Research ArticleStochastic P-Bifurcation of a Bistable Viscoelastic Beam withFractional Constitutive Relation under Gaussian White Noise

Yajie Li 12 ZhiqiangWu 12 Guoqi Zhang12 and FengWang 12

1Department of Mechanics School of Mechanical Engineering Tianjin University Tianjin 300072 China2Tianjin Key Laboratory of Nonlinear Dynamics and Chaos Control Tianjin University Tianjin 300072 China

Correspondence should be addressed to Zhiqiang Wu zhiqwutjueducn

Received 28 July 2018 Revised 26 October 2018 Accepted 5 November 2018 Published 2 December 2018

Academic Editor Dorota Mozyrska

Copyright copy 2018 Yajie Li et al This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

In this paper we study the stochastic P-bifurcation problem for axially moving of a bistable viscoelastic beam with fractionalderivatives of high order nonlinear terms under Gaussian white noise excitation First using the principle for minimum meansquare error we show that the fractional derivative term is equivalent to a linear combination of the damping force and restoringforce so that the original system can be simplified to an equivalent system Second we obtain the stationary Probability DensityFunction (PDF) of the systemrsquos amplitude by stochastic averaging and using singularity theory we find the critical parametriccondition for stochastic P-bifurcation of amplitude of the system Finally we analyze the types of the stationary PDF curves of thesystem qualitatively by choosing parameters corresponding to each region within the transition set curveWe verify the theoreticalanalysis and calculation of the transition set by showing the consistency of the numerical results obtained byMonteCarlo simulationwith the analytical results The method used in this paper directly guides the design of the fractional order viscoelastic materialmodel to adjust the response of the system

1 Introduction

Fractional calculus is a generalization of integer-order cal-culus it extends the order of calculus operation from thetraditional integer order to the case of noninteger orderand it has a history of more than 300 years as so far Dueto the limitation of the definition of integer-order deriva-tive it cannot express the memory property of viscoelasticsubstances The definition of fractional derivative containsconvolution which can well express the memory effect andshow the cumulative effect over time Compared with thetraditional integer-order calculus fractional calculus hasmore advantages and is a suitable mathematical tool fordescribing the memory characteristics [1ndash11] and in recentyears it has become the powerful mathematical tool in manydisciplines especially in the study of viscoelastic materials

The fractional derivative can accurately describe theconstitutive relation of viscoelastic materials with fewerparameters so the studies of fractional differential equationson the typical mechanical properties and the influences offractional order parameters on the system are very necessary

and have important significance In recent years manyscholars have done a lot of work and achieved fruitful resultsin this field Li and Tang studied the nonlinear parametricvibration of an axially moving string made by rubber-likematerials a new nonlinear fractional mathematical modelgoverning transverse motion of the string is derived basedon Newtonrsquos second law the Euler beam theory and theLagrangian strain and the principal parametric resonanceis analytically investigated via applying the direct multiscalemethod [12] Liu et al introduced a transfer entropy andsurrogate data algorithm to identify the nonlinearity levelof the system by using a numerical solution of nonlinearresponse of beams the Galerkin method was applied todiscretize the dimensionless differential governing equationof the forced vibration and then the fourth-order Runge-Kutta method was used to obtain the time history responseof the lateral displacement [13] Liu et al investigated thestochastic stability of coupled viscoelastic system with non-viscously damping driven by white noise through momentLyapunov exponents and Lyapunov exponents obtained thecoupled Ito stochastic differential equations of the norm of

HindawiDiscrete Dynamics in Nature and SocietyVolume 2018 Article ID 6935095 10 pageshttpsdoiorg10115520186935095

2 Discrete Dynamics in Nature and Society

the response and angles process by using the coordinatetransformation and discussed the effects of various physicalquantities of stochastic coupled system on the stochasticstability [14] Nutting Gemant and Scott-Blair et al [15ndash17]first proposed the fractional derivative models to study theconstitutive relation of viscoelastic materials and the researchon the viscoelastic materials with fractional derivative isalso increasing and so far it is still a research hotspot [18ndash25] Rodr Guez et al calculated the correlation functionof transverse wave in linear and homogeneous viscoelasticliquid by the Generalized Langevin Equation (GLE) methodand the influence of fractional correlation function on thedynamic behavior of the system is analyzed [26] Bagley andTorvik used fractional calculus to study the dynamic behaviorof viscoelastic damping structure and the responses of thesystem under general load as well as step load are analyzedrespectively [27 28] Pakdemirli and Ulsoy studied theprimary parametric resonance and combined resonance ofthe axial acceleration rope based on the discrete perturbationmethod and the multiscale method [29] Zhang and Zhuanalyzed the stability and dynamic response of viscoelasticbelt under parametric excitation by the multiscale method[30 31] Chen et al studied the dynamic behavior and steady-state response of axially accelerating viscoelastic beam by theGalerkin method [32ndash35] derived the differential equationof nonlinear vibration for axially moving viscoelastic ropeand then pointed out that the damping of viscoelastic ropeonly exists in the nonlinear term [36 37] Leung et alstudied the steady-state response of a simply supportedviscoelastic column under the axial harmonic excitationbased on the fractional derivative constitutive model ofcubic nonlinear and derived the generalizedMathieu-Duffingequation with time delay by the Galerkin discrete methodthen the bifurcation behavior of the system caused by theorder of the fractional derivative is analyzed [38] GhayeshandMoradian developed the Kelvin-Voigt viscoelastic modelof the axially moving and the tensile belt and then foundthe existence of nontrivial limit cycle in this system [39]Liu et al studied the dynamic response of an axially movingviscoelastic beamunder randomdisorder periodic excitationthe first order expression of the solution is obtained by themultiscale method and the stochastic jump phenomenonbetween the steady-state solutions is carried out [40] Yangand Fang derived the system equation based on Newtonrsquossecond law and the fractional Kelvin constitutive relation andthen studied the stability of the axially moving beam underthe parametric resonance condition [41] Leung et al studiedthe singlemode dynamic characteristics of the nonlinear archwith the fractional derivative the steady-state solution of thesystem is obtained based on the residual harmonic homotopymethod and the influence of the parametric variation on thedynamic behaviors of the viscoelastic damping material isanalyzed [42] Galucio et al obtained the fractional derivativemodel to describe the viscoelasticity of the system based onthe Timoshenko theory and Euler-Bernoulli hypothesis andproposed a finite element formula for analyzing the sandwichbeam of viscoelastic material with fractional derivative andthe results were verified numerically [43]

Due to the complexity of fractional derivative the analysismethod of it becomes more difficult the study on the vibra-tion characteristics of the parameters can only be qualitativelyanalyzed and the critical conditions of the parametric influ-ences cannot be found which affect the analysis and designof such systems as well as the stochastic P-bifurcation ofbistability for the viscoelastic beamwith fractional derivativesof high order nonlinear terms under random noise excitationhas not been reported In view of the above situationthe nonlinear vibration of viscoelastic beam with fractionalconstitutive relation under Gaussian white noise excitation istaken as an example the transition set curve of the fractionalorder system as well as the critical parametric conditionfor stochastic P-bifurcation of the system is obtained by thesingularity theory and then the types of stationary PDFcurves of the system in each region in the parametric planedivided by the transition set are analyzed By the method ofMonte Carlo simulation the numerical results are comparedwith the analytical results obtained in this paper it can beseen that the numerical solutions are in good agreementswith the analytical solutions and thus the correctness of thetheoretical analysis in this paper is verified

2 Equation of Axially MovingViscoelastic Beam

There are many definitions of fractional derivatives andthe Riemann-Liouville derivative and Caputo derivative arecommonly used The initial conditions corresponding to theRiemann-Liouville derivative have no physical meaningshowever the initial conditions of the systems described bythe Caputo derivative have clear physical meanings and theirforms are the same as the initial conditions for the differentialequations of integer order So in this paper the Caputo-typefractional derivative is adopted as follows

119862119886119863119901 [119909 (119905)] = 1

Γ (119898 minus 119901) int119905

119886

119909(119898) (119906)(119905 minus 119906)1+119901minus119898119889119906 (1)

where 119898 minus 1 lt 119901 le 119898119898 isin 119873 119905 isin [119886 119887] Γ(119898) is the EulerGamma function and119909(119898)(119905) is themorder derivative of119909(119905)

For a given physical system due to the fact that the initialmoment of the oscillator is 119905 = 0 the following form of theCaputo derivative is often used

1198620119863119901 [119909 (119905)] = 1

Γ (119898 minus 119901) int119905

0

119909(119898) (119906)(119905 minus 119906)1+119901minus119898119889119906 (2)

In this paper the transverse vibration 119910(119909 119905) of a vis-coelastic simply supported beam under lateral excitation119865(119909 119905) as shown in Figure 1 is considered applying thedrsquoAlambert principle the governing equation can be writtenas [42]

120597Q120597119909 = 119865 (119909 119905) minus 12058811986012059721199101205971199052

120597119872120597119909 = Q minus 119873120597119910120597119909

(3)

Discrete Dynamics in Nature and Society 3

y

O

F (x t) =Asin(xL)(t)

y (x t)

L

M

N

X

Q

Fdx

M+ M

xdx

N+ N

xdx

Q+ Q

xdx

Figure 1 Schematic stress diagram of viscoelastic beam

where 120588 is the mass density of the beam119860 is the area of cross-section119872 is the bending moment Q is the lateral force and119873 is the horizontal force From (3) we have

12059721198721205971199092 + 11987312059721199101205971199092 + 1205881198601205971199102

1205971199052 minus 119865 (119909 119905) = 0 (4)

Assuming the material of the beam obeys a fractional deriva-tive viscoelastic constitutive relation

120590 (119909 119911 119905) = 1198640120576 (119909 119911 119905) + 1198641 1198620119863119901 [120576 (119909 119911 119905)]= 1198640 (1 + 120578 1198620119863119901) 120576 (119909 119911 119905) ≜ 119878120576 (119909 119911 119905) (5)

where119901 is the order of fractional derivative 1198620119863119901[120576(119909 119911 119905)] asis defined in (2) 120578 = 11986411198640 is the material modulus ratio and120576(119909 119911 119905) is axial strain component

When the deformation of the beam is small the axialstrain 120576 and lateral displacement 119910(119909 119905) satisfy the relation-ship as follows

120576 (119909 119911 119905) = minus1199111205972119910 (119909 119905)1205971199092 (6)

Substituting (6) into (5) yields

120590 (119909 119911 119905) = 119878 [minusz1205972119910 (119909 119905)1205971199092 ] (7)

The relationship between bending moment119872(119909 119905) and axialstress 120590(119909 119911 119905) of the beam can be expressed as follows

119872(119909 119905) = intℎ2minusℎ2

119911120590 (119909 119911 119905) 119889119911 (8)

where ℎ is the thickness of the beamFrom (7) and (8) the expression of the bending moment119872 can be obtained as follows

119872(119909 119905) = 119868 [1198640 (12059721199101205971199092) + 1198641 sdot 1198620119863119901 (1205972119910

1205971199092)]

= 119868119878(12059721199101205971199092)

(9)

where 119868 = ℎ312The expression of the horizontal tension is

119873 = 11986401198602119871 int1198710[(120597119910

120597119909)2 minus (120597119910

120597119909)4]119889119909

+ 11986411198602119871 int1198710

1198620119863119901 [(120597119910

120597119909)2 minus (120597119910

120597119909)4]119889119909

= 1198602119871119878int119871

0[(120597119910

120597119909)2 minus (120597119910

120597119909)4]119889119909

(10)

Substituting (9) and (10) into system (4) system (4) can berewritten as

119868119878(12059741199101205971199094)

+ 119878 119860211987112059721199101205971199092 int

119871

0

1198620119863119901 [(120597119910

120597119909)2 minus (120597119910

120597119909)4]119889119909

+ 12058811986012059721199101205971199052 minus 119865 (119909 119905) = 0

(11)

The boundary conditions are

12059721199101205971199092 = 0 when 119909 = 0 119909 = 119871 (12)

According to the boundary conditions (12) the solution ofsystem (11) can be expressed as the Fourier series

119910 (119909 119905) = infinsum119899=1

119906119899 (119905) sin (119899120587119871 119909) (13)

4 Discrete Dynamics in Nature and Society

Assume that the initial transverse vibration of the system is1199100(119909) = 0 and the transverse load of the system satisfies thefollowing form

119865 (119909 119905) = 120588119860 sdot sin (120587119871119909) 120585 (119905) (14)

where 120585(119905) is the Gaussian white noise which satisfies119864[120585(119905)] = 0 119864[120585(119905)120585(119905 minus 120591)] = 2119863120575(120591)119863 represents the noiseintensity and 120575(120591) is the Dirac function

By the discrete format based on Galerkin method system(11) can be reduced to the fractional differential equation asfollows

+ 1198961119906 + 1205781198961 1198620119863119901119906 minus 11989631199063 minus 1205781198963119906 1198620119863119901 (1199062) + 11989651199065+ 1205781198965119906 1198620119863119901 (1199064) = 120585 (119905) (15)

where 1198620119863119901119905 119906 is the 119901 (0 lt 119901 lt 1) order Caputo derivative of119906(119905) as is defined in (2) and

1198961 = (120587119871)4 1198681198640120588119860

1198963 = (120587119871)4 11986404120588

1198965 = (120587119871)6 3119864016120588

(16)

For convenience system (15) can be represented as follows

+ 1199082119906 + 1205781199082 1198620119863119901119906 minus 11989631199063 minus 1205781198963119906 1198620119863119901 (1199062)+ 11989651199065 + 1205781198965119906 1198620119863119901 (1199064) = 120585 (119905) (17)

where119908 = radic1198961The fractional derivative term has contributions to both

damping and restoring forces [44ndash47] hence introducing thefollowing equivalent system

+ 1199082119906 + 1205781199082 [119888 (119886) + 1199082 (119886) 119906] minus 11989631199063minus 1205781198963119906 1198620119863119901 (1199062) + 11989651199065 + 1205781198965119906 1198620119863119901 (1199064)

= 120585 (119905) (18)

where 119888(119886) 1199082(119886) are the coefficients of equivalent dampingand restoring forces of fractional derivative 1198620119863119901119906 respec-tively

The error between system (17) and (18) is

119890 = 1205781199082 [119888 (119886) + 1199082 (119886) 119906 minus 1198620119863119901119906] (19)

Thenecessary conditions forminimummean square error are[48]

120597119864 [1198902]120597 (119888 (119886)) = 0120597119864 [1198902]120597 (119908 (119886)) = 0

(20)

Substituting (19) into (20) yields

119864 [119888 (119886) 2 + 1199082 (119886) 119906 minus 1198620119863119901119906]= lim119879997888rarrinfin

1119879 int1198790(119888 (119886) 2 + 1199082 (119886) 119906 minus 1198620119863119901119906) 119889119905

= 0119864 [119888 (119886) 119906 + 1199082 (119886) 1199062 minus 119906 1198620119863119901119906]

= lim119879997888rarrinfin

1119879 int1198790(119888 (119886) 119906 + 1199082 (119886) 1199062 minus 119906 1198620119863119901119906) 119889119905

= 0

(21)

Assume that the solution of system (18) has the followingform

119906 (119905) = 119886 (119905) cos120593 (119905)120593 (119905) = 119908119905 + 120579 (22)

where119908 is the natural frequency of system (17)Based on (22) we can obtain

(119905) = minus119886 (119905) 119908 sin 120593 (119905) (23)

Substituting (22) and (23) into (21) yields

lim119879997888rarrinfin

1119879 int1198790(119888 (119886) 2 + 1199082 (119886) 119906 minus 1198620119863119901119906) 119889119905

= lim119879997888rarrinfin

1119879 int1198790(119888 (119886) 11988621199082sin2120593 minus 11988621199081199082 (119886) sin 120593

sdot cos120593 minus 119886119908 sin 120593 1198620119863119901119906) 119889119905 asymp 119888 (119886) 119886211990822+ 1Γ (1 minus 119901) lim

119879997888rarrinfin

1119879 int119879119900[(119886119908 sin 120593)

sdot int1199050

(119905 minus 120591)120591120572 119889120591] 119889119905 = 119888 (119886) 119886211990822minus 1Γ (1 minus 119901) lim

119879997888rarrinfin

1119879 int119879119900(11988621199082 sin 120593

sdot int1199050

sin 120593 cos (119908120591) minus cos120593 sin (119908120591)120591120572 )119889119905 = 0

(24)

lim119879997888rarrinfin

1119879 int1198790(119888 (119886) 119906 + 1199082 (119886) 119906119906 minus 119906 1198620119863119901119906) 119889119905

= lim119879997888rarrinfin

1119879 int1198790(minus119888 (119886) 1198862119908 sin 120593 cos120593 + 11988621199082 (119886)

sdot cos2120593 minus 119886 cos120593 1198620119863119901119906) 119889119905 asymp 11988621199082 (119886)2

Discrete Dynamics in Nature and Society 5

0 1090807060504030201p

0

02

04

06

08

1

12

a

stable limit cycleunstable limit cyclenumerical result

Figure 2 Bifurcation diagram of amplitude of system (28) at119863 = 0

minus 1Γ (1 minus 119901) lim

119879997888rarrinfin

1119879 int119879119900[(119886 cos120593)

sdot int1199050

119909 (119905 minus 120591)120591120572 119889120591] 119889119905 = 11988621199082 (119886)2 + 1Γ (1 minus 119901)

sdot lim119879997888rarrinfin

1119879 int119879119900(1198862119908 cos120593

sdot int1199050

sin 120593 cos (119908120591) minus cos120593 sin (119908120591)120591120572 )119889119905 = 0

(25)

To simplify (24) and (25) further asymptotic integrals areintroduced as follows

int1199050

cos (119908120591)120591119901 119889120591 = 119908119901minus1 (Γ (1 minus 119901) sin (1199011205872 )+ sin (119908119905)

(119908119905)119901 + 119900 ((119908119905)minus119901minus1))int1199050

sin (119908120591)120591119901 119889120591 = 119908119901minus1 (Γ (1 minus 119901) cos (1199011205872 )minus cos (119908119905)

(119908119905)119901 + 119900 ((119908119905)minus119901minus1))

(26)

Substituting (26) into (24) and (25) and averaging themacross 120593 produce the ultimate forms of 119888(119886) and 1199082(119886) asfollows

119888 (119886) = 119908119901minus1 sin (1199011205872 ) 1199082 (119886) = 119908119901 cos (1199011205872 )

(27)

Therefore the equivalent system associated with system (18)can be expressed as follows

+ 11990820119906 + 120574 minus 11989631199063 minus 1205781198963119906 1198620119863119901 (1199062) + 11989651199065+ 1205781198965119906 1198620119863119901 (1199064) = 120585 (119905) (28)

where11990820 = 1199082[1+120578119908119901 cos(1199011205872)] and 120574 = 120578119908119901+1 sin(1199011205872)

Next we consider the stochastic P-bifurcation of system(28) which comprises the fractional derivatives of high ordernonlinear terms and analyze the influence of parametricvariation on the system response

3 The Stationary PDF of Amplitude

For the system (28) the material modulus ratio is given as120578 = 05 coefficients of nonlinear terms are given as 1198963 = 781198965 = 59 respectively and nature frequency is given as119908 = 1For the convenience to discuss the parametric influence thebifurcation diagram of amplitude of the limit cycle along withvariation of the fractional order 119901 is shown in Figure 2 when119863 = 0

As can be seen from Figure 2 the solution correspond-ing to the solid line is almost completely coincided withthe numerical solution which proves the correctness andaccuracy of the approximate analytical result of the deter-ministic system at the same time it shows that the solutioncorresponding to the solid line is stable and the solutioncorresponding to the dotted line is unstable And it also can beseen that there is 1 attractor in the system when 119901 changes inthe interval [0 06817] equilibrium as shown in Figure 3(a)there are 2 attractors when 119901 changes in the interval [068181] equilibrium and limit cycle as shown in Figure 3(b)

In order to obtain the stationary PDF of the amplitude ofsystem (28) the following transformation is introduced

119906 (119905) = 119886 (119905) cosΦ (119905) (119905) = minus119886 (119905) 1199080 sinΦ (119905)Φ (119905) = 1199080119905 + 120579 (119905)

(29)

where 1199080 is the natural frequency of the equivalent system(28) 119886(119905) and 120579(119905) represent the amplitude process and thephase process of the system response respectively and theyare all random processes

Substituting (29) into (28) we can obtain

119889119886119889119905 = 11986511 (119886 120579) + G11 (119886 120579) 120585 (119905)

6 Discrete Dynamics in Nature and Society

minus1

minus05

0

05

1

dxd

t

p=04

0 05 1minus05minus1x

(a) Equilibrium

minus1

minus05

0

05

1

dxd

t

p=08

0 05 1minus05minus1x

(b) Equilibrium and limit cycle

Figure 3 Phase diagrams of system (28) at119863 = 0

119889120579119889119905 = 11986521 (119886 120579) + 11986621 (119886 120579) 120585 (119905) (30)

in which

11986511 = minusΔ sinΦ119908011986521 = minusΔ cosΦ119886119908011986611 = minus sinΦ119908011986621 = minuscosΦ1198861199080

(31)

and

Δ = 120578119908119901+11198861199080 sin (1199011205872 ) sinΦ minus 11989631198863cos3Φ minus 1205781198963sdot 11988632 (21199080)119901 cosΦ cos (2Φ + 119901120587

2 ) minus 11989651198865cos5Φminus 12057811989651198865 cosΦ[18 (41199080)119901 cos (4Φ + 119901120587

2 )+ 12 (21199080)119901 cos (2Φ + 119901120587

2 )]

(32)

Equation (30) can be regarded as a Stratonovich stochasticdifferential equation by adding the corresponding Wong-

Zakai correction term it can be transformed into thefollowing Ito stochastic differential equation

119889119886 = [11986511 (119886 120579) + 11986512 (119886 120579)] 119889119905+ radic2119863G11 (119886 120579) 119889119861 (119905)

119889120579 = [11986521 (119886 120579) + 11986522 (119886 120579)] 119889119905+ radic211986311986621 (119886 120579) 119889119861 (119905)

(33)

where 119861(119905) is a standard Wiener process and

11986512 (119886 120579) = 11986312059711986611120597119886 11986611 + 11986312059711986611120597120579 1198662111986522 (119886 120579) = 11986312059711986621120597119886 11986611 + 11986312059711986621120597120579 11986621

(34)

By the stochastic averaging method averaging (33) regardingΦ the following averaged Ito equation can be obtained asfollows

119889119886 = 1198981 (119886) 119889119905 + 1205901 (119886) 119889119861 (119905)119889120579 = 1198982 (119886) 119889119905 + 1205902 (119886) 119889119861 (119905) (35)

where

1198981 (119886) = minus12057811989652119901119908119901minus10 1198865 sin (1199011205872)8

minus 12057811989632119901119908119901minus10 1198863 sin (1199011205872)8

minus 120578119908119901+1119886 sin (1199011205872)2 + 119863211988611990820

12059012 (119886) = 119863119908201198982 (119886) = 1198965 (2120578 sdot 21199011199081199010 cos (1199011205872) + 5) 1198864

161199080

Discrete Dynamics in Nature and Society 7

+ 1198963 (120578 sdot 21199011199081199010 cos (1199011205872) + 3) 119886281199080

12059022 (119886) = 119863119886211990820 (36)

Equations (35) and (36) show that the averaged Ito equationfor 119886(119905) is independent of 120579(119905) and the random process 119886(119905) isa one-dimensional diffusion processThe corresponding FPKequation associated with 119886(119905) can be written as

120597119901120597119905 = minus 120597120597119886 [1198981 (119886) 119901] + 12 12059721205971198862 [12059012 (119886) 119901] (37)

The boundary conditions are

119901 = 119888 119888 isin (minusinfin +infin) when 119886 = 0

119901 997888rarr 0 120597119901120597119886 997888rarr 0 when 119886 997888rarr infin

(38)

Thus based on these boundary conditions (38) we can obtainthe stationary PDF of amplitude 119886 as follows

119901 (119886) = 11986212059012 (119886) exp [int119886

0

21198981 (119906)12059012 (119906) 119889119906] (39)

where C is a normalized constant which satisfies

119862 = [intinfin0

( 112059012 (119886) exp [int119886

0

21198981 (119906)12059012 (119906) 119889119906])119889119886]minus1 (40)

Substituting (36) into (39) the explicit expression for thestationary PDF of amplitude 119886 can be obtained as follows

119901 (119886) = 11986211988611990820119863 exp[[minus1205781198862 sin (1199011205872) (11989652119901+11198864119908119901+10 + 3119896321199011198862119908119901+10 + 2411990820119908119901+1)48119863 ]

] (41)

where11990820 = 1199082[1 + 120578119908119901 cos(1199011205872)]4 Stochastic P-Bifurcation

Stochastic P-bifurcation refers to the changes of the numberof peaks in the PDF curve in order to obtain the critical para-metric condition for stochastic P-bifurcation the influencesof the parameters for stochastic P-bifurcation of the systemare analyzed by using the singularity theory below

