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.

Extending the Modified Inertia Representation to Constrained Rigid Multibody Systems

The inertia representation of a constrained system includes the formulation of the kinetic energy and its corresponding mass matrix, given the coordinates of the system. The way to find a proper inertia representation achieving better numerical performance is largely unexplored. This paper extends the modified inertia representation (MIR) to the constrained rigid multibody systems. By using the orthogonal projection, we show the possibility to derive the MIR for many types of non-minimal coordinates. We present examples of the derivation of the MIR for both planar and spatial rigid body systems. Error estimation shows that the MIR is different from the traditional inertia representation (TIR) in that its parameter γ can be used to reduce the kinetic energy error. With preconditioned γ, numerical results show that the MIR consistently presents significantly higher numerical accuracy and faster convergence speed than the TIR for the given variational integrator. The idea of using different inertia representations to improve the numerical performance may go beyond constrained rigid multibody systems to other systems governed by differential algebraic equations.

Analysis of Constraint Forces in Multibody Systems Based on the Differential Null Space Matrix Method

This paper treats a problem to determine constraint forces in rigid mutibody systems. One of the most often applied algorithms for determination of constraint forces is based on the use of recursive Newton-Euler formalism. Another algorithm often applied for determination of constraint forces is based on the use of Lagrange multipliers. This paper presents a new method to determine constraint forces in rigid multibody systems. First relative displacements which violate the constraints, called anti-constraint relative displacements, are introduced, and governing equations which involve the constraint forces explicitly are derived. In the derived equations, the constraint forces appear independently from the Lagrange multipliers. Then, a method is proposed to determine the constraint forces by eliminating the Lagrange multipliers based on the methods proposed in previous papers by the author. The method is extended to have ability to treat systems with redundant constraints. Finally, validity of the proposed method is confirmed via numerical examples.