AN ALTERNATIVE MODEL FOR STATIC AND DYNAMIC FRICTION IN DYNAMIC SYSTEM SIMULATION

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1 Preprints st FAC-conference on Mechatronic systems, September 8-2, 2, Darmstadt, Germany, Vol.2, pp AN ALTENATVE MODEL FO STATC AND DYNAMC FCTON N DYNAMC SYSTEM SMULATON Peter C. Breedveld Cornelis J. Drebbel nstitute of Systems Engineering University of Twente, EL/T, P.O. Box 27, 75 AE Enschede, Netherlands, tel.: , fax: , P.C.Breedveld@el.utwente.nl Abstract: A benchmark problem is defined to test potential solutions for re-usable submodels representing static ( stick ) and dynamic dry ( Coulomb ) friction in dynamic system simulation that are based on relevant physical port behavior. Next an alternative submodel is proposed with several advantages over other possible implementations. t is shown that dynamic causality and dynamic model structure can always be prevented and that some solutions for numerical computation can be completely embedded in a one-port resistor, if necessary externally modulated by the normal force between the contact surfaces. This makes these solutions highly re-usable, although they create a trade-off between accuracy and numerical stiffness. The key issue is that the generic expression for the contact force derived originally to solve the case with dynamic structure, results in a numerically robust and efficient submodel with static port causality and static model structure that does not create this numerical stiffness. Copyright 2 FAC. Keywords: (dry, Coulomb) friction, modeling, simulation, bond graphs, dynamic systems, models. NTODUCTON Many studies have been devoted to the phenomenon of (dry) friction between two rubbing and sliding contact surfaces. Consequently, there is an abundance of literature on this topic, also if one constrains this area of research to its role in dynamic system simulation and control (Armstrong-Helouvry et al., 994). The aim of this study is, after having introduced a demanding benchmark to be able to test and to compare re-usable submodel implementations, to introduce a generic re-usable implementation of static and dynamic friction. This study is made from the point of view of a port-based approach of the aspects of the physical processes involved that are relevant for the problem under consideration. 2. PHYSCAL BACKGOUND Friction is primarily an entropy-producing or dissipative phenomenon. Consequently, a key insight that follows from a port-based approach is that the constitutive port behavior that is to be identified is primarily a relation between the contact force and the relative velocity between the contact points. This leaves other parasitic effects, like pre-slip elastic displacement, open for modeling by separate submodels. Note that this does not mean that no states can be part of the friction submodel. Specific effects like frictional memory and rising static friction do involve states, but not energy states. For frictional memory, i.e. the effect that the friction depends on whether the contact surfaces are accelerating or decelerating with respect to each other, it is sufficient to have information about whether the absolute value of the relative velocity is increasing or decreasing, which requires a binary state. For rising static friction it is sufficient to keep track of another binary - ternary in case of anisotropy - state that allows distinction between slip and stick. Although the rising process itself has the nature of a relaxation process, it only modulates the value of the breakaway force, thus not having any impact on the power dissipated by the resistive port, which remains zero in the stick mode. Viscous friction, and other effects that have a continuous constitutive relation, can be added separately as regular one-port resistors. Therefore, this study concentrates on a classification of the different forms and implementations of the force-velocity relationship that describes the phenomena of static (characterized by a breakaway force) and dynamic friction (characterized by the so-called Coulomb force) in an idealized form. t is also shown, how the settling effect of the breakaway force or rising static friction, characterized by a so-called dwell-time, may be added to port-based friction submodel implementations. 3. A BENCHMAK POBLEM The port-based approach to the friction-modeling problem is not new. Karnopp (985) introduced a popular model and illustrated his derivation with a port-based notation called bond graphs (Karnopp and osenberg, 974). However, his model can only be applied in case of absolute velocities, i.e. the stick -mode is constrained to the case in which one of the two rubbing or sliding objects is at zero speed with respect to the inertial reference. Note that no straightforward extension of Karnopp s model is possible, as the model is a fixed combination of an -port and an - port of which the state (momentum) is also used to monitor the binary state (stick or slip). By contrast, the aim of this paper is to be able to deal with the following, more generic, situation that can be considered a benchmark: Two non-identical objects, e.g. flywheels with different inertia, are able to move frictionless with respect to the inertial frame (ideal bearings). However, they rub and slide against each other in such a way that static and dynamic friction at their mutual interface cannot be neglected. Note that the momentum of the two objects can still change while in stick mode, which shows that the momentum state cannot be used for keeping track of the slip-stick mode, like Karnopp s model does. Existing benchmarks seem to have been inspired by models being able to represent stickslip induced oscillations (limit cycle behavior) that are often deliberately generated in measurement set-ups. 77

