A short remark on He’s frequency formulation

Abstract

The unprecedented frequency formulation proposed by Chinese mathematician, Ji-Huan He, has been caught much attention due to its extreme simplicity and remarkable accuracy for nonlinear oscillators. This short remark uses an example without a linear term arising in ball-bearing oscillating to show its effectiveness and reliability.

Introduction

Nonlinear oscillators arise everywhere from nano/micro devices to trains.^1–8 Even the oldest device called as Fangzhu, which was designed to collect water from air, works according to the low frequency property of the absorbed water drop. ⁹ Generally an exact solution to a nonlinear equation is extremely difficult, and an approximate one is much needed, though many analytical methods are available in open literature and new ones are appearing, in engineering, a fact estimation of the periodic property of a nonlinear oscillator is much needed, the simpler, the better, as claimed by Chinese mathematician, Dr Ji-Huan He, who suggested the simplest method for a conservative nonlinear oscillator,^10,11 shedding a bright light on the nonlinear vibration theory. This short remark wants to verify He’s frequency formulation using a nonlinear oscillator without a linear term, such a problem makes the solution process impossible by the traditional perturbation method even when the nonlinear term is very weak.

He’s frequency formulation

He’s frequency formulation is extremely simple and it is accessible to everyone. This paper uses the well-known Duffing equation to elucidate the solution process

$$u^{″}+f(u)=0,\hspace{1em}u(0)=A,\hspace{1em}{u}^{\prime }(0)=0$$

(1)

where f(u) is a function of u, for nonlinear oscillators, it requires that f(u)/u > 0.

He’s frequency formulation predicts the square of the frequency in the form

$${\omega }^2=f^{\hspace{0.17em}\hspace{0.17em}\prime }(u){|}_{u=\tilde{u}}$$

(2)

where $\tilde{u}$ is always chosen as $\tilde{u}=\frac{A}{2}$ .

For Duffing equation, we have

$$f(u)=u+\epsilon u^3$$

(3)

It is easy to calculate its derivative with respect to u, that is

$$f^{\hspace{0.17em}\hspace{0.17em}\prime }(u)=1+3\epsilon u^2$$

(4)

The frequency can be obtained as

$$\omega =\sqrt{f^{\hspace{0.17em}\prime }(u){|}_{u=\frac{A}{2}}}=\sqrt{1+3\epsilon \frac{A^2}{4}}$$

(5)

The result is same as those obtained by the homotopy perturbation method, the variational iteration method or other methods. Considering the simple solution process, this method has the obvious advantages over all analytical methods.

An example

This example considers the motion of a ball-bearing oscillating in a glass tube that is bent into a curve such that the restoring force depends upon the cube of the displacement $u$ , the governing equation.

$$u^{″}+\epsilon u^3=0,\hspace{1em}u(0)=A,\hspace{1em}{u}^{\prime }(0)=0$$

(6)

This equation has no linear terms, and the perturbation method becomes invalid ever for the small $\epsilon$ , i.e. $0<\epsilon <<1$ . Here $f(u)=\epsilon u^3$ , and its frequency reads

$$\omega =\sqrt{f^{\prime }(u){|}_{u=A/2}}=\frac{\sqrt{3\epsilon }}{2}A$$

(7)

Its exact frequency is

$${\omega }_{\mathit{exact}}={[\Gamma (3/4)]}^2{\pi }^{-1/2}{\mu }^{1/2}A$$

(8)

The relative error is given by

$$\left|\frac{\omega -{\omega }_{\mathit{exact}}}{{\omega }_{\mathit{exact}}}\right|\times 100\%=2.2\%$$

(9)

Considering the simple operation, the 2.2% relative error is acceptable in most engineering applications, making the nonlinear vibration theory much accessible to engineers who have little knowledge on mathematics.

Conclusion

An example is enough to show the effectiveness and accuracy and reliability of He’s frequency. As Dr Ji-Huan He has been emphasized, the simpler is the better. In engineering applications, a fast and effective estimation of a nonlinear vibration problem is needed, and He’s frequency formation becomes a universal tool for this purpose.