Published: 30 June 2014

A revised averaging method and general forms of approximate solution for nonlinear oscillator with only polynomial-type displacement nonlinearity

Yongjun Shen1
Yongxiang Zhao2
Shaopu Yang3
Haijun Xing4
1, 2, 3, 4Department of Mechanical Engineering, Shijiazhuang Tiedao University, Shijiazhuang, 050043, China
Corresponding Author:
Yongjun Shen
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Abstract

In this paper a revised averaging method is presented, that does not need the detuning factor in the solving procedure. Comparison with the traditional averaging method shows that it has the similar solving procedure and the same result as the primary resonance of the traditional averaging method. Then the nonlinear oscillator with only polynomial-type displacement nonlinearity is studied, and the general forms of the first-order approximate solution by this revised averaging method, and by the traditional averaging method for the super-harmonic resonance and sub-harmonic resonance are established. At last, the Duffing oscillator is investigated as an example, and the comparison of the analytical and numerical results proves the validity and simplicity of the presented method.

1. Introduction

Averaging method is an effective analytical method for the approximate solution of the weakly nonlinear system. Since its presentation [1, 2], averaging method had been revised and improved to process a lot of nonlinear problems, and some relevant advances were summarized in the monographs [1-4].

Averaging method has been applied in a great deal theoretic and engineering problems extensively [5-13]. For example, Wang and Hu [5] reduced the delay differential equation of infinite dimensional to an ordinary differential equation by averaging method. Roy [6] developed the averaging method to find the periodic solutions of strongly nonlinear oscillators with harmonic excitations. Chatterjee [7] improved the averaging method based on harmonic balance, where the closed form solutions to the unperturbed problem were not needed, so as to research some strongly nonlinear system. Kumar and Datta [8] used the stochastic averaging technique to research the probability density function of the response for strongly nonlinear system subject to both multiplicative and additive random excitations. Ji and Hansen [9] constructed a valid asymptotic expansion solution of a nonlinear oscillator composed of a weakly nonlinear and a linear differential equation based on the averaging method and the continuity condition. Then they [10] extended this strategy to an approximate solution for the super-harmonic resonance of a periodically excited nonlinear oscillator with a piecewise nonlinearity, and the validity of the developed analysis was confirmed by comparing the approximate solutions with the results of direct numerical integration of the original equation. Yang, Tang, Chen and Lim [11] applied the averaging method to analyze the instability phenomena caused by sub-harmonic and combination resonance of transverse parametric vibrations of an axially accelerating tensioned Timoshenko beam on simple supports. Li, Ji and Hansen [12] studied a system composed of two Van der Pol oscillators with delayed position and velocity coupling by the averaging method, where the stability and the number of periodic solutions in 1:1 internal resonance were researched. Yang and Chen [13] investigated the parametrical vibration of a nonlinear oscillator with Davidenkov’s hysteretic nonlinearity by the averaging method and singularity theory, and the universal unfolding was also obtained.

In the above researches, the averaging method were generally used to obtain the approximate solution according to the relationship between the excitation frequency and the natural frequency, so as to induce different responses, such as the primary resonance, sub-harmonic resonance and super-harmonic resonance, etc. In this paper, we revise the traditional averaging method, and find that the revised averaging method could obtain the entirely same form as the primary resonance of the traditional averaging method when researching the first-order approximate solution. And the presented method does not need to introduce the detuning factor as the traditional averaging method. Then the general forms of the approximate solutions of nonlinear system with only polynomial-type displacement nonlinearity are established, which may make the research on this kind of nonlinear system more convenient. At last the validity of the presented method is proved by comparison of the approximate solution and the numerical one of the Duffing oscillator.

2. The revised averaging method

The nonlinear system considered here is:

1
x¨t+ω02xt=px,x˙+Fcosωt,

where ω0 is the natural frequency, p(x,x˙) includes linear damping force, nonlinear damping force and nonlinear restoring force, F and ω is the amplitude and frequency of external excitation respectively.

