Multibody Dynamics Explain

8/3/2019 Multibody Dynamics Explain

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MULTIBODY DYNAMICS: BRIDGINGFOR MULTIDISCIPLINARYAPPLICATIONS

Jorge A.C. Ambr sioIDMEC Instituto Superior T cnico, Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal

Abstract Simple or complex systems characterized by large relative motions be-tween their components find in the multibody dynamics formalisms themost general and efficient computational tools for their analysis. Ini-tially restricted to the treatment of rigid bodies, the multibody methodsare now widely used to describe the system components deformations,regardless of their linear or nonlinear nature. The ease of including inthe multibody models different descriptions of the contact problems,control paradigms or equations of equilibrium of other disciplines isdemonstrated here to show the suitability of these approaches to beused in multidisciplinary applications

Keywords: Flexible multibody dynamics, contact, biomechanics, vehicle dynamics,railway dynamics, crashworthiness.

1. Introduction

The design requirements of advanced mechanical and structural sys-tems and the real-time simulation of complex systems exploit the easeof use of the powerful computational resources available today to cre-ate virtual prototyping environments. These advanced simulation facili-

ties play a fundamental role in the study of systems that undergo largerigid body motion while their components experience material or geo-metric nonlinear deformations, such as vehicles, deployable structures,space satellites, machines operating at high speeds or robot manipula-tors. Some examples of engineering and biological systems for whichthe large overall motion is of fundamental importance are exemplifiedin Fig. 1. If on the one hand the nonlinear finite element method is themost powerful and versatile procedure to describe the flexibility of thesystem components, on the other hand the multibody dynamic formula-tions are the basis for the most efficient computational techniques that

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W. Gutkowski and T.A. Kowalewski (eds.), Mechanics of the 21st Century , 6188. 2005 Springer. Printed in the Netherlands.


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Figure 1. Natural biological and artificial engineering systems for which multibodydynamics provides irreplaceable modeling methodologies.

deal with large overall motion. Therefore, it is no surprise that manyof the most recent formulations on flexible multibody dynamics and on

nite element methods with large rotations share some common features.In multibody dynamics methods, the body-fixed coordinate frames are

generally adopted to position each one of the system components andto allow for the specification of the kinematic constraints that representthe restrictions on the relative motion between the bodies. Several for-malisms are published suggesting the use of different sets of coordinates,such as Cartesian [1], natural [2] and relative coordinates [3]. Depend-ing on the type of applications, each of these types of coordinates hasadvantages and disadvantages. Due to the ease of the computationalimplementation, their physical meaning and the widespread knowledgeof their features, all the formalisms presented in this work are based onthe use of Cartesian coordinates.

he methodological structure of the equations of motion of the multi-body system obtained allows the incorporation of the equilibrium equa-tions of a large number of disciplines and their solution in a combinedform. The description of the structural deformations exhibited by thesystem components by using linear [5] or non-linear finite elements [6] in

the framework of multibody dynamics is an example of the integrationof the equations of equilibrium of different specialties. Of particularimportance for the applications pursued with the methodologies pro-

osed is the treatment of contact and impact, which is introduced inthe multibody systems equations by using either unilateral constraints


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Multibody Dynamics: Bridging for Multidisciplinary Applications 63

[7] or a continuous contact force model [8]. The availability of the statevariables in the multibody formulation allows for the use of differentcontrol paradigms in the framework of vehicle dynamics, biomechanics

or robotics and their integration with the multibody equations [9]. Thecoupling between the fluid and structural dynamics equations allows forthe development of applications, where the fluid-structure interaction isanalyzed, especially for cases with large absolute or relative rotations inthe system components, are of importance [10, 11].

The research carried at IDMEC provides the examples offered in thiswork. Application cases involving the modeling of realistic mechanisms,passive safety of road and rail vehicles, impact and human locomotionbiomechanics, automotive and railway dynamics are used to demonstratethe developments reviewed here.

2. Rigid Multibody Dynamics

A multibody system is defined as a collection of rigid and/or flexiblebodies constrained by kinematic joints and eventually acted upon bya set of internal and/or external forces. The position and orientation ofeach body i in the space is described by a position vector r and a set ofrotational coordinates pi, which are organized in a vector as [1]:

qi = [rT, pT T. 1)

According to this definition, a multibody system with nb bodies isdescribed by a set of coordinates in the form:

q = [qT1 , qT2 , . . . , q

Tnb]

T. 2)

The dependencies among system coordinates, which result from the

existence of mechanical joints interconnecting several bodies, are definedthrough the introduction of kinematic relationships written as [1]:

(q, t) = 0, 3)

where is the time variable, which is used only for the driving con-straints. The second time-derivative of Eq. (3) with respect to timeyields:

(q, q, q, t = 0 D = , 4)

where D is the Jacobian matrix of the constraints, q is the accelerationvector and is the vector that depends on the velocities and time.

