He’s frequency–amplitude formulation for nonlinear oscillator with damping

Abstract

He’s frequency–amplitude formulation takes an active role in the analysis of nonlinear conservative oscillators. This paper shows that the formulation can be extended into damping cases, and the result is the same as that obtained by the homotopy perturbation method.

Introduction

In this study, we intend to seek approximate frequency of the following nonlinear oscillator with damping ¹ through He’s frequency–amplitude formulation

$$u^{″}+\frac{\mathit{ku}}{1+c{u^{\prime }}^2}+\mathit{bu}=0,u(0)=A,u^{\prime }(0)=0$$

(1)where b, c and k are constants, the middle fraction represents damping effect. This equation describes a kind of damped membrane vibration or damped string vibration.

It was hardly possible for scientists all over the world to solve nonlinear oscillators accurately in early stage of nonlinear science. Now the condition is changed, some effective methods have been developed for the analyses of nonlinear oscillators, such as the variational approach,^2–6 the homotopy perturbation methods,^7–13 the variational iteration method,^14–17 He’s frequency–amplitude formulation,^18–21 the parameter expansion method, ²² the Hamiltonian approach, ²³ the stability analysis, ²⁴ the max-min approach, ²⁵ the energy balance method, ²⁶ and so on. Although most of the methods are suitable for conservative oscillators, there is some literature focusing on damping cases. Gheorghe and Cristian reported the stability of solutions to damped nonlinear oscillators.^27,28 Ene et al. employed the optimal homotopy perturbation method for nonlinear oscillators with non-viscous exponential damping. ²⁹ Tian et al. studied the effects of fractional-order intrinsic damping, fractional-order external damping and nonlinear noise on resonant behaviors.^30,31 There are other references^32–34 on the investigation of properties of nonlinear oscillators with random damping. This paper will demonstrate the effectiveness and simplicity of He’s frequency–amplitude formulation for approximate frequency of nonlinear oscillator with damping.

He’s frequency–amplitude formulation

He’s frequency–amplitude formulation was originated from an ancient Chinese mathematics method. Below is an algebraic equation

$$F(x)=0$$

(2)

Let x₁ and x₂ be its two trial solutions, which produce two residuals F(x₁) and F(x₂), respectively; according to the ancient Chinese mathematics method, an approximate solution of the algebraic equation can be written as

$$x=\frac{x_{2}F(x_1)-x_{1}F(x_2)}{F(x_1)-F(x_2)}$$

(3)

Equation (3) generally gives an accurate solution, although with the extremely simple calculating procedure.

He’s frequency–amplitude formulation was proposed by Prof. Ji-Huan He for nonlinear oscillators^35,36 as below

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

(4)in which A is its amplitude, and $u(0)=A$ , $u^{\prime }(0)=0$ are initial conditions. If we put two trial functions $u_1=A\cos {\omega }_{1}t$ and $u_2=A\cos {\omega }_{2}t$ into equation (4), then two residuals are produced

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

(5)and

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

(6)

Meanwhile, $u_1=A\cos {\omega }_{1}t$ and $u_2=A\cos {\omega }_{2}t$ are, respectively, the exact solutions to the following two linear differential equations

$$u^{″}+{\omega }_1^{2}u=0$$

(7)and

$$u^{″}+{\omega }_2^{2}u=0$$

(8)where ω₁ and ω₂ are the frequencies of the two linear differential equations. According to He’s frequency–amplitude formulation, an approximate frequency of the nonlinear oscillator of equation (4) can be expressed by

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

(9)where the phases of the residuals R(t₁) and R(t₂) rest on location points t₁ and t₂.

Solving process

We rewrite equation (1) in the form

$$u^{″}+(b+k)u+c{u^{\prime }}^2(u^{″}+\mathit{bu})=0$$

(10)

Here, we substitute a trial function $u=A\cos \omega t$ , which is the solution to linear equation $u^{″}+{\omega }^{2}u=0$ , into equation (10), we get a residual

$$R(t)=(b+k-{\omega }^2)A\cos \omega t+{\omega }^{2}c{A}^3(b-{\omega }^2){\sin }^2\omega t\cos \omega t$$

(11)

When ω = 1, it becomes

$$R_1(t)=(b+k-1)A\cos t+c{A}^3(b-{\omega }^2){\sin }^{2}t\cos t$$

(12)

According to He’s frequency–amplitude formulation in equation (9), an approximate frequency of equation (10) can be calculated by

$${\omega }^2=\frac{{\omega }^2{R}_1(t)-R(t)}{R_1(t)-R(t)}$$

(13)

Geng and Cai set the phase of residuals by $t=\frac{T}{12}$ , ³⁷ where T ( $=\frac{2\pi }{\omega }$ ) is the period, that is to say that the phases of equations (11) and (12) are both $\frac{\pi }{6}$ .

$$R\left(\frac{T}{12}\right)=\frac{\sqrt{3}}{2}(b+k-{\omega }^2)A+\frac{\sqrt{3}}{8}{\omega }^{2}c{A}^3(b-{\omega }^2)$$

(14)and

$$R_1\left(\frac{T}{12}\right)=\frac{\sqrt{3}}{2}(b+k-1)A+\frac{\sqrt{3}}{8}c{A}^3(b-1)$$

(15)

Substituting equations (14) and (15) into equation (13), we get

$${\omega }^2=\frac{\frac{1}{4}c{A}^2{\omega }^2+b+k}{-\frac{1}{4}c{A}^2(b-{\omega }^2-1)+1}$$

(16)

Equation (16) is an algebraic equation, we rewrite it as

$$c{A}^2{\omega }^4+(4-\mathit{bc}{A}^2){\omega }^2-4(b+k)=0$$

(17)which is the same as the algebraic equation obtained through homotopy perturbation method.¹ The approximate frequency of equation (1) can be computed from equation (17)

$$\omega =\sqrt{\frac{\mathit{bc}{A}^2-4+\sqrt{{(\mathit{bc}{A}^2-4)}^2+16c{A}^2(b+k)}}{2c{A}^2}}$$

(18)which is the same as that obtained by the homotopy perturbation method,¹ showing the reliability of He’s frequency–amplitude formulation.

Summary

After the brief solving procedure shown in the context above, we obtained the reliable relationship between the frequency and the amplitude of damped nonlinear vibration, which includes the main parameters affecting the frequency property. He’s frequency–amplitude formulation provides an efficient method for the analysis of nonlinear oscillators with damping.