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7 CONTACT PROBLEMS IN

MULTIBODY DYNAMICS

A REVIEWFriedrich Pfeiffer and Christoph Glocker

Lehrstuhl B fur Mechanik

Technische Universitat Munchen

D-85747 Garching, Germany

[email protected]

[email protected]

Abstract: The interest in problems of contact mechanics came up in the sixtieswith questions of statics and elastomechanics. Most of the mathematical toolswere developed and applied in those fields. With increasing pressure from thepractical side of vibration and noise connected with contact processes meth-ods of multibody dynamics with unilateral constraints have been elaborated.The paper gives a review of these activities mainly based on findings from theauthors’ institute.

7.1 INTRODUCTION

Contact events appear in dynamical systems very often due to the fact that theworld of dynamics usually happens to be as much unilateral as it is bilateral.Walking, grasping, climbing are typically unilateral processes, the operation ofmachines and mechanisms includes a large variety of unilateral aspects. Fromthis there emerges a need to extend multibody theory by contact phenomena.

All contact processes have some characteristic features in common. If acontact is closed, a motion changes from slip to stick, we come out with someadditional constraints generating constraint forces. We then call the contactactive. Otherwise it is passive. Obviously transitions in such contacts dependon the dynamics of the system under consideration. The beginning of such acontact event is indicated by kinematical magnitudes like relative distances orrelative velocities, the end by kinetic magnitudes like normal force or friction

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R.P. Gilbert et al. (eds.), From Convexity to Nonconvexity, 85-109. © 2001 Kluwer Academic Publishers. Printed in the Netherlands.

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85

86 FROM CONVEXITY TO NONCONVEXITY

force surplus. This will deliver a basis for the mathematical formulation tofollow.

A large variety of possibilities exists in modeling local contact physics, fromNewton’s, Poisson’s and Coulomb’s laws to a discretization of local behaviorby FE- or BE-methods. But, simulations of large dynamical systems requirecompact contact laws. Therefore, we shall concentrate on the first types of lawswhich inspite of their simple structure still are able to describe realistically alarge field of applications.

Literature covers aspects like contact laws, FEM- and BEM-analysis, con-tact statics, contact dynamics and a large body of various applications. Withrespect to multibody systems with multiple unilateral contacts most of themathematical fundamentals, though firstly regarding statical problems only,were laid down by European scientists. First considerations were started byMoreau and his school [11], which in the meantime continues his efforts in aremarkable way [1]. Moreau introduced convex analysis into multibody dy-namics and reformulated the classical equations of motion in terms of measuredifferential inclusions in order to cover both, impact free motion and shocks asthey appear in frictional contact problems.

The scientific community includes in addition scientists like Panagiotopou-los and Lotstedt who developed powerful methods for statical and dynam-ical problems of contact mechanics [12], [9]. Especially Panagiotopoulosestablished a general theory on unilateral problems in mechanics. Lotstedtdeveloped an advanced index-two-type integration algorithm for planar contactproblems in rigid body systems [9]. The Swedish school in that field is contin-ued with remarkable results by Klarbring, who focusses his work to problemsof FEM- and BEM-modeling [7].

At the autor’s insitute research has been performed in that field for morethan ten years which is mostly summarized in the book by Pfeiffer, Glocker[15]. Some newer results on nonlinear complementarity problems may be foundin Wosle [20]. As the author’s institute is one of engineering mechanics, mostof the research work deals with a transfer of the demanding mathematicalfundamentals to an engineering and application-friendly level.

7.2 THE EVOLUTION OF A THEORY

Woodpecker Toy

The first example with structure-variant properties, which was analysed at theauthor’s institute was a woodpecker toy which operates by self-excited vibra-tions (Fig. 7.1 and [13]). The theory applied 1984 was at that time incompletebecause no reasonable concept for impacts with friction was available. There-fore friction was included empirically. The results compared very well withmeasurements. In the meantime an impact-with-friction-theory exists [3], [4].

Its application to the woodpecker example confirms the simpler theory at thesame time achieving improvements with respect to friction modeling. Figure7.2 depicts some results in the form of phase portraits [3]. The data used forthe woodpecker may be seen from [15].

CONTACT PROBLEMS IN MULTIBODY DYNAMICS 87

Figure 7.1 Woodpecker’s self-excited vibration.

Figure 7.2 Phase Space Portraits [3].

The main phases of motion are (Fig. 7.2): (1 – 2) sliding, (2) loss of contact,(2 – 3 ) free fall with high frequency oscillation of 73 Hz, (3) inelastic impactof the upper edge of the sleeve, (4) beak impact, (5 – 6) free fall downward, (6– 7) transition to sleeve sticking and self-locking, (7 – 1) woodpecker angularmotion with a low frequency oscillation of around 9 Hz, which has also beenapproved by measurements.

Gear Rattling

Gear rattling is a noise problem appearing in gear stages not under load. Ex-amples are change-over gears or gear-driven balancers. Figure 7.3 illustrates afive-stage gear configuration where the driving shaft moves the countershaft,and according to the switched gear, the countershaft drives the main shaft. Allgear wheels mesh. Switching is performed by synchronizers connecting the ap-propriate wheel with the main shaft. The wheel is loaded and all other wheelsare not under load and can rattle due to backlash in the mesh of the gears.The driving shaft is excited by the torsional vibrations of the drive system,


which are more or less harmonical. These excitations are transported to thecountershaft where it generates rattling in the loose gear wheels. Figure 7.3shows the case with the fourth stage switched, which for this type of gearboxresults in a direct connection of the driving and main shafts. The countershaftand all gear wheels are not under load. The harmonic excitation leads to animpact driven vibration of the countershaft and, for example, in the fifth gearto chaotic vibrations of the fifth gear wheel.

