The simpler,the better: Analytical methods for nonlinear oscillators and fractional oscillators

Abstract

In engineering, a fast estimation of the periodic property of a nonlinear oscillator is much needed. This paper reviews some simplest methods for nonlinear oscillators, including He’s frequency formulation, the max-min approach and the homotopy perturbation method. A mathematical insight into He’s frequency formulation is given, and the weighted average is introduced to further improve the estimated accuracy of the frequency. Fractional oscillators are also discussed.

Keywords

He’s frequency formulation homotopy perturbation method variational approach Hamiltonian approach weighed average fractal calculus fractional oscillator

Introduction

Nonlinear oscillations arise everywhere in our everyday life and engineering. As an exact solution might be too complex to be used for a practical application, many analytical methods have been used in open literature, for example, the variational iteration method,^1–7 the homotopy perturbation method,^8–20 He–Laplace method,^21–23 the variational approach^24–29 and the Hamiltonian approach.^30,31 The most important property of a nonlinear oscillator is the relationship between the frequency and its amplitude, the simplest method to estimate the frequency–amplitude relationship might be He’s frequency formulation^32–34 and the max-min approach,^35,36 which are still under development and many modifications were proposed to improve the accuracy.^37–45

Consider a nonlinear oscillator in the form

u^{″} + f (u) = 0, u (0) = A, u^{'} (0) = 0

(1)

The simplest criterion for the existence of a periodic solution of equation (1) is¹

\frac{f (u)}{u} > 0

(2)

This criterion can be understood by a pendulum-like oscillator

u^{″} = - {\frac{f (u)}{u}} u

(3)

where

u^{″}

is the acceleration, u is the displacement. When the acceleration is positive/negative, its displacement must be negative/positive, otherwise the displacement might tend to be either infinity or zero, and no periodic motion might occur. As an example, we consider the following equation

u^{″} + sign (u) = 0, u (0) = A, u^{'} (0) = 0

(4)

where

sign (u)

is a sign function defined as

sign (u) = {\begin{array}{l} 1, u > 0 \\ 0, u = 0 \\ - 1, u < 0 \end{array}

(5)

It is obvious that we have

\frac{sign (u)}{u} > 0

(6)

So equation (4) has a periodic solution.

In this paper, we will review some simplest methods for fast calculation of the frequency–amplitude relationship.

He’s frequency formulation

He’s frequency formulation was initially inspired by an ancient Chinese algorithm³⁶; now this formulation has gained much attention and has been under development, with various modifications appearing in open literature.^37–45

We define a residual function R(t) as

R (t) = u^{″} + f (u)

(7)

He’s frequency formulation is to choose two trial solutions

u_{1} (t) = A \cos ω_{1} t

(8)

u_{2} (t) = A \cos ω_{2} t

(9)

where

ω_{1}

and

ω_{2}

are the arbitrarily chosen frequencies. The residuals are, respectively, as

R_{1} (t) = - A ω_{1}^{2} \cos ω_{1} t + f (A \cos ω_{1} t)

(10)

R_{2} (t) = - A ω_{2}^{2} \cos ω_{2} t + f (A \cos ω_{2} t)

(11)

The residuals are time-dependent, and we can use their weighted average

{\tilde{R}}_{1} = \int_{0}^{T / 4} R_{1} (t) w (t) d t

(12)

{\tilde{R}}_{2} = \int_{0}^{T / 4} R_{2} (t) w (t) d t

(13)

where w(t) is a weighted function. He’s frequency formulation is

ω^{2} = \frac{{\tilde{R}}_{2} ω_{1}^{2} - {\tilde{R}}_{1} ω_{2}^{2}}{{\tilde{R}}_{2} - {\tilde{R}}_{1}}

(14)

If the weighted function is chosen as

w (t) = \cos ω t

(15)

we have the original He’s formulation. If we choose w(t)=1, we have Ren–Hu’s modification.⁴¹ If we choose

w (t) = R (t)

, we have

{\tilde{R}}_{1} = \int_{0}^{T / 4} R_{1}^{2} (t) d t

(16)

{\tilde{R}}_{2} = \int_{0}^{T / 4} R_{2}^{2} (t) dt

(17)

For simplicity, we can use discrete weighted average

{\tilde{R}}_{1} = \frac{\sum_{i = 1}^{N} {[- A ω_{1}^{2} \cos ω_{1} t_{i} + f (A \cos ω_{1} t_{i})] w (ω_{2} t_{i})}}{\sum_{i = 1}^{N} w (ω_{2} t_{i})}

