a simple approach for dealing with autonomous conservative oscillator under initial velocity

The current study is involved to analytical solution of nonlinear oscillators under initial velocity. by using energy conservation principle, system initial condition converts to condition which oscillator’s velocity become zero. when oscillator’s speed is zero and placed out of movement’s origin, the relation between frequency and amplitude could be extracted. by paying attention to energy conservation principle and relation between the initial velocity and amplitude, the frequency-amplitude relation is extended to frequency-initial velocity relationship. in order to demonstrate the effectiveness of proposed method, duffing oscillator with cubic nonlinearity and oscillator with discontinuity are considered. comparison of results with numerical solution shows good agreement. the proposed method is simple and efficient enough to achieve the analytical approximation of nonlinear autonomous conservative oscillator with initial velocity.