The homotopy perturbation method for a nonlinear oscillator with a damping

Abstract

The homotopy perturbation method is extended to solve nonlinear oscillators with damping terms, and an explicit relationship between the frequency and amplitude is obtained, the main factor affecting the periodic property is revealed.

Introduction

In this paper, we study the following nonlinear oscillator

$$\ddot{\mu }+\frac{\mathit{ku}}{1+c\dot{u}}+\mathit{bu}=0,u(0)=A,\dot{u}(0)=0$$

(1)

where k, b and c are constants. The middle term in equation (1) is a damping term.

Equation (1) can describe the string vibration or the membrane vibration under damping. There are many analytical methods for solving effectively nonlinear oscillators, for example, the variational iteration method,^1–4 the variational approach,^5–8 the homotopy perturbation method,^9–18 He’s frequency formulation^19–23 and others, but most methods are limited to conservative oscillators, and it is difficult to solve nonlinear oscillators with damping terms. This paper shows that the homotopy perturbation method is also valid for solving equation (1) analytically.

Homotopy perturbation method

The homotopy perturbation method can be powerfully applied to any nonlinear conservative oscillators^9–18; in this paper, we will extend it to nonlinear oscillators with a damping term. We write equation (1) in the form

$$\ddot{u}+\left(b+k\right)u+c{\dot{u}}^{\text{2}}\left(\ddot{u}+\mathit{bu}\right)=0$$

(2)

and construct a homotopy equation in the form

$$\ddot{u}+\left(b+k\right)u+\mathit{pc}{\dot{u}}^{\text{2}}\left(\ddot{u}+\mathit{bu}\right)=0$$

(3)

Following the requirement of the homotopy perturbation method, we expand the solution and the coefficient of the linear term in the forms

$$u=u_0+p{u}_{\text{1}}+p^{\text{2}}{u}_{\text{2}}+···$$

(4)

And

$$b+k={\omega }^{\text{2}}+p{\omega }_{\text{1}}+p^{\text{2}}{\omega }_{\text{2}}+···$$

(5)

We can obtain a series of linear equations

$${\ddot{u}}_0+{\omega }^{\text{2}}{u}_0=0,u_0\left(0\right)=A,{\dot{u}}_0\left(0\right)=0$$

(6)

$${\ddot{u}}_1+{\omega }^2{u}_1+{\omega }_1{u}_0+c{\dot{u}}_0^2\left({\ddot{u}}_0+b{u}_0\right)=0,u_1(0)=0,{\dot{u}}_1(0)=0$$

(7)

The solution of equation (6) is

$$u_0=A\text{cos}\omega t$$

(8)

Using this result, equation (7) becomes

$${\ddot{u}}_1+{\omega }^{\text{2}}{u}_{\text{1}}+{\omega }_{\text{1}}A\text{cos}\omega t+c{A}^{\text{2}}{\omega }^{\text{2}}{\text{sin}}^{\text{2}}\omega t\left(-A{\omega }^{\text{2}}\text{cos}\omega t+\mathit{bA}\text{cos}\omega t\right)=0$$

(9)

Simplifying equation (9) results in

$${\ddot{u}}_1+{\omega }^2{u}_1+A\left({\omega }_1+\frac{1}{4}c{A}^2{\omega }^2(b-{\omega }^2)\right)\cos \omega t-\frac{1}{4}c{A}^3{\omega }^2(b-{\omega }^2)\cos 3\omega t=0$$

(10)

No secular term should be included in the solution, this requires

$${\omega }_1+\frac{1}{4}c{A}^2{\omega }^2(b-{\omega }^2)=0$$

(11)

If the first-order approximate solution is enough, from equation (5), we have

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

(12)

Solving the frequency from equation (12) yields the result

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

(13)

The frequency and amplitude relationship given in equation (13) is important for optimization of the vibration process in the practical applications. The main parameters affecting the frequency can be seen in equation (13).

Conclusion

By a simple calculation as shown above, we obtain the needed frequency, the explicit formula given in equation (13) shows explicitly the effect of damping on the periodic property.