the transversal creeping vibrations of a fractional...

THE TRANSVERSAL CREEPING VIBRATIONS OFA FRACTIONAL DERIVATIVE ORDER CONSTITUTIVERELATION OF NONHOMOGENEOUS BEAM

KATICA (STEVANOVIC) HEDRIH

Received 25 January 2005; Accepted 17 March 2005

We considered the problem on transversal oscillations of two-layer straight bar, which isunder the action of the lengthwise random forces. It is assumed that the layers of the barwere made of nonhomogenous continuously creeping material and the correspondingmodulus of elasticity and creeping fractional order derivative of constitutive relation ofeach layer are continuous functions of the length coordinate and thickness coordinates.Partial fractional differential equation and particular solutions for the case of natural vi-brations of the beam of creeping material of a fractional derivative order constitutiverelation in the case of the influence of rotation inertia are derived. For the case of naturalcreeping vibrations, eigenfunction and time function, for different examples of boundaryconditions, are determined. By using the derived partial fractional differential equation ofthe beam vibrations, the almost sure stochastic stability of the beam dynamic shapes, cor-responding to the nth shape of the beam elastic form, forced by a bounded axially noiseexcitation, is investigated. By the use of S. T. Ariaratnam’s idea, as well as of the averagingmethod, the top Lyapunov exponent is evaluated asymptotically when the intensity ofexcitation process is small.

Copyright © 2006 Katica (Stevanovic) Hedrih. This is an open access article distributedunder the Creative Commons Attribution License, which permits unrestricted use, dis-tribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

The use of composite beams is now a real trend in many engineering applications. Thistrend calls for the development of efficient tools, suitable for the analysis of beams ex-hibiting three-dimensional effects, for which the classical beam theory assumptions areno more valid.

Transversal vibration beam problem is classical, but in current university books onvibrations we can find only Euler-Bernoulli’s classical partial differential equation (see[24, 31]) for describing transversal beam vibrations. In some monographs [25–27], wecan find a nonlinear partial differential equation for describing transversal vibrations of

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2006, Article ID 46236, Pages 1–18DOI 10.1155/MPE/2006/46236

http://dx.doi.org/10.1155/S1024123X06462360

2 Creeping vibrations of a nonhomogeneous beam

the beam with nonlinear constitutive stress-strain relation of the beam ideal elastic ma-terial. Recently new models of the constitutive stress-strain relations of the rheologicalnew beam materials [10] were found in books [5] and as applications in journal papers[2, 4, 19–22]. In the university book by Raskovic [31], extended partial differential equa-tion of the transversal ideally elastic beam vibration is presented with members by whichinfluences of the inertia rotation of the beam cross-section and shear of the cross-sectionby transversal forces are presented.

In the paper by Tabaddor [36], compare the experimentally and theoretically obtainedsingle-mode responses of a cantilever beam. The analytical portion involves solving anintegro-differential equation via the method of multiple scales. For the single-mode re-sponse, a large discrepancy is found between theory and experiment for an assumed idealclamp model.

Bypassing the complexity of a full three-dimensional elasticity analysis, Crespo da Silvaderived nonlinear equations governing the dynamics of 3D motions of beams.

The purpose of the paper by Fatmi and Zenzri [6] was to simplify the numerical im-plementation of the exact elastic beam theory in order to allow an inexpensive and largeuse of it. A finite element method is proposed for the computation of the beam operatorsinvolved in this theory. The discretization is reduced since only one element is required inthe longitudinal direction of the beam. The proposed method is applied to homogeneousand composite beams made of isotropic materials and to symmetric and antisymmet-ric laminated beams made of transversely isotropic materials. Structural beam rigidities,elastic couplings, warpings, and three-dimensional stresses are provided and comparedto available results.

The integral theory of analytical dynamics of discrete hereditary systems is presentedin the monograph by Goroshko and Hedrih [10] and their applications are published inthe following papers by Hedrih (Stevanovic) [12, 13, 16–20].

In the paper by Machado [37] we learn that the papers by Gemant [8] and Oldham[28], among other cited papers, contain the basic aspects of the fractional calculus theoryand the study of its properties can be addressed in these references, while research resultscan be found in papers by Osler [29], Ross [32], Campos [3], Samko [33], and others. Wemust also refer to Gorenflo and Mainardi’s [9].

