A Universal Nonlinear Analyzer for Rigid Multibody Systems Based on the Efficient Galerkin Averaging-Incremental Harmonic Balance Method

The bridge between the multibody dynamic modeling theory and nonlinear dynamic analysis theory is built for the first time in this work by introducing an efficient Galerkin averaging-incremental harmonic balance (EGA-IHB) method for steady-state nonlinear dynamic analysis of index-3 differential algebraic equations (DAEs) for general rigid multibody systems. The multibody dynamic modeling theory has made significant advances in generality and simplicity, and multibody systems are usually governed by DAEs. Since the fast Fourier transform and EGA are used, the EGA-IHB method has excellent robustness and computational efficiency. Since the Floquet theory cannot be directly used for stability analysis of periodic responses of DAEs, a new stability analysis procedure is developed, where perturbed, linearized DAEs are reduced to ordinary differential equations with use of independent generalized coordinates. A modified arc-length continuation method with a scaling strategy is used for calculating response curves and conducting parameter studies. Three examples are used to show the performance and capability of the current method. Periodic solutions of DAEs from the EGA-IHB method show excellent agreement with those from numerical integration methods. Amplitude-frequency and amplitude-parameter response curves are generated, and stability and period-doubling bifurcations are analyzed. The EGA-IHB method can be used as a universal solver and nonlinear analyzer for obtaining steady-state periodic responses of DAEs for general multibody systems.

Dwelling on the Connection Between SO(3) and Rotation Matrices in Rigid Multibody Dynamics – Part 1: Description of an Index-3 DAE Solution Approach

In rigid multibody dynamics simulation using absolute coordinates, a choice must be made in relation to how to keep track of the attitude of a body in 3D motion. The commonly used choices of Euler angles and Euler parameters each have drawbacks, e.g., singularities, and carrying along extra normalization constraint equations, respectively. This contribution revisits an approach that works directly with the orientation matrix and thus eschews the need for generalized coordinates used at each time step to produce the orientation matrix A. The approach is informed by the fact that rotation matrices belong to the SO(3) Lie matrix group. The numerical solution of the dynamics problem is anchored by an implicit first order integration method that discretizes, without index reduction, the index 3 Differential Algebraic Equations (DAEs) of multibody dynamics. The approach handles closed loops and arbitrary collections of joints. Our main contribution is the outlining of a systematic way for computing the first order variations of both the constraint equations and the reaction forces associated with arbitrary joints. These first order variations in turn anchor a Newton method that is used to solve both the Kinematics and Dynamics problems. The salient observation is that one can express the first order variation of kinematic quantities that enter the kinematic constraint equations, constraint forces, external forces, etc., in terms of Euler infinitesimal rotation vectors. This opens the door to a systematic approach to formulating a Newton method that provides at each iteration an orthonormal rotation matrix A. The Newton step calls for repeatedly solving linear systems of the form Gδ = e, yet evaluating the iteration matrix G and residuals e is inexpensive, to the point where in the Part 2 companion contribution the proposed formulation is shown to be two times faster for Kinematics and Dynamics analysis when compared to the Euler parameter and Euler angle approaches in conjunction with a set of four mechanisms.