Nonlinear dynamics of flexible nanopositioning systems with geometrical imperfection

Abstract

In this paper, the nonlinear dynamics of flexible beams with geometrical imperfection and concentrated end-mass is investigated. Based on the Euler–Bernoulli beam theory and nonlinear strain–displacement relations, governing equations are obtained. The Galerkin method is employed to discretize the nonlinear partial differential equations of motion. The effect of system parameters on both the linear and nonlinear dynamics of the system are studied based on the numerical solution of the set of nonlinear ODEs with coupled terms. The results show that dynamic behavior of the system is affected significantly by the geometrical imperfection. It is revealed that the geometrical imperfection results in jump instability and sub-harmonics. Since, geometrical imperfections are inevitable in manufacturing process of these equipments, the effects of imperfections should be considered in design prior to the manufacturing of nanopositioning systems.