Period/frequency estimation of a nonlinear oscillator

Abstract

Fast estimation of a nonlinear oscillator is very much needed in engineering, the perturbation method is valid only for weakly nonlinear oscillators. This paper gives some simple formula for this purpose, only simple calculation is needed to have a relatively high accuracy of the estimated frequency.

Introduction

Recently, García ¹ and Suárez ² gave some simple formulae for estimation of the period of a nonlinear oscillators. Considering a nonlinear oscillator in the form

$$u^{″}(t)+f(u)=0,u\prime (0)=0,u(0)=A$$

(1)where f(u)/u > 0 and f(0)=0, García obtained the following period estimation ¹

$$T_{Garcia}=2\pi \sqrt{\frac{A^{N+2}}{(N+2){\displaystyle {\int }_0^A{u}^{N}f(u)\text{d}u}}}$$

(2)where $N$ is called as García number for optimal identification. The optimal value of $N$ is 3 for $f(u)=-u^q,q\in {\Re }^{+}$ . Suárez Antola suggested the following period formula ²

$$T_{Antola}=2\pi \sqrt{\frac{{\displaystyle {\int }_0^{A}W(u)u^2\text{d}u}}{{\displaystyle {\int }_0^{A}W(u)f(u)u\text{d}u}}}$$

(3)where W is a weighing function. When we choose $W(u)=u^{N-1}$ , equation (3) reduces to equation (2).

Both formulae given in equations (2) and (3) are simple and useful for practical applications. This mini review article summarizes some even simpler period/frequency formulae mainly based on previous work on this special direction.

He’s frequency formulation

For the nonlinear oscillator given in equation (1), we can choose two arbitrary trial frequencies ${\omega }_1$ and ${\omega }_{\text{2}}$ for two trial solutions

$$u_1(t)=A\cos {\omega }_{1}t$$

(4)

$$u_{\text{2}}(t)=A\cos {\omega }_{\text{2}}t$$

(5)which lead to the following residuals, respectively

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

(6)

$$R_{\text{2}}(t)=-A{\omega }_{\text{2}}^2\cos {\omega }_{\text{2}}t+f(A\cos {\omega }_{\text{2}}t)$$

(7)

He’s frequency formula reads ³

$${\omega }_{He}^2=\frac{{\omega }_1^2{R}_2(0)-{\omega }_2^2{R}_1(0)}{R_2(0)-R_1(0)}=\frac{f(A)}{A}$$

(8)

For Duffing equation

$$u^{″}+u+\epsilon u^3=0,u\prime (0)=0,u(0)=A$$

(9)

Choosing ${\omega }_{\text{1}}=\text{1}$ and ${\omega }_{\text{2}}=\text{2}$ , we have

$${\omega }^{\text{2}}=\text{1}+\epsilon A^2$$

(10)

Note that ${\omega }_{\text{1}}$ and ${\omega }_{\text{2}}$ can be arbitrarily chosen, for example ${\omega }_{\text{1}}=\text{1}$ and ${\omega }_{\text{2}}=\text{200}$ , and all cases end in the same result. When $\epsilon =\text{0}$ , equation (10) is the exact frequency; while when $\epsilon \to \infty$ , the exact frequency is ${\omega }_{ex}=0.9318\sqrt{\epsilon A^2}$ , the accuracy of equation (10) reaches 6.82%.

Geng and Cai ⁴ suggested a modification for equation (8)

$${\omega }_{Geng-Cai}^2=\frac{{\omega }_1^2{R}_2(t_2)-{\omega }_2^2{R}_1(t_1)}{R_2(t_2)-R_1(t_1)}=\frac{f(A\cos \frac{\pi }{6})}{A\cos \frac{\pi }{6}}$$

(11)where $t_1=T_1/12=\pi /(6{\omega }_1)$ , $t_{\text{2}}=T_{\text{2}}/12=\pi /(6{\omega }_{\text{2}})$ .

The frequency formula was furthered improved in He^5,6

$${\omega }_{He}^2=\frac{{\omega }_1^2{\tilde{R}}_2-{\omega }_2^2{\tilde{R}}_1}{{\tilde{R}}_2-{\tilde{R}}_1}=\frac{\text{4}}{\pi A}{\displaystyle {\int }_0^{\frac{\pi }{2}}f(A\cos u)}\cdot \cos u\text{d}u$$

(12)where ${\tilde{R}}_1$ and ${\tilde{R}}_{\text{2}}$ are calculated by

$${\tilde{R}}_1=\frac{\text{4}}{T_1}{\displaystyle {\int }_0^{T_1/4}{R}_1(t)\cos \left(\frac{2\pi }{T_1}t\right)\text{d}t}$$

(13)

$${\tilde{R}}_{\text{2}}=\frac{\text{4}}{T_{\text{2}}}{\displaystyle {\int }_0^{T_{\text{2}}/4}{R}_{\text{2}}(t)\cos \left(\frac{2\pi }{T_{\text{2}}}t\right)\text{d}t}$$

(14)

For the Duffing equation

$${\omega }^{\text{2}}=\text{1}+\frac{\text{3}}{\text{4}}\epsilon A^2$$

(15)

The accuracy of the frequency given in equation (15) is 7% when $\epsilon \to \infty$ . Applications of the frequency are available in literature.^7–20

Max-min approach to nonlinear oscillators

We can re-write equation (1) in the form

$$u^{″}+F(u)u=0$$

(16)where $F(u)=f(u)/u$ .

