Amplitude–frequency relationship obtained using Hamiltonian approach for oscillators with sum of non-integer order nonlinearities

In this paper the Hamiltonian approximate analytical approach is extended for solving vibrations of conservative oscillators with the sum of integer and/or non-integer order strong nonlinearities. Solution of the nonlinear differential equation is assumed in the form of a trigonometric function with unknown frequency. The frequency equation is obtained based on the hypothesis that the derivative of the Hamiltonian in amplitude of vibration is zero. The accuracy of the approximate solution is treated with two different approaches: comparing the analytical value for period of vibration with the period obtained numerically and developing an error estimation method based on the ratio between the averaged residual function and the total constant energy of system. The procedures given in the paper are applied for two types of examples: an oscillator with a strong nonlinear term and an oscillator where the nonlinearity is of polynomial type.