7 CONTACT PROBLEMS IN

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1 7 CONTACT PROBLEMS IN MULTIBODY DYNAMICS A REVIEW Friedrich Pfeiffer and Christoph Glocker Lehrstuhl B für Mechanik Technische Universität München D Garching, Germany pfeiffer@lbm.mw.tu-muenchen.de glocker@lbm.mw.tu-muenchen.de Abstract: The interest in problems of contact mechanics came up in the sixties with questions of statics and elastomechanics. Most of the mathematical tools were developed and applied in those fields. With increasing pressure from the practical side of vibration and noise connected with contact processes methods of multibody dynamics with unilateral constraints have been elaborated. The paper gives a review of these activities mainly based on findings from the authors institute. 7.1 INTRODUCTION Contact events appear in dynamical systems very often due to the fact that the world of dynamics usually happens to be as much unilateral as it is bilateral. Walking, grasping, climbing are typically unilateral processes, the operation of machines and mechanisms includes a large variety of unilateral aspects. From this there emerges a need to extend multibody theory by contact phenomena. All contact processes have some characteristic features in common. If a contact is closed, a motion changes from slip to stick, we come out with some additional constraints generating constraint forces. We then call the contact active. Otherwise it is passive. Obviously transitions in such contacts depend on the dynamics of the system under consideration. The beginning of such a contact event is indicated by kinematical magnitudes like relative distances or relative velocities, the end by kinetic magnitudes like normal force or friction

2 86 FROM CONVEXITY TO NONCONVEXITY force surplus. This will deliver a basis for the mathematical formulation to follow. A large variety of possibilities exists in modeling local contact physics, from Newton s, Poisson s and Coulomb s laws to a discretization of local behavior by FE- or BE-methods. But, simulations of large dynamical systems require compact contact laws. Therefore, we shall concentrate on the first types of laws which inspite of their simple structure still are able to describe realistically a large field of applications. Literature covers aspects like contact laws, FEM- and BEM-analysis, contact statics, contact dynamics and a large body of various applications. With respect to multibody systems with multiple unilateral contacts most of the mathematical fundamentals, though firstly regarding statical problems only, were laid down by European scientists. First considerations were started by Moreau and his school [11], which in the meantime continues his efforts in a remarkable way [1]. Moreau introduced convex analysis into multibody dynamics and reformulated the classical equations of motion in terms of measure differential inclusions in order to cover both, impact free motion and shocks as they appear in frictional contact problems. The scientific community includes in addition scientists like Panagiotopoulos and Lötstedt who developed powerful methods for statical and dynamical problems of contact mechanics [12], [9]. Especially Panagiotopoulos established a general theory on unilateral problems in mechanics. Lötstedt developed an advanced index-two-type integration algorithm for planar contact problems in rigid body systems [9]. The Swedish school in that field is continued with remarkable results by Klarbring, who focusses his work to problems of FEM- and BEM-modeling [7]. At the autor s insitute research has been performed in that field for more than ten years which is mostly summarized in the book by Pfeiffer, Glocker [15]. Some newer results on nonlinear complementarity problems may be found in Wösle [20]. As the author s institute is one of engineering mechanics, most of the research work deals with a transfer of the demanding mathematical fundamentals 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 the author s institute was a woodpecker toy which operates by self-excited vibrations (Fig. 7.1 and [13]). The theory applied 1984 was at that time incomplete because no reasonable concept for impacts with friction was available. Therefore friction was included empirically. The results compared very well with measurements. In the meantime an impact-with-friction-theory exists [3], [4]. Its application to the woodpecker example confirms the simpler theory at the same time achieving improvements with respect to friction modeling. Figure 7.2 depicts some results in the form of phase portraits [3]. The data used for the woodpecker may be seen from [15].

3 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 impact of 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 angular motion with a low frequency oscillation of around 9 Hz, which has also been approved by measurements. Gear Rattling Gear rattling is a noise problem appearing in gear stages not under load. Examples are change-over gears or gear-driven balancers. Figure 7.3 illustrates a five-stage gear configuration where the driving shaft moves the countershaft, and according to the switched gear, the countershaft drives the main shaft. All gear wheels mesh. Switching is performed by synchronizers connecting the appropriate wheel with the main shaft. The wheel is loaded and all other wheels are 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,

4 88 FROM CONVEXITY TO NONCONVEXITY which are more or less harmonical. These excitations are transported to the countershaft where it generates rattling in the loose gear wheels. Figure 7.3 shows the case with the fourth stage switched, which for this type of gearbox results in a direct connection of the driving and main shafts. The countershaft and all gear wheels are not under load. The harmonic excitation leads to an impact driven vibration of the countershaft and, for example, in the fifth gear to 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 woodpecker example in an early stage [13]. For rattling a multibody approach with multiple impacts turns out to be sufficient. Following [15] we describe a multibody configuration with constraints in normal direction by M q h W N λ N = 0; ġ N = W T N q + ŵ N (7.1) where q R f are generalized coordinates, M R f,f is the mass matrix, h includes all forces, λ N R ns are constraint forces in normal contact direction, W N R f,ns is the constraint matrix, q N R ns the relative velocities in the contacts and ŵ N are usually excitation terms. Assuming very short impacts we reduce the force-acceleration level to an impulse-velocity level: te M( q E q A ) W N Λ N = 0 with Λ N = lim λ N dt (7.2) t E t A t A (indices A and E for the beginning and the end of the impact, respectively). Introducing Newton s law of impact ġ NE = ɛ ġ NA (7.3)

5 CONTACT PROBLEMS IN MULTIBODY DYNAMICS 89 (ɛ matrix of coefficients of restitution), we may combine the equations (7.1), (7.2), (7.3) to give ġ NE = ġ NA + G N Λ N Λ N = G 1 N (E + ɛ N )ġ NA q E = q A M 1 W N G 1 N (E + ɛ N)ġ NA G N = W T NM 1 W N. (7.4) It turns out that an excellent measure for the noise intensity of gear rattling can be achieved by a sum over all impulses Λ N Noise Intensity Λ N, (7.5) which represents a relative measure, not an absolute one. Many applications to real gears confirm however the fact, that the measure (7.5) describes all parameter 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 has been 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 impact theory with certain extensions to friction. The operation of a turbine blade damper is completely based on stick-slip processes, which was at that time a step into new research efforts [5], [14]. Turbine blade dampers are parabolic sheet steel devices often used in gas turbines with the requirement to damp the blade vibrations by relative motion through dry friction. Figure 7.5 shows a typical configuration. The dampers are located between the blade platforms of two neighboring blades and pressed against the oblique plane by centrifugal forces. The system was approximated in this case by a two-dimensional model with seven degrees of freedom and two contact zones (see Fig. 7.5). The blades are

6 90 FROM CONVEXITY TO NONCONVEXITY Figure 7.5 Turbine blade damper and its mechanical model. excited approximately harmonically by gasdynamic forces. For such a model the following movements are possible: Sliding at both contact points K 1 and K 2 (5 DOF), Sliding at K 1 and stiction at K 2 (4 DOF), Stiction at K 1 and sliding at K 2 (4 DOF), Stiction at K 1 and K 2 (3 DOF). Stiction means for the 4 DOF case rolling without sliding, for the 3 DOF case a reduction of the (q 1, q 3, q 5 ) coordinates to one DOF. For each configuration a set of equations of motion can be established, which describe the motion as long as no transition occurs. In the case of a blade damper this transitions follow typical stick-slip behavior: Transition from sliding to stiction: Relative tangential velocity in one of the contact becomes zero, and at the same time the static friction force must be larger than the tangential constraint 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 motion has been selected for further evaluation up to the next transition [5]. For such evaluation processes computing time can be reduced significantly by first determining the frictionless equilibria, which may be achieved by solving a set of nonlinear algebraic equations resulting from the equations of motion with velocities and accelerations equal to zero [15]. Theory has been compared with measurements. For these experiments the centrifugal force is replaced by a spring force, the excitation is realized elec-

7 CONTACT PROBLEMS IN MULTIBODY DYNAMICS 91 Figure 7.6 Test-set-up for a turbine blade damper. tromagnetically, and the blades are represented by two bars. The geometry of the damper and the blade platforms is the same as for real dampers, but on a larger scale. The amplitudes of the damper and the blades are measured by six inductive displacement transducers and the spring force by a strain gauge arrangement. Figure 7.6 illustrates the setup and the measurement scheme to experimentally determine the stick-slip motion. One must keep in mind that the centrifugal forces in gas turbines are so large that the dampers move only occasionally. With respect to some time unit they move only about 10% of that time. Therefore, a slight improvement is worth an effort. One important magnitude of a damper is the excitation force amplitude F E necessary to break away from a stiction situation in both contact points K 1, K 2 in Fig These results give a measure for the damping capability of the devices. Figure 7.7 shows three curves for three different values of the spring force (centrifugal force) and for a contact angle of γ = 50 o. The excitation force amplitude F E [N] must be augmented for larger centrifugal (spring) forces F S. The very small F E values for an excitation frequency at 61 Hz can be explained in the following way. The value of f E = 61 Hz corresponds to an eigenfrequency of the blocked blade-damper system. This means that an excitation with these frequencies leads to large amplitudes of the blades and as a consequence to a considerable reduction of stiction. Therefore, the damper devices move from stick 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 configuration and its elasticity and by the main and nose carriages. The practical

8 92 FROM CONVEXITY TO NONCONVEXITY Figure 7.7 Comparison theory (,- - -, - -) and measurement (,, ). example of a landing airplane was the first one where the different sets of equations of motion could not be writen down explicitly, at least not with reasonable effort, but where some theoretical algorithm had to be established selecting the appropriate equations automatically [2], [17]. The internal structure of the carriage systems leads to impacts and to stick-slip phenomena, which influence the 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 For the shock absorbers it will be assumed that their bending deformation is homogeneous over the variable length, which means that the two components of the absorber do not bend in a different way. It turns out that this will be a sufficient approximation. During operation we have to consider various continuous and discontinuous nonlinearities. The tires possess a stiffness characteristic which increases progressively with the tire deformation. Moreover, the spin-up process of the wheels is governed by a highly nonlinear friction-slippage relationship. The shock absorber itself reveals some nonlinear features. The two gas chambers follow a polytropic compression and expansion law, the fluid flow losses induce a quadratic damping behavior, and the friction forces from ground reaction forces are nonlinear also. Discontinuous nonlinearities enter through the upper and lower stops within the shock absorber and through stick-slip processes between the two shock absorber cylinders (Fig. 7.9). Mechanically and mathematically they follow the theory as presented in the next section (see also [15]). The degrees of freedom resulting from this model include rigid degrees of freedom (number f r ) and elastic degrees of freedom (number f e ). With q r

9 CONTACT PROBLEMS IN MULTIBODY DYNAMICS 93 Figure 7.8 Planar mechanical model of a landing elastic airplane. R fr and q e R fe we obtain q T = [q T r, q T e ] = [x, y, α, y 22, y 23, ϕ 2, ϕ 3, q T 1, q T 2, q T 3 ] R f=fr+fe q T i = [q i1, q i2,..., q inai ], i = 1, 2, 3 n ai : number of shape functions of body i f : number of minimal coordinates. (7.6) The q i R nai are the elastic coordinates, (x, y, α) the three rigid degrees of freedom of the fuselage, (y 22, y 23 ) the translational and (ϕ 2, ϕ 3 ) the rotational carriage deflections. The derivation of the equations of motion follows the theory presented below and in [15]. Details may also be found in [2], [17]. Simulations consider a landing impact on a plane surface with a medium general aviation propeller-driven airplane consisting of a rigid fuselage and rigid carriages. The touchdown velocity is 4.3 m/s, and the horizontal velocity is 60 m/s. The landing angle of attack is α = 13 o, and the angular velocity is α = 0. The initial values of the geometric magnitudes y, y 22, y 23 are chosen in such a way that touchdown of the main carriage wheels starts with zero forces. The nose wheel has no ground contact for these initial values (α = 13 o ) [2], [15]. Elasticity has been considered by a Ritz approach and by taking into account the first five eigenfrequencies for the fuselage and the first two eigenfrequencies for the main and nose carriages. All other data were the same as for the rigid airplane.

10 94 FROM CONVEXITY TO NONCONVEXITY 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 landing impact. The center of mass moves about 300 mm (curve 1), and its velocity has a maximum at the beginning (curve 2). The forces acting on the fuselage from the main and nose carriages reach a maximum of about 170 kn for the main absorbers (curve 3) and a peak value of about 450 kn for the nose, where the nose load builds up later (curve 4).

11 CONTACT PROBLEMS IN MULTIBODY DYNAMICS 95 Figure 7.11 Elastic airplane landing impact carriage results [2]. The displacements are more or less the same as in the rigid case, the diagrams of which are not given here (see [15]). The nose wheel performs an internal impact at the upper stop (curve 7), and in the force and stroke velocity diagrams the lowest elastic fuselage eigenfrequency ( 7 Hz) can be detected. The elasticity neither increases nor decreases the tendencies for impacts and stick-slip (curves 8 in Fig. 7.11), but it reduces significantly the maximum load on the nose carriage. Assembly Processes Mating parts together involves various and sometimes quickly changing combinations of impacts with and without friction, of stick-slip processes and even of jamming. As these combinations and the transitions between them cannot be predicted beforehand because they depend on the current state, we need a theory which is able to deal with such processes. At the autor s institute this 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 theories we usually work on different levels: The event of a transition between contact configurations is either indicated by magnitudes of relative kinematics (at the beginning) or by those of constraint forces (at the end). The relative kinematical magnitudes undergo a metamorphosis from indicators to constraint equations as components of the constraint matrices valid for the contact configuration to come. The transition itself is governed by the complementarity of relative kinematics and constraint forces: If one group is zero the other is not. Considerations of that kind in combination with the accompanying mathematics [12], [15] allow an efficient modeling of assembly processes as described above. As an example we regard the peg-in-hole insertion performed by a manipulator [10], [18]. A robotic manipulator is usually considered as a tree-like multibody system with rigid and elastic components. Links very often are modelled as rigid

12 96 FROM CONVEXITY TO NONCONVEXITY Figure 7.12 Mechanical models of links and joints. τ A = c(γ M /i G γ A )+d( γ M /i G γ A ), where τ A is the torque at robot arm. γ M, γ A are the coordinates of motor and arm, respectively, c the joint stiffness, d the joint damping coefficient, and i G the gear ratio. bodies, only for special tasks as elastic bodies. Joints may ideally follow a given 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 freedom into motor or internal degrees of freedom (n M ), external ones (n A ) and elastic ones (n C ), which results in γ = γ M γa γ C R n, n = n M + n A + n C. (7.7) The equations of motion are then derived by applying Jourdain s principle which says that constraint forces in the joints are perpendicular to the direction of motion and do not affect power. With given kinematics of the manipulator under consideration, the mobility is specified by the Jacobians J T = v/ γ, J R = ω/ γ, where v, ω are the translational and rotational velocities, respectively. When we premultiply Newton/Euler s equations for each individual body with respective Jacobians the equations of motion can recursively be generated in the final form: M(γ) γ + ĥ(γ, γ) = Bû (7.8) with the inertia matrix M R n,n, centrifugal, coriolis and gravitational forces ĥ R n and the control input Bû R n.

13 CONTACT PROBLEMS IN MULTIBODY DYNAMICS 97 Because most mating tasks in assembly cells are limited to a small area of workpiece interaction the robot motion will be slow and centrifugal and coriolis forces may in the following be neglected compared to gravitational and inertia forces. Hence, the robot dynamics can be linearized around a given system state (γ 0, γ 0 ) = 0 and deviations are given by q = q M qa q C = γ γ 0, γ 0 = I G γ A,0 γ A,0, (7.9) 0 where the matrix I G R n M,n A represents the gear ratio that transforms the arm coordinates of the reference position onto the respective motor coordinates. The linearized equations of motion for the open loop dynamics then are M q + P q + Qq = h + Bu, (7.10) where P R n,n is the velocity proportional matrix containing damping terms of joints and compliance and Q R n,n is the stiffness matrix. The vector h R n contains the entire nominal dynamics and feedforward control with τ M0, where the motor drive torque results from τ M = τ M0 + u, u being the feedback torque. Bu R n is feedback control with the controller matrix B. For PD position control the feedback term typically has the form u = K P (q M q M0 ) K D ( q M q M0 ). (7.11) During assembly the workpiece held by the robot comes into various contact situations with the complementary mating part. This is illustrated in Fig for a planar peg-in-hole insertion task. In this process one or more contacts may arise which constrains the gripper motion and thus the robot s mobility by 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 theory for various configurations of assembly processes and on a consideration

14 98 FROM CONVEXITY TO NONCONVEXITY of disturbances during the peg-in-hole insertion. We first consider the perturbed motion during a planar peg-in-hole insertion task where forces only develop in the presence of uncertainties. For the verification of our theoretical approach experiments with a five degrees-of-freedom laboratory robot with a force-torque sensor were carried out. The fixture housing the complementary part is equipped with six distance sensors that measure the gripper s position and orientation and in addition disposes of two translational and three rotational adjustments which allow to produce definite positioning errors with respect to the parts that are to be mated. In a first experiment we investigated the insertion process when there was a lateral error between the peg and the hole. The respective numerical and experimental results are depicted in Fig The peaks in the force history at t = 0.9 s result from the impact when the peg hits the chamfer. When twopoint-contact is reached at t = 1.2 s the manipulator s motion is slowed down very fast until the jamming condition is reached. Only when the controller torques become large enough the motion is continued with sliding in two and finally even one contact point. This cycle represented by the oscillations in the force history repeats until the end of the trajectory and demonstrates the phenomenon of changing constraints during an assembly operation. Figure 7.14 Insertion force F x during assembly due to lateral offset. In a second experiment we investigated the permissible tolerance range of lateral and angular errors of the peg so that a maximum force of 5 N is not exceeded during the insertion phase. In this context we did not take into consideration the forces that develop while the peg is guided along the chamfer since this automatically involves the question of chamfer design. Fig contrasts the numerical results with some data obtained by experiment. The respective lines identified by theory correspond to the existence of different contact configurations where the maximum insertion force of 5 N is reached and limit the range of permissible positioning tolerances. The difference between numerical and experimental data mainly results from backlash in the last joint of the robot which was not modeled. Note that in both experiments the sensor

15 CONTACT PROBLEMS IN MULTIBODY DYNAMICS 99 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 PD joint controller was used. 7.3 PRESENT MATHEMATICAL FORMULATION Extensive experience with practical engineering problems has brought up the theory to a level which allows a very straightfoward formulation of contact problems in multibody systems. The formulation is kept as close as possible to the classical theory used in applied dynamics. This enables engineers to capture the main ideas of multivalued force laws quickly, to understand the theory behind them intuitively and to apply them in a correct manner. Even teaching this subject in a graduate course has turned out to be successful. In the following we will give a brief overview of the structure of the equations in the case of spatial Coulomb friction. Geometry of Surfaces We consider two bodies moving with v P i, Ω i (i = 1, 2) where v P i denotes the velocity of the body-fixed point P of body i and Ω i its angular velocity. Both bodies are assumed to be convex, at least in a neighborhood of a potential contact point, and may be represented by a parametrization of their surfaces in the form r P Σi = r P Σi (q, ξ i, η i ), see Figure 7.16 [10]. Note that r P Σ only depends on the surface parameters ξ and η when stated in the corresponding body-fixed frame whereas an additional dependence on the generalized coordinates q is observed when using arbitrary frames. For each of the bodies one defines an orthonormal basis (u, v, n) via the tangents in order to obtain s := ξ r P Σ, t := η r P Σ (7.12) n := s t s t, u := s, v := n u (7.13) s

16 100 FROM CONVEXITY TO NONCONVEXITY 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 ṙ P Σ and the trihedral ( u, v, ṅ) are ṙ P Σ = Ω r P Σ + R ζ with R = ζ r P Σ = (s, t) k = Ω k + K ζ with K = ζ k (7.14) where ζ = (ξ, η) T and (k, K) is any of (u, U), (v, V ), (n, N). The velocity v Σ of a point moving on the surface in terms of the rigid body motion (v P, Ω) is thus given by v Σ = v P + ṙ P Σ which yields v Σ = v C + R ζ with v C = v P + Ω r P Σ (7.15) Here, v C denotes the rigid body velocity portion of v Σ. Analogously we may find the acceleration a C = v C by differentiation of the second equation in (7.15), a C = a Q + Ω R ζ with a Q = a P + Ω r P Σ + Ω (Ω r P Σ ). (7.16) Finally we express v C and a C by the generalized magnitudes q and q, v C = J C q + ˆι C a C = J C q + ῑ C, ῑ C = J C q + ˆι C, (7.17) which involves the Jacobian J C that will later turn out to be the Jacobian of the contact point. Contact Kinematics Within the framework as presented in the foregoing section one is able to derive the main equations of contact kinematics in dynamics which are: The definition of the contact points, their distance, their relative velocities with respect

17 CONTACT PROBLEMS IN MULTIBODY DYNAMICS 101 to the normal and the tangential directions, and the corresponding relative accelerations. In order to do the first step one introduces a distance vector r D := r OΣ2 r OΣ1, (7.18) cp. [3], where r OΣi are vectors pointing from a common inertially fixed point O to the surfaces under consideration. Next, the two tangent planes spanned by (u i, v i ) are adjusted parallel to each other 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 n T 1 u 2 = 0, n T 1 v 2 = 0, r T Du 1 = 0, r T Dv 1 = 0 (7.19) and are called the contact parameters; the corresponding points at the surfaces given by r P Σi (ζ i ) are called the contact points. When the bodies are moving the conditions (7.19) for the contact points should not change [3], [10], thus ṅ T 1 u 2 + n T 1 u 2 = 0, ṙ T Du 1 + r T D u 1 = 0 ṅ T 1 v 2 + n T 1 v 2 = 0, ṙ T Dv 1 + r T D v 1 = 0 (7.20) must hold which constitutes a system of four linear equations in ζ i when substituting (ṅ i, u i, v i ) from (7.14) and v Σi from (7.15) with ṙ D = v Σ2 v Σ1. After the solution (ζ 1, ζ 2) of (7.19) has been found the distance g N of the contact points can be calculated as g N = n T 1 r D (7.21) with values greater than zero for separation and values less than zero for overlapping. The relative velocities of the contact points with respect to the three directions (u 1, v 1, n 1 ) are then given by ġ K = (v C2 v C1 ) T k 1, ( K, k 1 ) = ( U, u 1 ), ( V, v 1 ), ( N, n 1 ) (7.22) where it can be easily verified that differentiation of (7.21) with respect to time results in (7.22) for ( K, k 1 ) = ( N, n 1 ). The temporal changes of the relative velocities ġ K in (7.22) are given by g K = (a C2 a C1 ) T k 1 + (v C2 v C1 ) T k1, (7.23) where a Ci is known from (7.16), k 1 may be taken from (7.14), and ζ i involved in a Ci and k 1 is the solution of (7.20). Substituting v Ci and a Ci from (7.17) into (7.22), (7.23) yields a representation of the relative velocities and accelerations in the form ġ K = w T K q + ŵ K with w K = (J C2 J C1 ) T k 1 ; ŵ K = (ˆι C2 ˆι C1 ) T k 1 g K = w T K q + w K with w K = ẇ T K q + ŵ K. (7.24)

18 102 FROM CONVEXITY TO NONCONVEXITY Finally we arrange the two tangential relative velocities in the vector ġ T, i.e. ġ T T := (ġ U, ġ V ) which gives us the sliding direction of the contact points in the common tangent plane. With the abbreviations W T := (w U, w V ), ŵ T T = (ŵ U, ŵ V ), w T T = ( w U, w V ) we rewrite (7.24) in the form ġ N = w T N q + ŵ N, g N = w T N q + w N, ġ T = W T T q + ŵ T g T = W T T q + w T (7.25) which will serve as the contact kinematic equations in standard form in the sequel. Kinetics The dynamics of a multibody system with n bodies and one contact is described by the Newton-Euler equations in the form n i=1 [ J T S (ṗ F ) + J T R( L ] M) J T C1F C1 J T C2F C2 = 0, (7.26) i see e.g. [3], [15]. The terms p i and L i denote the momentum and the moment of 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 C 1 and C 2 are denoted by F C1 and F C2. In order to perform a projection into the configuration space all terms have to be premultiplied by the corresponding Jacobians. In particular, J C1 and J C2 are the Jacobians of the contact points which have already been introduced in equation (7.17). Decomposing the contact forces into components with respect to trihedral (u 1, v 1, n 1 ) admits, together with the cutting principle, a unique representation in the form F C2 = F C1 =: n 1 λ N + u 1 λ U + v 1 λ V. (7.27) Putting (7.27) into (7.26) and using the abbreviations w K = (J C2 J C1 ) T k 1 and W T = (w U, w V ) from equation (7.24) and below we arrive with M q h w N λ N W T λ T = 0, (7.28) where the vector λ T of the tangential contact forces is defined to be λ T T = (λ U, λ V ). The sum in (7.26) is taken into account by the term (M q h) with the symmetric and positive definite mass matrix M of the system. In the case of n A contacts eq. (7.28) clearly transforms to M q h n A i=1 which has still to be completed by the contact laws. w Ni λ Ni W T i λ T i = 0 (7.29)

19 Contact Laws CONTACT PROBLEMS IN MULTIBODY DYNAMICS 103 Generally, every force occuring in a rigid multibody system may be expressed in terms of some relative kinematic magnitudes such as relative displacements and velocities. Classical applied forces are represented by continuous functions whereas classical bilateral constraints already require a representation via setvalued mappings. Contact laws as used in the sequel belong to the latter class and constitute mixed-type force charateristics, acting in some areas as classical applied forces, in other areas as constraints. With g Ni being the distance of the contact points and λ Ni the corresponding normal force one might express impenetrability of rigid bodies by the Signorini- Fichera condition [1], [3], [7], [9], [12] g Ni 0, λ Ni 0, g Ni λ Ni = 0 (7.30) which might be combined, for example, with Coulomb-friction in the tangential directions [1], [7], [11], [12], i.e. ġ T i = 0 λ T i µ 0i λ Ni ġ T i 0 λ T i = µ i λ Ni, λ T i = α i ġ T i, α i > 0, (7.31) where µ i (ġ T i ) denotes the coefficient of friction for contact i and µ 0i = µ i (0). By introducing the index sets [3], [15], [20] I A = {1, 2,..., n A } I C = {i I A : g Ni = 0, ġ Ni 0} I N = {i I C : ġ Ni = 0} I T = {i I N : ġ T i = 0} (7.32) and by the use of certain continuity assumptions on the trajectories q(t) one is also able to express the contact laws (7.30), (7.31) on the acceleration level in order to finally solve equation (7.29) for the unknowns q. This yields, for the contact law in the normal direction (7.30) for i I A \ I N : λ Ni = 0 for i I N : g Ni 0, λ Ni 0, g Ni λ Ni = 0 (7.33) where the complementarity condition in the second line of (7.33) might also be expressed by one of the variational inequalities, cp. [3], [7], [12], [20], i I N g Ni (λ Ni λ Ni) 0, λ Ni 0, λ Ni 0 i I N ( g Ni g Ni) λ Ni 0, g Ni 0, g Ni 0 (7.34)

20 104 FROM CONVEXITY TO NONCONVEXITY for i I N. Similarly we obtain from (7.31) the friction law on the acceleration level in the form [20] for i I A \ I N : λ T i = 0 for i I N \ I T : λ T i = e i µ i λ Ni g T i = 0 λ T i µ 0i λ Ni for i I T : g T i 0 λ T i = µ 0i λ Ni, λ T i = α i g T i, α i > 0, (7.35) with e i being the unit vector of the sliding direction, i.e. e i = ġ T i / ġ T i. We recall that, due to the dependence on the normal forces, only quasivariational inequalities are available for the third portion of the contact law (7.35) which read [20] g T T i(λ T i λ T i ) 0, λ T i µ 0i λ Ni, λ T i : λ T i µ 0i λ Ni i I T ( g T i g T i ) T λ T i i I T µ 0i λ Ni ( g T i g T i ), g (7.36) T i. i I T Equations (7.21), (7.25), (7.29), (7.33) and (7.35) provide a complete description of impact free non-smooth rigid body motion under the influence of Coulomb friction in the framework of non-smooth analysis. The equations considered above thus allow any further discussion and evaluation by applying available theoretical results and algorithms from this field [1], [7], [11], [12]. Impacts have been excluded. They can be treated in an analogous manner by rewriting (7.26) as an equality of measures [11] and solving them for the impact times. This yields, together with appropriate impact laws, a set of relations similar to those considered here, with force magnitudes playing then the role of impulsions, and accelerations which have to be expressed by jumps in the velocities [3], [4], [15]. Application: Vibratory Feeders Vibratory feeders are just one example in machine dynamics on which the theory of spatial contact problems comes to a fruitful application [19]. They are the most common devices used to feed small parts in automatic assembly lines. Compared to other machines vibratory feeders seem to be quite simple. The high number of different types of devices based on the vibratory feeding principle, and the large amount of applications suggests that we are looking at a well developed and reliable tool. This impression is wrong. Problems in automated assembly are mostly caused by malfunctions of part feeders. These errors result from purely experimental tuning of the feeders, which is often done without a theoretical background, especially as far as the mechanics of the 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, mostly represented by an electro magnetic excited oscillator, and the dynamics of the

21 CONTACT PROBLEMS IN MULTIBODY DYNAMICS 105 Figure 7.17 Vibratory bowl feeder and mechanical model. parts transportation process. This yields a coupled system in the sense that the parts to be transported are affected by the vibrations of the track and vice versa. The base device including the drives and the drive mechanisms can be modeled as a bilaterally coupled system with well-known standard techniques. Nevertheless, this may result in an extensive job. Due to the highly sensitive contact mechanics, all oscillating frequencies and flexible structures have to be analysed very well. Furthermore, interactions with an oscillating environment must be taken into account. After having computed the vibrations of the base device which now serve as an excitation source for the parts one may concentrate on the modeling of the transportation process. The changes in the contact configurations between each of the parts on one side, and between the parts and the track on the other side are characteristic properties of the feeding process. Closed contacts either occur for entire time intervals or only for discrete points in time, the latter event being the usual outcome of an impact. Friction ist fundamental for the transportation of the parts: Indeed, it is the very feature that makes the feeder work. Consequently, the modeling of the process must be done with respect to friction effects. This leads to a structure variant multibody system with spatial contact 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 piecewise approximated by planes. Therefore, unilateral constraints result in point contacts, i.e. contacts between corners and planes or between lines and lines. Contact areas occuring at parallel lines and planes are composed of single point contacts. As one example of intense design parameter studies we present the transportation velocity, because of its outstanding importance for the basic function of the feeder. Not only fast transport but also robustness with respect to unsafe parameters are design criteria.

22 106 FROM CONVEXITY TO NONCONVEXITY Simulation has shown that the results obtained by using planar and spatial models nearly coincide. Even feeding with many parts, which was calculated for the planar model, does not influence the feeding rate. Therefore the averaged results for the transportation rate in the plane and spatial case as well as for one or for many parts are nearly the same allowing us to perform the layoutsimulations with one part only. In the following some results for the spatial model are presented. The model under consideration applies to Fig with only one rectangular block of 1 cm x 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 inclination have been chosen to be 100 Hz, 10 o, and 20 o, respectively. Fig shows the average transportation velocity depending on the excitation amplitude for different friction- and impact-coefficients. Every point of the curves results from one simulation which was performed until a stable average value was found. Positive velocities mean transportation upwards; negative downwards. Figure 7.18 Transportation rates of a vibratory feeder. In the left diagram of Fig we see the results for a friction coefficient µ = 0.2 and four different impact coefficients ε. Small amplitudes cause the conveyor not to work, resulting in pure sliding downwards of the parts. By increasing 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 coefficient of friction. In this case the transportation rate rises, and the conveyor is working even with bigger impact coefficients. These results can serve as a basis for the feeder design which allows to choose appropriate materials for the track, to find the best excitation parameters, and to estimate the expected feeding rate. The experimental verification of the presented results is the subject of current activities.

23 7.4 CONCLUSIONS REFERENCES 107 In the area of multibody theory regarding unilateral contacts the methodology has reached a state, which allows an application in nearly all fields of mechanical engineering. As unilateral contacts are realized in machines and mechanisms to a very large extend, the use of the corresponding theories is just at the beginning. The paper gives some examples, which demonstrate the typical features of unilateral constraints in combination with machines: They might be applied to achieve some functional performance like the transportation rate in vibratory feeders, or they might generate unwanted vibrations, noise and wear like rattling in gears or wear in chains. The main problem limiting today s applications is of numerical nature. To describe a machine like the vibration conveyor includes tedious numerical evaluations solving the complementarity problem at each time step. Even with modern high-speed computers we come out with a severe computing time problem. At the time being there are three groups of algorithms available to treat numerics. The enumerative methods try to find a solution by a kind of intelligent trial and error procedures. They do not satisfy. The pivot-algorithms are related to the well-known simplex-algorithm. The Lemke-method is an example. We apply it with good success, but with large computing times. Within the framework of iterative methods a modified Newton-approach seems to be very promising, even for the nonlinear complementarity problem. At the time being we are working on it. A second problem, also closely connected with numerics, consists in the solution of nonlinear complementarities as emerging from spatial unilateral contacts. Up to now we have used the following methods: linearization of the friction cones, the augmented Lagrangian methods and NCP-functions as they appear in Mangasarian s theorem. All methods work satisfactorily, all have to be 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 to understand much better problems including unilateral features, which in former times were treated only in a very simple way. We are able to analyse machines with unilateral contacts in a way, which five years ago we could not have thought of. Scientists like Professor Fichera have contributed to this development very significantly. References [1] Alart, P. and Curnier, A. (1991). A Mixed Formulation for Frictional Contact Problems Prone to Newton Like Solution Methods, Comp. Meth. Appl. Mech. Eng. 92 (3), pp [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 Starrkörpersystemen mit Reibung und Stößen. Fortschrittberichte VDI, Reihe 18, Nr. 182, VDI-Verlag,

24 108 FROM CONVEXITY TO NONCONVEXITY Düsseldorf. [4] Glocker, Ch. and Pfeiffer, F. (1995). Multiple Impacts with Friction in Rigid Multibody Systems, Nonlinear Dynam., Vol. 7, pp [5] Hajek, M. (1990). Reibungsdämpfer für Turbinenschaufeln. Fortschrittberichte VDI, Reihe 11, Nr. 128, VDI-Verlag, Düsseldorf. [6] Karagiannis, K. and Pfeiffer, F. (1991). Theoretical and experimental investigation of gear-rattling, Nonlinear Dynam., Vol. 2, pp [7] Klarbring, A. and Björkman, G. (1988). A mathematical programming approach to contact problems with friction and varying contact surface, Computers & Structures, Vol. 30, No. 5, pp [8] Küçükay, F. and Pfeiffer, F. (1986). Über Rasselschwingungen in Kfz- Schaltgetrieben, Ing.-Arch., Vol. 56, pp [9] Lötstedt, P. (1982). Numerical simulation of time-dependent contact and friction problems in rigid body mechanics. Technical Report TRITA-NA- 8214, Dept. of Numerical Analysis and Computing Science, The Royal Institute of Technologie, Sweden. [10] Meitinger, Th. (1997). Dynamik automatisierter Montageprozesse. Docotoral Thesis at the Dept. of Mech. Eng., Inst. B f. Mech. Tech. Univ. of Munich. [11] Moreau, J.J. (1988). Unilateral contact and dry friction in finite freedom dynamics, CISM courses and lectures, Nonsmooth mechanics and applications, Springer-Verlag, Wien, New York. [12] Panagiotopoulos, P.D. (1993). Hemivariational Inequalities Applications in Mechanics and Engineering. Springer-Verlag, Berlin, Heidelberg, New York. [13] Pfeiffer, F. (1984). Mechanische Systeme mit unstetigen Übergängen, Ing.-Arch., Vol. 54, pp [14] Pfeifer, F. and Hajek, M. (1992). Stick-slip motion of turbine blade dampers, Phil. Trans. R. Soc. London, A 338, pp [15] Pfeiffer, F. and Glocker, Ch. (1996). Multibody Danamics with Unilateral Contacts. John Wiley, New York. [16] Seyfferth, W. (1993). Modellierung unstetiger Montageprozesse mit Robotern. Fortschrittberichte VDI, Reihe 11, Nr. 199, VDI-Verlag, Düsseldorf. [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 Manipulationsvorgängen. Fortschrittberichte VDI, Reihe 2, Nr. 304, VDI-Verlag, Düsseldorf. [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.

25 REFERENCES 109 [20] Wösle, M. (1997). Dynamik von räumlichen strukturvarianten Mehrkörpersystemen. Doctoral Thesis at the Dept. of Mech. Eng. Inst. B f. Mech., Tech. Univ. of Munich.