dynamics part one

Chapter 4

Dynamics

Dynamics is the branch of mechanics that is concerned with the study of motionand the relation between the forces and motion. The central focus of our study isthe dynamics of systems of rigid bodies and its application to technical problems.Furthermore, we are basically concerned with the computer aided dynamics of rigidbodies to give an insight into the contemporary classical dynamics from the compu-tational point of view. This should familiarise the reader (and user of this book) withthe basic concepts of todays computational dynamics realized in various programpackages [4], [5].

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 handapproaches. 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 the computational tools available within the variousprogram packages [17].

The experience shows that for an accurate and effective computation, the me-chanical and mathematical models of the given engineering problem have to beproperly established. The computational model should contain all the necessarypieces of information considering the mechanical phenomena under investigation.It also should be formulated properly to suit the computational method that is in-tended to be utilized to obtain the final solution. On the other hand, many compu-tational methods for the various kinds of problems are at the users disposal today.Among them the appropriate ones for the problem at hand should be chosen andapplied.

In this chapter our main goal is to provide the basic principles of the contempo-rary computational dynamics of rigid body systems as well as the necessary theo-retical background. The starting point is the question: What should be considered

190

CHAPTER 4. DYNAMICS 191

in establishing a proper computational model that should be successfully solved ?

4.1 Introduction

4.1.1 Issues of applied dynamicsDynamics can be classified into the 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 and thefocus of the intended dynamical analysis, the sub-domain whose approach is bestsuited to the problem at hand should be chosen . In the sequel of the chapter, anoverview of the characteristics of the sub-domains and problems of the contempo-rary dynamics is given.

Multibody dynamics

Multibody dynamics deals with the mechanical systems of interconnected rigid bod-ies that undergo large displacements and rotations [4], [5], [19]. The bodies areinterconnected by kinematical constraint elements and coupling elements. Bothvisco-elastic and inertia properties of the real technical system are discretised dur-ing the process of shaping of the systems mechanical model [18].

The mathematical modelling of the established discretised mechanical modelleads to the ordinary differential equations (ODE) (minimal form mathematicalmodels) or to the differential-algebraic equations (DAE) (mathematical models indescriptor form) [5].

The concepts of the multibody dynamics can be successfully utilized withinthe framework of the following technical applications: vehicle systems, aircraftsubsystems, robotic systems, various kind of mechanisms, biomechanical systems,mechatronics.

Structural dynamics

Structural dynamics deals with the deformable mechanical structures whose seg-ments generally do not undergo large displacements and rotations (not kinematicalchains). The mass and visco-elastic properties of the system are distributed alongthe structure [18].

The basic mathematical modelling generally leads to partial differential equa-tions (PDE). The discretisation of the system that is usually performed in the sequelof the mathematical modelling procedure yields a mathematical model in the formof ODE. By using finite element approach [1], very powerful computational proce-


dures are available for tackling the problems of structural dynamics, see also chapter5.

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 [19].

Flexible multibody dynamics typically deals with non-linear structures whosesegments undergo large rigid body motion superposed by flexible deformations[10]. Modelling and computational procedures of the multibody dynamics andstructural dynamics are being combined in order to formulate efficient proceduresfor problems of this kind. The methods of the flexible multibody dynamics aresubjects of extensive ongoing research activities [5].

The applications of flexible multibody dynamics systems can be found in vari-ous multibody systems with connected rigid and flexible segments like aircraft ro-tary wings, flexible robots, biomechanical systems, high-speed mechanisms.

Problems of dynamics

In dynamics various classes of problems can be distinguished.

Inverse dynamicsThe inverse dynamics deals with the determination of the applied and con-straint forces and torques for a mechanical system whose motion is prescribed[16].Beside the full dynamic approach, in which all the forces of the systemare considered in the computation, the quasi-static approach of the inversedynamics can be applied. Within the framework of the quasi-static approach,the inertia forces of the system are neglected.In most of the cases, an inverse dynamics problem leads to a set of algebraicequations.

Forward dynamicsThe forward dynamics deals with the determination of the motion of the sys-tem that is subjected to prescribed applied forces and torques [4].In most of the technical applications (systems bodies undergo large displace-ments and rotations, coupling elements of the system possess non-linear char-acteristics), a forward dynamics problem leads to solving non-linear ordinary


Dynamics

Dynamics of MBS

Dynamial behaviour(stability tests) Inverse dynamics Forward dynamics Optimisation

Structural dynamics

Dynamical modelling

Deriving dynamical equations

NewtonEulerapproach

Lagrange approach,Jourdains 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

analysisForwarddynamics

Forwarddynamics

Figure 4.1: Issues of applied dynamics

differential equations. Depending on the formulation of the mathematicalmodel, additional algebraic equations may be imposed on the system.

VibrationsIn most of the cases in linear domain [13] [15], solving the vibrational prob-lem leads to the determination of the system eigenvalues and modes (Chapter5). The system stability problem can also be mentioned in this context.


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

OptimisationThe problem of the optimisation of the mechanical systems (weight, costs,deformation and stresses, dynamical trajectories etc.) is very important inengineering and lies far out of the scope of this book [2], [6].However, it can be stated that specialised methods that allow for the optimi-sation of the mechanical systems according to the specified criteria may beapplied. Another possibility is to look for an improved design via repeatedsimulations and variations of the systems parameters.

In Figure 4.1 issues of applied mechanics, problems and solving methods ofsolution are depicted schematically.

4.1.2 Modelling of Mechanical SystemsThe modelling of mechanical systems has two major steps that are illustrated inFigure 4.2.

Figure 4.2: Steps of modelling

The first step is the mapping of the reality (technical system) into a set of simpli-fied entities in order to establish a mechanical model [18]. The mechanical modelshould include at least the effects under consideration, but not more, i.e. the modelsshould be as complex as necessary but as simple as possible (A. Einstein: Ev-erything should be made as simple as possible, but not simpler). The mechanicalmodelling is not an unique but an iterative process. It needs a lot of engineering


experience since proper analogies between the reality and the model, dependent onthe goals of the analysis, have to be established [5].

Once the mechanical model is built, in the second step a mathematical model, i.e. a set of the governing equations which describe the models dynamical behavior,has to be formulated [7]. The mathematical modelling is also not an unique process.It depends on the goals of the analysis and the computational procedures and toolsas well 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 characteristicsof the real objects are important, but only within the scope of the given task andintended analysis.

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

The crucial modelling criterion is that the mechanical model should be able todescribe (take into account) those mechanical properties of the real system that areunder the consideration with the desired accuracy [16], [18].

Mathematical modelling

Mathematical modelling is a process of formulating a mathematical text (a set ofthe equations of motion, for example), referred to the established mechanical modelby following physical laws and principles (Newtonian classical mechanics, smoothor non-smooth theory).