In traditional averaging method and other perturbation-based methods [14-15], such as multi-scale method, KBM method, it is necessary to consider that whether the excitation frequency ω is close to the natural frequency, associated with the multiple or fraction of the natural frequency ω0. Hence the solving procedure could be divided into some cases such as the primary resonance, secondary resonance including super-harmonic resonance and sub-harmonic resonance, etc. And some detuning factors should be introduced in the traditional averaging method.

Here we present the revised averaging method, where it is unnecessary to distinguish the above cases and introduce the detuning factor.

Letting the solution of Eq. (1) as:

2a
x=acosωt+θ,

and

2b
x˙=-aωsinωt+θ.

By differentiating Eq. (2a) to t, one could obtain:

3
x˙=a˙cosϕ-aω+θ˙sinϕ,

where ϕ=ωt+θ. Combined with Eq. (2b), it yields:

4
a˙cosϕ-aθ˙sinϕ=0.

By differentiating Eq. (2b) to t, one could obtain:

5
x¨=-a˙ωsinϕ-aωω+θ˙cosϕ.

Combined Eq. (5) with Eq. (1), it will yield:

6
a˙sinϕ+aθ˙cosϕ=-1ωFcosϕ-θ+pacosϕ,-aωsinϕ+aω2-ω02cosϕ.

From Eq. (4) and Eq. (6), and by solving the system of equations with a˙ and θ˙ as unknowns, one could get:

7a
a˙=-1ωPa,θsinϕ,

and

7b
aθ˙=-1ωPa,θcosϕ,

where P(a,θ)=Fcos(ϕ-θ)+p(acosϕ,-aωsinϕ)+a(ω2-ω02)cosϕ.

Applying the averaging procedure to Eq. (7) in a periodic T (here T=2π), that means:

8a
a˙=-1Tω0TPa,θsinϕdϕ,
8b
θ˙=-1aTω0TPa,θcosϕdϕ.

Moreover, one could obtain the simpler form as:

9a
a˙=-F2ωsinθ-1Tω0Tpa,θsinϕdϕ,
9b
θ˙=-F2aωcosθ-ω2-ω022ω-1aTω0Tpa,θcosϕdϕ.

3. The traditional averaging method

In traditional averaging method, one must consider the approximation degree of the excitation frequency ω to the natural frequency ω0, associated with the multiple or the fraction of natural frequency ω0.

3.1. Primary resonance

This case means the excitation frequency ω is close to the natural frequency ω0. In order to illustrate the approximation degree, one should introduce:

10
ω2=ω02+εσ,

where ε is a small dimensionless parameter and σ is the detuning factor. The nonlinear restoring force and the damping force p(x,x˙), the amplitude of excitation force F should also be re-scaled as:

11
px,x˙=εp1x,x˙, F=εf.

Then Eq. (1) could be transformed into:

12
x¨t+ω2xt=εp1x,x˙+εσx+εfcosωt.

Letting the solution has the form as Eq. (2a) and Eq. (2b), and after the similar deducing procedure, one could obtain:

13a
a˙=-1ωQx,x˙,tsinϕ,
13b
aθ˙=-1ωQx,x˙,tcosϕ,

where Q(x,x˙,t)=ε[p1(x,x˙)+σacosϕ+fcos(ϕ-θ)].

Averaging the right-hand side of Eq. (13) in a periodic T (here T=2π), it would be:

14a
a˙=-εf2ωsinθ-εTω0Tp1a,θsinϕdϕ,
14b
θ˙=-εf2aωcosθ-εσ2ω-εaTω0Tp1a,θcosϕdϕ.

Based on the scale relationship shown in Eq. (11), it could be concluded that Eq. (13) and Eq. (14) are entirely the same as Eq. (7) and Eq. (9). Accordingly, the revised averaging method is totally equal to the primary resonance of the traditional averaging method. Nevertheless, it is unnecessary to introduce the detuning factor in the revised method as the traditional averaging method, which could make the deducing procedure much simpler.

3.2. Super-harmonic resonance

In traditional averaging method, super-harmonic resonance means that the multiple of the excitation frequency is close to the natural frequency, i.e. ω0nω, where n is a natural number unequal to 1. In order to illustrate the approximation degree of nω to ω0, introducing:

15
(nω)2=ω02+εσ,

where ε is a small dimensionless parameter and σ is the detuning factor. The nonlinear restoring force and the damping force p(x,x˙) should be rescaled as:

16
px,x˙=εp1x,x˙,

that yields:

17
x¨t+nω2xt=εp1x,x˙+εσx+Fcosωt.

