An Energy-Based Approach to Parameterizing Parasitic Elements for Eliminating Derivative Causality

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1 An Energy-Based Approach to Parameterizing Parasitic Elements for Eliminating Derivative ausality Geoff ideout and Jeffrey L. Stein Automated Modeling Laboratory Department of Mechanical Engineering University of Michigan Ann Arbor, M USA Keywords constraints, parasitic elements, causality, activity. ABSTAT Stiff coupling springs can be used at mechanism joints to eliminate derivative causality in multibody system models. A tradeoff exists between having parasitic springs that are stiff enough to resist excessive deflection and thus enforce the constraints, and minimizing the numerical stiffness of the mathematical model. Previous parameter selection techniques have ensured that parasitic springinduced eigenvalues are greater by a suitable factor than the largest physically meaningful natural frequency. One of the difficulties associated with these spectral techniques is that they require the system be linearized, taking into account the variation of eigenvalues as the equilibrium configuration changes. Further, severe inputs may cause excessive joint separation due to static or dynamic forces, regardless of the decoupling of the parasitic and system modes. A physical domain technique is proposed to quantitatively assess the contribution of parasitic elements to overall system response, and thus the extent to which the elements allow spurious motion. The activity (time integral of the absolute value of power flow) of parasitic springs is calculated and interpreted in light of physical performance to ensure suitable approximation of the kinematic constraints without creating undue computational burden. An illustrative example demonstrates the potential of the proposed technique. NTODUTON To successfully simulate the response of a multibody mechanical system (MBS) is to overcome computational challenges involving a considerable number of variables and equations with a high degree of nonlinearity. Dependencies generally exist among momenta in a rigidly-constrained MBS, resulting in an implicit set of differential equations and constraint equations that are nonlinear functions of geometric variables. The modeler must decide on one of the following options to extract useful simulation output from the model [osenberg and Mcalla, 99]:. integrate the governing equations directly using an implicit integrator 2. introduce parasitic stiffness and/or resistance elements to remove dependencies among energy storage elements, and then formulate and solve explicit differential equations 3. combine local element effects to eliminate dependent energy variables, or attempt to eliminate dependent variables symbolically. Despite advances in the robustness and efficiency of implicit numerical methods since Option 2 was formally proposed by Karnopp and Margolis (979), there is still justification for Options 2 and 3 depending on the modeling objective and the complexity of the system. This is because it is difficult to ascertain the extent to which internal constraints are being enforced in the numerical analysis when directly integrating implicit equations [osenberg and Mcalla, 99]. Option 3, removing derivative causality, can sometimes be accomplished using methods in which variables are transformed to produce explicit energy storage fields [osenberg and Mcalla (99), Allen (979), and Karnopp (992)]. However, Option 2 may still be desirable if the symbolic transformation is extremely complex. A model augmented with parasitic elements does not bring with it the additional cost of mapping the transformed variables back to the physical system for design or physical-domain model reduction. Augmentation with parasitic elements is obviously justified if actual bearing stiffnesses are fairly low and contribute eigenvalues with a frequency on the order of other system natural frequencies. The stiffnesses may be calculated from the principles of solid mechanics. f, however, the constraining elements are assumed rigid, the compliant elements become a numerical analysis tool and must be chosen carefully. The stiffnesses must be artificially low (compared to infinity) but not so low as to generate spurious relative motion in the joints that corrupt the predictions of the modeled system states. Near-infinite stiffnesses introduce excessively high-frequency dynamics and require very small integration time steps. Stiff integrators are typically not robust for systems with discontinuities in the system elements or inputs [iley,