The system kinematic constraints are added to the equations of mo-tion using the Lagrange multipliers technique [1]. Denoting by the


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vector of the unknown Lagrange multipliers, the equations of motion fora mechanical system are written as

M DT

D 0

q

=

f

(5)

here M is the global mass matrix, containing the mass and momentsof inertia of all bodies, and is the force vector, containing all forcesand moments applied to the system bodies plus the gyroscopic forces.

he Lagrange multipliers, associated with the kinematic constraints,are physically related with the reaction forces generated between the

odies interconnected by kinematic joints, given by [1]

c) = Tq, (6)

he usual procedures to handle the integration of sets of differential-algebraic equations must still be applied in this case in order to eliminatethe constraint drift or to maintain it under control [1, 2].

Forward Dynamics

he computational strategy used to solve the forward dynamics of

the system, represented by Eq. (5), is outlined in Fig. 2. The solutionprocedure starts by the determination of the initial positions and velo-cities of the system components. Next, the system inertia, the Jacobianmatrices, the forces and the right-hand-side of the kinematic accelera-tion constraint equations vectors, are calculated and assembled in theequations of motion. Equation (5) is then solved to find the system ac-celerations, and in the process the Lagrange multipliers. By integratingthe current velocities and the system accelerations, at time , the newpositions and velocities for time t + t are calculated by using a variable

Figure 2. Solution of the forward dynamic analysis of a multibody system.


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order, variable time-step integration procedure [1, 2, 4]. The forward dy-namics simulation proceeds until the previously set final time is reached.

The procedure outlined in Fig. 2 is used in general purpose multibody

dynamics codes, such as DAP-3D [1]. Throughout this work it is demon-strated that all engineering applications foreseen here are implemented,either by developing specific kinematic constraints or by implementingforce models in Eq. (5).

Application Example of a Roller Coaster. hen a body travelsalong a guide, not only its path has to be followed, but also its spa-tial orientation has to be prescribed, according to spatial characteristicsof the curve. The formulation adopted to implement these kinematicconstraints, using the moving Frenet frame associated with the trackcenterline based on the work by Pombo and Ambr sio [12], is outlinednext.

Prescribed Motion Constraint. he objective here is to define theconstraint equations that enforce that a point of a rigid body follows thereference path [12]. Consider a point R, located on a rigid body i, thathas to follow the specified path depicted in Fig. 3. The path is defined

by a parametric curve g(L , which is controlled by a global parameterL that represents the length travelled along the curve until the currentlocation of point R. The kinematic constraint is

pmc,3) = 0 rRi g(L) = 0, 7)

where rRi = ri +AisRi epresents the coordinates of point R with respect

to the global coordinate system (x, y, z), ri is the vector that definesthe location of the body-fixed coordinate system (, , ) , Ai is the

transformation matrix from the body i fixed coordinates to the globalreference frame, and s Ri represents the coordinates of point R with re-

Figure 3. ocal frame alignment constraint.


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spect to the body-fixed reference frame. For notational purposes ()means that () is expressed in body-fixed coordinates.

he second part of the constraint ensures that the spatial orientation

of body i remains unchanged with respect to the moving frame of the ref-erence path represented in Fig. 3. Consider that (u, u, u) representthe unit vectors associated with the axis of the body-fixed coordinatesystem (, , )i. Let the Frenet frame of the general parametric curveg(L) be defined by the principal unit vectors (t, n, b)L. At the ini-tial time of analysis, the relative orientation between the body vectors(u , u, u)i and the local frame (t, n, b)L leads to

lfac,3) = 0

nT ubT unT u

a

bc

= 0. (8)

his kinematic constraint ensures that the alignment remains constanthroughout the analysis. The transformation matrix from the body ixed coordinates to the global coordinate system is:

A = [u u u]i (9)

defining the following unit vectors as:u1 1 0 0}

T; u2 0 1 0}T; u3 0 0 1}

T. (10)

Equation (8) is now rewritten in a more usable form as:

(lfac,3) = 0

nTA u1bTA u1nTA u3

abc

= 0, (11)

hich constitutes the second part of the path following constraint.