Figure 7.3 Five-stage change-over-gear, fourth gear switched.

The problem is analyzed with the same theory as being applied to the wood-pecker example in an early stage [13]. For rattling a multibody approach withmultiple impacts turns out to be sufficient. Following [15] we describe a multi-body configuration with constraints in normal direction by

Mq − h−W NλN = 0; gN = W TN q + wN (7.1)

where q ∈ <f are generalized coordinates, M ∈ <f,f is the mass matrix, hincludes all forces, λN ∈ <ns are constraint forces in normal contact direction,W N ∈ <f,ns is the constraint matrix, qN ∈ <ns the relative velocities in thecontacts and wN are usually excitation terms. Assuming very short impactswe reduce the force-acceleration level to an impulse-velocity level:

M(qE − qA)−W NΛN = 0 with ΛN = limtE→tA

∫ tE

tA

λNdt (7.2)

(indices A and E for the beginning and the end of the impact, respectively).Introducing Newton’s law of impact

gNE = −ε gNA (7.3)


(ε matrix of coefficients of restitution), we may combine the equations (7.1),(7.2), (7.3) to give

gNE = gNA + GNΛN

ΛN = −G−1N (E + εN )gNA

qE = qA −M−1W NG−1N (E + εN )gNA

GN = W TNM−1W N .

(7.4)

It turns out that an excellent measure for the noise intensity of gear rattlingcan be achieved by a sum over all impulses ΛN

Noise Intensity ∼∑

|ΛN |, (7.5)

which represents a relative measure, not an absolute one. Many applicationsto real gears confirm however the fact, that the measure (7.5) describes allparameter dependencies correctly.

Figure 7.4 Rattling theory versus measurements.

Figure 7.4 depicts two examples for the change-over-gears of a pick-up-car(left) and an upper middle-class car (right). Only one measurement point hasbeen considered for adapting a proportionality factor for eq. (7.5).

Turbine Blade Damper

The two examples of the woodpecker toy and gear rattling afford only impacttheory with certain extensions to friction. The operation of a turbine bladedamper is completely based on stick-slip processes, which was at that time astep into new research efforts [5], [14].

Turbine blade dampers are parabolic sheet steel devices often used in gasturbines with the requirement to damp the blade vibrations by relative motionthrough dry friction. Figure 7.5 shows a typical configuration. The dampersare located between the blade platforms of two neighboring blades and pressedagainst the oblique plane by centrifugal forces.

The system was approximated in this case by a two-dimensional model withseven degrees of freedom and two contact zones (see Fig. 7.5). The blades are


Figure 7.5 Turbine blade damper and its mechanical model.

excited approximately harmonically by gasdynamic forces. For such a modelthe following movements are possible:

Sliding at both contact points K1 and K2 (5 DOF),Sliding at K1 and stiction at K2 (4 DOF),Stiction at K1 and sliding at K2 (4 DOF),Stiction at K1 and K2 (3 DOF).

Stiction means for the 4 DOF case rolling without sliding, for the 3 DOF casea reduction of the (q1, q3, q5) coordinates to one DOF. For each configurationa set of equations of motion can be established, which describe the motion aslong as no transition occurs. In the case of a blade damper this transitionsfollow typical stick-slip behavior:

Transition from sliding to stiction:Relative tangential velocity in one of the contact becomes zero, and atthe same time the static friction force must be larger than the tangentialconstraint force.

Transition from stiction to sliding:The tangential constraint force becomes equal to the static friction force,and at the same time the tangential acceleration starts to be non-zero.

After such a transition event the corresponding set of equations of motionhas been selected for further evaluation up to the next transition [5]. Forsuch evaluation processes computing time can be reduced significantly by firstdetermining the frictionless equilibria, which may be achieved by solving a setof nonlinear algebraic equations resulting from the equations of motion withvelocities and accelerations equal to zero [15].

Theory has been compared with measurements. For these experiments thecentrifugal force is replaced by a spring force, the excitation is realized elec-


Figure 7.6 Test-set-up for a turbine blade damper.

tromagnetically, and the blades are represented by two bars. The geometry ofthe damper and the blade platforms is the same as for real dampers, but ona larger scale. The amplitudes of the damper and the blades are measured bysix inductive displacement transducers and the spring force by a strain gaugearrangement. Figure 7.6 illustrates the setup and the measurement scheme toexperimentally determine the stick-slip motion.

One must keep in mind that the centrifugal forces in gas turbines are so largethat the dampers move only occasionally. With respect to some time unit theymove only about 10% of that time. Therefore, a slight improvement is worth aneffort. One important magnitude of a damper is the excitation force amplitudeFE necessary to break away from a stiction situation in both contact points K1,K2 in Fig. 7.5. These results give a measure for the damping capability of thedevices. Figure 7.7 shows three curves for three different values of the springforce (centrifugal force) and for a contact angle of γ = 50o. The excitation forceamplitude FE [N ] must be augmented for larger centrifugal (spring) forces FS .The very small FE values for an excitation frequency at 61 Hz can be explainedin the following way. The value of fE = 61 Hz corresponds to an eigenfrequencyof the blocked blade-damper system. This means that an excitation with thesefrequencies leads to large amplitudes of the blades and as a consequence to aconsiderable reduction of stiction. Therefore, the damper devices move fromstick to slip at very small excitation force amplitudes.