(18)

{\tilde{R}}_{2} = \frac{\sum_{i = 1}^{N} {[- A ω_{2}^{2} \cos ω_{2} t_{i} + f (A \cos ω_{2} t_{i})] w (ω_{2} t_{i})}}{\sum_{i = 1}^{N} w (ω_{2} t_{i})}

(19)

If we write the trial solution as

\tilde{u} (t) = A \cos ω t

(20)

to optimally identify the frequency, this trequires the residual is minimal

R (\tilde{u}) = - ω^{2} A \cos ω t + f (\tilde{u}) = - ω^{2} \tilde{u} + f (\tilde{u}) \to \min

(21)

This implies

\frac{dR (\tilde{u})}{d \tilde{u}} = - ω^{2} + \frac{df (\tilde{u})}{d \tilde{u}} = 0

(22)

We, therefore, obtain another simple frequency formulation, which reads³⁴

ω^{2} = \frac{df}{du} (u = \tilde{u})

(23)

In some previous publications,³⁴ we choose $\tilde{u} = A / 2$

ω^{2} = \frac{df}{dx} (\frac{A}{2})

(24)

We can use the weighted average of equation (23) as discussed above.

If we write

\frac{df}{du} \approx \frac{f (\tilde{u}) - f (0)}{\tilde{u} - 0} = \frac{f (\tilde{u})}{\tilde{u}}

(25)

equation (23) becomes³⁵

ω^{2} = \frac{f (\tilde{u})}{\tilde{u}}

(26)

This formulation is widely used in the max-min approach³⁵

ω^{2} = {\frac{f (A \cos ω t)}{A \cos ω t} |}_{ω t = T / N}

(27)

We can also use instead of equation (27) the weighted average of equation (26).

Variational approach

The variational approach^24,29 is to establish a variational principle for equation (1)

J (u) = \int_{0}^{T / 4} {\frac{1}{2} {u^{'}}^{2} - F (u)} d t

(28)

where

\partial F / \partial u = f (u)

. If we choose an initial guess in the form

\tilde{u} (t) = A \cos ω t

(29)

The functional equation (28) becomes

J (A) = \int_{0}^{T / 4} {\frac{1}{2} A^{2} ω^{2} \sin^{2} ω t - F (A \cos ω t)} d t

(30)

By setting

\frac{dJ (A)}{dA} = 0

(31)

we can obtain the needed frequency.

Hamiltonian approach

The Hamiltonian approach³⁰ is a further extension of the variational approach. In equation (28), ${u^{'}}^{2} / 2$ is the kinetic energy and $F (u)$ is the potential energy. According to energy conservation, we have

\frac{1}{2} {u^{'}}^{2} + F (u) = \frac{1}{2} {u^{'}}_{0}^{2} + F (u_{0})

(32)

If we begin with $u (t) = A \cos ω t$ , we have the following equation

\frac{1}{2} A^{2} ω^{2} \sin^{2} ω t + F (A \cos ω t) - F (A) = 0

(33)

ω^{2} = \frac{2 (F (A) - F (A \cos ω t))}{A^{2} \sin^{2} ω t}

(34)

Using the weighted average, the frequency can be easily determined. The most simple calculation is to set $ω t = T / N$ , where T is the period, N is a constant

ω^{2} = {\frac{2 (F (A) - F (A \cos ω t))}{A^{2} \sin^{2} ω t} |}_{ω t = T / N}

(35)

Homotopy perturbation method

The homotopy perturbation method^8,9 is to change gradually from a homotopy equation when p = 0 to its original one when p = 1

u^{″} + ω^{2} u + p [f (u) - ω^{2} u] = 0

(36)

When p = 0, we have a linear oscillator with a frequency of $ω$ , which is to be determined later. When p = 1, equation (36) becomes the original one.

We can also construct the following homotopy equations in case the Duffing equation’s frequency is known

u^{″} + u + ε u^{3} + p [f (u) - u - ε u^{3}] = 0

(37)

u^{″} + u + (ε_{0} + ε_{1} p + ε_{2} p^{2} + \dots) u^{3} + p [f (u) - u - ε u^{3}] = 0

(38)

where

ε_{0} + ε_{1} + ε_{2} + \dots = ε

(39)

Li and He¹⁸ suggested a modification of the homotopy perturbation method coupled with the enhanced perturbation method, which is extremely effective for forced nonlinear oscillators; Adamu et al.²⁰ proposed the parameterized homotopy perturbation method, which gives a high accurate solution for a nonlinear oscillator.