The solution should be as:

18
x=acosϕ+Bcosωt,
x˙=-anωsinϕ-Bωsinωt,

where:

ϕ=nωt+θ, B=F(n2-1)ω2.

Based on the standard procedure of averaging method, one can obtain:

19a
a˙=-εnωPa,ϕsinϕ,
19b
aθ˙=-εnωPa,ϕcosϕ,

where P(a,ϕ)=p1(a,ϕ)+σ[acosϕ+Bcos[(ϕ-θ)/n]].

Averaging the right-hand side of Eq. (19) in a periodic T (here T=2nπ), it would be:

20a
a˙=-εTnω0TPa,ϕsinϕdϕ,
20b
θ˙=-εTanω0TPa,ϕcosϕdϕ,

that could be simplified as:

21a
a˙=-εTnω0Tp1a,ϕsinϕdϕ,
21b
θ˙=-εσ2nω-εTanω0Tp1a,ϕcosϕdϕ.

3.3. Sub-harmonic resonance

In traditional averaging method, sub-harmonic resonance means that the fraction of the excitation frequency is close to the natural frequency, i.e. ω0ω/n, where n is a natural number unequal to 1. In order to illustrate the approximation degree, introducing:

22
ωn2=ω02+εσ,

where ε is a small dimensionless parameter and σ is the detuning factor. The nonlinear restoring force and the damping force p(x,x˙) should be rescaled as:

23
px,x˙=εp1x,x˙,

that yields:

24
x¨t+ωn2xt=εp1x,x˙+εσx+Fcosωt.

The solution should be as:

25
x=acosϕ+Bcosωt,
x˙=-aωnsinϕ-Bωsinωt,

where:

ϕ=ωnt+θ, B=n2F(1-n2)ω2.

Based on the standard procedure of averaging method, one can obtain:

26a
a˙=-εnωPa,ϕsinϕ,
26b
aθ˙=-εnωPa,ϕcosϕ,

where P(a,ϕ)=p1(a,ϕ)+σ[acosϕ+Bcos[n(ϕ-θ)]].

Averaging the right-hand side of Eq. (26) in a periodic T (here T=2π), it would be:

27a
a˙=-εnTω0TPa,ϕsinϕdϕ,
27b
θ˙=-εnTaω0TPa,ϕcosϕdϕ,

that could be simplified as:

28a
a˙=-εnTω0Tp1a,ϕsinϕdϕ,
28b
θ˙=-εnσ2ω-εnTaω0Tp1a,ϕcosϕdϕ.

4. The general solution of nonlinear system with only polynomial-type displacement nonlinearity

In this section we consider a kind of nonlinear system, where the nonlinear part is only composed of polynomial-type nonlinear restoring force. That means:

29
px,x˙=-cx˙-αxn.

4.1. The revised averaging method

From Eq. (29), one could obtain:

30
pa,θ=caωsinϕ-αancosnϕ.

In order to obtain the explicit form for Eq. (9), we introduce the expansion forms for the even and odd powers of trigonometric function shown as page 28 in [16] or page 80-82 in [17]:

31
cos2lβ=122l-1k=0l-1C2lkcos2l-2kβ+12C2ll,
cos2l+1β=122lk=0lC2l+1kcos2l-2k+1β,

where l is natural number, C2lk=(2l)!/k!(2l-k)! is the binominal coefficient (0!=1).

Based on Eq. (31) and the orthogonality of trigonometric function, the integral in Eq. (9a) is:

32
02πp(a,θ)sinϕdϕ=02π(caωsinϕ-αancosnϕ)sinϕdϕ=caωπ.

The integral in Eq. (9b) is:

33
02πp(a,θ)cosϕdϕ=02π(caωsinϕ-αancosnϕ)cosϕdϕ=0,

when n is even, and:

34
02π(caωsinϕ-αancosnϕ)cosϕdϕ=-C2llanαπ2n,

when n is odd (supposing n=2l-1).