Landing Impact of an Airplane

The landing impact of an elastic airplane is governed by the airplane config-uration and its elasticity and by the main and nose carriages. The practical


Figure 7.7 Comparison theory (—,- - -, -·-) and measurement (2, 4, ).

example of a landing airplane was the first one where the different sets of equa-tions of motion could not be writen down explicitly, at least not with reasonableeffort, but where some theoretical algorithm had to be established selecting theappropriate equations automatically [2], [17]. The internal structure of the car-riage systems leads to impacts and to stick-slip phenomena, which influencethe landing dynamics significantly.

A plane elastic model was established with the following bodies [15]:

i = 1 → elastic fuselage,i = 2 → elastic main carriage,i = 3 → elastic nose carriage,i = 4 → rigid main carriage wheels,i = 5 → rigid nose carriage wheels.

The principal configuration is depicted in Fig. 7.8. For the shock absorbersit will be assumed that their bending deformation is homogeneous over thevariable length, which means that the two components of the absorber do notbend in a different way. It turns out that this will be a sufficient approximation.

During operation we have to consider various continuous and discontinu-ous nonlinearities. The tires possess a stiffness characteristic which increasesprogressively with the tire deformation. Moreover, the spin-up process of thewheels is governed by a highly nonlinear friction-slippage relationship. Theshock absorber itself reveals some nonlinear features. The two gas chambersfollow a polytropic compression and expansion law, the fluid flow losses inducea quadratic damping behavior, and the friction forces from ground reactionforces are nonlinear also.

Discontinuous nonlinearities enter through the upper and lower stops withinthe shock absorber and through stick-slip processes between the two shockabsorber cylinders (Fig. 7.9). Mechanically and mathematically they followthe theory as presented in the next section (see also [15]).

The degrees of freedom resulting from this model include rigid degrees offreedom (number fr) and elastic degrees of freedom (number fe). With qr ∈


Figure 7.8 Planar mechanical model of a landing elastic airplane.

<fr and qe ∈ <fe we obtain

qT = [qTr , qT

e ]= [x, y, α, y22, y23, ϕ2, ϕ3, q

T1 , qT

2 , qT3 ] ∈ <f=fr+fe

qTi = [qi1, qi2, . . . , qinai

], i = 1, 2, 3nai : number of shape functions of body i

f : number of minimal coordinates.

(7.6)

The qi ∈ <nai are the elastic coordinates, (x, y, α) the three rigid degrees offreedom of the fuselage, (y22, y23) the translational and (ϕ2, ϕ3) the rotationalcarriage deflections. The derivation of the equations of motion follows thetheory presented below and in [15]. Details may also be found in [2], [17].

Simulations consider a landing impact on a plane surface with a mediumgeneral aviation propeller-driven airplane consisting of a rigid fuselage and rigidcarriages. The touchdown velocity is 4.3 m/s, and the horizontal velocity is 60m/s. The landing angle of attack is α = 13o, and the angular velocity is α = 0.The initial values of the geometric magnitudes y, y22, y23 are chosen in such away that touchdown of the main carriage wheels starts with zero forces. Thenose wheel has no ground contact for these initial values (α = 13o) [2], [15].

Elasticity has been considered by a Ritz approach and by taking into accountthe first five eigenfrequencies for the fuselage and the first two eigenfrequenciesfor the main and nose carriages. All other data were the same as for the rigidairplane.


Figure 7.9 Two-chamber shock absorber model.

Figure 7.10 Elastic airplane landing impact — airplane results [2].

Figure 7.10 illustrates the situation of the airplane for the above landingimpact. The center of mass moves about 300 mm (curve 1), and its velocityhas a maximum at the beginning (curve 2). The forces acting on the fuselagefrom the main and nose carriages reach a maximum of about 170 kN for themain absorbers (curve 3) and a peak value of about 450 kN for the nose, wherethe nose load builds up later (curve 4).


Figure 7.11 Elastic airplane landing impact — carriage results [2].

The displacements are more or less the same as in the rigid case, the di-agrams of which are not given here (see [15]). The nose wheel performs aninternal impact at the upper stop (curve 7), and in the force and stroke veloc-ity diagrams the lowest elastic fuselage eigenfrequency (∼7 Hz) can be detected.The elasticity neither increases nor decreases the tendencies for impacts andstick-slip (curves 8 in Fig. 7.11), but it reduces significantly the maximum loadon the nose carriage.

Assembly Processes

Mating parts together involves various and sometimes quickly changing com-binations of impacts with and without friction, of stick-slip processes and evenof jamming. As these combinations and the transitions between them cannotbe predicted beforehand because they depend on the current state, we needa theory which is able to deal with such processes. At the autor’s institutethis exactly has been the stage to enter into the methods of complementarity,sub-differentials and convex analysis [3], [15], [16].