Roller-Coaster Dynamics. Let the roaller-coaster rail be definedwith the spatial geometry described in Fig. 4. The path-following con-straint is used to enforce the vehicles to follow the rail for the prescribedgeometry.

he roller coaster vehicle consists of a train with three cars that areinterconnected by linking bars, represented in Fig. 5. The multibodymodel of the vehicle is assembled using eleven rigid bodies, corresponding

to 3 car bodies, 6 wheelsets and 2 connection bars.he complete vehicle model only has 1 d.o.f., which is the longitudinalmotion of the cars. The motion of the vehicle is guided by the dynamicsdescribed by Eq. (5). A view of the motion of the roller coaster is dis-played in Fig. 6 and the details of the analysis are found in reference [13].


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Figure 4. View of the roller coaster as used in the simulation.

igure 5. Multibody model of the roller coaster vehicle.

Figure 6. Snap shots of the roller coaster motion as observed from the second car.

Note that the study of these vehicles only requires the use of the path-following constraint. The contact forces are not explicitly used but theycan be calculated from the Lagrange multipliers associated to the pathconstraint.

Inverse Dynamics

In many applications all external forces are known and the motion ofthe system is also known. Therefore, the only unknowns are the internalforces. Let the first row of Eq. (5) be re-written as

M + DT = g (12)

z

1 2 4 5

7

x

9

6

8

10

11


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which is

DT = g M. (13)

Equation (13) emphasizes that the only unknowns of the system arethe Lagrange multipliers. The reaction forces at the joints are given by:

g c) = DT. (14)

he solution of the equations of motion in inverse dynamics can beused to solve for the internal forces of the human body, i.e., muscle andanatomical joint reaction forces, that develop for known motions.

Application to Biomechanics: Gait AnalysisFor biomechanical applications in gait a three-dimensional model, pre-

sented in Fig. 7, is used [14]. It has a kinematic structure made of thirty-three rigid bodies, interconnected by revolute and universal joints, insuch a way that sixteen anatomical segments are represented.

11

16

5

12

13

7

4

6

1

10

15

14

3

4

5

6

8

9

10

11

12

1

3

14

15

16

v2

3

3

v7

v8

v6

6

v5

v5

v4

v4

v21

v 0

11

1122

v 2

v 2

v13

v 3

v15v 4

v15

616

7

v20

v 0

v 7

v 8

v 9

14

15

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1718

19

0

21

22

23

7

6

8

45

2

3

1

10

11

12

13

1

32 25

24 26

27

28

29

30

33

Figure 7. The biomechanical model, its kinematic structure and a detail of the

ankle joint.

Joint Moments-of-Force: A Determinate Problem. o drive

the biomechanical model in the inverse dynamic analysis, joint actuatorssuch as the one represented in Fig. 8 for the knee joint, are specified. Theactuators are the kinematic constraints in which the angle between twoadjacent segments is a known function of time. These additional equa-tions are added to the system kinematic equation so that the number of


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m

m2

m3

Om1

m1 = Om2

m2=

m3

Im3

Figure 8. Joint actuator associated with the knee joint and muscle actuator.

non-redundant constraint equations becomes equal to the number of co-ordinates. Equation (13) is solved to obtain the Lagrange multipliers as-sociated with the joint actuators, representing the net moments-of-forceof the muscles that cross those joints. The inverse dynamics problem,as stated here, is totally determined.

Muscle Forces: A Redundant Problem. he solution of theinverse dynamics problem with muscle actuators introduces indetermi-nacy in the biomechanical system, since it involves more unknowns thanequations of motion. By using optimization techniques to find the mus-cle forces that minimize a prescribed objective function, a solution forthe problem is obtained. The optimization problem is stated as:

inimize F0 u )

subject to( i = 0, j = 1,...,nec,( i 0, j = ( ec + 1) , . . . , ntc,

loweri

upperi i = 1, . . . , nsv

(15)

where i are the state variables bounded respectively by lower anduupper, F0( i is the objective or cost function to minimize and f (u )are constraint equations that restrain the state variables.

he minimization of the cost functions simulate the physiological cri-

teria adopted by the central nervous system when deciding which musclesto recruit and what level of activation to obtain the adequate motion.Several cost functions have been proposed for the study of the redundantproblem in biomechanics [15]. The minimization of the sum of the cubeof the muscle stresses [16] is often used in applications involving human


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Figure 9. Lower extremity muscle apparatus.

Figure 10. Muscle forces for the hamstrings and triceps surae.


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locomotion

F0 =n a

m=1

Fml F

ml

Fm2am

3

(16)

where ma are the number of muscle actuators and is the specificmuscle strength with a constant value of 31.39 N/cm2 [17]. The humanlocomotion apparatus, represented in Fig. 9, is modeled having the mus-cles with the physiological data described in Yamaguchi [17]. The statevariables associated with muscle actuators represent muscle activationsthat can only assume values between 0 and 1.

