Stochastic jump and bifurcation of Duffing oscillator with fractional derivative damping under combined harmonic and white noise excitations

International Journal of Non-Linear Mechanics 46 (2011) 1324–1329

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Stochastic jump and bifurcation of Duffing oscillator with fractional derivative damping under combined harmonic and white noise excitations Lincong Chen a, Weiqiu Zhu b,n a b

College of Civil Engineering, Huaqiao University, 361021 Xiamen, China Departments of Mechanics, State Key Laboratory of Fluid Power Transmission and Control, Zhejiang University, Hangzhou 310027, China

a r t i c l e i n f o

a b s t r a c t

Article history: Received 22 September 2009 Received in revised form 3 July 2011 Accepted 6 July 2011 Available online 21 July 2011

The stochastic jump and bifurcation of Duffing oscillator with fractional derivative damping of order a (0 o a o 1) under combined harmonic and white noise excitations are studied. First, the system state is approximately represented by two-dimensional time-homogeneous diffusive Markov process of amplitude and phase difference using the stochastic averaging method. Then, the method of reduced Fokker–Plank–Kolmogorov (FPK) equation is used to predict the stationary response of the original system. The phenomenon of stochastic jump and bifurcation as the fractional orders’ change is examined. & 2011 Elsevier Ltd. All rights reserved.

Keywords: Fractional derivative damping Duffing oscillator Stochastic jump Bifurcation Combined harmonic and white noise excitations

1. Introduction The phenomena of stochastic jump and bifurcation are often observed in a non-linear oscillators with a harden spring subjected to combined deterministic and random excitations or narrow-band noise. Over the years, there have been a number of studies conducted for such systems, see, e.g. [1–11] and the references therein. At the same time, some approximate methods, such as the equivalent linearization the perturbation method, the stochastic averaging method, the multiple scale method, the Monte Carlo simulation and the path integral method have been adapted for analyses of such phenomena. However, the stochastic jump and bifurcation of harden spring oscillator with fractional derivative damping subjected to combined harmonic and white noise excitations have never been studied so far. Several authors examined the complicated dynamical behaviors of non-linear systems with fractional derivative damping in the last three decades. On one hand, the literature for the deterministic analysis is rather extensive. For example, Padovan and Sawicki [12] discussed the long time behavior of Duffing oscillator with fractional derivative damping using the perturbation method. Refs. [13,14] examined the chaos in non-linear oscillators with fractional derivative damping numerically. More

n

Corresponding author. Tel.: þ86 571 87953102. E-mail address: [email protected] (W. Zhu).

0020-7462/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijnonlinmec.2011.07.002

recently, Pa´lfalvi [15] provided a computationally efficient solution method for the fractionally damped vibration equation using the Adomian decomposition method and Taylor series. Leung and Guo [16] proposed an improved harmonic balance method for autonomous and non-autonomous systems with fractional derivative damping and examined the interaction among the excitation frequency, fractional order, amplitude, phase angle and the frequency amplitude response. Rossikhin and Shitikova [17] have summarized the novel trends and recent results in the latest ten years on application of fractional calculus for dynamics problems of linear and non-linear hereditary mechanics of solids. Furthermore, Rossikhin [18] presented his own group’s retrospective on fractional calculus viscoelastic application in mechanics of solids and structures. On the other hand, attempts to the studies of those systems under random excitations are proved to be unfruitful. Spanos and Zeldin [19] presented a frequency-domain approach for the random vibration of fractionally damped systems. Agrawal [20,21] presented an analytical scheme for stochastic dynamic systems with fractional derivatives damping using the eigenvector expansion method and the properties of Laplace transforms of convolution integrals. Ye et al. [22] used the method of the Fourier-transform-based technique and the Duhamel integraltype expression to determine the stochastic seismic response of ¨ structures with viscoelastic dampers. Rudinger [23] proposed a frequency-domain approach for tuned mass damper with fractional derivative and applied it to a linear SDOF system under

