A simplified He’s frequency–amplitude formulation for nonlinear oscillators

Abstract

Derived from an ancient Chinese algorithm, He’s frequency–amplitude formulation is an effective approach to finding an approximate solution of a nonlinear oscillator. In this article, based on He’s formulation, a simplified formulation is proposed. Some nonlinear oscillators are adopted as examples to demonstrate the solving process using this simplified formulation. Through the demonstration, it can be seen that the solving process is simplified.

Keywords

He’s frequency–amplitude formulation ancient Chinese mathematics nonlinear oscillators

Introduction

Nonlinear behaviours can be observed in plenty of real-world phenomena. And nonlinear oscillation has been a hot research topic for many years and involved in applications of different areas, such as automotive, sensing, fluid–solid interaction, aerospace, micro- and nano-scale and bioengineering. The significance of nonlinear oscillation theory has many reasons, such as developing new devices in micro- and nano-scales, analysing real-world cases with the consideration of nonlinearity to better insight into the oscillatory devices, having uncertainties in the model parameters and improving the design of nonlinear microelectromechanical systems. Therefore, researchers from different fields have explored the nature of nonlinear oscillators for decades.¹

Currently, more and more analytical and approximate methods have been developed to find approximate solutions of nonlinear oscillators, for instance, He’s variational iteration method^2–8; the homotopy perturbation method^9–24; the variational method²⁵; the parameter-expanding method^26,27 and other methods.²⁸ Researchers worldwide have sought a concise and practical approach to solve various nonlinear oscillators for a long time.

In 2006, a Chinese mathematician, Prof. Ji-Huan He, suggested a simple approach to finding an approximate solution of a nonlinear oscillator in his review article.²⁸ This method has been further developed in Refs. 29,30 and is now named after Prof. He, called He’s frequency–amplitude formulation.

This formulation is famous for its accuracy and convenience: only a few calculations can result in an accurate solution for the whole solution domain. He’s frequency–amplitude formulation is derived from an ancient Chinese algorithm. In this article, we will review the ancient Chinese algorithm and He’ frequency–amplitude formulation. Furthermore, a simplified formulation will be proposed and demonstrated.

He’s Frequency–Amplitude Formulation

The oldest method of approximating the real roots of a nonlinear equation is suggested by chapter 7 of Nine Chapters (九章数学) in China. To illustrate the idea of this algorithm, we can start by considering the following equation

\begin{matrix} f (x) = 0 \end{matrix}

(1)

Let $x_{1}$ and $x_{2}$ be the approximate solutions of equation (1), which lead to the remainders $f (x_{1})$ and $f (x_{2})$ respectively, and the following result will be acquired through the ancient Chinese algorithm

\begin{matrix} x = \frac{x_{2} f (x_{1}) - x_{1} f (x_{2})}{f (x_{1}) - f (x_{2})} \end{matrix}

(2)

Some application of this algorithm can be found in Refs. 31–35. The modern development of this algorithm led to the widely used He’s frequency–amplitude formulation.^28–30

He’s frequency–amplitude formulation can be derived by considering a generalized nonlinear oscillator with initial conditions in the form

\begin{matrix} μ^{″} + f (μ) = 0, μ (0) = A, μ^{'} (0) = 0 \end{matrix}

(3)

Then we can use two trial functions

\begin{matrix} μ_{1} (t) = A \cos ω_{1} t \end{matrix}

(4)

\begin{matrix} μ_{2} (t) = A \cos ω_{2} t \end{matrix}

(5)

which are respectively the solutions of the following equations

\begin{matrix} μ^{″} + ω_{1}^{2} μ = 0, μ (0) = A, μ^{'} (0) = 0 \end{matrix}

(6)

\begin{matrix} μ^{″} + ω_{2}^{2} μ = 0, μ (0) = A, μ^{'} (0) = 0 \end{matrix}

(7)

where

ω_{1}

and

ω_{2}

are the frequencies of the linear oscillators. And we will have the following remainders

\begin{matrix} R_{1} (t) = f (μ_{1}) - ω_{1}^{2} A \cos ω_{1} t \end{matrix}

(8)

\begin{matrix} R_{2} (t) = f (μ_{2}) - ω_{2}^{2} A \cos ω_{2} t \end{matrix}

(9)

Like the Chinese ancient algorithm, the approximate solution $μ = A \cos ω t$ can be denoted as the following equation

\begin{matrix} μ = \frac{μ_{2} R_{1} (t) - μ_{1} R_{2} (t)}{R_{1} (t) - R_{2} (t)} \end{matrix}

(10)

Expand the approximate solution $μ = A \cos ω t$ in a power series

\begin{matrix} μ = A \cos ω t = A (1 - \frac{1}{2!} ω^{2} t^{2} + \frac{1}{4!} ω^{4} t^{4} - \dots) \end{matrix}

