An efficient Galerkin averaging-incremental harmonic balance method for nonlinear dynamic analysis of rigid multibody systems governed by differential–algebraic equations

An efficient Galerkin averaging-incremental harmonic balance (EGA-IHB) method is developed 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. The bridge between the multibody dynamic modeling theory and nonlinear dynamic analysis theory is built for the first time in this work, and the EGA-IHB method can be used as a universal solver for obtaining steady-state periodic responses of DAEs for general multibody systems. Since the fast Fourier transform and EGA are used, the EGA-IHB method has excellent robustness. 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 proposed for calculating response curves and conducting parameter studies. Several 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 current method shows excellent computational efficiency and robustness in solving high-dimensional DAEs.