A good and effective mathematical model has to reflect the type and characterof the analysis that is to be performed (linear or non-linear analysis, for example),but also has to be properly formulated to suit the computational procedures andalgorithms that are intended to be used for the manipulation and evaluation of thegenerated equations [7].

In some special cases, the solution of the established mathematical model maybe found analytically, where 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 the numerical solutions(Chapter 4.3.1).

In the past three decades numerous computational techniques and algorithms


have been established to generate the governing equations for the 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 the various program packages that are offered in the market today [18], [19].

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

4.2 Mechanical ModellingAs it was mentioned in Chapter 4.1.2, the mechanical modelling is a process of themapping of the reality to a set of simplified elements. The established set of theelements (mechanical model) has to be able to describe those mechanical propertiesof the real system which influence the dynamical phenomena under consideration.

Given the goals of the analysis and characteristics of the real system whose dy-namical behaviour has to be investigated, a first step toward establishing a propermechanical model is a decision whether the system is to be modelled as the multi-body system or the modelling principles of structural dynamics are to be applied[18], [10].

Many technical systems consist of the large number of bodies interconnected bythe constraint elements such as the joints, bearings, springs, dampers or actuators.These systems can be successfully modelled as multibody systems. It can be statedgenerally that if the bodies in the system undergo large motion and small vibrations,a very powerful tool is the modelling using the multibody system approach [19].

If the multibody system concept is adopted for modelling purposes, a real sys-tem will be discretised by means of the elements that will be reviewed in the sequelof the chapter. The discussion will be confined to the modelling principles of theclassical multibody dynamics (the models are established as the systems of inter-connected rigid bodies) and flexibility of the segments is not considered. Somephenomena of the dynamic behaviour of the elastic bodies are addressed in Chapter5.

Since the kinematical structure of the system determines its characteristics to agreat extent, the types and the character of the kinematical constraints and the waythey determine the behaviour of the system will be discussed in detail. The classifi-cation of the forces that appear in the multibody systems will be also overviewed.

Another part of the mechanical modelling is the idealised description of the realload. It may be introduced in the model as concentrated forces and moments aswell as forces and moments distributed over line, surface or volume [18], [1]. Anappropriate modelling of the load is also dependent on the particular task and the


established model itself.Once the mechanical model is established, the corresponding mathematical model

has to be formulated.

4.2.1 Elements of Multibody SystemsAs it is depicted in the Fig. 4.3, multibody systems consist of elements with iner-tia and constraint elements and coupling elements without inertia [18]. Inertia isrepresented by 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 the system caused by this type of the actuators is generally notknown. It is a subject of the forward dynamic analysis of the system.

Actuators that prescribe the motion of the system, i.e. prescribe the particulardisplacements or rotations of the systems bodies as the functions of time(displacement actuators).The forces imposed by the actuators of this type are generally not known.These forces are the subject of the inverse dynamic analysis of the system[16].Since these actuators prescribe the systems motion (the system is constrainedto evolve in time in the specific way), the actuators of this type can be con-sidered as the kinematical constraints. Consequently, the forces imposed bythem are classified as the constraint forces (see classification of forces andkinematical constraints in the sequel of the chapter). The actuators of thistype are also called kinematical drivers.

4.2.2 System forcesThe forces that appear in the multibody systems can be classified into the categoriesas discussed in the sequel [12], [18]. The classification of the sytem forces is im-portant due to the fact that the different types of forces play a different role inthe process of establishing of the mathematical model of the system (see Chapter4.3.2).


Passive elements

mass point

spring

damper

rod

support, bearings, joints

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

Coupling elements Constraint elements

P

Coupling elements Constraint elements

Active elements

rigid body withnodal points Pi andcenter of gravity C

Figure 4.3: Elements of multibody system

External and internal forcesThis classification is based on the individual choice of the systems boundary.The external forces act from the outside of the boundary.The internal forces act inside the boundary of the system. The internal forcesalways appear in pairs.

Applied and constraint forcesThe applied forces are forces imposed on the system by the coupling elementsas well as forces which can be described by physical laws. They influence theway how the system evolves in time (as well as the system constraint forces).Some examples of applied forces are: gravity force, force actuators, springs,dampers, forces due to the magnetic field, etc.


The constraint forces are imposed to the system by the kinematical constraintelements (the joints, bearings, actuators that prescribe motion of the system).In the case of the ideal constraints, these forces are collinear to the directionof the restricted motion (see discussion on the kinematical constraints below).They influence the possible motion of the system.

In Figure 4.5 system forces that appear in double pendulum are analysed and clas-sified by deriving the free body diagram of the system.

4.2.3 Kinematical constraintsThe kinematical constraints are the mechanical entities that are imposed by thejoints, bearings and the system prescribed motions (kinematical drivers) [21]. Theyrestrict the system motion and reduce its degrees of freedom and are represented bythe equations that describe the kinematical restrictions imposed on the system.

The kinematical constraints are independent if these equations are linearly inde-pendent [16], [21] (the number of the independent kinematical constraints is equalto the number of the linear independent equations between the constraint forces,rank of the matrix in the equation (4.82)).

The kinematical constraints can be independent of time (scleronomic constraints)or can prescribe the motion of the system as a function of time (rheonomic con-straints).

If the kinematical constraints are represented by equations comprising only dis-placements and rotations i.e. the constraints are at the position level since there areno velocities or accelerations in the equations, the constraints are called holonomicconstraints. If the constraint equations are at the velocity level (containing timederivatives of position coordinates) which can be directly transformed by integra-tion into the position level, they are also holonomic.

If the constraint equations are at the velocity level and they can not be directlytransformed into the position level, they are called non-holonomic constraints [18].

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 of theconstraint 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 degree of freedom

In the case of a totally unconstrained free system of p rigid bodies, the degreesof freedom (DOF) of the system are 6p. It stems from the fact that 6p independentcoordinates are necessary to describe the kinematical configuration (the position


and orientation of the systems bodies) uniquely [4]. In Table 4.1 typical constraintelements and their characteristics are depicted.

Table 4.1 Types and valency of bearings

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

If all q constraints are independent, the degrees of freedom of the system aref = 6p q.

If only r of the q constraints are independent, the degree of freedom of thesystem is f = 6p r. The number r of independent constraints is equal tothe rank of matrix Q in equation (4.82).

If the system possesses f DOF, there are f independent coordinates necessary todescribe the configuration of the system uniquely. These coordinates are calledgeneralised coordinates and can be choosen in different ways appropriate to theparticular problem. The choice of a set of generalised coordinates may strongly in-fluence the process of mathematical modelling as well as the process of solving theequations (see Chapter 4.3). Degree of freedom of double pendulum is determinedin Figure 4.4.