Accordingly, when the nonlinear part is only composed of polynomial-type nonlinear restoring force, Eq. (9a) and Eq. (14a) could be as:

35a
a˙=-F2ωsinθ-ca2,

and Eq. (9b) and Eq. (14b) could be as:

35b
θ˙=-F2aωcosθ-ω2-ω022ω,when n is even,-F2aωcosθ-ω2-ω022ω+αan-1C2ll2n+1ω,when n is odd andn=2l-1.

4.2. The super-harmonic resonance

From Eq. (29), one could obtain:

36
p1a,θ=canωsinϕ+Bωsinωt-α[acosϕ+Bcos(ωt)]n.

In order to compute the integral in Eq. (21), the binomial theorem is adopted:

37
(a1+a2)n=k=0nCnka1n-ka2k,

where a1, a2 is complex number and Cnk is the binomial coefficient.

Based on the aforementioned the expansion form for the even and odd powers of trigonometric function and the orthogonality of trigonometric function, the integral in Eq. (21a) is:

38
02nπp1(a,ϕ)sinϕdϕ=can2ωπ-αnBnπ2n-1sinθ.

The integral in Eq. (21b) is:

39
02nπp1a,ϕcosϕdϕ=-αnBnπ2n-1cosθ,

when n is even, and:

40
02nπp1a,ϕcosϕdϕ=-nπαanC2ll2n-αnπ2nl2=1l-1an-2l2B2l2Cn2l2Cn-2l2+1l-l2C2l2l2
-nπαBn2n-1cosθ,

when n is odd (supposing n=2l-1).

Accordingly, Eq. (21) should be as:

41a
a˙=-εca2+εαBn2nnωsinθ,
41b
θ˙=-εσ2nω+εαBn2nanωcosθ,when n is even,-εσ2nω+εα2n+1anωanC2ll+2Bncosθ+l2=1l-1an-2l2B2l2Cn2l2Cn-2l2+1l-l2C2l2l2,when n is odd .

4.3. The sub-harmonic resonance

From Eq. (29), one could obtain:

42
p1a,θ=caωnsinϕ+Bωsinωt-α[acosϕ+Bcos(ωt)]n.

Similarly as the super-harmonic resonance, the integral in Eq. (28a) is:

43
02πp1a,ϕsinϕdϕ=caπωn-nαπan-1B2n-1sinnθ.

The integral in Eq. (28b) is:

44
02πp1(a,ϕ)cosϕdϕ=-nαBπan-12n-1cosnθ,

when n is even, and:

45
02πp1(a,ϕ)cosϕdϕ=-παanC2ll2n-απ2nl2=1l-1an-2l2B2l2Cn2l2Cn-2l2+1l-l2C2l2l2-nπα  an-1B2n-1cosnθ,

when n is odd.

Accordingly, Eq. (28) should be as:

46a
a˙=-εca2+εαn2an-1B2nωsinnθ,
θ˙=-εnσ2ω+εn2αBan-22nωcosnθ,

when n is even, and:

46b
θ˙=-εnσ2ω+εn  α2n+1aωanC2ll+2nan-1Bcosnθ+l2=1l-1an-2l2B2l2Cn2l2Cn-2l2+1l-l2C2l2l2,

when n is odd.

Here only single item of the polynomial-type displacement nonlinearity is considered, but it is still very useful in nonlinear analysis. When several items of polynomial-type displacement nonlinearity exist in the nonlinear system, they could be treated similarly as the above procedure separately.

5. An example – Duffing oscillator

In this section the classical Duffing oscillator is selected as an example to verify the aforementioned results. The Duffing oscillator is as:

47
x¨t+ω02xt+εcx˙t+εαx3t=Fcosωt,

where p(x,x˙)=-εcx˙(t)-εαx3(t).

5.1. The steady-state solution

Based on Eq. (35), one could obtain the standard equations about the amplitude and phase of the approximate solution as:

48
a˙=-F2ωsinθ-εca2,θ˙=-F2aωcosθ-ω2-ω022ω+3εαa28ω.

