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The development of simple and yet accurate formulations of frequency–amplitude relationships for non-conservative nonlinear oscillators is an important issue. The present paper is concerned with integral-type frequency–amplitude formulas in the dimensionless time domain and time domain to accurately determine vibrational frequencies of nonlinear oscillators. The novel formulation is a balance of kinetic energy and the work during motion of the nonlinear oscillator within one period; its generalized formulation permits a weight function to appear in the integral formula. The exact values of frequencies can be obtained when exact solutions are inserted into the formulas. In general, the exact solution is not available; hence, low-order periodic functions as trial solutions are inserted into the formulas to obtain approximate values of true frequencies. For conservative nonlinear oscillators, a powerful technique is developed in terms of a weighted integral formula in the spatial domain, which is directly derived from the governing ordinary differential equation (ODE) multiplied by a weight function, and integrating the resulting equation after inserting a general trial ODE to acquire accurate frequency. The free parameter is involved in the frequency–amplitude formula, whose optimal value is achieved by minimizing the absolute error to fulfill the periodicity conditions. Several examples involving two typical non-conservative nonlinear oscillators are explored to display the effectiveness and accuracy of the proposed integral-type formulations.

Purpose To investigate the simple and effective method for solving nonlinear vibration equations, a nonlinear Helmholtz–Duffing oscillator with small amplitude is considered. In this paper, we apply the Harmonic balance method with the help of the Gamma function to solve the Helmholtz–Duffing oscillator with quadratic and cubic nonlinear terms. Method Based on the harmonic balance method, the amplitude formula can be easily solved using the Gamma function with only one key step. Results Some numerical experiments show that the couple of the harmonic balance method and the Gamma function is greatly simple and accurate compared to the approximate analytical solutions obtained by the energy balance method and the He’s frequency–amplitude formula. Conclusion There are many methods for solving nonlinear vibration equations in existing literatures. It is worth noting that, in this work, based on the framework of the harmonic balance method, the introduction of special functions (such as the Gamma function) can make the calculation process of nonlinear Helmholtz–Duffing oscillator more conveniently. The comparisons with the existing methods reveal that the proposed method is more precise and effective to compute the small amplitude oscillation. In addition, employing to the idea of special functions, this type of method provides a new creative scope for solving such nonlinear oscillation equations.