F

Figure 4.4: Degree of freedom of mechanical system: double pendulum

Types of mechanical systems

The mechanical systems can be classified in terms of the number of its DOF andhow the imposed kinematical constraints are arranged (Figure 4.6 and Figure 4.7).


systemsboundary

constraintexternal internal applied

Force

Figure 4.5: Forces in mechanical system: a double pendulum

Statical determinationIf all q constraints are independent, the system is a statically determined. Onthe other hand, if only r of the q constraints are independent, than n= q rconstraints are superflous. The system is statically n times overdetermined.In this case the constraint forces can not be calculated without introducingfurther modelling assumptions (elastic properties). In Figure 4.6 and Figure4.7, the systems 4.6 d) and 4.7 d) are statically overdetermined.


(Slider)

(Pendulum)

a)

b)

c)

d)

e)

y

x

Figure 4.6: Systems with various constraints and DOF: beams with various supports

Kinematical determinationA system is kinematically determined, f = 0, if the displacements and rota-tions of all its members are completely determined by the constraints.If all kinematical constraints do not depend on time, the system is a staticalone. The kinematical constraints that do not depend on time are called sclero-nomic constraints [3]. In Figure 4.7, the systems c) and d) are kinematicallydetermined (statical systems).Otherwise, if at least one constraint is dependent on time, the system doesnot have a fixed configuration but evolves in time and can be considered as akinematical or dynamical system. The kinematical constraints that depend on


statically and kinematicallydetermined support

statically overdeterminedsupport

y

x

a)

b)

c)

d)

Figure 4.7: Systems with various constraints and DOF: structures and mechanisms

time are called rheonomic constraints [3].If the kinematical configuration of the system is not fully constrained by thekinematical constraints, i.e. f = 6p r > 0, the system has f degrees offreedom. All examples presented in Fig. 4.6 as well as the examples a) andb) in Fig. 4.7, are kinematically undetermined.Note: mechanisms are kinematically undetermined f => 0 as long as theirmotion is not prescribed. For example, the well known four-bar linkageposses 1 DOF if there is no rheonomic constraint which determines its motion


(if so,f= 0).

Types of statical and dynamical analysis

Depending on the kinematical structure of the mechanical system (Figure 4.6 andFigure 4.7), different kinds of analysis can be undertaken.

Kinematically determined systemIn the case of the kinematically determined system, a static analysis (Chapter3) or an inverse dynamic analysis can be performed (it depends on whetherthe system is a statical one or its structure evolves in time).In both cases the geometrical configuration of the system is not dependent onthe applied forces that are imposed on the system. The motion of the systemis completely defined by the kinematical constraints [16].

Kinematically undetermined systemIf the system is not fully kinematically constrained but posseses f degree offreedom, the time evolution of the systems kinematical configuration is notfully determined by the kinematical constraints and it is dependent on thesystems applied forces [4]. To determine the systems motion, the forwarddynamical analysis must be performed.If the systems constraint forces are of the interest, they can be calculatedduring forward dynamic analysis or subsequently after the motion of the sys-tem is determined (depending on the formulation of the systems governingequations).

4.3 Mathematical modelling

4.3.1 Introduction to mathematical modellingBefore formulating of the governing equations of multibody systems, we surveydynamics of particles and rigid bodies based on the laws of classical mechanics.The vectorial entities like displacements, velocities, forces and torques, possesinga magnitude and direction, come into account represented by vectorial componentsor by the scalar magnitudes (coordinates) related to the vector bases.

In computational mechanics vector entities are represented by arranging the co-ordinates in one-dimensional arrays called matrices or in a sloppy manner vec-tors. Similarly, tensors are arranged in multi-dimensional arrays. The aim ofadopting matrix representations is to allow for performing required vector/tensor


operations by using operations with matrices that can be easily utilised in computerapplications.

Starting from the classical vector representation, the matrix equations will bederived.

Dynamics of particles

x

f

r

vOy

z

Figure 4.8: Motion of particle

By applying Newtons second law the equation of motion of a particle depictedin Figure 4.8 can be written [20]

f = m

r = m

_

v = ma =

d

dt

(mv) ; (4.1)

where mv is the linear momentum of a particle. The angular momentum of a parti-cle with respect to O is

h

O

= r mv : (4.2)EQUATIONS OF MOTION OF SYSTEM OF PARTICLES

If the system of p particles shown in Figure 4.9 is considered, the Newtons lawfor the i-th particle yields

f

i

= f

e

i

+ f

i

i

= m

i

a

i

: (4.3)

where fi

e denotes the resultant of the external forces acting on the i-th particle andthe resultant of the internal forces f

i

i is given by the equationp

X

j=1

f

ij

= f

i

i

: (4.4)


rp

f i12f1e

f2e

fpe

f i21

f i1p f ip1

f i2p

f ip2

r2

r1

x

Oy

z

Figure 4.9: System of p particles

In the equation (4.4), fij

are the system internal forces which acts between thebodies i and j and according to the Newtons third law [12] it is

f

ij

= f

ji

: (4.5)When summing up over the entire system of p particles, it can be written

p

X

i=1

f

e

i

+

p

X

i=1

f

i

i

=

p

X

i=1

m

i

a

i

;

p

X

i=1

f

i

i

= 0 ; (4.6)

p

X

i=1

f

e

i

=

p

X

i=1

m

i

a

i

: (4.7)

The equation (4.7) can be elaborated by utilizing the relations (4.8 - 4.11). Theposition r

C

of the system mass centre C is defined by

mr

C

=

p

X

i=1

m

i

r

i

: (4.8)

where m =P

p

i=1

m

i

is the total mass of the system. Differentiation of (4.8) withrespect to time leads to the linear momentum of the system of particles

mv

C

=

p

X

i=1

m

i

v

i

: (4.9)

The second differentiation of (4.8) yields

ma

C

=

p

X

i=1

m

i

a

i

; (4.10)


which can be introduced into (4.7)p

X

i=1

f

e

i

= ma

C

: (4.11)

According to (4.11), the mass centre of the system of particles moves as if theentire mass of the system were concentrated at that point and all the external forceswere applied there [20].

ANGULAR MOMENTUM OF SYSTEM OF PARTICLES

firAi

vi

vA

ith particle,

x

Oy

z

Figure 4.10: Angular momentum of i-th particle

The angular momentum of the i-th particle about an arbitrary moving point A(Figure 4.10) is

h

Ai

= r

Ai

m

i

v

i

(4.12)and differentiation leads to

_

h

Ai

=

_

r

Ai

m

i

v

i

+ r

A

i

m

i

_

v

i

: (4.13)If A coincides with the fixed point O ( _r

A

i

v

i

= v

i

v

i

= 0), then it can bewritten

_

h

Ai

= r

Ai

m

i

_

v

i

= r

Ai

f

i

; (4.14)or

_

h

Ai

= l

Ai

; (4.15)where l

Ai

is the resultant torque with respect to A (Figure 4.10).The angular momentum of the system of particles about the moving point A is

h

A

=

p

X

i=1

r

Ai

m

i

v

i

: (4.16)


After differentiation of (4.16) with respect to time and some algebraic operationsand substitutions, it can be shown that equation (4.13) for the system of particleshas the form

_

h

A

=

_

r

AC

mv

C

+

p

X

i=1

r

Ai

f

e

i

; (4.17)

where rAC

is the position vector of the mass centre of the system of particles withrespect to point A.