Letting a˙=0 and θ˙=0, one could obtain the frequency-response equation about the steady-state amplitude as:

49
a-2εcω2+ω2-ω02-3εαa-242=F2.

It could be found that there maybe exist one or three plus real solution when ω is changed, that is the reason of jump hysteresis phenomena of the amplitude and had been found in many literatures.

Considering the super-harmonic resonance, then:

50
p1x,x˙=-cx˙t-αx3t,

and based on Eq. (41), one could obtain the standard equations about the amplitude and phase of free oscillation in the approximate solution as:

51
a˙=-εca2+εαB324ωsinθ,θ˙=-εσ6ω+εα24aω(3a3+B3cosθ+6aB2).

Letting a˙=0 and θ˙=0, one could obtain the frequency-response equation about the steady-state amplitude and phase for free oscillation for super-harmonic resonance as:

52a
a-2[144c2ω2+(4σ-3αa-2-6αB2)2] =α2B6,
52b
θ-=arcsin12ωca-αB3.

Combining with Eq. (18), one could obtain the super-harmonic response of the Duffing oscillator as:

x=a-cos3ωt+θ-+Bcosωt.

Considering the sub-harmonic resonance, then:

53
p1x,x˙=-cx˙t-αx3t.

Based on Eq. (46), one could obtain the standard equations about the amplitude and phase for free oscillation in the approximate solution as:

54
a˙=-εca2+9εαa2B8ωsin(3θ),θ˙=-3εσ2ω+9εα8ω[a2+aBcos(3θ)+2B2].

Letting a˙=0 and θ˙=0, one could obtain the frequency-response equation about the non-zero steady-state amplitude and phase of free oscillation for sub-harmonic resonance as:

55a
16c2ω2+(12σ-9αa-2-18αB2)2=81α2a-2B2,
55b
θ-=13arcsin4cω9αa-B.

Combining with Eq. (25), one could obtain the sub-harmonic response of the Duffing oscillator as:

x=a-cosωt3+θ-+Bcosωt.

5.2. Numerical simulation

An illustrative example system is studied herein as defined by the basic system parameters ω0=1, ε=0.05, c=0.2, α=1.3, F=1.5. Fig. 1 illustrates a comparison of the peak values of the approximately analytical solution by the presented method and the numerical solution in the frequency range ω[1, 6]. The values of the approximately analytical solution are denoted by the solid line, while the values of the numerical results are indicated by the circle. It could be found that the approximately analytical solution agrees very well with the numerical results in the given frequency range except two small regions. The first region is the frequency range for super-harmonic resonance around ω=1/3, where the presented method could not open out the jump and hysteresis phenomenon in this range. But the range is small, and the peak values in this range are not large compared with those of the whole amplitude-frequency curves. The second region is the frequency range for the sub-harmonic resonance larger than ω=3, where the presented method could not interpret the multi-value phenomenon of the amplitude, and the amplitude of the sub-harmonic resonance by the numerical integration is large. Nevertheless, the existing range for the sub-harmonic is small, so that one could omit it in many practical engineering.

The effects of some important system parameters on the amplitude-frequency are also researched. For example, when the damping coefficient is selected as c=0.02 and c=0.6, the results are shown in Fig. 2. It could be found that when the damping coefficient is smaller, the difference of the results between the approximately analytical solution and the numerical integration would be larger, and the sub-harmonic resonance would be more distinct. If the damping coefficient is larger, the sub-harmonic resonance would vanish and the two results would agree very well with each other, and the difference of the results between the approximately analytical solution and the numerical integration would be neglectable.