In [9, 34] fractional calculus is mathematically based on corresponding integral andfractional order differential equations and in [5] fractional calculus is coupled with con-stitutive relation of real creeping material. In [2, 4] the authors presented new results ofthe stability and creeping and dynamical stability of viscoelastic column with fractionalderivative constitutive relation of rod material. Papers by Hedrih (Stevanovic) [19, 20]are in relation to the transversal vibrations of the beam of the hereditary material and thestochastic stability of the beam dynamic shapes, corresponding to the nth shape of the beamelastic form. Also, in [19] the transversal vibrations of the beam of the new models of theconstitutive stress-strain relations material in the form of a fractional derivative order con-stitutive relation beam are studied, and as well, the stochastic stability of the beam dynamicshapes, corresponding to the nth shape of the beam elastic form, is examined by using ideasof S. T. Ariaratnam [1]. By Isayev and Mamedov [23] some results on dynamic stabilityof nonhomogenous bars are presented.

Katica (Stevanovic) Hedrih 3

Foster and Berdichevsky [7] apply the quantitative method to estimate the violation ofSaint-Venant’s principle in the problem of flexural vibration of a two-dimensional strip.A probabilistic approach is used to determine the relative magnitude of the penetratingstress state and the results of computations are presented as a function of frequency. Theresults are not dependent on material properties except for the Poisson ratio. The ma-jor conclusion of these papers is that over a wide range of frequencies, the maximumpropagating stress is always small compared with the maximum applied stress; hence,Saint-Venant’s principle may be said to apply to this problem. An interesting outcomeof the study is that the accuracy of engineering theories for flexural vibrations is muchhigher than for longitudinal vibrations.

In [1] the stochastic stability of viscoelastic systems under bounded noise excitationby S. T. Ariaratnam is investigated and some new interesting results for applicationsare found. The paper by Parks and Pritchard [30] is a contribution on the construc-tion and the use of the Lyapunov functionals. The monograph by Stratonovich [35] isthe monograph with topics in the theory of random noise used in [1] and in this paper.Asymptotic method of averaging applied to the nonstationary nonlinear processes andto the nonlinear vibrations of deformable bodies is the topic of the three monographs[25–27]. Krilov-Bogolyubov-Mitropol’skiı’s method is presented in the book by Hedrih(Stevanovic) [14, 15]. This paper contains new results on transversal vibrations on non-homogeneous beams based on the contents of the cited references and books.

2. Model of creeping rheological body

For modeling processes of solidification and relaxation, models of Kelvin’s viscous-elasticmaterial and Maxwell’s ideal-elastic-viscous fluid are being used. In their paper, Goroshkoand Puchko [11] have used model of standard hereditary body to modeling dynamics ofmechanical systems with rheological links. Studying elements of mechanics of heredi-tary systems in their monograph, G. N. Savin and Yu. Ya. Ruschisky gave survey of bothstructure and analysis of the rheological models of simple and complex laws for linear de-formable hereditary-elastic media, as well as theory of growing old of hereditary-elasticsystems.

Recently, there is a noticeable interest in using fractional derivatives to describe creepbehavior of material. In solid mechanics particularly for describing problems related tomaterial creep behavior including viscoelastic and viscoplastic effects, fractional deriva-tives have a longer history (see [5, 9]). Mathematical basis of the fractional derivative andshort complete of fractional calculus are presented in the monograph paper by Gorenfloand Mainardi [9].

The paper by Dli et al. [4] contains the consideration of dynamical stability of vis-coelastic column with fractional derivative constitutive relation. The paper by Baclic andAtanackovic [2] considered stability and creep of a fractional derivative order viscoelasticrod.

We introduce that material of the one layer beam is a creeping material. Parameters ofthe beam creep material are the following: α is proper material constant of the character-istic creep law of material, E0 and Eα are modulus of elasticity and creeping properties ofmaterial.


By using stress-strain relation from the cited references, a single-axis stress state of thecreep hereditary-type material is described by fractional order time derivative differentialrelation in the form of three-parameter model. For line element of beam creep material,constitutive stress-strain state relation is expressed by fractional derivative constitutiverelation in the following form:

σz(z, y, t)= y{E0∂ϕ(z, t)∂z

+EαDαt

[∂ϕ(z, t)∂z

]}, (A)

where Dαt [·] is notation of the fractional derivative operator defined by the following

expression:

Dαt

[y∂ϕ(z, t)∂z

]= y

Γ(1−α)d

dt

∫ t0

[∂ϕ(z,τ)/∂z

](t− τ)α

dτ = yDαt

[∂ϕ(z, t)∂z

], (B)

where α is ratio number from interval 0 < α < 1; σz(z, y, t) is normal stress in the point ofcross-section of the line element, at distance z from the left beam end and at point withdistance y from neutral axis—bending beam axis; ϕ(z, t) is turn angle of the beam cross-section for pure bending; and εz(z, y, t)= y(∂ϕ(z, t)/∂z) is dilatation of the line element.