When point A coincides with the fixed origin O ( _rAC

=

_

r

C

= v

C

, vC

mv

C

=

0 ) or point A coincides with the centre of mass C, equation (4.17) reduces to

_

h

A

=

p

X

i=1

r

Ai

f

e

i

: (4.18)

So, if A coincides with C, equation (4.17) can also be written as

_

h

C

=

p

X

i=1

r

Ci

f

i

e

; (4.19)

or_

h

C

= l

C

; (4.20)where

l

C

=

p

X

i=1

r

Ci

f

i

e (4.21)

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 uniquely de-scribed by the Newtons equation (4.3). The introduction of an angular momentum(Eq. 4.13) does not bring any new information into account. It has been introducedhere as a pre-stage to dynamics of rigid body, where the consideration of angularmomentum leads to the essential Eulers equation.

Dynamics of rigid body

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

BASIC KINEMATICAL RELATIONSIn Figure 4.11 the following notation is used:

! angular velocity of a body,r

A

position vector of the point A (body-fixed reference point),(x; y; z) inertial coordinate system k,(x

0

; y

0

; z

0

) coordinate system fixed to the body k0.


rA

vA

zy

x

rC

rC

r

r

x

Oy

z

Figure 4.11: Rigid body

Since the body (Figure 4.11) is rigid, r 0 is a body-fixed position vector and doesnot change its magnitude but only its orientation due to the rotation of body. Its timederivative with respect to the inertial system k can be expressed as

_

r

0

= ! r

0

: (4.22)With the previously introduced notation, r can be written as

r = r

A

+ r

0

and the velocity is obtained as

v = v

A

+

_

r

0

= v

A

+ ! r

0

: (4.24)The orientation of a body in the inertial coordinate system can be determined via

the Euler angles '; #; that specify the orientation of the fixed body system k0 withrespect to the inertial system k. Other possibilities to describe the orientation ofthe body-fixed system include Bryant (cardan) angles, Euler parameters, Rodriguezparameters etc. [3], [16].

The relation between ! and the derivatives of the Euler angles _xR

= [ _'

_

#

_

]

T

can be expressed in matrix form

! = H

R

_

x

R

: (4.25)LINEAR MOMENTUM

The mass of a body is given by

m =

Z

m

dm ; (4.26)


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

r

C

=

1

m

Z

r dm : (4.27)

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

r

AC

= r

C

0

=

1

m

Z

r

0

dm : (4.28)

From (4.23), the linear momentum of the body can be written in the formZ

m

(v

A

+ ! r

0

) dm = v

A

Z

m

dm+ !

Z

m

r

0

dm ; (4.29)

orZ

m

(v

A

+ ! r

0

) dm = v

A

m+ ! mr

C

0

= m(v

A

+ ! r

C

0

)

= mv

C

= m

_

r

C

: (4.30)ANGULAR MOMENTUM

The absolute angular momentum of a body with respect to O (origin of theinertial coordinate system) is determined by [3]

h

O

=

Z

m

r

_

r dm (4.31)

or after introducingr = r

A

+ r

0

; (4.32)

h

O

=

Z

m

(r

A

+ r

0

) (v

A

+ ! r

0

) dm

= r

A

(v

A

+ ! r

C

0

)m+ r

C

0

v

A

m+

Z

m

r

0

(! r

0

) dm : (4.33)

The termR

m

r

0

(! r

0

) dm can be written in the form

Z

m

r

0

(! r

0

) dm =

Z

m

(r

0

2

E r

0

r

0

) dm !

= I

A

! ; (4.34)


where IA

is the inertia tensor of the body with respect to A

I

A

=

Z

m

(r

0

2

E r

0

r

0

) dm ; (4.35)

E is the unit vector and r0r0 denotes dyadic product.Finally, if the bodys centre of mass C is chosen as the reference point A (r

C

0

=

0 ; vA

! v

C

; rA

! r

C

) and equation (4.34) is taken into account, (4.33) becomes

h

O

= r

C

v

C

m+ I

C

! : (4.36)

EQUATIONS OF MOTION OF RIGID BODYThe Newtons equation determines dynamics of the bodys translational motion

[12]d

dt

(mv

C

) = f ; (4.37)where:mv

C

is the linear momentum of the rigid body,f is the resultant of all forces acting on the body.

If the mass of the body is constant (dm=dt = 0), the equation (4.37) becomes

ma

C

= f : (4.38)

The Eulers equation determines the dynamics of the bodys rotational motion

_

h

O

= l

O

; (4.39)

where:h

O

is the absolute angular momentum with respect to thefixed reference point O in the inertial space,

l

O

is the resultant torque with respect to O.The result of the derivative of equation (4.36) with respect to time is

_

h

O

= v

C

v

C

m+ r

C

ma

C

+ I

C

_

! + ! I

C

! ; (4.40)

and since vC

v

C

m = 0 ,

_

h

O

= r

C

ma

C

+ I

C

_

! + ! I

C

!: (4.41)

By substituting equation (4.41) into equation (4.39) and by considering equation(4.38), the Eulers equation can be written in form

I

C

_

! + ! I

C

! = l

O

r

C

f = l

C

; (4.42)


or in shortI

C

_

! + ! I

C

! = l

C

: (4.43)NEWTON-EULER EQUATIONS IN MATRIX FORMBy following the rules of matrix algebra, the vector-valued equations (4.37) and(4.42) can be written in the matrix form [16]

ma

C

= f ; (4.44)

I

C

+

~

!I

C

! = l

C

: (4.45)The matrix stands for the bodys angular acceleration

=

_

! (4.46)

and the vector product is performed using the skew-symetric matrix ~!.The matrix equation (4.44) is derived from the non-coordinate expression (in-

variant form) (4.37) by using the inertial coordinate system k. On the other hand, thematrix equation (4.45) is derived from invariant form (4.43) by using the body-fixedcoordinate system k0. By using a body-fixed coordinate system the components ofthe inertia tensor of the body with respect to C remain constant. This is very con-venient from the computational point of view.

4.3.2 Mathematical models and proceduresMathematical models

As a result of the mathematical modelling via different methods for the formulationof the governing equations, the two basic forms of the mathematical models can bedistinguished: 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 [5].

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 freebody diagram, 6pcartesian 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 4.12: 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: closed form solu-tion and numerical (approximate) solution. If the methods for obtaining numericalsolution have to be applied (this is the case for the most technical applications), thiscan be done using either symbolic or numerical approach to computational proce-dures [4].

Closed form solutionSearching for the closed form solution pays off if there are indications thata solution of the established mathematical model can be found by using pureanalytical methods (the result is expressed in the form of functions). Thissolution is exact under the assumptions that have been made during the me-chanical and mathematical modelling of the system (the obtained solutionwould be free of numerical errors of any kind).