2 Preprints st FAC-conference on Mechatronic systems, September 8-2, 2, Darmstadt, Germany, Vol.2, pp Consequently, they have springs attached between the objects or between an object and some external (e.g. inertial) point(s). n contrast with these benchmarks no springs are attached to the objects in the benchmark proposed above. This is a deliberate choice in order to be able to make the benchmark as demanding as possible on the potential solutions. As springs can be considered to have an additional stabilizing effect that may hide numerical instabilities caused by the implementation of the submodel, a model without springs is likely to give a better insight in the behavior of the friction submodel during simulation. Of course, springs may always be added later as separate submodels in order to obtain models that represent practical measurement situations involving limit cycle behavior. The benchmark experiment is as follows: The initial velocities of the two objects are slightly different (cf. table ). Shortly after starting the simulation both objects will come into stick mode. f a braking force (or torque) is applied next, two situations can occur. The resulting contact force between the two objects does ) not exceed the breakaway force that characterizes the static friction and consequently both objects are decelerated while keeping the same speed (they remain in stick mode) 2) exceed the breakaway force and consequently the object the braking force is applied to, is decelerated more than the other one. After releasing the brake, the second object that is still at a higher speed will decelerate while accelerating the first object until both reach the same speed again. This means that the objects go from stick into slip and finally into stick mode again. This case can be divided into the two situations where the braking force is applied such that the first object has or has not come to a full stop before the brake is released. Table : parameter values used in the benchmark element or parameter value units port J Nms 2 initial momentum Nms 2 J 2 Nms 2 initial momentum. Nms µ st.2 - µ c. - τ dwell. s v preslip. rad/s F N Nm M (brake) µ st.2 - µ c. - τ dwell. s v preslip. rad/s F N (brake t[2,3] (start, peak.5,.,.5; s input) and stop times of the triangle inputs).4,.5,.6 slope 3 Nms - Simulation Total time 4 s All friction models discussed herein have been exposed to a test in which a braking force is applied twice: first a brake action that corresponds to case and after the system has returned to stationary operation a brake action that corresponds to case 2, while the first object comes to a full stop before the brake is released. One may further think of the problem described above as two flywheels of different inertia on ideal bearings that have a rubbing or sliding mutual contact surface (think of two clutch plates). Braking torques can be applied separately to each of the flywheels, but in the benchmark only the brake on object becomes active and needs to be modeled. Note that the submodel is independent of the domain: it works both for translation and rotation. This explains the use of symbols for forces and velocities, which are considered torques and angular velocities in the example. Key elements of a bond graph model of this benchmark problem are two -junctions representing the two angular velocities of the wheels and their torque balances. To these -junctions the respective -elements representing the storage of angular momentum and kinetic energy of the flywheels are connected. A -junction is connected in between in such a way that the relative velocity is computed and this velocity itself is represented by a third -junction to which the resistive port of a friction submodel can be connected that represents the phenomenon under study, viz. the friction between the contact surfaces of the wheels. A braking torque can be represented by a modulated effort source, but is modeled by a similar friction submodel modulated and activated by the normal force in order to prevent accelerating torques. This also means that this benchmark checks on potential undesired (numerical) interaction between two friction ports. n contrast with regular bond graphs, the -junction representing the inertial reference velocity is shown explicitly, in order to be able to re-use all proposed implementations of the friction submodel. The brake activation is not realized by rather unrealistic step-like changes, but by a linearly increasing and decreasing normal force (triangle-shaped inputs), with the