Modeling and dynamic analysis of a planetary mechanism with an elastic belt

This paper presents a study of dynamic characteristics of a differential, planetary, path-generating mechanism with a timing belt. By accounting for the belt elasticity and employing Lagrange’s equations, a general model of the mechanism is obtained in the form of three nonlinear differential equations. The system’s stiffness is shown to be a periodic time-varying parameter and parametric vibrations are in order. A sample analysis is provided for a particular case, in which the mechanism generates a straight line. Parametric stability of the sample mechanism is investigated based on solving one of the three equations, both in its nonlinear and linearized (Mathieu–Hill’s) forms. Solution is carried out numerically based on expanding the sought-for function into Taylor series on every calculation step. The remaining two equations are then used to find the driving torques. Effects of belt drive speed ratio, belt material damping, and planet link balancing, on the mechanism’s dynamic behavior are investigated.