Unfortunately, in most of the cases (except for some linear models and thesimpler tasks of small dimensionality), it is not possible to find the closedform solution and a numerical procedure has to be applied to obtain the so-lution of the model (the numerical procedure may be launched immediatelyafter the mathematical model is established, or some symbolic manipulationsand simplifications can be performed prior to the numerical calculations).

Symbolic approachThe symbolic mathematical operations consist of the manipulations with themathematical entities without assigning their numerical values.If a computational tool gives possibilities for the 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 the mathematical model using symbolic formalismsis established, it can be used for repeated numerical calculations e.g. in nu-merical integration schemes [11].However, the extent to which the efficiency may be improved using the sym-bolic tools is dependent on the task (mathematical model) at hand. The sym-bolic manipulations are often computationally more costly than the numericalprocedures and for some types of problems very efficient numerical proce-dures can be utilized e.g. the sparse matrices techniques).Although symbolic procedures are much in use in todays computation, thedesign and implementation of symbolic algorithms are the topics of the on-going research activities.

Numerical approachBy using this approach, the numerical values are assigned to the symbolicitems as soon as the mathematical model is established and the whole com-putational process deals with the numerical values.The majority of the computational packages on todays market are numeri-cally oriented, especially packages and tools that are designed for a generaluse [4].

4.3.3 Formulation of governing equations of mechanical systemsWhen performing dynamic analysis of a given mechanical system, the 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.).


The derived mathematical model serves as a basic set of equations by meansof which the systems motion and constraint forces can be determined [18]. In themost of the cases, the basic set of equations will have to be manipulated further tosuit the intended analysis and computational procedure.

As it was already explained, an output of the different formalisms consists ofmathematical models shaped in different forms which require different numericalprocedures and algorithms in order to obtain the final solution [4].

In the sequel of the chapter the methods for formulating the governing equa-tions of mechanical systems, that are most commonly used in the computationaldynamics today, are given briefly. The main characteristics of each method as wellas the application properties are provided concisely. The computational procedures,appropriate for handling specific tasks and based on the formulation methods givenbelow, are discussed in Chapter 4.4. In Chapter 4.5 some illustrative examples aregiven.

Multibody systems of free bodies

Prior to the investigation of constrained mechanical system, a system of free bodiesis considered to prescribe the nature of underlying dynamics. The multibody systemof free bodies, shown in Figure 4.12, is a mechanical system of rigid bodies whosemotion is not constrained by kinematical constraints of any kind. Therefore, if thesystem consists of p bodies, it posseses 6p degrees of freedom (DOF) [19]. The

l1

f i2p

f ip2

f1e

1

f2el2

2

fpe

lp

f i1pf ip1

p

f i12

f i21x

O

yz v1

vp

v2

Figure 4.13: Free-body diagram of multibody system of free bodies

determination of the absolute position and orientation of the i-th body of the system


is given by the vector of the body mass centre e.g. expressed in the inertial Cartesiancoordinate system (other coordinate systems can also be chosen)

x

Ti

= [x

i

y

i

z

i

]

T

; (4.47)and e.g. the Euler angles of the bodys absolute orientation

x

Ri

= ['

i

#

i

i

]

T

: (4.48)(Note: In Eq. (4.46), (4.47) and subsequent text the index C is omitted sincecentre of mass will always be reffered to describe a position of the body) By group-ing equations (4.46) and (4.47) together, the body absolute position vector can beintroduced in the form

x

i

= [x

T

Ti

x

T

Ri

]

T

: (4.49)Newton-Euler equations of i-th body

The Newton-Euler equations are basic equations of rigid body dynamics, seeChapter 4.1. The Newton equation determines dynamics of the bodys translationalmotion, while the bodys rotational motion is determined by the Euler equation [16].

The Newton equation is given by

m

i

a

i

= f

i

; (4.50)or in the matrix form

m

i

x

Ti

= f

i

: (4.51)The Euler equation is expressed by

I

i

_

!

i

+ !

i

I

i

!

i

= l

i

; (4.52)or, following the rules of the matrix algebra,

I

i

i

+

~

!

i

I

i

!

i

= l

i

; (4.53)where the bodys angular acceleration is given by the equation

i

=

_

!

i

: (4.54)The relation between the bodys angular velocity !

i

and the time derivatives of Eulerangles x

Ri

= ['

i

#

i

i

]

T

, by means of which the absolute orientation of the bodyin the inertial coordinate system is specified, can be given in the form [3]

!

i

= H

Ri

_

x

Ri

; (4.55)and the differentiation with respect to time using the chain rule yields

i

= H

Ri

x

Ri

+

i

: (4.56)


In equation (4.55), all terms in which the second derivative appear linearly are ex-pressed in the product H

Ri

x

Ri

and all others are grouped in i

[18]. By taking intoaccount equations (4.55), equation (4.52) can be written in the form

I

i

H

Ri

x

Ri

+ I

i

i

+

~

!

i

I

i

!

i

= l

i

: (4.57)Futhermore, the equations (4.50) and (4.56) can be grouped together to form the

Newton-Euler equations of the i-th body in the matrix form:

m

i

E 0

0 I

i

E 0

0 H

Ri

x

Ti

x

Ri

+

0

I

i

i

+

~

!

i

I

i

!

i

=

f

i

l

i

; (4.58)

or in shortM

i

H

i

x

i

+ q

v

i

= q

a

i

: (4.59)The dimensions of the matrices in equation (4.58) are

dim[M

i

] = 6 6; dim[H

i

] = 6 6; dim[

x

i

] = 6 1 ;

dim[q

v

i

] = 6 1; dim[q

a

i

] = 6 1 :

Newton-Euler equations of p bodiesBy formulating equation (4.58) for each body in the system (i = 1:::p), the

Newton-Euler equations of the multibody system of free bodies [16] can be ob-tained in the form:

MH

x + q

v

= q

a

; (4.62)where the matrices are specified as follows

x = [x

T

1

x

T

2

::: x

T

p

]

T

; dim[x] = 6p 1 ; (4.63)

M =

2

6

6

6

6

6

6

6

6

4

m

1

E 0 0 0 0 0

0 I

1

0 0 0 0

0 0 m

2

E 0 0 0

0 0 0 I

2

0 0

0 0 0 0 m

p

E 0

0 0 0 0 0 I

p

3

7

7

7

7

7

7

7

7

5

; dim[M] = 6p 6p ; (4.64)

H =

2

6

6

6

6

6

6

6

6

4

E 0 0 0 0 0

0 H

R1

0 0 0 0

0 0 E 0 0 0

0 0 0 H

R2

0 0

0 0 0 0 E 0

0 0 0 0 0 H

Rp

3

7

7

7

7

7

7

7

7

5

; dim[H] = 6p 6p ; (4.65)


q

v

=

2

6

6

6

6

6

6

6

6

4

0

I

1

1

+

~

!