Fig. 1Comparison of the peak values of the approximately analytical solution by the presented method and the direct numerical integration for the basic system parameters: the solid line for the approximately analytical solution and the circles for the direct numerical integration

Comparison of the peak values of the approximately analytical solution by the presented  method and the direct numerical integration for the basic system parameters: the solid line for  the approximately analytical solution and the circles for the direct numerical integration

Fig. 2Comparison of the peak values of the approximately analytical solution by the presented method and the direct numerical integration where the solid line is for the approximately analytical solution and the circles are for the direct numerical integration with different damping coefficients: a) c=0.02; b) c=0.6

Comparison of the peak values of the approximately analytical solution by the presented method and the direct numerical integration where the solid line is for the approximately analytical solution  and the circles are for the direct numerical integration with different damping coefficients:  a) c=0.02; b) c=0.6

a)

Comparison of the peak values of the approximately analytical solution by the presented method and the direct numerical integration where the solid line is for the approximately analytical solution  and the circles are for the direct numerical integration with different damping coefficients:  a) c=0.02; b) c=0.6

b)

Fig. 3Comparison of the peak values of the approximately analytical solution by the presented method and the direct numerical integration where the solid line is for the approximately analytical solution and the circles are for the direct numerical integration with different amplitudes of excitation force: a) F=0.5; b) F=2.5

Comparison of the peak values of the approximately analytical solution by the presented  method and the direct numerical integration where the solid line is for the approximately analytical  solution and the circles are for the direct numerical integration with different amplitudes  of excitation force: a) F=0.5; b) F=2.5

a)

Comparison of the peak values of the approximately analytical solution by the presented  method and the direct numerical integration where the solid line is for the approximately analytical  solution and the circles are for the direct numerical integration with different amplitudes  of excitation force: a) F=0.5; b) F=2.5

b)

When the amplitude of excitation force is changed, for example, F=0.5 and F=2.5, the amplitude-frequency curves obtained by the presented method and the numerical integration are shown in Fig. 3. It could be found that when the amplitude of excitation force is small enough, the sub-harmonic resonance disappears and the super-harmonic resonance is so small as to be neglectable. If the amplitude of excitation force is large enough, the sub-harmonic and super-harmonic resonance would all be obvious. But the amplitude of super-harmonic resonance and the existing range of sub-harmonic resonance are all small.

If the nonlinear stiffness coefficient is changed, for example, α=0.5 and α=2.1, the amplitude-frequency curves obtained by the presented method and the numerical integration are shown in Fig. 4. It could be found that when the nonlinear stiffness coefficient is small, the sub-harmonic resonance would disappear and the super-harmonic resonance is very small. If the nonlinear stiffness coefficient is large enough, the sub-harmonic and super-harmonic resonance would all be clear. But the amplitude of super-harmonic resonance and the existing range of sub-harmonic resonance are still small.

Fig. 4Comparison of the peak values of the approximately analytical solution by the presented method and the direct numerical integration where the solid line is for the approximately analytical solution and the circles are for the direct numerical integration with different nonlinear stiffness coefficients: a) α=0.5; b) α=2.1

Comparison of the peak values of the approximately analytical solution by the presented  method and the direct numerical integration where the solid line is for the approximately analytical  solution and the circles are for the direct numerical integration with different nonlinear stiffness coefficients: a) α=0.5; b) α=2.1

a)

Comparison of the peak values of the approximately analytical solution by the presented  method and the direct numerical integration where the solid line is for the approximately analytical  solution and the circles are for the direct numerical integration with different nonlinear stiffness coefficients: a) α=0.5; b) α=2.1

b)

Comparatively speaking, the amplitude-frequency curves obtained by the presented method agree well with the numerical integration, which could make us research the nonlinear problems more convenient.

6. Conclusions

In this paper a revised averaging method is presented, which is entirely equal to the results of the primary resonance of the traditional averaging method. It is unnecessary to introduce the small dimensionless parameter and the detuning factor, which makes the solving procedure more convenient. Moreover, the general forms of the approximate solution of nonlinear system with only polynomial-type displacement nonlinearity are obtained. The Duffing oscillator, as an example, is researched and the results show the validity of the presented method.

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About this article

Received
21 March 2014
Accepted
26 May 2014
Published
30 June 2014
Keywords
averaging method
nonlinear oscillator
approximate solution
polynomial-type displacement nonlinearity
Acknowledgements

The authors are grateful to the support by National Natural Science Foundation of China (No. 11072158 and 11372198), the Program for New Century Excellent Talents in University (NCET-11-0936), and the Cultivation Plan for Innovation Team and Leading Talent in Colleges and Universities of Hebei Province (LJRC018).