Dynamic analyses of a flexible quick-return mechanism by the fixed and variable finite-difference grids

The finite difference method (FDM) with fixed and variable grids is proposed to approximate the numerical solutions of a flexible quick-return mechanism. In the dynamic analysis and simulation, the flexible rod is divided into two regions. Each region with time-dependent length is modeled by Euler-beam theory. Sufficient stability and convergence conditions are established for these finite difference schemes. It is found that for the fixed-grid method, numerical divergence occurs when the moving boundary moves across any of the neighboring nodes. The possibility of break down can be avoided via the variable-grid method, in which a coordinate transformation is employed to fix the moving boundary. Numerical results are discussed and provided to justify the stability and convergence.