Dynamics analysis of linear elastic planar mechanisms

This paper presents a formulation for the dynamics analysis of an elastic mechanism and integrating a stiff system using efficient numerical methods. Because all the elastic degrees of freedom are included in the vector of generalized variables, the size of the equations is much larger than that obtained using either the assumed mode or the distributed parameter finite element approach. However, the resulting system matrix is sparse and the elastic coordinates are absent from the system matrix, and these are useful properties for subsequent numerical analysis. Techniques for solving a system of linear time-variant equations are applied to the dynamics equations, assuming that the system matrix is slow-changing, and thus, may be approximated by a series of piecewise constant matrices. It is argued that the problem of determining the integration time step is transformed into the problem of computing the exponential of the system matrix with automatic time scaling. A numerical example is given to show that the behavior of the rigid coordinates converges to that of an all-rigid-body model by artificially increasing the Young's modulus of the elastic components, despite the very-high-frequency vibrations of the elastic coordinates induced by the increment of the stiffness.