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

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. INTRODUCTION 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 [Rosenberg and McCalla, 1991]: 1. 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 (1979), 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 [Rosenberg and McCalla, 1991]. Option 3, removing derivative causality, can sometimes be accomplished using methods in which variables are transformed to produce explicit energy storage fields [Rosenberg and McCalla (1991), Allen (1979), and Karnopp (1992)]. 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. If, 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 [Riley,