3. Partial fractional differential equation

The formulation of the problem of stochastic stability of nonhomogenous creeping barsof a fractional order derivative constitutive relation of material is assumed to be a con-tinuous function of the length coordinate. Let us consider the problem on transversaloscillations of two-layer straight bar, which is under the action of the lengthwise randomforces. The excitation process is a bounded noise excitation.

It is assumed that the layers of the bar were made of continuously creeping non-homogenous material and the corresponding modulus of elasticity and creeping frac-tional order derivative constitutive relation of each layer are continuous functions of thelength coordinate and thickness coordinates and are changed under the following laws(see Figure 3.1):

E(1)e (z, y)= E(1)

0 f (1)e (z) f (11)

e (y),

E(2)e (z, y)= E(2)

0 f (2)e (z) f (22)

e (y),

E(1)α (z, y)= E(1)

0α f(1)α (z) f (11)

α (y),

E(2)α (z, y)= E(2)

0α f(2)α (z) f (22)

α (y),

0≤ α≤ 1, 0≤ z ≤ �, −h1 ≤ y ≤ h2.

(3.1)

In this case connection between increments of stresses and deformations in each layeris represented in view:

Δσ (1)z = E(1)

e Δε(1)z +E(1)

α Dαt

[Δε(1)

z

], −h1 ≤ y ≤ 0,

Δσ (2)z = E(2)

e Δε(2)z +E(2)

α Dαt

[Δε(2)

z

], 0≤ y ≤ h2,

(3.2)


y

h2X

σzh1

Z

dA

yx

C

Figure 3.1

where

Dαt

[εz(t)

]= dαεz(t)dtα

= ε(α)z (t)= 1

Γ(1−α)d

dt

∫ t0

εz(τ)(t− τ)α

dτ. (3.3)

Here h1 and h2 are thicknesses of the corresponding layers.Dilatations are

εz = y∂ϕ(z, t)∂z

, Δεz = y∂Δϕ(z, t)

∂z, (3.4)

where ϕ(z, t) is angle of pure bending. The normal stress of pure bending is

dσ (1)z =E(1)

0 f (1)e (z) f (11)

e (y)dy∂ϕ(z, t)∂z

+E(1)0α f

(1)α (z) f (11)

α (y)Dαt

[dy

∂ϕ(z, t)∂z

], −h1≤ y≤0,

dσ (2)z = E(2)

0 f (2)e (z) f (22)

e (y)dy∂ϕ(z, t)∂z

+E(2)0α f

(2)α (z) f (22)

α (y)Dαt

[dy

∂ϕ(z, t)∂z

], 0≤ y ≤ h2.

(3.5)

From the equilibrium conditions we can write

N∑i=1

�Fi = 0,N∑i=1

�M�Fi0 = �Mf x = �M�Fi

f x (3.6)

or ∫∫A′′σ (1)z dxdy +

∫∫A′′σ (2)z dxdy = 0,

∫∫A′′σ (1)z xdxdy +

∫∫A′′σ (2)z xdxdy ∼= 0,

∫∫A′′σ (1)z y dxdy +

∫∫A′′σ (2)z y dxdy =Mf x

(3.7)

or∫ b/2−b/2

∫ 0

−h1

{E(1)

0 f (1)e (z) f (11)

e (y)y∂ϕ(z, t)∂z

+E(1)0α f

(1)α (z) f (11)

α (y)Dαt

[y∂ϕ(z, t)∂z

]}dxdy

−∫ b/2−b/2

∫ h2

0

{E(2)

0 f (2)e (z) f (22)

e (y)y∂ϕ(z, t)∂z

+E(2)0α f

(2)α (z) f (22)

α (y)Dαt

[y∂ϕ(z, t)∂z

]}dxdy = 0,


∫ b/2−b/2

∫ 0

−h1

{E(1)

0 f (1)e (z) f (11)

e (y)y2 ∂ϕ(z, t)∂z

+E(1)0α f

(1)α (z) f (11)

α (y)Dαt

[y2 ∂ϕ(z, t)

∂z

]}dxdy

−∫ b/2−b/2

∫ h2

0

{E(2)

0 f (2)e (z) f (22)

e (y)y2 ∂ϕ(z, t)∂z

+E(2)0α f

(2)α (z) f (22)

α (y)Dαt

[y2 ∂ϕ(z, t)

∂z

]}dxdy =Mf x.