Within the practical application of these to a certain extend abstract the-ories we usually work on different levels: The event of a transition betweencontact configurations is either indicated by magnitudes of relative kinematics(at the beginning) or by those of constraint forces (at the end). The relativekinematical magnitudes undergo a metamorphosis from indicators to constraintequations as components of the constraint matrices valid for the contact con-figuration to come. The transition itself is governed by the complementarityof relative kinematics and constraint forces: If one group is zero the other isnot. Considerations of that kind in combination with the accompanying math-ematics [12], [15] allow an efficient modeling of assembly processes as describedabove. As an example we regard the peg-in-hole insertion performed by amanipulator [10], [18].

A robotic manipulator is usually considered as a tree-like multibody systemwith rigid and elastic components. Links very often are modelled as rigid


Figure 7.12 Mechanical models of links and joints. τA = c(γM/iG−γA)+d( ˙γM/iG−˙γA), where τA is the torque at robot arm. γM , γA are the coordinates of motor and arm,

respectively, c the joint stiffness, d the joint damping coefficient, and iG the gear ratio.

bodies, only for special tasks as elastic bodies. Joints may ideally follow agiven torque law, or they may be elastic with own degrees of freedom (see Fig.7.12). Therefore we may subdivide our total number n of degrees of freedominto motor or internal degrees of freedom (nM ), external ones (nA) and elasticones (nC), which results in

γ =

γM

γA

γC

∈ <n, n = nM + nA + nC . (7.7)

The equations of motion are then derived by applying Jourdain’s principlewhich says that constraint forces in the joints are perpendicular to the directionof motion and do not affect power. With given kinematics of the manipulatorunder consideration, the mobility is specified by the Jacobians JT = ∂v/∂γ,JR = ∂ω/∂γ, where v,ω are the translational and rotational velocities, re-spectively. When we premultiply Newton/Euler’s equations for each individualbody with respective Jacobians the equations of motion can recursively be gen-erated in the final form:

M(γ) γ + h(γ, γ) = Bu (7.8)

with the inertia matrix M ∈ <n,n, centrifugal, coriolis and gravitational forcesh ∈ <n and the control input Bu ∈ <n.


Because most mating tasks in assembly cells are limited to a small area ofworkpiece interaction the robot motion will be slow and centrifugal and coriolisforces may in the following be neglected compared to gravitational and inertiaforces. Hence, the robot dynamics can be linearized around a given systemstate (γ0, γ0) = 0 and deviations are given by

q =

qM

qA

qC

= γ − γ0, γ0 =

IGγA,0

γA,0

0

, (7.9)

where the matrix IG ∈ <nM ,nA represents the gear ratio that transforms thearm coordinates of the reference position onto the respective motor coordinates.The linearized equations of motion for the open loop dynamics then are

Mq + P q + Qq = h + Bu, (7.10)

where P ∈ <n,n is the velocity proportional matrix containing damping termsof joints and compliance and Q ∈ <n,n is the stiffness matrix. The vectorh ∈ <n contains the entire nominal dynamics and feedforward control withτM0 , where the motor drive torque results from τM = τM0 + u, u being thefeedback torque. Bu ∈ <n is feedback control with the controller matrix B.For PD position control the feedback term typically has the form

u = −KP (qM − qM0)−KD(qM − qM0

). (7.11)

During assembly the workpiece held by the robot comes into various contactsituations with the complementary mating part. This is illustrated in Fig. 7.13for a planar peg-in-hole insertion task. In this process one or more contactsmay arise which constrains the gripper motion and thus the robot’s mobilityby closing the kinematical loop of the manipulator.

Figure 7.13 Various contact configurations for the planar peg-in-hole problem.

The experimental investigations are focussed on a verification of the the-ory for various configurations of assembly processes and on a consideration


of disturbances during the peg-in-hole insertion. We first consider the per-turbed motion during a planar peg-in-hole insertion task where forces onlydevelop in the presence of uncertainties. For the verification of our theoreticalapproach experiments with a five degrees-of-freedom laboratory robot with aforce-torque sensor were carried out. The fixture housing the complementarypart is equipped with six distance sensors that measure the gripper’s posi-tion and orientation and in addition disposes of two translational and threerotational adjustments which allow to produce definite positioning errors withrespect to the parts that are to be mated.

In a first experiment we investigated the insertion process when there wasa lateral error between the peg and the hole. The respective numerical andexperimental results are depicted in Fig. 7.14. The peaks in the force historyat t = 0.9 s result from the impact when the peg hits the chamfer. When two-point-contact is reached at t = 1.2 s the manipulator’s motion is slowed downvery fast until the jamming condition is reached. Only when the controllertorques become large enough the motion is continued with sliding in two andfinally even one contact point. This cycle represented by the oscillations inthe force history repeats until the end of the trajectory and demonstrates thephenomenon of changing constraints during an assembly operation.

Figure 7.14 Insertion force Fx during assembly due to lateral offset.

In a second experiment we investigated the permissible tolerance range oflateral and angular errors of the peg so that a maximum force of 5 N is notexceeded during the insertion phase. In this context we did not take intoconsideration the forces that develop while the peg is guided along the chamfersince this automatically involves the question of chamfer design. Fig. 7.15contrasts the numerical results with some data obtained by experiment. Therespective lines identified by theory correspond to the existence of differentcontact configurations where the maximum insertion force of 5 N is reached andlimit the range of permissible positioning tolerances. The difference betweennumerical and experimental data mainly results from backlash in the last jointof the robot which was not modeled. Note that in both experiments the sensor


Figure 7.15 Permissible positioning errors for a maximum insertion force of 5N .

served only as a force measuring device for data-processing, since a simple PDjoint controller was used.

