The conservative Helmholtz-Duffing oscillator is analyzed by means of three analytical techniques. The max-min, second-order of the Hamiltonian, and the global error minimization approaches are applied to achieve natural frequencies. The obtained results are compared with the homotopy perturbation method and numerical solutions. The results show that second-order of the global error minimization method is very accurate, so it can be widely applicable in engineering problems. 1. Introduction Mathematical modeling and frequency analysis of the nonlinear vibrational systems are an important and interesting field of mechanics. A lot of researchers have worked in this field and have proposed a lot of methods for demonstrating the dynamics responses of these systems [1–4]. They have developed this field of science and have analyzed the responses of the nonlinear vibration problems such as Duffing oscillators [5–10], nonlinear dynamics of a particle on a rotating parabola [11], nonlinear oscillators with discontinuity [12], oscillators with noninteger order nonlinear connection [13], the plasma physics equation [14], and van der Pol oscillator [15, 16]. The Helmholtz-Duffing equation is a nonlinear problem with the quadratic and cubic nonlinear terms. Surveying the literature shows that this equation has wide applications in the engineering problems. For example, due to different vibration behavior of functionally graded materials (FGMs) at positive and negative amplitudes, the governing equations of FGM beams, plates, and shells are conduced to a second-order nonlinear ordinary equation with quadratic and cubic nonlinear terms [17–20]. Moreover, Sharabiani and Yazdi [21] obtained a Helmholtz-Duffing type equation within studying of nonlinear free vibrations of functionally graded nanobeams with surface effects. On the other hand, they revealed application of this equation in FG nanostructures. In this paper, the frequency-amplitude relationship of the conservative Helmholtz-Duffing oscillator is obtained by means of the max-min [22–26], Hamiltonian [27–31], and global error minimization methods [32–35]. The Hamiltonian approach is a kind of energy method and is proposed by He [27]. It is a simple method and can be used for the conservative nonlinear equations. Recently, it is applied for dynamic analysis of an electromechanical resonator [36]. Moreover, Akbarzade and Khan [37] employed the second-order Hamiltonian approach for nonlinear dynamic analysis of conservative coupled systems of mass-spring. The max-min approach is made on the base of Chengtian’s
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