(11)

In the same way, we will have

\begin{matrix} μ_{1} = A \cos ω_{1} t = A (1 - \frac{1}{2!} ω_{1}^{2} t^{2} + \frac{1}{4!} ω_{1}^{4} t^{4} - \dots) \end{matrix}

(12)

\begin{matrix} μ_{2} = A \cos ω_{2} t = A (1 - \frac{1}{2!} ω_{2}^{2} t^{2} + \frac{1}{4!} ω_{2}^{4} t^{4} - \dots) \end{matrix}

(13)

We can take the first three terms in the power series

\begin{matrix} μ = A \cos ω t = A (1 - \frac{1}{2!} ω^{2} t^{2} + \frac{1}{4!} ω^{4} t^{4}) \end{matrix}

(14)

and similarly

\begin{matrix} μ_{1} = A \cos ω_{1} t = A (1 - \frac{1}{2!} ω_{1}^{2} t^{2} + \frac{1}{4!} ω_{1}^{4} t^{4}) \end{matrix}

(15)

\begin{matrix} μ_{2} = A \cos ω_{2} t = A (1 - \frac{1}{2!} ω_{2}^{2} t^{2} + \frac{1}{4!} ω_{2}^{4} t^{4}) \end{matrix}

(16)

Then substituting equations (14)–(16) in equation (10)

\begin{matrix} \begin{array}{l} A - \frac{1}{2!} A t^{2} ω^{2} + \frac{1}{4!} A t^{4} ω^{4} \\ = A - \frac{1}{2!} A t^{2} [\frac{ω_{2}^{2} R_{1} (t_{1}) - ω_{1}^{2} R_{2} (t_{2})}{R_{1} (t_{1}) - R_{2} (t_{2})}] + \frac{1}{4!} A t^{4} [\frac{ω_{2}^{4} R_{1} (t_{1}) - ω_{1}^{4} R_{2} (t_{2})}{R_{1} (t_{1}) - R_{2} (t_{2})}] \end{array} \end{matrix}

(17)

In the light of the terms of $t^{2}$ , we will have

\begin{matrix} ω^{2} = \frac{ω_{2}^{2} R_{1} (t_{1}) - ω_{1}^{2} R_{2} (t_{2})}{R_{1} (t_{1}) - R_{2} (t_{2})} \end{matrix}

(18)

which has been used in many previous works.^36–48

In the light of the $t^{4}$ terms, we will have

\begin{matrix} ω^{4} = \frac{ω_{2}^{4} R_{1} (t_{1}) - ω_{1}^{4} R_{2} (t_{2})}{R_{1} (t_{1}) - R_{2} (t_{2})} \end{matrix}

(19)

which is also feasible in solving nonlinear oscillators.⁴⁹ Besides, an alternative modification of He’s formulation has been proposed in Ref. 50.

A Simplified Frequency–Amplitude Formulation

Inspired by the ancient Chinese algorithm (2), a simplified frequency–amplitude formulation can be envisaged. If we replace all $ω^{2}$ in equation (18) with $ω$ , the following formulation can be obtained

\begin{matrix} ω = \frac{ω_{2} R_{1} (t_{1}) - ω_{1} R_{2} (t_{2})}{R_{1} (t_{1}) - R_{2} (t_{2})} \end{matrix}

(20)

which is very similar to equation (2).

In the following part of this article, we will use some simple examples to demonstrate the calculation process of using this formulation. Through the demonstration, hopefully, its efficiency and convenience can be shown.

Examples

We can consider the Duffing equation as an example

\begin{matrix} μ^{″} + μ + ε μ^{3} = 0, μ (0) = A, μ^{'} (0) = 0 \end{matrix}

(21)

Two trial functions can be chosen

\begin{matrix} μ_{1} (t) = A \cos t \end{matrix}

(22)

\begin{matrix} μ_{2} (t) = A \cos ω t \end{matrix}

(23)

By substituting equations (22) and (23) in (21), the following two residuals can be obtained

\begin{matrix} R_{1} (t_{1}) = ε A^{3} \cos^{3} t_{1} \end{matrix}

(24)

\begin{matrix} R_{2} (t_{2}) = ε A^{3} \cos^{3} ω t_{2} + (1 - ω^{2}) A \cos ω t_{2} \end{matrix}

(25)

According to equation (20), the following approximate frequency can be obtained

\begin{matrix} ω = \frac{ω R_{1} (t_{1}) - R_{2} (t_{2})}{R_{1} (t_{1}) - R_{2} (t_{2})} = \frac{ω ε A^{3} \cos^{3} t_{1} - ε A^{3} \cos^{3} ω t_{2} - (1 - ω^{2}) A \cos ω t_{2}}{ε A^{3} \cos^{3} t_{1} - ε A^{3} \cos^{3} ω t_{2} - (1 - ω^{2}) A \cos ω t_{2}} \end{matrix}

