Frequency and solution of an oscillator with a damping

Abstract

The frequency is an intrinsic mathematical index for a nonlinear oscillator. The variational iteration method and He’s frequency formulation are used to handle the special oscillator $\ddot{x}+\left(1+{\dot{x}}^2\right)x=0$ for its approximate frequency and solutions. The Laplace transform and Ren–Hu’s average residual are adopted to simplify the solution process.

Introduction

Nonlinear oscillators have an imperative role in practical applications and theoretical analysis. He and Kong ¹ found that the oscillation property of a moving jet during the electro-spinning process^2–4 plays an important role in fiber’s morphology and properties. Li and He ⁵ established an attachment oscillator on nano-scale by the geometric potential theory,^6–8 showing the nano-scale oscillation is the key factor of the creature’s smart adhesion. Zhao et al. ⁹ studied needle’s vibration in electro-spinning, revealing that the vibration can control the nano-fiber’s morphology. Nonlinear vibration theory sheds a bright light on the nanotechnology and nonlinear science as well.

Recently, much concern was referred to an oscillator in the form^10–13

$$\ddot{x}+\left(1+{\dot{x}}^2\right)x=0,x\left(0\right)=A,\text{ and }\dot{x}\left(0\right)=0$$

(1)

with a velocity-dependent damping.

This oscillator can be solved by various methods,^10–13 among which the variational iteration method accompanied by Laplace transform ¹⁴ and the modified He’s amplitude–frequency formulation ¹⁵ have been caught much attention due to their simple solution process and high accurate solutions.

Variational iteration method with Laplace transform

The variational iteration method was proposed in the 1990s,^16–18 the main difficulty is to effectively identify He–Lagrange multiplier involved in the iteration algorithm; however, the condition was changed, a simple and effective identification process by Laplace transform was suggested by Anjum and He. ¹⁴

Rewrite the problem (1) as

$$\ddot{x}+{\text{ω}}^{2}x+g(x)=0$$

(2)

with $g\left(x\right)=\left(1+{\dot{x}}^2\right)x-{\text{ω}}^{2}x$ .

According to the variational iteration method with Laplace transform, ¹⁴ we have the iteration algorithm

$$\begin{array}{l}\text{L}\left[x_{n+1}\right]=\text{L}\left[x_n\right]-\text{L}\left[{\displaystyle {\int }_0^t\frac{1}{\omega }\left({\ddot{x}}_n\left(\xi \right)+x_n\left(\xi \right)\left(1+{\dot{x}}_n^2\left(\xi \right)\right)\right)\sin \omega \left(t-\xi \right)\text{d}\xi }\right]\\ =\text{L}\left[x_n\right]-\frac{1}{\omega }\text{L}\left[\sin \omega t\right]\cdot \text{L}\left[{\ddot{x}}_n\left(t\right)+x_n\left(t\right)\left(1+{\dot{x}}_n^2\left(t\right)\right)\right]\end{array}$$

(3)

where L is the operator of the Laplace transform.

Assume the initial approximation be

$$x_0=A\cos \omega t$$

(4)

Then we have

$$\begin{array}{l}\text{L}\left[x_1\left(t\right)\right]=\text{L}\left[A\text{cosω}t\right]-\frac{1}{\text{ω}}\text{L}\left[\sin \text{ω}t\right]\cdot \text{L}\left[-{A\text{ω}}^2\text{cosω}t+A\left(1+{\text{ω}}^2{A}^2{\text{sin}}^2\text{ω}t\right)\text{cosω}t\right]\\ =\text{L}\left[A\text{cosω}t\right]-\frac{1}{\text{ω}}\text{L}\left[\sin \text{ω}t\right]\text{L}\left[\left(A-{A\text{ω}}^2+\frac{A^3{\text{ω}}^2}{4}\right)\text{cosω}t-\frac{A^3{\text{ω}}^2}{4}\text{cos}3\text{ω}t\right]\end{array}$$

(5)

Imposing the inverse Laplace transform in equation (5), there holds the first-order approximation

$$x_1\left(t\right)=\left(A-\frac{A^3}{32}\right)\cos \omega t-\frac{t}{2\omega }\left(A-A{\omega }^2+\frac{A^3{\omega }^2}{4}\right)\sin \omega t+\frac{A^3}{32}\cos 3\omega t$$

(6)

No secular term in equation (6) requires that

$$A-A{\omega }^2+\frac{A^3{\omega }^2}{4}=0$$

(7)

which yields the result

$${\omega }^2=\frac{4}{4-A^2}$$

(8)