An example

We consider a nonlinear oscillator in the form

u^{″} + \frac{u^{3}}{1 + u^{2}} = 0, u (0) = A, u^{'} (0) = 0

(40)

We write equation (40) in the form

u^{″} + (\frac{u^{2}}{1 + u^{2}}) u = 0

(41)

The coefficient of u in equation (41) is larger than zero, so equation (41) has a periodic solution.

When $u ≪ 1$ , equation (40) becomes

u^{″} + u^{3} = 0, u (0) = A, u^{'} (0) = 0

(42)

The frequency of equation (42) by the homotopy perturbation method is

ω = \frac{\sqrt{3}}{2} A

(43)

When $u ≫ 1$ , equation (41) becomes a linear oscillator

u^{″} + u = 0

(44)

Its frequency is

ω = 1

(45)

Now we check the correctness of He’s frequency formulation discussed above.

f (u) = \frac{u^{3}}{1 + u^{2}}

(46)

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

f^{'} (u) = \frac{3 u^{2} + u^{4}}{{(1 + u^{2})}^{2}}

(47)

Setting u = A/2, we have

ω^{2} = \frac{\frac{3}{4} A^{2} + \frac{1}{16} A^{4}}{{(1 + \frac{1}{4} A^{2})}^{2}}

(48)

When $A ≪ 1$ , equation (48) is equivalent to equation (43); while when $A ≫ 1$ , $ω^{2} = 1$ as predicted in equation (45).

The max-min approach leads to the following result

ω^{2} = {\frac{A^{2} \cos^{2} ω t}{1 + A^{2} \cos^{2} ω t} |}_{ω t = T / 12} = \frac{\frac{3}{4} A^{2}}{1 + \frac{3}{4} A^{2}}

(49)

The variational principle of equation (40) reads

J (u) = \int_{0}^{T / 4} {- \frac{1}{2} {u^{'}}^{2} + \frac{1}{2} u^{2} - \frac{1}{2} \ln (1 + u^{2})} d t

(50)

Its Hamiltonian can be written as

\frac{1}{2} {u^{'}}^{2} + \frac{1}{2} u^{2} - \frac{1}{2} \ln (1 + u^{2}) = \frac{1}{2} A^{2} - \frac{1}{2} \ln (1 + A^{2})

(51)

We choose $u (t) = A \cos ω t$ and setting $ω t = T / 12$ , we have

\begin{array}{l} ω^{2} = {\frac{A^{2} - A^{2} \cos^{2} ω t - \ln (1 + A^{2}) + \ln (1 + A^{2} \cos^{2} ω t)}{A^{2} \sin^{2} ω t} |}_{ω t = T / 12} \\ = \frac{A^{2} - \frac{3}{4} A^{2} - \ln (1 + A^{2}) + \ln (1 + \frac{3}{4} A^{2})}{\frac{1}{4} A^{2}} \\ = \frac{\frac{1}{4} A^{2} - \ln (1 + A^{2}) + \ln (1 + \frac{3}{4} A^{2})}{\frac{1}{4} A^{2}} \end{array}

(52)

The accuracy can be improved by weighted average discussed above.

Discussion and conclusions

The above methods are also valid for fractional oscillators.^46–49 We consider a nonlinear oscillator with a fractal derivative^50,51

\frac{d^{2} u}{d t^{2 α}} + \cdot f (u) = 0

(53)

The fractal derivative $du / d t^{α}$ is defined in a fractal time,^50,51 where $α$ is the value of the fractal dimensions. The fractional oscillator implies an intermittent change in time, for example, a stock will not be changed at non-working days, if a week has two non-working days, then the fractal dimensions can be calculated as

α = \frac{\ln 5}{\ln 7}

(54)

For equation (54), we assume that the minimal scale of time is a day, and scales less than one day are not considered. If we observe the stock change in a scale larger than two days, for example, a week or a year, we can consider continuity of a stock change. We assume that the large time scale is s, according to the fractal theory, we have^50,51

s = t^{α}

(55)

Under a large time scale, equation (53) becomes

\frac{d^{2} u}{d s^{2}} + \cdot f (u) = 0

(56)

Equation (56) can be solved analytically by various analytical methods including those discussed above.

