Dynamic responses of axially moving viscoelastic beam under a randomly disordered periodic excitation

Journal of Sound and Vibration 331 (2012) 4045–4056

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Dynamic responses of axially moving viscoelastic beam under a randomly disordered periodic excitation Di Liu n, Wei Xu, Yong Xu Department of Applied Mathematics, Northwestern Polytechnical University, Xi’an, Shaanxi 710072, China

a r t i c l e i n f o

abstract

Article history: Received 14 January 2012 Received in revised form 1 April 2012 Accepted 5 April 2012 Handling Editor L.N. Virgin Available online 5 May 2012

We investigate dynamic responses of axially moving viscoelastic beam subject to a randomly disordered periodic excitation. The method of multiple scales is used to derive the analytical expression of first-order uniform expansion of the solution. Based on the largest Lyapunov exponent, the almost sure stability of the trivial steady-state solution is examined. Meanwhile, we obtain the first-order and the second-order steady-state moments for the non-trivial steady-state solutions. Specially, we discuss the first mode theoretically and numerically. Results show that under the same conditions of the parameters, as the intensity of the random excitation increases, non-trivial steady-state solution fluctuation will become strenuous, which will result in the non-trivial steady-state solution lose stability and the trivial steady-state solution can be a possible. In the case of parametric principal resonance, the stochastic jump is observed for the first mode, which indicates that the stationary joint probability density concentrates at the non-trivial solution branch when the random excitation is small, but with the increase of intensity of the random excitation, the probability of the trivial steady-state solution will become larger. This phenomenon of stochastic jump can be defined as a stochastic bifurcation. & 2012 Elsevier Ltd. All rights reserved.

1. Introduction The transverse vibration analysis of axially moving beams has attracted a number of attentions of researchers over many years and it continues to be a hot topic today [1–8]. More specifically, Wickert and Mote [9] analyzed the nonlinear vibrations of axially moving continua using a modal analysis and Green’s function method. In [10], Pakdemirli and Ulsoy investigated principal parametric resonances and combination resonances of an axially accelerating string using the discretization perturbation techniques and the multiple scales method. Zhang and Zu [11,12] applied the method of multiple scales to consider the dynamic response and analyze the stability of parametrically excited viscoelastic belts. Chen et al. [13–16] applied Galerkin method to investigate the axially accelerating viscoelastic beam model which can be represented by the governing equation with partial differential form. In [17], Multidimensional Lindstedt–Poincare method was extended to the nonlinear vibration analysis of axially moving systems. Zhang et al. [18–21] combined the theory of normal form and Melnikov method or Shilnikov method to find the chaotic dynamics of the axially moving viscoelastic belt. Recently, Ghayesh [22,23] studied the vibrations and stability of axially moving system including the time-dependent tension and axial speed, or energy dissipation mechanisms.

n

Corresponding author. E-mail addresses: [email protected] (D. Liu), [email protected] (W. Xu), [email protected] (Y. Xu).

0022-460X/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jsv.2012.04.005

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D. Liu et al. / Journal of Sound and Vibration 331 (2012) 4045–4056

It is well known that engineering structures are often subjected to time dependent loadings of both deterministic and random fluctuations, which are generally described by the stochastic differential equations. Many approximate approaches has been employed to solve this kind of stochastic differential equations. For example, Rajan and Davies [24] used the two methods of multiple time scaling and stochastic averaging to study the random primary responses of a Duffing oscillator to narrow-band random excitation. Subsequently, Davies and Rajan [25] investigated the third-order super-harmonic and one third-order sub-harmonic response of a Duffing oscillator under narrow-band random excitation by the method of multiple time scales. Nayfeh and Serhan [26] proposed a second-order closure method to analyze the mean square responses and their local stability of a Duffing–Rayleigh oscillator excited by the sum of a deterministic harmonic component and a random component. Rong et al. [27–29] extended the multiple scaling methods to the analysis of nonlinear systems under random excitations. However, previous works only analyzed the steady-state response of the axially moving viscoelastic beam without disturbing factors. When researchers considered the disturbing factors, they often neglected the viscosity and elasticity of material. In order to address the lack in the above researches, this investigation concentrates on the dynamic responses of axially moving viscoelastic beam under a disturbing factor, i.e., a randomly disordered periodic excitation. We use the method of multiple scales to determine the first-order expansion of the solution of equations. This paper not only gives the random dynamic equations of the amplitude and phase angles, but also explores the stability of trivial and non-trivial steady-state solution. Moreover, the stochastic jump, the so-called stochastic bifurcation will be found. The paper is organized as follows. In Section 2, we introduce the axially moving viscoelastic beam and randomly disordered periodic excitation. In Section 3, the method of multiple scales is carried out to determine the equation of amplitude and phase, and then the first-order approximate solution for the principal parameter resonance is obtained. In Section 4, the almost sure stability of trivial steady-state solution is studied by the largest Lyapunov exponent and the stochastic jump of the first modal is observed. Finally, some conclusions are given in Section 5.

