A short remark on He’s frequency formulation

Abstract

Recently Shen suggested a modification of He’s frequency formulation using Lagrange interpolation with great success for the cubic-quintic Duffing oscillator. This paper proposed an alternative modification of He’s frequency formulation by dividing the oscillators into two extreme conditions when the amplitude is either extremely small or remarkably large, and He’s frequency formulation is used for each case, and a final frequency formulation is obtained by matching the two extreme conditions. Comparison of the approximate frequency with the exact one for various amplitudes shows good agreement.

Introduction

Recently Shen studied the following cubic-quintic Duffing equation ¹

$$\frac{d^{2}x}{d{t}^2}+x+x^3+x^5=0,x\left(0\right)=A,\frac{dx}{dt}\left(0\right)=0$$

(1)

and obtain the following frequency-amplitude relationship

$${\omega }^2={\frac{dg\left(x\right)}{dx}|}_{x=0.5A}={\left(1+3x^2+5x^4\right)|}_{x=0.5A}=1+\frac{3}{4}{A}^2+\frac{5}{16}{A}^4$$

(2)

where $g\left(x\right)=x+x^3+x^5$ . Eq.(2) is called as He’s frequency formulation.^2–4 Eq.(2) gives large error when the amplitude tends to infinity, Shen ¹ overcame the difficult by choosing different location points:

$${\omega }^2={\frac{dg\left(x\right)}{dx}|}_{x=kA}$$

(3)

where $0<k<1$ . By choosing different values of k and using the Lagrange interpolation, Shen ¹ obtained a much better result. He’s frequency formulation has been widely applied and extended to fractal vibration systems,^5–8 many modifications were appeared.^9–12 He’s frequency formulation can give a very fast insight into the physical properties of a complex oscillator, e.g. the MEMS oscillator.^13–15

Modification of He’s frequency formulation

In this paper we consider two extreme conditions when $x<<1$ and $x>>1$ . Eq.(1) can be simplified as

$$\frac{d^{2}x}{d{t}^2}+x+x^3=0,x\left(0\right)=A,\frac{dx}{dt}\left(0\right)=0,x<<1$$

(4)

and

$$\frac{d^{2}x}{d{t}^2}+x^5=0,x\left(0\right)=A,\frac{dx}{dt}\left(0\right)=0,x>>1$$

(5)

Eq.(4) is the standard Duffing oscillator. By He’s frequency formulation,^2–4 we have

$${\omega }_1=\sqrt{1+\frac{3}{4}{A}^2},A<<1$$

(6)

$${\omega }_2=\sqrt{\frac{5}{16}{A}^4},A>>1$$

(7)

Eq.(7) leads to a large error when $A>>1.$ Fortunately Eq.(5) can be solved exactly with ease. We write Eq.(5) in the form

$$\frac{d^{2}x}{d{t}^2}+f\left(x\right)=0$$

(8)

We introduce a transform

$$\frac{dx}{dt}=y$$

(9)

Eq.(8) becomes

$$y\frac{dy}{dx}+f\left(x\right)=0$$

(10)

Eq.(10) can be written in the form

$$ydy+f\left(x\right)dx=0$$

(11)

Integrating Eq.(11) gives the following result

$$\frac{1}{2}{y}^2+F\left(x\right)=h$$

(12)

where $h=F\left(x_0\right),\frac{d}{dx}F\left(x\right)=f\left(x\right)$ .

Solving y from Eq.(12), we have

$$\frac{dx}{dt}=y=\pm \sqrt{2F\left(x_0\right)-2F\left(x\right)}$$

(13)

We write Eq.(13) in the form

$$dt=\pm \frac{dx}{\sqrt{2F\left(x_0\right)-2F\left(x\right)}}$$

(14)

Its period can be calculated as

$$T=4{\displaystyle {\int }_0^{x_0}\frac{dx}{\sqrt{2F\left(x_0\right)-2F\left(x\right)}}}=2\sqrt{2}{\displaystyle {\int }_0^{x_0}\frac{dx}{\sqrt{{\displaystyle {\int }_0^{x_0}f\left(s\right)ds}}}}=2\sqrt{2}{\displaystyle {\int }_0^A\frac{dx}{\sqrt{{\displaystyle {\int }_0^{A}f\left(s\right)ds}}}}$$

(15)

where $x_0=x\left(0\right)=A$ .