(3.8)

If we introduce the following notations:

a(1)(1)e =

∫ 0

−h1

f (11)e (y)ydy, a(2)(1)

e =∫ h2

0f (22)e (y)ydy,

a(1)(1)α =

∫ 0

−h1

f (11)α (y)ydy, a(2)(1)

α =∫ h2

0f (22)α (y)ydy,

a(1)(2)e =

∫ 0

−h1

f (11)e (y)y2dy, a(2)(2)

e =∫ h2

0f (22)e (y)y2dy,

a(1)(2)α =

∫ 0

−h1

f (11)α (y)y2dy, a(2)(2)

α =∫ h2

0f (22)α (y)y2dy,

(3.9)

or set the following form:

a(1)(n)α =

∫ 0

−h1

f (11)α (y)yn dy, a(2)(n)

α =∫ h2

0f (22)α (y)yn dy, n= 0,1,2, (3.10)

we can write the previous equilibrium conditions in the following relations:

a(1)(1)e E(1)

0 f (1)e (z)− a(2)(1)

e E(2)0 f (2)

e (z)= 0=⇒ f (2)e (z)= f (1)

e (z)E(1)

0

E(2)0

a(1)(1)e

a(2)(1)e

,

a(1)(1)α E(1)

0α f(1)α (z)− a(2)(1)

α E(2)0α f

(2)α (z)= 0=⇒ f (2)

α (z)= f (1)α (z)

E(1)0α

E(2)0α

a(1)(1)α

a(2)(1)α

(3.11)

and write the following expression for bending moment:

Mf x(z, t)= b∂ϕ(z, t)∂z

{E(1)

0 a(1)(2)e f (1)

e (z) +E(2)0 a(2)(2)

e f (2)e (z)

}

+ bDαt

[∂ϕ(z, t)∂z

]{E(1)

0α a(1)(2)α f (1)

α (z) +E(2)0α a

(2)(2)α f (2)

α (z)}

,

(3.12)


also with respect to the relations (3.10) we can write (3.12) in the following form:

Mf x(z, t)= E(1)0 b

∂ϕ(z, t)∂z

f (1)e (z)

{a(1)(2)e + a(2)(2)

ea(1)(1)e

a(2)(1)e

}

+E(1)0α b f

(1)α (z)Dα

t

[∂ϕ(z, t)∂z

]{a(1)(2)α + a(2)(2)

αa(1)(1)α

a(2)(1)α

}.

(3.13)

We take into account the rotatory inertia of cross-section and we can write the followingequations of bar dynamics:

dJx∂2ϕ(z, t)∂t2

=−dMf (z, t) +FT(z, t)dz+FN (Ξ,z, t)dv(z, t),

dm∂2v(z, t)∂t2

= dFT(z, t).

(3.14)

If we introduce

dm= ρ1A1 + ρ2A2, dJx =[ρ1I(1)

x + ρ2I(2)x

]dz, (3.15)

we can write

[ρ1I(1)

x + ρ2I(2)x

]∂2ϕ(z, t)∂t2

= E(1)0 b

[a(1)(2)e + a(2)(2)

ea(1)(1)e

a(2)(1)e

]∂

∂z

[∂ϕ(z, t)∂z

f (1)e (z)

]

+E(1)0α b

[a(1)(2)α + a(2)(2)

αa(1)(1)α

a(2)(1)α

]∂

∂z

{f (1)α (z)Dα

t

[∂ϕ(z, t)∂z

]}+ FT +FN

∂v(z, t)∂z

,

(ρ1A1 + ρ2A2

)∂2v(z, t)∂t2

= ∂FT(z, t)∂z

.

(3.16)

After applying derivative with respect to time, we can write

∂ϕ(z, t)∂z

= ∂2v(z, t)∂z2

,∂3ϕ(z, t)∂z∂t2

= ∂4v(z, t)∂z2∂t2

, (3.17)

[ρ1I(1)

x + ρ2I(2)x

]∂3ϕ(z, t)∂t2∂z

= E(1)0 b

[a(1)(2)e + a(2)(2)

ea(1)(1)e

a(2)(1)e

]∂2

∂z2

[∂ϕ(z, t)∂z

f (1)e (z)

]

+E(1)0α b

[a(1)(2)α + a(2)(2)

αa(1)(1)α

a(2)(1)α

]∂2

∂z2

{f (1)α (z)Dα

t

[∂ϕ(z, t)∂z

]}

+∂FT∂z

+∂

∂z

[FN

∂v(z, t)∂z

].

