Available online at www.sciencedirect.com

Journal of the Franklin Institute 354 (2017) 7917–7945 www.elsevier.com/locate/jfranklin

Higher-order stochastic averaging for a SDOF

fractional viscoelastic system under bounded noise

excitation

Jian Deng

Department of Civil Engineering, Lakehead University, Thunder Bay, Ontario P7B 5E1, Canada

Received 24 November 2016; received in revised form 1 September 2017; accepted 20 September 2017 Available online 28 September 2017

Abstract

The moment Lyapunov exponents and stochastic stability of a single-degree-of-freedom (SDOF) frac- tional viscoelastic system under bounded noise excitation are studied by using the method of higher-order stochastic averaging. A realistic example of such a system is the transverse vibration of a viscoelastic column under the excitation of stochastic axial compressive load. The excitation is modeled as a bounded noise, which is a realistic model of stochastic fluctuation in engineering applications. The viscoelastic material is assumed to follow a fractional Kelvin–Voigt constitutive relation. The method of higher-order stochastic averaging is used to approximate the fractional stochastic differential equation of motion, and then moment Lyapunov exponents are determined for the system with small damping and weak random

fluctuation. The approximate results are confirmed by Monte-Carlo simulations. It is found that conver- gence of moment Lyapunov exponents depends on the width of power spectral density of the bounded noise process. For this viscoelastic structure, second-order averaging analysis is adequate for stability analysis. The effects of various parameters on the stochastic stability of the system are discussed and possible explanations are explored. © 2017 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

h0

E-mail address: jian.deng@lakeheadu.ca .

ttps://doi.org/10.1016/j.jfranklin.2017.09.019 016-0032/© 2017 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

7918 J. Deng / Journal of the Franklin Institute 354 (2017) 7917–7945

1. Introduction

Viscoelasticity is referred to as a mechanical behavior that combines liquidlike and solid- like characteristics. Viscoelasticity has been observed in many engineering materials, such

as polymers, concrete, metals at elevated temperatures, and rocks, which have been used

extensively in industrial world [1] . Most theories on viscoelasticity are devoted to traditionalbinary viscoelastic models allowing for only two possible states for its components: pure elas-ticity and pure viscosity, where Hoek’s spring and Newton’s dash-pot are used to representthe purely elastic and purely viscous behaviors, respectively. Several conventional mechanical models for viscoelasticity were proposed. For example, the Maxwell model is represented

by a dash-pot and a spring connected in series, and the Kelvin–Voigt model consists of adash-pot and a spring in parallel.

However, many real-world materials are composed of multi-state components, which have various performance levels between elasticity and viscosity, with various effects on the entire system performance. By using the theory of fractional calculus ( Appendix A ), another com-ponent can be formulated with property between spring and dash-pot, which is introduced

in Appendix B . Application of fractional calculus in viscoelasticity [2] and stability analysisof fractional-order systems can be found in theoretical and practical studies [3] . However, loadings from earthquakes, explosion, wind, and ocean waves can be described satisfactorily

only in probabilistic domain, which results in the fact that the equation of motion of the vis-coelastic system under such excitations is usually governed by stochastic differential equation

and the response and stability of the system are difficult to be obtained exactly. The methodof stochastic averaging has been widely used to approximately solve stochastic differential equations containing a small parameter. Under certain conditions, the stochastic averaging is able to reduce the dimension of problems and then greatly simplify the solution [4] . Thepopularity of stochastic averaging can be felt from the overwhelming number of papers inthe literature [5,6] .

The basic idea of stochastic averaging method is to approximate the original Stratonovich

stochastic system by an averaged Itô stochastic system (diffusive Markov process). The Itôsystem is presumably easier to study, and we can infer properties of the dynamics of theoriginal system by the understanding of the dynamics of the averaged system. It has beenshown that the ordinary stochastic averaging is a first order approximation method [4] . Denget al. [7] investigated stochastic stability of a fractional viscoelastic column under bounded

noise excitation using a first-order stochastic averaging method. A systematic and unified

approach of second-order stochastic averaging based on the Stratonovich–Khasminskii limit theorem was presented [8] . Huang and Xie found that the second-order averaging resultsagree better with those estimated by Monte Carlo simulation than the first-order averaging

[9] . More recently, moment Lyapunov exponents of Hill equation were studied by the methodof stochastic averaging, both the first-order and the second-order [10] .

The aim of the present paper is to study the stability of fractional viscoelastic systemsunder bounded noise excitation by using the method of higher-order stochastic averaging. The paper is an extention of [7] from stochastic averaging of first-order to higher-order. Italso extends [9–11] from white noise to bounded noise.

The paper is structured as follows. Section 2 derives the stochastic equation of motionand uses the method of higher order stochastic averaging to obtain Itô stochastic differen- tial equations. Results and discussions are given in Section 3 . Conclusions are presented inSection 4 .

J. Deng / Journal of the Franklin Institute 354 (2017) 7917–7945 7919

2

2

c

o

a

w

A

t

a

M

T

wL

E

ρ

f

v

S

o

t

q

w

β

b

W

a

r

η

w

a

. Stochastic averaging

.1. Formulation

Consider the stability problem of a column of uniform cross section under dynamic axialompressive load F ( t ). The study of such as vibrating column would yield interesting insightsn the behavior of a much wider class of structures [12] . The equation of motion is deriveds [4]

∂ 2 M(x, t )

∂x 2 = ρA

∂ 2 v(x, t )

∂t 2 + β0

∂v(x, t )

∂t + F (t )

∂ 2 v(x, t )

∂x 2 , (2.1)

here L is the length of the column, ρ is the mass density per unit volume of the column, is the cross-sectional area, v(x, t ) is the transverse displacement of the central axis, β0 is

he damping constant. The moment M ( x , t ) at the cross-section x and the geometry relationre

(x, t ) =

∫ A σ (x , t ) z d A, ε(x , t ) = −∂ 2 v(x, t )

∂x 2 z. (2.2)

he viscoelastic material is supposed to follow fractional constitutive model in Eq. (B.1) ,hich can be recast as σ (t ) = [ E + η RL

0 D

μt ] ε(t ) . Here RL

0 D

μt is an operator for Riemann–

iouville(RL) fractional derivative of order μ, which is defined in Eq. (A.8) . Substituting intoq. (2.2) and then to Eq. (2.1) yields

A

∂ 2 v(x, t )

∂t 2 + β0

∂v(x, t )

∂t + E I

∂ 4 v(x, t )

∂x 4 + I η RL

0 D

μt

[∂ 4 v(x, t )

∂x 4

]+ F (t )

∂ 2 v(x, t )

∂x 2 = 0. (2.3)

If the column is simply supported, the solution of Eq. (2.3) can be written in the followingorm:

(x, t ) = q(t ) sin

πx

L

. (2.4)

ubstitute Eq. (2.4) into Eq. (2.3) , suppose the damping, viscoelastic effect, and the amplitudef load are all small, and introduce a small parameter 0 < ε�1. If the function F ( t ) is takeno be a stochastic process, then the equation of motion is given by

¨ (t ) + 2εβ ˙ q (t ) + ω

2 [1 − εξ (t ) + ετε RL 0 D

μt

]q(t ) = 0, (2.5)

here

=

β0

2ρA

, ω

2 =

E I

ρA

(π

L

)4 , P = E I

(π

L

)2 , ξ (t ) =

F (t )

P

, τε =

η

E

. (2.6)

Mathematically, random excitations can be described as stochastic processes, either wide-and noise processes or narrow-band noise processes. A typical wide-band noise process ishite noise which is not a physically realizable process. A bounded noise η( t ) is a realistic

nd versatile narrow-band stochastic fluctuation in engineering applications and is normallyepresented as

(t ) = ζ cos [νt + ε 1 / 2 σW (t ) + θ

], (2.7)

here ζ is the noise amplitude, σ is the noise intensity, W ( t ) is the standard Wiener process,nd θ is a random variable uniformly distributed in the interval [0, 2 π ]. It overcomes the

7920 J. Deng / Journal of the Franklin Institute 354 (2017) 7917–7945

shortcoming of Ornstein–Uhlenbeck process, which is not bounded. The inclusion of the phase angle θ makes the bounded noise η( t ) a stationary process. Eq. (2.7) may be written as

η(t ) = ζ cos Z (t ) , d Z (t ) = νt + σd W (t ) , (2.8)

where the initial condition of Z ( t ) is Z(0) = θ . This process is of course bounded between−ζ and + ζ for all time t and hence is a bounded stochastic process. The auto-correlationfunction of η( t ) is given by

R(τ ) = E [ η(t ) η(t + τ ) ] =

1

2

ζ 2 cos ντ exp

(−σ 2

2

| τ | )

, (2.9)

and the spectral density function of η( t ) is

S(ω) =

∫ + ∞

−∞

R(τ ) e −iωτ d τ =

ζ 2 σ 2

(ω

2 + ν2 +

1

4

σ 4

)

2

[(ω + ν) 2 +

1

4

σ 4

][(ω − ν) 2 +

1

4

σ 4

] . (2.10)

When the noise intensity σ is small, the bounded noise can be used to model a processabout frequency ν. Hence, the bounded noise is a very good realistic model of narrow-bandnoise and is widely used in many engineering applications. In the limit as σ approaches zero,the bounded noise reduces to a deterministic sinusoidal function. On the other hand, in thelimit as σ approaches infinite, the bounded noise becomes a white noise of constant spectraldensity. However, since the mean-square value is fixed at 1

2 , this constant spectral densitylevel reduces to zero in the limit. The bounded noise process has been widely applied inengineering [4] , such as wind turbulence and earthquake load.

The moment stability of a stochastic dynamical system with state vector x ( t ) is determinedby moment Lyapunov exponents, which are defined by Xie [4]

�(p) = lim

t → ∞

1

t log E

[‖ x(t ) ‖ p ], (2.11)

where E [ ·] denotes the expected value and ‖ · ‖ represents a suitable vector norm. The p thmoments are asymptotically stable if �( p ) < 0. Moreover, �( p ) is a convex function of p and�′ ( p ) is equal to the largest Lyapunov exponent λ, which is defined by

λ = lim

t → ∞

1

t log ‖ x(t ) ‖ , (2.12)

and describes the almost-sure or sample stability of the system. As shown in Fig. 1 , thealmost-sure stability or instability cannot always guarantee the moment stability, which can

be explained by the theory of large deviation [4] . Thus, it is important to obtain the momentLyapunov exponents of stochastic systems so that the complete properties of dynamic stability

can be described. However, it is difficult to obtain the exact moment Lyapunov exponents of the stochastic

system concerned in Eq. (2.5) . In this paper, the method of higher-order stochastic averagingwill be used to obtain the differential equations governing the p th moment. The momentstability of viscoelastic system (2.5) can then be determined by solving the averaged equations.

2.2. First-order stochastic averaging

Because higher-order stochastic averaging is based on lower-order, for completeness, this subsection introduces first-order stochastic averaging, which was used in [7] . To apply the

J. Deng / Journal of the Franklin Institute 354 (2017) 7917–7945 7921

Fig. 1. Almost sure stability and moment stability.

ab

q

S

a

a

w

o

P

ϕ

w

a

R

d

ξ

S

veraging method, the transformation is made to the amplitude and phase variable a and ϕ

y means of the relations

(t ) = a(t ) cos �(t ) , ˙ q (t ) = −ω a(t ) sin �(t ) , �(t ) =

ν

2

t + ϕ(t ) . (2.13)

ubstituting Eq. (2.13) into Eq. (2.5) yields

˙ cos �(t ) − a ̇ ϕ sin �(t ) = −ε�a sin �(t ) ,

˙ sin �(t ) + a ̇ ϕ cos �(t ) = ε�a cos �(t ) − 2εaβ sin �(t )

+ εωξ (t ) a cos �(t ) + εωτε RL 0 D

μt q(s) , (2.14)

here ε� = ω − 1 2 ν. Solving Eq. (2.14) and assuming P = a

p , the p th norm of system (2.5) ,ne can obtain

˙ (t ) = εpP

[−2β sin

2 �(t ) +

1

2

ωξ(t ) sin 2�(t ) + ωτε U

ss

],

˙ (t ) = ε [� − β sin 2�(t ) + ω cos 2 �(t ) ξ (t ) + ωτε U

cs ], (2.15)

here

U

ss = −ω sin �(t )

�(1 − μ)

∫ t 0

[P (s)

P (t )

]1 /p sin �(s)

(t − s) μd s,

U

cs = −ω cos �(t )

�(1 − μ)

∫ t 0

[P (s)

P (t )

]1 /p sin �(s)

(t − s) μd s,

nd for ease of presentation, the fractional derivative in Eq. (A.8) is recast as

L 0 D

μt

[f (τ ) g(t )

] =

g(t )

�(1 − μ)

∫ t

0

f ′ (τ )

(t − τ ) μd τ. (2.16)

The bounded noise is recast as, by assuming that the magnitude is small and then intro-ucing a small parameter ε1/2 ,

(t ) = ζ cos [ νt + ψ(t ) ] , ψ(t ) = ε 1 / 2 ϑW (t ) + θ.

ubstituting ξ ( t ) in Eq. (2.15) yields

7922 J. Deng / Journal of the Franklin Institute 354 (2017) 7917–7945

˙ P (t ) = εpP

{−β[1 − cos 2�] +

ζω

2

cos [ νt + ψ(t )] sin 2� + ωτε U

ss

},

˙ ϕ (t ) = ε {� − β sin 2� + ωζ cos [ νt + ψ(t )] cos 2 � + ωτε U

cs },

˙ ψ (t ) = ε 1 / 2 ϑ

˙ W (t ) . (2.17)

Eqs. (2.17) are exactly equivalent to Eq. (2.5) and still cannot be accurately solved. However,it is observed the presence of a small parameter ε means both P and ϕ change slowly.Therefore, one can expect to obtain reasonably accurate results by averaging the response over one period. This may be done by applying the averaging operator given by

M

τ(·) = lim

T →∞

1

T

∫ τ+ T

τ

(·) d τ,

where ( · ) represents the equation of motion. When applying the averaging operator the inte-gration is performed over explicitly appearing τ only. The averaging method can be applied

to obtain the averaged equations as follows, without distinction between the averaged and theoriginal non-averaged variables P and ϕ. When applying the averaging operation, P ( t ) andϕ( t ) are treated as unchanged, i.e., they are replaced by the corresponding averaged variablesdirectly.

˙ P (t ) = εpP

[−β +

1

4

ζω sin (2ϕ − ψ) + ωτε M

t

{U

ss }]

,

˙ ϕ (t ) = ε

[� +

1

4

ζω cos (2ϕ − ψ) + ωτε M

t

{U

cs }]

, (2.18)

where some averaged identities have been used,

M

t ( cos 2�) = M

t ( sin 2�) = 0,

M

t

{ cos [ νt + ψ(t )] sin 2�

} =

1

2

sin (2ϕ − ψ) ,

M

t

{ cos [ νt + ψ(t )] cos 2 �

} =

1

4

cos (2ϕ − ψ) . (2.19)

Applying the transformation τ = t − s and changing the order of integration lead to

M

t

{U

ss } = −ω

2

H

c (ν

2

), M

t

{U

cs } =

ω

2

H

s (ν

2

), (2.20)

where

H

c (ν

2

)=

1

�(1 − μ)

∫ ∞

0 τ−μ cos

ντ

2

d τ =

(ν

2

)μ−1 sin

μπ

2

,

H

s (ν

2

)=

1

�(1 − μ)

∫ ∞

0 τ−μ sin

ντ

2

d τ =

(ν

2

)μ−1 cos

μπ

2

. (2.21)

Substituting Eqs. (2.20) into Eq. (2.18) yields

˙ P (t ) = εpP

[− ˆ β +

1

4

ζω sin (2ϕ − ψ)

],

˙ ϕ (t ) = ε

[ˆ � +

1

4

ζω cos (2ϕ − ψ)

], (2.22)

J. Deng / Journal of the Franklin Institute 354 (2017) 7917–7945 7923

w

β

d

d

d

w

m

f

S

t

d

w

F

h

c

a

d

C

t

w

m

fi

e

here

ˆ = β +

1

2

ωτε H

c (ν

2

), ˆ � = � +

1

2

ωτε H

s (ν

2

). (2.23)

Introducing the transformation � = ϕ − 1 2 ψ and Itô Lemma results in three Itô stochastic

ifferential equations (SDE)

d P (t ) = m P P d t,

�(t ) = m � d t − ε 1 / 2 ϑ

2

d W (t ) ,

ψ(t ) = ε 1 / 2 ϑ d W (t ) , (2.24)

here

P = εp

[− ˆ β +

1

4

ζω sin (2�)

], m � = ε

[ˆ � +

1

4

ζω cos (2�)

]. (2.25)

The eigenvalue problem governing the p th moment Lyapunov exponent can be formulatedrom these SDEs. Applying a linear stochastic transformation [13]

= T (�) P, P = T −1 (�) S, 0 ≤ � ≤ π, (2.26)

he Itô equation for the transformed p th norm process S is given by,

S =

(m P T + m �T � +

εϑ

2

8

T ��

)P d t − ε 1 / 2 ϑ

2

T �P d W (t )

=

1

T

(m P T + m �T � +

εϑ

2

8

T ��

)S d t − ε 1 / 2 ϑ

2

T �T

S d W (t ) , (2.27)

here T � and T �� are the first and second derivatives of T with respect to �, respectively.or bounded and non-singular transformation T ( �), both processes P and S are expected toave the same stability behavior, as Eq. (2.26) is a linear transformation. Therefore, T ( �) ishosen so that the drift term of the Itô Eq. (2.27) is independent of �, i.e.,

1

T

(m P T + m �T � +

εϑ

2

8

T ��

)= �,

nd

S = �S d t − ε 1 / 2 ϑ

2

T �T

S d W (t ) . (2.28)

omparing the drift terms in Eqs. (2.27) and (2.28) , such a transformation T ( �) is given byhe following differential equation

ϑ

2

8

T �� + m �1 T � + m p1 T =

ˆ �T , (2.29)

here

�1 =

m �

ε =

ˆ � +

1

4

ζω cos (2�) , m p1 =

m p

ε = p

[− ˆ β +

1

4

ζω sin (2�)

]. (2.30)

T ( �) is a periodic function in � of period π , and �q(t ) (p) = �(p) = ε ̂ �(p) . Eq. (2.29) de-nes an eigenvalue problem for a second-order differential operator with

ˆ �(p) being theigenvalue and T ( �) the associated eigenfunction.

7924 J. Deng / Journal of the Franklin Institute 354 (2017) 7917–7945

Taking the expected value of Eq. (2.28) leads to d E [ S] = �E [ S] d t, from which it can beseen that � is the Lyapunov exponent of the transformed p th moment E [ S] of Eqs. (2.24) orEq. (2.5) . Since both processes P and S have the same stability behavior, � is the Lyapunovexponent of p th moment E [ P ] . The remaining task for determining the moment Lyapunovexponents is to solve the eigenvalue problem.

2.3. Determination of moment Lyapunov exponents

Since the coefficients in Eq. (2.29) are periodic with period π , it is reasonable to considera Fourier series expansion of the eigenfunction T ( �) in the form

T (�) = C 0 +

K ∑

k=1

( C k cos 2k� + S k sin 2k�) . (2.31)

To solve Eq. (2.29) , substituting the expansion into eigenvalue problem (2.29) and equating

the coefficients of like trigonometric terms sin 2 k � and cos 2 k �, k = 0, 1 , . . . , K yield a setof homogeneous linear algebraic equations with infinitely many equations for the unknown

coefficients C 0 , C k and S k ,

Constant : − (c 0 +

ˆ �) C 0 + b 0 S 1 = 0,

cos 2� : − (c 1 +

ˆ �) C 1 + d 1 S 1 + b 1 S 2 = 0,

sin 2� : 2a 1 C 0 + d 1 C 1 + b 1 C 2 + (c 1 +

ˆ �) S 1 = 0,

cos 2k� : a k S k−1 + d k S k + b k S k+1 − (c k +

ˆ �) C k = 0,

sin 2k� : a k C k−1 + d k C k + b k C k+1 + (c k +

ˆ �) S k = 0,

cos 2K � : a K S K−1 + d K S K − (c K +

ˆ �) C K = 0,

sin 2K � : a K C K−1 + d K C K + (c K +

ˆ �) S K = 0, (2.32)

where

k = 2, 3 , . . . , K − 1 ,

a n =

1

8

ζω(2n − 2 − p) , b n =

1

8

ζω(2n + 2 + p) ,

c n =

1

2

σ 2 n

2 + p ̂

β, d n = 2n ̂

�, n = 0, 1 , 2, . . . , K.

To have a non-trivial solution of the C 0 , C k and S k , it is required that the determinantof the coefficient matrix of Eqs. (2.32) equal zero to yield a polynomial equation of degree2K + 1 for ˆ �(K ) (p) ,

e (K ) 2K+1

[ ˆ �(K )

] 2K+1 + e (K )

2K

[ ˆ �(K )

] 2K + · · · + e (K )

1 ˆ �(K ) + e (K )

0 = 0, (2.33)

where ˆ �(K ) denotes the approximate moment Lyapunov exponent. Then, one may approximate the moment Lyapunov exponent of the system by

�q(t ) (p) ≈ ε ̂ �(K ) (p) . (2.34)

For K = 1 , Eq. (2.33) reduces to a cubic equation with

J. Deng / Journal of the Franklin Institute 354 (2017) 7917–7945 7925

e

e

e

a

�

i

W

a

2

P

w

d

P

w

f

E

{

a

A

(1) 3 = 1 , e (1)

2 = σ 2 + 3 p ̂

β,

(1) 1 = 4 ̂

�2 +

σ 4

4

− 1

32

ζ 2 ω

2 p(p + 2) + 3 ̂

β2 p

2 + 2σ 2 ˆ βp,

(1) 0 =

ˆ β3 p

3 + σ 2 ˆ β2 p

2 + p ̂

β

[4 ̂

�2 +

σ 4

4

− 1

32

ζ 2 ω

2 p(p + 2)

]− 1

64

ζ 2 ω

2 σ 2 p(p + 2) ,

nd an approximate expression of the moment Lyapunov exponent is given by

ˆ (1) (p) =

A 2

6

− 2(3 e 1 − e 2 2 )

3 A 2 − e 2

3

,

A 2 =

(12A 1 − 108 e 0 + 36 e 1 e 2 − 8 e 3 2

)1 / 3 ,

A 1 =

(81 e 2 0 + 12e 3 1 − 54e 0 e 1 e 2 + 12e 0 e

3 2 − 3 e 2 1 e

2 2

)1 / 2 , (2.35)

n which the superscript “(1)” in the coefficients e ′ s is dropped for clarity of presentation.hen K > 1, there are no analytical solutions for the polynomial Eq. (2.33) and numerical

pproaches must be applied to solve it.

.4. Second-order stochastic averaging

Rewrite the transformed equation of motion in Eq. (2.17)

˙ (t ) = ε f 1 (P, ϕ, t ) , ˙ ϕ (t ) = ε f 2 (P, ϕ, t ) , ˙ ψ (t ) = ε 1 / 2 σ ˙ W (t ) , (2.36)

here

f 1 (P, ϕ, t ) = pP

{−β[ 1 − cos 2�(t ) ] +

ζω

2

cos [ νt + ψ(t ) ] sin 2� + ωτε U

sc

},

f 2 (P, ϕ, t ) = � − β sin 2� + ωζ cos [ νt + ψ(t ) ] cos 2 � + ωτε U

cc . (2.37)

To formulate a second-order stochastic averaging, the near-identity transformation is intro-uced [8]

(t ) = P̄ (t ) + ε P 1 (P̄ , ϕ̄ , t

), ϕ(t ) = ϕ̄ (t ) + ε ϕ 1

(P̄ , ϕ̄ , t

), (2.38)

here P̄ (t ) and ϕ̄ (t ) stand for nonoscillatory amplitude and phase angle, respectively. Dif-erentiating Eq. (2.38) and equating each result with the corresponding functions fromq. (2.36) give

˙ P̄

˙ ϕ̄

}

+ ε

⎧ ⎪ ⎨

⎪ ⎩

∂P 1 ∂ ̄P

˙ P̄ +

∂P 1 ∂ ̄ϕ

˙ ϕ̄ +

∂P 1 ∂t

∂ϕ 1 ∂ ̄P

˙ P̄ +

∂ϕ 1 ∂ ̄ϕ

˙ ϕ̄ +

∂ϕ 1 ∂t

⎫ ⎪ ⎬

⎪ ⎭

= ε

⎧ ⎪ ⎨

⎪ ⎩

f 1 (

P̄ + ε P 1 , ϕ̄ + ε ϕ 1 , t )

f 2 (

P̄ + ε P 1 , ϕ̄ + ε ϕ 1 , t )⎫ ⎪ ⎬

⎪ ⎭

. (2.39)

Substituting the Taylor expansion

f j (

P̄ + ε P 1 , ϕ̄ + ε ϕ 1 , t )

= f j ( ̄P , ϕ̄ , t ) + ε P 1 ∂ f j ∂ P̄

+ ε ϕ 1 ∂ f j ∂ ϕ̄

+ o (ε) , j = 1 , 2, (2.40)

nd the inverse matrix

=

⎡

⎣

1 + ε ∂P 1 ∂ ̄P

ε ∂P 1 ∂ ̄ϕ

ε ∂ϕ 1 ∂ ̄P

1 + ε ∂ϕ 1 ∂ ̄ϕ

⎤

⎦ , A

−1 =

⎡

⎣

1 − ε ∂P 1 ∂ ̄P

−ε ∂P 1 ∂ ̄ϕ

−ε ∂ϕ 1 ∂ ̄P

1 − ε ∂ϕ 1 ∂ ̄ϕ

⎤

⎦ + o (ε) (2.41)

7926 J. Deng / Journal of the Franklin Institute 354 (2017) 7917–7945

into Eq. (2.39) yields

˙ P̄ = ε u 1 + ε 2 u 2 + o

(ε 2 ), ˙ ϕ̄ = ε v 1 + ε 2 v 2 + o

(ε 2 ), (2.42)

where

u 1 = f 1 − ∂P 1

∂t , u 2 =

∂ f 1 ∂ P̄

P 1 +

∂ f 1 ∂ ϕ̄

ϕ 1 − ∂P 1

∂ P̄

f 1 − ∂P 1

∂ ϕ̄

f 2 +

∂P 1

∂t

∂P 1

∂ P̄

+

∂ϕ 1

∂t

∂P 1

∂ ϕ̄

,

v 1 = f 2 − ∂ϕ 1

∂t , v 2 =

∂ f 2 ∂ P̄

P 1 +

∂ f 2 ∂ ϕ̄

ϕ 1 − ∂ϕ 1

∂ P̄

f 1 − ∂ϕ 1

∂ ϕ̄

f 2 +

∂P 1

∂t

∂ϕ 1

∂ P̄

+

∂ϕ 1

∂t

∂ϕ 1

∂ ϕ̄

. (2.43)

Now efforts are being made to obtain u 1 , u 2 , v 1 and v 2 , and their averaged values. Firstly,to obtain u 1 and v 1 , one can assume that

∂P 1

∂t = f 1 − M

t { f 1 } , ∂ϕ 1

∂t = f 2 − M

t { f 2 } , (2.44)

then the terms f 1 − (∂ P 1 /∂ t ) and f 2 − (∂ ϕ 1 /∂ t ) contain only nonoscillatory terms. It alsosuggests that P 1 and ϕ1 are oscillatory functions. Substituting Eqs. (2.20) into the averaged

Eq. (2.37) yields

u 1 = M

t { f 1 } = pP

[− ˆ β +

ζω

4

sin (2ϕ − ψ)

],

v 1 = M

t { f 2 } =

ˆ � +

ζω

4

cos (2ϕ − ψ) , (2.45)

where P and ϕ are treated as a constant under the averaging operation. Substituting

Eqs. (2.37) and (2.45) into Eq. (2.44) yields

∂P 1

∂t =

1

4

p ̄P

[ 4β cos (νt + 2 ̄ϕ ) + ωζ sin (ψ − 2 ̄ϕ ) + 2ωτε H

s (ν

2

)sin (νt + 2 ̄ϕ )

+ 2ωτε H

c (ν

2

)cos (νt + 2 ̄ϕ ) + 2ωζ cos (νt + ψ) sin (νt + 2 ̄ϕ )

] ,

∂ϕ 1

∂t = −β sin ( νt + 2 ̄ϕ ) +

1

2

ωτε H

s (ν

2

)cos ( νt + 2 ̄ϕ ) − 1

2

ωτε H

c (ν

2

)sin (νt + 2 ̄ϕ )

− 1

4

ωζ cos (ψ − 2 ̄ϕ ) +

1

2

ζω cos (νt + ψ) +

1

2

ωζ cos (νt + ψ) cos (νt + 2 ̄ϕ ) . (2.46)

During the derivation of Eq. (2.46) , the following equations must be used,

U

sc = − ν

2�(1 − μ)

∫ t

0

sin

(ν

2

t + ϕ̄

)sin

(ν

2

s + ϕ̄

)(t − s) μ

d s

= − ν

2�(1 − μ)

[ ∫ ∞

0

sin

(νt 2 + ϕ̄

)sin

(νt

2

− ντ

2

+ ϕ̄

)τμ

d τ

−∫ ∞

t

sin

(νt

2

+ ϕ̄

)sin

(νt

2

− ντ

2

+ ϕ̄

)τμ

d τ

]

J. Deng / Journal of the Franklin Institute 354 (2017) 7917–7945 7927

T

U

ϕ

r

w

U

=

ν

4�(1 − μ)

∫ ∞

0

cos (νt − ν

2

τ + 2 ̄ϕ

)τμ

d τ − 1

2

H

c (ν

2

)

= −1

2

H

c (ν

2

)+

1

2

[ cos (νt + 2 ̄ϕ ) H

c (ν

2

)+ sin (νt + 2 ̄ϕ ) H

s (ν

2

)] . (2.47)

he third equation exists when t → ∞ . Similarly, one may obtain

cc =

1

2

H

s (ν

2

)− 1

2

[ sin (νt + 2 ̄ϕ ) H

c (ν

2

)− cos (νt + 2 ̄ϕ ) H

s (ν

2

)] ,

U

ss =

1

2

H

s (ν

2

)+

1

2

[ sin (νt + 2 ̄ϕ ) H

c (ν

2

)− cos (νt + 2 ̄ϕ ) H

s (ν

2

)] ,

U

cs =

1

2

H

c (ν

2

)+

1

2

[ cos (νt + 2 ̄ϕ ) H

c (ν

2

)+ sin (νt + 2 ̄ϕ ) H

s (ν

2

)] . (2.48)

Solving these ordinary first-order differential equations in Eqs. (2.46) results in

P 1 =

p ̄P

8 ν

{4

[ 2β + ωτε H

s (ν

2

)] sin (νt + 2 ̄ϕ ) − 4ωτε H

s (ν

2

)cos (νt + 2 ̄ϕ )

−ωζ cos (2νt + ψ + 2 ̄ϕ )

},

1 =

1

8 ν

{4

[ 2β + ωτε H

c (ν

2

)] cos (νt + 2 ̄ϕ ) + 4ωζ sin (νt + ψ)

+4ωτε H

s (ν

2

)sin (νt + 2 ̄ϕ ) + ωζ sin (2νt + ψ + 2 ̄ϕ )

}. (2.49)

Secondly, to obtain u 2 and v 2 , the following derivatives of f 1 and f 2 in Eqs. (2.37) areequired

∂ f 1 ∂ P̄

= −2βp sin

2 � +

p

2

ωζ cos (νt + ψ) sin 2� + ωτε

(pU

sc + P̄ V

sc ),

∂ f 1 ∂ ϕ̄

= p ̄P

[−2β sin 2� + ωζ cos (νt + ψ) cos2� + ωτε ( U

cc − U

ss ) ],

∂ f 2 ∂ P̄

=

ωτε V

cc

p

,

∂ f 2 ∂ ϕ̄

= −2β cos 2� − ωζ cos (νt + ψ) sin 2� − ωτε ( U

cs + U

sc ) , (2.50)

here

cc =

RL 0 D

μt

{[ P̄ (s)

P̄ (t )

] 1 /p cos (ωt + ϕ̄ ) cos (ωs + ϕ̄ )

},

U

ss =

RL 0 D

μt

{[ P̄ (s)

P̄ (t )

] 1 /p sin (ωt + ϕ̄ ) sin (ωs + ϕ̄ )

},

U

cs =

RL 0 D

μt

{[P̄ (s)

P̄ (t )

]1 /p

cos (ωt + ϕ̄ ) sin (ωs + ϕ̄ )

},

7928 J. Deng / Journal of the Franklin Institute 354 (2017) 7917–7945

U

sc =

RL 0 D

μt

{[P̄ (s)

P̄ (t )

]1 /p

sin (ωt + ϕ̄ ) cos (ωs + ϕ̄ )

},

V

cc =

RL 0 D

μt

{[P̄ (s)

P̄ (t )

]1 /p [ 1

P̄ (s) − 1

P̄ (t )

] cos (ωt + ϕ̄ ) cos (ωs + ϕ̄ )

},

V

sc =

RL 0 D

μt

{[P̄ (s)

P̄ (t )

]1 /p [ 1

P̄ (s) − 1

P̄ (t )

] sin (ωt + ϕ̄ ) cos (ωs + ϕ̄ )

},

V

ss =

RL 0 D

μt

{[P̄ (s)

P̄ (t )

]1 /p [ 1

P̄ (s) − 1

P̄ (t )

] sin (ωt + ϕ̄ ) sin (ωs + ϕ̄ )

},

V

cc =

RL 0 D

μt

{[P̄ (s)

P̄ (t )

]1 /p [ 1

P̄ (s) − 1

P̄ (t )

] cos (ωt + ϕ̄ ) cos (ωs + ϕ̄ )

}, (2.51)

and

∂U

sc

∂ P̄

=

V

sc

p

, ∂U

cc

∂ P̄

=

V

cc

p

, ∂U

cc

∂ ϕ̄

= −U

sc − U

cs , ∂U

sc

∂ ϕ̄

= U

cc − U

ss . (2.52)

Substituting P 1 and ϕ1 into Eq. (2.43) leads to the expressions of u 2 and v 2 , but they aretoo tedious to list here. Our main interest is the averaged values. Performing averaging onthem gives:

M

t { u 2 } = M

t

{∂ f 1 ∂ P̄

P 1 +

∂ f 1 ∂ ϕ̄

ϕ 1 − ∂P 1

∂ P̄

M

t { f 1 } − ∂P 1

∂ ϕ̄

M

t { f 2 }

}

= − p ̄P ωζ

4ν

{[ 2β + ωτε H

c (ν

2

)] cos (ψ − 2 ̄ϕ ) − ωτε H

s (ν

2

)sin (ψ − 2 ̄ϕ )

}

= − p ̄P ζω

2ν

[ ˆ β cos (ψ − 2 ̄ϕ ) + ˆ α sin (ψ − 2 ̄ϕ )

] ,

M

t { v 2 } = M

t

{ ∂ f 2 ∂ P̄

P 1 +

∂ f 2 ∂ ϕ̄

ϕ 1 − ∂ϕ 1

∂ P̄

M

t { f 1 } − ∂ϕ 1

∂ ϕ̄

M

t { f 2 }

}

= − 1

32ν

{8 ζω

2 τε H

s (ν

2

)cos (ψ − 2 ̄ϕ ) + 8 ωζ

[ 2β + ωτε H

c (ν

2

)] sin (ψ − 2 ̄ϕ )

+ ω

2 ζ 2 + 8

[ ωτε H

s (ν

2

)] 2 + 8

[ ωτε H

c (ν

2

)] 2 + 32ωβτε H

c (ν

2

)+ 32β2

}

=

1

2ν

[ ζω ̂ α cos (ψ − 2 ̄ϕ ) − ˆ βωζ sin (ψ − 2 ̄ϕ ) +

ˆ δ] , (2.53)

where

ˆ α = −1

2

ωτε H

s (ν

2

),

ˆ δ = −{

1

16

ω

2 ζ 2 +

1

2

[ ωτε H

s (ν

2

)] 2 +

1

2

[ ωτε H

c (ν

2

)] 2 + 2ωβτε H

c (ν

2

)+ 2β2

}.

Applying the transformation � = ϕ − 1 2 ψ and Itô Lemma results in three Itô stochastic

differential equations

J. Deng / Journal of the Franklin Institute 354 (2017) 7917–7945 7929

d

d

w

m

m

m

m

I

S

t

i

d

T

w

c

g

C

(C(C

2

C

d P (t ) =

(ε m p1 + ε 2 m p2

)P d t,

�(t ) =

(ε m �1 + ε 2 m �2

)d t − ε 1 / 2

σ

2

d W (t ) ,

ψ(t ) = ε 1 / 2 σ d W (t ) , (2.54)

here

p1 =

1

P

M

t

{u 1 } = p

[− ˆ β +

ζω

4

sin (2�)

],

p2 =

1

P

M

t

{u 2 } = − pζω

2ν

[ ˆ β cos (2�) − ˆ α sin (2�)

] ,

�1 = M

t

{v 1 } =

ˆ � +

ζω

4

cos (2�) ,

�2 = M

t

{v 2 } =

1

2ν

[ ζω ̂ α cos (2�) +

ˆ βωζ sin (2�) +

ˆ δ] . (2.55)

t is observed that Eqs. (2.54) are also coupled. Introducing a linear stochastic transformation

= T (�) P, P = T −1 (�) S, 0 ≤ � ≤ π,

he Itô equation for the transformed p th norm process S is given by, again using Itô Lemman the vector case

S =

[(ε m p1 + ε 2 m p2

)T +

(ε m �1 + ε 2 m �2

)T � +

εσ 2

8

T ��

]P d t − ε 1 / 2 σ

2

T �P d W (t )

=

1

T

[(ε m p1 + ε 2 m p2

)T +

(ε m �1 + ε 2 m �2

)T � +

εσ 2

8

T ��

]S d t − ε 1 / 2 σ

2

T �T

S d W (t ) .

(2.56)

he eigenvalue problem governing the p th moment Lyapunov exponent is

σ 2

8

T �� + ( m �1 + εm �2 ) T � +

(m p1 + εm p2

)T =

ˆ �T , (2.57)

here T ( �) is a periodic function in � of period π , and �q(t ) (p) = �(p) = ε ̂ �(p) . Substituting the expansion in Eq. (2.31) into eigenvalue problem (2.57) and equating the

oefficients of like trigonometric terms sin 2 k � and cos 2 k �, k = 0, 1 , . . . yield a set of homo-eneous linear algebraic equations with infinitely many equations for the unknown coefficients 0 , C k and S k , k = 1 , 2, . . .

Constant terms:

c 0 +

ˆ �)C 0 − b 0 S 1 + ε

2b 0

υ

(ˆ βC 1 − ˆ αS 1

)= 0,

oefficients for cos 2 �:

c 1 +

ˆ �)C 1 − d 1 S 1 − b 1 S 2 + ε

1

υ

(−e 1 S 1 − 2 ̂ αb 1 S 2 − 4a 1 ̂ βC 0 + 2 ̂

βb 1 C 2

)= 0,

oefficients for sin 2 �:

a 1 C 0 + d 1 C 1 + b 1 C 2 +

(c 1 +

ˆ �)

S 1 + ε 1

υ

(e 1 C 1 + 2 ̂ αb 1 C 2 + 4a 1 ̂ βC 0 + 2 ̂

βb 1 S 2

)= 0,

oefficients for cos 2 k �:

7930 J. Deng / Journal of the Franklin Institute 354 (2017) 7917–7945

a k S k−1 −(

c k +

ˆ �)C k + d k S k + b k S k+1

+ ε 1

υ

(2a k ̂ αS k−1 + e k S k + 2b k ̂ αS k+1 + 2a k ̂ αC k−1 − 2b k ˆ βC k+1

)= 0,

k = 2, 3 , . . . , K − 1 ,

Coefficients for sin 2 k �:

a k C k−1 + d k C k + b k C k+1 +

(c k +

ˆ �)

S k

+ ε 1

υ

(2a k ̂ αC k−1 + e k C k + 2b k ̂ αC k+1 − 2a k ̂ αS k−1 + 2b k ˆ βS k+1

)= 0,

k = 2, 3 , . . . , K − 1 ,

Coefficients for cos 2 K �:

a K S K−1 −(

c K +

ˆ �)C K + d K S K + ε

1

υ

(2a k ̂ αS K−1 + e k S K + 2a k ˆ βC K−1

)= 0,

Coefficients for sin 2 K �:

a K C K−1 + d K C K +

(c K +

ˆ �)

S K + ε 1

υ

(2a k ̂ αC K−1 + e k C K + 2a k ˆ βS K−1

)= 0, (2.58)

where

a k =

1

8

ζω(2k − 2 − p) , b k =

1

8

ζω(2k + 2 + p) ,

c k =

1

2

σ 2 k 2 + p ̂

β, d k = 2k ˆ �, e k = k ̂ δ, k = 0, 1 , 2, . . . . (2.59)

As in the first-order stochastic averaging, Eqs. (2.58) is reduced to a set of 2K + 1 homo-geneous linear equations for C 0 , C k and S k , k = 1 , 2, . . . , K . The determinant is set to zero toyield a polynomial equation of degree 2K + 1 for ˆ �K (p) . Solving the polynomial equationfor the largest root yields the approximate moment Lyapunov exponents. The terms H

s (

1 2 ν)

and H

c (

1 2 ν)

depend upon the material’s constitutive model.

2.5. Higher-Order stochastic averaging

Following the procedure of Section 2.4 , higher-order stochastic averaging analysis can be carried out. Let’s take the three-order stochastic averaging as an example of higher-order method. Introducing a near-identity transformation as in Eq. (2.38) , but up to order ε2

P (t ) = P̄ (t ) + ε P 1 + ε 2 P 2 , ϕ(t ) = ϕ̄ (t ) + ε ϕ 1 + ε 2 ϕ 2 , (2.60)

where P 1 = P 1 ( ̄P , ϕ̄ , t ) , P 2 = P 2 ( ̄P , ϕ̄ , t ) , ϕ 1 = ϕ 1 ( ̄P , ϕ̄ , t ) , and ϕ 2 = ϕ 2 ( ̄P , ϕ̄ , t ) are unknownfunctions. When P 2 = 0 and ϕ 2 = 0, it is reduced to second-order averaging.

Differentiating Eq. (2.60) and equating each result with the corresponding functions from

Eq. (2.36) gives

{ ˙ P̄

˙ ϕ̄

}

+ ε

⎧ ⎪ ⎨

⎪ ⎩

∂P 1 ∂ ̄P

˙ P̄ +

∂P 1 ∂ ̄ϕ

˙ ϕ̄ +

∂P 1 ∂t

∂ϕ 1 ∂ ̄P

˙ P̄ +

∂ϕ 1 ∂ ̄ϕ

˙ ϕ̄ +

∂ϕ 1 ∂t

⎫ ⎪ ⎬

⎪ ⎭

+ ε 2

⎧ ⎪ ⎨

⎪ ⎩

∂P 2 ∂ ̄P

˙ P̄ +

∂P 2 ∂ ̄ϕ

˙ ϕ̄ +

∂P 2 ∂t

∂ϕ 2 ∂ ̄P

˙ P̄ +

∂ϕ 2 ∂ ̄ϕ

˙ ϕ̄ +

∂ϕ 2 ∂t

⎫ ⎪ ⎬

⎪ ⎭

= ε

{

f 1 (P̄ + ε P 1 + ε 2 P 2 , ϕ̄ + ε ϕ 1 + ε 2 ϕ 2 , t

)f 2 (P̄ + ε P 1 + ε 2 P 2 , ϕ̄ + ε ϕ 1 + ε 2 ϕ 2 , t

)}

, (2.61)

J. Deng / Journal of the Franklin Institute 354 (2017) 7917–7945 7931

o

A

w

A

A

w

A

A

A

A

P

w

μ

μ

r in matrix form

{ ˙ P̄

˙ ϕ̄

}

= ε

{

f 1 (P̄ + ε P 1 + ε 2 P 2 , ϕ̄ + ε ϕ 1 + ε 2 ϕ 2 , t

)− ∂P 1 ∂t

f 2 (P̄ + ε P 1 + ε 2 P 2 , ϕ̄ + ε ϕ 1 + ε 2 ϕ 2 , t

)− ∂ϕ 1 ∂t

}

− ε 2

{ ∂P 2 ∂t

∂ϕ 2 ∂t

}

, (2.62)

here

=

⎡

⎣

1 + ε ∂P 1 ∂ ̄P

+ ε 2 ∂P 2 ∂ ̄P

ε ∂P 1 ∂ ̄ϕ

+ ε 2 ∂P 2 ∂ ̄ϕ

ε ∂ϕ 1 ∂ ̄P

+ ε 2 ∂ϕ 2 ∂ ̄P

1 + ε ∂ϕ 1 ∂ ̄ϕ

+ ε 2 ∂ϕ 2 ∂ ̄ϕ

⎤

⎦ . (2.63)

Using the expansion of 1/| A | to the order of ε2 , the inverse matrix of A is obtained as

−1 =

1

| A |

⎡

⎣

1 + ε ∂ϕ 1 ∂ ̄ϕ

+ ε 2 ∂ϕ 2 ∂ ̄ϕ

−ε ∂P 1 ∂ ̄ϕ

− ε 2 ∂P 2 ∂ ̄ϕ

−ε ∂ϕ 1 ∂ ̄P

− ε 2 ∂ϕ 2 ∂ ̄P

1 + ε ∂P 1 ∂ ̄P

+ ε 2 ∂P 2 ∂ ̄P

⎤

⎦ =

[ A 11 A 12

A 21 A 22

] , (2.64)

here

11 = 1 − ε ∂P 1

∂ P̄

+ ε 2

[

∂P 1

∂ ϕ̄

∂ϕ 1

∂ P̄

− ∂P 2

∂ P̄

+

(∂P 1

∂ P̄

)2 ]

,

12 = − ε ∂P 1

∂ ϕ̄

+ ε 2 [∂P 1

∂ ϕ̄

∂ϕ 1

∂ ϕ̄

− ∂P 2

∂ ϕ̄

+

∂P 1

∂ P̄

∂P 1

∂ ϕ̄

],

21 = − ε ∂ϕ 1

∂ P̄

+ ε 2 [∂ϕ 1

∂ P̄

∂ϕ 1

∂ ϕ̄

− ∂ϕ 2

∂ P̄

+

∂P 1

∂ P̄

∂ϕ 1

∂ P̄

],

22 = 1 − ε ∂ϕ 1

∂ ϕ̄

+ ε 2

[

∂P 1

∂ ϕ̄

∂ϕ 1

∂ P̄

− ∂ϕ 2

∂ ϕ̄

+

(∂ϕ 1

∂ ϕ̄

)2 ]

.

Considering the Taylor expansion of a function with multi-variables

f j (P̄ + ε P 1 + ε 2 P 2 , ϕ̄ + ε ϕ 1 + ε 2 ϕ 2 , t

) = f j (P̄ , ϕ̄ , t

)+ ε

{P 1

∂ f j (P̄ , ϕ̄ , t

)∂ P̄

+ ϕ 1 ∂ f j (P̄ , ϕ̄ , t

)∂ ϕ̄

}

+ ε 2 {

P 2 ∂ f j (P̄ , ϕ̄ , t

)∂ P̄

+ ϕ 2 ∂ f j (P̄ , ϕ̄ , t

)∂ ϕ̄

+

1

2

P

2 1

∂ 2 f j (P̄ , ϕ̄ , t

)∂ P̄

2

+

1

2

ϕ

2 1

∂ 2 f j (P̄ , ϕ̄ , t

)∂ ϕ̄

2 + P 1 ϕ 1

∂ 2 f j (P̄ , ϕ̄ , t

)∂ P̄ ∂ ϕ̄

}+ o

(ε 2 ), j = 1 , 2. (2.65)

Substituting Eqs. (2.64) and (2.65) into Eq. (2.62) yields

˙ ̄= ε μ1 + ε 2 μ2 + ε 3 μ3 + o

(ε 3 ), ˙ ϕ̄ = ε ν1 + ε 2 ν2 + ε 3 ν3 + o

(ε 3 ), (2.66)

here

1 = f 1 − ∂P 1

∂t , μ2 =

∂ f 1 ∂ P̄

P 1 +

∂ f 1 ∂ ϕ̄

ϕ 1 − ∂P 1

∂ P̄

(f 1 − ∂P 1

∂t

)− ∂P 1

∂ ϕ̄

(f 2 − ∂ϕ 1

∂t

)− ∂P 2

∂t ,

3 =

∂ f 1 ∂ P̄

P 2 +

∂ f 1 ∂ ϕ̄

ϕ 2 +

P

2 1

2

∂ 2 f 1 ∂ P̄

2 +

ϕ

2 1

2

∂ 2 f 1 ∂ ϕ̄

2 + P 1 ϕ 1

∂ 2 f 1 ∂ P̄ ∂ ϕ̄

7932 J. Deng / Journal of the Franklin Institute 354 (2017) 7917–7945

− ∂P 1

∂ P̄

(P 1

∂ f 1 ∂ P̄

+ ϕ 1 ∂ f 1 ∂ ϕ̄

− ∂P 2

∂t

)− ∂P 1

∂ ϕ̄

(P 1

∂ f 2 ∂ P̄

+ ϕ 1 ∂ f 2 ∂ ϕ̄

− ∂ϕ 2

∂t

)

+

(f 1 − ∂P 1

∂t

)[

∂P 1

∂ ϕ̄

∂ϕ 1

∂ P̄

−∂P 2

∂ P̄

+

(∂P 1

∂ P̄

)2 ]

+

(f 2 −∂ϕ 1

∂t

)(∂P 1

∂ ϕ̄

∂ϕ 1

∂ ϕ̄

−∂P 2

∂ ϕ̄

+

∂P 1

∂ P̄

∂P 1

∂ ϕ̄

),

(2.67)

ν1 = f 2 − ∂ϕ 1

∂t , ν2 =

∂ f 2 ∂ P̄

P 1 +

∂ f 2 ∂ ϕ̄

ϕ 1 − ∂ϕ 1

∂ P̄

(f 1 − ∂P 1

∂t

)− ∂ϕ 1

∂ ϕ̄

(f 2 − ∂ϕ 1

∂t

)− ∂ϕ 2

∂t ,

ν3 =

∂ f 2 ∂ P̄

P 2 +

∂ f 2 ∂ ϕ̄

ϕ 2 +

P

2 1

2

∂ 2 f 2 ∂ P̄

2 +

ϕ

2 1

2

∂ 2 f 2 ∂ ϕ̄

2 + P 1 ϕ 1

∂ 2 f 2 ∂ P̄ ∂ ϕ̄

− ∂ϕ 1

∂ P̄

(P 1

∂ f 1 ∂ P̄

+ ϕ 1 ∂ f 1 ∂ ϕ̄

− ∂P 2

∂t

)− ∂ϕ 1

∂ ϕ̄

(P 1

∂ f 2 ∂ P̄

+ ϕ 1 ∂ f 2 ∂ ϕ̄

− ∂ϕ 2

∂t

)

+

(f 2 − ∂ϕ 1

∂t

)[

∂P 1

∂ ϕ̄

∂ϕ 1

∂ P̄

− ∂ϕ 2

∂ ϕ̄

+

(∂ϕ 1

∂ ϕ̄

)2 ]

+

(f 1 − ∂P 1

∂t

)(∂P 1

∂ P̄

∂ϕ 1

∂ P̄

− ∂ϕ 2

∂ P̄

+

∂ϕ 1

∂ P̄

∂ϕ 1

∂ ϕ̄

). (2.68)

Since third-order averaging can be reduced to the second order averaging, it is naturalto assume that P 1 and ϕ1 have the same forms as in Eqs. (2.49) . Following the same ideaof deriving Eqs. (2.49) , one can integrate μ2 with respect to t , solve the result for P 2 , andremove any terms from P 2 that are linear in ϕ̄ (in order for the near-identity transformation(2.60) to be uniformly valid in the large t limit). It can be assumed that

∂P 2

∂t = u 2 − M

t

{u 2 },

∂ϕ 2

∂t = v 2 − M

t

{v 2 }. (2.69)

Substituting u 2 and v 2 in Eq. (2.43) and their averaging values in Eq. (2.53) into Eq. (2.69)and solving the differential equations yields the functions P 2 and ϕ2 , whose expressions, however, are too long to list here.

To obtain μ̄3 and ν̄3 in Eq. (2.66) , the derivatives of f 1 and f 2 in Eq. (2.50) are neededand some second order derivatives must be used,

∂ 2 f 1 ∂ P̄

2 =

∂

∂ P̄

[ −2βp sin

2 � +

p

2

ωζ cos (νt + ψ) sin 2� + ωτε

(pU

sc + P̄ V

sc )]

= ω τε V

sc + ω τε P̄

∂V

sc

∂ P̄

,

∂ 2 f 1 ∂ P̄ ∂ ϕ̄

=

∂

∂ ϕ̄

[ −2βp sin

2 � +

p

2

ωζ cos (νt + ψ) sin 2� + ωτε

(pU

sc + P̄ V

sc )]

= − 2βp sin 2� + pωζ cos (νt + ψ) cos 2� + ωτε p ( U

cc − U

ss ) + ωτε P̄ ( V

cc − V

ss ) ,

∂ 2 f 1 ∂ ϕ̄

2 =

∂

∂ ϕ̄

p ̄P

[−2β sin 2� + ωζ cos (νt + ψ) cos2� + ωτε ( U

cc − U

ss ) ]

= − 2p ̄P

[2β cos 2� + ωζ cos (νt + ψ) sin2� + ωτε ( U

sc + U

cs ) ],

∂ 2 f 2 ∂ P̄

2 =

∂

∂ P̄

[ωτε V

cc

p

]

J. Deng / Journal of the Franklin Institute 354 (2017) 7917–7945 7933

w

a

t

M

w

γ

λ̂

d

d

d

w

m

m

m

=

ωτε

p

RL 0 D

μt

{[P̄ (s)

P̄ (t )

]1 /p [ (

1

p

− 1

)1

P̄ (s) 2 −

2

p

1

P̄ (t ) ̄P (s) +

(1

p

+ 1

)1

P̄ (t ) 2

]

cos (ωt + ϕ̄ ) cos (ωs + ϕ̄ )

},

∂ 2 f 2 ∂ P̄ ∂ ϕ̄

=

∂

∂ ϕ̄

[ωτε V

cc

p

]= −ωτε

p

( V

sc + V

cs ) ,

∂ 2 f 2 ∂ ϕ̄

2 =

∂

∂ ϕ̄

[−2β cos 2� − ωζ cos (νt + ψ) sin 2� − ωτε ( U

cs + U

sc ) ]

= 4β sin 2� − 2ωζ cos (νt + ψ) cos 2� − 2ωτε ( U

cc − U

ss ) , (2.70)

here Eq. (2.52) and

∂U

ss

∂ ϕ̄

= U

cs + U

sc , ∂U

cs

∂ ϕ̄

= −U

ss + U

cc , ∂V

sc

∂ ϕ̄

= V

cc − V

ss , ∂V

cc

∂ ϕ̄

= −V

sc − V

cs , (2.71)

re used, and U

sc , U

cc , U

cs , U

ss , V

ss , V

cc , V

sc , and V

cs are defined in Eq. (2.51) . Substituting these derivatives and P 1 , ϕ1 , P 2 , and ϕ2 into Eqs. (2.67) and (2.68) yields

he values of μ3 and ν3 . Their averaged values are

t

{μ3

} =

p ̄P ζω

256 ν2

{ˆ γ sin (ψ − 2 ̄ϕ ) − 128 ̂ α ˆ β cos (ψ − 2 ̄ϕ )

},

M

t

{v 3 } =

−ζω

256 ν2

{ˆ γ cos (ψ − 2 ̄ϕ ) + 128 ̂ α ˆ β sin (ψ − 2 ̄ϕ ) − ˆ λ

}, (2.72)

here

ˆ = 7 ω

2 ζ 2 − 128 ̂ α2 , ˆ α = −1

2

ωτε H

s (ν

2

),

= 128 ω τε ˆ �

{ω τε

[ H

c (ν

2

)] 2 + 4βH

c (ν

2

)+ ωτε

[ H

s (ν

2

)] 2 }− 8 ̂ α

(64β2 − 3 ω

2 ζ 2 )+ 8�

(64β2 + ω

2 ζ 2 ).

Applying the transformation � = ϕ − 1 2 ψ and Itô Lemma results in three Itô stochastic

ifferential equations

d P (t ) =

(ε m p1 + ε 2 m p2 + ε 3 m p3

)P d t,

�(t ) =

(ε m �1 + ε 2 m �2 + ε 3 m �3

)d t − ε 1 / 2

σ

2

d W (t ) ,

ψ(t ) = ε 1 / 2 σ d W (t ) , (2.73)

here

p1 =

1

P

M

t

{u 1 } = p

[ − ˆ β +

ζω

4

sin (2�) ] ,

p2 =

1

P

M

t

{u 2 } = − pζω

2ν

[ ˆ β cos (2�) − ˆ α sin (2�)

] ,

p3 =

1

P

M

t

{μ3

} =

pζω

256 ν2

{ˆ γ sin (ψ − 2 ̄ϕ ) − 128 ̂ α ˆ β cos (ψ − 2 ̄ϕ )

},

7934 J. Deng / Journal of the Franklin Institute 354 (2017) 7917–7945

m �1 = M

t

{v 1 } =

ˆ � +

ζω

4

cos (2�) ,

m �2 = M

t

{v 2 } =

1

2ν

[ ζω ̂ α cos (2�) +

ˆ βωζ sin (2�) +

ˆ δ] ,

m �3 = M

t

{v 3 } =

−ζω

256 ν2

{ˆ γ cos (ψ − 2 ̄ϕ ) + 128 ̂ α ˆ β sin (ψ − 2 ̄ϕ ) − ˆ λ

}. (2.74)

It is observed that Eqs. (2.73) are also coupled. Introducing a linear stochastic transforma-tion

S = T (�) P, P = T −1 (�) S, 0 ≤ � ≤ π,

the Itô equation for the transformed p th norm process S is given by, again using Itô Lemmain the vector case

d S =

[ (ε m p1 + ε 2 m p2 + ε 3 m p3

)T +

(ε m �1 + ε 2 m �2 + ε 3 m �3

)T � +

ε σ 2

8

T ��

] P d t

− ε 1 / 2 σ

2

T �P d W (t )

=

1

T

[ (ε m p1 + ε 2 m p2 + ε 3 m p3

)T +

(ε m �1 + ε 2 m �2 + ε 3 m �3

)T � +

ε σ 2

8

T ��

] S d t

− ε 1 / 2 σ

2

T �T

S d W (t ) . (2.75)

The eigenvalue problem governing the p th moment Lyapunov exponent is

σ 2

8

T �� +

(m �1 + εm �2 + ε 2 m �3

)T � +

(m p1 + εm p2 + ε 2 m p3

)T =

ˆ �T , (2.76)

where T ( �) is a periodic function in � of period π , and �q(t ) (p) = �(p) = ε ̂ �(p) . Substituting the expansion in Eq. (2.31) into eigenvalue problem (2.76) and equating the

coefficients of like trigonometric terms sin 2 k � and cos 2 k �, k = 0, 1 , . . . yield a set of homo-geneous linear algebraic equations with infinitely many equations for the unknown coefficients C 0 , C k and S k , k = 1 , 2, . . .

Constant terms:

(c 0 +

ˆ �) C 0 − b 0 S 1 + ε 2b 0

υ

( ˆ βC 1 − ˆ αS 1 )+ ε 2

a 1

8 υ2

(ˆ λC 1 − ˆ γ S 1 ) = 0,

Coefficients for cos 2 �:

(c 1 +

ˆ �) C 1 − d 1 S 1 − b 1 S 2 + ε 1

υ

(− e 1 S 1 − 2 ˆ αb 1 S 2 − 4a 1 ˆ βC 0 + 2

ˆ βb 1 C 2 )

+ ε 2 1

32υ2

(− 4a 3 ˆ γ S 2 − p

ˆ λC 0 + 4a 3 ˆ λC 2 ) = 0,

Coefficients for sin 2 �:

2a 1 C 0 + d 1 C 1 + b 1 C 2 + (c 1 +

ˆ �) S 1 + ε 1

υ

(e 1 C 1 + 2 ˆ αb 1 C 2 + 4a 1 ˆ βC 0 + 2

ˆ βb 1 S 2 )

+ ε 2 1

32υ2

(4a 3 ̂ λS 2 + p ˆ γC 0 + 4a 3 ˆ γC 2

) = 0,

Coefficients for cos 2 k �:

J. Deng / Journal of the Franklin Institute 354 (2017) 7917–7945 7935

a

C

C

a

C

a

w

a

o

i

p

T

3

s

m

e

c

k S k−1 − (c k +

ˆ �) C k + d k S k + b k S k+1

+ ε 1

υ

(2a k ˆ αS k−1 + e k S k + 2b k ˆ αS k+1 + 2a k ˆ αC k−1 − 2b k ˆ βC k+1

)+ ε 2

1

8 υ2

(− b k−2 ˆ γ S k−1 − a k+2 ˆ γ S k+1

− b k−2 ˆ λC k−1 + a k+2 ˆ λC k+1 ) = 0, k = 2, 3 , . . . , K − 1 ,

oefficients for sin 2 k �:

a k C k−1 + d k C k + b k C k+1 + (c k +

ˆ �) S k

+ ε 1

υ

(2a k ˆ αC k−1 + e k C k + 2b k ˆ αC k+1 − 2a k ˆ αS k−1 + 2b k ˆ βS k+1

)+ ε 2

1

8 υ2

(− b k−2 ˆ λS k−1 + a k+2 ˆ λS k+1

+ b k−2 ˆ γC k−1 + a k+2 ˆ γC k+1

)= 0, k = 2, 3 , . . . , K − 1 ,

oefficients for cos 2 K �:

K S K−1 − (c K +

ˆ �) C K + d K S K + ε 1

υ

(2a k ˆ αS K−1 + e k S K + 2a k ˆ βC K−1

)+ ε

1

8 υ2

(b k−2 ˆ γ S K−1 + b k−2 ˆ λC K−1

) = 0,

oefficients for sin 2 K �:

K C K−1 + d K C K + (c K +

ˆ �) S K + ε 1

υ

(2a k ˆ αC K−1 + e k C K + 2a k ˆ βS K−1

)+ ε

1

8 υ2

(b k−2 ˆ γC K−1 − b k−2 ˆ λS K−1

) = 0, (2.77)

here

k =

1

8

ζω(2k − 2 − p) , b k =

1

8

ζω(2k + 2 + p) ,

c k =

1

2

σ 2 k 2 + p ̂

β, d k = 2k ̂ �, e k = k ̂ δ, k = 0, 1 , 2, . . . .

As in the first-order and second-order stochastic averaging, Eq. (2.77) is reduced to a setf 2K + 1 homogeneous linear equations for C 0 , C k and S k , k = 1 , 2, . . . , K . The determinants set to zero to yield a polynomial equation of degree 2K + 1 for ˆ �K (p) . Solving theolynomial equation for the largest root yields the approximate moment Lyapunov exponents.he terms H

s (

1 2 ν)

and H

c (

1 2 ν)

depend upon the material’s constitutive model.

. Results and discussions

The moment Lyapunov exponent is the nontrivial solution of Eq. (2.33) , where the Fouriereries expansion for T ( �) is truncated to the K th order. Theoretically, the set of approximateoment Lyapunov exponents obtained by this procedure converges to the corresponding true

xponent values as K → ∞ . Practically, Fig. 2 illustrates that moment Lyapunov momentsonverge so quickly with the increase of Fourier expansion order K that results of K = 3 are

7936 J. Deng / Journal of the Franklin Institute 354 (2017) 7917–7945

Fig. 2. Effect of Fourier series expansion on moment Lyapunov exponents.

Fig. 3. Fourier series expansion on moment Lyapunov exponents.

accurate enough. However, this convergence depends on the width of power spectral density

of the bounded noise process. Fig. 3 shows when ζ = 5 , even K = 12 is not adequate. Thisresult is different from [14] , where real noise was considered. From Fig. 4 , it is seen that forsmall σ or large ζ the bounded noise is a narrow-band process; whereas when σ is increased

or ζ is decreased, the power spectral density function becomes flat. Hence, if the power spectral density function of the bounded noise is relatively sharp, which means it is a narrowband process, a large number of sinusoidal terms in the Fourier series Eq. (2.33) must betaken to obtain a good approximation of the moment Lyapunov exponent �( p ). On the otherhand, if the power spectral density function of the bounded noise is relatively flat, whichmeans it is a wide band process, fewer sinusoidal terms in the Fourier series Eq. (2.33) canyield a satisfactory approximation of �( p ).

Effect of damping β on moment Lyapunov exponents is shown in Fig. 5 , in which theparameter β plays a stabilizing role in the system stability. The approximate moment Lyapunov

J. Deng / Journal of the Franklin Institute 354 (2017) 7917–7945 7937

Fig. 4. Power spectral density of a bounded noise process.

Fig. 5. Effect of damping β on moment Lyapunov exponents.

e

t

i

o

1

f

H

T

v

e

xponents are compared with the results from Monte-Carlo simulation [7] , which shows thathe two results agree well, although discrepancy is found when p is large.

Effect of fractional orders μ on moment Lyapunov exponents is shown in Fig. 6 , whichndicates that the parameter μ plays an important role in the system stability. The increasef the fractional order μ would enhance the system stability. When μ changes from 0 to, the system gradually becomes stabilized. As mentioned in Appendix B , when μ shiftsrom 0 to 1, the fractional Newton element in a fractional Kelvin–Voigt model moves from aooke element to an integer-value Newton element, i.e., from pure elasticity to pure viscosity.herefore, the fractional order μ enables a subtle and smooth transition from elasticity toiscosity in one element, which abandons the conventional binary theory that only allow anlement either to function totally elastic (a spring) or perform perfect viscosity (a dashpot).

7938 J. Deng / Journal of the Franklin Institute 354 (2017) 7917–7945

Fig. 6. Effect of fractional orders μ on moment Lyapunov exponents.

Fig. 7. Effect of σ of bounded noise on moment Lyapunov exponents.

This multi-state fractional component has various performance levels and creates various viscoelastic modes. Fig. 6 clearly shows fractional components have vital effects on the entiresystem’s performance.

Fig. 7 shows that with the increase of the noise intensity parameter σ , the stability of thesystem increases. Three-dimensionally, Fig. 8 show that the smaller the noise intensity σ , themore significant the parametric resonance. Effect of noise amplitude ( ζ ) on moment Lyapunov

exponents is shown in Fig. 9 , which indicates the de-stabilizing effect of noise amplitude onthe system stability. Great effect of ζ on parametric resonance is observed in Fig. 10 . Largeramplitude would result in prominent parametric resonance. One possible explanation is that from the power spectrum density function of bounded noise in Eq. (2.10) , Fig. 4 shows thelarger of σ or the smaller of ζ , the wider the frequency band of power spectrum, which

J. Deng / Journal of the Franklin Institute 354 (2017) 7917–7945 7939

Fig. 8. Parametric resonance of SDOF fractional viscoelastic system.

Fig. 9. Effect of noise amplitude ζ on moment Lyapunov exponents.

7940 J. Deng / Journal of the Franklin Institute 354 (2017) 7917–7945

Fig. 10. Parametric resonance of a SDOF fractional viscoelastic system.

Fig. 11. Effect of retardation τ ε on moment Lyapunov exponents.

J. Deng / Journal of the Franklin Institute 354 (2017) 7917–7945 7941

s

f

s

t

p

o

a

a

4

n

s

t

t

T

l

t

f

a

v

a

h

f

a

p

A

u

uggests that the power of the noise not be concentrated in the neighborhood of the centralrequency ν, which reduces the effect of the primary parametric resonance.

Fig. 11 illustrates that the system retardation factor ( τ ε) plays a stabilizing role in momenttability, for the stability index increases with the retardation factor. The stability index ishe non-trivial zero δ of the moment stability �( p ), i.e., �(δ) = 0, which characterizes therobability with which the response of an almost-surely system exceeds a threshold.

The above figures illustrate that there is some difference between first-order and higher-rder averaging. However, the third-order averaging is almost identical to the second-orderveraging, which suggests that for viscoelastic structures, second-order averaging analysis isdequate.

. Conclusions

The stochastic stability of a fractional viscoelastic column under the excitation of a boundedoise is studied by using the method of higher order stochastic averaging. The higher ordertochastic averaging is based on lower order stochastic averaging and is formulated by in-roducing a near identity transformation. We investigate the stochastic stability of the systemhrough moment Lyapunov moment, which is an ideal avenue to characterize moment stability.he moment Lyapunov exponent is determined numerically by solving a set of homogeneous

inear equations, which is obtained from an eigenvalue problem by using linear stochasticransformation and Itô Lemma. It is found that there is some difference between the resultsrom the first-order and higher-order averaging. However, Results from the third-order aver-ging are almost identical to those of the second-order averaging, which suggests that for thisiscoelastic structure, second-order averaging analysis is adequate. The approximate resultsre compared well to Monte Carlo simulations, which shows that the tedious derivation forigher-order stochastic averaging is correct in this paper. The convergence issue and the ef-ects of various parameters on the stochastic stability of the system are discussed in detailnd possible explanations are explored. These results pave a road for utilizing or controllingarametric resonances in engineering applications.

ppendix A. Fractional calculus

Consider the first three repeated integration and reduce to a single integral by using thesual change of variables formula

f (−1) (t ) =

∫ t

0 f (τ ) d τ,

f (−2) (t ) =

∫ t

0 d τ2

∫ τ2

0 f (τ1 ) d τ1 =

∫ t

0 f (τ1 ) d τ1

∫ t

τ1

d τ2 =

∫ t

0 (t − τ1 ) f (τ1 ) d τ1 ,

f (−3) (t ) =

∫ t

0 d τ3

∫ τ3

0 d τ2

∫ τ2

0 f (τ1 ) d τ1 =

∫ t

0 d τ3

∫ t 3

0 (t 3 − τ1 ) f (τ1 ) d τ1

=

∫ t

0 f (τ1 ) d τ1

∫ t

τ 1

(t 3 − τ1 ) d τ3 =

1

2

∫ t

0 (t − τ1 )

2 f (τ1 ) d τ1 . (A.1)

7942 J. Deng / Journal of the Franklin Institute 354 (2017) 7917–7945

.

One can continue in this way to find that

f (−n) (t ) =

d

−n f (t )

d t =

1

(n − 1)!

∫ t

0 (t − τ ) n−1 f (τ ) d τ, n ∈ N

+ , (A.2)

for almost all x with −∞ ≤ 0 < t ≤ ∞ , and N

+ is the set of positive integers. By replacing the factorial expression for its gamma function equivalent, one can generalize

the formula for repeated integration in Eq. (A.2) for all n ∈ R

+ ,

f (−μ) (t ) =

RL 0 D

−μt

[f (t )

] =

1

�(μ)

∫ t

0 (t − τ ) μ−1 f (τ ) d τ, μ ∈ R

+ , (A.3)

where μ is the fractional coefficient, R

+ is positive real numbers, RL 0 D

−μt

[ · ] is an operator for Riemann–Liouville fractional integral of order μ, �( μ) is the gamma function,

�(μ) =

∫ ∞

0 e −t t μ−1 d t . (A.4)

If μ is a positive integer, then �(μ) = (μ − 1)! , so this definition of f (−μ) (t ) agrees with(A.2) when μ is a positive integer. Eq. (A.3) is the definition of Riemann–Liouville (RL)fractional integral [15] .

In a similar way, fractional derivative can be defined from repeated derivative of a function.However, it is more convenient to formulate a definition of fractional derivative directly usingthe fractional integral. For a fixed n ≥1 and integer m ≥n the (m − n) th derivative of thefunction f ( t ) can be written as

f (m−n) (t ) = f (m) [

f (−n) (t ) ] =

1

�(n) D

m

∫ t

0 (t − τ ) n−1 f (τ ) d τ, m ∈ N

+ , (A.5)

where the symbol D

m ( m ≥0) denotes m th iterated differentiations. Substituting Eq. (A.3) intoEq. (A.5) leads to

RL 0 D

m−αt

[f (t )

] =

RL 0 D

m

t

[RL 0 D

−αt f (t )

] =

1

�(α)

d

m

d t m

∫ t

0 (t − τ ) α−1 f (τ ) d τ, 0 < α ≤ 1 , m ∈ N

+

(A.6)

Denoting μ = m − α, one can write fractional derivative as

RL 0 D

μt

[f (t )

] =

1

�(m − μ)

d

m

d t m

∫ t

0

f (τ )

(t − τ ) −(m−μ)+1 d τ, m − 1 ≤ μ < m, (A.7)

where RL a D

μt

[ · ] is an operator for Riemann–Liouville fractional derivative of order μ. It can be seen that RL

0 D

0 t

[f (t )

] = f (t ) and

RL 0 D

m

t

[f (t )

] = f (m) (t ) . Specifically, for 0 < μ≤1,assuming f ( t ) is absolutely continuous for t > 0 , one has (see Eq. (2.1.28) of [15] )

RL 0 D

μt

[f (t )

] =

1

�(1 − μ)

d

d t

∫ t

0

f ( τ )

( t − τ ) μd τ =

1

�(1 − μ)

[ f (0)

t μ+

∫ t

0

f ′ (τ )

(t − τ ) μd τ] . (A.8)

Further assuming f (0) = 0 gives the fractional derivative in Caputo (C) form

C

0 D

μt

[f (t )

] =

1

�(1 − μ)

∫ t

0

f ′ ( τ )

( t − τ ) μd τ =

1

�(1 − μ)

∫ t

0

f ′ (t − τ )

τμd τ. (A.9)

Fig. A.12 gives a diagram for a simple function f (t ) = t and its fractional calculus asa showcase RL

0 D

μt

[t ] = t 1 −μ�(2) / �(2 − μ) . The fractional calculus of arbitrary real order μ

can be considered as an interpolation of the sequence of integer-order calculus.

J. Deng / Journal of the Franklin Institute 354 (2017) 7917–7945 7943

Fig. A.12. Fractional calculus of a power function.

Fig. B.13. Schematic of a fractional element between a spring ( μ = 0) and a dashpot ( μ = 1 ).

A

s

e

a

m

m

σ

w

v

ppendix B. Fractional viscoelasticity

Fractional calculus provides a perfect tool to construct a fractional viscoelastic componenthown in Fig. B.13 . With the various values of μ from 0 to 1, property of the fractionallement changes from elastic to viscous.

Fractional Viscoelasticity may be formulated by combining Hook springs, fractional pots,nd dash pots in series or in parallel. The fractional Kelvin–Voigt viscoelastic mechanicalodel consists of a fractional Newton dash-pot and a Hooke spring in parallel, which isathematically defined as

(t ) = E ε(t ) + ηd

με(t )

d t μ= E ε(t ) + η RL

0 D

μt

[ε(t )

], (B.1)

here E is the Young’s elastic modulus and η is the viscosity modulus or coefficient ofiscosity.

7944 J. Deng / Journal of the Franklin Institute 354 (2017) 7917–7945

Fig. B.14. Creep and recovery for a fractional viscoelastic material.

When μ = 0, the fractional Newton part becomes a constant η, which is the case forthe Hooke model. It is further observed that, only when μ = 0, this model exhibits transientelasticity at t = 0. When μ = 1 , the relaxation modulus of the fractional Newton part becomesa Dirac delta function, i.e., σ (t ) = η δ(t ) , which is the case of the integer Newton model.

Hence, the fractional Newton element is a continuum between the spring model and theNewton model ( Fig. B.13 ). A typical relationship of creep and recovery for a fractionalviscoelastic material is displayed in Fig. B.14 , which clearly shows that the extra degree-of-freedom from the fractional order can improve the performance of traditional viscoelastic elements. Specifically, when μ = 1 , this fractional model reduces to the ordinary integer Kelvin–Voigt model.

References

[1] R. Mainardi , Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models,Imperial College Press, London, 2010 .

[2] H. Xu , X. Jiang , Creep constitutive models for viscoelastic materials based on fractional derivatives, Comput.Math. Appl. 73 (6) (2017) 1377–1384 .

[3] Q. Song , X. Yang , C. Li , T. Huang , X. Chen , Stability analysis of nonlinear fractional-order systems withvariable-time impulses, J. Frankl. Inst. 354 (2017) 2959–2978 .

[4] W.-C. Xie , Dynamic Stability of Structures, Cambridge University Press, Cambridge, 2006 . [5] Q. Ling , X.L. Jin , Z.L. Huang , Response and stability of SDOF viscoelastic system under wideband noise

excitations, J. Frankl. Inst. 348 (2011) 2026–2043 . [6] Y. Xu , B. Pei , J.L. Wu , Stochastic averaging principle for differential equations with non-Lipschitz coefficients

driven by fractional Brownian motion, Stoch. Dyn. 17 (2017) 1750013 . [7] J. Deng , W.-C. Xie , M.D. Pandey , Stochastic stability of a fractional viscoelastic column under bounded noise

excitation, J Sound Vib. 333 (6) (2014) 1629–1643 . [8] M. Hijawi , N. Moshchuk , R.A. Ibrahim , Unified second-order stochastic averaging approach, ASME J. Appl.

Mech. 64 (2) (1997) 281–291 . [9] Q. Huang , W.-C. Xie , Stability of SDOF linear viscoelastic system under the excitation of wideband noise,

ASME J. Appl. Mech. 75 (2) (2008) 021012 . [10] G. Janevski , P. Kozic , I. Pavlovic , Moment Lyapunov exponents of the parametrical Hill’s equation under the

excitation of two correlated wideband noises, Struct. Eng. Mech. 52 (3) (2014) 525–540 .

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