1

I

1

!

1

0

I

2

2

+

~

!

2

I

2

!

2

0

I

p

p

+

~

!

p

I

p

!

p

3

7

7

7

7

7

7

7

7

5

; dim[q

v

] = 6p 1 ; (4.66)

q

a

=

2

6

6

6

6

6

6

6

6

4

f

1

l

1

f

2

l

2

f

p

l

p

3

7

7

7

7

7

7

7

7

5

; dim[q

a

] = 6p 1 : (4.67)

In the case of the multibody system of free bodies, the Newton-Euler equations(4.59) are the equations of motion of the system.

Equation (4.59) represents 6p dimensional ODE system. It can be integrated intime for specified initial conditions x

0

;

_

x

0

to determine the systems motion (vari-ables x ; _x ; x).

Some computational issuesThe inertia matrix MH in Eq.(4.59) has non-symetrical properties which may de-crese significantly the 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 the efficiency of the procedure, it may be helpful to sym-metrise the matrix MH. This can be done by premultiplying Eq.(4.59) by HT ,which reads

H

T

MH

x+H

T

q

v

= H

T

q

a

: (4.68)In many applications the integration of (4.65) requires a less computer-power thanintegration of (4.59).

Constrained multibody systems

MATHEMATICAL MODEL IN FULL DESCRIPTOR FORMThe constrained multibody system (Figure 4.13) is a mechanical system of rigidbodies whose motion is constrained by kinematical constraints. If the system con-sists of p bodies whose motion is constrained by q kinematical constraints, the sys-


tem possess f = 6p q DOF [21].

f1e1

f1ek

l

2e1

l

2eh

Figure 4.14: Constrained multibody system (mechanical model)

The following notation is used:f

e

j

i

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

e

j

i

: : : j-th (j = 1:::h) applied external torque that acts on the i-th body (i = 1:::p).In Figure 4.13 this is ilustrated at the body i = 1.

Forces in constrained multibody systemApplied forces

In Figure 4.14, the free-body diagram of constrained multibody system is de-rived. The resultant applied force that acts on the i-th body is

f

i

= f

e

i

+ f

i

i

; (4.69)

where the resultant force of the external applied forces, reduced to the centre ofmass C

i

, is

f

e

i

=

k

X

j=1

f

e

j

i

(4.70)

and the 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 appliedforces between the bodies i and j

f

i

i

=

p

X

j=1

f

i

ij

: (4.71)


fc

12

f1e

l

1c

l

1

f2e

l

2c

l

2

fpe

l

pc

l

p

fc

21

fi

1p

fi

p1

fi

2p fi

p2

fc

p0

Figure 4.15: Free-body diagram of constrained multibody system

The reduction of forces with different application points to one specific point im-plies the formulation of an equivalent couple of force and torque acting at this spe-cific point.

The resultant torque about the centre of mass Ci

of the applied forces andtorques that act on the i-th body (Figure 4.14) is

l

i

=

h

X

j=1

l

e

j

i

+

p

X

j=1

l

i

ij

+ l

red

i

; (4.72)

where l iij

is an internal applied torque (internal torsional spring, for example)that acts between bodies i and j and l

red

i

is the torque due to the reduction of theforces f e

i

and f ii

to the centre of mass Ci

.

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

mass Ci

(Figure 4.14), is given by the equation

f

c

i

=

p

X

j=0

f

c

ij

; (4.73)

where f cij

is a constraint force that acts between bodies i and j (i; j = 0:::p, theindex 0 stands for the body of external world).


If an index i or j is equal to zero, the constraint force is of the external type(a force due to a kinematical constraint with the external world, a bearing forexample). If none of indices i; j is zero, the constraint force is of the internal type(due to a kinematical constraint that restricts relative motion of the bodies, revolutejoint for example).

The resultant torque about the centre of mass Ci

of the constraint forces andtorques that act on the i-th body can be given in the form

l

c

i

=

p

X

j=0

l

c

ij

+ l

c

red

i

; (4.74)

where l cij

is a constraint torque that acts between bodies i and j or a body and theexternal world (i; j = 0:::p) and l c

red

i

is a reduction torque of the forces f cij

.

Newton-Euler equations of i-th bodyAs previously mentioned, the Newton equation determines the dynamics of a

rigid bodys translational motion, while the bodys rotational motion is determinedby the Euler equation. When a body is kinematically constrained, the constraintforces and torques also influence the motion of a body and have to be considered inthe framework of the Newton-Euler equations together with the applied forces andtorques [19].

The Newton equation is given by

m

i

a

i

= f

i

+ f

c

i

; (4.75)or in the matrix form

m

i

x

Ti

= f

i

+ f

c

i

: (4.76)The Euler equation is expressed by

I

i

_

!

i

+ !

i

I

i

!

i

= l

i

+ l

c

i

; (4.77)or following the rules of the matrix algebra

I

i

i

+

~

!

i

I

i

!

i

= l

i

+ l

c

i

: (4.78)By considering (4.55), equation (4.74) can be expressed in the form

I

i

H

Ri

x

Ri

+ I

i

i

+

~

!

i

I

i

!

i

= l

i

+ l

c

i

: (4.79)After introduction of the body absolute position vector (4.60), equations (4.73) and(4.76) can be grouped together to form the Newton-Euler equations of the i-th body

m

i

E 0

0 I

i

E 0

0 H

Ri

x

Ti

x

Ri

+

0

I

i

i

+

~

!

i

I

i

!

i

=

f

i

l

i

+

f

c

i

l

c

i

;

(4.80)


or in shortM

i

H

i

x

i

+ q

v

i

= q

a

i

+ q

c

i

: (4.81)The dimensions of the matrices in equation (4.78) are

dim[M

i

] = 6 6; dim[H

i

] = 6 6; dim[

x

i

] = 6 1 ;

dim[q

v

i

] = 6 1; dim[q

a

i

] = 6 1; dim[q

c

i

] = 6 1 :

Newton-Euler equations of constrained system of p bodiesBy formulating equation (4.78) for each body in the system (i = 1:::p), the

Newton-Euler equations of the constrained multibody system can be arranged inthe form

MH

x+ q

v

= q

a

+ q

c

: (4.84)As it will be described in the sequel, the system constraint forces qc in the equation(4.79) can be expressed via kinematical constraint equations and additional param-eters.

