Contact of Planar Flexible Multibody Systems Using a Linear Comple-mentarity Formulation

Saeed Ebrahimi∗ and Peter Eberhard∗∗

Institute B of Mechanics, University of Stuttgart,Pfaffenwaldring 9, 70569 Stuttgart, Germany

Descriptions leading to Linear Complementarity Problems (LCPs) are well established in the contact modeling of rigid multi-body systems and have a very strong mathematical basis. These approaches yield exact solutions for contact problems con-sisting of contact (force/ acceleration level) and impact (impulse/ velocity level). By utilizing these methods, also frictionalcontact can be handled appropriately, see [1] and [2].

In this paper a formulation for extending these methods for consideration of planar deformable bodies is given. In thecase of deformable bodies, a finite wave propagation speed is inherent and, thus, impact computations on impulse/ velocitylevel which lead to jumps in velocity are no longer required. In this paper only the case of continual contact is taken intoaccount. For this purpose, the complementarity relations are reformulated in such a way that contact modeling of constrainedand non-constrained planar deformable bodies is possible, too.

In this formulation, deformable bodies are modeled based on the moving frame of reference approach with modal coordi-nates which is frequently used for the simulation of deformable multibody systems [3].

1 Introduction

With increasing importance of the simulation of multibody systems in industry and engineering, there are more and moreapplications which deal with contact problems and make it an essential and very demanding topic in multibody dynamics.Therefore, many investigations focus on this topic and many theoretical and mathematical methods are developed for thispurpose. Linear complementarity problems resulting from one of these methods were frequently used for contact modeling ofrigid bodies, see e.g. [1]. In this approach the contact problem of rigid bodies is formulated in a complementarity form for thecases of continual contact and impact.

In applications where the flexibility of contacting bodies is not negligible, rigid bodies contact can not be used anymoreand the deformability of contacting bodies must be taken into account. Therefore, in this work, it has been tried to reformulatethis method in order to be able to consider the deformation of contacting bodies. In doing so, the moving frame of referenceapproach is utilized in order to introduce the elastic coordinates into the equations of motion in a modal description. Maybeit is important to emphasize that this paper only considers the continual contact case of deformable bodies and for impactcalculation an extended formulation has to be used.

2 Contact Formulation

Deformable bodies are modeled here using the well-known moving frame of reference approach. In this approach, by in-troducing the rigid and elastic coordinates, the movements of bodies are separated into two independent parts of rigid bodymovement and elastic body movement. In other words, it is supposed that the movement of deformable bodies consists of asmall deformation around a large rigid body motion. In this approach the deformation of bodies is assumed to be small.

Considering two deformable bodies i and j, one can calculate the velocity of each arbitrary point pi and pj located on thebodies i and j, respectively, in terms of generalized velocities

vpi = Li · qi , vpj = Lj · qj , (1)

where Li and Lj are matrices which project the generalized velocities qi and qj onto the velocities in the reference coordinatesystem. In this relationship, the generalized velocities qi and qj include the rigid (R, θ) and elastic (qf ) generalized velocities

qi = (Ri, θi, qfi) , qj = (Rj , θj, qfj ) . (2)

Then, the relative velocities in the normal and tangential directions for any arbitrary contact point k can be written as

gkN = nk

i · (Lki · qi − Lk

j · qj) , gkT = tk

i · (Lki · qi − Lk

j · qj) . (3)

∗ e-mail: ebrahimi@mechb.uni-stuttgart.de, Phone: +49 711 685 6530, Fax: +49 711 685 6400∗∗ e-mail: eberhard@mechb.uni-stuttgart.de, Phone: +49 711 685 6388, Fax: +49 711 685 6400

PAMM · Proc. Appl. Math. Mech. 5, 197–198 (2005) / DOI 10.1002/pamm.200510076

© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

The relative acceleration in the normal and tangential directions is calculated by taking the derivative of the relative velocities

gkN = (Wk

N )T · q + (wkN )T · q , gk

T = (WkT )T · q + (wk

T )T · q , (4)

where

q =[

qi

qj

], q =

[qi

qj

], Wk

N =[

(nki · Lk

i )T

−(nki · Lk

j )T

], Wk

T =[

(tki · Lk

i )T

−(tki · Lk

j )T

],

wkN =

[(nk

i · Lki + nk

i · Lki )T

−(nki · Lk

j + nki · Lk

j )T

], wk

T =[

(tki · Lk

i + tki · Lk

i )T

−(tki · Lk

j + tki · Lk

j )T

].