7.3 PRESENT MATHEMATICAL FORMULATION

Extensive experience with practical engineering problems has brought up thetheory to a level which allows a very straightfoward formulation of contactproblems in multibody systems. The formulation is kept as close as possibleto the classical theory used in applied dynamics. This enables engineers tocapture the main ideas of multivalued force laws quickly, to understand thetheory behind them intuitively and to apply them in a correct manner. Eventeaching this subject in a graduate course has turned out to be successful. Inthe following we will give a brief overview of the structure of the equations inthe case of spatial Coulomb friction.

Geometry of Surfaces

We consider two bodies moving with vPi,Ωi (i = 1, 2) where vPi denotes thevelocity of the body-fixed point P of body i and Ωi its angular velocity. Bothbodies are assumed to be convex, at least in a neighborhood of a potentialcontact point, and may be represented by a parametrization of their surfacesin the form rPΣi = rPΣi(q, ξi, ηi), see Figure 7.16 [10].

Note that rPΣ only depends on the surface parameters ξ and η when statedin the corresponding body-fixed frame whereas an additional dependence onthe generalized coordinates q is observed when using arbitrary frames. Foreach of the bodies one defines an orthonormal basis (u,v,n) via the tangents

s := ∂ξrPΣ, t := ∂ηrPΣ (7.12)

in order to obtain

n :=s× t

|s× t|, u :=

s

|s|, v := n× u (7.13)


Figure 7.16 The spatial contact problem.

where we agree that (ξ, η) are ordered such that n becomes the outward normal.The (absolute) changes in time of the contour vector rPΣ and the trihedral(u, v, n) are

rPΣ = Ω× rPΣ + Rζ with R = ∂ζrPΣ = (s, t)k = Ω× k + Kζ with K = ∂ζk

(7.14)

where ζ = (ξ, η)T and (k,K) is any of (u,U), (v,V ), (n,N). The velocity vΣ

of a point moving on the surface in terms of the rigid body motion (vP ,Ω) isthus given by vΣ = vP + rPΣ which yields

vΣ = vC + Rζ with vC = vP + Ω× rPΣ (7.15)

Here, vC denotes the rigid body velocity portion of vΣ. Analogously we mayfind the acceleration aC = vC by differentiation of the second equation in(7.15),

aC = aQ + Ω×Rζ with aQ = aP + Ω× rPΣ + Ω× (Ω× rPΣ). (7.16)

Finally we express vC and aC by the generalized magnitudes q and q,

vC = JC q + ιC

aC = JC q + ιC , ιC = JC q + ˙ιC ,(7.17)

which involves the Jacobian JC that will later turn out to be the Jacobian ofthe contact point.

Contact Kinematics

Within the framework as presented in the foregoing section one is able to derivethe main equations of contact kinematics in dynamics which are: The defini-tion of the contact points, their distance, their relative velocities with respect


to the normal and the tangential directions, and the corresponding relativeaccelerations. In order to do the first step one introduces a distance vector

rD := rOΣ2 − rOΣ1, (7.18)

cp. [3], where rOΣi are vectors pointing from a common inertially fixed pointO to the surfaces under consideration.

Next, the two tangent planes spanned by (ui,vi) are adjusted parallel to eachother such that the distance vector becomes parallel to each of the normals.The surface parameters (ζ?

1, ζ?2) corresponding with that situation solve the

four nonlinear equations

nT1 u2 = 0, nT

1 v2 = 0, rTDu1 = 0, rT

Dv1 = 0 (7.19)

and are called the contact parameters; the corresponding points at the surfacesgiven by rPΣi(ζ?

i ) are called the contact points. When the bodies are movingthe conditions (7.19) for the contact points should not change [3], [10], thus

nT1 u2 + nT

1 u2 = 0, rTDu1 + rT

Du1 = 0nT

1 v2 + nT1 v2 = 0, rT

Dv1 + rTDv1 = 0

(7.20)

must hold which constitutes a system of four linear equations in ζ?

i when sub-stituting (ni, ui, vi) from (7.14) and vΣi from (7.15) with rD = vΣ2 − vΣ1.

After the solution (ζ?1, ζ

?2) of (7.19) has been found the distance gN of the

contact points can be calculated as

gN = nT1 rD (7.21)

with values greater than zero for separation and values less than zero for over-lapping. The relative velocities of the contact points with respect to the threedirections (u1,v1,n1) are then given by

gK = (vC2 − vC1)T k1, (K ,k1) = (U ,u1), (V ,v1), (N ,n1) (7.22)

where it can be easily verified that differentiation of (7.21) with respect to timeresults in (7.22) for (K ,k1) = (N ,n1). The temporal changes of the relativevelocities gK in (7.22) are given by

gK = (aC2 − aC1)T k1 + (vC2 − vC1)T k1, (7.23)

where aCi is known from (7.16), k1 may be taken from (7.14), and ζ?

i involvedin aCi and k1 is the solution of (7.20).