2 2], and artificially high damping may be required for convergence if the parasitic dynamics are too fast. Previous approaches to systematic selection of the parasitic parameters will be shown in the next section to have limitations. A time-domain measure of the physical contribution of the parasitic elements to the overall system dynamics is proposed as a tuning criterion MOTVATON FO NEW SELETON METHOD Previous Methods Guidelines for selecting parasitic element parameters in the general case have focused on the eigenvalues of a linearized, constant-coefficient state-space representation of the system. The system natural frequencies can be identified and tracked as parasitic spring stiffnesses are increased. As the lowest parasitic eigenfrequency exceeds the system natural frequencies, the system modes become decoupled from the coupling spring modes and the system behavior approaches that of the ideally constrained [Karnopp and Margolis, 979]. For an explicit nonlinear system, the unforced state equations are amenable to linearization and one can quickly compute the eigenvalues resulting from the stiff springs. Karnopp and Margolis (979) successfully simulated a system augmented with parasitic compliances that introduced a lowest spring-generated frequency 2.4 times the actual system frequency (this factor for spectral decoupling was not suggested as a general rule of thumb). The fixed integration time step was chosen based on the highest parasitic frequency. Zeid (989) proposed setting the parasitic spring stiffnesses by first normalizing the bond graph and getting a linearized, constant state matrix. The bound on the largest imaginary part of the eigenvalues of the system with coupling springs was computed as a function of the parasitic stiffnesses, which were then set so that the eigenvalues were ten times the highest natural frequency of the original system. Zeid (989) also noted that using parasitic resistances in place of springs would introduce no new states, but if the initial joint displacements were nonzero then no restoring forces would be generated. Gradual separation of the joint might still occur even if the initial joint displacements were zero. Eigenvalue-based parasitic element tuning is primarily limited in that it does not take into account the possibility that a severe forcing function may induce inappropriately large coupling spring states even if the parasitic dynamics are much faster than those of the original rigidly-constrained system. Even if the spectral sizing methods consider the frequency content of the input as well as the system natural frequencies, changes in input amplitude can increase joint deformations without changing the input spectrum. Another disadvantage of eigenvalue-based tuning is that it requires careful and repeated system linearizations to compute the eigenvalues, which will change as a function of the equilibrium configuration. ase Study Slider rank System With Severe nput The following case study will demonstrate that severe inputs can cause excessive parasitic spring deflections even when parasitic eigenvalues and system natural frequencies are decoupled. onsider the slider-crank mechanism of Figure, which employs parasitic springs K S in the x- and y- directions at the crank-link pin, and another parasitic spring in the y-direction to constrain the slider. K S (x-direction) K S2 (y-direction) Figure. Slider-rank Schematic The system bond graph is shown in Figure 2. The slidercrank parameters are given in Table below. Table. Slider-rank Parameters Parameter Value J.24 kg-m 2 a.5 m m 8 kg J kg-m 2 b 2. m K N/m 5 N-s/m S K S3 (y-direction) The bond graph of this system was implemented in the 2SM bond graph simulation environment [2SM, 22]. A rigidly-constrained system model was simulated with an implicit integrator to give the baseline against which the explicit system s performance was compared. The first simulation was run with no parasitic damping, and with 6 N/m coupling springs. To ensure the parasitic and system dynamics were uncoupled, the system was first linearized about crank angles ranging from to 2π radians. The system natural frequency, i.e., the natural frequency due to the interaction of the inertias and the restoring spring K, was computed and is shown in Figure 3 to vary as a function of operating point. The Decoupling Factor the ratio of the lowest parasitic natural frequency to the system frequency - varies significantly, with a low of about 7 when crank angle is around nπ radians (Figure 3). n the absence of a welldefined lower bound for the Decoupling Factor, the simulation was run with 6 N/m coupling springs and the parasitic dynamics assumed decoupled.

3 J MSe Torque rank w _asin acos vx vy stiff KS Parasitic Elements _b2sin b2cos w2 m_x J2 _b2sin2 b2cos2 Slider Elements damper spring KS2 stiff2 Parasitic Elements Link m_y KS3 stiff3 Parasitic Elements Figure 2. Slider-rank Bond Graph Figure 3. System Eigenvalue and Parasitic Decoupling Factor vs. onfiguration Applied Torque [N-m] DEOUPLNG FATO ANK ANGLE [rad] time {s} Figure 4. Torque nput to rank Having selected parasitic springs based on eigenvalue decoupling, the forced response of the system was then studied to evaluate their performance. The system was subjected to the input torque shown in Figure 4, which generates rapid anti-clockwise crank rotation starting from standstill. The initial crank angle was 45 degrees, at which point the spring was undeflected. An arbitrary amount of damping SYSTEM EGENVALUE DEOUPLNG FATO SYSTEM EGENVALUE ( N-s/m) was added in parallel with the parasitic springs to dissipate high-frequency vibration, without overdamping the parasitic modes and thereby increasing the spectral radius. Figure 5 shows the divergence of the ideallyconstrained and parasitic-spring-augmented predictions of crank velocity due to inadequate joint constraint enforcement. The maximum coupling spring displacements were 9.6, 8., and 7.5 mm - errors on the order of 4% of the crank length. The input should be taken into account in this system, and the parasitic springs selected so that not only are their dynamics decoupled in the frequency domain, but also their deflections are low enough to satisfy the modeler that a rigid constraint is adequately approximated. By comparison, arbitrarily increasing the spring stiffnesses to 8 N/m renders the crank velocity time series virtually identical. rank Velocity w/ Parasitic Springs deal rank Velocity time {s} Figure 5. rank Velocity [rad/s], K S = 6

4 A power-based technique previously used for physical-domain model reduction will next be applied to the problem of parasitic spring and damper parameter selection. The joint deflections resulting from the system dynamics and the input are correlated with the power-based metric. The technique can be applied to nonlinear systems and is proposed as a guide to aid the modeler in sizing each of the parasitic elements separately. PAAST SPNG TUNNG USNG ATVTY The activity metric proposed by Louca et al. [997] correlates the time integrated magnitude of power flow through a lumped-parameter element with that element s contribution to the overall system dynamics. For a given time interval, the activity of each element is calculated and compared to the activities of the other elements. Activity over the time interval [t i, t f ] is formally defined as in Equation. tf A= effort flow dt () ti n the traditional application of the activitybased model reduction algorithm, the modeler would eliminate elements in order of activity contribution from lowest to highest until model prediction degraded beyond an acceptable level. Typically a threshold is set such that lowestactivity elements are removed until a certain percentage of the system activity is eliminated. n this study, low activity of a parasitic coupling element is taken as a sufficient condition for the element to adequately enforce the joint constraint. Low activity is not a necessary condition, as the coupling springs may be so soft as to generate negligible force (effort) and thus have low activity at this point of course, their efficacy in enforcing the constraints would be called into question by the astute modeler. As in the application of MOA [Louca et al., 997], and indeed the spectral method of sizing parasitic elements, the threshold below which an element s activity is considered negligible is not immediately known. A given activity threshold cannot be analytically mapped back to bounds on the response of physicallymeaningful system variables. The utility of activity lies in its direct application to the nonlinear system, and in its correlation with the coupling spring velocities and forces induced by not only system free response but also the external forcing function. Table 2 demonstrates the correlation between high coupling spring stiffness, low activity, and accurate system response as indicated by the coupling spring states. Note that increasing the spring stiffnesses from 6 to 8 N/m increased the computation time with a variable-step stiff-system integrator from 3.6 to 25.6 seconds. The 6 N-m springs have an activity on the order of the lowest-activity physical system element (crank rotational inertia J see Appendix for list of system element activities), and can be expected to affect system output. ncreasing the stiffness forces the activities an order of magnitude below that of the system elements, where their effect on system output will be reduced drastically. ncreasing the input amplitude excites the joints to the extent that the coupling springs are not even the lowest-activity elements in the system. For this input, activity signals that the springs must be again resized. The complex interactions within the system, and the very nature of the activity metric, preclude inverting the parasitic element activities to determine the amount of physical constraint violation, or even setting a generally appropriate threshold activity percentage. Equal activities of the coupling springs do not result in equal peak displacements. Stiffnesses K S [N/m] Table 2. Stiff Spring Activity and Deflection (No Parasitic Damping) Activity atio Max.Deflection [mm] (K Si Act./Lowest System Element Act.) K S K S2 K S3 K S K S2 K S (nput Torque Quadrupled)

5 The modeler may only be assured that the farther the parasitic spring activities are below the lowest original system element activity, the smaller the effect of those parasitic elements will be on the system response. The smaller the effect, the better, in light of the springs role in approximating a rigid constraint; however, one may not desire overly-conservative stiff elements and the attendant computational penalty. Nonetheless, it is proposed that activity can be used to ensure that parasitic coupling springs approximate rigid kinematic constraints. f the modeler is interested first and foremost in approximating an ideal constraint with coupling elements that do not affect the dynamics of the rest of the system, then a time-domain simulation and activity calculation can be performed and the parasitic element parameters chosen so that their activity is, say, an order of magnitude lower than the activity of the least active system element. The modeler can vary the individual stiffnesses until this is achieved. The advantages of this method are ) it can be directly applied to the nonlinear system model and the check performed by simply doing a timedomain simulation of the explicit system; 2) the effect of the input is considered, in that the activity of the elements is affected by both free and forced response; and 3) the elements can easily be tuned individually, rather than all being assigned a single value that exceeds the lowest system frequency (which varies as configuration changes) by a certain factor. llustrative Example For a.75 second simulation with the torque input of Figure 4, the spring stiffnesses were manually adjusted until the activity of each was approximately % of the lowest-activity system element (Activity atio of.). The final values and maximum spring deflections are shown in Table 3. Table 3. oupling Spring Stiffness For Activity % of Lowest System Element Spring Stiffness [N/m} Max. Deflection [mm] Activity atio (per Table 2) K S 6.67x K S2.2x K S3.629x Figure 6 shows the crank velocity prediction for the system with parasitic spring values from Table 3. The rigidly-constrained system results are shown for comparison. The velocity peak magnitudes and times are more accurately predicted than in Figure 5. Figure 7 compares the deflection of K S in meters when the stiffness is selected according to the activity method versus the spectral method in Section 2.2. For this example, the maximum deflections of K S are approximately 8 times smaller when the spring stiffness is governed by the activity threshold. The reductions in the deflections of K S2 and K S3 are significant but not as great, reflecting their lower original activity ratios when all stiffnesses were 6 N/m. Larger joint forces, whether from free or forced system response, increase the activity of the associated parasitic spring. For the example slider-crank with a torque input the same shape as that in Figure 4, the activity values confirm that the greatest joint separation forces occur horizontally at the crank-link pin. rank Velocity w/ Parasitic Springs deal rank Velocity time {s} Figure 6. Predicted rank Velocity omparison Spring Displacement - K S rank Angle [rad] Figure 7. Deflection of Spring K S vs. rank Angle. K S = 6.67 MN/m K S =. MN/m

6 DSUSSON Previous research, especially by Louca et al. (997), has shown the effectiveness of activity as a predictor of an element s contribution to overall system dynamics and response. This recommends activity as a measure of when an element such as a parasitic spring does not contribute to system dynamics. The greater the separation between the parasitic spring activities and the system element activities, the less the parasitic springs will contribute to the dynamic system response and therefore the better they will approximate a rigid constraint. The reader is cautioned that the order-of-magnitude separation of parasitic and system element activities in Section 3. is not presented as the definitive criterion for selecting parasitic elements that will behave unobtrusively. The modeler may in practice settle on values of 5%, or %, of the lowest system element activity. The activity method does not at this point solve the threshold selection problem that is also inherent in the eigenvalue-based method, but the fact remains that the activity method will not be as easily undermined by the input. Of course, if one were interested in studying high frequency vibration or fatigue life of the links, then use of the actual physical joint stiffnesses would dictate the stiffness values. The coupling stiffnesses would no longer be parasitic - included only to preclude derivative causality - and the