2. Axially moving viscoelastic beam modals with randomly disordered periodic excitation As shown in Fig. 1, let L be the total length of a uniform viscoelastic beam and W the axial transport speed and is constant. Denote the density by r and the cross-sectional area by A. The initial tension and the damping coefficient are denoted by P0 and C n. So, applying Newton’s second law of motion, the equation of axially moving viscoelastic beam to an imperfectly parameter periodic excitation is given as [7,14,23] !   2 @2 U @2 U @ @U @2 MðX,TÞ 2@ U n @U þ þ ðP þ rA þ 2 W W þ A s ðX,TÞÞ þh0 Uf ðTÞ, (1) ¼ C L 0 2 2 @X@T @T @X @X @T @X @X 2 where X and T denote the axial coordinate and the time respectively, U(X,T) describes the transverse displacement, sL(X,T) represents the axial disturbed stress, M(X,T) is the bending moment and f(T) is the imperfectly periodic excitation. As the moving viscoelastic beams adopt viscoelastic materials in practical engineering, the Kelvin model is chosen to describe the viscoelastic property of the linear material with the constitution relation as

sL ðX,TÞ ¼ E0 eL ðX,TÞ þ Z

@eL ðX,TÞ , @T

(2)

where E0 is the stiffness constant, Z is the damping coefficient of the dynamic viscosity, and eL(X,T) denotes the Lagrangian strain   1 @U 2 eL ðX,TÞ ¼ , (3) 2 @X which is the axial direction related to the transverse displacement. For the slender beam with small flexural stiffness ðE0 I 5 1Þ, the linear moment–curvature relationship is defined as MðX,TÞ ¼ E0 I

@2 U @X

2

þ ZI

@3 U @X 2 @T

:

Fig. 1. The model of axially moving viscoelastic beam.

(4)

D. Liu et al. / Journal of Sound and Vibration 331 (2012) 4045–4056

Substituting Eqs. (2)–(4) into Eq. (1), the axially accelerating of viscoelastic beams yields ! 2 @2 U @2 U @2 U @4 U @5 U 2@ U þW rA þ 2W P 0 2 þE0 I 4 þ ZI 4 @X@T @T 2 @X 2 @X @X @X @T  2 2   3 @U 3 @U @ U @U @2 U @U @U @ U þ E0 þ ¼ Cn þ h0 Uf ðT Þ: þ 2 Z Z @T 2 @X @X @X@T @X @X @X 2 @T @X 2

4047

(5)

Introduce the dimensionless variables and parameters as sffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffi U X P0 rA E0 I , g0 ¼ W : , kf ¼ u ¼ pffiffiffi , x ¼ , t ¼ T L P0 eL P 0 L2 rAL2 sffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffi IZ E0 A Cn L2 h0 L 2 , h ¼ pffiffiffi , a ¼ pffiffiffiffiffiffiffiffiffiffiffi , k1 ¼ , C¼ P0 e rAP0 P0 e rAP0 where e is a small dimensionless parameter accounting for the fact that the linear external damping coefficient, transverse displacement and the viscosity coefficient are very small. So, Eq. (5) becomes   4 @2 u @2 u @2 u 3 2 @2 u @u 2 @u @5 u 2@ u þ ðg20 1Þ 2 þ kf ea2 þ2g0 ¼ ek1 2 þ eC þ ehuf ðt Þ, (6) 2 4 @t@x 2 @x @t @x @x @x @t@x4 @t where f(t) is a random disordered periodic process governed by the following equation f ðtÞ ¼ cosðjðtÞÞ,

jðtÞ ¼ Ot þ bWðtÞ þ Y, j_ ðtÞ ¼ O þ bxðtÞ

with O 40 being the frequency of the random excitation and Y being a constant, W(t) being the standard Wiener process and x(t) being a Gaussian white noise of unit intensity, which describes random temporal deviations of the excitation frequency from its expected or mean value O. The power spectrum density of random process f(t) was given in [30] as   4 b2 O2 þ o2 þ b4 1 : Sx ðoÞ ¼   4 2 2 O2 o2 þ b4 þ o2 b4 Hence, the random process f(t) can be assumed as a narrow-band noise. When b-N, the power spectrum density Sx(o) becomes the power spectrum of white noise. When b-0, the randomly disordered periodic excitation becomes a deterministic harmonic excitation. 3. The first-order approximate solution We introduce the mass, gyroscopic and the linear stiffness operators as follows: M ¼ I,