Governing equations of constrained multibody systemsKinematical constraints equations

The Newton-Euler equations (4.79) are part of the governing equations of theconstrained multibody system. Since motion of the system bodies is kinematicallyconstrained, the components of the system position vector x are not independentbut satisfy a set of q kinematical constraint equations, which can be put in the form[16]

g(x; t) = 0 ; dim[g] = q : (4.85)By differentiation of (4.80) with respect to time, the equation that expresses therelation between the system velocities is obtained as

@g

@x

_

x+

@g

@t

= 0 ; (4.86)or in the short form

Q

_

x =

@g

@t

; (4.87)where the matrix Q is defined as

Q(x; t) =

@g

@x

; dim[Q] = q 6p : (4.88)If (4.80) is differentiated twice, the equation that expresses dependency betweensystem accelerations can be formulated. After application of the chain rule of dif-ferentiation, the kinematical constraint equations at the level of acceleration can bewritten in the short form

Q

x =

c : (4.89)


Constraint forces via kinematical constraintsIt can be shown [18] that the system constraint forces qc which are caused by

the ideal kinematical constraints (the friction is not considered, the constraint forcesq

c are orthogonal to the directions of the imposed kinematical constraints) can beexpressed via matrix Q and q unknowns

i

, (i = 1:::q) that are usually calledLagrange multipliers.

If the vector of the Lagrange multipliers is introduced in the form

= [

1

2

:::

q

]

T

; dim[] = q 1 ; (4.90)the system constraint forces can be expressed by the equation

q

c

= Q

T

: (4.91)In the context of the equation (4.86) it can be stated here, without going to thedetails, that the directions of the system constraint forces qc are expressed by thecolumns of the transposed matrix QT while the magnitudes of the constraint forcesare given by the Lagrange multipliers vector .

The matrix Q, defined in (4.83), by means of which the system constrainedforces qc are expressed in (4.86), is usually called system constraint matrix [4].The system constraint matrix is the one of the most important matrices in the do-main of dynamics of constrained multibody systems. By checking its rank it can beexamined if the system is properly constrained i.e. if all constraints imposed on thesystem are independent or some of them are superflous. The number of independentconstraints is equal to the rank of Q (see Chapter 4.2.3).

System governing equationsAfter insertion of (4.86) into (4.79), the Newton-Euler equations of the con-

strained multibody system can be written in the form

MH

x+ q

v

= q

a

+Q

T

: (4.92)Unlike the Newton-Euler equations of the free multibody system (4.14), the set ofthe equation (4.87) can not be solved and integrated in time directly since it containsq additional algebraic unknowns [16].

To make a set of the equations (4.87) complete and solvable, the kinematicalconstraint equations have to be added to the mathematical model and consideredsimultaneously with the Newton-Euler equations.

With this aim in view, the Newton-Euler equations (4.87) and q kinematicalconstraint equation (4.80) are grouped together, forming the governing equations ofthe constrained multibody systems

MH

x+ q

v

= q

a

+Q

T


g(x; t) = 0 : (4.94)The equation (4.88) is 6p + q dimensional DAE system (DAE of index 3) that

can be solved and integrated in time to obtain the motion of the system (variablesx ;

_

x ;

x) and the systems constraint forces qc = QT. For the numerical timeintegration, the specific solving procedures for DAE systems have to be used [5].

To utilize more convenient numerical procedure for the integration, the govern-ing equations of the constrained multibody systems are very often formulated by us-ing the kinematical constraint equations at the acceleration level (equation (4.84)),instead of the constraints at the position level (equation (4.80)).

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

MH

x+ q

v

= q

a

+Q

T

Q

x =

c : (4.96)The set of equations (4.88) as well as the set (4.89) represent the governing

equations of the constrained multibody system expressed in the full descriptor form[5].

As it was already explained, by integrating (4.88) or (4.89) for the specifiedinitial conditions x

0

;

_

x

0

the systems motion as well as the systems constraintforces can be determined.

Because of inherent numerical instability of DAE system presented by (4.87)and (4.88), numerical time integration of these equations is a challenging task whichhas to be treated very carefully [8].

Similarlly as it was the case with Eq. (4.59), the inertia matrixMH in Eq. (4.88)is a non-symetric one which makes time integration of (4.89) less efficient. As itwas explained, premultiplication of (4.89) by HT symmetrise the inertia matrixMH and brings (4.87) in the form

H

T

MH

x +H

T

q

v

= H

T

q

a

+H

T

Q

T

; (4.97)

which can be integrated more efficiently.

Characteristics of mathematical model in full descriptor form Basic equations: Newtons and Eulers equations, the constraint forces are

included

Absolute coordinates, 6 coordinates per body: e.g. cartesian coordinates ofthe bodys mass centers and bodys Euler angles (or other parameters)

Easy-to-obtain mathematical model, straightforward universal approach [16]


Once the mathematical model of the multibody system at hand is established,the model can be easily re-formulated if the kinematical structure of the sys-tem is changedSince the kinematical structure of the system is reflected within the govern-ing equations only through the kinematical constraints equations g(x; t) = 0and the system constraint matrix Q (equation (4.83)), just these terms haveto be changed/redefined if a new kinematical configuration of the system isintroduced (see also Fig. 4.12)

Appropriate for forward and inverse dynamical problem

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

Forward dynamicsModel allows for determination of the system motion and constraintforces simultaneouslySince model is based on the six coordinates per body, position and ori-entation of each body are automatically calculated in the course of sim-ulation.

Appropriate for computer algorithmsThe model is easy to establish by using standard matrix algebra operationsIt is suitable for implementation in general purpose multibody algorithms [4]

Characteristics of the equationsThe mathematical model in the full descriptor is expressed by DAE equationsDAE equations are generally more difficult to solve than ODE systems, con-straint violation stabilisation procedures are generally needed [22].

If it is required, the governing equations expressed in the full descriptor form can bereduced to the mathematical model in the minimal form (that represents the equa-tions of motion of the constrained multibody system) which will be described in thesequel.

MATHEMATICAL MODEL IN MINIMAL FORM

Reduction of model from full descriptor to minimal form


In order to shape the mathematical model in the minimal form i.e. to establishequations of motion of the constrained multibody system, a minimal set of coordi-nates y, dim[y] = f (f= number of DOF), have to be chosen. By means of y, thekinematical configuration of the system is uniquely described [18].

The relation between the system full descriptor absolute coordinates

x = [x

T

1

x

T

2

::: x

T

p

]

T

; dim[x] = 6p 1

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

x = f(y; t) ; (4.99)(an implicit formulation '(x; _x; t) = 0 is also possible). The corresponding equa-tion for velocities takes the form

_

x =

@f

@y

_

y +

@f

@t

= J

_

y +

@f

@t

; (4.100)

and the equation for the acceleration level reads as

x = J

y +

a ; (4.101)where the matrix J is given in the form

J =

@f

@y

; dim[J] = 6p f : (4.102)

The matrix J is usually called Jacobian matrix. It is not unique but depends on thechosen set of coordinates by means of which the system kinematical configurationis described.

It can be shown [18] that the Jacobian matrix J has a property of being orthogonal-complementary matrix to the system constraint matrix Q. The relation betweenthese two matrices can be expressed by

QJ = J

T

Q

T

= 0 : (4.103)The orthogonality between the matrices J and Q stems from the fact that these

matrices span different subspaces which are mutually orthogonal. Namely, thecolumns of the transposed constraint matrix QT as well as the Jacobian matrix Jare vectors which form the basis of the two vectorial subspaces: the q-dimensionalsubspace of the system constraint forces is spanned by Q and the f -dimensionalsubspace of the system velocities is spanned by J.