Substituting vCi and aCi from (7.17) into (7.22), (7.23) yields a representa-tion of the relative velocities and accelerations in the form

gK = wTK q + wK with wK = (JC2 − JC1)T k1; wK = (ιC2 − ιC1)T k1

gK = wTK q + wK with wK = wT

K q + ˙wK .(7.24)


Finally we arrange the two tangential relative velocities in the vector gT , i.e.gT

T := (gU , gV ) which gives us the sliding direction of the contact points inthe common tangent plane. With the abbreviations W T := (wU ,wV ), wT

T =(wU , wV ), wT

T = (wU , wV ) we rewrite (7.24) in the form

gN = wTN q + wN , gT = W T

T q + wT

gN = wTN q + wN , gT = W T

T q + wT

(7.25)

which will serve as the contact kinematic equations in standard form in thesequel.

Kinetics

The dynamics of a multibody system with n bodies and one contact is describedby the Newton-Euler equations in the form

n∑i=1

[JT

S (p− F ) + JTR(L−M)

]i− JT

C1F C1 − JTC2F C2 = 0, (7.26)

see e.g. [3], [15]. The terms pi and Li denote the momentum and the momentof momentum of body i, and F i and M i are the applied forces and moments,respectively. The contact forces acting at the contact points C1 and C2 aredenoted by F C1 and F C2. In order to perform a projection into the configura-tion space all terms have to be premultiplied by the corresponding Jacobians.In particular, JC1 and JC2 are the Jacobians of the contact points which havealready been introduced in equation (7.17).

Decomposing the contact forces into components with respect to trihedral(u1,v1,n1) admits, together with the cutting principle, a unique representationin the form

F C2 = −F C1 =: n1λN + u1λU + v1λV . (7.27)

Putting (7.27) into (7.26) and using the abbreviations wK = (JC2 − JC1)T k1

and W T = (wU ,wV ) from equation (7.24) and below we arrive with

Mq − h−wNλN −W T λT = 0, (7.28)

where the vector λT of the tangential contact forces is defined to be λTT =

(λU , λV ). The sum in (7.26) is taken into account by the term (Mq−h) withthe symmetric and positive definite mass matrix M of the system. In the caseof nA contacts eq. (7.28) clearly transforms to

Mq − h−nA∑i=1

wNiλNi −W TiλTi = 0 (7.29)

which has still to be completed by the contact laws.


Contact Laws

Generally, every force occuring in a rigid multibody system may be expressedin terms of some relative kinematic magnitudes such as relative displacementsand velocities. Classical applied forces are represented by continuous functionswhereas classical bilateral constraints already require a representation via set-valued mappings. Contact laws as used in the sequel belong to the latter classand constitute mixed-type force charateristics, acting in some areas as classicalapplied forces, in other areas as constraints.

With gNi being the distance of the contact points and λNi the correspondingnormal force one might express impenetrability of rigid bodies by the Signorini-Fichera condition [1], [3], [7], [9], [12]

gNi ≥ 0, λNi ≥ 0, gNiλNi = 0 (7.30)

which might be combined, for example, with Coulomb-friction in the tangentialdirections [1], [7], [11], [12], i.e. gTi = 0 ⇒ |λTi| ≤ µ0iλNi

gTi 6= 0 ⇒ |λTi| = µiλNi,λTi = −αi gTi, αi > 0,

(7.31)

where µi(gTi) denotes the coefficient of friction for contact i and µ0i = µi(0).By introducing the index sets [3], [15], [20]

IA = 1, 2, . . . , nAIC = i ∈ IA : gNi = 0, gNi ≥ 0IN = i ∈ IC : gNi = 0IT = i ∈ IN : gTi = 0

(7.32)

and by the use of certain continuity assumptions on the trajectories q(t) one isalso able to express the contact laws (7.30), (7.31) on the acceleration level inorder to finally solve equation (7.29) for the unknowns q. This yields, for thecontact law in the normal direction (7.30)

for i ∈ IA \ IN : λNi = 0for i ∈ IN : gNi ≥ 0, λNi ≥ 0, gNiλNi = 0 (7.33)

where the complementarity condition in the second line of (7.33) might also beexpressed by one of the variational inequalities, cp. [3], [7], [12], [20],

∑i∈IN

gNi(λ?Ni − λNi) ≥ 0, λNi ≥ 0, ∀λ?

Ni ≥ 0∑i∈IN

(g?Ni − gNi) λNi ≥ 0, gNi ≥ 0, ∀g?

Ni ≥ 0(7.34)


for i ∈ IN . Similarly we obtain from (7.31) the friction law on the accelerationlevel in the form [20]

for i ∈ IA \ IN : λTi = 0for i ∈ IN \ IT : λTi = −eiµiλNi

for i ∈ IT :

gTi = 0 ⇒ |λTi| ≤ µ0iλNi

gTi 6= 0 ⇒ |λTi| = µ0iλNi,λTi = −αi gTi, αi > 0,

(7.35)

with ei being the unit vector of the sliding direction, i.e. ei = gTi/|gTi|. Werecall that, due to the dependence on the normal forces, only quasivariationalinequalities are available for the third portion of the contact law (7.35) whichread [20]∑

i∈IT

gTTi(λ

?Ti − λTi) ≥ 0, |λTi| ≤ µ0iλNi, ∀λ?

Ti : |λ?Ti| ≤ µ0iλNi∑

i∈IT

(g?Ti − gTi)T λTi ≥

∑i∈IT

µ0iλNi(|gTi| − |g?Ti|), ∀g?

