Effect of manufacturing tolerances on dynamic equilibria of multibody systems undergoing prescribed rotational motion

Abstract

A general formulation is developed for the tolerance analysis of dynamic equilibria in a multibody system undergoing prescribed rotational motion, with applications including robots, spacecraft, propulsion and power generation systems, and sensors and actuators. In a state of dynamic equilibrium, a subset of the generalized coordinates assumes constant values while the remaining coordinates vary and respond in time. Manufacturing tolerances can be mathematically represented by probabilistic distributions or statistical variables through either an analytical approach or a Monte Carlo simulation. In the present tolerance work, the tolerances of design parameters including lengths, stiffnesses, inertias, and attachment positions are examined. In order to analytically calculate the statistical response of the dynamic equilibrium positions to such tolerances, the first-order sensitivities of the equilibria with respect to parameters are calculated. To illustrate the method’s accuracy and computational efficiency, two numerical examples are considered, and the statistical results obtained analytically for the equilibria are compared with those calculated through Monte Carlo simulation. In some cases, an equilibrium configuration can have an operating condition for which the response has zero standard deviation to perturbations of a design parameter. That condition can be a useful design point to the extent that typical manufacturing tolerances or other sources of variation would have no effect on the dynamic equilibrium configuration.