The fractal derivative can be used to search for a variational principle for an equation with a damping term

\frac{d^{2} u}{d t^{2}} + μ \cdot \frac{du}{dt} + f (u) = 0

(57)

The Hamiltonian invariant can be obtained as

\frac{1}{2} {(\frac{du}{dt})}^{2} + \frac{1}{2} μ {(\frac{du}{d t^{1 / 2}})}^{2} + F (u) = H_{0}

(58)

where

H_{0}

is the Hamiltonian invariant, which depends upon the initial conditions and

\partial F / \partial u = f

To conclude, this paper explains some simplest frequency formulae for nonlinear oscillators, the mathematical insight into the formulae reveal all frequency formulae can be improved by choosing a suitable weighted function. This paper can be used as a paradigm for practical applications.

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.

References

JH.

Some asymptotic methods for strongly nonlinear equations. Int J Mod Phys B 2006; 20: 1141–1199.

JH.

Variational iteration method – a kind of non-linear analytical technique: some examples. Int J Nonlinear Mech 1999; 34: 699–708.

JH.

Variational iteration method – some recent results and new interpretations. J Comput Appl Math 2007; 207: 3–17.

Tao

Chen

YH.

Variational iteration method with matrix Lagrange multiplier for nonlinear oscillators. J Low Freq Noise Vibr Act Control 2019; DOI:10.1177/1461348418817868.

Anjum

JH.

Laplace transform: making the variational iteration method easier. Appl Math Lett 2019: 134–138.

Ahmad

Variational iteration algorithm-I with an auxiliary parameter for wave-like vibration equations. J Low Freq Noise Vibr Act Control 2019; DOI: 10.1177/1461348418823126.

The VIM-Padé technique for strongly nonlinear oscillators with cubic and harmonic restoring force. J Low Freq Noise Vibr Act Control 2019; DOI:10.1177/1461348418813612.

JH.

Homotopy perturbation method with an auxiliary term. Abstr Appl Anal 2012; Article Number: 857612.

JH.

Homotopy perturbation method with two expanding parameters. Indian J Phys 2014; 88: 193–196.

10.

Adamu

Ogenyi

Parameterized homotopy perturbation method. Nonlinear Sci Lett A 2017; 8: 240–243.

11.

El-Dib

YO.

Multiple scales homotopy perturbation method for nonlinear oscillators. Nonlinear Sci Lett A 2017; 8: 352–364.

12.

JH.

Homotopy perturbation method for nonlinear oscillators with coordinate dependent mass. Results Phys 2018; 10: 270–271.

13.

Liu

, et al. Hybridization of homotopy perturbation method and Laplace transformation for the partial differential equations. Therm Sci 2017; 21: 1843–1846.

14.

Shqair

Solution of different geometries reflected reactors neutron diffusion equation using the homotopy perturbation method. Results Phys 2019; 12: 61–66.

15.

Song

HY.

A modification of homotopy perturbation method for a hyperbolic tangent oscillator arising in nonlinear packaging system. J Low Freq Noise Vibr Act Control; DOI: 10.1177/1461348418822135.

16.

Pasha

Nawaz

Arif

MS.

The modified homotopy perturbation method with an auxiliary term for the nonlinear oscillator with discontinuity. J Low Freq Noise Vibr Act Control 2019; DOI: 10.1177/0962144X18820454.

17.

Garcıa

AG.

Homotopy perturbation method with an auxiliary parameter for nonlinear oscillators. J Low Freq Noise Vibr Act Control 2019; DOI: 10.1177/1461348418811028

18.

CH.

Homotopy perturbation method coupled with the enhanced perturbation method. J Low Freq Noise Vibr Act Control 2019; DOI: 10.1177/1461348418800554.

19.

Kuang

Wang

Huang

, et al. Homotopy perturbation method with an auxiliary term for the optimal design of a tangent nonlinear packaging system. J Low Freq Noise Vibr Act Control 2019; DOI: 10.1177/1461348418821204.

20.

Adamu

Ogenyi

Tahir

AG.

Analytical solutions of nonlinear oscillator with coordinate-dependent mass and Euler–Lagrange equation using the parameterized homotopy perturbation method . J Low Freq Noise Vibr Act Control 2019; DOI: 10.1177/1461348418821222.

21.

Nadeem M and Li

He–Laplace method for nonlinear vibration systems and nonlinear wave equations. J Low Freq Noise Vibr Act Control 2019; DOI: 10.1177/1461348418818973

22.

Nadeem

He–Laplace method for nonlinear vibration in shallow water waves. J Low Freq Noise Vibr Act Control 2019: 10.1177/1461348418817869

23.

Suleman

Yue

, et al. He–Laplace method for general nonlinear periodic solitary solution of vibration equations. J Low Freq Noise Vibr Act Control 2019; DOI: 10.1177/1461348418816266.

24.

JH.