Ti.(7.36)

Equations (7.21), (7.25), (7.29), (7.33) and (7.35) provide a complete de-scription of impact free non-smooth rigid body motion under the influence ofCoulomb friction in the framework of non-smooth analysis. The equations con-sidered above thus allow any further discussion and evaluation by applyingavailable theoretical results and algorithms from this field [1], [7], [11], [12].Impacts have been excluded. They can be treated in an analogous manner byrewriting (7.26) as an equality of measures [11] and solving them for the impacttimes. This yields, together with appropriate impact laws, a set of relationssimilar to those considered here, with force magnitudes playing then the roleof impulsions, and accelerations which have to be expressed by jumps in thevelocities [3], [4], [15].

Application: Vibratory Feeders

Vibratory feeders are just one example in machine dynamics on which thetheory of spatial contact problems comes to a fruitful application [19]. Theyare the most common devices used to feed small parts in automatic assemblylines. Compared to other machines vibratory feeders seem to be quite simple.The high number of different types of devices based on the vibratory feedingprinciple, and the large amount of applications suggests that we are lookingat a well developed and reliable tool. This impression is wrong. Problems inautomated assembly are mostly caused by malfunctions of part feeders. Theseerrors result from purely experimental tuning of the feeders, which is oftendone without a theoretical background, especially as far as the mechanics ofthe transportation process is concerned.

The transportation process of a vibratory feeder (linear or bowl feeder, Fig.7.17) is based on a micro ballistic principle that is driven by an oscillating track.The mechanical model can be split in the dynamics of the base device, mostlyrepresented by an electro magnetic excited oscillator, and the dynamics of the


Figure 7.17 Vibratory bowl feeder and mechanical model.

parts transportation process. This yields a coupled system in the sense thatthe parts to be transported are affected by the vibrations of the track and viceversa.

The base device including the drives and the drive mechanisms can be mod-eled as a bilaterally coupled system with well-known standard techniques. Nev-ertheless, this may result in an extensive job. Due to the highly sensitive contactmechanics, all oscillating frequencies and flexible structures have to be analysedvery well. Furthermore, interactions with an oscillating environment must betaken into account.

After having computed the vibrations of the base device which now serveas an excitation source for the parts one may concentrate on the modeling ofthe transportation process. The changes in the contact configurations betweeneach of the parts on one side, and between the parts and the track on the otherside are characteristic properties of the feeding process. Closed contacts eitheroccur for entire time intervals or only for discrete points in time, the latterevent being the usual outcome of an impact. Friction ist fundamental for thetransportation of the parts: Indeed, it is the very feature that makes the feederwork. Consequently, the modeling of the process must be done with respect tofriction effects. This leads to a structure variant multibody system with spatialcontact laws consisting of unilateral constraints and planar Coulomb friction,the theory of which has been presented in the preceeding section.

The parts to be transported are modeled by rigid bodies with surfaces piece-wise approximated by planes. Therefore, unilateral constraints result in pointcontacts, i.e. contacts between corners and planes or between lines and lines.Contact areas occuring at parallel lines and planes are composed of single pointcontacts. As one example of intense design parameter studies we present thetransportation velocity, because of its outstanding importance for the basicfunction of the feeder. Not only fast transport but also robustness with respectto unsafe parameters are design criteria.


Simulation has shown that the results obtained by using planar and spatialmodels nearly coincide. Even feeding with many parts, which was calculated forthe planar model, does not influence the feeding rate. Therefore the averagedresults for the transportation rate in the plane and spatial case as well as forone or for many parts are nearly the same allowing us to perform the layout-simulations with one part only.

In the following some results for the spatial model are presented. The modelunder consideration applies to Fig. 7.17 with only one rectangular block of 1 cmx 2 cm x 5 cm and no devices to orientate the parts. The excitation frequency,the angle between excitation direction and the track, and the track’s inclinationhave been chosen to be 100 Hz, 10o, and 20o, respectively.

Fig. 7.18 shows the average transportation velocity depending on the excita-tion amplitude for different friction- and impact-coefficients. Every point of thecurves results from one simulation which was performed until a stable averagevalue was found. Positive velocities mean transportation upwards; negativedownwards.

Figure 7.18 Transportation rates of a vibratory feeder.

In the left diagram of Fig. 7.18 we see the results for a friction coefficientµ = 0.2 and four different impact coefficients ε. Small amplitudes cause theconveyor not to work, resulting in pure sliding downwards of the parts. Byincreasing the excitation amplitude the upwards feeding process starts working,but only in certain regions for small impact coefficients, and based on a ballistic(that means: impact-controlled) mechanism.

The right diagram shows the same investigations but with a bigger coefficientof friction. In this case the transportation rate rises, and the conveyor is workingeven with bigger impact coefficients.

These results can serve as a basis for the feeder design which allows to chooseappropriate materials for the track, to find the best excitation parameters, andto estimate the expected feeding rate. The experimental verification of thepresented results is the subject of current activities.

REFERENCES 107

7.4 CONCLUSIONS

In the area of multibody theory regarding unilateral contacts the methodologyhas reached a state, which allows an application in nearly all fields of mechanicalengineering. As unilateral contacts are realized in machines and mechanismsto a very large extend, the use of the corresponding theories is just at thebeginning. The paper gives some examples, which demonstrate the typicalfeatures of unilateral constraints in combination with machines: They mightbe applied to achieve some functional performance like the transportation ratein vibratory feeders, or they might generate unwanted vibrations, noise andwear like rattling in gears or wear in chains.