G ¼ 2g0

@ , @x

K ¼ ðg20 1Þ

4 @2 2 @ þkf 4 : 2 @x @x

(7)

So, Eq. (6) will be written as M

  @2 u @u @u 3 2 @2 u @u 2 @5 u þ Ku ¼ eC þ ek1 2 þG ea2 þ ehu cosðOt þ bWðtÞ þ YÞ: 2 @t @t 2 @x @x @t@x4 @t

(8)

In the method of multiple scales, the uniformly approximate solution of Eq. (8) is sought in the form uðx,t, eÞ ¼ u0 ðx,T 0 ,T 1 Þ þ eu1 ðx,T 0 ,T 1 Þ þ    ,

(9)

where T0 ¼t is a fast time scale and T1 ¼ et is a slow time scale. By denoting the differential operators D0 ¼q/qT0 and D1 ¼@/@T1, we get @ ¼ D0 þ eD1 þ    , @t

@2 ¼ D20 þ 2eD0 D1 þ    : @t 2

(10)

Substituting Eqs. (9) and (10) into Eq. (8) and comparing coefficients of e with equal powers, we obtain the following equations: MD20 u0 þ GD0 u0 þKu0 ¼ 0, MD20 u1 þGD0 u1 þ Ku1 ¼ C

  @u0 3 2 @2 u0 @u0 2 2 @5 u0 þ k1 a þhu cosðOT 0 þ bWðT 1 Þ þ YÞ: @T 0 2 @x @x2 @T 0 @x4

(11) (12)

It should be pointed out that we used the statistical property of the standard Wiener process bWðT 0 Þ ¼ pbffiffie WðeT 0 Þ ¼

bWðT 1 Þ in Eq. (12).

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D. Liu et al. / Journal of Sound and Vibration 331 (2012) 4045–4056

Under the simple support boundary conditions

@2 u @x2

uð0,tÞ ¼ uð1,tÞ ¼ 0,

x¼0

@2 u ¼ 2 @x

¼ 0,

(13)

x¼1

according to Wickert and Mote [31], the general solution of Eq. (11) consists of one infinite sequence of modes corresponding to the infinite sequence of frequencies, that is u0 ðx,T 0 ,T 1 Þ ¼

1 X

fn ðxÞAn ðT 1 Þeion T 0 þcc,

(14)

n¼1

where fn and on represent the nth complex mode function and the nth natural frequency, and An(T1) is the nth slowly varying amplitude of the response. cc represents the complex conjugate of its preceding terms. Under the small flexural stiffnessðnamely : kf {1Þ, Kong and Parker [32] took the perturbation techniques and the phase closure principle to derive the nth natural frequency as " # 2 kf n2 p2 ðg40 þ 6g20 þ1Þ 2 on ¼ np 1g0 þ , (15) 2 and the mode function as

fn ðxÞ ¼

r 22,n r 23,n r 23,n r 21,n

eir1,n x þ eir2,n x þ

r 21,n r 22,n r 23,n r 21,n

eir3,n x þ

r 22,n r 23,n r 23,n r 21,n

eir1,n

r 22,n r 21,n r 24,n r 24,n

eir4,n ðx1Þ ,

(16)

where ri,n(i¼1,2,3,4) satisfy the following equation 2

kf r 4i,n þ ð1g20 Þr 2i,n þ 2g0 on r i,n o2n ¼ 0:

(17)

In this paper, we only consider the case that the nth mode is directly excited with the assumption that there are no interactions among the modes (that is, the system is far from internal resonance). Thus, the solution of nth mode can be expressed as u0 ¼ fn ðxÞAn ðT 1 Þeion T 0 þ cc:

(18)

Substituting Eq. (18) into Eq. (12), we obtain 0 MD20 u1 þ GD0 u1 þ Ku1 ¼ 2ion fn A_ n eion T 0 2g0 fn A_ n eion T 0 þ iC on fn An eion T 0

ia2 on fn An eion T 0 þ ð4Þ

þ

00 3 2 0 0 00 0 k ð2fn fn fn A2n An þ n fn ðf Þ2 An A2n Þeion T 0 2 1

i 3 2 00 0 2 3ion T 0 h h k1 fn ðfn Þ e þ fn An eiðon T 0 þ OT 0 þ bW þ YÞ þ fn An eiðOT 0 on T 0 þ bW þ YÞ : 2 2

(19)

In order to examine the nth mode principal parametric resonance, we introduce the detuning frequency parameter s to describe the nth mode principal parametric resonance as follows:

O ¼ 2on þ es,

(20)

and so ðOon ÞT 0 ¼ on T 0 þ es: According to [33], Eq. (19) has a solution only if the right side of Eq. (19) is orthogonal to every solution of the homogeneous problem, that is, D 0 ð4Þ 2ion fn A_ n eion T 0 2g0 fn A_ n eion T 0 þ iC on fn An eion T 0 ia2 on fn An eion T 0