Variational approach for nonlinear oscillators. Chaos Solitons Fractals 2007; 34: 1430–1439.

25.

Tao

Chen

GH.

An effective modification of Ji–Huan He’s variational approach to nonlinear singular oscillator. J Low Freq Noise Vibr Act Control 2019; DOI: 10.1177/1461348418816265.

26.

JH.

Generalized equilibrium equations for shell derived from a generalized variational principle. Appl Math Lett 2017; 64: 94–100.

27.

JH.

An alternative approach to establishment of a variational principle for the torsional problem of piezo elastic beams. Appl Math Lett 2016; 52: 1–3.

28.

JH.

A remark on Samuelson’s variational principle in economics. Appl Math Lett 2018; 84: 143–147.

29.

JH.

An elementary introduction to recently developed asymptotic methods and nanomechanics in textile engineering. Int J Mod Phys B 2008; 22: 3487–3578.

30.

JH.

Hamiltonian approach to nonlinear oscillators. Phys Lett A 2010; 374: 2312–2314.

31.

Zhang

Locally exact discretization method for nonlinear oscillation systems. J Low Freq Noise Vibr Act Control 2019.DOI: 10.1177/1461348418817125

32.

JH.

An improved amplitude-frequency formulation for nonlinear oscillators. Int J Nonlinear Sci Numer Simul 2008; 9: 211–212.

33.

JH.

An approximate amplitude-frequency relationship for a nonlinear oscillator with discontinuity. Nonlinear Sci Lett A 2016; 7: 77–85.

34.

JH.

Amplitude–frequency relationship for conservative nonlinear oscillators with odd nonlinearities. Int J Appl Comput Math 2017; 3: 1557–1560.

35.

JH.

Max-min approach to nonlinear oscillators. Int J Nonlinear Sci Numer Simul 2008; 9: 207–210.

36.

Liu

JH.

On relationship between two ancient Chinese algorithms and their application to flash evaporation. Results Phys 2017; 7: 320–322.

37.

Ren

Theoretical basis of He frequency–amplitude formulation for nonlinear oscillators. Nonlinear Sci Lett A 2018; 9: 86–90.

38.

Tian

Liu

Period/frequency estimation of a nonlinear oscillator. J Low Freq Noise Vibr Act Control 2019; DOI: 10.1177/1461348418756013.

39.

García

An amplitude-period formula for a second order nonlinear oscillator. Nonlinear Sci Lett A 2017; 8: 340–347.

40.

Ren

ZY.

The frequency–amplitude formulation with ω4 for fast insight into a nonlinear oscillator. Results Phys 2018; 11: 1052–1053.

41.

Ren

GF.

He’s frequency–amplitude formulation with average residuals for nonlinear oscillators. J Low Freq Noise Vibr Act Control 2019; DOI: 10.1177/1461348418812327.

42.

Wang

Yao

SW.

A complement to period/frequency estimation of a nonlinear oscillator. J Low Freq Noise Vibr Act Control 2019; DOI: 10.1177/1461348418822162.

43.

JH.

Nanoscale adhesion and attachment oscillation under the geometric potential, Part 1: the formation mechanism of nanofiber membrane in the electrospinning. Results Phys 20–19; 12: 1405–2.

44.

Zhao

Liu

Needle’s vibration in needle-disk electrospinning process: theoretical model and experimental verification. J Low Freq Noise Vibr Act Control 2019; DOI: 10.1177/1461348418817703.

45.

Qiang

Wan

, et al. The effect of sonic vibration on electrospun fiber mats. J Low Freq Noise Vibr Act Control 2019; DOI: 10.1177/1461348418813256.

46.

Wang

JY.

Amplitude–frequency relationship to a fractional Duffing oscillator arising in microphysics and tsunami motion. J Low Freq Noise Vibr Act Control 2019; DOI: 10.1177/1461348418795813.

47.

Tian

, et al. A fractal modification of the surface coverage model for an electrochemical arsenic sensor. Electrochim Acta 2019; 296: 491–493.

48.

Wang

Shi

, et al. Fractal calculus and its application to explanation of biomechanism of polar bear hairs. Fractals 2018; 26: Article Number:1850086.

49.

Wang

Deng

Fractal derivative model for tsunami travelling. Fractals 2019; DOI: 10.1142/S0218348X19500178.

50.

JH.

Fractal calculus and its geometrical explanation. Results Phys 2018; 10: 272–276.

51.

JH.

A tutorial review on fractal spacetime and fractional calculus. Int J Theor Phys 2014; 53: 3698–3718.