Accurate approximate analytical solutions for nonlinear free vibration of systems with serial linear and nonlinear stiffness

This paper deals with free vibration of a nonlinear system having combined linear and nonlinear springs in series. The conservative oscillation system is formulated as a nonlinear ordinary differential equation having linear and nonlinear stiffness components. The governing equation is linearized and associated with the harmonic balance method to establish new and accurate higher-order analytical approximate solutions. Unlike the perturbation method which is restricted to nonlinear conservative systems with a small perturbed parameter and also unlike the classical harmonic balance method which results in a complicated set of algebraic equations, the new approach yields simple approximate analytical expressions valid for small as well as large amplitudes of oscillation. Some examples are solved and compared with numerical integration solutions and published results. New solutions to the nonlinear systems are also presented and discussed.