For the sake of convenience we can write 119901(119886) as follows119901 (119886) = 119862119877 (119886119863 119908 120578 119901 1198963 1198965)

sdot exp [119876 (119886119863 119908 120578 119901 1198963 1198965)] (42)

in which

119877 (119886119863119908 120578 119901 1198963 1198965) = 11988611990820119863119876(119886119863119908 120578 119901 1198963 1198965)= minus1205781198862 sin (1199011205872) (11989652119901+11198864119908119901+10 + 3119896321199011198862119908119901+10 + 2411990820119908119901+1)48119863

(43)

According to the singularity theory the stationary PDF needsto satisfy the following two conditions

120597119901 (119886)120597119886 = 0

1205972119901 (119886)1205971198862 = 0

(44)

Substituting (42) into (44) we can obtain the followingcondition [49]

119867= 1198771015840 + 1198771198761015840 = 0 11987710158401015840 + 211987710158401198761015840 + 11987711987610158401015840 + 11987711987610158402 = 0 (45)

where 119867 represents the critical condition for the changes ofthe number of peaks in the PDF curve

Substituting (43) into (45) we can get the critical para-metric condition for stochastic P-bifurcation of the system asfollows

119863 = 1812057811989652119901+1119908119901+10 sin (1199011205872 ) 1198866

+ 1812057811989632119901+1119908119901+10 sin (1199011205872 ) 1198864+ 12057811990820119908119901+1 sin (1199011205872 ) 1198862

(46)

where11990820 = 1199082[1 + 120578119908119901 cos(1199011205872)]And amplitude 119886 satisfies311989652119901119908119901+10 1198864 + 211989632119901119908119901+10 1198862 + 411990820119908119901+1 = 0 (47)

Taking the parameters as 120578 = 05 1198963 = 78 1198965 = 59and 119908 = 1 according to (46) and (47) the transition setfor stochastic P-bifurcation of the system with the unfoldingparameters 119901 and 119863 can be obtained as shown in Figure 4

According to the singularity theory the topological struc-tures of the stationary PDF curves of different points (119901119863)in the same region are qualitatively the same Taking a point(119901119863) in each region all varieties of the stationary PDFcurves which are qualitatively different could be obtainedIt can be seen that the unfolding parameter 119901 minus 119863 planeis divided into two subregions by the transition set curve

8 Discrete Dynamics in Nature and Society

1 Monostable region

2 Bistable region

0001002003004005006007008009

01

D01 02 03 04 05 06 07 08 09 10

p

Figure 4 The transition set curve of system (28)

p=03D=01

Monte Carlo simulationAnalytical solution

0

02

04

06

08

1

12

14

p(a)

02 04 06 08 1 12 14 16 18 20a

(a) parameter (119901119863) in region 1 of Figure 4

p=07D=003

Monte Carlo simulationAnalytical solution

0

02

04

06

08

1

12

14

p(a)

02 04 06 08 1 12 14 16 18 20a

(b) parameter (119901119863) in region 2 of Figure 4

Figure 5 The stationary PDF curves for the amplitude of system (28)

for convenience each region in Figure 4 is marked with anumber

Taking a given point (119901119863) in each of the two subregionsof Figure 4 the characteristics of stationary PDF curvesare analyzed and the corresponding results are shown inFigure 5

As can be seen from Figure 4 the parametric regionwhere the PDF curve appears bimodal is surrounded by anapproximately triangular region When the parameter (119901119863)is taken in the region 1 the PDF curve has only a distinctpeak as shown in Figure 5(a) in region 2 the PDF curvehas a distinct peak near the origin but the probability isobviously not zero far away from the origin there are boththe equilibrium and limit cycle in the system simultaneouslyas shown in Figure 5(b)

The analysis results above show that the stationaryPDF curves of the system amplitude in any two adjacentregions in Figure 4 are qualitatively different No matterthe values of the unfolding parameters cross any line inthe figure the system will occur stochastic P-bifurcationbehaviors so the transition set curve is just the criticalparametric condition for the stochastic P-bifurcation of thesystem and the analytic results in Figure 5 are in goodagreement with the numerical results by Monte Carlo simu-lation which further verify the correctness of the theoreticalanalysis

5 Conclusion

In this paper we studied the stochastic P-bifurcation foraxially moving of a bistable viscoelastic beam model withfractional derivatives of high order nonlinear terms underGaussian white noise excitation According to the minimummean square error principle we transformed the originalsystem into an equivalent and simplified system and obtainedthe stationary PDF of the system amplitude using stochasticaveraging In addition we obtained the critical parametriccondition for stochastic P-bifurcation of the system usingsingularity theory based on this the system response canbe maintained at the monostability or small amplitudenear the equilibrium by selecting the appropriate unfoldingparameters providing theoretical guidance for system designin practical engineering and avoiding the instability anddamage caused by the large amplitude vibration or nonlinearjump phenomenon of the system Finally the numericalresults by Monte Carlo simulation of the original systemalso verify the theoretical results obtained in this paper Weconclude that the order 119901 of fractional derivative and thenoise intensity 119863 can both cause stochastic P-bifurcation ofthe system and the number of peaks in the stationary PDFcurve of system amplitude can change from 2 to 1 by selectingthe appropriate unfolding parameters (119901119863) it also showsthat the method used in this paper is feasible to analyze

Discrete Dynamics in Nature and Society 9

the stochastic P-bifurcation behaviors of viscoelastic materialsystem with fractional constitutive relation

Data Availability

Data that support the findings of this study are includedwithin the article (and its additional files) All other dataare available from the corresponding author upon reasonablerequest

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work was supported by the National 973 Project ofChina (Grant no 2014CB046805) and the Natural ScienceFoundation of China (Grant no 11372211)

References

[1] W Tan and M Xu ldquoPlane surface suddenly set in motion in aviscoelastic fluid with fractionalMaxwell modelrdquoActaMechan-ica Sinica English Series The Chinese Society of Theoretical andApplied Mechanics vol 18 no 4 pp 342ndash349 2002

[2] J Sabatier O P Agrawal and J A Machado Advance in Frac-tional Calculus Theoretical Developments and Applications inPhysics and Engineering Springer DordrechtThe Netherlands2007

[3] I Podlubny ldquoFractional order systems andPIDcontrollerrdquo IEEETransactions on Automatic Control vol 44 no 1 pp 208ndash2141999

[4] C A Monje Y Chen B M Vinagre D Xue and V FeliuFractional-Order Systems and Controls Fundamentals andApplications Advances in Industrial Control Springer LondonUK 2010

[5] Q Li Y L Zhou X Q Zhao and X Y Ge ldquoFractional orderstochastic differential equation with application in Europeanoption pricingrdquo Discrete Dynamics in Nature and Society vol2014 Article ID 621895 12 pages 2014

[6] S Nie H Sun Y Zhang et al ldquoVertical distribution ofsuspended sediment under steady flow existing theories andfractional derivative modelrdquo Discrete Dynamics in Nature andSociety vol 2017 Article ID 5481531 11 pages 2017

[7] X Liu L Hong H Dang and L Yang ldquoBifurcations andsynchronization of the fractional-order Bloch systemrdquo DiscreteDynamics in Nature and Society vol 2017 Article ID 469430510 pages 2017

[8] W Li and Y Pang ldquoAsymptotic solutions of time-space frac-tional coupled systems by residual power series methodrdquoDiscrete Dynamics in Nature and Society vol 2017 Article ID7695924 10 pages 2017

[9] Z Liu T Xia and JWang ldquoImage encryption technology basedon fractional two-dimensional triangle function combinationdiscrete chaotic map coupled with Menezes-Vanstone ellipticcurve cryptosystemrdquo Discrete Dynamics in Nature and Societyvol 2018 Article ID 4585083 24 pages 2018

[10] C Sanchez-Lopez V H Carbajal-Gomez M A Carrasco-Aguilar and ICarro-Perez ldquoFractional-OrderMemristor Emu-lator Circuitsrdquo Complexity vol 2018 Article ID 2806976 10pages 2018

[11] K Rajagopal A Karthikeyan P Duraisamy and R Weldegior-gis ldquoBifurcation and Chaos in Integer and Fractional OrderTwo-Degree-of-Freedom Shape Memory Alloy OscillatorsrdquoComplexity vol 2018 9 pages 2018

[12] Ying Li and Ye Tang ldquoAnalytical Analysis on Nonlinear Para-metric Vibration of an Axially Moving String with FractionalViscoelastic Dampingrdquo Mathematical Problems in Engineeringvol 2017 Article ID 1393954 9 pages 2017

[13] G Liu C Ye X Liang Z Xie and Z Yu ldquoNonlinear DynamicIdentification of Beams Resting on Nonlinear ViscoelasticFoundations BASed on the Time-DELayed TRAnsfer Entropyand IMProved SurrogateDATaAlgorithmrdquoMathematical Prob-lems in Engineering vol 2018 Article ID 6531051 14 pages 2018

[14] D Liu Y Wu and X Xie ldquoStochastic Stability Analysis of Cou-pled Viscoelastic SYStems with Nonviscously Damping DRIvenby White NOIserdquo Mathematical Problems in Engineering vol2018 Article ID 6532305 16 pages 2018

[15] P G Nutting ldquoAnew general law of deformationrdquo Journal ofTheFranklin Institute vol 191 no 5 pp 679ndash685 1921

[16] A Gemant ldquoA method of analyzing experimental resultsobtained fromelasto-viscous bodiesrdquo Journal of Applied Physicsvol 7 no 8 pp 311ndash317 1936

[17] G S Blair and J Caffyn ldquoVI An application of the theoryof quasi-properties to the treatment of anomalous strain-stressrelationsrdquo The London Edinburgh and Dublin PhilosophicalMagazine and Journal of Science vol 40 no 300 pp 80ndash942010

[18] CMote ldquoDynamic Stability of Axially MovingMateralsrdquo Shockand Vibration vol 4 no 4 pp 2ndash11 1972

[19] AUlsoy andCMote ldquoBand SawVibration and Stabilityrdquo Shockand Vibration vol 10 no 11 pp 3ndash15 1978

[20] J A Wickert and C D Mote Jr ldquoCurrent research on thevibration and stability of axially-moving materialsrdquo Shock andVibration vol 20 no 1 pp 3ndash13 1988

[21] K W Wang and S P Liu ldquoOn the noise and vibration of chaindrive systemsrdquo Shock and Vibration Digest vol 23 no 4 pp 8ndash13 1991

[22] F Pellicano and F Vestroni ldquoComplex dynamics of high-speedaxially moving systemsrdquo Journal of Sound and Vibration vol258 no 1 pp 31ndash44 2002

[23] N-H Zhang and L-Q Chen ldquoNonlinear dynamical analysis ofaxially moving viscoelastic stringsrdquo Chaos Solitons amp Fractalsvol 24 no 4 pp 1065ndash1074 2005

[24] M H Ghayesh H A Kafiabad and T Reid ldquoSub- and super-critical nonlinear dynamics of a harmonically excited axiallymoving beamrdquo International Journal of Solids and Structuresvol 49 no 1 pp 227ndash243 2012

[25] M H Ghayesh ldquoStability and bifurcations of an axially movingbeam with an intermediate spring supportrdquo Nonlinear Dynam-ics vol 69 no 1-2 pp 193ndash210 2012

[26] R Rodrıguez E Salinas-Rodrıguez and J Fujioka ldquoFractionalTime Fluctuations in Viscoelasticity A Comparative Study ofCorrelations and Elastic Modulirdquo Entropy vol 20 no 1 p 282018

[27] R L Bagley and P J Torvik ldquoFractional calculus-a differentapproach to the analysis of viscoelastically damped structuresrdquoAIAA Journal vol 21 no 5 pp 741ndash748 2012

10 Discrete Dynamics in Nature and Society

[28] R L Bagley and P J Torvik ldquoFractional calculus in the transientanalysis of viscoelastically damped structuresrdquo AIAA Journalvol 23 no 6 pp 918ndash925 1985

[29] M Pakdemirli and A G Ulsoy ldquoStability analysis of an axiallyaccelerating stringrdquo Journal of Sound andVibration vol 203 no5 pp 815ndash832 1997

[30] L Zhang and J W Zu ldquoNonlinear vibration of parametricallyexcited moving belts part I Dynamic responserdquo Journal ofApplied Mechanics vol 66 no 2 pp 396ndash402 1999

[31] L Zhang and J W Zu ldquoNonlinear vibration of parametricallyexcited viscoelastic moving belts part II Stability analysisrdquoJournal of Applied Mechanics vol 66 no 2 pp 403ndash409 1999

[32] L Q Chen X D Yang and C J Cheng ldquoDynamic stability ofan axially accelerating viscoelastic beamrdquo European Journal ofMechanic vol 23 no 4 pp 659ndash666 2004

[33] L Q Chen and X D Yang ldquoSteady-state response of axiallymoving viscoelastic beams with pulsating speed comparisonof two nonlinear modelsrdquo International Journal of Solids andStructures vol 42 no 1 pp 37ndash50 2005

[34] L Q Chen and X D Yang ldquoTransverse nonlinear dynamicsof axially accelerating viscoelastic beams based on 4-termGalerkin truncationrdquo Chaos Soliton and Fractal vol 27 no 3pp 748ndash757 2006

[35] H Ding and L-Q Chen ldquoGalerkin methods for natural fre-quencies of high-speed axially moving beamsrdquo Journal of Soundand Vibration vol 329 no 17 pp 3484ndash3494 2010

[36] L-Q Chen J Wu and J W Zu ldquoAsymptotic nonlinear behav-iors in transverse vibration of an axially acceleratingviscoelasticstringrdquo Nonlinear Dynamics vol 35 no 4 pp 347ndash360 2004

[37] L-Q Chen N-H Zhang and J W Zu ldquoThe regular andchaotic vibrations of an axially moving viscoelastic string basedon fourth order Galerkin truncationrdquo Journal of Sound andVibration vol 261 no 4 pp 764ndash773 2003

[38] A Y T Leung H X Yang and J Y Chen ldquoParametricbifurcation of a viscoelastic column subject to axial harmonicforce and time-delayed controlrdquo Computers amp Structures vol136 pp 47ndash55 2014

[39] MH Ghayesh andNMoradian ldquoNonlinear dynamic responseof axially moving stretched viscoelastic stringsrdquo Archive ofApplied Mechanics vol 81 no 6 pp 781ndash799 2011

[40] D Liu W Xu and Y Xu ldquoDynamic responses of axially movingviscoelastic beamunder a randomly disordered periodic excita-tionrdquo Journal of Sound and Vibration vol 331 no 17 pp 4045ndash4056 2012

[41] T Z Yang and B Fang ldquoStability in parametric resonance of anaxially moving beam constituted by fractional order materialrdquoArchive of AppliedMechanics vol 82 no 12 pp 1763ndash1770 2012

[42] A Y T Leung H X Yang P Zhu and Z J Guo ldquoSteady stateresponse of fractionally damped nonlinear viscoelastic archesby residue harmonic homotopyrdquo Computers amp Structures vol121 pp 10ndash21 2013

[43] A C Galucio J-F Deui and R Ohayon ldquoFinite elementformulation of viscoelastic sandwich beams using fractionalderivative operatorsrdquo Computational Mechanics vol 33 no 4pp 282ndash291 2004

[44] L Chen W Wang Z Li and W Zhu ldquoStationary responseof Duffing oscillator with hardening stiffness and fractionalderivativerdquo International Journal of Non-Linear Mechanics vol48 pp 44ndash50 2013

[45] L C Chen Z S Li Q Zhuang and W Q Zhu ldquoFirst-passagefailure of single-degree-of-freedom nonlinear oscillators with

fractional derivativerdquo Journal of Vibration and Control vol 19no 14 pp 2154ndash2163 2013

[46] Y Shen S Yang H Xing and G Gao ldquoPrimary resonance ofDuffing oscillator with fractional-order derivativerdquo Communi-cations in Nonlinear Science and Numerical Simulation vol 17no 7 pp 3092ndash3100 2012

[47] Y G Yang W Xu Y H Sun and X D Gu ldquoStochastic responseof van der Pol oscillator with two kinds of fractional derivativesunder Gaussian white noise excitationrdquo Chinese Physics B vol25 no 2 pp 13ndash21 2016

[48] L Chen and W Zhu ldquoStochastic response of fractional-ordervan der Pol oscillatorrdquoTheoretical amp Applied Mechanics Lettersvol 4 no 1 pp 68ndash72 2014

[49] W Li M Zhang and J Zhao ldquoStochastic bifurcations of gen-eralized Duffingndashvan der Pol system with fractional derivativeunder colored noiserdquoChinese Physics B vol 26 no 9 pp 62ndash692017

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Submit your manuscripts atwwwhindawicom

Discrete Dynamics in Nature and Society 3

y

O

F (x t) =Asin(xL)(t)

y (x t)

L

M

N

X

Q

Fdx

M+ M

xdx

N+ N

xdx

Q+ Q

xdx

Figure 1 Schematic stress diagram of viscoelastic beam

where 120588 is the mass density of the beam119860 is the area of cross-section119872 is the bending moment Q is the lateral force and119873 is the horizontal force From (3) we have

12059721198721205971199092 + 11987312059721199101205971199092 + 1205881198601205971199102

1205971199052 minus 119865 (119909 119905) = 0 (4)

Assuming the material of the beam obeys a fractional deriva-tive viscoelastic constitutive relation

120590 (119909 119911 119905) = 1198640120576 (119909 119911 119905) + 1198641 1198620119863119901 [120576 (119909 119911 119905)]= 1198640 (1 + 120578 1198620119863119901) 120576 (119909 119911 119905) ≜ 119878120576 (119909 119911 119905) (5)

where119901 is the order of fractional derivative 1198620119863119901[120576(119909 119911 119905)] asis defined in (2) 120578 = 11986411198640 is the material modulus ratio and120576(119909 119911 119905) is axial strain component

When the deformation of the beam is small the axialstrain 120576 and lateral displacement 119910(119909 119905) satisfy the relation-ship as follows

120576 (119909 119911 119905) = minus1199111205972119910 (119909 119905)1205971199092 (6)

Substituting (6) into (5) yields

120590 (119909 119911 119905) = 119878 [minusz1205972119910 (119909 119905)1205971199092 ] (7)

The relationship between bending moment119872(119909 119905) and axialstress 120590(119909 119911 119905) of the beam can be expressed as follows

119872(119909 119905) = intℎ2minusℎ2

119911120590 (119909 119911 119905) 119889119911 (8)

where ℎ is the thickness of the beamFrom (7) and (8) the expression of the bending moment119872 can be obtained as follows

119872(119909 119905) = 119868 [1198640 (12059721199101205971199092) + 1198641 sdot 1198620119863119901 (1205972119910

1205971199092)]

= 119868119878(12059721199101205971199092)

(9)

where 119868 = ℎ312The expression of the horizontal tension is

119873 = 11986401198602119871 int1198710[(120597119910

120597119909)2 minus (120597119910

120597119909)4]119889119909

+ 11986411198602119871 int1198710

1198620119863119901 [(120597119910

120597119909)2 minus (120597119910

120597119909)4]119889119909

= 1198602119871119878int119871

0[(120597119910

120597119909)2 minus (120597119910

120597119909)4]119889119909

(10)

Substituting (9) and (10) into system (4) system (4) can berewritten as

119868119878(12059741199101205971199094)

+ 119878 119860211987112059721199101205971199092 int

119871

0

1198620119863119901 [(120597119910

120597119909)2 minus (120597119910

120597119909)4]119889119909

+ 12058811986012059721199101205971199052 minus 119865 (119909 119905) = 0

(11)

The boundary conditions are

12059721199101205971199092 = 0 when 119909 = 0 119909 = 119871 (12)

According to the boundary conditions (12) the solution ofsystem (11) can be expressed as the Fourier series

119910 (119909 119905) = infinsum119899=1

119906119899 (119905) sin (119899120587119871 119909) (13)

4 Discrete Dynamics in Nature and Society

Assume that the initial transverse vibration of the system is1199100(119909) = 0 and the transverse load of the system satisfies thefollowing form

119865 (119909 119905) = 120588119860 sdot sin (120587119871119909) 120585 (119905) (14)

where 120585(119905) is the Gaussian white noise which satisfies119864[120585(119905)] = 0 119864[120585(119905)120585(119905 minus 120591)] = 2119863120575(120591)119863 represents the noiseintensity and 120575(120591) is the Dirac function

By the discrete format based on Galerkin method system(11) can be reduced to the fractional differential equation asfollows

+ 1198961119906 + 1205781198961 1198620119863119901119906 minus 11989631199063 minus 1205781198963119906 1198620119863119901 (1199062) + 11989651199065+ 1205781198965119906 1198620119863119901 (1199064) = 120585 (119905) (15)

where 1198620119863119901119905 119906 is the 119901 (0 lt 119901 lt 1) order Caputo derivative of119906(119905) as is defined in (2) and

1198961 = (120587119871)4 1198681198640120588119860

1198963 = (120587119871)4 11986404120588

1198965 = (120587119871)6 3119864016120588

(16)

For convenience system (15) can be represented as follows

+ 1199082119906 + 1205781199082 1198620119863119901119906 minus 11989631199063 minus 1205781198963119906 1198620119863119901 (1199062)+ 11989651199065 + 1205781198965119906 1198620119863119901 (1199064) = 120585 (119905) (17)

where119908 = radic1198961The fractional derivative term has contributions to both

damping and restoring forces [44ndash47] hence introducing thefollowing equivalent system

+ 1199082119906 + 1205781199082 [119888 (119886) + 1199082 (119886) 119906] minus 11989631199063minus 1205781198963119906 1198620119863119901 (1199062) + 11989651199065 + 1205781198965119906 1198620119863119901 (1199064)

= 120585 (119905) (18)

where 119888(119886) 1199082(119886) are the coefficients of equivalent dampingand restoring forces of fractional derivative 1198620119863119901119906 respec-tively

The error between system (17) and (18) is

119890 = 1205781199082 [119888 (119886) + 1199082 (119886) 119906 minus 1198620119863119901119906] (19)

Thenecessary conditions forminimummean square error are[48]

120597119864 [1198902]120597 (119888 (119886)) = 0120597119864 [1198902]120597 (119908 (119886)) = 0

(20)

Substituting (19) into (20) yields

119864 [119888 (119886) 2 + 1199082 (119886) 119906 minus 1198620119863119901119906]= lim119879997888rarrinfin

1119879 int1198790(119888 (119886) 2 + 1199082 (119886) 119906 minus 1198620119863119901119906) 119889119905

= 0119864 [119888 (119886) 119906 + 1199082 (119886) 1199062 minus 119906 1198620119863119901119906]

= lim119879997888rarrinfin

1119879 int1198790(119888 (119886) 119906 + 1199082 (119886) 1199062 minus 119906 1198620119863119901119906) 119889119905

= 0

(21)

Assume that the solution of system (18) has the followingform

119906 (119905) = 119886 (119905) cos120593 (119905)120593 (119905) = 119908119905 + 120579 (22)

where119908 is the natural frequency of system (17)Based on (22) we can obtain

(119905) = minus119886 (119905) 119908 sin 120593 (119905) (23)

Substituting (22) and (23) into (21) yields

lim119879997888rarrinfin

1119879 int1198790(119888 (119886) 2 + 1199082 (119886) 119906 minus 1198620119863119901119906) 119889119905

= lim119879997888rarrinfin

1119879 int1198790(119888 (119886) 11988621199082sin2120593 minus 11988621199081199082 (119886) sin 120593

sdot cos120593 minus 119886119908 sin 120593 1198620119863119901119906) 119889119905 asymp 119888 (119886) 119886211990822+ 1Γ (1 minus 119901) lim

119879997888rarrinfin

1119879 int119879119900[(119886119908 sin 120593)

sdot int1199050

(119905 minus 120591)120591120572 119889120591] 119889119905 = 119888 (119886) 119886211990822minus 1Γ (1 minus 119901) lim

119879997888rarrinfin

1119879 int119879119900(11988621199082 sin 120593

sdot int1199050

sin 120593 cos (119908120591) minus cos120593 sin (119908120591)120591120572 )119889119905 = 0

(24)

lim119879997888rarrinfin

1119879 int1198790(119888 (119886) 119906 + 1199082 (119886) 119906119906 minus 119906 1198620119863119901119906) 119889119905

= lim119879997888rarrinfin

1119879 int1198790(minus119888 (119886) 1198862119908 sin 120593 cos120593 + 11988621199082 (119886)

sdot cos2120593 minus 119886 cos120593 1198620119863119901119906) 119889119905 asymp 11988621199082 (119886)2