To illustrate the type of results obtained for the muscle forces in a caseof normal cadence gait of a 50%ile male, the muscle forces for the ham-

strings and triceps surae are presented in Fig. 10.

3. Contact and Impact

Let a triangular patch, where point of the body shown in Fig. 11will impact, be defined by points i, j and l. The normal to the outsidesurface of the contact patch is defined as = ij jl . The position ofthe point k with respect to point i of the surface is

rik = rk r (17)

which is decomposed in a tangential and a normal component, given by

rtk = rik

rTikn

n; rnik =

rTikn

n. (18)

The necessary conditions for contact are that node k enetrates thefront surface of the patch, but not through its back surface, withwhich a thickness e is associated. These conditions are written as

0 rTikn e. (19)

n

jl

ir

jr

k

nk

( ) nrr iktik =

ij

l

k

k

Figure 11. Contact detection between a finite element node and a surface.


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he remaining necessary conditions for contact result from the needor the node to be inside of the triangular patch. These three extra

conditions are

rtikrijT

n 0;

rtkrjlT

n 0 and

rtikrkiT

n 0. (20)

Equations (19) and (20) are necessary conditions for contact. How-ever, depending on the contact force model actually used, they may notbe sufficient to ensure effective contact.

Unilateral Constraints

If contact between a node and a surface is detected, a kinematic con-straint is imposed. For flexible bodies let us assume a fully plastic nodalcontact, i.e., the normal components of the node k velocity and acceler-ation, with respect to the surface, are null during contact

qk = q( )k

q

( )Tk n

n; qk = q

)k

q

)Tk n

n (21)

here q( )k and q

( )k represent the nodal velocity and acceleration imme-

diately before impact respectively . The kinematic constraint implied by

Eqs. (21) is removed when the normal reaction force between the nodeand the surface becomes opposite to the surface normal, i.e.,

nk =

Tk n > 0. (22)

It should be noted that the contact force is related to the Lagrangemultiplier associated by the kinematic constraint defined by Eqs. (22).Therefore, the change of sign of the force is in fact the change of sign ofhe multiplier.

This contact model is not suitable to be used directly with rigid body

contact. The sudden change of the rigid body velocity and accelerationwould imply that the velocity and acceleration equations resulting fromhe kinematic constraints would not be fulfilled. Other forms of this

contact model can be found in the work by Pfeiffer and Glocker [7].

Continuous Contact Force Model

An alternative description of contact considers this to be a continuousevent where the contact force is a function of the penetration between

the surfaces. This leads to the continuous force contact model, proposedby Lankarani and Nikravesh [8], and briefly described here. Let thecontact force between two bodies be written as

s,i Kn + D u (23)


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where is the pseudo-penetration, is the pseudo-velocity of penetra-tion, K is the equivalent stiffness, D is a damping coefficient and u isa unit vector normal to the impacting surfaces. Using the hysteresis dis-sipation model and the equivalent stiffness, calculated for instance forHertzian elastic contact [18], the nonlinear contact force is

s,i Kn

1 +

3 1 e2

)

u (24)

where ( ) is the initial contact velocity and e is the restitution coeffi-cient. Note that K is a function of the geometry and material properties

of the impacting surfaces.

Application to Railway Dynamics The Wheel-RailContact Problem

ne of the interesting applications of multibody dynamics with con-tact mechanics is the description of the wheel-rail contact in railwaydynamics, represented in Fig. 12. The stability of the running vehicledepends ultimately on the rail-wheel contact and on the vehicle primarysuspension. Therefore, methodologies that provide accurate models torepresent the phenomena are of particular importance.

In a general case of a railway vehicle one or two points of each wheelare in contact with the rail, as shown in Fig. 12. The diametric sec-

tion that contains the wheel flange contact point makes an angle sfRwwith the diametric section that contains the wheel tread contact point.The possibility of detecting contact in different diametric sections allowspredicting derailment and it is, therefore, of utmost importance.

Let the generalized geometry of the rail and wheel be described by

generalized surfaces resulting from sweeping the rail profile along the rail

Figure 12. Two points of contact in the rail and wheel surfaces: lead contact.

FlaFlaFlaFla(LeLLL

TrTrTrTr

conconconcon

cttttt))))


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centerlines and the wheel profiles around the base circle of the wheel.In order to ensure that the search for the contact points is betweenconvex surfaces, the wheel profile is divided in treat and flange profiles.

he contact between the rail and one of the wheel surfaces is describedgenerically in Fig. 13, where the mating surfaces are represented as freesurfaces.

