The harmonic balance method for Yao–Cheng oscillator

Abstract

Recently Yao and Cheng suggested a nonlinear oscillator with a damping, and an analytical solution was obtained by the homotopy perturbation method. This paper shows that the oscillator can also be effectively solved by the harmonic balance method.

Introduction

Recently analytical approaches to nonlinear vibrations have been caught much attention, for examples, the harmonic balance method, ¹ He’s frequency formulation,^2–7 and the variational iteration method.^8–11 In this paper we will study Yao–Cheng oscillator in the form ¹²

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

(1)where k, b, and c are constants.

Equation (1) can be effectively solved by the homotopy perturbation method, ¹² which is generally effective for oscillators without a damping term.^13–18

The harmonic balance method

In order to make the calculation simple, equation (1) is equivalently written in the form

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

(2)

The harmonic balance method ¹ is to search for a solution in the form

$$u(t)=A\hspace{0.17em}\cos \hspace{0.17em}\omega t$$

(3)where $\omega$ is the unknown frequency to be determined later.

The use of equation (2) in equation (1) results in the following residual

$$\begin{array}{c}R(t)=u^{″}+(b+k)u+c{u^{\prime }}^2(u^{″}+\mathit{bu})=-A{\omega }^2\cos \hspace{0.17em}\omega t+A(b+k)\cos \hspace{0.17em}\omega t+c{A}^2{\omega }^2{\sin }^2\omega t(-A{\omega }^2\cos \hspace{0.17em}\omega t+\mathit{bA}\cos \hspace{0.17em}\omega t)\end{array}$$

(4)

Simplification of equation (4) gives

$$R(t)=A(-{\omega }^2+(b+k)+\frac{1}{4}c{A}^2{\omega }^2(b-{\omega }^2))\cos \hspace{0.17em}\omega t-\frac{1}{4}c{A}^3{\omega }^2(b-{\omega }^2){\cos }^3\omega t$$

(5)

Setting the coefficient of $\cos \hspace{0.17em}\omega t$ to be zero results in

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

(6)which gives the following frequency and amplitude relationship

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

(7)

Equation (7) is exactly same as that by the homotopy perturbation method. ¹²

A fractal modification of Yao–Cheng oscillator

Yao–Cheng oscillator can model an oscillator with a damping term, when the velocity is zero, the damping force is maximal, and while the velocity reaches its maximum, the damping force is minimal. Such a case can be seen in an attachment oscillation in gecko’s smart adhesion, ¹⁹ considering the fractal property of the gecko’s attachment system, the attachment oscillator ¹⁹ can be modified as

$$\frac{d}{d{t}^{\alpha }}\left(\frac{\mathit{du}}{d{t}^{\alpha }}\right)+\frac{\mathit{ku}}{1+c{\left(\frac{\mathit{du}}{d{t}^{\alpha }}\right)}^2}+\frac{\lambda }{u^{2\beta +1}}=0,u(0)=A,\frac{\mathit{du}}{d{t}^{\alpha }}(0)=0$$

(8)where $d/d{t}^{\alpha }$ is the fractal derivative, its definition and its applications are referred to literature.^20–24 When $\alpha$  = 1 and $\beta$  = 0, equation (8) becomes Yao–Cheng oscillator. When $\alpha$  = 1 and c = 0, equation (8) becomes He–Li oscillator for attachment vibration. ¹⁹

Conclusion

This paper shows that the harmonic balance method ¹ is as effective as the homotopy perturbation method^10–13 for Yao–Cheng oscillator. The advantages of the harmonic balance method are: (1) no need for construction of the homotopy equation, which is a must in the homotopy perturbation method; (2) no expansion for the solution and the parameter involved in the linear term.