An accurate relationship between frequency and amplitude to nonlinear oscillations

ABSTRACT A weighted global error minimization (WGEM) is proposed in this study. The goal is to improve the accuracy of the global error minimization (GEM) based on a weighting function. In addition to the first-order approximation, the fourth-order approximation for the Duffing oscillator is demonstrated by coupling a proper weighting function with the GEM. In order to exhibit the advantage of this modification, the obtained result is compared with both the exact frequency and the outcome of the GEM. The corollary outstandingly reveals that approximations using the WGEM have a lower relative error than those from the GEM in the first-order approximation. Also the modified approach can preserve its accuracy in the fourth-order approximation. The WGEM can be promisingly utilized to other resembling nonlinear problems.


Introduction
The accurate estimation of the relationship between frequency and amplitude of nonlinear oscillators is of significant importance for many fields of applied science and engineering. Surveys of the literature express there are different approximate analytical techniques for dealing with the nonlinear problems. Among them, one may attract attention to the weighted linearization technique [1], the energy balance method [2][3][4], the optimal homotopy asymptotic method [5], the linearized harmonic balance method [6], the global residue harmonic balance method [7], the homotopy analysis method [8,9], Max-Min approach [10,11], Hamiltonian approach [12,13], the variational approach [14][15][16], the variational iteration method [17], the rational harmonic balance method [18], the Chebyshev polynomial approximation [19] and so on [20][21][22].
The aim of this investigation is to reach a more accurate relationship between frequency and amplitude to nonlinear vibrating systems in conjunction with the global error minimization (GEM) [23][24][25]. The precision of the GEM has been improved by combining with an appropriate weighting function. The weighting function plays a special role in acquiring an accurate estimation. Therefore, a suitable weighting function is chosen based on related works and author knowledge [26,27]. Whereas it is integrable and nonnegative over the interval. It should be noted that an arbitrary weighting function is inappropriate for the procedure.
The rest of the manuscript is organized as follows. The outline of the proposed modification is presented in Section 2. The relationship between the frequency and the initial amplitude of the Duffing oscillator is provided via either first-and fourth-order approximations in Section 3. This study ends with a brief conclusion in the last section.

The weighted GEM
This section gives the basic idea of the weighted global error minimization (WGEM). Consider a general nonlinear oscillator as follows: By defining a functional as follows: where W(t) is a weighting function and hereon it is selected as follows: Also by assuming F(u) is an odd function. One may employ an approximate trial function in the form of so that And the unknown parameters (i.e. a (2n+1) & ω) have been obtained through the following conditions: To demonstrate the practicality and effectiveness of the aforementioned procedure, a cubic Duffing oscillator with strong nonlinearity is taken into consideration in the present study. The results are illustrated in the next section.

The cubic Duffing oscillator
This section investigates the accuracy of the approach by the cubic Duffing oscillator. The governing equation for this type oscillator is: (7) Equation (7) is a mathematical model of a conservative system. The exact frequency of this oscillator is given as: Where K(m) is the complete elliptic integral of first kind and defined as follows: The first-and fourth-order approximations for this nonlinear model are given in the following context.

First-order approximation
Based on Section 2, the WGEM is implemented properly. The minimization problem of Equation (7) can be rewritten as: for the first-order approximation, the trial function is: Where a 1 = A, substituting Equation (11) into Equation (10) yields: For more convenience, in current work the value of μ is selected equal to one (μ = 1). Maximum relative error of Equation (13) is as low as 0.22% in whole of the domain. Furthermore, for large values of the nonlinearity parameter (λ = ε A 2 ), the relative error of the first-order approximation is as low as 0.01%, viz., In comparison to the GEM [23], the accuracy of the WGEM is excellent.

Fourth-order approximation
To illustrate the accuracy of the approach in higherorder approximations, the fourth-order approximation is applied to the Duffing oscillator. In this section, the following trial function is considered: u 4 (t) = a 1 cos(ωt) + a 3 cos(3ωt) + a 5 cos(5ωt) + a 7 cos(7ωt).
Based on the WGEM algorithm and using Equation (15) as the trial function, the minimization problem of Equation (7) is determined. Thereupon by applying the conditions of Equation (6), the relative error of the approximate frequency and the coefficients of Equation (15) for different values of the nonlinearity parameter are achieved. The results are presented in Table 1. As can be seen, the fourth-order approximation of the algorithm gives an excellent accuracy for whole of the domain. Moreover, in this case the relative error of the WGEM is lower than the GEM slightly.

Conclusions
Excellent agreement of the approximate frequency with the exact one has been displayed. This study scrutinizes the accuracy of the modified global error minimization by the Duffing oscillator. This applicable technique successfully provides an accurate approximate frequency for the first-order approximation. Also the higher-order estimation using this approach is more accurate for both small and large amplitudes.The general perspective of this straightforward approach indicates the technique is reliable, simple, powerful and accurate for conservative nonlinear oscillators.

Disclosure statement
No potential conflict of interest was reported by the author.