to be formulated,if equations of motion are to be derived. Therefore, equations of motion (1.98) ofa constrained system can be derived straightforwardly by formulating the matricesdirectly by means of (1.99), (1.100) and (1.101).

Symmetrising inertia matrixInertia matrix MH′J in Eq.(1.96) has non-symetrical properties which may de-

crease significantly an efficiency of computation, as it was explained. Therefore,prior to elimination of constraint forces and reduction of dimension of (1.98), itmay be advisable to symmetrise inertia matrix to improve an efficiency of the inte-gration procedure.

This can be done by premultiplying Eq.(1.96) by HT . After elimination of con-straint forces (additional premultiplication of (1.96) by JT ), Eq. (1.97) reads as

JTHTMH′Jy + JTHT (MH′a + qv ′) = JTHTqa′ . (1.102)

Because of symmetric properties of inertia matrix JTHTMH′, the equations of mo-tion of a constrained multibody system (1.102) can be integrated more efficiently.

Lagrange equations of second kindMathematical model in minimal form can also be derived using Lagrange equa-

tions of second kind. With this aim in view, a kinetic energy of the system

T =

p∑

i=1

(1

2miv

Ti vi +

1

2ωT

i Iiωi

)

(1.103)

has to be determined. The expression in the bracket expresses a kinetic energy ofthe i-th body in a system [5].

By introduction of absolute position vector of the i-th body

xi = [xTTi xT

Ri]T (1.104)

and vector ti of the applied forces and torques, reduced to the i-th body mass centre

ti = [fTi lTi ]T , (1.105)

Lagrange equations of second kind are given by

d

dt

(∂T

∂y

)

−∂T

∂y= qa

gen . (1.106)

In equation (1.106), a vector of generalized applied forces qagen has the form

qagen = [q1 q2 ... qf ]

T , (1.107)


with its coordinates

qi =

p∑

i=1

tTi

∂xi

∂yi

(1.108)

while y is a vector of the system minimal coordinates.By utilizing (1.106), mathematical model in minimal form (f -dimensional ODE

system), which is equivalent to symmetrised minimal form (1.102) that is derivedfrom the full descriptor form, can be obtained straightforwardly.

Full descriptor form– 6p dynamical equations of

the free–body diagram, 6p“cartesian” coordinates x

– q kinematical constraintsequations

��

Minimal form– f equations of motion,

f generalised coordinates y

Minimal form– f * equations of motion,

f *generalised coordinates y*

Governing equation(holonomic system)

p...bodies, q...constraints, f...DOF

+ q* constraints

– q + q* constraints

– f * = 6p – (q + q*)

��

��

�

��

� ��

��

� ��

� ��

Figure 1.16: Modelling of system with additional kinematical constraints

Characteristics of mathematical model in minimal form

• Minimal number of generalised coordinates (the same number as DOF)

The coordinates may be of absolute or relative type.

• Problem-dependent mathematical modelling

A set of minimal coordinates appropriate to the problem at hand has to bechosen [15], [5].


A proper choice of coordinates gives opportunity for a more ’elegant’ mod-elling process as well as the model of a simpler mathematical structure. Ifsolution has to be found numerically, a simpler structure of the model maylead to more accurate results.

• In order to formulate equations of motion, kinematical constraints are to beintroduced and considered at the early stage of mathematical modelling.

As a consequence, if equations of motion of a system with the changed kine-matical structure (described by a different vector of the minimal coordinatesy) have to be formulated, a new relation has to be established and new equa-tions of motion completely re-derived, even if only a small change has beenintroduced [5]. This is illustrated in Fig. 1.16.

• Appropriate for forward dynamical problem. Appropriate for inverse dynam-ical problem, if only applied forces have to be determined.

– Forward dynamics

Model allows for determination of a motion of the system.

Since model is based on generalised coordinates given in minimal form,additional calculations are needed in order to determine position andorientation variables of each body in a system (additional calculationis based on the values of generalised coordinates and kinematical con-straints equations).

– Inverse dynamics

If only applied forces have to be determined, an utilization of mini-mal form model is plausible since computational procedure will not beunnecessarily burdened by the superfluous coordinates and constraintforces.

Since constraint forces do not appear in equations of motion, the equa-tions of kinematical constraints have to be used if these forces are to bedetermined.

• Computer algorithms

– Lagrangian equations of second kind: less appropriate

During generation of equation of motion, differentiation of the systemenergy terms is needed.

This procedure is not so appropriate as to be efficiently incorporatedto a computational procedure (this holds especially for the large-scalesystems).


– Newton-Euler equations and application of d’Alembert’s or Jourdain’sprinciple: reduction from full descriptor form is appropriate and leadsto very effective algorithm

– The application of Gauss’s principle: appropriate, but not widely used

• Characteristics of equations

Mathematical model in minimal form is expressed by ODE.

Theory of ODE systems is very well established and there are numerous in-tegration methods available for the particular simulation task [9]. Integrationof ODE is generally a simpler computational task than integration of DAEsystems. This is the main advantage of minimal form compared to the fulldescriptor form formulation.

However, although integration of ODE systems can be considered as a straight-forward procedure, ODE integration algorithm should be chosen with care toget a proper solution [9], [19].

BIBLIOGRAPHY 299

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[7] N. Gershenfeld.The Nature of Mathematical Modelling. University Press,Cambridge, 1999.

[8] E. Hairer, S. P. Nørsett, and G. Wanner.Solving Ordinary Differential Equa-tions I. Springer-Verlag, Berlin, Germany, 1987.

[9] E. Hairer and G. Wanner. Solving Ordinary Differential Equations II.Springer-Verlag, Berlin, Germany, 1991.

[10] J. L. Junkins and Youdan Kim.Introduction to Dynamics and Control of Flex-ible Structures. AIAA, American Institute of Aeronautics and Astronautics,Inc., Reston, VA, USA, 1993.

[11] K. Kreuzer and W. Schiehlen.NEWEUL. 1988. (in German).

[12] K. Magnus and H. H. Müller.Grundlagen der Technischen Mechanik. 1974.(in German).

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[14] Nattick. Control Toolbox Users Guide. The MathWorks, 1996.

[15] D. E. Newland.Mechanical Vibration Analysis and Computation. LongmanScientific and Technical, Essex, England, 1989.

[16] P. E. Nikravesh.Computer-Aided Analysis of Mechanical Systems. Prentice-Hall International Editions, Englewood Cliffs, New Jersey, USA, 1988.

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[19] W. Schiehlen. Multibody system dynamics: Roots and perspectives.Multi-body System Dynamics, 1:149–188, 1997.

[20] A. A. Shabana.Dynamics of Multibody Systems. John Wiley & Sons, NewYork, USA, 1989.

[21] A. A. Shabana. Flexible multibody dynamics: Review of past and recentdevelopments.Multibody System Dynamics, 1(3):189–222, 1997.

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