(3.18)


By introducing derivatives (3.17) into (3.18) we obtain the following partial fractionaldifferential equation:

[ρ1I(1)

x + ρ2I(2)x

]∂4v(z, t)∂t2∂z2

= E(1)0 b

[a(1)(2)e + a(2)(2)

ea(1)(1)e

a(2)(1)e

]∂2

∂z2

[∂2v(z, t)∂z2

f (1)e (z)

]

+E(1)0α b

[a(1)(2)α + a(2)(2)

αa(1)(1)α

a(2)(1)α

]∂2

∂z2

{f (1)α (z)Dα

t

[∂2v(z, t)∂z2

]}

+(ρ1A1 + ρ2A2

)∂2v(z, t)∂t2

+∂

∂z

[FN

∂v(z, t)∂z

],

∂2v(z, t)∂t2

+E(1)

0 b[a(1)(2)e + a(2)(2)

e(a(1)(1)e /a(2)(1)

e)]

(ρ1A1 + ρ2A2

) ∂2

∂z2

[∂2v(z, t)∂z2

f (1)e (z)

]

+1(

ρ1A1 + ρ2A2) ∂∂z

[FN

∂v(z, t)∂z

]+E(1)

0α b[a(1)(2)α + a(2)(2)

α(a(1)(1)α /a(2)(1)

α)]

(ρ1A1 + ρ2A2

)

× ∂2

∂z2

{f (1)α (z)Dα

t

[∂2v(z, t)∂z2

]}+

[ρ1I(1)

x + ρ2I(2)x]

(ρ1A1 + ρ2A2

) ∂4v(z, t)∂t2∂z2

= 0.

(3.19)

By introducing the following notations:

c 20x =

E (1)0

ρi 2xe =

E(1)0 b[a(1)(2)e + a(2)(2)

e(a(1)(1)e /a(2)(1)

e)]

(ρ1A1 + ρ2A2

) ,

c 20xα =

E (1)0α

ρi 2xα =

E(1)0α b[a(1)(2)α + a(2)(2)

α(a(1)(1)α /a(2)(1)

α)]

(ρ1A1 + ρ2A2

) ,

i 2x =

[ρ1I(1)

x + ρ2I(2)x]

(ρ1A1 + ρ2A2

) , i 2x =

[ρ1I(1)

x + ρ2I(2)x]

Aρ,

i 2xe =

b[a(1)(2)e + a(2)(2)

e(a(1)(1)e /a(2)(1)

e)]

A, i 2

xα =b[a(1)(2)α + a(2)(2)

α(a(1)(1)α /a(2)(1)

α)]

A,

(3.20)

we obtain the following partial fractional differential equation of transversal vibrationsof creeping of two-layer straight bar, which is under the action of the lengthwise randomforces:

∂2v(z, t)∂t2

+ c 20x∂2

∂z2

[∂2v(z, t)∂z2

f (1)e (z)

]+

1(ρ1A1 + ρ2A2

) ∂∂z

[FN

∂v(z, t)∂z

]

+ c 20xα

∂2

∂z2

{f (1)α (z)Dα

t

[∂2v(z, t)∂z2

]}− i 2

x∂4v(z, t)∂t2∂z2

= 0.

(3.21)


We study the following special case: from (3.21), we exclude members which containaxial forces, or we suppose that axial forces are equal to zero, and we solve the followingequation:

∂2v(z, t)∂t2

+ c 20x∂2

∂z2

[∂2v(z, t)∂z2

f (1)e (z)

]+ c 2

0xα∂2

∂z2

{f (1)α (z)Dα

t

[∂2v(z, t)∂z2

]}− i 2

x∂4v(z, t)∂t2∂z2

= 0

(3.22)

when

f (1)e (z)= f (1)

α (z)= f (z) (3.23)

and we can write

∂2v(z, t)∂t2

+ c 20x∂2

∂z2

[∂2v(z, t)∂z2

f (z)]

+ c 20xα

∂2

∂z2

{f (z)Dα

t

[∂2v(z, t)∂z2

]}− i 2

x∂4v(z, t)∂t2∂z2

= 0.

