Steady-state Rotary Periodic Solutions of Rigid and Flexible Mechanisms with Large Spatial Rotations Using the Incremental Harmonic Balance Method for Differential-Algebraic Equations

Abstract Steady-state rotary periodic responses of mechanisms lead to stress cycling in flexible structures or connecting joints, which in turn can result in structural fatigue. A general approach is developed to study rotary periodic solutions of rigid and flexible mechanisms with large spatial rotations based on the incremental harmonic balance (IHB) method. The challenge in analyzing such dynamic systems emanates from the non-commutativity of the spatial rotation and the non-superposition nature of the rotational coordinates. The generally used rotational coordinates, such as Euler angles, cannot be expanded into Fourier series, which prevents direct usage of the IHB method. To overcome the problem, the natural coordinates method and absolute nodal coordinate formulation are used herein for the dynamic modeling of the rigid and flexible bodies, respectively. The absolute positions and gradients are used as generalized coordinates and rotational coordinates are naturally avoided. Equations of motions of the system are differential-algebraic equations (DAEs), and they are solved by the IHB method to obtain the steady-state rotary periodic solutions. The effectiveness of the proposed approach is verified by the simulation of rigid and flexible examples with spatial rotations. The approach is general and robust and it has the potential to be further extended for other extensive multibody dynamic systems.