Discrete Dynamics in Nature and Society 5

0 1090807060504030201p

0

02

04

06

08

1

12

a

stable limit cycleunstable limit cyclenumerical result

Figure 2 Bifurcation diagram of amplitude of system (28) at119863 = 0

minus 1Γ (1 minus 119901) lim

119879997888rarrinfin

1119879 int119879119900[(119886 cos120593)

sdot int1199050

119909 (119905 minus 120591)120591120572 119889120591] 119889119905 = 11988621199082 (119886)2 + 1Γ (1 minus 119901)

sdot lim119879997888rarrinfin

1119879 int119879119900(1198862119908 cos120593

sdot int1199050

sin 120593 cos (119908120591) minus cos120593 sin (119908120591)120591120572 )119889119905 = 0

(25)

To simplify (24) and (25) further asymptotic integrals areintroduced as follows

int1199050

cos (119908120591)120591119901 119889120591 = 119908119901minus1 (Γ (1 minus 119901) sin (1199011205872 )+ sin (119908119905)

(119908119905)119901 + 119900 ((119908119905)minus119901minus1))int1199050

sin (119908120591)120591119901 119889120591 = 119908119901minus1 (Γ (1 minus 119901) cos (1199011205872 )minus cos (119908119905)

(119908119905)119901 + 119900 ((119908119905)minus119901minus1))

(26)

Substituting (26) into (24) and (25) and averaging themacross 120593 produce the ultimate forms of 119888(119886) and 1199082(119886) asfollows

119888 (119886) = 119908119901minus1 sin (1199011205872 ) 1199082 (119886) = 119908119901 cos (1199011205872 )

(27)

Therefore the equivalent system associated with system (18)can be expressed as follows

+ 11990820119906 + 120574 minus 11989631199063 minus 1205781198963119906 1198620119863119901 (1199062) + 11989651199065+ 1205781198965119906 1198620119863119901 (1199064) = 120585 (119905) (28)

where11990820 = 1199082[1+120578119908119901 cos(1199011205872)] and 120574 = 120578119908119901+1 sin(1199011205872)

Next we consider the stochastic P-bifurcation of system(28) which comprises the fractional derivatives of high ordernonlinear terms and analyze the influence of parametricvariation on the system response

3 The Stationary PDF of Amplitude

For the system (28) the material modulus ratio is given as120578 = 05 coefficients of nonlinear terms are given as 1198963 = 781198965 = 59 respectively and nature frequency is given as119908 = 1For the convenience to discuss the parametric influence thebifurcation diagram of amplitude of the limit cycle along withvariation of the fractional order 119901 is shown in Figure 2 when119863 = 0

As can be seen from Figure 2 the solution correspond-ing to the solid line is almost completely coincided withthe numerical solution which proves the correctness andaccuracy of the approximate analytical result of the deter-ministic system at the same time it shows that the solutioncorresponding to the solid line is stable and the solutioncorresponding to the dotted line is unstable And it also can beseen that there is 1 attractor in the system when 119901 changes inthe interval [0 06817] equilibrium as shown in Figure 3(a)there are 2 attractors when 119901 changes in the interval [068181] equilibrium and limit cycle as shown in Figure 3(b)

In order to obtain the stationary PDF of the amplitude ofsystem (28) the following transformation is introduced

119906 (119905) = 119886 (119905) cosΦ (119905) (119905) = minus119886 (119905) 1199080 sinΦ (119905)Φ (119905) = 1199080119905 + 120579 (119905)

(29)

where 1199080 is the natural frequency of the equivalent system(28) 119886(119905) and 120579(119905) represent the amplitude process and thephase process of the system response respectively and theyare all random processes

Substituting (29) into (28) we can obtain

119889119886119889119905 = 11986511 (119886 120579) + G11 (119886 120579) 120585 (119905)

6 Discrete Dynamics in Nature and Society

minus1

minus05

0

05

1

dxd

t

p=04

0 05 1minus05minus1x

(a) Equilibrium

minus1

minus05

0

05

1

dxd

t

p=08

0 05 1minus05minus1x

(b) Equilibrium and limit cycle

Figure 3 Phase diagrams of system (28) at119863 = 0

119889120579119889119905 = 11986521 (119886 120579) + 11986621 (119886 120579) 120585 (119905) (30)

in which

11986511 = minusΔ sinΦ119908011986521 = minusΔ cosΦ119886119908011986611 = minus sinΦ119908011986621 = minuscosΦ1198861199080

(31)

and

Δ = 120578119908119901+11198861199080 sin (1199011205872 ) sinΦ minus 11989631198863cos3Φ minus 1205781198963sdot 11988632 (21199080)119901 cosΦ cos (2Φ + 119901120587

2 ) minus 11989651198865cos5Φminus 12057811989651198865 cosΦ[18 (41199080)119901 cos (4Φ + 119901120587

2 )+ 12 (21199080)119901 cos (2Φ + 119901120587

2 )]

(32)

Equation (30) can be regarded as a Stratonovich stochasticdifferential equation by adding the corresponding Wong-

Zakai correction term it can be transformed into thefollowing Ito stochastic differential equation

119889119886 = [11986511 (119886 120579) + 11986512 (119886 120579)] 119889119905+ radic2119863G11 (119886 120579) 119889119861 (119905)

119889120579 = [11986521 (119886 120579) + 11986522 (119886 120579)] 119889119905+ radic211986311986621 (119886 120579) 119889119861 (119905)

(33)

where 119861(119905) is a standard Wiener process and

11986512 (119886 120579) = 11986312059711986611120597119886 11986611 + 11986312059711986611120597120579 1198662111986522 (119886 120579) = 11986312059711986621120597119886 11986611 + 11986312059711986621120597120579 11986621

(34)

By the stochastic averaging method averaging (33) regardingΦ the following averaged Ito equation can be obtained asfollows

119889119886 = 1198981 (119886) 119889119905 + 1205901 (119886) 119889119861 (119905)119889120579 = 1198982 (119886) 119889119905 + 1205902 (119886) 119889119861 (119905) (35)

where

1198981 (119886) = minus12057811989652119901119908119901minus10 1198865 sin (1199011205872)8

minus 12057811989632119901119908119901minus10 1198863 sin (1199011205872)8

minus 120578119908119901+1119886 sin (1199011205872)2 + 119863211988611990820

12059012 (119886) = 119863119908201198982 (119886) = 1198965 (2120578 sdot 21199011199081199010 cos (1199011205872) + 5) 1198864

161199080

Discrete Dynamics in Nature and Society 7

+ 1198963 (120578 sdot 21199011199081199010 cos (1199011205872) + 3) 119886281199080

12059022 (119886) = 119863119886211990820 (36)

Equations (35) and (36) show that the averaged Ito equationfor 119886(119905) is independent of 120579(119905) and the random process 119886(119905) isa one-dimensional diffusion processThe corresponding FPKequation associated with 119886(119905) can be written as

120597119901120597119905 = minus 120597120597119886 [1198981 (119886) 119901] + 12 12059721205971198862 [12059012 (119886) 119901] (37)

The boundary conditions are

119901 = 119888 119888 isin (minusinfin +infin) when 119886 = 0

119901 997888rarr 0 120597119901120597119886 997888rarr 0 when 119886 997888rarr infin

(38)

Thus based on these boundary conditions (38) we can obtainthe stationary PDF of amplitude 119886 as follows

119901 (119886) = 11986212059012 (119886) exp [int119886

0

21198981 (119906)12059012 (119906) 119889119906] (39)

where C is a normalized constant which satisfies

119862 = [intinfin0

( 112059012 (119886) exp [int119886

0

21198981 (119906)12059012 (119906) 119889119906])119889119886]minus1 (40)

Substituting (36) into (39) the explicit expression for thestationary PDF of amplitude 119886 can be obtained as follows

119901 (119886) = 11986211988611990820119863 exp[[minus1205781198862 sin (1199011205872) (11989652119901+11198864119908119901+10 + 3119896321199011198862119908119901+10 + 2411990820119908119901+1)48119863 ]

] (41)

where11990820 = 1199082[1 + 120578119908119901 cos(1199011205872)]4 Stochastic P-Bifurcation

Stochastic P-bifurcation refers to the changes of the numberof peaks in the PDF curve in order to obtain the critical para-metric condition for stochastic P-bifurcation the influencesof the parameters for stochastic P-bifurcation of the systemare analyzed by using the singularity theory below

For the sake of convenience we can write 119901(119886) as follows119901 (119886) = 119862119877 (119886119863 119908 120578 119901 1198963 1198965)

sdot exp [119876 (119886119863 119908 120578 119901 1198963 1198965)] (42)

in which

119877 (119886119863119908 120578 119901 1198963 1198965) = 11988611990820119863119876(119886119863119908 120578 119901 1198963 1198965)= minus1205781198862 sin (1199011205872) (11989652119901+11198864119908119901+10 + 3119896321199011198862119908119901+10 + 2411990820119908119901+1)48119863

(43)

According to the singularity theory the stationary PDF needsto satisfy the following two conditions

120597119901 (119886)120597119886 = 0

1205972119901 (119886)1205971198862 = 0

(44)

Substituting (42) into (44) we can obtain the followingcondition [49]

119867= 1198771015840 + 1198771198761015840 = 0 11987710158401015840 + 211987710158401198761015840 + 11987711987610158401015840 + 11987711987610158402 = 0 (45)

where 119867 represents the critical condition for the changes ofthe number of peaks in the PDF curve

Substituting (43) into (45) we can get the critical para-metric condition for stochastic P-bifurcation of the system asfollows

119863 = 1812057811989652119901+1119908119901+10 sin (1199011205872 ) 1198866

+ 1812057811989632119901+1119908119901+10 sin (1199011205872 ) 1198864+ 12057811990820119908119901+1 sin (1199011205872 ) 1198862

(46)

where11990820 = 1199082[1 + 120578119908119901 cos(1199011205872)]And amplitude 119886 satisfies311989652119901119908119901+10 1198864 + 211989632119901119908119901+10 1198862 + 411990820119908119901+1 = 0 (47)

Taking the parameters as 120578 = 05 1198963 = 78 1198965 = 59and 119908 = 1 according to (46) and (47) the transition setfor stochastic P-bifurcation of the system with the unfoldingparameters 119901 and 119863 can be obtained as shown in Figure 4

According to the singularity theory the topological struc-tures of the stationary PDF curves of different points (119901119863)in the same region are qualitatively the same Taking a point(119901119863) in each region all varieties of the stationary PDFcurves which are qualitatively different could be obtainedIt can be seen that the unfolding parameter 119901 minus 119863 planeis divided into two subregions by the transition set curve

8 Discrete Dynamics in Nature and Society

1 Monostable region

2 Bistable region

0001002003004005006007008009

01

D01 02 03 04 05 06 07 08 09 10

p

Figure 4 The transition set curve of system (28)

p=03D=01

Monte Carlo simulationAnalytical solution

0

02

04

06

08

1

12

14

p(a)

02 04 06 08 1 12 14 16 18 20a

(a) parameter (119901119863) in region 1 of Figure 4

p=07D=003

Monte Carlo simulationAnalytical solution

0

02

04

06

08

1

12

14

p(a)

02 04 06 08 1 12 14 16 18 20a

(b) parameter (119901119863) in region 2 of Figure 4

Figure 5 The stationary PDF curves for the amplitude of system (28)

for convenience each region in Figure 4 is marked with anumber

Taking a given point (119901119863) in each of the two subregionsof Figure 4 the characteristics of stationary PDF curvesare analyzed and the corresponding results are shown inFigure 5

As can be seen from Figure 4 the parametric regionwhere the PDF curve appears bimodal is surrounded by anapproximately triangular region When the parameter (119901119863)is taken in the region 1 the PDF curve has only a distinctpeak as shown in Figure 5(a) in region 2 the PDF curvehas a distinct peak near the origin but the probability isobviously not zero far away from the origin there are boththe equilibrium and limit cycle in the system simultaneouslyas shown in Figure 5(b)

The analysis results above show that the stationaryPDF curves of the system amplitude in any two adjacentregions in Figure 4 are qualitatively different No matterthe values of the unfolding parameters cross any line inthe figure the system will occur stochastic P-bifurcationbehaviors so the transition set curve is just the criticalparametric condition for the stochastic P-bifurcation of thesystem and the analytic results in Figure 5 are in goodagreement with the numerical results by Monte Carlo simu-lation which further verify the correctness of the theoreticalanalysis

5 Conclusion

In this paper we studied the stochastic P-bifurcation foraxially moving of a bistable viscoelastic beam model withfractional derivatives of high order nonlinear terms underGaussian white noise excitation According to the minimummean square error principle we transformed the originalsystem into an equivalent and simplified system and obtainedthe stationary PDF of the system amplitude using stochasticaveraging In addition we obtained the critical parametriccondition for stochastic P-bifurcation of the system usingsingularity theory based on this the system response canbe maintained at the monostability or small amplitudenear the equilibrium by selecting the appropriate unfoldingparameters providing theoretical guidance for system designin practical engineering and avoiding the instability anddamage caused by the large amplitude vibration or nonlinearjump phenomenon of the system Finally the numericalresults by Monte Carlo simulation of the original systemalso verify the theoretical results obtained in this paper Weconclude that the order 119901 of fractional derivative and thenoise intensity 119863 can both cause stochastic P-bifurcation ofthe system and the number of peaks in the stationary PDFcurve of system amplitude can change from 2 to 1 by selectingthe appropriate unfolding parameters (119901119863) it also showsthat the method used in this paper is feasible to analyze

Discrete Dynamics in Nature and Society 9

the stochastic P-bifurcation behaviors of viscoelastic materialsystem with fractional constitutive relation

Data Availability

Data that support the findings of this study are includedwithin the article (and its additional files) All other dataare available from the corresponding author upon reasonablerequest

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work was supported by the National 973 Project ofChina (Grant no 2014CB046805) and the Natural ScienceFoundation of China (Grant no 11372211)

References

[1] W Tan and M Xu ldquoPlane surface suddenly set in motion in aviscoelastic fluid with fractionalMaxwell modelrdquoActaMechan-ica Sinica English Series The Chinese Society of Theoretical andApplied Mechanics vol 18 no 4 pp 342ndash349 2002

[2] J Sabatier O P Agrawal and J A Machado Advance in Frac-tional Calculus Theoretical Developments and Applications inPhysics and Engineering Springer DordrechtThe Netherlands2007

[3] I Podlubny ldquoFractional order systems andPIDcontrollerrdquo IEEETransactions on Automatic Control vol 44 no 1 pp 208ndash2141999

[4] C A Monje Y Chen B M Vinagre D Xue and V FeliuFractional-Order Systems and Controls Fundamentals andApplications Advances in Industrial Control Springer LondonUK 2010

[5] Q Li Y L Zhou X Q Zhao and X Y Ge ldquoFractional orderstochastic differential equation with application in Europeanoption pricingrdquo Discrete Dynamics in Nature and Society vol2014 Article ID 621895 12 pages 2014

[6] S Nie H Sun Y Zhang et al ldquoVertical distribution ofsuspended sediment under steady flow existing theories andfractional derivative modelrdquo Discrete Dynamics in Nature andSociety vol 2017 Article ID 5481531 11 pages 2017

[7] X Liu L Hong H Dang and L Yang ldquoBifurcations andsynchronization of the fractional-order Bloch systemrdquo DiscreteDynamics in Nature and Society vol 2017 Article ID 469430510 pages 2017

[8] W Li and Y Pang ldquoAsymptotic solutions of time-space frac-tional coupled systems by residual power series methodrdquoDiscrete Dynamics in Nature and Society vol 2017 Article ID7695924 10 pages 2017

[9] Z Liu T Xia and JWang ldquoImage encryption technology basedon fractional two-dimensional triangle function combinationdiscrete chaotic map coupled with Menezes-Vanstone ellipticcurve cryptosystemrdquo Discrete Dynamics in Nature and Societyvol 2018 Article ID 4585083 24 pages 2018

[10] C Sanchez-Lopez V H Carbajal-Gomez M A Carrasco-Aguilar and ICarro-Perez ldquoFractional-OrderMemristor Emu-lator Circuitsrdquo Complexity vol 2018 Article ID 2806976 10pages 2018

[11] K Rajagopal A Karthikeyan P Duraisamy and R Weldegior-gis ldquoBifurcation and Chaos in Integer and Fractional OrderTwo-Degree-of-Freedom Shape Memory Alloy OscillatorsrdquoComplexity vol 2018 9 pages 2018

[12] Ying Li and Ye Tang ldquoAnalytical Analysis on Nonlinear Para-metric Vibration of an Axially Moving String with FractionalViscoelastic Dampingrdquo Mathematical Problems in Engineeringvol 2017 Article ID 1393954 9 pages 2017

[13] G Liu C Ye X Liang Z Xie and Z Yu ldquoNonlinear DynamicIdentification of Beams Resting on Nonlinear ViscoelasticFoundations BASed on the Time-DELayed TRAnsfer Entropyand IMProved SurrogateDATaAlgorithmrdquoMathematical Prob-lems in Engineering vol 2018 Article ID 6531051 14 pages 2018

[14] D Liu Y Wu and X Xie ldquoStochastic Stability Analysis of Cou-pled Viscoelastic SYStems with Nonviscously Damping DRIvenby White NOIserdquo Mathematical Problems in Engineering vol2018 Article ID 6532305 16 pages 2018

[15] P G Nutting ldquoAnew general law of deformationrdquo Journal ofTheFranklin Institute vol 191 no 5 pp 679ndash685 1921

[16] A Gemant ldquoA method of analyzing experimental resultsobtained fromelasto-viscous bodiesrdquo Journal of Applied Physicsvol 7 no 8 pp 311ndash317 1936

[17] G S Blair and J Caffyn ldquoVI An application of the theoryof quasi-properties to the treatment of anomalous strain-stressrelationsrdquo The London Edinburgh and Dublin PhilosophicalMagazine and Journal of Science vol 40 no 300 pp 80ndash942010

[18] CMote ldquoDynamic Stability of Axially MovingMateralsrdquo Shockand Vibration vol 4 no 4 pp 2ndash11 1972

[19] AUlsoy andCMote ldquoBand SawVibration and Stabilityrdquo Shockand Vibration vol 10 no 11 pp 3ndash15 1978

[20] J A Wickert and C D Mote Jr ldquoCurrent research on thevibration and stability of axially-moving materialsrdquo Shock andVibration vol 20 no 1 pp 3ndash13 1988

[21] K W Wang and S P Liu ldquoOn the noise and vibration of chaindrive systemsrdquo Shock and Vibration Digest vol 23 no 4 pp 8ndash13 1991

[22] F Pellicano and F Vestroni ldquoComplex dynamics of high-speedaxially moving systemsrdquo Journal of Sound and Vibration vol258 no 1 pp 31ndash44 2002

[23] N-H Zhang and L-Q Chen ldquoNonlinear dynamical analysis ofaxially moving viscoelastic stringsrdquo Chaos Solitons amp Fractalsvol 24 no 4 pp 1065ndash1074 2005

[24] M H Ghayesh H A Kafiabad and T Reid ldquoSub- and super-critical nonlinear dynamics of a harmonically excited axiallymoving beamrdquo International Journal of Solids and Structuresvol 49 no 1 pp 227ndash243 2012

[25] M H Ghayesh ldquoStability and bifurcations of an axially movingbeam with an intermediate spring supportrdquo Nonlinear Dynam-ics vol 69 no 1-2 pp 193ndash210 2012

[26] R Rodrıguez E Salinas-Rodrıguez and J Fujioka ldquoFractionalTime Fluctuations in Viscoelasticity A Comparative Study ofCorrelations and Elastic Modulirdquo Entropy vol 20 no 1 p 282018

[27] R L Bagley and P J Torvik ldquoFractional calculus-a differentapproach to the analysis of viscoelastically damped structuresrdquoAIAA Journal vol 21 no 5 pp 741ndash748 2012

10 Discrete Dynamics in Nature and Society

[28] R L Bagley and P J Torvik ldquoFractional calculus in the transientanalysis of viscoelastically damped structuresrdquo AIAA Journalvol 23 no 6 pp 918ndash925 1985

[29] M Pakdemirli and A G Ulsoy ldquoStability analysis of an axiallyaccelerating stringrdquo Journal of Sound andVibration vol 203 no5 pp 815ndash832 1997

[30] L Zhang and J W Zu ldquoNonlinear vibration of parametricallyexcited moving belts part I Dynamic responserdquo Journal ofApplied Mechanics vol 66 no 2 pp 396ndash402 1999

[31] L Zhang and J W Zu ldquoNonlinear vibration of parametricallyexcited viscoelastic moving belts part II Stability analysisrdquoJournal of Applied Mechanics vol 66 no 2 pp 403ndash409 1999

[32] L Q Chen X D Yang and C J Cheng ldquoDynamic stability ofan axially accelerating viscoelastic beamrdquo European Journal ofMechanic vol 23 no 4 pp 659ndash666 2004

[33] L Q Chen and X D Yang ldquoSteady-state response of axiallymoving viscoelastic beams with pulsating speed comparisonof two nonlinear modelsrdquo International Journal of Solids andStructures vol 42 no 1 pp 37ndash50 2005

[34] L Q Chen and X D Yang ldquoTransverse nonlinear dynamicsof axially accelerating viscoelastic beams based on 4-termGalerkin truncationrdquo Chaos Soliton and Fractal vol 27 no 3pp 748ndash757 2006

[35] H Ding and L-Q Chen ldquoGalerkin methods for natural fre-quencies of high-speed axially moving beamsrdquo Journal of Soundand Vibration vol 329 no 17 pp 3484ndash3494 2010

[36] L-Q Chen J Wu and J W Zu ldquoAsymptotic nonlinear behav-iors in transverse vibration of an axially acceleratingviscoelasticstringrdquo Nonlinear Dynamics vol 35 no 4 pp 347ndash360 2004

[37] L-Q Chen N-H Zhang and J W Zu ldquoThe regular andchaotic vibrations of an axially moving viscoelastic string basedon fourth order Galerkin truncationrdquo Journal of Sound andVibration vol 261 no 4 pp 764ndash773 2003

[38] A Y T Leung H X Yang and J Y Chen ldquoParametricbifurcation of a viscoelastic column subject to axial harmonicforce and time-delayed controlrdquo Computers amp Structures vol136 pp 47ndash55 2014

[39] MH Ghayesh andNMoradian ldquoNonlinear dynamic responseof axially moving stretched viscoelastic stringsrdquo Archive ofApplied Mechanics vol 81 no 6 pp 781ndash799 2011

[40] D Liu W Xu and Y Xu ldquoDynamic responses of axially movingviscoelastic beamunder a randomly disordered periodic excita-tionrdquo Journal of Sound and Vibration vol 331 no 17 pp 4045ndash4056 2012

[41] T Z Yang and B Fang ldquoStability in parametric resonance of anaxially moving beam constituted by fractional order materialrdquoArchive of AppliedMechanics vol 82 no 12 pp 1763ndash1770 2012

[42] A Y T Leung H X Yang P Zhu and Z J Guo ldquoSteady stateresponse of fractionally damped nonlinear viscoelastic archesby residue harmonic homotopyrdquo Computers amp Structures vol121 pp 10ndash21 2013

[43] A C Galucio J-F Deui and R Ohayon ldquoFinite elementformulation of viscoelastic sandwich beams using fractionalderivative operatorsrdquo Computational Mechanics vol 33 no 4pp 282ndash291 2004

[44] L Chen W Wang Z Li and W Zhu ldquoStationary responseof Duffing oscillator with hardening stiffness and fractionalderivativerdquo International Journal of Non-Linear Mechanics vol48 pp 44ndash50 2013

[45] L C Chen Z S Li Q Zhuang and W Q Zhu ldquoFirst-passagefailure of single-degree-of-freedom nonlinear oscillators with

fractional derivativerdquo Journal of Vibration and Control vol 19no 14 pp 2154ndash2163 2013

[46] Y Shen S Yang H Xing and G Gao ldquoPrimary resonance ofDuffing oscillator with fractional-order derivativerdquo Communi-cations in Nonlinear Science and Numerical Simulation vol 17no 7 pp 3092ndash3100 2012

[47] Y G Yang W Xu Y H Sun and X D Gu ldquoStochastic responseof van der Pol oscillator with two kinds of fractional derivativesunder Gaussian white noise excitationrdquo Chinese Physics B vol25 no 2 pp 13ndash21 2016

[48] L Chen and W Zhu ldquoStochastic response of fractional-ordervan der Pol oscillatorrdquoTheoretical amp Applied Mechanics Lettersvol 4 no 1 pp 68ndash72 2014

[49] W Li M Zhang and J Zhao ldquoStochastic bifurcations of gen-eralized Duffingndashvan der Pol system with fractional derivativeunder colored noiserdquoChinese Physics B vol 26 no 9 pp 62ndash692017

Hindawiwwwhindawicom Volume 2018

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4 Discrete Dynamics in Nature and Society

Assume that the initial transverse vibration of the system is1199100(119909) = 0 and the transverse load of the system satisfies thefollowing form

119865 (119909 119905) = 120588119860 sdot sin (120587119871119909) 120585 (119905) (14)

where 120585(119905) is the Gaussian white noise which satisfies119864[120585(119905)] = 0 119864[120585(119905)120585(119905 minus 120591)] = 2119863120575(120591)119863 represents the noiseintensity and 120575(120591) is the Dirac function