Figure 13. Candidates to contact points between two parametric surfaces.

he geometric conditions for contact between the convex surfaces aredefined by vector products defined between the surfaces. The first condi-ion is that the surfaces normals ni and nj at the candidates to contact

points have to be parallel. This condition means that nj has null pro-jections over the tangent vectors tu and tw:

nj ni = 0 nTj t

ui 0,

nT

j twi = 0.

(25)

he second condition is that the vector d, which represents the dis-tance between the candidates to contact points, has to be parallel to thenormal vector ni. This condition is mathematically written as:

d n = 0 dTtu = 0,

dTtw = 0.(26)

he geometric conditions (25) and (26) provide four nonlinear equationswith four unknowns, the four parameters , , s and hat define thetwo surfaces. This system of equations provides solutions for the locationof the candidates to contact points that have to be sorted out.

x


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The coordinates of the candidates to contact points are determinedby solving an optimal problem and the distance between such points iscalculated in the process. The points are in contact if

dTnj 0. (27)

hen contact is detected, the normal force is calculated using Eq. (24)and the tangential forces are evaluated using the Kalker theory, thePolach formulation or the Heuristic nonlinear creep model. It has beenfound that the Polach formulation provides the best approach for thetangent forces, and it is used hereafter [13].

The wheel-rail contact model outlined here is used to model the ML95

trainset, shown in Fig. 14, which is used by the Lisbon subway company(ML) for passengers traffic.The multibody model of the trailer vehicle of the train, developed

in the work by Pombo [13], is composed of the car shell suspended bya set of springs, dampers and other rigid connecting elements on thebogies. This assembly of connective elements constitutes the secondarysuspension, sketched in Fig. 15, which is the main one responsible forthe passengers comfort.

The connections between the bogies chassis and the wheelsets, also

achieved by another set of springs, dampers and rigid connecting ele-

Figure 14. Schematic representation of the ML95 trainset.

Figure 15. Secondary suspension model of the ML95 trailer vehicle.


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Figure 16. Primary suspension model of the ML95 trailer bogie: a) Three-dimensional spring-damper elements; b) Suspension model with springs and dampers.

Figure 17. Lift of the right wheel of the leading wheelset for vehicle forward velo-cities of 10 and 20 m/s, using the Kalker linear theory.

ments, constitute the primary suspension represented in Fig. 16. Theprimary suspension is the main suspension responsible for the vehiclerunning stability.

he simulation results of the vehicle, running in a circular track witha radius of 200 m with velocities of 10 and 20 m/s, show that the predic-tion of flange contact is of fundamental importance. Fig. 17 shows thatcontact forces obtained with the Kalker linear theory originate the lift ofhe outer wheel of the front wheelset at the entrance of the curve. De-

spite this wheel lift, derailment does not occur and the analysis proceedsup to end. Nevertheless, such results are not realistic since the existenceof flange contact involves high creepages, which makes the Kalker lineartheory inappropriate to compute the creep forces. Therefore only theHeuristic and the Polach creep force models must be considered.

Another aspect to note is that flange contact is detected with all creepforce models. Even when running at the speed of 10 m/s, where the cen-trifugal forces effect is balanced by the track cant, flange contact occurs.Lateral flange forces develop on the wheels of both wheelsets of the frontbogie as presented in Fig. 18 for a vehicle forward velocity of 10 m/s and

a)

Three-dimensional

spring-damper elements

Bogie frame Axlebox

Wheelset

b)


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using the Polach creep force model. During curve negotiation, the outerwheel of the leading wheelset and the inner wheel of the rear wheelsethave permanent flange contact.

Referring to Fig. 19, for the velocity of 10 m/s, the flange contact oc-curs on the outer and in the inner wheels of the vehicle. For the velocityof 20 m/s, only the outer wheels have flange contact. This is explainedby the fact that, when running at 20 m/s, the vehicle negotiates thecurve with a velocity higher than the balanced speed.

4. Flexible Multibody Dynamics with PlasticHinges

Many applications of multibody dynamics require the description ofthe flexibility of its components. For structural crashworthiness it is

-5000

0

5000

10000

15000

0000

5000

0 3 6 9 12 15 18 21 24 27 30

Time [s]

LateralFlangeF

orce

[N

]

Left Ws 3 (P olach)

Right W s 3 (Polach)

Left Ws 4 (P olach)

Right W s 4 (Polach)

Figure 18. Lateral flange forces on the wheels of both wheelsets of the front bogiefor a vehicle forward velocity of 10 m/s, using the Polach creep force model.