(26)

Choosing the location where $t_{1} = t_{2} = 0$ , we will have

\begin{matrix} ω = \frac{ω ε A^{3} - ε A^{3} - (1 - ω^{2}) A}{ε A^{3} - ε A^{3} - (1 - ω^{2}) A} = \frac{(ω - 1) ε A^{3} + (ω^{2} - 1) A}{(ω^{2} - 1) A} = 1 + \frac{ε A^{2}}{1 + ω} \end{matrix}

(27)

By rearranging equation (27), the following frequency–amplitude relationship can be obtained

\begin{matrix} ω^{2} = 1 + ε A^{2} \end{matrix}

(28)

which is the same as the relationship in Ref. 28.

We can also choose the location where $t_{1} = T_{1} / 12$ and $t_{2} = T_{2} / 12$ . According to equation (20), the following equation can be obtained

\begin{matrix} ω = \frac{ω ε A^{3} \cos^{3} (π / 6) - ε A^{3} \cos^{3} (π / 6) - (1 - ω^{2}) A \cos (π / 6)}{ε A^{3} \cos^{3} (π / 6) - ε A^{3} \cos^{3} (π / 6) - (1 - ω^{2}) A \cos (π / 6)} = 1 + \frac{3 ε A^{2}}{4 (ω + 1)} \end{matrix}

(29)

The result can be simplified as

\begin{matrix} ω^{2} = \frac{3}{4} ε A^{2} + 1 \end{matrix}

(30)

which is the same as the result in Ref. 36.

At last, a golden mean location where $t_{1} = 0.0955 T_{1}$ and $t_{2} = 0.0955 T_{2}$ can also be considered. According to equation (20), the frequency can be denoted as

ω = \frac{ω ε A^{3} \cos^{3} 0.191 π - ε A^{3} \cos^{3} 0.191 π - (1 - ω^{2}) A \cos 0.191 π}{ε A^{3} \cos^{3} 0.191 π - ε A^{3} \cos^{3} 0.191 π - (1 - ω^{2}) A \cos 0.191 π} = 1 + \frac{0.6811 ε A^{2}}{(ω + 1)}

(31)

The result can be expressed as follows

\begin{matrix} ω^{2} = 0.6811 ε A^{2} + 1 \end{matrix}

(32)

which is the same as that in Ref. 37.

We can also consider the following nonlinear oscillator

\begin{matrix} μ^{″} + \frac{1}{1 + ε μ^{2}} = 0, μ (0) = A, μ^{'} (0) = 0 \end{matrix}

(33)

Two trial solutions can be the same as equations (22) and (23). By substituting equations (22) and (23) in (33), we can get the following residuals

\begin{matrix} R_{1} (t) = \frac{A \cos t}{1 + ε A^{2} \cos^{2} t} - A \cos t \end{matrix}

(34)

\begin{matrix} R_{2} (t) = \frac{A \cos ω t}{1 + ε A^{2} \cos^{2} ω t} - A ω^{2} \cos ω t \end{matrix}

(35)

According to equation (20), the following equation can be obtained

\begin{matrix} ω = \frac{ω ((A \cos t_{1} / (1 + ε A^{2} \cos^{2} t_{1})) - A \cos t_{1}) - ((A \cos ω t_{2} / (1 + ε A^{2} \cos^{2} ω t_{2})) - A ω^{2} \cos ω t_{2})}{((A \cos t_{1} / (1 + ε A^{2} \cos^{2} t_{1})) - A \cos t_{1}) - ((A \cos ω t_{2} / (1 + ε A^{2} \cos^{2} ω t_{2})) - A ω^{2} \cos ω t_{2})} \end{matrix}

(36)

When $t_{1} = t_{2} = 0$ , we will have

\begin{matrix} ω^{2} = \frac{1}{1 + ε A^{2}} \end{matrix}

(37)

When $t_{1} = T_{1} / 12$ and $t_{2} = T_{2} / 12$ , we will have

\begin{matrix} ω^{2} = \frac{1}{1 + (3 / 4) ε A^{2}} \end{matrix}

(38)

When $t_{1} = 0.0955 T_{1}$ and $t_{2} = 0.0955 T_{2}$ , we will have

\begin{matrix} ω^{2} = \frac{1}{1 + 0.6811 ε A^{2}} \end{matrix}

(39)

These results are precisely the same as the results obtained in Ref. 37.

Conclusion

In this article, a simplified frequency–amplitude formulation has been proposed based on He’s frequency–amplitude formulation. This simplified formulation is more convenient and concise than He’s formulation and is feasible in practical applications. By applying the simplified formulation, the calculation process can be simplified without the loss of accuracy.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Zi-Yin Ren

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