Elastodynamics of Planar Mechanisms Using Planar Actual Finite Line Elements, Lumped Mass Systems, Matrix-Exponential Method, and the Method of “Critical-Geometry-Kineto-Elasto-Statics” (CGKES)

Abstract

The article presents a general method for the elastodynamic analysis of planar mechanisms. It uses planar actual finite line elements (regular and irregular elements given in a companion article) and lumped mass systems to formulate the equations of motion of a mechanism. Damping coefficient matrix can incorporate time dependent viscous or coulomb damping coefficients in addition to the coefficients of velocity dependent internal damping. The forcing vector can incorporate any externally applied time dependent force or torque, inertial forces and inertial torques, any nonlinear viscous or Coulomb damping forces and torques. The matrix exponential method is introduced for the numerical solution of the equations of motion. Matrix displacement method of determining dynamic stresses using the generalized coordinate displacements is given. Elastodynamic analysis of a plane four-bar mechanism is performed for several cycles of kinematic motion, and the dynamic stresses are compared with those obtained by experiments. The method of “Critical-Geometry-Kineto-Elasto-Statics” (CGKES) is proposed for the computation of dynamic stress magnitudes making use of the critical geometry of the mechanism. It requires the analysis of a mechanism at the critical geometry position of the mechanism which is defined by the lowest fundamental frequency of the mechanism. The results predicted by the method of CGKES compares within two percent with the experimental results.