Front

wheelset

(Ws 4)

Flange

ontact

Rearwheelset

(Ws 3)Flange

contact

Figure 19. Contact configuration during curve negotiation.


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often unfeasible to use large nonlinear finite element models. The useof multibody dynamics with plastic hinges is an alternative formulationthat allows building insightful models for crashworthiness.

Formulation of Plastic Hinges

In many impact situations, the individual structural members areoverloaded giving rise to plastic deformations in highly localized regions,called plastic hinges. These deformations, presented in Fig. 20 for stru-ctural bending, develop at points where maximum bending momentsoccur, load application points, joints or locally weak areas [19]. Multi-body models obtained with this method are relatively simple, which

makes the procedure adequate for the early phases of vehicle design.he methodology described herein is known in industry as conceptual

modeling [20].

Figure 20. ocalized deformations on a beam and a plastic hinge.

he plastic hinge concept has been developed by using generalized

spring elements to represent constitutive characteristics of localized plas-tic deformation of beams and kinematic joints to control the deforma-tion kinematics [21], as illustrated in Fig. 21. The characteristics ofthe spring-damper that describes the properties of the plastic hinge areobtained by experimental component testing, finite element nonlinearanalysis or simplified analytical methods.

he plastic hinge constitutive equation can be modified to accountfor the strain rate sensitivity of some materials. A dynamic correctionfactor is used to account for the strain rate sensitivity given by [21].

P P = 1 + 0.07V0.82, (28)

here P and Ps are the dynamic and static forces, respectively, and V isthe relative velocity between the adjacent bodies. The force or moment


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AnalyticalTest

Rotation ( Rad)

Moment(kN

m)

0 .05 .10 .15 .20 .2 0

1

3

4

6

Figure 21. Plastic hinge bending moment and its constitutive relationship.

to apply due to the plastic hinge is multiplied by the ratio calculated inEq. (28) before it is used in the force vector of the multibody equationsof motion.

Application of the Plastic Hinge Approach toCrashworthiness of Surface Vehicles

The multibody of an off-road vehicle with three occupants, shown inFig. 22, is used to demonstrate the plastic hinge approach to complexcrash events. The model includes all moving components of the vehicle,suspension systems and wheels, and a tire model [16]. The biomechanicalmodels for the occupants are similar to those described in Fig. 7.

The three occupants, with a 50%tile, integrated in the vehicle areseated. Two occupants in the front of the vehicle have shoulder and lap

Figure 22. nitial position of the vehicle and occupants for the rollover.


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seat belts while the occupant seated in the back of the vehicle has noseatbelt.

he rollover situation for the simulation is such that the initial condi-

tions correspond to experimental conditions where the vehicle moves ona cart with a lateral velocity of 13.41 m/s until the impact with a water-

lled decelerator system occurs. The vehicle is then ejected with a rollangle of 23 degrees.

he results of the simulation are pictured in Fig. 23 by several framesof the animation. The vehicle first impacts the ground with its left tires.At this point the rear occupant is ejected. The rollover motion of thevehicle proceeds with an increasing angular velocity, mainly due to theground tire contact friction forces. The occupants in the front of thevehicle are held in place by the seat belts. Upon continuing its rollmotion, the vehicle impacts the ground with its rollbar cage, while theejection of the rear occupant is complete. Bouncing from the invertedposition, the vehicle completes another half turn and impacts the groundwith the tires again. The HICs for all occupants largely exceed 1000,which indicates a very high probability of fatal injuries for the occupantsunder the conditions simulated.

An experimental test of the vehicle was carried out at the Transporta-

tion Research Center of Ohio [22], being an overview of the footage ob-tained shown in Fig. 24. The outcome of the experimental test, whichis rather similar to the outcome of the simulation, is further used tovalidate the vehicle model [21].

Figure 23. Views of the outcome of the rollover simulation of a vehicle with threeoccupants.

Figure 24. View of the experimental test for the truck rollover.


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5. Flexible Multibody Dynamics with FiniteElements

General Equations of Motion of a Single BodyLet the principle of the virtual works be used to express the equi-

librium of the flexible body in the current configuration t+t and letan updated Lagrangean formulation be used to obtain the equations ofmotion of the flexible body [23]. Let the finite element method be usedto represent the equations of motion of the flexible body. Referring toFig. 25, the assembly of all finite elements used in the discretization ofa single flexible body results in its equations of motion written as [6]

Mrr Mrf MrfMr M Mf

Mf r Mf Mf f

r

u

=

grg

g f

srs

s f

00

0 0 0

0 0 0

0 0 KL + KN L

0

0

u

(29)

where r and are respectively the translational and angular accelera-tions of the body-fixed reference frame and u denotes the nodal accele-rations measured in body fixed coordinates. The local coordinate frame attached to the flexible body, is used to represent the gross motionof the body and its deformation.