For Eq.(5), $f\left(x\right)=x^5$ , so we have

$$T=2\sqrt{2}{\displaystyle {\int }_0^A\frac{dx}{\sqrt{{\displaystyle {\int }_x^A{s}^{5}ds}}}}=2\sqrt{2}{\displaystyle {\int }_0^A\frac{dx}{\sqrt{\frac{1}{6}\left(A^6-x^6\right)}}}=4\sqrt{3}{\displaystyle {\int }_0^A\frac{dx}{\sqrt{A^6-x^6}}}$$

(16)

By introducing a transform, $x=A\cos t$ , the above equation becomes

$$\begin{array}{l}T=4\sqrt{3}{\displaystyle {\int }_0^{\frac{\pi }{2}}\frac{A\sin tdt}{\sqrt{A^6-A^6{\cos }^{6}t}}}=\frac{4\sqrt{3}}{A^2}{\displaystyle {\int }_0^{\frac{\pi }{2}}\frac{\sin tdt}{\sqrt{1-{\cos }^{6}t}}}\\ =\frac{4\sqrt{3}}{A^2}{\displaystyle {\int }_0^{\frac{\pi }{2}}\frac{dt}{\sqrt{1+{\cos }^{2}t+{\cos }^{4}t}}}=\frac{4\sqrt{3}}{A^2}\times 1.2143\\ =\frac{8.4131}{A^2}\end{array}$$

(17)

The frequency of Eq.(5) can be exactly calculated as

$${\omega }_2=\frac{2\pi }{T_2}=0.7468A^2$$

(18)

To match the two extremes, we give the following frequency formulation

$$\omega =\left(1-p\right){\omega }_1+p{\omega }_2=\left(1-p\right)\sqrt{1+\frac{3}{4}{A}^2+0.7468A^{2}p}$$

(19)

where $p$ is a matching parameter, satisfying the following conditions

$$p=p\left(A\right)=\{\begin{array}{l}0,A<<1\\ 1,A>>1\end{array}$$

(20)

The matching parameter is also used in the homotopy perturbation method.^16–19 In this paper we suggest that p can be expressed in the form

$$p\left(A\right)=1-{\left(1-e^{-a_1{A}^{-a_2}}\right)}^{a_3}$$

(21)

where $a_1,a_2,a_3$ are constants. In this paper we choose

$$a_1=2.7587,a_2=1.3916,a_3=3.7555$$

(22)

The matching parameter can be written as

$$p\left(A\right)=1-{\left(1-e^{-2.7587A^{-1.3916}}\right)}^{3.7555}$$

(23)

As we can see from Figure 1, Eq.(23) meets the basic properties of Eq.(20).

Figure 1.

The property of the matching parameter versus the amplitude.

The approximate frequency is

$$\omega ={\left(1-e^{-2.7587A^{-1.3916}}\right)}^{3.7555}\sqrt{1+\frac{3}{4}{A}^2}+0.7468A^2\left[1-{\left(1-e^{-2.7587A^{-1.3916}}\right)}^{3.7555}\right]$$

(24)

Table 1 and Figure 2 show high accuracy of Eq.(4), when $A\to \infty$ , the error tends to zero.

Table 1.

The approximate results of frequencies.

A	Exact frequency ${\omega }_{exa}$	Approximate FrequencyEq.(24)	Relative error %	Approximate FrequencyRef[1]	Relative error of Ref.[1](%)
0.1	1.00377	1.00374	0.00299	1.00503	0.12525
0.3	1.03554	1.03320	0.22597	1.04703	1.10937
0.5	1.10654	1.08876	1.60681	1.13843	2.88222
1	1.52359	1.33263	12.53356	1.62770	6.83301
3	7.26863	7.00768	3.59009	7.47316	2.81386
5	19.1815	18.9012	1.46130	19.3652	0.95745
8	48.2946	47.9380	0.73839	48.4140	0.24734
10	75.1774	74.7884	0.51745	75.2393	0.08236
20	299.22	298.86	0.12031	298.83	0.13228
50	1867.57	1867.25	0.01714	1864.06	0.18775
70	3659.98	3659.56	0.01148	3652.92	0.19299
100	7468.83	7468.12	0.00951	7454.23	0.19546
300	67,215.57	67,207.88	0.01144	67,082.71	0.19766
500	186,709.04	186,685.66	0.01252	186,339.67	0.19783
700	365,949.25	365,901.47	0.01306	365,225.11	0.19788
1000	746,834.69	746,733.98	0.01348	745,356.66	0.19791

Figure 2.

Comparison of the approximate frequency with the exact one.

Discussion and conclusion

The matching theory is very famous in economics, and the matching technology in this paper reminders the well-known Nobel Matching, which is relative to the Nobel prizes to Al Roth and Lloyd Shapley. ²⁰ This paper suggests a totally new modification of He’s frequency formulation, which considers two extremes of A<<1 and A>>1. The late case leads to an exact frequency, while the former case sees a high accuracy when He’s frequency formulation is applied. The two extremes are then matched using a homotopy parameter. The results show the our results are much better than those given in Ref.[1]. The largest error occurs at A=1, this is the turning point for A<<1 and A>>1. The present modification can be applied to nonlinear oscillators with more than one nonlinear terms.^21,22