Based on the above relationships, it is clear that effects of deformabilities are introduced to the kinematical relationships ofthe relative normal and tangential accelerations (gN and gT ) of contact points through the matrices Li and Lj . Although Eq.(4) looks very similar to the rigid body case, its computation is very different. This equation which holds for contact point kcan be used to obtain the matrix form of the relative normal and tangential accelerations (gN and gT ) for all the nc contactpoints between nb bodies

gN = WTN · q + wT

N · q , gT = WTT · q + wT

T · q , (5)

where

WN =[

W1N . . . Wk

N . . .Wnc

N

], WT =

[W1

T . . . WkT . . .Wnc

T

],

wN =[

w1N . . . wk

N . . .wnc

N

], wT =

[w1

T . . . wkT . . .wnc

T

].

The equations of motion of deformable multibody systems including constrained and non-constrained bodies can be writtenin the following matrix form, see [3],(

M CTq

Cq 0

)︸ ︷︷ ︸

Mc

·(

qλc

)︸ ︷︷ ︸

qc

=(

Fγ

)︸ ︷︷ ︸

hc

+(

Fc

0

). (6)

where M is the mass matrix, Cq is the Jacobian matrix of all constraints except the contact constraints, λc are the Lagrangianmultipliers corresponding to the constraint forces, F is the vector of generalized forces including external forces, Coriolisforces, stiffness and damping forces, γ is a vector resulting from differentiation of constraints yielding multiplication of theJacobian matrix Cq and the generalized accelerations q and Fc is the vector of contact forces. The above equations of motioncan be reformulated by using two different vectors λN and λH considering sliding and sticking contact

Mc · qc − hc −(

WN + WG · µG WH

0 0

)︸ ︷︷ ︸

WNH

·(

λN

λH

)︸ ︷︷ ︸

λ

= 0 . (7)

In this equation which is similar to the ones in [1], WG and WH are matrices extracted from the matrix WT which correspondto the sliding and sticking parts, respectively. From this equation by multiplying the inverse of mass matrix (supposing thatthere is no dependent constraint in the system such that the mass matrix Mc is regular), we can find qc as a function of theLagrangian multipliers λ

qc = M−1c · hc + M−1

c · WNH · λ . (8)

In the next step, the vector qc can be substituted in the relative accelerations of all contact points as follows

g =(

gN

gH

)=

(WT

N 0WT

H 0

)︸ ︷︷ ︸

WT

·(

qλc

)︸ ︷︷ ︸

qc

+(

wTN

wTH

)· q

︸ ︷︷ ︸w

= WT ·M−1c · hc + WT · M−1

c ·WNH · λ + w . (9)

The vector wH is a part of the vector wT which corresponds to the sticking contact. By following the same procedure asdescribed in [1] and by using Eq. (9) one can construct the complementarity form of equations of motion and then simulatethe continual contact problem of planar deformable bodies, too. Several examples which have been tested show the validityof this approach and its implementation.

References

[1] F. Pfeiffer and C. Glocker: Multibody Dynamics with Unilateral Contacts, New York: J. Wiley & Sons, 1996.[2] P. Eberhard: Kontaktuntersuchungen durch hybride Mehrkorpersystem/ Finite Elemente Simulationen, Aachen: Shaker Verlag, 2000.[3] A.A. Shabana: Dynamics of Multibody Systems, New York: J. Wiley & Sons, 1998.

© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

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