Figure 25. General motion of a flexible body.

ed updated

tion

b

tt

t

0


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Linear Deformations of Flexible Bodies

In many situations it is enough to consider that the components of

the multibody system experience only linear elastic deformations. Fur-thermore, assume that the mode superposition technique can be used.hen, the flexible part of the body is described by a sum of selected

modes of vibration asu Xw (30)

here the vector w represents the contributions of the vibration modestowards the nodal displacements and X is the modal matrix. Due to theeference conditions, the modes of vibration used here are constrained

modes. Due to the assumption of linear elastic deformations the modalmatrix is invariant. The reduced equations of motion for a linear flexibleody are [5]

Mr M fX

XTMf r I

qr

w

gr

XTgf

sr

XTsf

0

w(31)

here is a diagonal matrix with the squares of the natural frequenciesassociated with the modes of vibration selected. For a more detailed

discussion on the selection of the modes used the interested reader isreferred to [5].

he methodology is demonstrated through the application to the si-ulation of the unfolding of a satellite antenna, the Synthetic Aperture

Radar (SAR) antenna that is a part of the European research satellite

Figure 26. The European satellite with the folded and unfolded configurations ofthe antenna.


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ERS-1, represented in Fig. 26. During the transportation the antenna isfolded, in order to occupy as little space as possible. After unfolding,the mechanical components take the configuration shown in Fig. 26(a).

The SAR antenna consists in two identical subsystems, each withthree coupled four-bar links that unfold two panels on each side. Thecentral panel is attached to the main body of the satellite. Each unfold-ing system has two degrees of freedom, driven individually by actuatorslocated in joints A and B. In the first phase of the unfolding processthe panel 3 is rolled out, around an axis normal to the main body, bya rotational spring-damper-actuator in joint A, while the panel 2 is helddown by blocking the joints D and E. The second phase begins withthe joint A blocked, next the panels 2 and 3 are swung out to the finalposition by a rotational spring-damped-actuator.

The model used for one half of the folding antenna, schematicallydepicted Fig. 27, is composed of 12 bodies, 16 spherical joints and 3revolute joints. The central panel is attached to the satellite, definedas body 1, which has much higher mass and inertia. The data for thisantenna is reported in the work of Anantharamann [24].

Figure 27. The SAR antenna: a) half unfolded state b) folded antenna; c) multibody

model.

In the first phase of the unfolding antenna, the rotational spring-damped-actuator is applied in the revolute joint R3. For the secondphase, the revolute joint R3 is blocked and the system is moved tothe next equilibrium position by a spring-actuator-damped positionedin joint R1. The unfolding processes for rigid and flexible models areshown in Fig. 28, only for its first phase.

The different behavior between the rigid and the flexible models is

noticeable in Fig. 28. Though not shown here, the rotational actuatormoment responsible for the start of the unfolding is not correctly pre-dicted by the rigid multibody model. Being a very light and flexiblestructure, the discrepancies, if not identified during the design stage,would lead to the failure of the unfolding process.

1.31.

a) b)b)

Panel 3 (B3) Panel 2 (B2))

c)c)c)c)


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Figure 28. First phase of the unfolding of the SAR antenna (rigid and flexiblemodels).

Nonlinear Deformations in Multibody Systems

For flexible multibody systems experiencing nonlinear geometric andmaterial deformations, the equations of motion for a flexible body aregiven by Eq. (29). However, due to the time variance of all its coefficients,Eq. (29) is not efficient for computational implementation. Instead, byconsidering a lumped mass formulation for the mass matrix and referringthe nodal accelerations to the inertial frame, the equations of motion fora single flexible body take the form of [6]

mI + AMAT AMS 0

AMT

J + STM S 00 0 Mff

r

qf

=

+ AC n T ITC

g f KL + KN L u

(32)

where the absolute nodal displacements are written as

qkf

d

k

= uk +

AT

xk + k

0 I

r

+

( k + k)

+ 2 k k

(33)

ith k being the position of node in the reference configuration. In

Eq. (32) M

is a diagonal mass matrix containing the mass of the noundary nodes,

AT [A . . . A]T , S =

x1 +

1

T. . .

xn +

n

T T


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and I [I . . . I]T where A is the transformation matrix from the bodyfixed to global coordinate coordinates and k denotes the position ofnode k. Vectors and C

represent respectively the reaction forceand moment of the flexible part of the body over the rigid part, givenby

= g

F (KL + KN L) (KL + KN L) ,

= g

F (KL + KN L) (KL + KN L .

