The harmonic balance method is used to construct approximate frequency–amplitude relations and periodic solutions to the relativistic oscillator. By combining linearization of the governing equation with the harmonic balance method, we construct analytical approximations to the oscillation frequencies and periodic…

A more accurate analytical approximate expression for the slope at any point of the elastic curve of a slender cantilever column is obtained by using a heuristic but pedagogical derivation. This derivation is based on the linearization of the nonlinear…

The analytical approximate technique developed by Wu et al. for conservative oscillators with odd nonlinearity is used to construct approximate frequency-amplitude relations and periodic solutions to the relativistic oscillator. By combining Newton's method with the method of harmonic balance, analytical…

The second-order harmonic balance method is used to construct three approximate frequency-amplitude relations for a conservative nonlinear singular oscillator in which the restoring force is inversely proportional to the dependent variable. Two procedures are used to solve the nonlinear differential…

The asymptotic behaviours for small and large amplitudes, A, of the period for a nonlinear oscillator, where the square of the angular frequency depends quadratically on the velocity, are obtained. These asymptotic expressions are compared with the exact period, T(A),…

Two Lidstedt-Poincaré perturbation based methods are used to solve the nonlinear differential equation of a nonlinear oscillator having the square of the angular frequency depends quadratically on the velocity. Mickens published two interesting papers [1, 2] about this oscillator and…

The first-order harmonic balance method via the first Fourier coefficient is used to construct two approximate frequency–amplitude relations for a conservative nonlinear oscillatory system in which the restoring force has an irrational form. This system corresponds to the motion of…

A cubication procedure of the nonlinear differential equation for conservative nonlinear oscillators is analyzed and discussed. This scheme is based on the Chebyshev series expansion of the restoring force and this allows us to approximate the original nonlinear differential equation…

In their comment, Qing-Yin and Pei derived another analytical approximate expression for the large-angle pendulum period, which they compare with other expressions previously published. Most of these approximate formulas are based on the approximation of the original nonlinear differential equation…

An analytical approximate technique for conservative nonlinear oscillators is proposed. This method is a modification of the generalized harmonic balance method in which analytical approximate solutions have rational form. This approach gives us not only a truly periodic solution but…