The main problem limiting today’s applications is of numerical nature. Todescribe a machine like the vibration conveyor includes tedious numerical eval-uations solving the complementarity problem at each time step. Even withmodern high-speed computers we come out with a severe computing time prob-lem. At the time being there are three groups of algorithms available to treatnumerics.

The enumerative methods try to find a solution by a kind of intelligenttrial and error procedures. They do not satisfy. The pivot-algorithms are re-lated to the well-known simplex-algorithm. The Lemke-method is an example.We apply it with good success, but with large computing times. Within theframework of iterative methods a modified Newton-approach seems to be verypromising, even for the nonlinear complementarity problem. At the time beingwe are working on it.

A second problem, also closely connected with numerics, consists in the so-lution of nonlinear complementarities as emerging from spatial unilateral con-tacts. Up to now we have used the following methods: linearization of thefriction cones, the augmented Lagrangian methods and NCP-functions as theyappear in Mangasarian’s theorem. All methods work satisfactorily, all have tobe improved, or a better one must be developed. We are also working on it.

As a conclusion, theories and methods available to-day help the engineers tounderstand much better problems including unilateral features, which in formertimes were treated only in a very simple way. We are able to analyse machineswith unilateral contacts in a way, which five years ago we could not have thoughtof. Scientists like Professor Fichera have contributed to this development verysignificantly.

References

[1] Alart, P. and Curnier, A. (1991). “A Mixed Formulation for FrictionalContact Problems Prone to Newton Like Solution Methods,” Comp. Meth.Appl. Mech. Eng. 92 (3), pp. 353–357.

[2] Braun, J. (1989). Dynamik und Regelung elastischer Flugzeugfahrwerke.Thesis at the Dept. of Mech. Eng., Inst. B f. Mech., Tech. Univ. of Munich.

[3] Glocker, Ch. (1995). Dynamik von Starrkorpersystemen mit Reibungund Stoßen. Fortschrittberichte VDI, Reihe 18, Nr. 182, VDI-Verlag,


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[4] Glocker, Ch. and Pfeiffer, F. (1995). “Multiple Impacts with Friction inRigid Multibody Systems,” Nonlinear Dynam., Vol. 7, pp. 471–497.

[5] Hajek, M. (1990). Reibungsdampfer fur Turbinenschaufeln. Fortschrit-tberichte VDI, Reihe 11, Nr. 128, VDI-Verlag, Dusseldorf.

[6] Karagiannis, K. and Pfeiffer, F. (1991). “Theoretical and experimentalinvestigation of gear-rattling,” Nonlinear Dynam., Vol. 2, pp. 367–387.

[7] Klarbring, A. and Bjorkman, G. (1988). “A mathematical programmingapproach to contact problems with friction and varying contact surface,”Computers & Structures, Vol. 30, No. 5, pp. 1185–1198.

[8] Kucukay, F. and Pfeiffer, F. (1986). “Uber Rasselschwingungen in Kfz-Schaltgetrieben,” Ing.-Arch., Vol. 56, pp. 25–37.

[9] Lotstedt, P. (1982). Numerical simulation of time-dependent contact andfriction problems in rigid body mechanics. Technical Report TRITA-NA-8214, Dept. of Numerical Analysis and Computing Science, The RoyalInstitute of Technologie, Sweden.

[10] Meitinger, Th. (1997). Dynamik automatisierter Montageprozesse. Doco-toral Thesis at the Dept. of Mech. Eng., Inst. B f. Mech. Tech. Univ. ofMunich.

[11] Moreau, J.J. (1988). “Unilateral contact and dry friction in finite free-dom dynamics,” CISM courses and lectures, Nonsmooth mechanics andapplications, Springer-Verlag, Wien, New York.

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[13] Pfeiffer, F. (1984). “Mechanische Systeme mit unstetigen Ubergangen,”Ing.-Arch., Vol. 54, pp. 232–240.

[14] Pfeifer, F. and Hajek, M. (1992). “Stick-slip motion of turbine bladedampers,” Phil. Trans. R. Soc. London, A 338, pp. 503–517.

[15] Pfeiffer, F. and Glocker, Ch. (1996). Multibody Danamics with UnilateralContacts. John Wiley, New York.

[16] Seyfferth, W. (1993). Modellierung unstetiger Montageprozesse mit Robo-tern. Fortschrittberichte VDI, Reihe 11, Nr. 199, VDI-Verlag, Dusseldorf.

[17] Wapenhans, H. (1989). Dynamik und Regelung von Flugzeugfahrwerken.Thesis at the Dept. of Mech. Eng., Inst. B f. Mech., Tech. Univ. of Munich.

[18] Wapenhans, H. (1994). Optimierung von Roboterbewegungen bei Manipu-lationsvorgangen. Fortschrittberichte VDI, Reihe 2, Nr. 304, VDI-Verlag,Dusseldorf.

[19] Wolfsteiner, P. and Pfeiffer, F. (1997). “Dynamics of a Vibratory Feeder,”Proc. of the 1996 ASME 16th Biennal Conference on Vibration and Noise,Sacramento, California.

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