L. Chen, W. Zhu / International Journal of Non-Linear Mechanics 46 (2011) 1324–1329

white noise excitation. Huang and Jin [24] have studied the response and stability of single-freedom-of-degree systems with fractional derivative damping under white noise using the stochastic averaging method, and also obtained an analytical scheme to determine the statistical behavior of a stochastic dynamical system including two terms of fractional derivative damping with real, arbitrary and fractional orders [25]. The present authors [26,27] extended the stochastic averaging method to the non-linear oscillator with fractional derivative damping under combined harmonic and white noise excitations, real noise excitation and studied the first passage failure of Duffing oscillator. More recently, Spanos and Evangelatos [28] have obtained the random response of a SDOF non-linear system with fractional derivatives damping using time domain simulation and statistical linearization technique. In this paper, the stochastic jump and bifurcation of Duffing oscillator involving fractional derivative damping under both harmonic and white noise excitations is investigated based on the stochastic averaging method and reduced FPK equation. The stochastic jump and P-bifurcation as the fractional orders’ change can be observed under some conditions and explained briefly using the stationary marginal probability density of amplitude and the samples of displacement of original system.

2. Averaged Itˆ o equations The motion equation of Duffing oscillator with small fractional derivative damping under combined harmonic and white noise excitations is of the form X€ ðtÞ þ bDa XðtÞ þ o20 X þ a0 X 3 ¼ E cos Ot þ XW1 ðtÞ þW2 ðtÞ

ð1Þ

where o0,a0, b, E and O are positive constants denoting the natural frequency of degenerate linear oscillator, intensity of non-linearity, small damping coefficient, amplitude and frequency of harmonic excitation, respectively; Wk(t)(k¼1.2) are independently Gaussian white noises with correlation functions E[Wk(t)Wk(t þ t)] ¼2Dkld(t). There are many definitions for fractional derivatives [29,30]. The following Riemann–Liouville definition is adopted: Z 1 d t XðttÞ Da XðtÞ ¼ dt ð2Þ Gð1aÞ dt 0 ta where G() is gamma function. Obviously, the integral definition (2) implies that fractional derivative has the history dependence and the associated response of generalized displacement and velocity is not Markovian process. The response of system (1) can be considered as random spread of periodic solutions of conservative Duffing oscillator in _ surrounding (0,0). The solution can be whole phase plane ðx, xÞ assumed of the following form: X1 ¼ XðtÞ ¼ A cos YðtÞ X2 ¼ X_ ðtÞ ¼ AnðA, YÞsin YðtÞ

ð3Þ

where qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A ¼ ½ð4a0 H þ o40 Þ1=2 o20 =a0

1 X

b2n ðAÞcos 2nY

ð5Þ

n¼0

where b2n ðAÞ ¼

1 2p

Z 2p

nðA, YÞcos 2nYdY

0

Z ¼ a0 A2 =ð4o20 þ 3a0 A2 Þ r 1=3

ð6Þ

The averaged frequency o(A)¼b0(A). Y(t) in Eq. (4) can be approximated as

YðtÞ  oðAÞt þ GðtÞ

ð7Þ

Treating Eq. (3) as a generalized van der Pol transformation from X, X_ to A, G one can obtain the following equations: dA ¼ F11 ðA, YÞ þ F12 ðA, Y, OtÞ þ G1k ðA, YÞWk ðtÞ dt dG ¼ F21 ðA, YÞ þF22 ðA, Y, OtÞ þ G2k ðA, YÞWk ðtÞ, dt

k ¼ 1,2

ð8Þ

where

nðA, YÞsin Y a nðA, YÞsin Y cos Ot D ðA cos YÞ,F12 ¼  o20 þ a0 A2 o20 þ a0 A2 nðA, YÞcos Y a nðA, YÞcos Y cos Ot F21 ¼ 2 D ðA cos YÞ,F22 ¼  o0 A þ a0 A3 o20 A þ a0 A3 AnðA, YÞsin Y cos Y nðA, YÞsin Y G11 ¼  ,G12 ¼  o20 þ a0 A2 o20 þ a0 A2 nðA, YÞcos2 Y nðA, YÞcos Y G21 ¼  ,G22 ¼  2 o20 þ a0 A2 o0 A þ a0 A3