Equation (8) is the same as that in Zhao et al. ⁹ and Xu. ¹⁰

He’s frequency formulation

He’s frequency formulation ¹⁹ was widely applied to nonlinear oscillators due to its simplicity and effectiveness.^20–22

According to He’s frequency–amplitude formulation, ¹⁹ we take a trial function

$$x_1=\mathit{A}\text{cos}{\omega }_{1}t$$

(9)

Submitting equation (9) into equation (1) results in the residual

$$\begin{array}{l}{R}_1=\left(1+{\omega }_1^2{A}^2{\mathrm{sinn}}^2{\omega }_{1}A\right)A\cos {\omega }_{1}t-{\omega }_1^{2}A\cos {\omega }_{1}t\\ =\left(1+{\omega }_1^2{A}^2\frac{1-\cos 2{\omega }_{1}t}{2}\right)A\cos {\omega }_{1}t-{\omega }_1^{2}A\cos {\omega }_{1}t\end{array}$$

(10)

Let ${\tilde{R}}_1$ be Ren–Hu’s average residual defined as ¹⁵

$${\tilde{R}}_1=\frac{4}{T}{\displaystyle {\int }_0^{\frac{T}{4}}{R}_1\left(t\right)\text{d}t}$$

(11)

with T the period of the oscillator.

Submitting equation (10) into equation (11), we have

$$\begin{array}{l}{\tilde{R}}_1=\frac{4}{\text{T}}{\displaystyle {\int }_0^{\frac{T}{4}}\left[\left(1+{\text{ω}}_1^2{A}^2\frac{1-\cos 2{\text{ω}}_{1}t}{2}\right)A\cos {\text{ω}}_{1}t-{\text{ω}}_1^{2}A\cos {\text{ω}}_{1}t\right]\text{d}t}\\ =\frac{2}{\text{π}}\left(A-{\text{ω}}_1^{2}A\right)+\frac{{\text{ω}}_1^2{A}^3}{3\text{π}}\end{array}$$

(12)

Let $x_2=\mathit{Acos}{\omega }_{2}t$ with ${\omega }_1\ne {\omega }_2$ , by the same procedure as above, we have

$${\tilde{R}}_2=\frac{2}{\text{π}}\left(A-{\text{ω}}_2^{2}A\right)+\frac{{\text{ω}}_2^2{A}^3}{3\text{π}}$$

(13)

Using He’s frequency formulation, ¹⁹ we have

$$\begin{array}{l}{\text{ω}}^2=\frac{{\tilde{R}}_2{\text{ω}}_1^2-{\tilde{R}}_1{\text{ω}}_2^2}{{\tilde{R}}_2-{\tilde{R}}_1}\\ =\frac{\left[\frac{2}{\text{π}}\left(A-{\text{ω}}_2^{2}A\right)+\frac{{\text{ω}}_2^2{A}^3}{3\text{π}}\right]{\text{ω}}_1^2-\left[\frac{2}{\text{π}}\left(A-{\text{ω}}_1^{2}A\right)+\frac{{\text{ω}}_1^2{A}^3}{3\text{π}}\right]{\text{ω}}_2^2}{\frac{2}{\text{π}}\left(A-{\text{ω}}_2^{2}A\right)+\frac{{\text{ω}}_2^2{A}^3}{3\text{π}}-\left[\frac{2}{\text{π}}\left(A-{\text{ω}}_1^{2}A\right)+\frac{{\text{ω}}_1^2{A}^3}{3\text{π}}\right]}\\ =\frac{6}{6-A^2}\end{array}$$

(14)

Equations (8) and (14) give an almost same result when A≪1. Equation (14) is a new angular frequency expression for the oscillator.

Conclusions

This paper shows the last development of the variational iteration method and He’s frequency formulation for nonlinear oscillators with a damping; the simple solution process and high accurate results make both methods famous for practical applications for fast and effective estimation about the frequency–amplitude relationship of a nonlinear oscillator. The Laplace transform makes free of the identification of the He–Lagrange multiplier in the variational iteration method. Ren–Hu’s average residual makes the calculation process much simpler. Both methods result in acceptable accuracy of the solutions with simple solution process, and we conclude both methods can be widely applied to nonlinear vibration systems.

The couple of the Laplace transform with either the variational iteration method^16–18 or the homotopy perturbation method^23–25 now is called as He–Laplace method^26–29 and it has obvious advantages in the simple solution process. Both methods discussed in this paper are also powerful tools for fractal calculus and fractional calculus.^30–33