The orthogonality given by (4.93) holds only for the ideal mechanical systemsand can be briefly explained by the fact that the system velocities are always orthog-onal to the system constraints. In analytical mechanics this is related to Jourdains


principle which is based on virtual power. Other principles are dAlemberts princi-ple formulated by Lagrange on the base of virtual work and Gausss principle basedon minimal constraints. The details of the introduced vectorial subspaces as well asthe strict mathematical proof are not given here and interested reader is refered tothe literature [18], [4].

By introducing equations (4.91), (4.92) and (4.93) into equation (4.86), theNewton-Euler equations of constrained systems can be expressed via the minimalset of coordinates y [18]

MH

0

(J

y +

a) + q

v

0

= q

a

0

+Q

0

T

(4.104)where matrices that are expressed by the new set of coordinates are denoted by.

Furthermore, if the equation (4.96) is multiplied from the left side by the trans-posed Jacobian matrix JT , two important effects will be achieved:

elimination of the constraint forces by means of the orthogonality relation(4.95) (principle of virtual work),

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

In this way, it can be written

J

T

MH

0

J

y + J

T

(MH

0

a + q

v

0

) = J

T

q

a

0

; (4.105)or in the short form

M

gen

y + q

v

gen

= q

a

gen

: (4.106)In equation (4.96), the generalised mass matrix is defined as

M

gen

= J

T

MH

0

J ; dim[M

gen

] = f f ; (4.107)the vector of centrifugal, Coriolis and gyroscopic terms reads as

q

v

gen

= J

T

(MH

0

a + q

v

0

) ; dim[q

v

gen

] = f 1 (4.108)and the vector of generalized applied forces is

q

a

gen

= J

T

q

a

0

; dim[q

a

gen

] = f 1 : (4.109)The set of equations (4.98) represents the equations of motion of the constrained

multibody system (mathematical model in minimal form, f -dimensional ODE sys-tem). It can be integrated in time for given initial conditions y

0

, _y0

to obtain thesystems motion.

Since the term of constraint forces QT vanishes from the governing equationsdue to the left multiplication by JT , it is obvious that this term must not be formu-lated, if equations of motion are to be derived. Therfore, the equations of motion


(4.98) of the constrained system can be derived straightforwardly by formulatingthe matrices directly by means of (4.99), (4.100) and (4.101).

Symmetrising inertia matrixThe inertia matrix MH0J in Eq.(4.96) has non-symetrical properties which may

decrese significantly the efficiency of computation, as it was explained. Therefore,prior to elimination of constraint forces and reduction of dimenison of (4.98), it maybe advisable to symmetrise the inertia matrix to improve the efficiency of integrationprocedure.

This can be done by premultiplying Eq.(4.96) by HT . After elimination of con-straint forces (additional premultiplication of (4.96) by JT ), Eq. (4.97) reads as

J

T

H

T

MH

0

J

y + J

T

H

T

(MH

0

a+ q

v

0

) = J

T

H

T

q

a

0

: (4.110)

Because of symmetric properties of inertia matrix JTHTMH0, equations of motionof constrained multibody system (4.102) can be integrated more efficiently.

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

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

T =

p

X

i=1

1

2

m

i

v

T

i

v

i

+

1

2

!

T

i

I

i

!

i

(4.111)

has to be determined. The expression in the bracket expresses the kinetic energy ofthe i-th body in the system [3].

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

x

i

= [x

T

Ti

x

T

Ri

]

T (4.112)

and vector ti

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

t

i

= [f

T

i

l

T

i

]

T

; (4.113)

the Lagrange equations of second kind are given by

d

dt

@T

@

_

y

@T

@y

= q

a

gen

: (4.114)

In equation (4.104), a vector of the generalized applied forces qagen

has the form

q

a

gen

= [q

1

q

2

::: q

f

]

T

; (4.115)


with its coordinates

q

i

=

p

X

i=1

t

T

i

@x

i

@y

i

(4.116)

and y is a vector of the system minimal coordinates.By utilizing (4.106) the mathematical model in the minimal form (f -dimensional

ODE system), which is equivalent to the simmetrised minimal form model (4.102)derived from the full descriptor form, can be obtained straightforwardly.

Full descriptor form 6p dynamical equations of

the freebody diagram, 6pcartesian 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 4.16: Modelling of the system with additional kinematical constraints

Characteristics of mathematical model in minimal form Minimal number of the generalised coordinates (the same number as DOF)

The coordinates may be of the absolute or relative type.

Problem-dependent mathematical modellingA set of the minimal coordinates appropriate to the problem at hand has to bechosen [13], [3].


A proper choice of the coordinates gives opportunity for a more elegantmodelling process as well as the models of a simpler mathematical structure.If the solution has to be found numerically, a simpler structure of the modelmay lead to more accurate results.

In order to formulate the equations of motion, the kinematical constraintsare to be introduced and considered at the early stage of the mathematicalmodelling .As a consequence, if the equations of motion of the system with the changedkinematical structure (described by a different vector of the minimal coordi-nates y) have to be formulated, the new relation has to be established and thenew equations of motion completely re-derived, even if only a small changehas been introduced [3].

Appropriate for the forward dynamical problem. Appropriate for the inversedynamical problem, if only applied forces have to be determined.

Forward dynamicsThe model allows for determination of the motion of the system.Since model is based on the generalised coordinates given in minimalform, the additional calculations are needed in order to determine posi-tion and orientation variables of the each body in the system (the addi-tional calculation is based on the values of the generalised coordinatesand the kinematical constraints equations).

Inverse dynamicsIf only the applied forces have to be determined, an utilization of theminimal form model is plausible since computational procedure will notbe unnecessarily burdened by the superflous coordinates and constraintforces.Since constraint forces are not contained in the equations of motion, theequations of the kinematical constraints have to be used, if these forceshave to be determined.

Computer algorithms

Lagrangian equations of second kind: not so appropriateDuring generation of the equation of motion, differentiation of the sys-tem energy terms is needed.This procedure is not so appropriate as to be efficiently incorporatedto the computational procedure (this holds specially for the large-scalesystems).


Newton-Euler equations and application of dAlemberts or Jourdainsprinciple (reduction from the full descriptor form: appropriate and leadsto very effective algorithm

The application of Gausss principle: appropriate, but not widely used

Characteristics of the equationsMathematical model in the minimal form is expressed by ODE.Theory of ODE systems is very well established and there are numerous inte-gration methods at ones disposal for the particular simulation task [8]. Inte-gration of ODE is generally a simpler computational task than integration ofDAE systems. This is the main advantage of the minimal form compared tothe full descriptor 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 [8], [17].

dynamics part one

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