A Numerically Efficient Algorithm for Steady-State Response of Flexible Mechanism Systems

Presented in this work is a numerically efficient algorithm for treating the periodic steady-state response of flexible mechanisms as the solution to separated two-point boundary value problems. The finite element method is applied to discretize continuous elastic mechanisms systems and a set of second-order ordinary differential equations is obtained with periodically time-varying coefficient matrices and forcing vectors. Modal analysis techniques are employed to decouple these equations into a number of single scalar ordinary differential equations in modal basis. The periodic time-boundary conditions at both ends of a fundamental period equal to a cycle of input motion are mathematically separated by introducing auxiliary variables, thus resulting in a so-called almost-block-diagonal matrix for linear algebraic systems of equations. Solving such a system is computationally less expensive than solving a general linear algebraic system. Examples are included to illustrate the procedures applied to a four-bar linkage through which computing time is compared with other approaches.