By the discrete format based on Galerkin method system(11) can be reduced to the fractional differential equation asfollows

+ 1198961119906 + 1205781198961 1198620119863119901119906 minus 11989631199063 minus 1205781198963119906 1198620119863119901 (1199062) + 11989651199065+ 1205781198965119906 1198620119863119901 (1199064) = 120585 (119905) (15)

where 1198620119863119901119905 119906 is the 119901 (0 lt 119901 lt 1) order Caputo derivative of119906(119905) as is defined in (2) and

1198961 = (120587119871)4 1198681198640120588119860

1198963 = (120587119871)4 11986404120588

1198965 = (120587119871)6 3119864016120588

(16)

For convenience system (15) can be represented as follows

+ 1199082119906 + 1205781199082 1198620119863119901119906 minus 11989631199063 minus 1205781198963119906 1198620119863119901 (1199062)+ 11989651199065 + 1205781198965119906 1198620119863119901 (1199064) = 120585 (119905) (17)

where119908 = radic1198961The fractional derivative term has contributions to both

damping and restoring forces [44ndash47] hence introducing thefollowing equivalent system

+ 1199082119906 + 1205781199082 [119888 (119886) + 1199082 (119886) 119906] minus 11989631199063minus 1205781198963119906 1198620119863119901 (1199062) + 11989651199065 + 1205781198965119906 1198620119863119901 (1199064)

= 120585 (119905) (18)

where 119888(119886) 1199082(119886) are the coefficients of equivalent dampingand restoring forces of fractional derivative 1198620119863119901119906 respec-tively

The error between system (17) and (18) is

119890 = 1205781199082 [119888 (119886) + 1199082 (119886) 119906 minus 1198620119863119901119906] (19)

Thenecessary conditions forminimummean square error are[48]

120597119864 [1198902]120597 (119888 (119886)) = 0120597119864 [1198902]120597 (119908 (119886)) = 0

(20)

Substituting (19) into (20) yields

119864 [119888 (119886) 2 + 1199082 (119886) 119906 minus 1198620119863119901119906]= lim119879997888rarrinfin

1119879 int1198790(119888 (119886) 2 + 1199082 (119886) 119906 minus 1198620119863119901119906) 119889119905

= 0119864 [119888 (119886) 119906 + 1199082 (119886) 1199062 minus 119906 1198620119863119901119906]

= lim119879997888rarrinfin

1119879 int1198790(119888 (119886) 119906 + 1199082 (119886) 1199062 minus 119906 1198620119863119901119906) 119889119905

= 0

(21)

Assume that the solution of system (18) has the followingform

119906 (119905) = 119886 (119905) cos120593 (119905)120593 (119905) = 119908119905 + 120579 (22)

where119908 is the natural frequency of system (17)Based on (22) we can obtain

(119905) = minus119886 (119905) 119908 sin 120593 (119905) (23)

Substituting (22) and (23) into (21) yields

lim119879997888rarrinfin

1119879 int1198790(119888 (119886) 2 + 1199082 (119886) 119906 minus 1198620119863119901119906) 119889119905

= lim119879997888rarrinfin

1119879 int1198790(119888 (119886) 11988621199082sin2120593 minus 11988621199081199082 (119886) sin 120593

sdot cos120593 minus 119886119908 sin 120593 1198620119863119901119906) 119889119905 asymp 119888 (119886) 119886211990822+ 1Γ (1 minus 119901) lim

119879997888rarrinfin

1119879 int119879119900[(119886119908 sin 120593)

sdot int1199050

(119905 minus 120591)120591120572 119889120591] 119889119905 = 119888 (119886) 119886211990822minus 1Γ (1 minus 119901) lim

119879997888rarrinfin

1119879 int119879119900(11988621199082 sin 120593

sdot int1199050

sin 120593 cos (119908120591) minus cos120593 sin (119908120591)120591120572 )119889119905 = 0

(24)

lim119879997888rarrinfin

1119879 int1198790(119888 (119886) 119906 + 1199082 (119886) 119906119906 minus 119906 1198620119863119901119906) 119889119905

= lim119879997888rarrinfin

1119879 int1198790(minus119888 (119886) 1198862119908 sin 120593 cos120593 + 11988621199082 (119886)

sdot cos2120593 minus 119886 cos120593 1198620119863119901119906) 119889119905 asymp 11988621199082 (119886)2

Discrete Dynamics in Nature and Society 5

0 1090807060504030201p

0

02

04

06

08

1

12

a

stable limit cycleunstable limit cyclenumerical result

Figure 2 Bifurcation diagram of amplitude of system (28) at119863 = 0

minus 1Γ (1 minus 119901) lim

119879997888rarrinfin

1119879 int119879119900[(119886 cos120593)

sdot int1199050

119909 (119905 minus 120591)120591120572 119889120591] 119889119905 = 11988621199082 (119886)2 + 1Γ (1 minus 119901)

sdot lim119879997888rarrinfin

1119879 int119879119900(1198862119908 cos120593

sdot int1199050

sin 120593 cos (119908120591) minus cos120593 sin (119908120591)120591120572 )119889119905 = 0

(25)

To simplify (24) and (25) further asymptotic integrals areintroduced as follows

int1199050

cos (119908120591)120591119901 119889120591 = 119908119901minus1 (Γ (1 minus 119901) sin (1199011205872 )+ sin (119908119905)

(119908119905)119901 + 119900 ((119908119905)minus119901minus1))int1199050

sin (119908120591)120591119901 119889120591 = 119908119901minus1 (Γ (1 minus 119901) cos (1199011205872 )minus cos (119908119905)

(119908119905)119901 + 119900 ((119908119905)minus119901minus1))

(26)

Substituting (26) into (24) and (25) and averaging themacross 120593 produce the ultimate forms of 119888(119886) and 1199082(119886) asfollows

119888 (119886) = 119908119901minus1 sin (1199011205872 ) 1199082 (119886) = 119908119901 cos (1199011205872 )

(27)

Therefore the equivalent system associated with system (18)can be expressed as follows

+ 11990820119906 + 120574 minus 11989631199063 minus 1205781198963119906 1198620119863119901 (1199062) + 11989651199065+ 1205781198965119906 1198620119863119901 (1199064) = 120585 (119905) (28)

where11990820 = 1199082[1+120578119908119901 cos(1199011205872)] and 120574 = 120578119908119901+1 sin(1199011205872)

Next we consider the stochastic P-bifurcation of system(28) which comprises the fractional derivatives of high ordernonlinear terms and analyze the influence of parametricvariation on the system response

3 The Stationary PDF of Amplitude

For the system (28) the material modulus ratio is given as120578 = 05 coefficients of nonlinear terms are given as 1198963 = 781198965 = 59 respectively and nature frequency is given as119908 = 1For the convenience to discuss the parametric influence thebifurcation diagram of amplitude of the limit cycle along withvariation of the fractional order 119901 is shown in Figure 2 when119863 = 0

As can be seen from Figure 2 the solution correspond-ing to the solid line is almost completely coincided withthe numerical solution which proves the correctness andaccuracy of the approximate analytical result of the deter-ministic system at the same time it shows that the solutioncorresponding to the solid line is stable and the solutioncorresponding to the dotted line is unstable And it also can beseen that there is 1 attractor in the system when 119901 changes inthe interval [0 06817] equilibrium as shown in Figure 3(a)there are 2 attractors when 119901 changes in the interval [068181] equilibrium and limit cycle as shown in Figure 3(b)

In order to obtain the stationary PDF of the amplitude ofsystem (28) the following transformation is introduced

119906 (119905) = 119886 (119905) cosΦ (119905) (119905) = minus119886 (119905) 1199080 sinΦ (119905)Φ (119905) = 1199080119905 + 120579 (119905)

(29)

where 1199080 is the natural frequency of the equivalent system(28) 119886(119905) and 120579(119905) represent the amplitude process and thephase process of the system response respectively and theyare all random processes

Substituting (29) into (28) we can obtain

119889119886119889119905 = 11986511 (119886 120579) + G11 (119886 120579) 120585 (119905)

6 Discrete Dynamics in Nature and Society

minus1

minus05

0

05

1

dxd

t

p=04

0 05 1minus05minus1x

(a) Equilibrium

minus1

minus05

0

05

1

dxd

t

p=08

0 05 1minus05minus1x

(b) Equilibrium and limit cycle

Figure 3 Phase diagrams of system (28) at119863 = 0

119889120579119889119905 = 11986521 (119886 120579) + 11986621 (119886 120579) 120585 (119905) (30)

in which

11986511 = minusΔ sinΦ119908011986521 = minusΔ cosΦ119886119908011986611 = minus sinΦ119908011986621 = minuscosΦ1198861199080

(31)

and

Δ = 120578119908119901+11198861199080 sin (1199011205872 ) sinΦ minus 11989631198863cos3Φ minus 1205781198963sdot 11988632 (21199080)119901 cosΦ cos (2Φ + 119901120587

2 ) minus 11989651198865cos5Φminus 12057811989651198865 cosΦ[18 (41199080)119901 cos (4Φ + 119901120587

2 )+ 12 (21199080)119901 cos (2Φ + 119901120587

2 )]

(32)

Equation (30) can be regarded as a Stratonovich stochasticdifferential equation by adding the corresponding Wong-

Zakai correction term it can be transformed into thefollowing Ito stochastic differential equation

119889119886 = [11986511 (119886 120579) + 11986512 (119886 120579)] 119889119905+ radic2119863G11 (119886 120579) 119889119861 (119905)

119889120579 = [11986521 (119886 120579) + 11986522 (119886 120579)] 119889119905+ radic211986311986621 (119886 120579) 119889119861 (119905)

(33)

where 119861(119905) is a standard Wiener process and

11986512 (119886 120579) = 11986312059711986611120597119886 11986611 + 11986312059711986611120597120579 1198662111986522 (119886 120579) = 11986312059711986621120597119886 11986611 + 11986312059711986621120597120579 11986621

(34)

By the stochastic averaging method averaging (33) regardingΦ the following averaged Ito equation can be obtained asfollows

119889119886 = 1198981 (119886) 119889119905 + 1205901 (119886) 119889119861 (119905)119889120579 = 1198982 (119886) 119889119905 + 1205902 (119886) 119889119861 (119905) (35)

where

1198981 (119886) = minus12057811989652119901119908119901minus10 1198865 sin (1199011205872)8

minus 12057811989632119901119908119901minus10 1198863 sin (1199011205872)8

minus 120578119908119901+1119886 sin (1199011205872)2 + 119863211988611990820

12059012 (119886) = 119863119908201198982 (119886) = 1198965 (2120578 sdot 21199011199081199010 cos (1199011205872) + 5) 1198864

161199080

Discrete Dynamics in Nature and Society 7

+ 1198963 (120578 sdot 21199011199081199010 cos (1199011205872) + 3) 119886281199080

12059022 (119886) = 119863119886211990820 (36)

Equations (35) and (36) show that the averaged Ito equationfor 119886(119905) is independent of 120579(119905) and the random process 119886(119905) isa one-dimensional diffusion processThe corresponding FPKequation associated with 119886(119905) can be written as

120597119901120597119905 = minus 120597120597119886 [1198981 (119886) 119901] + 12 12059721205971198862 [12059012 (119886) 119901] (37)

The boundary conditions are

119901 = 119888 119888 isin (minusinfin +infin) when 119886 = 0

119901 997888rarr 0 120597119901120597119886 997888rarr 0 when 119886 997888rarr infin

(38)

Thus based on these boundary conditions (38) we can obtainthe stationary PDF of amplitude 119886 as follows

119901 (119886) = 11986212059012 (119886) exp [int119886

0

21198981 (119906)12059012 (119906) 119889119906] (39)

where C is a normalized constant which satisfies

119862 = [intinfin0

( 112059012 (119886) exp [int119886

0

21198981 (119906)12059012 (119906) 119889119906])119889119886]minus1 (40)

Substituting (36) into (39) the explicit expression for thestationary PDF of amplitude 119886 can be obtained as follows

119901 (119886) = 11986211988611990820119863 exp[[minus1205781198862 sin (1199011205872) (11989652119901+11198864119908119901+10 + 3119896321199011198862119908119901+10 + 2411990820119908119901+1)48119863 ]

] (41)

where11990820 = 1199082[1 + 120578119908119901 cos(1199011205872)]4 Stochastic P-Bifurcation

Stochastic P-bifurcation refers to the changes of the numberof peaks in the PDF curve in order to obtain the critical para-metric condition for stochastic P-bifurcation the influencesof the parameters for stochastic P-bifurcation of the systemare analyzed by using the singularity theory below

For the sake of convenience we can write 119901(119886) as follows119901 (119886) = 119862119877 (119886119863 119908 120578 119901 1198963 1198965)

sdot exp [119876 (119886119863 119908 120578 119901 1198963 1198965)] (42)

in which

119877 (119886119863119908 120578 119901 1198963 1198965) = 11988611990820119863119876(119886119863119908 120578 119901 1198963 1198965)= minus1205781198862 sin (1199011205872) (11989652119901+11198864119908119901+10 + 3119896321199011198862119908119901+10 + 2411990820119908119901+1)48119863

(43)

According to the singularity theory the stationary PDF needsto satisfy the following two conditions

120597119901 (119886)120597119886 = 0

1205972119901 (119886)1205971198862 = 0

(44)

Substituting (42) into (44) we can obtain the followingcondition [49]

119867= 1198771015840 + 1198771198761015840 = 0 11987710158401015840 + 211987710158401198761015840 + 11987711987610158401015840 + 11987711987610158402 = 0 (45)

where 119867 represents the critical condition for the changes ofthe number of peaks in the PDF curve

Substituting (43) into (45) we can get the critical para-metric condition for stochastic P-bifurcation of the system asfollows

119863 = 1812057811989652119901+1119908119901+10 sin (1199011205872 ) 1198866

+ 1812057811989632119901+1119908119901+10 sin (1199011205872 ) 1198864+ 12057811990820119908119901+1 sin (1199011205872 ) 1198862

(46)

where11990820 = 1199082[1 + 120578119908119901 cos(1199011205872)]And amplitude 119886 satisfies311989652119901119908119901+10 1198864 + 211989632119901119908119901+10 1198862 + 411990820119908119901+1 = 0 (47)

Taking the parameters as 120578 = 05 1198963 = 78 1198965 = 59and 119908 = 1 according to (46) and (47) the transition setfor stochastic P-bifurcation of the system with the unfoldingparameters 119901 and 119863 can be obtained as shown in Figure 4

According to the singularity theory the topological struc-tures of the stationary PDF curves of different points (119901119863)in the same region are qualitatively the same Taking a point(119901119863) in each region all varieties of the stationary PDFcurves which are qualitatively different could be obtainedIt can be seen that the unfolding parameter 119901 minus 119863 planeis divided into two subregions by the transition set curve

8 Discrete Dynamics in Nature and Society

1 Monostable region

2 Bistable region

0001002003004005006007008009

01

D01 02 03 04 05 06 07 08 09 10

p

Figure 4 The transition set curve of system (28)

p=03D=01

Monte Carlo simulationAnalytical solution

0

02

04

06

08

1

12

14

p(a)

02 04 06 08 1 12 14 16 18 20a

(a) parameter (119901119863) in region 1 of Figure 4

p=07D=003

Monte Carlo simulationAnalytical solution

0

02

04

06

08

1

12

14

p(a)

02 04 06 08 1 12 14 16 18 20a

(b) parameter (119901119863) in region 2 of Figure 4

Figure 5 The stationary PDF curves for the amplitude of system (28)

for convenience each region in Figure 4 is marked with anumber

Taking a given point (119901119863) in each of the two subregionsof Figure 4 the characteristics of stationary PDF curvesare analyzed and the corresponding results are shown inFigure 5

As can be seen from Figure 4 the parametric regionwhere the PDF curve appears bimodal is surrounded by anapproximately triangular region When the parameter (119901119863)is taken in the region 1 the PDF curve has only a distinctpeak as shown in Figure 5(a) in region 2 the PDF curvehas a distinct peak near the origin but the probability isobviously not zero far away from the origin there are boththe equilibrium and limit cycle in the system simultaneouslyas shown in Figure 5(b)

The analysis results above show that the stationaryPDF curves of the system amplitude in any two adjacentregions in Figure 4 are qualitatively different No matterthe values of the unfolding parameters cross any line inthe figure the system will occur stochastic P-bifurcationbehaviors so the transition set curve is just the criticalparametric condition for the stochastic P-bifurcation of thesystem and the analytic results in Figure 5 are in goodagreement with the numerical results by Monte Carlo simu-lation which further verify the correctness of the theoreticalanalysis

5 Conclusion

In this paper we studied the stochastic P-bifurcation foraxially moving of a bistable viscoelastic beam model withfractional derivatives of high order nonlinear terms underGaussian white noise excitation According to the minimummean square error principle we transformed the originalsystem into an equivalent and simplified system and obtainedthe stationary PDF of the system amplitude using stochasticaveraging In addition we obtained the critical parametriccondition for stochastic P-bifurcation of the system usingsingularity theory based on this the system response canbe maintained at the monostability or small amplitudenear the equilibrium by selecting the appropriate unfoldingparameters providing theoretical guidance for system designin practical engineering and avoiding the instability anddamage caused by the large amplitude vibration or nonlinearjump phenomenon of the system Finally the numericalresults by Monte Carlo simulation of the original systemalso verify the theoretical results obtained in this paper Weconclude that the order 119901 of fractional derivative and thenoise intensity 119863 can both cause stochastic P-bifurcation ofthe system and the number of peaks in the stationary PDFcurve of system amplitude can change from 2 to 1 by selectingthe appropriate unfolding parameters (119901119863) it also showsthat the method used in this paper is feasible to analyze

Discrete Dynamics in Nature and Society 9

the stochastic P-bifurcation behaviors of viscoelastic materialsystem with fractional constitutive relation

Data Availability

Data that support the findings of this study are includedwithin the article (and its additional files) All other dataare available from the corresponding author upon reasonablerequest

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work was supported by the National 973 Project ofChina (Grant no 2014CB046805) and the Natural ScienceFoundation of China (Grant no 11372211)

References

[1] W Tan and M Xu ldquoPlane surface suddenly set in motion in aviscoelastic fluid with fractionalMaxwell modelrdquoActaMechan-ica Sinica English Series The Chinese Society of Theoretical andApplied Mechanics vol 18 no 4 pp 342ndash349 2002

[2] J Sabatier O P Agrawal and J A Machado Advance in Frac-tional Calculus Theoretical Developments and Applications inPhysics and Engineering Springer DordrechtThe Netherlands2007

[3] I Podlubny ldquoFractional order systems andPIDcontrollerrdquo IEEETransactions on Automatic Control vol 44 no 1 pp 208ndash2141999

[4] C A Monje Y Chen B M Vinagre D Xue and V FeliuFractional-Order Systems and Controls Fundamentals andApplications Advances in Industrial Control Springer LondonUK 2010

[5] Q Li Y L Zhou X Q Zhao and X Y Ge ldquoFractional orderstochastic differential equation with application in Europeanoption pricingrdquo Discrete Dynamics in Nature and Society vol2014 Article ID 621895 12 pages 2014

[6] S Nie H Sun Y Zhang et al ldquoVertical distribution ofsuspended sediment under steady flow existing theories andfractional derivative modelrdquo Discrete Dynamics in Nature andSociety vol 2017 Article ID 5481531 11 pages 2017

[7] X Liu L Hong H Dang and L Yang ldquoBifurcations andsynchronization of the fractional-order Bloch systemrdquo DiscreteDynamics in Nature and Society vol 2017 Article ID 469430510 pages 2017

[8] W Li and Y Pang ldquoAsymptotic solutions of time-space frac-tional coupled systems by residual power series methodrdquoDiscrete Dynamics in Nature and Society vol 2017 Article ID7695924 10 pages 2017

[9] Z Liu T Xia and JWang ldquoImage encryption technology basedon fractional two-dimensional triangle function combinationdiscrete chaotic map coupled with Menezes-Vanstone ellipticcurve cryptosystemrdquo Discrete Dynamics in Nature and Societyvol 2018 Article ID 4585083 24 pages 2018

[10] C Sanchez-Lopez V H Carbajal-Gomez M A Carrasco-Aguilar and ICarro-Perez ldquoFractional-OrderMemristor Emu-lator Circuitsrdquo Complexity vol 2018 Article ID 2806976 10pages 2018

[11] K Rajagopal A Karthikeyan P Duraisamy and R Weldegior-gis ldquoBifurcation and Chaos in Integer and Fractional OrderTwo-Degree-of-Freedom Shape Memory Alloy OscillatorsrdquoComplexity vol 2018 9 pages 2018

[12] Ying Li and Ye Tang ldquoAnalytical Analysis on Nonlinear Para-metric Vibration of an Axially Moving String with FractionalViscoelastic Dampingrdquo Mathematical Problems in Engineeringvol 2017 Article ID 1393954 9 pages 2017

[13] G Liu C Ye X Liang Z Xie and Z Yu ldquoNonlinear DynamicIdentification of Beams Resting on Nonlinear ViscoelasticFoundations BASed on the Time-DELayed TRAnsfer Entropyand IMProved SurrogateDATaAlgorithmrdquoMathematical Prob-lems in Engineering vol 2018 Article ID 6531051 14 pages 2018

[14] D Liu Y Wu and X Xie ldquoStochastic Stability Analysis of Cou-pled Viscoelastic SYStems with Nonviscously Damping DRIvenby White NOIserdquo Mathematical Problems in Engineering vol2018 Article ID 6532305 16 pages 2018

[15] P G Nutting ldquoAnew general law of deformationrdquo Journal ofTheFranklin Institute vol 191 no 5 pp 679ndash685 1921

[16] A Gemant ldquoA method of analyzing experimental resultsobtained fromelasto-viscous bodiesrdquo Journal of Applied Physicsvol 7 no 8 pp 311ndash317 1936

[17] G S Blair and J Caffyn ldquoVI An application of the theoryof quasi-properties to the treatment of anomalous strain-stressrelationsrdquo The London Edinburgh and Dublin PhilosophicalMagazine and Journal of Science vol 40 no 300 pp 80ndash942010

[18] CMote ldquoDynamic Stability of Axially MovingMateralsrdquo Shockand Vibration vol 4 no 4 pp 2ndash11 1972

[19] AUlsoy andCMote ldquoBand SawVibration and Stabilityrdquo Shockand Vibration vol 10 no 11 pp 3ndash15 1978

[20] J A Wickert and C D Mote Jr ldquoCurrent research on thevibration and stability of axially-moving materialsrdquo Shock andVibration vol 20 no 1 pp 3ndash13 1988

[21] K W Wang and S P Liu ldquoOn the noise and vibration of chaindrive systemsrdquo Shock and Vibration Digest vol 23 no 4 pp 8ndash13 1991

[22] F Pellicano and F Vestroni ldquoComplex dynamics of high-speedaxially moving systemsrdquo Journal of Sound and Vibration vol258 no 1 pp 31ndash44 2002

[23] N-H Zhang and L-Q Chen ldquoNonlinear dynamical analysis ofaxially moving viscoelastic stringsrdquo Chaos Solitons amp Fractalsvol 24 no 4 pp 1065ndash1074 2005

[24] M H Ghayesh H A Kafiabad and T Reid ldquoSub- and super-critical nonlinear dynamics of a harmonically excited axiallymoving beamrdquo International Journal of Solids and Structuresvol 49 no 1 pp 227ndash243 2012

[25] M H Ghayesh ldquoStability and bifurcations of an axially movingbeam with an intermediate spring supportrdquo Nonlinear Dynam-ics vol 69 no 1-2 pp 193ndash210 2012

[26] R Rodrıguez E Salinas-Rodrıguez and J Fujioka ldquoFractionalTime Fluctuations in Viscoelasticity A Comparative Study ofCorrelations and Elastic Modulirdquo Entropy vol 20 no 1 p 282018

[27] R L Bagley and P J Torvik ldquoFractional calculus-a differentapproach to the analysis of viscoelastically damped structuresrdquoAIAA Journal vol 21 no 5 pp 741ndash748 2012

10 Discrete Dynamics in Nature and Society

[28] R L Bagley and P J Torvik ldquoFractional calculus in the transientanalysis of viscoelastically damped structuresrdquo AIAA Journalvol 23 no 6 pp 918ndash925 1985

[29] M Pakdemirli and A G Ulsoy ldquoStability analysis of an axiallyaccelerating stringrdquo Journal of Sound andVibration vol 203 no5 pp 815ndash832 1997

[30] L Zhang and J W Zu ldquoNonlinear vibration of parametricallyexcited moving belts part I Dynamic responserdquo Journal ofApplied Mechanics vol 66 no 2 pp 396ndash402 1999