If we search for an approximate solution in the form

$$u=A\cos \omega t$$

(17)

Comparing equation (16) with a linear oscillator $u^{″}(t)+{\omega }^{2}u=0$ , we can obtain the following inequality for the square of the frequency

$$F_{\min }<{\omega }^{\text{2}}<F_{\max }$$

(18)

We write Duffing equation in the form

$$u^{″}+(1+\epsilon u^2)u=0$$

(19)

Accordingly we have

$$\frac{\text{1}}{\text{1}}<{\omega }^2<\frac{1+\epsilon A^2}{1}$$

(20)

Using He Chengtian’s average, ²¹ we have

$${\omega }^{\text{2}}=\frac{m+n(1+\epsilon A^2)}{m+n}=1+k\epsilon A^2$$

(21)where $m$ and $n$ are positive constants, $k=n/(m+n)<1$ . How to optimally identify the value of $k$ is given in He. ²¹ Applications of the max-min approach and some nonlinear oscillators are available in literature.^22–33

Taylor series ³⁴

To describe its trajectory of a nonlinear oscillator is extremely difficult for the whole solution domain, but it is relatively easy to give an accurate prediction of its trajectory at its initial stage [0, T/4], where T is its period. According to equation (1), we have

$$u\prime (0)=0,u(0)=A\text{and}{u}^{″}(0)=-f(u(0))=-f(A)$$

(22)

Differentiating equation (1) with respect to time, we can easily obtain the value for $u^{(n)}(0)$ , so we can obtain its initial trajectory by Taylor series

$$u(t)=u(0)+u^{\prime }(0)t+\frac{1}{2!}{u}^{″}(0)t^2+\frac{1}{3!}{u}^{‴}(0)t^3+\frac{1}{4!}{u}^{(4)}(0)t^4+\cdot \cdot \cdot ,t\in [0,T/4]$$

(23)

A higher order Taylor series predicts a higher accuracy of its initial trajectory. Considering its periodic property, we have

$$u(T/4)=0$$

(24)where T is the period of equation (1).

To elucidate the effectiveness of this simple approach, we consider a nonlinear oscillator in the form ³⁴

$$u^{″}+sign(u)=0,u(0)=A,u^{\prime }(0)=0$$

(25)

When $0<t<\frac{T}{4}$ , the solution is monotonely decreasing, and equation (25) becomes

$$u(0)=A,u^{\prime }(0)=0,u^{″}(0)=-1$$

(26)

Solving equation (26) results in

$$u(t)=A-\frac{1}{2}{t}^2,\text{0}<t<\frac{T}{4}$$

(27)

Matching $u(T/4)=0$ , we have

$$A-\frac{1}{2}{\left(\frac{T}{4}\right)}^2=0$$

(28)

Its period can be easily solved from equation (28), which reads

$$T=4\sqrt{2A}=5.656854\sqrt{A}$$

(29)which happens to be the exact period.

The simpler, the better

In He, ³⁵ a fast estimation of frequency was given, which is

$${\omega }^2={\frac{df(u(t))}{\text{d}u}|}_{u=u_i}$$

(30)or

$${\omega }^2={\frac{df(u(t))}{\text{d}u}|}_{t=T/N}$$

(31)

The usual location point is chosen as $u_i=A/2$ or $t=T/12$ . This is the simplest formula with relatively high accuracy, and can be used as a fast look into the periodic property of a nonlinear oscillator. For the Duffing equation, when we choose $u_i=A/2$ , we have

$${\omega }^2=1+{3\epsilon u^2|}_{u=A/2}=1+\frac{3}{4}\epsilon A^2$$

(32)

Another simple estimation is

$${\omega }^2=F(u(0))=\frac{f(u(0))}{u(0)}=\frac{f(A)}{A}$$

(33)

Using a weighting function, equation (33) can be modified as ²

$${\omega }^2=\frac{{\displaystyle {\int }_0^{A}W(u)f(u)\text{d}u}}{{\displaystyle {\int }_0^{A}W(u)u\text{d}u}}$$

(34)

If we choose $W(u)=u^n$ , we obtain the following result

$${\omega }^{\text{2}}=\frac{{\displaystyle {\int }_{\text{0}}^{\text{A}}{u}^n(u}+\epsilon u^3)\text{d}u}{{\displaystyle {\int }_0^A{u}^n\cdot u}\text{d}u}=\frac{\frac{1}{n+2}{A}^{n+2}+\frac{1}{n+4}\epsilon A^{n+4}}{\frac{1}{n+2}{A}^{n+2}}=1+\frac{n+2}{n+4}\epsilon A^2$$

(35)

The optimal value for $n$ for Duffing equation is $25<n<26$ .

When $n=25$ , ${\omega }^{\text{2}}=\text{1}+\frac{\text{27}}{\text{29}}\epsilon A^2$ with relative error of 3.55%,

When $n=26$ , ${\omega }^{\text{2}}=\text{1}+\frac{\text{14}}{\text{15}}\epsilon A^2$ with relative error of 3.68%.

The García number is defined as a natural number. However, the optimal value of n in equation (35) is n = 25.3255.

Conclusions

This paper gives some simple formulae for fast prediction of frequency of a nonlinear oscillator, and all obtained results given in this paper are valid for the whole solution domain $0<\epsilon <\infty$ with a relatively high accuracy of less than 7%, while the perturbation method depends upon a small parameter, its solution process is more complex than all formulae given in this paper, and further its results are only valid for the case when $\epsilon \ll 1$ .