F11 ¼

ð9Þ

Eq. (8) can be modeled as Strtonovich stochastic differential equations and then converted into Itˆo stochastic differential equations by adding the Wong–Zakai correction terms. The result is dA ¼ ½F11 ðA, YÞ þ F12 ðA, Y, OtÞ þ F13 ðA, YÞdt þ G1k ðA, YÞdBk ðtÞ dG ¼ ½F21 ðA, YÞ þ F22 ðA, Y, OtÞ þ F23 ðA, YÞdt þG2k ðA, YÞdBk ðtÞ, k ¼ 1,2

ð10Þ

where Bk(t) are standard Wiener processes, @G11 @G12 @G11 @G12 G11 þ D2 G12 þ D1 G21 þD2 G22 @A @A @G @G @G21 @G22 @G21 @G22 G11 þ D2 G12 þ D1 F23 ðA, YÞ ¼ D1 G21 þD2 G22 @A @A @G @G

F13 ðA, YÞ ¼ D1

ð11Þ Since the harmonic excitation has no effect on the response in the first approximation, the primary external resonance case is considered, i.e.,(O/o(A)) ¼1þ es, where es is the detuning parameter. Using the approximate relation (7) and Eq. (4) and introducing new variable D such that

D ¼ esFG ¼ OtY

ð12Þ

By using the Itˆo differential rule, Eq. (10) can be transformed into the following Itˆo equations: dA ¼ ½F11 ðA, YÞ þ F12 ðA, Y, Y þ DÞ þ F13 ðA, YÞdt þG1k ðA, YÞdBk ðtÞ dD ¼ fF21 ðA, YÞF22 ðA, Y, Y þ DÞF23 ðA, YÞ þ O nðA, YÞgdt þ G2k ðA, YÞdBk

ð13Þ

Then, completing the stochastic averaging with respect to Y, one can obtain the following averaged Itoˆ equations:

H ¼ X22 =2þ o20 X12 =2 þ a0 X14 =2

YðtÞ ¼ FðtÞ þ GðtÞ dF ¼ ½ðo20 þ3a0 A2 =4Þð1 þ Z cos 2YÞ1=2 nðA, YÞ ¼ dt

nðA, YÞ ¼

1325

ð4Þ

in which A, H, Y, F, G are all random processes, n(A,Y) is the instantaneous frequency of the oscillator and can be expanded into Fourier series, i.e.,

dA ¼ m1 ðA, DÞdt þ s11 ðA, DÞdB1 þ s12 ðA, DÞdB2 dD ¼ m2 ðA, DÞdt þ s21 ðA, DÞdB1 ðtÞ þ s22 ðA, DÞdB2 ðtÞ where m1 ¼ 

bAðo20 þ3a0 A2 =4Þsinðpa=2Þ a ðAÞ 2ðo20 þ a0 A2 Þb1 0

þ

Eð2b0 b2 ÞsinD 4ðo20 þ a0 A2 Þ

ð14Þ

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L. Chen, W. Zhu / International Journal of Non-Linear Mechanics 46 (2011) 1324–1329

a0 D1 Að3o20 þ3a0 A2 =2Þ D1 ðo20 þ 7a0 A2 =8Þ þ 4ðo20 þ a0 A2 Þ3 2ðo20 þ a0 A2 Þ2 A D2 o20 Aðo20 þ a0 A2 =2Þ D2 Aðo20 þ 7a0 A2 =8Þ þ þ 8ðo20 þ a0 A2 Þ3 4ðo20 þ a0 A2 Þ2 2 2 bðo0 þ3a0 A =4Þcosðpa=2Þ Eð2b0 þb2 ÞcosD þ þ Ob0 ðAÞ m2 ¼ a ðAÞ 4Aðo20 þ a0 A2 Þ 2ðo20 þ a0 A2 Þb1 0 D1 ðo20 þ 5a0 A2 =8Þ D2 A2 ðo20 þ 3a0 A2 =4Þ b11 ¼ s11 s11 þ s12 s12 ¼ þ ðo20 þ a0 A2 Þ2 4ðo20 þ a0 A2 Þ2 D1 ðo20 þ 7a0 A2 =8Þ D2 ð3o20 =4 þ 11a0 A2 =16Þ b22 ¼ s21 s21 þ s22 s22 ¼ þ A2 ðo20 þ a0 A2 Þ2 ðo20 þ a0 A2 Þ2 

b12 ¼ b21 ¼ 0

ð15Þ

in which b0 ¼ ðo20 þ3a0 A2 =4Þ1=2 ð1Z2 =16Þ b2 ¼ ðo20 þ 3a0 A2 =4Þ1=2 ðZ=2 þ 3Z3 =64Þ