[31] L Zhang and J W Zu ldquoNonlinear vibration of parametricallyexcited viscoelastic moving belts part II Stability analysisrdquoJournal of Applied Mechanics vol 66 no 2 pp 403ndash409 1999

[32] L Q Chen X D Yang and C J Cheng ldquoDynamic stability ofan axially accelerating viscoelastic beamrdquo European Journal ofMechanic vol 23 no 4 pp 659ndash666 2004

[33] L Q Chen and X D Yang ldquoSteady-state response of axiallymoving viscoelastic beams with pulsating speed comparisonof two nonlinear modelsrdquo International Journal of Solids andStructures vol 42 no 1 pp 37ndash50 2005

[34] L Q Chen and X D Yang ldquoTransverse nonlinear dynamicsof axially accelerating viscoelastic beams based on 4-termGalerkin truncationrdquo Chaos Soliton and Fractal vol 27 no 3pp 748ndash757 2006

[35] H Ding and L-Q Chen ldquoGalerkin methods for natural fre-quencies of high-speed axially moving beamsrdquo Journal of Soundand Vibration vol 329 no 17 pp 3484ndash3494 2010

[36] L-Q Chen J Wu and J W Zu ldquoAsymptotic nonlinear behav-iors in transverse vibration of an axially acceleratingviscoelasticstringrdquo Nonlinear Dynamics vol 35 no 4 pp 347ndash360 2004

[37] L-Q Chen N-H Zhang and J W Zu ldquoThe regular andchaotic vibrations of an axially moving viscoelastic string basedon fourth order Galerkin truncationrdquo Journal of Sound andVibration vol 261 no 4 pp 764ndash773 2003

[38] A Y T Leung H X Yang and J Y Chen ldquoParametricbifurcation of a viscoelastic column subject to axial harmonicforce and time-delayed controlrdquo Computers amp Structures vol136 pp 47ndash55 2014

[39] MH Ghayesh andNMoradian ldquoNonlinear dynamic responseof axially moving stretched viscoelastic stringsrdquo Archive ofApplied Mechanics vol 81 no 6 pp 781ndash799 2011

[40] D Liu W Xu and Y Xu ldquoDynamic responses of axially movingviscoelastic beamunder a randomly disordered periodic excita-tionrdquo Journal of Sound and Vibration vol 331 no 17 pp 4045ndash4056 2012

[41] T Z Yang and B Fang ldquoStability in parametric resonance of anaxially moving beam constituted by fractional order materialrdquoArchive of AppliedMechanics vol 82 no 12 pp 1763ndash1770 2012

[42] A Y T Leung H X Yang P Zhu and Z J Guo ldquoSteady stateresponse of fractionally damped nonlinear viscoelastic archesby residue harmonic homotopyrdquo Computers amp Structures vol121 pp 10ndash21 2013

[43] A C Galucio J-F Deui and R Ohayon ldquoFinite elementformulation of viscoelastic sandwich beams using fractionalderivative operatorsrdquo Computational Mechanics vol 33 no 4pp 282ndash291 2004

[44] L Chen W Wang Z Li and W Zhu ldquoStationary responseof Duffing oscillator with hardening stiffness and fractionalderivativerdquo International Journal of Non-Linear Mechanics vol48 pp 44ndash50 2013

[45] L C Chen Z S Li Q Zhuang and W Q Zhu ldquoFirst-passagefailure of single-degree-of-freedom nonlinear oscillators with

fractional derivativerdquo Journal of Vibration and Control vol 19no 14 pp 2154ndash2163 2013

[46] Y Shen S Yang H Xing and G Gao ldquoPrimary resonance ofDuffing oscillator with fractional-order derivativerdquo Communi-cations in Nonlinear Science and Numerical Simulation vol 17no 7 pp 3092ndash3100 2012

[47] Y G Yang W Xu Y H Sun and X D Gu ldquoStochastic responseof van der Pol oscillator with two kinds of fractional derivativesunder Gaussian white noise excitationrdquo Chinese Physics B vol25 no 2 pp 13ndash21 2016

[48] L Chen and W Zhu ldquoStochastic response of fractional-ordervan der Pol oscillatorrdquoTheoretical amp Applied Mechanics Lettersvol 4 no 1 pp 68ndash72 2014

[49] W Li M Zhang and J Zhao ldquoStochastic bifurcations of gen-eralized Duffingndashvan der Pol system with fractional derivativeunder colored noiserdquoChinese Physics B vol 26 no 9 pp 62ndash692017

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Discrete Dynamics in Nature and Society 5

0 1090807060504030201p

0

02

04

06

08

1

12

a

stable limit cycleunstable limit cyclenumerical result

Figure 2 Bifurcation diagram of amplitude of system (28) at119863 = 0

minus 1Γ (1 minus 119901) lim

119879997888rarrinfin

1119879 int119879119900[(119886 cos120593)

sdot int1199050

119909 (119905 minus 120591)120591120572 119889120591] 119889119905 = 11988621199082 (119886)2 + 1Γ (1 minus 119901)

sdot lim119879997888rarrinfin

1119879 int119879119900(1198862119908 cos120593

sdot int1199050

sin 120593 cos (119908120591) minus cos120593 sin (119908120591)120591120572 )119889119905 = 0

(25)

To simplify (24) and (25) further asymptotic integrals areintroduced as follows

int1199050

cos (119908120591)120591119901 119889120591 = 119908119901minus1 (Γ (1 minus 119901) sin (1199011205872 )+ sin (119908119905)

(119908119905)119901 + 119900 ((119908119905)minus119901minus1))int1199050

sin (119908120591)120591119901 119889120591 = 119908119901minus1 (Γ (1 minus 119901) cos (1199011205872 )minus cos (119908119905)

(119908119905)119901 + 119900 ((119908119905)minus119901minus1))

(26)

Substituting (26) into (24) and (25) and averaging themacross 120593 produce the ultimate forms of 119888(119886) and 1199082(119886) asfollows

119888 (119886) = 119908119901minus1 sin (1199011205872 ) 1199082 (119886) = 119908119901 cos (1199011205872 )

(27)

Therefore the equivalent system associated with system (18)can be expressed as follows

+ 11990820119906 + 120574 minus 11989631199063 minus 1205781198963119906 1198620119863119901 (1199062) + 11989651199065+ 1205781198965119906 1198620119863119901 (1199064) = 120585 (119905) (28)

where11990820 = 1199082[1+120578119908119901 cos(1199011205872)] and 120574 = 120578119908119901+1 sin(1199011205872)

Next we consider the stochastic P-bifurcation of system(28) which comprises the fractional derivatives of high ordernonlinear terms and analyze the influence of parametricvariation on the system response

3 The Stationary PDF of Amplitude

For the system (28) the material modulus ratio is given as120578 = 05 coefficients of nonlinear terms are given as 1198963 = 781198965 = 59 respectively and nature frequency is given as119908 = 1For the convenience to discuss the parametric influence thebifurcation diagram of amplitude of the limit cycle along withvariation of the fractional order 119901 is shown in Figure 2 when119863 = 0

As can be seen from Figure 2 the solution correspond-ing to the solid line is almost completely coincided withthe numerical solution which proves the correctness andaccuracy of the approximate analytical result of the deter-ministic system at the same time it shows that the solutioncorresponding to the solid line is stable and the solutioncorresponding to the dotted line is unstable And it also can beseen that there is 1 attractor in the system when 119901 changes inthe interval [0 06817] equilibrium as shown in Figure 3(a)there are 2 attractors when 119901 changes in the interval [068181] equilibrium and limit cycle as shown in Figure 3(b)

In order to obtain the stationary PDF of the amplitude ofsystem (28) the following transformation is introduced

119906 (119905) = 119886 (119905) cosΦ (119905) (119905) = minus119886 (119905) 1199080 sinΦ (119905)Φ (119905) = 1199080119905 + 120579 (119905)

(29)

where 1199080 is the natural frequency of the equivalent system(28) 119886(119905) and 120579(119905) represent the amplitude process and thephase process of the system response respectively and theyare all random processes

Substituting (29) into (28) we can obtain

119889119886119889119905 = 11986511 (119886 120579) + G11 (119886 120579) 120585 (119905)

6 Discrete Dynamics in Nature and Society

minus1

minus05

0

05

1

dxd

t

p=04

0 05 1minus05minus1x

(a) Equilibrium

minus1

minus05

0

05

1

dxd

t

p=08

0 05 1minus05minus1x

(b) Equilibrium and limit cycle

Figure 3 Phase diagrams of system (28) at119863 = 0

119889120579119889119905 = 11986521 (119886 120579) + 11986621 (119886 120579) 120585 (119905) (30)

in which

11986511 = minusΔ sinΦ119908011986521 = minusΔ cosΦ119886119908011986611 = minus sinΦ119908011986621 = minuscosΦ1198861199080

(31)

and

Δ = 120578119908119901+11198861199080 sin (1199011205872 ) sinΦ minus 11989631198863cos3Φ minus 1205781198963sdot 11988632 (21199080)119901 cosΦ cos (2Φ + 119901120587

2 ) minus 11989651198865cos5Φminus 12057811989651198865 cosΦ[18 (41199080)119901 cos (4Φ + 119901120587

2 )+ 12 (21199080)119901 cos (2Φ + 119901120587

2 )]

(32)

Equation (30) can be regarded as a Stratonovich stochasticdifferential equation by adding the corresponding Wong-

Zakai correction term it can be transformed into thefollowing Ito stochastic differential equation

119889119886 = [11986511 (119886 120579) + 11986512 (119886 120579)] 119889119905+ radic2119863G11 (119886 120579) 119889119861 (119905)

119889120579 = [11986521 (119886 120579) + 11986522 (119886 120579)] 119889119905+ radic211986311986621 (119886 120579) 119889119861 (119905)

(33)

where 119861(119905) is a standard Wiener process and

11986512 (119886 120579) = 11986312059711986611120597119886 11986611 + 11986312059711986611120597120579 1198662111986522 (119886 120579) = 11986312059711986621120597119886 11986611 + 11986312059711986621120597120579 11986621

(34)

By the stochastic averaging method averaging (33) regardingΦ the following averaged Ito equation can be obtained asfollows

119889119886 = 1198981 (119886) 119889119905 + 1205901 (119886) 119889119861 (119905)119889120579 = 1198982 (119886) 119889119905 + 1205902 (119886) 119889119861 (119905) (35)

where

1198981 (119886) = minus12057811989652119901119908119901minus10 1198865 sin (1199011205872)8

minus 12057811989632119901119908119901minus10 1198863 sin (1199011205872)8

minus 120578119908119901+1119886 sin (1199011205872)2 + 119863211988611990820

12059012 (119886) = 119863119908201198982 (119886) = 1198965 (2120578 sdot 21199011199081199010 cos (1199011205872) + 5) 1198864

161199080

Discrete Dynamics in Nature and Society 7

+ 1198963 (120578 sdot 21199011199081199010 cos (1199011205872) + 3) 119886281199080

12059022 (119886) = 119863119886211990820 (36)

Equations (35) and (36) show that the averaged Ito equationfor 119886(119905) is independent of 120579(119905) and the random process 119886(119905) isa one-dimensional diffusion processThe corresponding FPKequation associated with 119886(119905) can be written as

120597119901120597119905 = minus 120597120597119886 [1198981 (119886) 119901] + 12 12059721205971198862 [12059012 (119886) 119901] (37)

The boundary conditions are

119901 = 119888 119888 isin (minusinfin +infin) when 119886 = 0

119901 997888rarr 0 120597119901120597119886 997888rarr 0 when 119886 997888rarr infin

(38)

Thus based on these boundary conditions (38) we can obtainthe stationary PDF of amplitude 119886 as follows

119901 (119886) = 11986212059012 (119886) exp [int119886

0

21198981 (119906)12059012 (119906) 119889119906] (39)

where C is a normalized constant which satisfies

119862 = [intinfin0

( 112059012 (119886) exp [int119886

0

21198981 (119906)12059012 (119906) 119889119906])119889119886]minus1 (40)

Substituting (36) into (39) the explicit expression for thestationary PDF of amplitude 119886 can be obtained as follows

119901 (119886) = 11986211988611990820119863 exp[[minus1205781198862 sin (1199011205872) (11989652119901+11198864119908119901+10 + 3119896321199011198862119908119901+10 + 2411990820119908119901+1)48119863 ]

] (41)

where11990820 = 1199082[1 + 120578119908119901 cos(1199011205872)]4 Stochastic P-Bifurcation

Stochastic P-bifurcation refers to the changes of the numberof peaks in the PDF curve in order to obtain the critical para-metric condition for stochastic P-bifurcation the influencesof the parameters for stochastic P-bifurcation of the systemare analyzed by using the singularity theory below

For the sake of convenience we can write 119901(119886) as follows119901 (119886) = 119862119877 (119886119863 119908 120578 119901 1198963 1198965)

sdot exp [119876 (119886119863 119908 120578 119901 1198963 1198965)] (42)

in which

119877 (119886119863119908 120578 119901 1198963 1198965) = 11988611990820119863119876(119886119863119908 120578 119901 1198963 1198965)= minus1205781198862 sin (1199011205872) (11989652119901+11198864119908119901+10 + 3119896321199011198862119908119901+10 + 2411990820119908119901+1)48119863

(43)

According to the singularity theory the stationary PDF needsto satisfy the following two conditions

120597119901 (119886)120597119886 = 0

1205972119901 (119886)1205971198862 = 0

(44)

Substituting (42) into (44) we can obtain the followingcondition [49]

119867= 1198771015840 + 1198771198761015840 = 0 11987710158401015840 + 211987710158401198761015840 + 11987711987610158401015840 + 11987711987610158402 = 0 (45)

where 119867 represents the critical condition for the changes ofthe number of peaks in the PDF curve

Substituting (43) into (45) we can get the critical para-metric condition for stochastic P-bifurcation of the system asfollows

119863 = 1812057811989652119901+1119908119901+10 sin (1199011205872 ) 1198866

+ 1812057811989632119901+1119908119901+10 sin (1199011205872 ) 1198864+ 12057811990820119908119901+1 sin (1199011205872 ) 1198862

(46)

where11990820 = 1199082[1 + 120578119908119901 cos(1199011205872)]And amplitude 119886 satisfies311989652119901119908119901+10 1198864 + 211989632119901119908119901+10 1198862 + 411990820119908119901+1 = 0 (47)

Taking the parameters as 120578 = 05 1198963 = 78 1198965 = 59and 119908 = 1 according to (46) and (47) the transition setfor stochastic P-bifurcation of the system with the unfoldingparameters 119901 and 119863 can be obtained as shown in Figure 4

According to the singularity theory the topological struc-tures of the stationary PDF curves of different points (119901119863)in the same region are qualitatively the same Taking a point(119901119863) in each region all varieties of the stationary PDFcurves which are qualitatively different could be obtainedIt can be seen that the unfolding parameter 119901 minus 119863 planeis divided into two subregions by the transition set curve

8 Discrete Dynamics in Nature and Society

1 Monostable region

2 Bistable region

0001002003004005006007008009

01

D01 02 03 04 05 06 07 08 09 10

p

Figure 4 The transition set curve of system (28)

p=03D=01

Monte Carlo simulationAnalytical solution

0

02

04

06

08

1

12

14

p(a)

02 04 06 08 1 12 14 16 18 20a

(a) parameter (119901119863) in region 1 of Figure 4

p=07D=003

Monte Carlo simulationAnalytical solution

0

02

04

06

08

1

12

14

p(a)

02 04 06 08 1 12 14 16 18 20a

(b) parameter (119901119863) in region 2 of Figure 4

Figure 5 The stationary PDF curves for the amplitude of system (28)

for convenience each region in Figure 4 is marked with anumber

Taking a given point (119901119863) in each of the two subregionsof Figure 4 the characteristics of stationary PDF curvesare analyzed and the corresponding results are shown inFigure 5

As can be seen from Figure 4 the parametric regionwhere the PDF curve appears bimodal is surrounded by anapproximately triangular region When the parameter (119901119863)is taken in the region 1 the PDF curve has only a distinctpeak as shown in Figure 5(a) in region 2 the PDF curvehas a distinct peak near the origin but the probability isobviously not zero far away from the origin there are boththe equilibrium and limit cycle in the system simultaneouslyas shown in Figure 5(b)

The analysis results above show that the stationaryPDF curves of the system amplitude in any two adjacentregions in Figure 4 are qualitatively different No matterthe values of the unfolding parameters cross any line inthe figure the system will occur stochastic P-bifurcationbehaviors so the transition set curve is just the criticalparametric condition for the stochastic P-bifurcation of thesystem and the analytic results in Figure 5 are in goodagreement with the numerical results by Monte Carlo simu-lation which further verify the correctness of the theoreticalanalysis

5 Conclusion

In this paper we studied the stochastic P-bifurcation foraxially moving of a bistable viscoelastic beam model withfractional derivatives of high order nonlinear terms underGaussian white noise excitation According to the minimummean square error principle we transformed the originalsystem into an equivalent and simplified system and obtainedthe stationary PDF of the system amplitude using stochasticaveraging In addition we obtained the critical parametriccondition for stochastic P-bifurcation of the system usingsingularity theory based on this the system response canbe maintained at the monostability or small amplitudenear the equilibrium by selecting the appropriate unfoldingparameters providing theoretical guidance for system designin practical engineering and avoiding the instability anddamage caused by the large amplitude vibration or nonlinearjump phenomenon of the system Finally the numericalresults by Monte Carlo simulation of the original systemalso verify the theoretical results obtained in this paper Weconclude that the order 119901 of fractional derivative and thenoise intensity 119863 can both cause stochastic P-bifurcation ofthe system and the number of peaks in the stationary PDFcurve of system amplitude can change from 2 to 1 by selectingthe appropriate unfolding parameters (119901119863) it also showsthat the method used in this paper is feasible to analyze

Discrete Dynamics in Nature and Society 9

the stochastic P-bifurcation behaviors of viscoelastic materialsystem with fractional constitutive relation

Data Availability

Data that support the findings of this study are includedwithin the article (and its additional files) All other dataare available from the corresponding author upon reasonablerequest

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work was supported by the National 973 Project ofChina (Grant no 2014CB046805) and the Natural ScienceFoundation of China (Grant no 11372211)

References

[1] W Tan and M Xu ldquoPlane surface suddenly set in motion in aviscoelastic fluid with fractionalMaxwell modelrdquoActaMechan-ica Sinica English Series The Chinese Society of Theoretical andApplied Mechanics vol 18 no 4 pp 342ndash349 2002

[2] J Sabatier O P Agrawal and J A Machado Advance in Frac-tional Calculus Theoretical Developments and Applications inPhysics and Engineering Springer DordrechtThe Netherlands2007

[3] I Podlubny ldquoFractional order systems andPIDcontrollerrdquo IEEETransactions on Automatic Control vol 44 no 1 pp 208ndash2141999

[4] C A Monje Y Chen B M Vinagre D Xue and V FeliuFractional-Order Systems and Controls Fundamentals andApplications Advances in Industrial Control Springer LondonUK 2010

[5] Q Li Y L Zhou X Q Zhao and X Y Ge ldquoFractional orderstochastic differential equation with application in Europeanoption pricingrdquo Discrete Dynamics in Nature and Society vol2014 Article ID 621895 12 pages 2014

[6] S Nie H Sun Y Zhang et al ldquoVertical distribution ofsuspended sediment under steady flow existing theories andfractional derivative modelrdquo Discrete Dynamics in Nature andSociety vol 2017 Article ID 5481531 11 pages 2017

[7] X Liu L Hong H Dang and L Yang ldquoBifurcations andsynchronization of the fractional-order Bloch systemrdquo DiscreteDynamics in Nature and Society vol 2017 Article ID 469430510 pages 2017

[8] W Li and Y Pang ldquoAsymptotic solutions of time-space frac-tional coupled systems by residual power series methodrdquoDiscrete Dynamics in Nature and Society vol 2017 Article ID7695924 10 pages 2017

[9] Z Liu T Xia and JWang ldquoImage encryption technology basedon fractional two-dimensional triangle function combinationdiscrete chaotic map coupled with Menezes-Vanstone ellipticcurve cryptosystemrdquo Discrete Dynamics in Nature and Societyvol 2018 Article ID 4585083 24 pages 2018

[10] C Sanchez-Lopez V H Carbajal-Gomez M A Carrasco-Aguilar and ICarro-Perez ldquoFractional-OrderMemristor Emu-lator Circuitsrdquo Complexity vol 2018 Article ID 2806976 10pages 2018

[11] K Rajagopal A Karthikeyan P Duraisamy and R Weldegior-gis ldquoBifurcation and Chaos in Integer and Fractional OrderTwo-Degree-of-Freedom Shape Memory Alloy OscillatorsrdquoComplexity vol 2018 9 pages 2018

[12] Ying Li and Ye Tang ldquoAnalytical Analysis on Nonlinear Para-metric Vibration of an Axially Moving String with FractionalViscoelastic Dampingrdquo Mathematical Problems in Engineeringvol 2017 Article ID 1393954 9 pages 2017

[13] G Liu C Ye X Liang Z Xie and Z Yu ldquoNonlinear DynamicIdentification of Beams Resting on Nonlinear ViscoelasticFoundations BASed on the Time-DELayed TRAnsfer Entropyand IMProved SurrogateDATaAlgorithmrdquoMathematical Prob-lems in Engineering vol 2018 Article ID 6531051 14 pages 2018

[14] D Liu Y Wu and X Xie ldquoStochastic Stability Analysis of Cou-pled Viscoelastic SYStems with Nonviscously Damping DRIvenby White NOIserdquo Mathematical Problems in Engineering vol2018 Article ID 6532305 16 pages 2018

[15] P G Nutting ldquoAnew general law of deformationrdquo Journal ofTheFranklin Institute vol 191 no 5 pp 679ndash685 1921

[16] A Gemant ldquoA method of analyzing experimental resultsobtained fromelasto-viscous bodiesrdquo Journal of Applied Physicsvol 7 no 8 pp 311ndash317 1936

[17] G S Blair and J Caffyn ldquoVI An application of the theoryof quasi-properties to the treatment of anomalous strain-stressrelationsrdquo The London Edinburgh and Dublin PhilosophicalMagazine and Journal of Science vol 40 no 300 pp 80ndash942010

[18] CMote ldquoDynamic Stability of Axially MovingMateralsrdquo Shockand Vibration vol 4 no 4 pp 2ndash11 1972

[19] AUlsoy andCMote ldquoBand SawVibration and Stabilityrdquo Shockand Vibration vol 10 no 11 pp 3ndash15 1978

[20] J A Wickert and C D Mote Jr ldquoCurrent research on thevibration and stability of axially-moving materialsrdquo Shock andVibration vol 20 no 1 pp 3ndash13 1988

[21] K W Wang and S P Liu ldquoOn the noise and vibration of chaindrive systemsrdquo Shock and Vibration Digest vol 23 no 4 pp 8ndash13 1991

[22] F Pellicano and F Vestroni ldquoComplex dynamics of high-speedaxially moving systemsrdquo Journal of Sound and Vibration vol258 no 1 pp 31ndash44 2002

[23] N-H Zhang and L-Q Chen ldquoNonlinear dynamical analysis ofaxially moving viscoelastic stringsrdquo Chaos Solitons amp Fractalsvol 24 no 4 pp 1065ndash1074 2005

[24] M H Ghayesh H A Kafiabad and T Reid ldquoSub- and super-critical nonlinear dynamics of a harmonically excited axiallymoving beamrdquo International Journal of Solids and Structuresvol 49 no 1 pp 227ndash243 2012

[25] M H Ghayesh ldquoStability and bifurcations of an axially movingbeam with an intermediate spring supportrdquo Nonlinear Dynam-ics vol 69 no 1-2 pp 193ndash210 2012

[26] R Rodrıguez E Salinas-Rodrıguez and J Fujioka ldquoFractionalTime Fluctuations in Viscoelasticity A Comparative Study ofCorrelations and Elastic Modulirdquo Entropy vol 20 no 1 p 282018

[27] R L Bagley and P J Torvik ldquoFractional calculus-a differentapproach to the analysis of viscoelastically damped structuresrdquoAIAA Journal vol 21 no 5 pp 741ndash748 2012

10 Discrete Dynamics in Nature and Society

[28] R L Bagley and P J Torvik ldquoFractional calculus in the transientanalysis of viscoelastically damped structuresrdquo AIAA Journalvol 23 no 6 pp 918ndash925 1985

[29] M Pakdemirli and A G Ulsoy ldquoStability analysis of an axiallyaccelerating stringrdquo Journal of Sound andVibration vol 203 no5 pp 815ndash832 1997

[30] L Zhang and J W Zu ldquoNonlinear vibration of parametricallyexcited moving belts part I Dynamic responserdquo Journal ofApplied Mechanics vol 66 no 2 pp 396ndash402 1999

[31] L Zhang and J W Zu ldquoNonlinear vibration of parametricallyexcited viscoelastic moving belts part II Stability analysisrdquoJournal of Applied Mechanics vol 66 no 2 pp 403ndash409 1999

