Use of a path-following method for finding static equilibria of multibody systems modeled by the reduced transfer matrix method

Finding the stable static equilibrium position of multibody systems is a well-known problem. Dynamic relaxation methods are frequently utilized in engineering, however, they often require a significant amount of time. Alternatively, most commercial software employs the Newton–Raphson iterative method to solve a set of nonlinear equations to find the equilibrium position directly, in which the time derivatives of any quantity are set to zero. Nevertheless, this approach is highly dependent on initial conditions and can only find one equilibrium position for a specific initial condition, no matter how many degrees of freedom a system has. A path-following method is implemented in this paper to find the equilibrium position of the multibody system by using the reduced multibody system transfer matrix method to evaluate the acceleration functions and its Jacobian matrix, where the notion of direct differentiation is applied. The solution curves for changing generalized accelerations are then tracked using the arc-length method to obtain candidate equilibrium states if they vanish and identify the stable static equilibrium position. To demonstrate the effectiveness of the proposed method, numerical examples are presented, which provide a detailed overview of the complete computational flow.