advantage that proper transition from stick into slip mode can be better checked. Figure shows the resulting bond graph after regular causality assignment. Sf : M Se: Figure : Bond graph of the benchmark problem M in stick mode v= Sf : Se: Figure 2: Causal bond graph of benchmark in stick mode with differential causality 4. FCTON MODELS WTH DYNAMC POT CAUSALTY Both inertias in figure have an integral causality as long as the resistive port is allowed to have the force between the contact surfaces as output and the relative velocity as input. nterpretation of the stick mode as an imposed, zerovalued, relative velocity results in a change of causality. This in turn results in a differential causality of one of the -elements (dependent inertia in figure 2), i.e. the causality of the model becomes dynamic. f one would try to implement this model, this would require the ability to switch between models and the ability to detect the mode. This means that a binary state is needed to keep track of the 78

3 Preprints st FAC-conference on Mechatronic systems, September 8-2, 2, Darmstadt, Germany, Vol.2, pp mode (stick or slip) and to control the switching of causality and structure. n case of anisotropic materials the binary state becomes a ternary state (positive slip, stick or negative slip), because there is no symmetry around the origin then. However, the rest of this paper is constrained to the symmetric, isotropic case, as an extension to the anisotropic case is straightforward. Sf : M TS in stick mode v= TS Se: have to be re-initialized at a lower order. Another problem becomes clear if the model goes from stick into slip mode. The criterion for this transition is that the torque between the wheels reaches a breakaway level. However, this torque is not explicitly present in the model of the stick mode in figure 4. Therefore, the model of figure 2 with the dependent inertia is reexamined and some symbolic analysis to derive an expression for this torque is performed. Sf : M TS in stick mode? TS Se: in slip mode in slip mode Stick detection Figure 3: Causal bond graph with dynamic structure (intermediate stage) An efficient way to implement the required memory element to track this state is a time delay of a Boolean variable, e.g. named slip, with one computation time step. The previous value of slip may be used to switch from stick to slip vice versa. Next, this state has to be detected in such a way that the transition from stick to slip is based on a force criterion, the breakaway force, while the transition from slip to stick is primarily based on a velocity criterion as the causality has changed in principle. Figure 3 shows that this could be realized (it will not be the final solution!) by adding a slip detection submodel and two tumbleswitch-like -junctions ( TS ) that are switched by the binary mode. This means that the model structure becomes dynamic too. Furthermore, the states of the two models have to be transferred. For the -element that has an integral causality in both cases, this can be realized by creating a two-port -element with only one state of which the rates are switched by combining the two corresponding -elements in figure 3. However, the -element that gets a derivative causality not only makes the model implicit in the stick mode, but also has no state connected to it that can be directly manipulated. Most solutions will largely depend on the nature of the implicit integration routine used. The discussion below will clarify, why it hasn t even been considered to further implement the model along these lines, as it would be unnecessarily complex and not robust with respect to numerical routines. Closer inspection of figures 2 & 3 shows that the model in stick mode can be simplified into the bond graph of figure 4 that does not contain differential causalities. Consequently, port causality is not dynamic anymore, but the model structure is still dynamic. When the stick mode is now detected, the model has to be switched from the one in the lower part of figure 4, to the one in the upper part. This means that the momenta of the two inertias in the lower part have to be combined and transferred to the inertia in the upper part of the model in figure 4. For simulation, this requires the ability to reset integrator contents during simulation, which may become a complex process for some complex numerical integration procedures, especially in case of higher order methods that Stick detection Figure 4: Causal bond graph of the benchmark problem in stick mode without derivative causality (intermediate stage) 5. COMPUTATON OF THE NTEFACE FOCE O TOQUE n order to be able to decide whether or not two objects are in stick or in slip mode, it is obvious that the interface force (or torque) has to be compared with the breakaway force that parameterizes the static friction. The causality of the bond graph model in figure 2 shows that in the stick mode the two inertias of the moving objects are dependent. A straightforward symbolic analysis of the original model shows that the constraint that all velocities in the model are equal leads to the following expression for the tangential interface (or contact) force