[32] L Q Chen X D Yang and C J Cheng ldquoDynamic stability ofan axially accelerating viscoelastic beamrdquo European Journal ofMechanic vol 23 no 4 pp 659ndash666 2004

[33] L Q Chen and X D Yang ldquoSteady-state response of axiallymoving viscoelastic beams with pulsating speed comparisonof two nonlinear modelsrdquo International Journal of Solids andStructures vol 42 no 1 pp 37ndash50 2005

[34] L Q Chen and X D Yang ldquoTransverse nonlinear dynamicsof axially accelerating viscoelastic beams based on 4-termGalerkin truncationrdquo Chaos Soliton and Fractal vol 27 no 3pp 748ndash757 2006

[35] H Ding and L-Q Chen ldquoGalerkin methods for natural fre-quencies of high-speed axially moving beamsrdquo Journal of Soundand Vibration vol 329 no 17 pp 3484ndash3494 2010

[36] L-Q Chen J Wu and J W Zu ldquoAsymptotic nonlinear behav-iors in transverse vibration of an axially acceleratingviscoelasticstringrdquo Nonlinear Dynamics vol 35 no 4 pp 347ndash360 2004

[37] L-Q Chen N-H Zhang and J W Zu ldquoThe regular andchaotic vibrations of an axially moving viscoelastic string basedon fourth order Galerkin truncationrdquo Journal of Sound andVibration vol 261 no 4 pp 764ndash773 2003

[38] A Y T Leung H X Yang and J Y Chen ldquoParametricbifurcation of a viscoelastic column subject to axial harmonicforce and time-delayed controlrdquo Computers amp Structures vol136 pp 47ndash55 2014

[39] MH Ghayesh andNMoradian ldquoNonlinear dynamic responseof axially moving stretched viscoelastic stringsrdquo Archive ofApplied Mechanics vol 81 no 6 pp 781ndash799 2011

[40] D Liu W Xu and Y Xu ldquoDynamic responses of axially movingviscoelastic beamunder a randomly disordered periodic excita-tionrdquo Journal of Sound and Vibration vol 331 no 17 pp 4045ndash4056 2012

[41] T Z Yang and B Fang ldquoStability in parametric resonance of anaxially moving beam constituted by fractional order materialrdquoArchive of AppliedMechanics vol 82 no 12 pp 1763ndash1770 2012

[42] A Y T Leung H X Yang P Zhu and Z J Guo ldquoSteady stateresponse of fractionally damped nonlinear viscoelastic archesby residue harmonic homotopyrdquo Computers amp Structures vol121 pp 10ndash21 2013

[43] A C Galucio J-F Deui and R Ohayon ldquoFinite elementformulation of viscoelastic sandwich beams using fractionalderivative operatorsrdquo Computational Mechanics vol 33 no 4pp 282ndash291 2004

[44] L Chen W Wang Z Li and W Zhu ldquoStationary responseof Duffing oscillator with hardening stiffness and fractionalderivativerdquo International Journal of Non-Linear Mechanics vol48 pp 44ndash50 2013

[45] L C Chen Z S Li Q Zhuang and W Q Zhu ldquoFirst-passagefailure of single-degree-of-freedom nonlinear oscillators with

fractional derivativerdquo Journal of Vibration and Control vol 19no 14 pp 2154ndash2163 2013

[46] Y Shen S Yang H Xing and G Gao ldquoPrimary resonance ofDuffing oscillator with fractional-order derivativerdquo Communi-cations in Nonlinear Science and Numerical Simulation vol 17no 7 pp 3092ndash3100 2012

[47] Y G Yang W Xu Y H Sun and X D Gu ldquoStochastic responseof van der Pol oscillator with two kinds of fractional derivativesunder Gaussian white noise excitationrdquo Chinese Physics B vol25 no 2 pp 13ndash21 2016

[48] L Chen and W Zhu ldquoStochastic response of fractional-ordervan der Pol oscillatorrdquoTheoretical amp Applied Mechanics Lettersvol 4 no 1 pp 68ndash72 2014

[49] W Li M Zhang and J Zhao ldquoStochastic bifurcations of gen-eralized Duffingndashvan der Pol system with fractional derivativeunder colored noiserdquoChinese Physics B vol 26 no 9 pp 62ndash692017

Hindawiwwwhindawicom Volume 2018

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Submit your manuscripts atwwwhindawicom

6 Discrete Dynamics in Nature and Society

minus1

minus05

0

05

1

dxd

t

p=04

0 05 1minus05minus1x

(a) Equilibrium

minus1

minus05

0

05

1

dxd

t

p=08

0 05 1minus05minus1x

(b) Equilibrium and limit cycle

Figure 3 Phase diagrams of system (28) at119863 = 0

119889120579119889119905 = 11986521 (119886 120579) + 11986621 (119886 120579) 120585 (119905) (30)

in which

11986511 = minusΔ sinΦ119908011986521 = minusΔ cosΦ119886119908011986611 = minus sinΦ119908011986621 = minuscosΦ1198861199080

(31)

and

Δ = 120578119908119901+11198861199080 sin (1199011205872 ) sinΦ minus 11989631198863cos3Φ minus 1205781198963sdot 11988632 (21199080)119901 cosΦ cos (2Φ + 119901120587

2 ) minus 11989651198865cos5Φminus 12057811989651198865 cosΦ[18 (41199080)119901 cos (4Φ + 119901120587

2 )+ 12 (21199080)119901 cos (2Φ + 119901120587

2 )]

(32)

Equation (30) can be regarded as a Stratonovich stochasticdifferential equation by adding the corresponding Wong-

Zakai correction term it can be transformed into thefollowing Ito stochastic differential equation

119889119886 = [11986511 (119886 120579) + 11986512 (119886 120579)] 119889119905+ radic2119863G11 (119886 120579) 119889119861 (119905)

119889120579 = [11986521 (119886 120579) + 11986522 (119886 120579)] 119889119905+ radic211986311986621 (119886 120579) 119889119861 (119905)

(33)

where 119861(119905) is a standard Wiener process and

11986512 (119886 120579) = 11986312059711986611120597119886 11986611 + 11986312059711986611120597120579 1198662111986522 (119886 120579) = 11986312059711986621120597119886 11986611 + 11986312059711986621120597120579 11986621

(34)

By the stochastic averaging method averaging (33) regardingΦ the following averaged Ito equation can be obtained asfollows

119889119886 = 1198981 (119886) 119889119905 + 1205901 (119886) 119889119861 (119905)119889120579 = 1198982 (119886) 119889119905 + 1205902 (119886) 119889119861 (119905) (35)

where

1198981 (119886) = minus12057811989652119901119908119901minus10 1198865 sin (1199011205872)8

minus 12057811989632119901119908119901minus10 1198863 sin (1199011205872)8

minus 120578119908119901+1119886 sin (1199011205872)2 + 119863211988611990820

12059012 (119886) = 119863119908201198982 (119886) = 1198965 (2120578 sdot 21199011199081199010 cos (1199011205872) + 5) 1198864

161199080

Discrete Dynamics in Nature and Society 7

+ 1198963 (120578 sdot 21199011199081199010 cos (1199011205872) + 3) 119886281199080

12059022 (119886) = 119863119886211990820 (36)

Equations (35) and (36) show that the averaged Ito equationfor 119886(119905) is independent of 120579(119905) and the random process 119886(119905) isa one-dimensional diffusion processThe corresponding FPKequation associated with 119886(119905) can be written as

120597119901120597119905 = minus 120597120597119886 [1198981 (119886) 119901] + 12 12059721205971198862 [12059012 (119886) 119901] (37)

The boundary conditions are

119901 = 119888 119888 isin (minusinfin +infin) when 119886 = 0

119901 997888rarr 0 120597119901120597119886 997888rarr 0 when 119886 997888rarr infin

(38)

Thus based on these boundary conditions (38) we can obtainthe stationary PDF of amplitude 119886 as follows

119901 (119886) = 11986212059012 (119886) exp [int119886

0

21198981 (119906)12059012 (119906) 119889119906] (39)

where C is a normalized constant which satisfies

119862 = [intinfin0

( 112059012 (119886) exp [int119886

0

21198981 (119906)12059012 (119906) 119889119906])119889119886]minus1 (40)

Substituting (36) into (39) the explicit expression for thestationary PDF of amplitude 119886 can be obtained as follows

119901 (119886) = 11986211988611990820119863 exp[[minus1205781198862 sin (1199011205872) (11989652119901+11198864119908119901+10 + 3119896321199011198862119908119901+10 + 2411990820119908119901+1)48119863 ]

] (41)

where11990820 = 1199082[1 + 120578119908119901 cos(1199011205872)]4 Stochastic P-Bifurcation

Stochastic P-bifurcation refers to the changes of the numberof peaks in the PDF curve in order to obtain the critical para-metric condition for stochastic P-bifurcation the influencesof the parameters for stochastic P-bifurcation of the systemare analyzed by using the singularity theory below

For the sake of convenience we can write 119901(119886) as follows119901 (119886) = 119862119877 (119886119863 119908 120578 119901 1198963 1198965)

sdot exp [119876 (119886119863 119908 120578 119901 1198963 1198965)] (42)

in which

119877 (119886119863119908 120578 119901 1198963 1198965) = 11988611990820119863119876(119886119863119908 120578 119901 1198963 1198965)= minus1205781198862 sin (1199011205872) (11989652119901+11198864119908119901+10 + 3119896321199011198862119908119901+10 + 2411990820119908119901+1)48119863

(43)

According to the singularity theory the stationary PDF needsto satisfy the following two conditions

120597119901 (119886)120597119886 = 0

1205972119901 (119886)1205971198862 = 0

(44)

Substituting (42) into (44) we can obtain the followingcondition [49]

119867= 1198771015840 + 1198771198761015840 = 0 11987710158401015840 + 211987710158401198761015840 + 11987711987610158401015840 + 11987711987610158402 = 0 (45)

where 119867 represents the critical condition for the changes ofthe number of peaks in the PDF curve

Substituting (43) into (45) we can get the critical para-metric condition for stochastic P-bifurcation of the system asfollows

119863 = 1812057811989652119901+1119908119901+10 sin (1199011205872 ) 1198866

+ 1812057811989632119901+1119908119901+10 sin (1199011205872 ) 1198864+ 12057811990820119908119901+1 sin (1199011205872 ) 1198862

(46)

where11990820 = 1199082[1 + 120578119908119901 cos(1199011205872)]And amplitude 119886 satisfies311989652119901119908119901+10 1198864 + 211989632119901119908119901+10 1198862 + 411990820119908119901+1 = 0 (47)

Taking the parameters as 120578 = 05 1198963 = 78 1198965 = 59and 119908 = 1 according to (46) and (47) the transition setfor stochastic P-bifurcation of the system with the unfoldingparameters 119901 and 119863 can be obtained as shown in Figure 4

According to the singularity theory the topological struc-tures of the stationary PDF curves of different points (119901119863)in the same region are qualitatively the same Taking a point(119901119863) in each region all varieties of the stationary PDFcurves which are qualitatively different could be obtainedIt can be seen that the unfolding parameter 119901 minus 119863 planeis divided into two subregions by the transition set curve

8 Discrete Dynamics in Nature and Society

1 Monostable region

2 Bistable region

0001002003004005006007008009

01

D01 02 03 04 05 06 07 08 09 10

p

Figure 4 The transition set curve of system (28)

p=03D=01

Monte Carlo simulationAnalytical solution

0

02

04

06

08

1

12

14

p(a)

02 04 06 08 1 12 14 16 18 20a

(a) parameter (119901119863) in region 1 of Figure 4

p=07D=003

Monte Carlo simulationAnalytical solution

0

02

04

06

08

1

12

14

p(a)

02 04 06 08 1 12 14 16 18 20a

(b) parameter (119901119863) in region 2 of Figure 4

Figure 5 The stationary PDF curves for the amplitude of system (28)

for convenience each region in Figure 4 is marked with anumber

Taking a given point (119901119863) in each of the two subregionsof Figure 4 the characteristics of stationary PDF curvesare analyzed and the corresponding results are shown inFigure 5

As can be seen from Figure 4 the parametric regionwhere the PDF curve appears bimodal is surrounded by anapproximately triangular region When the parameter (119901119863)is taken in the region 1 the PDF curve has only a distinctpeak as shown in Figure 5(a) in region 2 the PDF curvehas a distinct peak near the origin but the probability isobviously not zero far away from the origin there are boththe equilibrium and limit cycle in the system simultaneouslyas shown in Figure 5(b)

The analysis results above show that the stationaryPDF curves of the system amplitude in any two adjacentregions in Figure 4 are qualitatively different No matterthe values of the unfolding parameters cross any line inthe figure the system will occur stochastic P-bifurcationbehaviors so the transition set curve is just the criticalparametric condition for the stochastic P-bifurcation of thesystem and the analytic results in Figure 5 are in goodagreement with the numerical results by Monte Carlo simu-lation which further verify the correctness of the theoreticalanalysis

5 Conclusion

In this paper we studied the stochastic P-bifurcation foraxially moving of a bistable viscoelastic beam model withfractional derivatives of high order nonlinear terms underGaussian white noise excitation According to the minimummean square error principle we transformed the originalsystem into an equivalent and simplified system and obtainedthe stationary PDF of the system amplitude using stochasticaveraging In addition we obtained the critical parametriccondition for stochastic P-bifurcation of the system usingsingularity theory based on this the system response canbe maintained at the monostability or small amplitudenear the equilibrium by selecting the appropriate unfoldingparameters providing theoretical guidance for system designin practical engineering and avoiding the instability anddamage caused by the large amplitude vibration or nonlinearjump phenomenon of the system Finally the numericalresults by Monte Carlo simulation of the original systemalso verify the theoretical results obtained in this paper Weconclude that the order 119901 of fractional derivative and thenoise intensity 119863 can both cause stochastic P-bifurcation ofthe system and the number of peaks in the stationary PDFcurve of system amplitude can change from 2 to 1 by selectingthe appropriate unfolding parameters (119901119863) it also showsthat the method used in this paper is feasible to analyze

Discrete Dynamics in Nature and Society 9

the stochastic P-bifurcation behaviors of viscoelastic materialsystem with fractional constitutive relation

Data Availability

Data that support the findings of this study are includedwithin the article (and its additional files) All other dataare available from the corresponding author upon reasonablerequest

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work was supported by the National 973 Project ofChina (Grant no 2014CB046805) and the Natural ScienceFoundation of China (Grant no 11372211)

References

[1] W Tan and M Xu ldquoPlane surface suddenly set in motion in aviscoelastic fluid with fractionalMaxwell modelrdquoActaMechan-ica Sinica English Series The Chinese Society of Theoretical andApplied Mechanics vol 18 no 4 pp 342ndash349 2002

[2] J Sabatier O P Agrawal and J A Machado Advance in Frac-tional Calculus Theoretical Developments and Applications inPhysics and Engineering Springer DordrechtThe Netherlands2007

[3] I Podlubny ldquoFractional order systems andPIDcontrollerrdquo IEEETransactions on Automatic Control vol 44 no 1 pp 208ndash2141999

[4] C A Monje Y Chen B M Vinagre D Xue and V FeliuFractional-Order Systems and Controls Fundamentals andApplications Advances in Industrial Control Springer LondonUK 2010

[5] Q Li Y L Zhou X Q Zhao and X Y Ge ldquoFractional orderstochastic differential equation with application in Europeanoption pricingrdquo Discrete Dynamics in Nature and Society vol2014 Article ID 621895 12 pages 2014

[6] S Nie H Sun Y Zhang et al ldquoVertical distribution ofsuspended sediment under steady flow existing theories andfractional derivative modelrdquo Discrete Dynamics in Nature andSociety vol 2017 Article ID 5481531 11 pages 2017

[7] X Liu L Hong H Dang and L Yang ldquoBifurcations andsynchronization of the fractional-order Bloch systemrdquo DiscreteDynamics in Nature and Society vol 2017 Article ID 469430510 pages 2017

[8] W Li and Y Pang ldquoAsymptotic solutions of time-space frac-tional coupled systems by residual power series methodrdquoDiscrete Dynamics in Nature and Society vol 2017 Article ID7695924 10 pages 2017

[9] Z Liu T Xia and JWang ldquoImage encryption technology basedon fractional two-dimensional triangle function combinationdiscrete chaotic map coupled with Menezes-Vanstone ellipticcurve cryptosystemrdquo Discrete Dynamics in Nature and Societyvol 2018 Article ID 4585083 24 pages 2018

[10] C Sanchez-Lopez V H Carbajal-Gomez M A Carrasco-Aguilar and ICarro-Perez ldquoFractional-OrderMemristor Emu-lator Circuitsrdquo Complexity vol 2018 Article ID 2806976 10pages 2018

[11] K Rajagopal A Karthikeyan P Duraisamy and R Weldegior-gis ldquoBifurcation and Chaos in Integer and Fractional OrderTwo-Degree-of-Freedom Shape Memory Alloy OscillatorsrdquoComplexity vol 2018 9 pages 2018

[12] Ying Li and Ye Tang ldquoAnalytical Analysis on Nonlinear Para-metric Vibration of an Axially Moving String with FractionalViscoelastic Dampingrdquo Mathematical Problems in Engineeringvol 2017 Article ID 1393954 9 pages 2017

[13] G Liu C Ye X Liang Z Xie and Z Yu ldquoNonlinear DynamicIdentification of Beams Resting on Nonlinear ViscoelasticFoundations BASed on the Time-DELayed TRAnsfer Entropyand IMProved SurrogateDATaAlgorithmrdquoMathematical Prob-lems in Engineering vol 2018 Article ID 6531051 14 pages 2018

[14] D Liu Y Wu and X Xie ldquoStochastic Stability Analysis of Cou-pled Viscoelastic SYStems with Nonviscously Damping DRIvenby White NOIserdquo Mathematical Problems in Engineering vol2018 Article ID 6532305 16 pages 2018

[15] P G Nutting ldquoAnew general law of deformationrdquo Journal ofTheFranklin Institute vol 191 no 5 pp 679ndash685 1921

[16] A Gemant ldquoA method of analyzing experimental resultsobtained fromelasto-viscous bodiesrdquo Journal of Applied Physicsvol 7 no 8 pp 311ndash317 1936

[17] G S Blair and J Caffyn ldquoVI An application of the theoryof quasi-properties to the treatment of anomalous strain-stressrelationsrdquo The London Edinburgh and Dublin PhilosophicalMagazine and Journal of Science vol 40 no 300 pp 80ndash942010

[18] CMote ldquoDynamic Stability of Axially MovingMateralsrdquo Shockand Vibration vol 4 no 4 pp 2ndash11 1972

[19] AUlsoy andCMote ldquoBand SawVibration and Stabilityrdquo Shockand Vibration vol 10 no 11 pp 3ndash15 1978

[20] J A Wickert and C D Mote Jr ldquoCurrent research on thevibration and stability of axially-moving materialsrdquo Shock andVibration vol 20 no 1 pp 3ndash13 1988

[21] K W Wang and S P Liu ldquoOn the noise and vibration of chaindrive systemsrdquo Shock and Vibration Digest vol 23 no 4 pp 8ndash13 1991

[22] F Pellicano and F Vestroni ldquoComplex dynamics of high-speedaxially moving systemsrdquo Journal of Sound and Vibration vol258 no 1 pp 31ndash44 2002

[23] N-H Zhang and L-Q Chen ldquoNonlinear dynamical analysis ofaxially moving viscoelastic stringsrdquo Chaos Solitons amp Fractalsvol 24 no 4 pp 1065ndash1074 2005

[24] M H Ghayesh H A Kafiabad and T Reid ldquoSub- and super-critical nonlinear dynamics of a harmonically excited axiallymoving beamrdquo International Journal of Solids and Structuresvol 49 no 1 pp 227ndash243 2012

[25] M H Ghayesh ldquoStability and bifurcations of an axially movingbeam with an intermediate spring supportrdquo Nonlinear Dynam-ics vol 69 no 1-2 pp 193ndash210 2012

[26] R Rodrıguez E Salinas-Rodrıguez and J Fujioka ldquoFractionalTime Fluctuations in Viscoelasticity A Comparative Study ofCorrelations and Elastic Modulirdquo Entropy vol 20 no 1 p 282018

[27] R L Bagley and P J Torvik ldquoFractional calculus-a differentapproach to the analysis of viscoelastically damped structuresrdquoAIAA Journal vol 21 no 5 pp 741ndash748 2012

10 Discrete Dynamics in Nature and Society

[28] R L Bagley and P J Torvik ldquoFractional calculus in the transientanalysis of viscoelastically damped structuresrdquo AIAA Journalvol 23 no 6 pp 918ndash925 1985

[29] M Pakdemirli and A G Ulsoy ldquoStability analysis of an axiallyaccelerating stringrdquo Journal of Sound andVibration vol 203 no5 pp 815ndash832 1997

[30] L Zhang and J W Zu ldquoNonlinear vibration of parametricallyexcited moving belts part I Dynamic responserdquo Journal ofApplied Mechanics vol 66 no 2 pp 396ndash402 1999

[31] L Zhang and J W Zu ldquoNonlinear vibration of parametricallyexcited viscoelastic moving belts part II Stability analysisrdquoJournal of Applied Mechanics vol 66 no 2 pp 403ndash409 1999

[32] L Q Chen X D Yang and C J Cheng ldquoDynamic stability ofan axially accelerating viscoelastic beamrdquo European Journal ofMechanic vol 23 no 4 pp 659ndash666 2004

[33] L Q Chen and X D Yang ldquoSteady-state response of axiallymoving viscoelastic beams with pulsating speed comparisonof two nonlinear modelsrdquo International Journal of Solids andStructures vol 42 no 1 pp 37ndash50 2005

[34] L Q Chen and X D Yang ldquoTransverse nonlinear dynamicsof axially accelerating viscoelastic beams based on 4-termGalerkin truncationrdquo Chaos Soliton and Fractal vol 27 no 3pp 748ndash757 2006

[35] H Ding and L-Q Chen ldquoGalerkin methods for natural fre-quencies of high-speed axially moving beamsrdquo Journal of Soundand Vibration vol 329 no 17 pp 3484ndash3494 2010

[36] L-Q Chen J Wu and J W Zu ldquoAsymptotic nonlinear behav-iors in transverse vibration of an axially acceleratingviscoelasticstringrdquo Nonlinear Dynamics vol 35 no 4 pp 347ndash360 2004

[37] L-Q Chen N-H Zhang and J W Zu ldquoThe regular andchaotic vibrations of an axially moving viscoelastic string basedon fourth order Galerkin truncationrdquo Journal of Sound andVibration vol 261 no 4 pp 764ndash773 2003

[38] A Y T Leung H X Yang and J Y Chen ldquoParametricbifurcation of a viscoelastic column subject to axial harmonicforce and time-delayed controlrdquo Computers amp Structures vol136 pp 47ndash55 2014

[39] MH Ghayesh andNMoradian ldquoNonlinear dynamic responseof axially moving stretched viscoelastic stringsrdquo Archive ofApplied Mechanics vol 81 no 6 pp 781ndash799 2011

[40] D Liu W Xu and Y Xu ldquoDynamic responses of axially movingviscoelastic beamunder a randomly disordered periodic excita-tionrdquo Journal of Sound and Vibration vol 331 no 17 pp 4045ndash4056 2012

[41] T Z Yang and B Fang ldquoStability in parametric resonance of anaxially moving beam constituted by fractional order materialrdquoArchive of AppliedMechanics vol 82 no 12 pp 1763ndash1770 2012

[42] A Y T Leung H X Yang P Zhu and Z J Guo ldquoSteady stateresponse of fractionally damped nonlinear viscoelastic archesby residue harmonic homotopyrdquo Computers amp Structures vol121 pp 10ndash21 2013

[43] A C Galucio J-F Deui and R Ohayon ldquoFinite elementformulation of viscoelastic sandwich beams using fractionalderivative operatorsrdquo Computational Mechanics vol 33 no 4pp 282ndash291 2004

[44] L Chen W Wang Z Li and W Zhu ldquoStationary responseof Duffing oscillator with hardening stiffness and fractionalderivativerdquo International Journal of Non-Linear Mechanics vol48 pp 44ndash50 2013

[45] L C Chen Z S Li Q Zhuang and W Q Zhu ldquoFirst-passagefailure of single-degree-of-freedom nonlinear oscillators with

fractional derivativerdquo Journal of Vibration and Control vol 19no 14 pp 2154ndash2163 2013

[46] Y Shen S Yang H Xing and G Gao ldquoPrimary resonance ofDuffing oscillator with fractional-order derivativerdquo Communi-cations in Nonlinear Science and Numerical Simulation vol 17no 7 pp 3092ndash3100 2012

[47] Y G Yang W Xu Y H Sun and X D Gu ldquoStochastic responseof van der Pol oscillator with two kinds of fractional derivativesunder Gaussian white noise excitationrdquo Chinese Physics B vol25 no 2 pp 13ndash21 2016