(34)

he coupling between the rigid body motion and the system deforma-tions is fully preserved. For a more detailed description of the formula-

tion, and the notation, the interested reader is referred to reference [6].As an application example of the nonlinear formulation for flexible

multibody systems, a sports vehicle with a front crash-box is analyzedfor various impact scenarios, represented in Fig. 29, where the angle of

Angle 10

no friction

Angle 20

friction = 0.5

Angle 10

friction = 0.5

10 cm ramp

Angle 20

no friction

a) (e)(d)

Figure 29. ifferent impact scenarios for the sports vehicle.

Figure 30. Motion of the vehicle for a 20 oblique impact without contact frictionand for impact with an oblique surface for a vehicle traveling over a ramp.


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impact and the topology of the road are different. The simulations arecarried until the vehicle reaches a full stop.

he vehicle motion, for the oblique impact scenario presented in

Fig. 29, is characterized by a slight rotation of the vehicle during im-act. This rotation is more visible in the case of frictionless impact. At

the simulated impact speed the influence of the car suspension elementson the deformation mechanism is minimal.

6. Conclusions

he multibody dynamics formalisms provide an extremely efficientramework to incorporate different disciplines. The behavior of a good

number of phenomena in different problems can be represented by kine-matic constraints (e.g., contact, muscle action, guidance) or by contactforces (e.g, impact phenomena, control, general interactions). However,different disciplines use different preferred numerical methods to solvetheir equilibrium equations which lead to difficulties in the co-simulationof different systems. The use of multibody formalisms in biomechanicspresents a strong increase due to the suitability to model contacts, mus-cles, anatomical joints, data processing, etc. The treatment of structuralcomponents with large rotations or of rotating bodies with structural de-

ormations finds in the flexible multibody dynamics efficient methods todeal with the problem. A continued effort to close the gap between the

exible multibody dynamics and the nonlinear finite element method isrequired. The need for more robust and efficient numerical methods tohandle the specific forms of the MBS equations and the discontinuitiesassociated to intermittent and fast behaviors are still required.

Acknowledgements

he contents of this work result from a team effort and collaborationswith many co-workers among which the contribution by Miguel Silva,Jo o Gon alves, Jo o Pombo, Manuel Seabra Pereira, Jo o Abrantes,Augusta Neto and Rog rio Leal are gratefully acknowledged.

References

[1] P. Nikravesh, Computer-Aided Analysis of Mechanical Systems, Prentice-Hall,Englewood Cliffs, New Jersey 1988.

[2] J. Garcia de Jalon, E. Bayo, Kinematic and Dynamic Simulation of MechanicalSystems The Real-Time Challenge, Springer-Verlag, Berlin 1994.

[3] P. Nikravesh and G. Gim, Systematic construction of the equations of motionfor multibody systems containing closed kinematic loops, Journal of MechanicalDesign, Vol. 115, No.1, pp.143149, 1993.


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[4] C.W. Gear, Numerical solution of differential-algebraic equations, IEEE Tran-sactions on Circuit Theory, Vol. CT-18, pp.8995, 1981.

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[7] F. Pfeiffer and C. Glocker, Multibody Dynamics with Unilateral Contacts, JohnWiley and Sons, New York 1996.

[8] H. Lankarani and P. Nikravesh, Continuous contact force models for impactanalysis in multibody systems, Nonlinear Dynamics, Vol. 5, pp.193207, 1994.

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[11] H. Mller, E. Lund, J. Ambr sio and J. Gon alves, Simulation of fluid loadedflexible multiple bodies, Multibody Systems Dynamics, Vol. 13, No.1, 2005.

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[14] M. Silva and J. Ambr sio, Kinematic data consistency in the inverse dynamicanalysis of biomechanical systems, Multibody System Dynamics, Vol. 8, pp.219239, 2002.

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[20] C.M. Kindervater, Aircraft and helicopter crashworthiness: design and simula-ion, [in:] Crashworthiness Of Transportation Systems: Structural Impact And

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21] P.E. Nikravesh, I.S. Chung, and R.L. Benedict, Plastic hinge approach to vehiclesimulation using a plastic hinge technique, J. Comp. Struct. Vol. 16, pp.385400,1983.

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[24] M. Anantharaman and M. Hiller, Numerical simulation of mechanical systemsusing methods for differential-algebraic equations, Int. J. Num. Meth. Eng.,Vol. 32, pp.15311542, 1991.

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