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In this work, we investigate and apply higher-order Hamiltonian approach (HA) as one of the novelty techniques to find out the approximate analytical solution for vibrating double-sided quintic nonlinear nano-torsional actuator. Periodic solutions are analytically verified, and consequently, the relationship between the initial amplitude and the natural frequency are obtained in a novel analytical way. The HA is then extended to the second-order to find more accurate results. To show the accuracy and applicability of the technique, the approximated results are compared with the homotopy perturbation method and numerical solution. According to the numerical results, it is highly remarkable that the second-order approximate solutions produce better than previously existing results and almost similar in comparing with the numerical solutions.

This paper aims to solve the nonlinear differential equation of the circular sector oscillator analytically via the modified harmonic balance method (MHBM). To assess the reliability and the precision of the present method, we have compared the obtained results with the global residue harmonic balance method, Akbari–Ganji’s method, and numerical Runge–Kutta method which reveals that the MHBM is more reliable than others methods.