[48] L Chen and W Zhu ldquoStochastic response of fractional-ordervan der Pol oscillatorrdquoTheoretical amp Applied Mechanics Lettersvol 4 no 1 pp 68ndash72 2014

[49] W Li M Zhang and J Zhao ldquoStochastic bifurcations of gen-eralized Duffingndashvan der Pol system with fractional derivativeunder colored noiserdquoChinese Physics B vol 26 no 9 pp 62ndash692017

Hindawiwwwhindawicom Volume 2018

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Applied MathematicsJournal of

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Mathematical PhysicsAdvances in

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Submit your manuscripts atwwwhindawicom

Discrete Dynamics in Nature and Society 7

+ 1198963 (120578 sdot 21199011199081199010 cos (1199011205872) + 3) 119886281199080

12059022 (119886) = 119863119886211990820 (36)

Equations (35) and (36) show that the averaged Ito equationfor 119886(119905) is independent of 120579(119905) and the random process 119886(119905) isa one-dimensional diffusion processThe corresponding FPKequation associated with 119886(119905) can be written as

120597119901120597119905 = minus 120597120597119886 [1198981 (119886) 119901] + 12 12059721205971198862 [12059012 (119886) 119901] (37)

The boundary conditions are

119901 = 119888 119888 isin (minusinfin +infin) when 119886 = 0

119901 997888rarr 0 120597119901120597119886 997888rarr 0 when 119886 997888rarr infin

(38)

Thus based on these boundary conditions (38) we can obtainthe stationary PDF of amplitude 119886 as follows

119901 (119886) = 11986212059012 (119886) exp [int119886

0

21198981 (119906)12059012 (119906) 119889119906] (39)

where C is a normalized constant which satisfies

119862 = [intinfin0

( 112059012 (119886) exp [int119886

0

21198981 (119906)12059012 (119906) 119889119906])119889119886]minus1 (40)

Substituting (36) into (39) the explicit expression for thestationary PDF of amplitude 119886 can be obtained as follows

119901 (119886) = 11986211988611990820119863 exp[[minus1205781198862 sin (1199011205872) (11989652119901+11198864119908119901+10 + 3119896321199011198862119908119901+10 + 2411990820119908119901+1)48119863 ]

] (41)

where11990820 = 1199082[1 + 120578119908119901 cos(1199011205872)]4 Stochastic P-Bifurcation

Stochastic P-bifurcation refers to the changes of the numberof peaks in the PDF curve in order to obtain the critical para-metric condition for stochastic P-bifurcation the influencesof the parameters for stochastic P-bifurcation of the systemare analyzed by using the singularity theory below

For the sake of convenience we can write 119901(119886) as follows119901 (119886) = 119862119877 (119886119863 119908 120578 119901 1198963 1198965)

sdot exp [119876 (119886119863 119908 120578 119901 1198963 1198965)] (42)

in which

119877 (119886119863119908 120578 119901 1198963 1198965) = 11988611990820119863119876(119886119863119908 120578 119901 1198963 1198965)= minus1205781198862 sin (1199011205872) (11989652119901+11198864119908119901+10 + 3119896321199011198862119908119901+10 + 2411990820119908119901+1)48119863

(43)

According to the singularity theory the stationary PDF needsto satisfy the following two conditions

120597119901 (119886)120597119886 = 0

1205972119901 (119886)1205971198862 = 0

(44)

Substituting (42) into (44) we can obtain the followingcondition [49]

119867= 1198771015840 + 1198771198761015840 = 0 11987710158401015840 + 211987710158401198761015840 + 11987711987610158401015840 + 11987711987610158402 = 0 (45)

where 119867 represents the critical condition for the changes ofthe number of peaks in the PDF curve

Substituting (43) into (45) we can get the critical para-metric condition for stochastic P-bifurcation of the system asfollows

119863 = 1812057811989652119901+1119908119901+10 sin (1199011205872 ) 1198866

+ 1812057811989632119901+1119908119901+10 sin (1199011205872 ) 1198864+ 12057811990820119908119901+1 sin (1199011205872 ) 1198862

(46)

where11990820 = 1199082[1 + 120578119908119901 cos(1199011205872)]And amplitude 119886 satisfies311989652119901119908119901+10 1198864 + 211989632119901119908119901+10 1198862 + 411990820119908119901+1 = 0 (47)

Taking the parameters as 120578 = 05 1198963 = 78 1198965 = 59and 119908 = 1 according to (46) and (47) the transition setfor stochastic P-bifurcation of the system with the unfoldingparameters 119901 and 119863 can be obtained as shown in Figure 4

According to the singularity theory the topological struc-tures of the stationary PDF curves of different points (119901119863)in the same region are qualitatively the same Taking a point(119901119863) in each region all varieties of the stationary PDFcurves which are qualitatively different could be obtainedIt can be seen that the unfolding parameter 119901 minus 119863 planeis divided into two subregions by the transition set curve

8 Discrete Dynamics in Nature and Society

1 Monostable region

2 Bistable region

0001002003004005006007008009

01

D01 02 03 04 05 06 07 08 09 10

p

Figure 4 The transition set curve of system (28)

p=03D=01

Monte Carlo simulationAnalytical solution

0

02

04

06

08

1

12

14

p(a)

02 04 06 08 1 12 14 16 18 20a

(a) parameter (119901119863) in region 1 of Figure 4

p=07D=003

Monte Carlo simulationAnalytical solution

0

02

04

06

08

1

12

14

p(a)

02 04 06 08 1 12 14 16 18 20a

(b) parameter (119901119863) in region 2 of Figure 4

Figure 5 The stationary PDF curves for the amplitude of system (28)

for convenience each region in Figure 4 is marked with anumber

Taking a given point (119901119863) in each of the two subregionsof Figure 4 the characteristics of stationary PDF curvesare analyzed and the corresponding results are shown inFigure 5

As can be seen from Figure 4 the parametric regionwhere the PDF curve appears bimodal is surrounded by anapproximately triangular region When the parameter (119901119863)is taken in the region 1 the PDF curve has only a distinctpeak as shown in Figure 5(a) in region 2 the PDF curvehas a distinct peak near the origin but the probability isobviously not zero far away from the origin there are boththe equilibrium and limit cycle in the system simultaneouslyas shown in Figure 5(b)

The analysis results above show that the stationaryPDF curves of the system amplitude in any two adjacentregions in Figure 4 are qualitatively different No matterthe values of the unfolding parameters cross any line inthe figure the system will occur stochastic P-bifurcationbehaviors so the transition set curve is just the criticalparametric condition for the stochastic P-bifurcation of thesystem and the analytic results in Figure 5 are in goodagreement with the numerical results by Monte Carlo simu-lation which further verify the correctness of the theoreticalanalysis

5 Conclusion

In this paper we studied the stochastic P-bifurcation foraxially moving of a bistable viscoelastic beam model withfractional derivatives of high order nonlinear terms underGaussian white noise excitation According to the minimummean square error principle we transformed the originalsystem into an equivalent and simplified system and obtainedthe stationary PDF of the system amplitude using stochasticaveraging In addition we obtained the critical parametriccondition for stochastic P-bifurcation of the system usingsingularity theory based on this the system response canbe maintained at the monostability or small amplitudenear the equilibrium by selecting the appropriate unfoldingparameters providing theoretical guidance for system designin practical engineering and avoiding the instability anddamage caused by the large amplitude vibration or nonlinearjump phenomenon of the system Finally the numericalresults by Monte Carlo simulation of the original systemalso verify the theoretical results obtained in this paper Weconclude that the order 119901 of fractional derivative and thenoise intensity 119863 can both cause stochastic P-bifurcation ofthe system and the number of peaks in the stationary PDFcurve of system amplitude can change from 2 to 1 by selectingthe appropriate unfolding parameters (119901119863) it also showsthat the method used in this paper is feasible to analyze

Discrete Dynamics in Nature and Society 9

the stochastic P-bifurcation behaviors of viscoelastic materialsystem with fractional constitutive relation

Data Availability

Data that support the findings of this study are includedwithin the article (and its additional files) All other dataare available from the corresponding author upon reasonablerequest

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work was supported by the National 973 Project ofChina (Grant no 2014CB046805) and the Natural ScienceFoundation of China (Grant no 11372211)

References

[1] W Tan and M Xu ldquoPlane surface suddenly set in motion in aviscoelastic fluid with fractionalMaxwell modelrdquoActaMechan-ica Sinica English Series The Chinese Society of Theoretical andApplied Mechanics vol 18 no 4 pp 342ndash349 2002

[2] J Sabatier O P Agrawal and J A Machado Advance in Frac-tional Calculus Theoretical Developments and Applications inPhysics and Engineering Springer DordrechtThe Netherlands2007

[3] I Podlubny ldquoFractional order systems andPIDcontrollerrdquo IEEETransactions on Automatic Control vol 44 no 1 pp 208ndash2141999

[4] C A Monje Y Chen B M Vinagre D Xue and V FeliuFractional-Order Systems and Controls Fundamentals andApplications Advances in Industrial Control Springer LondonUK 2010

[5] Q Li Y L Zhou X Q Zhao and X Y Ge ldquoFractional orderstochastic differential equation with application in Europeanoption pricingrdquo Discrete Dynamics in Nature and Society vol2014 Article ID 621895 12 pages 2014

[6] S Nie H Sun Y Zhang et al ldquoVertical distribution ofsuspended sediment under steady flow existing theories andfractional derivative modelrdquo Discrete Dynamics in Nature andSociety vol 2017 Article ID 5481531 11 pages 2017

[7] X Liu L Hong H Dang and L Yang ldquoBifurcations andsynchronization of the fractional-order Bloch systemrdquo DiscreteDynamics in Nature and Society vol 2017 Article ID 469430510 pages 2017

[8] W Li and Y Pang ldquoAsymptotic solutions of time-space frac-tional coupled systems by residual power series methodrdquoDiscrete Dynamics in Nature and Society vol 2017 Article ID7695924 10 pages 2017

[9] Z Liu T Xia and JWang ldquoImage encryption technology basedon fractional two-dimensional triangle function combinationdiscrete chaotic map coupled with Menezes-Vanstone ellipticcurve cryptosystemrdquo Discrete Dynamics in Nature and Societyvol 2018 Article ID 4585083 24 pages 2018

[10] C Sanchez-Lopez V H Carbajal-Gomez M A Carrasco-Aguilar and ICarro-Perez ldquoFractional-OrderMemristor Emu-lator Circuitsrdquo Complexity vol 2018 Article ID 2806976 10pages 2018

[11] K Rajagopal A Karthikeyan P Duraisamy and R Weldegior-gis ldquoBifurcation and Chaos in Integer and Fractional OrderTwo-Degree-of-Freedom Shape Memory Alloy OscillatorsrdquoComplexity vol 2018 9 pages 2018

[12] Ying Li and Ye Tang ldquoAnalytical Analysis on Nonlinear Para-metric Vibration of an Axially Moving String with FractionalViscoelastic Dampingrdquo Mathematical Problems in Engineeringvol 2017 Article ID 1393954 9 pages 2017

[13] G Liu C Ye X Liang Z Xie and Z Yu ldquoNonlinear DynamicIdentification of Beams Resting on Nonlinear ViscoelasticFoundations BASed on the Time-DELayed TRAnsfer Entropyand IMProved SurrogateDATaAlgorithmrdquoMathematical Prob-lems in Engineering vol 2018 Article ID 6531051 14 pages 2018

[14] D Liu Y Wu and X Xie ldquoStochastic Stability Analysis of Cou-pled Viscoelastic SYStems with Nonviscously Damping DRIvenby White NOIserdquo Mathematical Problems in Engineering vol2018 Article ID 6532305 16 pages 2018

[15] P G Nutting ldquoAnew general law of deformationrdquo Journal ofTheFranklin Institute vol 191 no 5 pp 679ndash685 1921

[16] A Gemant ldquoA method of analyzing experimental resultsobtained fromelasto-viscous bodiesrdquo Journal of Applied Physicsvol 7 no 8 pp 311ndash317 1936

[17] G S Blair and J Caffyn ldquoVI An application of the theoryof quasi-properties to the treatment of anomalous strain-stressrelationsrdquo The London Edinburgh and Dublin PhilosophicalMagazine and Journal of Science vol 40 no 300 pp 80ndash942010

[18] CMote ldquoDynamic Stability of Axially MovingMateralsrdquo Shockand Vibration vol 4 no 4 pp 2ndash11 1972

[19] AUlsoy andCMote ldquoBand SawVibration and Stabilityrdquo Shockand Vibration vol 10 no 11 pp 3ndash15 1978

[20] J A Wickert and C D Mote Jr ldquoCurrent research on thevibration and stability of axially-moving materialsrdquo Shock andVibration vol 20 no 1 pp 3ndash13 1988

[21] K W Wang and S P Liu ldquoOn the noise and vibration of chaindrive systemsrdquo Shock and Vibration Digest vol 23 no 4 pp 8ndash13 1991

[22] F Pellicano and F Vestroni ldquoComplex dynamics of high-speedaxially moving systemsrdquo Journal of Sound and Vibration vol258 no 1 pp 31ndash44 2002

[23] N-H Zhang and L-Q Chen ldquoNonlinear dynamical analysis ofaxially moving viscoelastic stringsrdquo Chaos Solitons amp Fractalsvol 24 no 4 pp 1065ndash1074 2005

[24] M H Ghayesh H A Kafiabad and T Reid ldquoSub- and super-critical nonlinear dynamics of a harmonically excited axiallymoving beamrdquo International Journal of Solids and Structuresvol 49 no 1 pp 227ndash243 2012

[25] M H Ghayesh ldquoStability and bifurcations of an axially movingbeam with an intermediate spring supportrdquo Nonlinear Dynam-ics vol 69 no 1-2 pp 193ndash210 2012

[26] R Rodrıguez E Salinas-Rodrıguez and J Fujioka ldquoFractionalTime Fluctuations in Viscoelasticity A Comparative Study ofCorrelations and Elastic Modulirdquo Entropy vol 20 no 1 p 282018

[27] R L Bagley and P J Torvik ldquoFractional calculus-a differentapproach to the analysis of viscoelastically damped structuresrdquoAIAA Journal vol 21 no 5 pp 741ndash748 2012

10 Discrete Dynamics in Nature and Society

[28] R L Bagley and P J Torvik ldquoFractional calculus in the transientanalysis of viscoelastically damped structuresrdquo AIAA Journalvol 23 no 6 pp 918ndash925 1985

[29] M Pakdemirli and A G Ulsoy ldquoStability analysis of an axiallyaccelerating stringrdquo Journal of Sound andVibration vol 203 no5 pp 815ndash832 1997

[30] L Zhang and J W Zu ldquoNonlinear vibration of parametricallyexcited moving belts part I Dynamic responserdquo Journal ofApplied Mechanics vol 66 no 2 pp 396ndash402 1999

[31] L Zhang and J W Zu ldquoNonlinear vibration of parametricallyexcited viscoelastic moving belts part II Stability analysisrdquoJournal of Applied Mechanics vol 66 no 2 pp 403ndash409 1999

[32] L Q Chen X D Yang and C J Cheng ldquoDynamic stability ofan axially accelerating viscoelastic beamrdquo European Journal ofMechanic vol 23 no 4 pp 659ndash666 2004

[33] L Q Chen and X D Yang ldquoSteady-state response of axiallymoving viscoelastic beams with pulsating speed comparisonof two nonlinear modelsrdquo International Journal of Solids andStructures vol 42 no 1 pp 37ndash50 2005

[34] L Q Chen and X D Yang ldquoTransverse nonlinear dynamicsof axially accelerating viscoelastic beams based on 4-termGalerkin truncationrdquo Chaos Soliton and Fractal vol 27 no 3pp 748ndash757 2006

[35] H Ding and L-Q Chen ldquoGalerkin methods for natural fre-quencies of high-speed axially moving beamsrdquo Journal of Soundand Vibration vol 329 no 17 pp 3484ndash3494 2010

[36] L-Q Chen J Wu and J W Zu ldquoAsymptotic nonlinear behav-iors in transverse vibration of an axially acceleratingviscoelasticstringrdquo Nonlinear Dynamics vol 35 no 4 pp 347ndash360 2004

[37] L-Q Chen N-H Zhang and J W Zu ldquoThe regular andchaotic vibrations of an axially moving viscoelastic string basedon fourth order Galerkin truncationrdquo Journal of Sound andVibration vol 261 no 4 pp 764ndash773 2003

[38] A Y T Leung H X Yang and J Y Chen ldquoParametricbifurcation of a viscoelastic column subject to axial harmonicforce and time-delayed controlrdquo Computers amp Structures vol136 pp 47ndash55 2014

[39] MH Ghayesh andNMoradian ldquoNonlinear dynamic responseof axially moving stretched viscoelastic stringsrdquo Archive ofApplied Mechanics vol 81 no 6 pp 781ndash799 2011

[40] D Liu W Xu and Y Xu ldquoDynamic responses of axially movingviscoelastic beamunder a randomly disordered periodic excita-tionrdquo Journal of Sound and Vibration vol 331 no 17 pp 4045ndash4056 2012

[41] T Z Yang and B Fang ldquoStability in parametric resonance of anaxially moving beam constituted by fractional order materialrdquoArchive of AppliedMechanics vol 82 no 12 pp 1763ndash1770 2012

[42] A Y T Leung H X Yang P Zhu and Z J Guo ldquoSteady stateresponse of fractionally damped nonlinear viscoelastic archesby residue harmonic homotopyrdquo Computers amp Structures vol121 pp 10ndash21 2013

[43] A C Galucio J-F Deui and R Ohayon ldquoFinite elementformulation of viscoelastic sandwich beams using fractionalderivative operatorsrdquo Computational Mechanics vol 33 no 4pp 282ndash291 2004

[44] L Chen W Wang Z Li and W Zhu ldquoStationary responseof Duffing oscillator with hardening stiffness and fractionalderivativerdquo International Journal of Non-Linear Mechanics vol48 pp 44ndash50 2013

[45] L C Chen Z S Li Q Zhuang and W Q Zhu ldquoFirst-passagefailure of single-degree-of-freedom nonlinear oscillators with

fractional derivativerdquo Journal of Vibration and Control vol 19no 14 pp 2154ndash2163 2013

[46] Y Shen S Yang H Xing and G Gao ldquoPrimary resonance ofDuffing oscillator with fractional-order derivativerdquo Communi-cations in Nonlinear Science and Numerical Simulation vol 17no 7 pp 3092ndash3100 2012

[47] Y G Yang W Xu Y H Sun and X D Gu ldquoStochastic responseof van der Pol oscillator with two kinds of fractional derivativesunder Gaussian white noise excitationrdquo Chinese Physics B vol25 no 2 pp 13ndash21 2016

[48] L Chen and W Zhu ldquoStochastic response of fractional-ordervan der Pol oscillatorrdquoTheoretical amp Applied Mechanics Lettersvol 4 no 1 pp 68ndash72 2014

[49] W Li M Zhang and J Zhao ldquoStochastic bifurcations of gen-eralized Duffingndashvan der Pol system with fractional derivativeunder colored noiserdquoChinese Physics B vol 26 no 9 pp 62ndash692017

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

8 Discrete Dynamics in Nature and Society

1 Monostable region

2 Bistable region

0001002003004005006007008009

01

D01 02 03 04 05 06 07 08 09 10

p

Figure 4 The transition set curve of system (28)

p=03D=01

Monte Carlo simulationAnalytical solution

0

02

04

06

08

1

12

14

p(a)

02 04 06 08 1 12 14 16 18 20a

(a) parameter (119901119863) in region 1 of Figure 4

p=07D=003

Monte Carlo simulationAnalytical solution

0

02

04

06

08

1

12

14

p(a)

02 04 06 08 1 12 14 16 18 20a

(b) parameter (119901119863) in region 2 of Figure 4

Figure 5 The stationary PDF curves for the amplitude of system (28)

for convenience each region in Figure 4 is marked with anumber

Taking a given point (119901119863) in each of the two subregionsof Figure 4 the characteristics of stationary PDF curvesare analyzed and the corresponding results are shown inFigure 5

As can be seen from Figure 4 the parametric regionwhere the PDF curve appears bimodal is surrounded by anapproximately triangular region When the parameter (119901119863)is taken in the region 1 the PDF curve has only a distinctpeak as shown in Figure 5(a) in region 2 the PDF curvehas a distinct peak near the origin but the probability isobviously not zero far away from the origin there are boththe equilibrium and limit cycle in the system simultaneouslyas shown in Figure 5(b)

The analysis results above show that the stationaryPDF curves of the system amplitude in any two adjacentregions in Figure 4 are qualitatively different No matterthe values of the unfolding parameters cross any line inthe figure the system will occur stochastic P-bifurcationbehaviors so the transition set curve is just the criticalparametric condition for the stochastic P-bifurcation of thesystem and the analytic results in Figure 5 are in goodagreement with the numerical results by Monte Carlo simu-lation which further verify the correctness of the theoreticalanalysis

5 Conclusion

In this paper we studied the stochastic P-bifurcation foraxially moving of a bistable viscoelastic beam model withfractional derivatives of high order nonlinear terms underGaussian white noise excitation According to the minimummean square error principle we transformed the originalsystem into an equivalent and simplified system and obtainedthe stationary PDF of the system amplitude using stochasticaveraging In addition we obtained the critical parametriccondition for stochastic P-bifurcation of the system usingsingularity theory based on this the system response canbe maintained at the monostability or small amplitudenear the equilibrium by selecting the appropriate unfoldingparameters providing theoretical guidance for system designin practical engineering and avoiding the instability anddamage caused by the large amplitude vibration or nonlinearjump phenomenon of the system Finally the numericalresults by Monte Carlo simulation of the original systemalso verify the theoretical results obtained in this paper Weconclude that the order 119901 of fractional derivative and thenoise intensity 119863 can both cause stochastic P-bifurcation ofthe system and the number of peaks in the stationary PDFcurve of system amplitude can change from 2 to 1 by selectingthe appropriate unfolding parameters (119901119863) it also showsthat the method used in this paper is feasible to analyze

Discrete Dynamics in Nature and Society 9

the stochastic P-bifurcation behaviors of viscoelastic materialsystem with fractional constitutive relation

Data Availability

Data that support the findings of this study are includedwithin the article (and its additional files) All other dataare available from the corresponding author upon reasonablerequest

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work was supported by the National 973 Project ofChina (Grant no 2014CB046805) and the Natural ScienceFoundation of China (Grant no 11372211)

References

[1] W Tan and M Xu ldquoPlane surface suddenly set in motion in aviscoelastic fluid with fractionalMaxwell modelrdquoActaMechan-ica Sinica English Series The Chinese Society of Theoretical andApplied Mechanics vol 18 no 4 pp 342ndash349 2002

[2] J Sabatier O P Agrawal and J A Machado Advance in Frac-tional Calculus Theoretical Developments and Applications inPhysics and Engineering Springer DordrechtThe Netherlands2007

[3] I Podlubny ldquoFractional order systems andPIDcontrollerrdquo IEEETransactions on Automatic Control vol 44 no 1 pp 208ndash2141999

[4] C A Monje Y Chen B M Vinagre D Xue and V FeliuFractional-Order Systems and Controls Fundamentals andApplications Advances in Industrial Control Springer LondonUK 2010

[5] Q Li Y L Zhou X Q Zhao and X Y Ge ldquoFractional orderstochastic differential equation with application in Europeanoption pricingrdquo Discrete Dynamics in Nature and Society vol2014 Article ID 621895 12 pages 2014

[6] S Nie H Sun Y Zhang et al ldquoVertical distribution ofsuspended sediment under steady flow existing theories andfractional derivative modelrdquo Discrete Dynamics in Nature andSociety vol 2017 Article ID 5481531 11 pages 2017

[7] X Liu L Hong H Dang and L Yang ldquoBifurcations andsynchronization of the fractional-order Bloch systemrdquo DiscreteDynamics in Nature and Society vol 2017 Article ID 469430510 pages 2017

[8] W Li and Y Pang ldquoAsymptotic solutions of time-space frac-tional coupled systems by residual power series methodrdquoDiscrete Dynamics in Nature and Society vol 2017 Article ID7695924 10 pages 2017

[9] Z Liu T Xia and JWang ldquoImage encryption technology basedon fractional two-dimensional triangle function combinationdiscrete chaotic map coupled with Menezes-Vanstone ellipticcurve cryptosystemrdquo Discrete Dynamics in Nature and Societyvol 2018 Article ID 4585083 24 pages 2018

[10] C Sanchez-Lopez V H Carbajal-Gomez M A Carrasco-Aguilar and ICarro-Perez ldquoFractional-OrderMemristor Emu-lator Circuitsrdquo Complexity vol 2018 Article ID 2806976 10pages 2018

[11] K Rajagopal A Karthikeyan P Duraisamy and R Weldegior-gis ldquoBifurcation and Chaos in Integer and Fractional OrderTwo-Degree-of-Freedom Shape Memory Alloy OscillatorsrdquoComplexity vol 2018 9 pages 2018