Z ¼ a0 A2 =ð4o20 þ 3a0 A2 Þ r1=3

ð16Þ

3. Reduced FPK equation The reduced FPK equation associated with averaged Itˆo Eq. (14) is 0¼

@ðm1 pÞ @ðm2 pÞ 1 @2 ðb11 pÞ 1 @2 ðb22 pÞ  þ þ @A @D 2 @A2 2 @D2

ð17Þ

where p ¼p(A,D) is the stationary joint probability density of amplitude A and phase difference D. The boundary conditions with respect to A are p ¼ finite as A ¼ 0

ð18Þ

p ¼ 0,@p=@A ¼ 0 as A-1

ð19Þ

Since p is the periodic function of D, it satisfies the following periodic boundary condition with respect to D: pðA, DÞ ¼ pðA,2p þ DÞ

ð20Þ

Furthermore, Eq. (17) satisfies the normalization condition, i.e., Z 2p pðA, DÞdAdD ¼ 1 ð21Þ

Z

1 0

0

The stationary marginal probability density of amplitude can be obtained as follows: Z 2p pðAÞ ¼ pðA, DÞdD ð22Þ 0

The reduced FPK Eq. (17) can be solved numerically with the boundary conditions (18)–(20) and normalization condition (21) using the finite difference method combined with the successive over-relaxation method.

4. Stochastic jump and bifurcation The long time behavior of the fractionally damped Duffing oscillator under harmonic excitation, especially the determination of the influence of fractional order on the frequency amplitude behavior, has been examined in detail by Padovan and Sawicki [12], and Leung and Guo [16], respectively. In this section, the stochastic jump and its bifurcation as the fractional orders’ change are examined using the stationary marginal probability

Fig. 1. Amplitude response curves of Duffing oscillator with fractional derivative damping under additive harmonic excitation for different values of fractional order a. The other parameters are b ¼ 0.1, a0 ¼ 2.0, E¼ 0.25 and o0 ¼ 1.0.

L. Chen, W. Zhu / International Journal of Non-Linear Mechanics 46 (2011) 1324–1329

1.5

1.2

0.9 p (A)

density of amplitude obtained from Eq. (17) and the Monte Carlo simulation of original system (1). First, the Duffing oscillator with fractional derivative damping under pure harmonic excitation, i.e., D1 ¼D2 ¼0 is examined briefly for comparison. The averaged Itˆo Eq. (14) are reduced to the ordinary differential equations. The stationary amplitude response curves of the system can be obtained by letting (dA/ dt) ¼(dD/dt)¼0 and are shown in Fig. 1(a)–(c). It is seen from these figures that the amplitude response of system is triple-value and two are stable while the other is unstable. The deterministic jump occurs at the two extreme values of the frequency interval of triple-value amplitude. Second, the stochastic jump phenomenon in Duffing oscillator with fractional derivative damping under combined harmonic and white noise excitations is examined. The stationary marginal probability density of amplitude shown in Figs.2 and 3 is bimodal. It is implies that there are two more probably motions in the response of Duffing oscillator with fractional derivative damping under combined harmonic and white noise excitations. According to the previous works [7–9,11] for stochastic jump of non-linear

1327

0.6

0.3

0

0

0.5

1 A

1.5

2

Fig. 4. Stationary marginal probability density of amplitude of system (1). The parameters are the same as those in Fig. 2 except fractional order a ¼ 0.15. Solid line denotes the analytical result; symbol ’ denotes the result from Monte Carlo simulation of original system.

1.4 1.2

p(A)

1 0.8 0.6 0.4 0.2 0

0

0.8 A

0.4

1.2

1.6

Fig. 2. Stationary marginal probability density of amplitude of system (1). The parameters are the same as those in Fig. 1(a) except for D1 ¼D2 ¼ 0.006. Solid line denotes the analytical result; symbol ’ denotes the result from Monte Carlo simulation of original system.

2

1.6

5. Conclusions

p(A)

1.2

0.8

0.4

0

oscillator with hardening spring subjected to narrow-band excitations, it is expected that stochastic jump may occur between two more probably motions. This is verified by Fig. 5a and b, which denote the samples of displacement of the system (1) obtained by using the Monte Carlo simulation directly. On the other hand, whether the stochastic jump occurs or not depends on the system’s parameters, such as the intensity of noise, the frequency rate, the amplitude of harmonic excitation, the strength of non-linear and the fractional order. Since the stochastic jump is related to the bimodal probability density, we call the appearance or disappearance of bimodal probability density as the bifurcation of the stochastic jump as the system’s parameters change. Here, the effect of the fractional order a on the stochastic jump is discussed briefly. When a ¼0.15 and provided that the other parameters are the same as those in Figs. 2 and 3, the stationary marginal probability density of amplitude shown in Fig. 4 is unimodal and no stochastic jump occurs, which is verified by Fig. 5c for the samples of displacement of the system (1) obtained using the Monte Carlo simulation directly. Finally, the stationary marginal probability density of amplitude of Duffing oscillator with linear damping under combined harmonic and white noise excitations is proposed for comparison in Fig. 6. It can be seen from Figs. 2–6 that the stochastic dynamical behavior of such oscillator is sensitive to the change in fractional order.

0

0.3

0.6

A

0.9

1.2

1.5

Fig. 3. Stationary marginal probability density of amplitude of system (1). The parameters are the same as those in Fig. 2 except fractional order a ¼0.5 and O ¼1.6. Solid line denotes the analytical result; symbol ’ denotes the result from Monte Carlo simulation of original system.

In the present paper, the stochastic jump and bifurcation of Duffing oscillators with fractional derivative damping subjected to combined harmonic and white noise excitations has been briefly studied using the stochastic averaging method and the reduced averaged FPK equation. The stochastic jump and its bifurcation as the fractional orders’ change have been observed for the first time in the Duffing oscillator with fractional derivative damping under both harmonic and white noise excitations through the stationary marginal probability density of amplitude and the samples for displacement of original system from the Monte Carlo simulation. Moreover, another significant conclusion can be inferred from the proposed numerical studied, i.e., the phenomena of stochastic jump and bifurcation as the fractional orders’ change may be also observed in a non-linear oscillator

1328

L. Chen, W. Zhu / International Journal of Non-Linear Mechanics 46 (2011) 1324–1329

with hardening spring and fractional derivative damping subjected to combined harmonic and real noise excitations or bounded noise excitation.

1.5 1

x1

0.5 Acknowledgments

0 -0.5 -1 -1.5 0

500

1,000 t

1,500

2,000

1.5

The work reported in this paper was supported by the National Natural Science Foundation of China under Grant nos. 11072212, 10932009 and 11002059, Fujian Province Natural Science Foundation of China under Grant no. 2010J05006, Specialized Research Fund for the Doctoral Program of Higher Education under Grant no. 20103501120003, Fundamental Research Funds for Huaqiao University under Grant no. JB-SJ1010 and Huaqiao University’s Research and Development Start under Grant no. 09BS622.

1 References

x1

0.5 0 -0.5 -1 -1.5

0

500

1,000 t

1,500

1

1.5

2 t

2.5

2,000

2

x1

1 0 -1 -2

3 x 104

Fig. 5. (a) A sample of displacement response from Monte Carlo simulation of original system (1) for fractional order a ¼0.8. The parameters are the same as those in Fig. 2. (b) A sample of displacement response from Monte Carlo simulation of original system (1) for fractional order a ¼0.5. The parameters are the same as those in Fig. 3. (c) A sample of displacement response from Monte Carlo simulation of original system (1) for fractional order a ¼ 0.15. The parameters are the same as those in Fig. 4.

Fig. 6. Stationary marginal probability density of amplitude of Duffing oscillator with linear damping using the Monte Carlo simulation directly. The parameters are the same as those in Fig. 2.

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