00 3 2 h 0 0 00 0 (21) þ k1 ð2fn fn fn An A2n þ n fn ðfn Þ2 An A2n Þeion T 0 þ fn An eiðOT 0 on T 0 þ bW þ YÞ , fn ¼ 0, 2 2 where the notation /U,US represents the inner product of two complex functions on [0,1]. Hence, Eq. (21) can be written as h 2 A_ n þC wn An þ a2 dn An þ k1 Bn An A2n þ mn An eiðs1 T 1 þ bWðT 1 Þ þ YÞ ¼ 0, 2

(22)

where

wn ¼  h

2 ion

Bn ¼

3

R1

R1 0

ion

R1 0

fn fn dx

fn fn dx þ g0

R1

0 0 fn fn dx

i,

dn ¼ h

R 1 00 0 2 0 fn ðfn Þ fn dx=2 h i , R1 R1 0 2 ion 0 fn fn dx þ g0 0 fn fn dx

0

f0n f0 n f00n fn dx þ 3

2 ion

R1 0

ion

R1 0

fð4Þ n fn dx

fn fn dx þ g0 R1

mn ¼ h

2 ion

R1 0

0

R1 0

i,

f0n fn dx

fn fn dx

fn fn dx þ g0

R1 0

i:

f0n fn dx

(23)

D. Liu et al. / Journal of Sound and Vibration 331 (2012) 4045–4056

4049

From the mode function which given by Eq. (16), Eq. (23) can be rewritten as

wn ¼ wnr þ iwni , dn ¼ dnr þ idni , Bn ¼ Bnr þ iBni , mn ¼ mnr þ imni , where wnr, dnr, Bnr, mnr represent the real parts, and wni, Expanding An in the polar form

(24)

dni, Bni, mni represent the imaginary parts of Eq. (23).

An ðT 1 Þ ¼ an ðT 1 Þeijn ðT 1 Þ ,

(25)

and substituting Eqs. (24) and (25) into Eq. (22), we see that Eq. (22) is equivalent to the following equations h 2 a_ n ¼ ðC wnr þ a2 dnr Þan k1 Bnr a3n  an ðmnr cos ðyn Þmni sin ðyn ÞÞ, 2 _ 2 an yn ¼ ðs þ2C wni þ2a2 dni Þan þ2k1 Bni a3n þ han ðmnr sin ðyn Þ þ mni cos ðyn ÞÞ þ ban W 0 ðT 1 Þ,

(26)

where yn ¼ sT 1 þ bWðT 1 Þ2jn ðT 1 Þ: Equivalently, Eq. (26) can be obtained as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h 2 a_ n ¼ ðC wnr þ a2 dnr Þan k1 Bnr a3n  an m2nr þ m2ni cos ðyn Þ, 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 an y_ n ¼ ðs þ2C wni þ2a2 dni Þan þ2k1 Bni a3n han m2nr þ m2ni sin ðyn Þ þ ban W 0 ðT 1 Þ,

(27)

where yn ¼ yn þ arctanðmni =mnr Þ: Therefore, solving Eq. (27) for the amplitude an and the phase yn, and combining Eq. (16), the first-order uniform expansion of the solution of Eq. (8) can be written as u0 ðx,tÞ ¼ fn ðxÞan ðetÞeiððOt=2Þðyn ðetÞ=2ÞÞ þ fn ðxÞan ðetÞeiððOt=2Þðyn ðetÞ=2ÞÞ þ OðeÞ:

(28)

4. Steady-state solution and their stability In this section, we present the steady-state solutions and their stability, including the trivial and non-trivial cases, as well as a new phenomenon as the stochastic jump will be observed. Numerical procedure will be carried out to verify the theoretical solution with good agreement. 4.1. Trivial steady-state solution and their stability It is obvious that Eq. (27) have a steady-state solution at an ¼0. In order to analyze the stability of this solution, we will neglect the nonlinear terms, and then the linearization equation of Eq. (27) at (0,0) can be written as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h a_ n ¼ ðC wnr þ a2 dnr Þan þ m2nr þ m2ni an cos ðyn Þ, 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi an y_ n ¼ ðs þ 2C wni þ 2a2 dni Þan h m2nr þ m2ni an sin ðyn Þ þ ban W 0 ðT 1 Þ: (29) Let nn ¼ lnðan Þ, according to Itˆo’s formula, Eq. (29) can be written as the following Itˆo stochastic differential equations qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dnn ¼ ðC wnr a2 dnr þ 2h m2nr þ m2ni cos yn ÞdT 1 , qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (30) dyn ¼ ðs þ 2C wni þ2a2 dni h m2nr þ m2ni sin yn ÞdT 1 þ bdW: Through Eq. (30), it is easy to find that the steady-state probability density function p(yn) is governed by the following Fokker–Plank equation 2

d p pffiffiffiffiffiffiffiffiffiffiffiffiffi

2h

2 dyn



i d h ðsh sin yn Þp ¼ 0, dyn

(31)

m2nr þ m2

ni : where s ¼ 2ðs þ 2C wni2 þ 2adni Þ , h ¼ b b2 Using the periodicity condition and the normality condition [34], we obtain the solution of Eq. (31) as Z esðyn þ pÞ þ h cos yn yn þ 2p syh cos y e dy, pðyn Þ ¼ 4p2 Iis ðhÞIis ðhÞ yn

(32)

where In(x) is the modified Bessel function of the first kind and n can be any real and complex number. According to Oseledec’s multiplicative ergodic theorem [35], the Lyapunov exponent of the trivial steady-state solution of Eq. (29) can be written as l ¼ lim T11 ln aannðTð0Þ1 Þ ¼ lim T11 ½nn ðT 1 Þnn ð0Þ T 1 -1

T 1 -1

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D. Liu et al. / Journal of Sound and Vibration 331 (2012) 4045–4056

Z qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 T1 m2nr þ m2ni lim cos yn ðtÞdt T 1 -1 T 1 0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  h ¼ ðC wnr þ a2 dnr Þ þ m2nr þ m2ni E cos yn 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z 2p h ¼ ðC wnr þ a2 dnr Þ þ m2nr þ m2ni cos yn pðyn Þdyn 2 0

¼ ðC wnr þ a2 dnr Þ þ

h ¼ ðC wnr þ a dnr Þ 4 2

h 2

( ) qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi I1is ðhÞ I1 þ is ðhÞ 2 2 : mnr þ mni þ Iis ðhÞ Iis ðhÞ

(33)

Specially, in this paper we consider the first mode (that is: n ¼1). When kf ¼0.03 and g0 ¼0.8, by finding the numerical values of Eqs. (15)–(17) and Eq. (23), we will derive o1 ¼1.2042, r1,1 ¼ 0.6689, r2,1 ¼  5.6230, r3,1 ¼2.4774þ20.5469i, r4,1 ¼2.4774–20.5469i, w1 ¼ 1.0017–0.0007i, d1 ¼  382.56þ 12.57i, B1 ¼  1.506þ620i and m1 ¼  0.3781 0.4388i. Note that the trivial steady-state solution of Eq. (29) is almost sure stable if and only if l o0. To understand the theoretical results, we consider the first mode with parameters a ¼0.02, k1 ¼0.1 and C¼ 0.5. Figs. 2(a) and 3(a) show the threedimensional plots of the largest Lyapunov exponent for different noise intensity, and the corresponding contour curves of l are presented in Figs. 2(b) and 3(b). Comparing Figs. 2 and 3, it can be seen that with the increase of excitation intensity, the graph surface of l  (h,s) will change. We find that the trivial solution is unstable (l 40, see Fig. 2(b)) when b ¼0.5 and the trivial solution is stable (l o0, see Fig. 3(b)) when b ¼2.5 at the point A. On the contrary, the trivial solution is stable (l o0, see Fig. 2(b)) when b ¼0.5 and the trivial solution is unstable (l 40, see Fig. 3(b)) when b ¼2.5 at the points B and C. Figs. 4–6 show the numerical simulation results of the trivial responses of the system to principal parametric response of the first mode at points A–C, which are in full agreement with the conclusions of stability above. 4.2. Non-trivial steady-state solution and their stability In this part, we will focus on the case of non-trivial steady-state solution of Eq. (27). To obtain the first-order and second-order steady-state moment, we use the Itˆo stochastic differential equation of Eq. (27) as follows: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 dan ¼ ðC wnr an a2 dnr an k1 Bnr a3n þ 2h m2nr þ m2ni an cos yn ÞdT 1 , qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 dyn ¼ ðs þ 2C wni þ2a2 dni þ 2k1 Bni a2n h m2nr þ m2ni sin yn ÞdT 1 þ bdWðT 1 Þ:

(34)

The perturbation technique will be taken to solve Eq. (34). When b ¼0, Eq. (34) becomes the following deterministic equation: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 a_ n ¼ C wnr an a2 dnr an k1 Bnr a3n þ 2h m2nr þ m2ni an cos yn , qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

y_ n ¼ s þ2C wni þ2a2 dni þ 2k21 Bni a2n h m2nr þ m2ni sin yn :

(35)

14 12

5

10

3

8

λ

2

C

λ=3

λ=2

h

4

B

6

1 15

0 10

10 5

0

σ

-5

5 -10

0

4 2 A λ=0

h 0

-12 -10

-8

-6

-4

-2

0

2

4

6

8

10

12

σ Fig. 2. Largest Lyapunov exponent l of the trivial solution of the system (29) of its first mode: a ¼0.02, k1 ¼0.1, C¼ 0.5 and b ¼0.5. (a) Mesh surface; (b) contour curves.

D. Liu et al. / Journal of Sound and Vibration 331 (2012) 4045–4056

4051

14 12

B

3.5 3

8

2

h

λ

2.5

C

λ=2

10

λ=1

1.5

6

1 4

15

0.5 0

10 10

5

h

5

0

-5

σ

-10

λ=0

2

A 0

0

-12 -10

-8

-6

-4

-2

0

2

4

6

8

10

12

σ Fig. 3. Largest Lyapunov exponent l of the trivial solution of the system (29) of its first mode: a ¼0.02, k1 ¼ 0.1, C ¼0.5 and b ¼2.5. (a) Mesh surface; (b) contour curves.

1.8

14 β = 0.5

β = 2.5

1.6

12

1.4 10 1.2 8

a1

a1

1 0.8

6

0.6 4 0.4 2 0

0.2

0

10

20

30

40

50

60

70

80

90

100

0

0

10

20

30

40

T1

50

60

70

80

90

100

T1

Fig. 4. The principal parametric response of the first mode at point A. (a) Unstable when b ¼ 0.5; (b) stable when b ¼ 2.5.

So the non-trivial steady-state solution can be denoted as an ¼an0 and yn ¼ yn0. Combining conditions a_ n ¼ 0, and ana0, we arrive at the following frequency–amplitude equation: C 1 a4n0 þC 2 a2n0 þ C 3 ¼ 0,

y_ n ¼ 0 (36)

where 4

C 1 ¼ 4k1 ðB2nr þ B2ni Þ, 2

2

C 2 ¼ 4k1 sBni þ 8k1 ðC Bnr wnr þ a2 Bnr dnr þ C Bni wni þ a2 Bni dni Þ, 2

C 3 ¼ 4ðC wnr þ a2 dnr Þ2 þ ðs þ 2C wnr þ 2a2 dnr Þ2 h ðm2nr þ m2ni Þ: For Eq. (36), there are one trivial solution and two possible steady-state non-trivial solutions with C 22 4 4C 1 C 3 given as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi112 0 C 2 7 C 22 4C 1 C 3 A, (37) an0 ¼ @ 2C 1 where for these two non-trivial solutions, one is stable and the other one is unstable.

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D. Liu et al. / Journal of Sound and Vibration 331 (2012) 4045–4056

10

x 109

4

β = 2.5

β = 0.5

9

3.5

8 3 7 2.5

6

2

a1

a1

5 4

1.5

3 1 2 0.5

1 0

0

10

20

30

40

50

60

70

80

90

0

100

0

10

20

30

40

T1

50

60

70

80

90

100

T1

Fig. 5. The principal parametric response of the first mode at point B. (a) Stable with b ¼0.5; (b) unstable with b ¼2.5.

4000

1.8

β = 2.5

β = 0.5

1.6

3500

1.4

3000

1.2

2500

a1

a1

1 2000

0.8 1500

0.6

1000

0.4

500

0.2 0

0

10

20

30

40

50

60

70

80

90

0

100

0

10

20

30

T1

40

50

60

70

80

90

100

T1

Fig. 6. The principal parametric response of the first mode at point C. (a) Stable with b ¼0.5; (b) unstable with b ¼2.5.

In the cases of C2 40 and C3 o0, there is a trivial solution, together with a non-trivial solution monotonically decreasing with s. This suggests that the mode here is softening type [36,37]. For C2 o0, the steady-state solution is composed of a qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi non-trivial solution and a trivial solution at C 2 o C 22 4C 1 C 3 , or it is made up of a trivial solution and two non-trivial qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi solution at C 2 4 C 22 4C 1 C 3 . Next, we determine the effects of the noise (that is, ba0). Thus, the solution of Eq. (34) can be defined as an ¼ an0 þ a~ n ,

yn ¼ yn0 þ y~ n ,

(38)

where an0, yn0 are determined by Eq. (35), and a~ n , y~ n are the perturbation terms. Substituting Eq. (38) into Eq. (34) and neglecting the nonlinear terms, we obtain the following Itˆo stochastic differential equations da~ n ¼ P11 a~ n dT 1 P12 y~ n dT 1 , dy~ n ¼ P 21 a~ n dT 1 P 22 y~ n dT 1 þ bdWðT 1 Þ, where 2