[12] Ying Li and Ye Tang ldquoAnalytical Analysis on Nonlinear Para-metric Vibration of an Axially Moving String with FractionalViscoelastic Dampingrdquo Mathematical Problems in Engineeringvol 2017 Article ID 1393954 9 pages 2017

[13] G Liu C Ye X Liang Z Xie and Z Yu ldquoNonlinear DynamicIdentification of Beams Resting on Nonlinear ViscoelasticFoundations BASed on the Time-DELayed TRAnsfer Entropyand IMProved SurrogateDATaAlgorithmrdquoMathematical Prob-lems in Engineering vol 2018 Article ID 6531051 14 pages 2018

[14] D Liu Y Wu and X Xie ldquoStochastic Stability Analysis of Cou-pled Viscoelastic SYStems with Nonviscously Damping DRIvenby White NOIserdquo Mathematical Problems in Engineering vol2018 Article ID 6532305 16 pages 2018

[15] P G Nutting ldquoAnew general law of deformationrdquo Journal ofTheFranklin Institute vol 191 no 5 pp 679ndash685 1921

[16] A Gemant ldquoA method of analyzing experimental resultsobtained fromelasto-viscous bodiesrdquo Journal of Applied Physicsvol 7 no 8 pp 311ndash317 1936

[17] G S Blair and J Caffyn ldquoVI An application of the theoryof quasi-properties to the treatment of anomalous strain-stressrelationsrdquo The London Edinburgh and Dublin PhilosophicalMagazine and Journal of Science vol 40 no 300 pp 80ndash942010

[18] CMote ldquoDynamic Stability of Axially MovingMateralsrdquo Shockand Vibration vol 4 no 4 pp 2ndash11 1972

[19] AUlsoy andCMote ldquoBand SawVibration and Stabilityrdquo Shockand Vibration vol 10 no 11 pp 3ndash15 1978

[20] J A Wickert and C D Mote Jr ldquoCurrent research on thevibration and stability of axially-moving materialsrdquo Shock andVibration vol 20 no 1 pp 3ndash13 1988

[21] K W Wang and S P Liu ldquoOn the noise and vibration of chaindrive systemsrdquo Shock and Vibration Digest vol 23 no 4 pp 8ndash13 1991

[22] F Pellicano and F Vestroni ldquoComplex dynamics of high-speedaxially moving systemsrdquo Journal of Sound and Vibration vol258 no 1 pp 31ndash44 2002

[23] N-H Zhang and L-Q Chen ldquoNonlinear dynamical analysis ofaxially moving viscoelastic stringsrdquo Chaos Solitons amp Fractalsvol 24 no 4 pp 1065ndash1074 2005

[24] M H Ghayesh H A Kafiabad and T Reid ldquoSub- and super-critical nonlinear dynamics of a harmonically excited axiallymoving beamrdquo International Journal of Solids and Structuresvol 49 no 1 pp 227ndash243 2012

[25] M H Ghayesh ldquoStability and bifurcations of an axially movingbeam with an intermediate spring supportrdquo Nonlinear Dynam-ics vol 69 no 1-2 pp 193ndash210 2012

[26] R Rodrıguez E Salinas-Rodrıguez and J Fujioka ldquoFractionalTime Fluctuations in Viscoelasticity A Comparative Study ofCorrelations and Elastic Modulirdquo Entropy vol 20 no 1 p 282018

[27] R L Bagley and P J Torvik ldquoFractional calculus-a differentapproach to the analysis of viscoelastically damped structuresrdquoAIAA Journal vol 21 no 5 pp 741ndash748 2012

10 Discrete Dynamics in Nature and Society

[28] R L Bagley and P J Torvik ldquoFractional calculus in the transientanalysis of viscoelastically damped structuresrdquo AIAA Journalvol 23 no 6 pp 918ndash925 1985

[29] M Pakdemirli and A G Ulsoy ldquoStability analysis of an axiallyaccelerating stringrdquo Journal of Sound andVibration vol 203 no5 pp 815ndash832 1997

[30] L Zhang and J W Zu ldquoNonlinear vibration of parametricallyexcited moving belts part I Dynamic responserdquo Journal ofApplied Mechanics vol 66 no 2 pp 396ndash402 1999

[31] L Zhang and J W Zu ldquoNonlinear vibration of parametricallyexcited viscoelastic moving belts part II Stability analysisrdquoJournal of Applied Mechanics vol 66 no 2 pp 403ndash409 1999

[32] L Q Chen X D Yang and C J Cheng ldquoDynamic stability ofan axially accelerating viscoelastic beamrdquo European Journal ofMechanic vol 23 no 4 pp 659ndash666 2004

[33] L Q Chen and X D Yang ldquoSteady-state response of axiallymoving viscoelastic beams with pulsating speed comparisonof two nonlinear modelsrdquo International Journal of Solids andStructures vol 42 no 1 pp 37ndash50 2005

[34] L Q Chen and X D Yang ldquoTransverse nonlinear dynamicsof axially accelerating viscoelastic beams based on 4-termGalerkin truncationrdquo Chaos Soliton and Fractal vol 27 no 3pp 748ndash757 2006

[35] H Ding and L-Q Chen ldquoGalerkin methods for natural fre-quencies of high-speed axially moving beamsrdquo Journal of Soundand Vibration vol 329 no 17 pp 3484ndash3494 2010

[36] L-Q Chen J Wu and J W Zu ldquoAsymptotic nonlinear behav-iors in transverse vibration of an axially acceleratingviscoelasticstringrdquo Nonlinear Dynamics vol 35 no 4 pp 347ndash360 2004

[37] L-Q Chen N-H Zhang and J W Zu ldquoThe regular andchaotic vibrations of an axially moving viscoelastic string basedon fourth order Galerkin truncationrdquo Journal of Sound andVibration vol 261 no 4 pp 764ndash773 2003

[38] A Y T Leung H X Yang and J Y Chen ldquoParametricbifurcation of a viscoelastic column subject to axial harmonicforce and time-delayed controlrdquo Computers amp Structures vol136 pp 47ndash55 2014

[39] MH Ghayesh andNMoradian ldquoNonlinear dynamic responseof axially moving stretched viscoelastic stringsrdquo Archive ofApplied Mechanics vol 81 no 6 pp 781ndash799 2011

[40] D Liu W Xu and Y Xu ldquoDynamic responses of axially movingviscoelastic beamunder a randomly disordered periodic excita-tionrdquo Journal of Sound and Vibration vol 331 no 17 pp 4045ndash4056 2012

[41] T Z Yang and B Fang ldquoStability in parametric resonance of anaxially moving beam constituted by fractional order materialrdquoArchive of AppliedMechanics vol 82 no 12 pp 1763ndash1770 2012

[42] A Y T Leung H X Yang P Zhu and Z J Guo ldquoSteady stateresponse of fractionally damped nonlinear viscoelastic archesby residue harmonic homotopyrdquo Computers amp Structures vol121 pp 10ndash21 2013

[43] A C Galucio J-F Deui and R Ohayon ldquoFinite elementformulation of viscoelastic sandwich beams using fractionalderivative operatorsrdquo Computational Mechanics vol 33 no 4pp 282ndash291 2004

[44] L Chen W Wang Z Li and W Zhu ldquoStationary responseof Duffing oscillator with hardening stiffness and fractionalderivativerdquo International Journal of Non-Linear Mechanics vol48 pp 44ndash50 2013

[45] L C Chen Z S Li Q Zhuang and W Q Zhu ldquoFirst-passagefailure of single-degree-of-freedom nonlinear oscillators with

fractional derivativerdquo Journal of Vibration and Control vol 19no 14 pp 2154ndash2163 2013

[46] Y Shen S Yang H Xing and G Gao ldquoPrimary resonance ofDuffing oscillator with fractional-order derivativerdquo Communi-cations in Nonlinear Science and Numerical Simulation vol 17no 7 pp 3092ndash3100 2012

[47] Y G Yang W Xu Y H Sun and X D Gu ldquoStochastic responseof van der Pol oscillator with two kinds of fractional derivativesunder Gaussian white noise excitationrdquo Chinese Physics B vol25 no 2 pp 13ndash21 2016

[48] L Chen and W Zhu ldquoStochastic response of fractional-ordervan der Pol oscillatorrdquoTheoretical amp Applied Mechanics Lettersvol 4 no 1 pp 68ndash72 2014

[49] W Li M Zhang and J Zhao ldquoStochastic bifurcations of gen-eralized Duffingndashvan der Pol system with fractional derivativeunder colored noiserdquoChinese Physics B vol 26 no 9 pp 62ndash692017

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Discrete Dynamics in Nature and Society 9

the stochastic P-bifurcation behaviors of viscoelastic materialsystem with fractional constitutive relation

Data Availability

Data that support the findings of this study are includedwithin the article (and its additional files) All other dataare available from the corresponding author upon reasonablerequest

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work was supported by the National 973 Project ofChina (Grant no 2014CB046805) and the Natural ScienceFoundation of China (Grant no 11372211)

References

[1] W Tan and M Xu ldquoPlane surface suddenly set in motion in aviscoelastic fluid with fractionalMaxwell modelrdquoActaMechan-ica Sinica English Series The Chinese Society of Theoretical andApplied Mechanics vol 18 no 4 pp 342ndash349 2002

[2] J Sabatier O P Agrawal and J A Machado Advance in Frac-tional Calculus Theoretical Developments and Applications inPhysics and Engineering Springer DordrechtThe Netherlands2007

[3] I Podlubny ldquoFractional order systems andPIDcontrollerrdquo IEEETransactions on Automatic Control vol 44 no 1 pp 208ndash2141999

[4] C A Monje Y Chen B M Vinagre D Xue and V FeliuFractional-Order Systems and Controls Fundamentals andApplications Advances in Industrial Control Springer LondonUK 2010

[5] Q Li Y L Zhou X Q Zhao and X Y Ge ldquoFractional orderstochastic differential equation with application in Europeanoption pricingrdquo Discrete Dynamics in Nature and Society vol2014 Article ID 621895 12 pages 2014

[6] S Nie H Sun Y Zhang et al ldquoVertical distribution ofsuspended sediment under steady flow existing theories andfractional derivative modelrdquo Discrete Dynamics in Nature andSociety vol 2017 Article ID 5481531 11 pages 2017

[7] X Liu L Hong H Dang and L Yang ldquoBifurcations andsynchronization of the fractional-order Bloch systemrdquo DiscreteDynamics in Nature and Society vol 2017 Article ID 469430510 pages 2017

[8] W Li and Y Pang ldquoAsymptotic solutions of time-space frac-tional coupled systems by residual power series methodrdquoDiscrete Dynamics in Nature and Society vol 2017 Article ID7695924 10 pages 2017

[9] Z Liu T Xia and JWang ldquoImage encryption technology basedon fractional two-dimensional triangle function combinationdiscrete chaotic map coupled with Menezes-Vanstone ellipticcurve cryptosystemrdquo Discrete Dynamics in Nature and Societyvol 2018 Article ID 4585083 24 pages 2018

[10] C Sanchez-Lopez V H Carbajal-Gomez M A Carrasco-Aguilar and ICarro-Perez ldquoFractional-OrderMemristor Emu-lator Circuitsrdquo Complexity vol 2018 Article ID 2806976 10pages 2018

[11] K Rajagopal A Karthikeyan P Duraisamy and R Weldegior-gis ldquoBifurcation and Chaos in Integer and Fractional OrderTwo-Degree-of-Freedom Shape Memory Alloy OscillatorsrdquoComplexity vol 2018 9 pages 2018

[12] Ying Li and Ye Tang ldquoAnalytical Analysis on Nonlinear Para-metric Vibration of an Axially Moving String with FractionalViscoelastic Dampingrdquo Mathematical Problems in Engineeringvol 2017 Article ID 1393954 9 pages 2017

[13] G Liu C Ye X Liang Z Xie and Z Yu ldquoNonlinear DynamicIdentification of Beams Resting on Nonlinear ViscoelasticFoundations BASed on the Time-DELayed TRAnsfer Entropyand IMProved SurrogateDATaAlgorithmrdquoMathematical Prob-lems in Engineering vol 2018 Article ID 6531051 14 pages 2018

[14] D Liu Y Wu and X Xie ldquoStochastic Stability Analysis of Cou-pled Viscoelastic SYStems with Nonviscously Damping DRIvenby White NOIserdquo Mathematical Problems in Engineering vol2018 Article ID 6532305 16 pages 2018

[15] P G Nutting ldquoAnew general law of deformationrdquo Journal ofTheFranklin Institute vol 191 no 5 pp 679ndash685 1921

[16] A Gemant ldquoA method of analyzing experimental resultsobtained fromelasto-viscous bodiesrdquo Journal of Applied Physicsvol 7 no 8 pp 311ndash317 1936

[17] G S Blair and J Caffyn ldquoVI An application of the theoryof quasi-properties to the treatment of anomalous strain-stressrelationsrdquo The London Edinburgh and Dublin PhilosophicalMagazine and Journal of Science vol 40 no 300 pp 80ndash942010

[18] CMote ldquoDynamic Stability of Axially MovingMateralsrdquo Shockand Vibration vol 4 no 4 pp 2ndash11 1972

[19] AUlsoy andCMote ldquoBand SawVibration and Stabilityrdquo Shockand Vibration vol 10 no 11 pp 3ndash15 1978

[20] J A Wickert and C D Mote Jr ldquoCurrent research on thevibration and stability of axially-moving materialsrdquo Shock andVibration vol 20 no 1 pp 3ndash13 1988

[21] K W Wang and S P Liu ldquoOn the noise and vibration of chaindrive systemsrdquo Shock and Vibration Digest vol 23 no 4 pp 8ndash13 1991

[22] F Pellicano and F Vestroni ldquoComplex dynamics of high-speedaxially moving systemsrdquo Journal of Sound and Vibration vol258 no 1 pp 31ndash44 2002

[23] N-H Zhang and L-Q Chen ldquoNonlinear dynamical analysis ofaxially moving viscoelastic stringsrdquo Chaos Solitons amp Fractalsvol 24 no 4 pp 1065ndash1074 2005

[24] M H Ghayesh H A Kafiabad and T Reid ldquoSub- and super-critical nonlinear dynamics of a harmonically excited axiallymoving beamrdquo International Journal of Solids and Structuresvol 49 no 1 pp 227ndash243 2012

[25] M H Ghayesh ldquoStability and bifurcations of an axially movingbeam with an intermediate spring supportrdquo Nonlinear Dynam-ics vol 69 no 1-2 pp 193ndash210 2012

[26] R Rodrıguez E Salinas-Rodrıguez and J Fujioka ldquoFractionalTime Fluctuations in Viscoelasticity A Comparative Study ofCorrelations and Elastic Modulirdquo Entropy vol 20 no 1 p 282018

[27] R L Bagley and P J Torvik ldquoFractional calculus-a differentapproach to the analysis of viscoelastically damped structuresrdquoAIAA Journal vol 21 no 5 pp 741ndash748 2012

10 Discrete Dynamics in Nature and Society

[28] R L Bagley and P J Torvik ldquoFractional calculus in the transientanalysis of viscoelastically damped structuresrdquo AIAA Journalvol 23 no 6 pp 918ndash925 1985

[29] M Pakdemirli and A G Ulsoy ldquoStability analysis of an axiallyaccelerating stringrdquo Journal of Sound andVibration vol 203 no5 pp 815ndash832 1997

[30] L Zhang and J W Zu ldquoNonlinear vibration of parametricallyexcited moving belts part I Dynamic responserdquo Journal ofApplied Mechanics vol 66 no 2 pp 396ndash402 1999

[31] L Zhang and J W Zu ldquoNonlinear vibration of parametricallyexcited viscoelastic moving belts part II Stability analysisrdquoJournal of Applied Mechanics vol 66 no 2 pp 403ndash409 1999

[32] L Q Chen X D Yang and C J Cheng ldquoDynamic stability ofan axially accelerating viscoelastic beamrdquo European Journal ofMechanic vol 23 no 4 pp 659ndash666 2004

[33] L Q Chen and X D Yang ldquoSteady-state response of axiallymoving viscoelastic beams with pulsating speed comparisonof two nonlinear modelsrdquo International Journal of Solids andStructures vol 42 no 1 pp 37ndash50 2005

[34] L Q Chen and X D Yang ldquoTransverse nonlinear dynamicsof axially accelerating viscoelastic beams based on 4-termGalerkin truncationrdquo Chaos Soliton and Fractal vol 27 no 3pp 748ndash757 2006

[35] H Ding and L-Q Chen ldquoGalerkin methods for natural fre-quencies of high-speed axially moving beamsrdquo Journal of Soundand Vibration vol 329 no 17 pp 3484ndash3494 2010

[36] L-Q Chen J Wu and J W Zu ldquoAsymptotic nonlinear behav-iors in transverse vibration of an axially acceleratingviscoelasticstringrdquo Nonlinear Dynamics vol 35 no 4 pp 347ndash360 2004

[37] L-Q Chen N-H Zhang and J W Zu ldquoThe regular andchaotic vibrations of an axially moving viscoelastic string basedon fourth order Galerkin truncationrdquo Journal of Sound andVibration vol 261 no 4 pp 764ndash773 2003

[38] A Y T Leung H X Yang and J Y Chen ldquoParametricbifurcation of a viscoelastic column subject to axial harmonicforce and time-delayed controlrdquo Computers amp Structures vol136 pp 47ndash55 2014

[39] MH Ghayesh andNMoradian ldquoNonlinear dynamic responseof axially moving stretched viscoelastic stringsrdquo Archive ofApplied Mechanics vol 81 no 6 pp 781ndash799 2011

[40] D Liu W Xu and Y Xu ldquoDynamic responses of axially movingviscoelastic beamunder a randomly disordered periodic excita-tionrdquo Journal of Sound and Vibration vol 331 no 17 pp 4045ndash4056 2012

[41] T Z Yang and B Fang ldquoStability in parametric resonance of anaxially moving beam constituted by fractional order materialrdquoArchive of AppliedMechanics vol 82 no 12 pp 1763ndash1770 2012

[42] A Y T Leung H X Yang P Zhu and Z J Guo ldquoSteady stateresponse of fractionally damped nonlinear viscoelastic archesby residue harmonic homotopyrdquo Computers amp Structures vol121 pp 10ndash21 2013

[43] A C Galucio J-F Deui and R Ohayon ldquoFinite elementformulation of viscoelastic sandwich beams using fractionalderivative operatorsrdquo Computational Mechanics vol 33 no 4pp 282ndash291 2004

[44] L Chen W Wang Z Li and W Zhu ldquoStationary responseof Duffing oscillator with hardening stiffness and fractionalderivativerdquo International Journal of Non-Linear Mechanics vol48 pp 44ndash50 2013

[45] L C Chen Z S Li Q Zhuang and W Q Zhu ldquoFirst-passagefailure of single-degree-of-freedom nonlinear oscillators with

fractional derivativerdquo Journal of Vibration and Control vol 19no 14 pp 2154ndash2163 2013

[46] Y Shen S Yang H Xing and G Gao ldquoPrimary resonance ofDuffing oscillator with fractional-order derivativerdquo Communi-cations in Nonlinear Science and Numerical Simulation vol 17no 7 pp 3092ndash3100 2012

[47] Y G Yang W Xu Y H Sun and X D Gu ldquoStochastic responseof van der Pol oscillator with two kinds of fractional derivativesunder Gaussian white noise excitationrdquo Chinese Physics B vol25 no 2 pp 13ndash21 2016

[48] L Chen and W Zhu ldquoStochastic response of fractional-ordervan der Pol oscillatorrdquoTheoretical amp Applied Mechanics Lettersvol 4 no 1 pp 68ndash72 2014

[49] W Li M Zhang and J Zhao ldquoStochastic bifurcations of gen-eralized Duffingndashvan der Pol system with fractional derivativeunder colored noiserdquoChinese Physics B vol 26 no 9 pp 62ndash692017

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

10 Discrete Dynamics in Nature and Society

[28] R L Bagley and P J Torvik ldquoFractional calculus in the transientanalysis of viscoelastically damped structuresrdquo AIAA Journalvol 23 no 6 pp 918ndash925 1985

[29] M Pakdemirli and A G Ulsoy ldquoStability analysis of an axiallyaccelerating stringrdquo Journal of Sound andVibration vol 203 no5 pp 815ndash832 1997

[30] L Zhang and J W Zu ldquoNonlinear vibration of parametricallyexcited moving belts part I Dynamic responserdquo Journal ofApplied Mechanics vol 66 no 2 pp 396ndash402 1999

[31] L Zhang and J W Zu ldquoNonlinear vibration of parametricallyexcited viscoelastic moving belts part II Stability analysisrdquoJournal of Applied Mechanics vol 66 no 2 pp 403ndash409 1999

[32] L Q Chen X D Yang and C J Cheng ldquoDynamic stability ofan axially accelerating viscoelastic beamrdquo European Journal ofMechanic vol 23 no 4 pp 659ndash666 2004

[33] L Q Chen and X D Yang ldquoSteady-state response of axiallymoving viscoelastic beams with pulsating speed comparisonof two nonlinear modelsrdquo International Journal of Solids andStructures vol 42 no 1 pp 37ndash50 2005

[34] L Q Chen and X D Yang ldquoTransverse nonlinear dynamicsof axially accelerating viscoelastic beams based on 4-termGalerkin truncationrdquo Chaos Soliton and Fractal vol 27 no 3pp 748ndash757 2006

[35] H Ding and L-Q Chen ldquoGalerkin methods for natural fre-quencies of high-speed axially moving beamsrdquo Journal of Soundand Vibration vol 329 no 17 pp 3484ndash3494 2010

[36] L-Q Chen J Wu and J W Zu ldquoAsymptotic nonlinear behav-iors in transverse vibration of an axially acceleratingviscoelasticstringrdquo Nonlinear Dynamics vol 35 no 4 pp 347ndash360 2004

[37] L-Q Chen N-H Zhang and J W Zu ldquoThe regular andchaotic vibrations of an axially moving viscoelastic string basedon fourth order Galerkin truncationrdquo Journal of Sound andVibration vol 261 no 4 pp 764ndash773 2003

[38] A Y T Leung H X Yang and J Y Chen ldquoParametricbifurcation of a viscoelastic column subject to axial harmonicforce and time-delayed controlrdquo Computers amp Structures vol136 pp 47ndash55 2014

[39] MH Ghayesh andNMoradian ldquoNonlinear dynamic responseof axially moving stretched viscoelastic stringsrdquo Archive ofApplied Mechanics vol 81 no 6 pp 781ndash799 2011

[40] D Liu W Xu and Y Xu ldquoDynamic responses of axially movingviscoelastic beamunder a randomly disordered periodic excita-tionrdquo Journal of Sound and Vibration vol 331 no 17 pp 4045ndash4056 2012

[41] T Z Yang and B Fang ldquoStability in parametric resonance of anaxially moving beam constituted by fractional order materialrdquoArchive of AppliedMechanics vol 82 no 12 pp 1763ndash1770 2012

[42] A Y T Leung H X Yang P Zhu and Z J Guo ldquoSteady stateresponse of fractionally damped nonlinear viscoelastic archesby residue harmonic homotopyrdquo Computers amp Structures vol121 pp 10ndash21 2013

[43] A C Galucio J-F Deui and R Ohayon ldquoFinite elementformulation of viscoelastic sandwich beams using fractionalderivative operatorsrdquo Computational Mechanics vol 33 no 4pp 282ndash291 2004

[44] L Chen W Wang Z Li and W Zhu ldquoStationary responseof Duffing oscillator with hardening stiffness and fractionalderivativerdquo International Journal of Non-Linear Mechanics vol48 pp 44ndash50 2013

[45] L C Chen Z S Li Q Zhuang and W Q Zhu ldquoFirst-passagefailure of single-degree-of-freedom nonlinear oscillators with

fractional derivativerdquo Journal of Vibration and Control vol 19no 14 pp 2154ndash2163 2013

[46] Y Shen S Yang H Xing and G Gao ldquoPrimary resonance ofDuffing oscillator with fractional-order derivativerdquo Communi-cations in Nonlinear Science and Numerical Simulation vol 17no 7 pp 3092ndash3100 2012

[47] Y G Yang W Xu Y H Sun and X D Gu ldquoStochastic responseof van der Pol oscillator with two kinds of fractional derivativesunder Gaussian white noise excitationrdquo Chinese Physics B vol25 no 2 pp 13ndash21 2016

[48] L Chen and W Zhu ldquoStochastic response of fractional-ordervan der Pol oscillatorrdquoTheoretical amp Applied Mechanics Lettersvol 4 no 1 pp 68ndash72 2014

[49] W Li M Zhang and J Zhao ldquoStochastic bifurcations of gen-eralized Duffingndashvan der Pol system with fractional derivativeunder colored noiserdquoChinese Physics B vol 26 no 9 pp 62ndash692017

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom