Ren’s frequency–amplitude formulation for nonlinear oscillators

Abstract

A fast insight into the frequency–amplitude relationship of a rehabilitation training device and other nonlinear vibrations as well is of great importance. This paper applies Ren’s frequency–amplitude formulation to obtain approximate frequencies of nonlinear oscillators with little calculation while keeping high relative accuracy. The golden mean location is used in the example procedures.

Introduction

Nonlinear vibration arises everywhere in engineering, such as electronics, biomechanics, physiology, textile engineering, and nanotechnology, though there are many analytical methods to approximately solve various nonlinear oscillators with remarkable accuracy, for examples, the variational iteration method,^1–3 the homotopy perturbation method,^4–6 the variational approach,^7,8 the Hamiltonian approach, ⁹ a complete reviews on analytical approaches to nonlinear oscillators are available in He. ¹⁰ Although much achievement in analytical methods has been made, it is of great importance to have a fast insight into the frequency and amplitude relation with simple calculation. For example, a fast estimation of the frequency of a rehabilitation training device is important for rehabilitation training. To this end, many frequency–amplitude formulae were applied, for examples, He’s max–min approach, ¹¹ He’s frequency–amplitude formulation and various modifications,^12–14 García’s amplitude-frequency formula, ¹⁵ Suárez-Antola’s amplitude–frequency formula. ¹⁶ Tian and Liu gave a systematic review on various amplitude–frequency formula. ¹⁷

He’s frequency–amplitude formulation was derived through ancient Chinese mathematics. ¹⁸ China is one of the most famous countries with brilliant ancient culture in the world. The ancient Chinese mathematics had made great contributions to the world human civilization, which is increasingly attracting worldwide attention and studies. According to an ancient Chinese mathematical method, the algebraic equation (equation (1)) can be solved as below

$$f(x)=0$$

(1)

Let x₁ and x₂ be its two approximate solutions, which lead to two remainders $f(x_1)$ and $f(x_2)$ , respectively. The approximate solution of the above algebraic equation can be expressed by equation (2)

$$x=\frac{x_2\hspace{0.17em}f(x_1)-x_1\hspace{0.17em}f(x_2)}{f(x_1)-f(x_2)}$$

(2)

The modern application of this algorithm has led to He’s frequency–amplitude formulation.^18–20 Recently, Zi-Yin Ren proposed another formulation ²¹ for nonlinear oscillators.

Ren’s frequency–amplitude formulation

Consider a generalized nonlinear oscillator with initial conditions in the below form

$$x^{″}+f\text{(}x\text{)}=0,x\text{(}0\text{)}=A,x^{\prime }\text{(}0\text{)}=0$$

(3)

We use two trial functions $x_1\text{(}t\text{)}=A\cos {\omega }_{1}t$ and $x_2\text{(}t\text{)}=A\cos {\omega }_{2}t$ , these are respectively solutions of the following linear oscillator equations

$$x^{″}+{\omega }_1^{2}x=0$$

(4)

and

$$x^{″}+{\omega }_2^{2}x=0$$

(5)

where ω₁ and ω₂ are the linear oscillator equations’ frequencies. Then we get two remainders from the trial functions

$$R_1\text{(}t\text{)}=f\text{(}{x}_1\text{)}-{\omega }_1^{2}A\text{cos}{\omega }_{1}t$$

(6)

and

$$R_2\text{(}t\text{)}=f\text{(}{x}_2\text{)}-{\omega }_2^{2}A\text{cos}{\omega }_{2}t$$

(7)

According to He’s frequency–amplitude formula, the approximate frequency of equation (3) can be written as

$${\omega }^2=\frac{{\omega }_2^2{R}_1(t_1)-{\omega }_1^2{R}_2(t_2)}{R_1(t_1)-R_2(t_2)}$$

(8)

Equation (8) is He’s frequency–amplitude formulation for nonlinear oscillators,^18–20 which was often used in literatures.^22–26

Zi-Yin Ren proposed another formulation ²¹ for nonlinear oscillators as shown in equation (9)

$${\omega }^4=\frac{{\omega }_2^4{R}_1(t_1)-{\omega }_1^4{R}_2(t_2)}{R_1(t_1)-R_2(t_2)}$$

(9)

This equation is called as Ren’s frequency–amplitude formulation, which is much effective to nonlinear oscillators.

Examples

Example 1

Now consider the below nonlinear oscillator

$$x^{″}+\frac{1}{x}=0,x(0)=A,x^{\prime }(0)=0$$

(10)

We substitute two trial functions $x_1(t)=A\cos t$ and $x_2\text{(}t\text{)}=A\text{cos}\omega t$ into it, we get two residuals

$$R_1(t)=-A\cos t+\frac{\epsilon }{A\cos t}$$

(11)

and

$$R_2(t)=-{\omega }^{2}A\cos \omega t+\frac{\epsilon }{A\cos \omega t}$$

(12)

According to equation (9), we have the approximate frequency

$${\omega }^4=\frac{{\omega }^4{R}_1\text{(}{t}_1\text{)}-R_2\text{(}{t}_2\text{)}}{R_1\text{(}{t}_1\text{)}-R_2\text{(}{t}_{\text{2}}\text{)}}$$

$$=\frac{-{\omega }^{4}A\cos t_1+\frac{{\omega }^4\epsilon }{A\cos t_1}+{\omega }^{2}A\cos \omega t_2-\frac{\epsilon }{A\cos \omega t_2}}{-A\cos t_1+\frac{\epsilon }{A\cos t_1}+{\omega }^{2}A\cos \omega t_2-\frac{\epsilon }{A\cos \omega t_2}}$$