P 11 ¼ ðC wnr þ a2 dnr þ 3k1 a2n0 Bnr 

h 2

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m2nr þ m2ni cos yn0 Þ,

(39)

D. Liu et al. / Journal of Sound and Vibration 331 (2012) 4045–4056

P 12 ¼

han0 2

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m2nr þ m2ni sin yn0 ,

2

P 21 ¼ 4k1 Bni an0 ,

4053

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P 22 ¼ h m2nr þ m2ni cos yn0 :

Due to [38], the first-order steady-state moment Ea~ n and second-order steady-state moment Ea~ 2n of Eq. (39) can be obtained by the moment method, where E denotes the mathematical expectation. Under the steady-state case, it is easy to obtain 2 2 dEa~ n dEy~ n dEa~ n dEy~ n dEa~ n y~ n ¼ ¼ ¼ ¼ ¼ 0, dT 1 dT 1 dT 1 dT 1 dT 1

which yields Ea~ n ¼ 0, and

Ey~ n ¼ 0,

(40)

8 2 > P11 Ea~ n þ P12 Eða~ n y~ n Þ ¼ 0, > > < 2 P21 Ea~ 2n ðP 11 þ P22 ÞEða~ n y~ n ÞP12 Ey~ n ¼ 0, > > > : 2P Eða~ y~ Þ2P Ey~ 2 þ b2 ¼ 0: n n 21 12 n

(41)

From Eq. (41), we may obtain 2 P 12 b , 2 2ðP 11 þP 11 P 22 þ P 11 P21 þP 12 P 21 Þ 2 P11 b , Eða~ n y~ n Þ ¼ 2 2ðP 11 þ P 11 P21 þ P11 P 22 þ P 12 P21 Þ 2

Ea~ n ¼

ðP 211 þ P11 P 22 þP 12 P 21 Þb

2

Ey~ n ¼

2

2ðP211 þ P11 P 21 þP 11 P22 þ P12 P 21 Þ

(42)

:

Combining Eqs. (37), (38) and (42), we have Ean ¼ Eðan0 þ a~ n Þ ¼ an0 , 2

Ea2n ¼ a2n0 þ Eða~ n Þ:

(43)

Through Eq. (43) and Fig. 7, we find that under the noise excitation, the increase of the noise intensity b leads to strenuous fluctuations of the non-trivial steady-state solution. This means that the random noise will affect the non-trivial steady-state solution at fixed points. 4.3. Stochastic jump and bifurcation In this subsection, we pay special attention to the statistical properties of the first mode. Fig. 8 shows the theoretical frequency–amplitude response on Eq. (43) and the numerical simulation result of Eq. (34), which verify the validity of the analyses method mentioned above. In Fig. 8, s is considered as a bifurcation parameter. With the increase of s, we find a steady-state trivial solution and two steady-state non-trivial solution with one is stable and the other is unstable in region 1, one unstable trivial solution and one stable non-trivial solution in region 2, and only one stable trivial solution in region 3. So, when b and h are fixed, the steady-state solution will jump as s changes from region 1 to region 2, and hence, the stochastic bifurcation will happen. When ba0, Eq. (34) is a two-dimensional Itoˆ stochastic differential equations, and thus the Fokker–Plank equation can be written as 2

@p @ @ b @2 ¼ ðf 1 ðan , yn ÞpÞ ðf 2 ðan , yn ÞpÞ þ ðpÞ, @T 1 @an @yn 2 @y2 n

(44) 2

density, f 1 ðan ,qn Þ ¼ C wnr an a2 dnr an k1 Bnr a3n þ where p ¼p(an,yn,Tn) is the transition probability qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 2 f 2 ðan , yn Þ ¼ s þ2C wni þ 2a dni þ2k1 Bni an h mnr þ m2ni sin yn . The initial condition of Eq. (44) is given as pðan , yn ,0Þ ¼ dðan an0 ÞUdðyn yn0 Þ:

h 2

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m2nr þ m2ni an cos yn and

(45)

The boundary condition for the an is written as pðan , yn ,T 1 Þ ¼ f inite at an ¼ 0, pðan , yn ,T 1 Þ,

@pðan , yn ,T 1 Þ=@an -0

at an -1:

(46)

The boundary condition for the yn is a periodic boundary, that is pðan , yn ,T 1 Þ ¼ pðan , yn þ 2mp,T 1 Þ:

(47)

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D. Liu et al. / Journal of Sound and Vibration 331 (2012) 4045–4056

10.2

10.4 β = 0.05

10.15 10.1

10.2

10.05

10.1

10

10

9.95

a

a

β = 0.1

10.3

9.9

9.9 9.8

9.85

9.7

9.8 9.6 9.75 9.7 20

9.5 40

60

80

100

120

140

160

180

200

20

40

60

80

100

120

140

160

180

200

T1

T1

Fig. 7. Non-trivial responses of the first mode: C ¼0.5, k1 ¼0.01, a ¼0.02, h ¼10.0 and s ¼  6.5. (a) b ¼ 0.05; (b) b ¼ 0.1. 180