integrally causal bond graph would be more complete as Karnopp and Margolis (979) suggest. Activity is preferred in general over sizing the springs based on a measure such as maximum joint deflection. While one could maximize the spring stiffnesses until some physical constraint was violated, it is not obvious which of innumerable metrics such as maximum spring state, integral of absolute value of spring state, etc. would best describe adequate joint behavior. The use of small peak joint deflection as a sizing criterion may not reveal the extent of the joint element s interaction with other elements, and does not evaluate the parasitic element in relation to the rest of the system. Damping in parallel with the parasitic springs is essential to prevent prolonged highfrequency components from being superimposed on the system response. The activity of resistive elements is typically well below that of energy storage elements such as masses and springs, and forcing the resistor activities to be the same as the parasitic spring activities is not recommended. The resistances used in generating Figures 5, 6, and 7 were high enough to damp out high-frequency joint transients, but not so high as to introduce severely overdamped parasitic modes that may increase integration time or render a fixed-step integrator unstable. Another approach to sizing parasitic elements is to start with parasitic dampers but no springs, and increase the resistances until system outputs are predicted with adequate accuracy. oupling springs may then be added to prevent drift in the joint positions. This was investigated for the slider-crank system. When the resistances were increased until the sum-ofsquared-error was comparable to that achieved by using stiff springs and nominal damping, the lowest parasitic natural frequency was over an order of magnitude higher than for the springs first approach. This is a serious disadvantage if the modeler wishes to use a fixed-step numerical integration scheme or does not have a sophisticated stiff-system integrator at his disposal. For this reason, as well as the fact that the compliant elements are a more intuitive representation of the physical joints, the springs first approach is preferred. SUMMAY AND ONLUSONS A previously published metric, activity, was proposed as a tool for assessing the contribution of parasitic joint elements to the overall response, both free and forced, of a nonlinear multibody system. t is shown that there is a correlation between low activity and compliant joint rigidity. By setting a threshold such as the activities of the parasitic elements must be an order of magnitude below the lowest-activity system element, the modeler can ensure that the parasitic elements do not contribute to the system response when they are supposed to approximate ideal constraints. The ease of implementation of the technique facilitates setting individual stiffness values rather than assigning an equal and possibly overlyconservative value to all. n contrast with eigenvalue-based parasitic spring tuning techniques, the activity method allows direct calculation of required quantities from a nonlinear simulation model in the time domain, and automatically takes into account the effect of the system inputs. Further work is required to systematically select the parameters of the dampers that are placed in parallel with the springs to attenuate high-frequency transients and increase the

7 stability, efficiency, and accuracy of the simulation. AKNOWLEDGEMENTS The authors would like to thank Dr. Loucas Louca of the University of Michigan Automated Modeling Laboratory for his valuable insights during the writing and refinement of this paper. EFEENES Allen,.. (979) Multiport epresentation of nertia Properties of Kinematic Mechanisms. J. Franklin nst., 38(3), September, pp Karnopp, D. (992) An Approach to Derivative ausality in Bond Graph Models of Mechanical Systems. J. Franklin nst., 329(), pp Karnopp, D. and Margolis, D. (979) Analysis and Simulation of Planar Mechanism Systems Using Bond Graphs. J. Dyn. Sys., Meas. and trl., April, pp Louca, L.S., Stein, J.L., Hulbert, G.M., and Sprague, J. (997) Proper Model Generation: An Energy-Based Methodology. Proc. GBM 97, Phoenix, AZ. iley, S.M. (2) Model eduction of Multibody Systems by the emoval of Generalized Forces of nertia. Ph.D. Thesis, Department of Mechanical Engineering, The University of Michigan. osenberg,.., and Mcalla, J. (99) Power to the User in Formulating Model Equations. Automated Modeling, ASME DS- Vol. 34, pp SM (22), v.3.2, ontrollab Products b.v., Enschede, The Netherlands Zeid, A. (989) Bond Graph Modeling of Planar Mechanisms With ealistic Joint Effects. J. Dyn. Sys., Meas. and trl., March, pp APPENDX Activity of System and Parasitic Elements K S = 6, K S2 = 6, K S3 = 6 m x 2252 m y 7749 J 2 22 K 283 J 963 K S 742 K S2 44 K S3 327 K S =6.67x 6, K S2 =.2x 6, K S3 =.629x 6 m x 2394 m y 779 J K 22 J 42 K S3 6 K S K S2 96 K S = 8, K S2 = 8, K S3 = 8 m x 254 m y 7544 J K 225 J 63 K S 57 K S2 5 K S3 4 K S = 8, K S2 = 8, K S3 = 8, nput Quadrupled m y 64 m x 84 J J 67 K S 4586 K 27 K S K S3 953