(13)

Choosing the location point $t_1=\omega t_2=0.3820\times \frac{\pi }{2}=0.6000$ , which is a golden mean location of $\frac{\pi }{2}$ , we have

$${\omega }^4=-{\omega }^2+\frac{\epsilon (1+{\omega }^2)}{A^2{\cos }^{2}0.6000}$$

(14)

so

$${\omega }^2=\frac{\epsilon }{A^2{\cos }^{2}0.6000}$$

(15)

The approximate period of equation (10) can be expressed as

$$T_{\mathrm{Appro}}=\frac{2\pi }{\omega }=5.1858\frac{A}{\sqrt{\epsilon }}$$

(16)

The exact period of equation (10) is ¹⁰

$$T_{\mathrm{Exact}}=2\sqrt{2\pi }\frac{A}{\sqrt{\epsilon }}=5.0133\frac{A}{\sqrt{\epsilon }}$$

(17)

The relative error of the approximate period of equation (16)

$$\frac{\left|T_{\mathrm{Exact}}-T_{\mathrm{Appro}}\right|}{T_{\mathrm{Exact}}}=3.44\%$$

(18)

can generally be acceptable.

Example 2

Now we consider another nonlinear oscillator below

$$x^{″}+\frac{x}{1+\epsilon x^2}=0,x(0)=A,x^{\prime }(0)=0$$

(19)

This equation can be rewritten as

$$x^{″}+\epsilon x^2{x}^{″}+x=0$$

(20)

Substituting the two trial functions, $x_1(t)=A\cos t$ and $x_2\text{(}t\text{)}=A\text{cos}\omega t$ , into equation (20), the two residuals are

$$R_1(t)=-\epsilon A^3{\cos }^{3}t$$

(21)

and

$$R_2(t)=(1-{\omega }^2)A\cos \omega t-\epsilon {\omega }^2{A}^3{\cos }^3\omega t$$

(22)

By Ren’s frequency–amplitude formulation, the approximate frequency can be expressed as

$${\omega }^4=\frac{{\omega }^4{R}_1\text{(}{t}_1\text{)}-R_2\text{(}{t}_2\text{)}}{R_1\text{(}{t}_1\text{)}-R_2\text{(}{t}_{\text{2}}\text{)}}$$

$$=\frac{-\epsilon {\omega }^4{A}^3{\cos }^3{t}_1+({\omega }^2-1)A\cos \omega t_2+\epsilon {\omega }^2{A}^3{\cos }^3\omega t_2}{-\epsilon A^3{\cos }^3{t}_1+({\omega }^2-1)A\cos \omega t_2+\epsilon {\omega }^2{A}^3{\cos }^3\omega t_2}$$

(23)

Still choosing the location point $t_1=\omega t_2=0.3820\times \frac{\pi }{2}=0.6000$ , we have

$${\omega }^4=\frac{-\epsilon {\omega }^2{A}^2{\cos }^{2}0.6000+1}{\epsilon A^2{\cos }^{2}0.6000+1}$$

(24)

Simplifying equation (24) results in

$${\omega }^2(\epsilon A^2{\cos }^{2}0.6000+1)=1$$

(25)

We, therefore, get

$${\omega }^2=\frac{1}{1+\epsilon A^2{\cos }^{2}0.6000}$$

(26)

We obtain the approximate period

$$T_{\text{Appro}}=\frac{2\text{π}}{\text{ω}}=2\text{π}\sqrt{1+0.6812\text{ε}{A}^2}$$

(27)

The exact period of equation (19) can be expressed as ¹⁰

$$T_{\mathrm{Exact}}=4\sqrt{\epsilon }{\displaystyle {\int }_0^A\frac{\text{d}x}{\sqrt{\ln \frac{1+\epsilon A^2}{1+\epsilon x^2}}}}$$

(28)

The relative error E_Relat of the approximate period can be calculated by

$$E_{\mathrm{Relat}}=\frac{\left|T_{\mathrm{Exact}}-T_{\mathrm{Appro}}\right|}{T_{\mathrm{Exact}}}$$

(29)

The comparison between the approximate periods T_Appro and the exact periods T_Exact is listed in Table 1 for different values of ε and A, a good agreement is obtained.

Table 1.

Relative errors of approximate period.

ε	A	T Exact	T Appro	E Relative
0.1	0.5	6.3417	6.3365	0.0008
0.1	1	6.5134	6.4937	0.0030
0.1	2	7.1497	7.0877	0.0087
0.1	5	10.446	10.330	0.0111
0.1	10	17.504	17.561	0.0033
1	0.1	6.3067	6.3045	0.0003
1	0.5	6.8407	6.7972	0.0064
1	1	8.2502	8.1468	0.0125
1	2	12.217	12.126	0.0074
1	5	26.307	26.679	0.0141
1	10	50.950	52.237	0.0253
10	0.1	6.5134	6.4937	0.0030
10	0.5	10.446	10.330	0.0111
10	1	17.504	17.561	0.0033
10	2	32.780	33.394	0.0187
10	5	79.896	82.235	0.0293

Conclusion

In this work, Ren’s frequency–amplitude formulation is demonstrated through examples, the solution process is very simple, and the results are fairly effective. Ren’s frequency–amplitude formulation provides an efficient alternative method for the analysis of nonlinear vibrations. Although He’s frequency–amplitude formulation is simple enough for practical applications, and Prof. Ji-Huan He has developed some even simpler amplitude-period formulae in his recent studies.^27,28