14

160

12

140

10

120

E(a)

E(a2)

8

6

100 80 60

region 1

4 region 1

region 2

2

region 2

20

σ2

σ1

0 -15

region 3

40

region 3

-10

-5

0

σ

5

σ1

0

10

15

-15

-10

-5

σ2

0

σ

5

10

15

Fig. 8. Frequency–amplitude response of the first mode: C¼ 0.5, k1 ¼0.01, a ¼ 0.02, h ¼10.0 b ¼ 0.1. ‘‘333’’numerical result, ‘‘—’’ theoretical result, ‘‘---’’ unstable solution.

Under the initial condition (Eq. (45)) and the boundary conditions (Eqs. (46) and (47)), the stationary joint probability will be obtained by solving the Fokker–Plank equation through the finite difference method. As is well known, the viscoelastic beam is of the softening type, as a result, under the disordered periodic excitation, the phenomenon of stochastic jumps of amplitude will happen at bifurcation parameter s forward or backward. Fig. 9 shows a change of the stationary joint probability density at different intensities of random excitation. Here we choose the bifurcation parameter s ¼ 6.5(s1 o s) so that this value belongs to region 2. Through Fig. 9(a), the stationary joint probability density concentrates at the non-trivial solution branch when b ¼0.1. But, with the increase of excitation intensity b, the diffusion phenomena occur of the non-trivial solution, meanwhile, the probability of the stationary solution transition from the non-trivial solution to the trivial solution increases. So, due to [39], the jump will happen when the random excitation changes. As the parameters change the qualitative behavior of the stationary solution will change and it can be regarded as a stochastic bifurcation. 5. Conclusions In this paper, the method of multiple scales is applied to investigate the principal parametric resonance of axially moving viscoelastic beam under a randomly disordered periodic excitation. Using the solvability condition, we give the largest Lyapunov exponent of the trivial steady-state solution to analyze the almost sure stability of the system, at the

D. Liu et al. / Journal of Sound and Vibration 331 (2012) 4045–4056

4055

0.8 2.5

0.6

p(a 1 , θ 1 )

p(a 1 , θ 1 )

2 1.5 1

0.4 0.2

0.5 4

0 20

0 20

2

15 10

10

θ1

0

a1

15

a1

5 0

5 0

-2

-2

-1

1

0

3

2

4

θ1

0.3 0.2

0.25 0.2

p(a 1 , θ 1 )

p(a 1 , θ 1 )

0.15 0.1 0.05

0.15 0.1 0.05 0 20

0 20 15 10

a1

5 0

-2

-1

0

1

2

θ1

3

4

15 10

a1

5 0

-2

-1

0

1

2

3

4

θ1

Fig. 9. Stationary joint probability density of the first mode to the different excitation intensity C ¼0.5, k1 ¼0.01, a ¼ 0.02, h ¼ 10.0 and s ¼  6.5. (a) b ¼ 0.1; (b) b ¼0.3; (c) b ¼ 0.5; (d) b ¼ 0.7.

same time we derived the first-order and the second-order steady-state moments for the non-trivial steady-state solutions. Theoretical analyses and numerical simulations for the first mode show that under the same conditions of the parameters, with the increase of the intensity of random excitation, non-trivial steady-state solution fluctuation will become strenuous. This cause the non-trivial steady-state solution may lose its stability and the system may have a trivial steady-state solution. In order to investigate the stochastic jumps between the non-trivial steady-state solution and the trivial steady-state solution, we solved the Fokker–Plank equation corresponds to it’s two-dimension Itˆo stochastic differential equations using the finite difference method. From the numerical simulation for the first mode, we find that the stationary joint probability density concentrates at the non-trivial solution branch when the random excitation increases is small, but with the increase of intensity of the random excitation, the probability of the trivial steady-state solution will become larger. Such phenomena indicate that stochastic jump occurs with the increase of the intensity of random excitation, which can be defined as a stochastic bifurcation.

Acknowledgment This work is supported by the National Natural Science Foundation of China under Grant Nos. 11172233, 10932009 and 10972181, NCET, Aoxing Star plan and FFR of NPU. The authors wish to express their appreciation to the anonymous reviewers for their insightful readings and helpful comments. References [1] C.D. Mote Jr., Dynamic stability of axially moving materials, Shock and Vibration Digest 4 (1972) 2–11. [2] A.G. Ulsoy, C.D. Mote Jr., Band saw vibration and stability, Shock and Vibration Digest 10 (1978) 3–15.

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[3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39]

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