Statics Analysis for Multibody Systems Containing Ball-and-Socket Joints Based on the Reduced Transfer Matrix Method

Abstract Figuring out the static equilibrium position of the system is necessary and essential for various multibody systems, either for providing an initial condition for transient dynamics analysis or facilitating the stability analysis of the system. Regardless of the strategy employed, the system needs to be analyzed first to establish its dynamics equations. The reduced multibody system transfer matrix method is a fully recursive dynamic method that uses joint coordinates and is an effective approach for analyzing system dynamics. The generalized accelerations can be quickly obtained, given the generalized coordinates and velocities of the system. For systems without constraint equations, the nonlinear equations for which the generalized accelerations are zero can be solved to obtain the static equilibrium position quickly. However, inherent singularities occur if Euler angles are used as the generalized coordinates for ball-and-socket joints. Therefore, Euler parameters are adopted as the generalized coordinates for ball-and-socket joints. Since Euler parameters are not independent and subject to constraint equation, the equilibrium cannot be obtained by solving only a system of nonlinear equations with zero generalized accelerations. In this paper, the reduced transfer equations of the ball-and-socket joint elements with Euler parameters are derived. Based on this, a static equilibrium solution for multibody systems containing ball-and-socket joints is established, focusing on deriving the Jacobian matrices. Numerical examples, along with comparative analyses of the dynamic method, validates the proposed method.