Approximate Analytical Solutions to Jerk Equations

Abstract

Nonlinear third-order differential equations, known as nonlinear jerk equations, involving the third temporal derivative of displacement are considered in this paper. This kind of equations is of much interest in analyzing some structures exhibiting rotating and translating motions such are robots or machine tools, where excessive jerk (defined as the time derivative of the acceleration) lead to accelerated wear in transmissions and bearing elements, noisy operations, and large contouring errors at discontinuities (such as corners) in the machining path. In this paper, we propose a new analytical technique called the Optimal Auxiliary Functions Method (OAFM) to analyze some particular cases of jerk functions involving cubic nonlinearity. Numerical simulations are also developed in order emphasize the accuracy of the obtained results. Several numerical examples show that the proposed procedure is simple and easy to use.