Discussion on the accuracies of He’s frequency–amplitude formulation and its modification with average residuals

Abstract

A complete numerical comparison is given between the He’s frequency–amplitude formulation and its modification with average residuals for both weak and strong nonlinear oscillators, showing both methods have good accuracy. This paper concludes that both methods provide a simple but effective tool to various nonlinear oscillators.

The accuracies of He’s frequency–amplitude formulation^1–4 and also its modification with average residuals ⁵ were discussed by Wang et al.; ⁶ although their finding is reasonable, their conclusion is doubtful.

We analysed the generalized Duffing oscillator in the form ⁵

$$u″+a_{0}u+a_{\text{1}}{u}^{\text{3}}+a_{\text{2}}{u}^{\text{5}}+a_{\text{3}}{u}^{\text{7}}+···+a_n{u}^{\text{2}n+\text{1}}=0,\hspace{1em}u\left(0\right)=A,\hspace{1em}u\prime \left(0\right)=0$$

(1)

when α₃ = α₄= α₅= ⋯ = α_n = 0, equation (1) becomes

$$u″+a_{0}u+a_{\text{1}}{u}^{\text{3}}+a_{\text{2}}{u}^{\text{5}}=0,\hspace{1em}u\left(0\right)=A,\hspace{1em}u\prime \left(0\right)=0$$

(2)

According to He’s frequency–amplitude formulation, the frequency–amplitude relationship can approximately be written as^1–4

$${\omega }_{\mathrm{Classic}}=\sqrt{{\alpha }_0+\frac{3}{4}{\alpha }_1{A}^2+\frac{5}{8}{\alpha }_2{A}^4}$$

(3)

while the approximate frequency–amplitude relationship from its modification with average residuals reads ⁵

$${\omega }_{\mathrm{Pr}\mathrm{esent}}=\sqrt{{\alpha }_0+\frac{2}{3}{A}^2{\alpha }_1+\frac{8}{15}{A}^4{\alpha }_2}$$

(4)

However, the two results given in equations (3) and (4) are a little different. When α₁ = α₂ = 0, both cases result in an exact result; Table 1 gives other cases for comparison with exact ones, showing both He’s formulation and its modification give ideal results; although the accuracy might be different for some special points, when α₁ or α₂ tends to infinity, equation (4) is much better than equation (3), showing that the modification ⁵ is good enough for practical applications.

Table 1.

Comparison of He’s formulation and its modification with the exact one for the case of A = 1.

α 0	α 1	α 2	ω Exact	Present process		Classic process
				ω Present	Relative error	ω Classic	Relative error
0.01	0.01	0.01	0.15236	0.14832	0.0265	0.15411	0.0115
0.01	0.01	1	0.75906	0.74162	0.0230	0.80156	0.0560
0.01	0.01	100	7.4696	7.3041	0.0222	7.9068	0.0585
0.01	1	0.01	0.85668	0.82583	0.0360	0.87536	0.0218
0.01	1	1	1.1364	1.1000	0.0320	1.1769	0.0356
0.01	1	100	7.5177	7.3491	0.0224	7.9536	0.0580
0.01	100	0.01	8.4731	8.1659	0.0363	8.6612	0.0222
0.01	100	1	8.5062	8.1982	0.0362	8.6968	0.0224
0.01	100	100	11.317	10.955	0.0320	11.726	0.0361
1	0.01	0.01	1.0068	1.0060	0.0008	1.0069	0.0001
1	0.01	1	1.2676	1.2410	0.0210	1.2777	0.0080
1	0.01	100	7.5430	7.3716	0.0227	7.9692	0.0565
1	1	0.01	1.3200	1.2931	0.0204	1.3252	0.0039
1	1	1	1.5236	1.4832	0.0265	1.5411	0.0115
1	1	100	7.5906	7.4162	0.0230	8.0156	0.0560
1	100	0.01	8.5339	8.2263	0.0360	8.7182	0.0216
1	100	1	8.5339	8.2583	0.0323	8.7536	0.0257
1	100	100	11.364	11.000	0.0320	11.769	0.0356
100	0.01	0.01	10.001	10.001	0.0000	10.001	0.0000
100	0.01	1	10.032	10.027	0.0005	10.032	0.0000
100	0.01	100	12.647	12.383	0.0209	12.748	0.0080
100	1	0.01	10.038	10.034	0.0004	10.038	0.0000
100	1	1	10.068	10.060	0.0008	10.069	0.0001
100	1	100	12.676	12.410	0.0210	12.777	0.0080
100	100	0.01	13.178	12.910	0.0203	13.229	0.0039
100	100	1	13.200	12.931	0.0204	13.252	0.0039
100	100	100	15.236	14.832	0.0265	15.411	0.0115

We conclude that both He’s formulation and its modification are suitable for fast estimation of the period property of any conservative oscillators, and there is no obvious evidence showing that He’s formulation is advantageous over its modification. We consider any modification of a known method will contribute to the development of the science and technology, and we consider that Wang et al.’s conclusion is doubtful.

Due to the simple calculation of the modification of He’s formula, the obtained frequency can be effectively used as an initial guess for the variational iteration method, ⁷ the homotopy perturbation method,^8–12 for a better accuracy if necessary. The modification can be also used for a recently hot topic on fractional oscillators. ¹³