F r : Fm 2 Fm 2 Fr = () m+ where F i is the external force (torque) on object i with mass (inertia) m i. Note that this is a generic expression that may be used for all sorts of hard contact situations, not only for tangential forces, but also for the case of normal forces at stops (e.g. backlash, etc). t is shown that this generic form converges to more a familiar expression in the special case that one of the objects (e.g. the second) is considered fixed to the inertial reference. This can be considered a limit operation in which the corresponding mass goes to infinity (physical origin of a source of constant velocity): Fm Fm 2 2 lim ( Fr ) = lim ( ) = F (2) m m 2 + However, in order to be able to reuse the submodel, the generic form () is used. The resulting submodel will not be implicit and can be given a dynamic structure in such a way that no dynamic port causality has to be implemented (figure 5). The reader is warned that this is an intermediate step in the process of obtaining the final submodel that will even have a static structure. 79

4 Preprints st FAC-conference on Mechatronic systems, September 8-2, 2, Darmstadt, Germany, Vol.2, pp M TS S_2 2 port slip detection Figure 5: Causal bond graph with dynamic structure and integral causality (intermediate stage) Se 6. FCTON MODELS WTH STATC POT CAUSALTY AND STATC STUCTUE 6. Friction models with pre-slip velocity n case of a static causality and structure the defined port causality is such, that the tangential contact force is always expressed as a function of the relative velocity between the contact surfaces. For instance, the dynamic or Coulomb friction can be implemented by using the discontinuous signum operator (sgn) or smooth approximations like a hyperbolic tangent (tanh) or an arctangent (arctan). n that case, no causal problems occur and the model can be implemented in a straightforward manner. n order to be able to result in (sustained) limit cycle behavior, a friction model should represent static friction too, as this creates a zone with a negative slope. This zone is sometimes called Stribeck zone, even though this zone may be small as the negative derivative may go to infinity in some implementations. n other cases the complete constitutive relation that contains this zone is called the Stribeck curve. ts implementation requires a transition behavior that is dependent on the force reaching the breakaway level or the velocity becoming (almost) zero, which requires some care in order to reach a robust solution. f the static friction phenomenon needs to be added while the force has to remain the output variable (i.e. static port causality), this can be done by creating a small region around the origin in which the force is allowed to build up to the breakaway level. This can be implemented in a piece-wise linear or nonlinear smooth way, but results in a relatively large differential resistance around the origin, which in turn may result in a numerically stiff set of differential equations. This means that ordinary, fixed step integration routines like the common unge-kutta 4 routine, result in time-consuming simulations, when the maximum value of the velocity that is allowed in the stick mode is small. However, variable step, variable order methods are able to deal with these kinds of models rather efficiently, such that they can be used successfully in many situations. f it is assumed that there is symmetry around the origin (isotropic materials are assumed), the relations can be derived on the basis of the absolute value of the velocity. This is demonstrated below using the signum operator for the piece-wise linear case. Note that the velocity at the previously computed time step is needed to deal with frictional memory (a delay operator has a state). n pseudo code: Fr( tk ) = if [ vt ( k) < vpreslip] not slip( tk )] vt ( k ) then ( ) v preslip else sgn( vt ( k)) end; slip( tk) = if [ vt ( k) > vpreslip] then true then falseelse slip( tk ) end end; where the parameter v preslip is the velocity margin ( earea ) around the origin before slip occurs, = µ st, = µ c, with F N the normal force on the contact surface, µ st the static friction coefficient and µ c the dynamic friction coefficient; t k is the time at the current, (3) else if vt ( k) <.* vpreslip (4) k th simulation step, whereas t k- is the time at the previous, (k-) th simulation step. Note that F N may be either a parameter, resulting in a regular nonlinear resistor, or a variable, resulting in an externally modulated nonlinear resistor. The peak to the breakaway level is only turned on during acceleration, not during deceleration. Note that the piecewise linear parts can be replaced by smooth approximations, e.g. by using a hyperbolic tangent or an arc tangent. This will not be discussed separately as the basics of the implementation remain the same. Sf Mscpl brake scpl Figure 6: Causal bond graph with static causality and structure with pre-slip velocity Benchmark simulation experiments with the fixed causality submodel (cf. figure 6) discussed above demonstrate that this kind of submodel result in conceptually simple (rest of the model remains standard) and reusable solutions that are numerically efficient as long as efficient numerical algorithms and/or simulation packages are used. Only in those cases where a fixed step integration algorithm is used together with a restriction to a relatively small pre-slip velocity, simulation may become inefficient. n that case a solution without pre-slip velocity should be considered. From the earlier discussion it seems reasonable to conclude that this seems to require a dynamic structure. However, additional use of relation () for the contact force leads to a surprisingly simple and efficient submodel with the advantages of static port causality and static structure. This is discussed next. 6.2 Friction model without pre-slip velocity The expression derived earlier for the contact force during stick can be given a much wider use than just testing the slip criterion: if the output of the resistive port is made equal to this expression in stick mode, both inertias, although independent, will have the same velocity as output: their rates of change are: 72

5 Preprints st FAC-conference on Mechatronic systems, September 8-2, 2, Darmstadt, Germany, Vol.2, pp Fm 2 Fm p 2 = F Fr = F = m+ m = ( F+ F2) m+ Fm 2 Fm p 2 2 = F2 + Fr = F2+ = m+ = ( F+ F2) m+ such that the accelerations are equal: p 2 F+ F2 p v 2 = = = = v (7) m+ m This means that if the velocities are equal at the moment the stick mode is entered (which is the criterion for this transition and the definition of the stick mode!), the velocities will remain the same while in stick mode. As the numerical integration process is identical for both - elements, non-negligible differences will be mainly the result of inaccuracies in the determination of the transition from slip into stick mode, because any remaining velocity difference at the moment of switching will be preserved. f this solution is implemented using event constructs to improve accuracy of the detection, one should be aware of the fact that an event construct is generally implemented to detect a zero crossing. This means that relaxation behavior where the relative velocity becomes equal to zero within the numerical accuracy without changing sign will not be detected. Checking whether or not the velocities remain constant provides a way out: n case the velocities are constant, it can be checked whether the relative velocity is within a certain error margin from zero in order to properly detect the transition into the stick mode. The restriction to constant velocities needs to be made, because always checking the transition with this error margin would make the zero-cross event detection less accurate. Mscd (5) (6) zero during stick, the power of this port remains zero too, thus showing that this relaxation process purely has a modulating nature and can be implemented by an exponential solution of which the start time t start is reset to the current time at the moment the stick mode is entered. While in slip mode, the start time is kept equal to the process time in order to immobilize this process (in pseudo code): t ( t ) = if slip( t ) start k k (8) then t else t ( ) end; k start t k where slip is the Boolean state that monitors the stick or slip mode. Naturally, the static friction has to be immobilized too when the normal force changes sign: = if > then if τdwell == then µ st else (9) ( t t )/ ( ( ) start τ µ dwell st µ st µ c e ) end else end; By setting the parameter of the dwell time to zero this effect can be switched off. v v 2 F N brake (input) Slip (Boolean state) computed points are shown F st with rising effect Figure 8: qualitative impression of 2-sim variable step (BDF) simulation result of model in figure7 Sf scd Figure 7: Causal bond graph with static causality and structure without pre-slip velocity Note that this new implementation of the friction submodel (figure 7) is inspired by dynamic causality and dynamic structure respectively, but has a static structure, due to the use of the symbolically derived expression for the contact force (). The price that is paid for eliminating the need for a pre-slip velocity to realize a static structure, is that the submodel is not completely contained in the resistive port: information about the external forces (i.e. modulating signals) and the inertias of the objects involved (i.e. parameter relation across submodel boundaries) remains necessary. 7. ADDTON OF SNG STATC FCTON Ability to represent rising static friction requires an addition of exponential, relaxation-like behavior to the submodel. The breakaway force increases to its maximum level with a time constant that is called dwell time. As the flow of the resistive port, viz. the relative velocity, remains 8. SMULATON EXPEMENTS Due to page limitations no extensive results can be shown. The reader will have to take for granted that implementations of the submodels and benchmarks discussed above have been fully implemented and tested. An exception is the first intermediate model with dynamic structure and derivative causality, implementation of which would have been a major task leading to a result that is doomed to poor performance compared to the implementation without derivative causality. The result in figure 8 gives an impression of the velocities of the two objects, the binary slip state, the brake activation and the breakaway criterion, demonstrating the effect of the dwell-time. The model allowing pre-slip velocity gives virtually identical results. The simulation window of the benchmarks is chosen large with respect to the disturbances and the actual dynamic behavior in order to emphasize the advantage of variable step-size methods (note the computed points at the binary slip signal ). n case of use of the BDF method, all simulations could be performed with high accuracy on reasonably standard PC s running Windows 98 or NT (ranging from Pentium 66 to Pentium 45) using 2-sim (2) with simulation times (including graphical /O) of the order of one second (i.e. four times real-time). Proper comparisons should be made 72

6 Preprints st FAC-conference on Mechatronic systems, September 8-2, 2, Darmstadt, Germany, Vol.2, pp on the basis of the CPU time on a stand-alone system, which is even a small fraction of the total time. The above numbers are just mentioned to give a crude impression. n case variable step integration is used not much computation time is gained by the newly derived model without a preslip velocity, although it is numerically more robust. However, in case fixed-step integration routines are used, the new model performs about ten times faster on the benchmark. The re-usability of the submodels is excellent. n both cases it was possible to obtain a running simulation within a few minutes of an extension of the benchmark from a chain of two flywheels to one of four slipping or sticking flywheels. Another minute is required to add linear -elements to add viscous friction to the model. Sf brake Mscpl scpl scpl scpl Figure 9: e-usability: simple extension of the benchmark to four slipping flywheels (the result is shown for the model with pre-slip, cf. figure 6) Mscd Sf "m" = zero scd scd scd m m3 m4 Figure : Modularity: straightforward extension of the case used in figure with viscous friction (the result is shown for the model without pre-slip, cf. figure 7) v v 2 v 3 v time Figure : Simulation result of figure 9 (same for scd) Figures 9-2 and give an impression of the results in both cases (with and without pre-slip) after a braking action that brings the first wheel to a complete stop while the normal forces between the wheels are getting less when going further away from the brake. n this case simulation with both types of submodels results in total simulation times of about ten times real-time, with the exception of the K4 method for the model with pre-slip. Simulation time is then of the order of real-time. need for resetable integrators. This implementation has proven itself in the benchmark test and other models and has the advantage over nonlinear resistors with a pre-slip velocity zone that it does not result in numerically stiff models. However, if the numerical stiffness is not a problem, the submodels with pre-slip velocity are preferred due to the fact that such a submodel needs no additional external information, which makes re-use very simple. This does not mean that the new submodel is hard to re-use: the only additional actions that have to be taken are that the submodel is provided with information about the external forces on the objects involved and about their inertias (cf. figure 7 and ). The port-based approach has thus proven to be crucial to obtain the presented submodel. Further improvements of the possibilities in numerical integration engines of simulation software to use event constructs in numerical routines may extend the possibilities to create robust re-usable submodels other than those with a static structure discussed herein. One can think of options like the detection of approach to zero within the numerical accuracy but without zero-crossing, or like combination of integrator resets with finite state machines using variable step methods v 2 v v 3 v Figure 2: Simulation result of figure (same for sclp). EFEENCES Armstrong-Helouvry, B., P. Dupont, and C. Canudas de Wit (994), A Survey of Models, Analysis Tools and Compensation Methods for the Control of Machines with Friction, Automatica, Vol. 3, No. 7, pp Karnopp, D.C. (985), Computer simulation of stick-slip friction in mechanical dynamic systems, ASME J. Dynamic Syst. Meas. & Control, 7 (), -3. Karnopp, D.C., and.c. osenberg (974), System Dynamics: A Bond Graph Approach, Wiley, N.Y. 2-sim (2), modeling and simulation software: all discussed models can be simulated with the free demo version obtainable from but only with fixed step methods and without speed-up by built-in compiler; (sub-)models upon request from the author. 9. DSCUSSON AND CONCLUSONS A new submodel for static and dynamic friction has been found by studying a port-based representation of the physical interactions and by drawing conclusions about the causal consequences: dynamic causality was replaced first by dynamic structure with derivative causality. Next derivative causality was eliminated by symbolic analysis. Finally, the key step was taken to use this symbolic result also to make the model structure static, thus preventing the 722