(3.24)

4. Solution of the partial fractional differential equation of the beam transversalvibrations with creep material properties

By using Bernoulli’s method for obtaining solution and for solution of the partial frac-tional differential equation (3.24), we can write a product of the two functions dependingon separate coordinate z and time t in the following form:

v(z, t)= Z(z)T(t). (4.1)

By introducing this solution into (3.24) we obtain

Z(z)T(t) + c 20xT(t)

d2

dz2

[Z′′(z) f (z)

]+ c 2

0xαd2

dz2

{Z′′(z) f (z)

}Dαt

[T(t)

]− i 2x Z

′′(z)T(t)= 0

(4.2)

or

Z(z) +d2

dz2

[Z′′(z) f (z)

]{c 2

0xT(t)T(t)

+ c 20xα

1T(t)

Dαt

[T(t)

]}− i 2x Z

′′(z)= 0 (4.3)

or we obtain two equations

c 20xT(t)T(t)

+ c 20xα

1T(t)

Dαt

[T(t)

]=− 1k4

,

Z(z)− d2

dz2

[Z′′(z) f (z)

] 1k4− i 2

x Z′′(z)= 0

(4.4)

or

T(t) + ω2αxD

αt

[(t)]

+ ω20xT(t)= 0, (4.5)

d2

dz2

[Z′′(z) f (z)

]+ i 2

x k4Z′′(z)− k4Z(z)= 0, (4.6)


where

ω20x = k4c 2

0x = k4E(1)0 b[a(1)(2)e + a(2)(2)

e(a(1)(1)e /a(2)(1)

e)]

(ρ1A1 + ρ2A2

) ,

ω2αx = k4c 2

0xα = k4E(1)0α b[a(1)(2)α + a(2)(2)

α(a(1)(1)α /a(2)(1)

α)]

(ρ1A1 + ρ2A2

) ,

i 2x =

[ρ1I(1)

x + ρ2I(2)x]

(ρ1A1 + ρ2A2

) .

(4.7)

We can obtain the solution of the fractional differential equation of system (4.5) byusing the Laplace transform, and by having in mind that, in initial moment, dα−1T(t)/dtα−1|t=0 = 0. Solutions for special cases when α = 0 and α = 1, and for beam kineticparameters: ω0 > (1/2)ω2

1 for soft creep and ω0 < (1/2)ω21 for strong creep, are solutions

of classical ordinary differential equation. It is the same for α= 1 and ω0x = (1/2)ω21x.

For the general case when ω20x �= 0, the Laplace transform of the solution L{T(t)} of

the fractional differential equation of system (4.5) can be developed, in two steps, intoseries with respect to binoms (pα +ω2

0x/ω2αx), and with respect to pα. Then we obtain the

following expression:

L{T(t)

}=(T0 +

T0

p

)1p

∞∑k=0

(−1)kω2kαx

p2k

k∑j=0

(k

j

)pαjω

2( j−k)αx

ω2 jox

. (4.8)

By using inverse of the Laplace transform of the solution L{T(t)}, for general case,when beam material parameter is from interval 0≤ α≤ 1, for solution of the fractional-differential equation of system (4.5), we obtain the following expression in the form ofpotential series of the time t:

T(t)= L−1{T(t)}

=∞∑k=0

(−1)kω2kαxt

2kk∑j=0

(k

j

)ω

2 jαxt−αj

ω2 jox

[T0

Γ(2k+ 1−αj) +T0t

Γ(2k+ 2−αj)

].

(4.9)

And in that case there are special cases when ω20x = 0 for α= 0 and for α= 1.

In Figure 4.1 numerical simulations and graphical presentation of the solution of thefractional differential equation of system (4.5) are presented. Time functions T(t,α) sur-faces for the different beam transversal vibrations kinetic and creep material parametersin the space (T(t,α), t,α) for interval 0≤ α≤ 1 are visible in (a) for (ωαx/ω0x)= 1, (b) for(ωαx/ω0x)= 1/4, (c) for (ωαx/ω0x)= 1/3, and (d) for (ωαx/ω0x)= 3.

In Figure 4.2 the time functions T(t,α) surfaces and curves families for the differentbeam transversal vibrations kinetic and discrete values of the creeping material parame-ters 0≤ α≤ 1 are presented in (a) and (c) for (ωαx/ω0x)= 1, (b) and (d) for (ωαx/ω0x)=1/4, (e) for (ωαx/ω0x)= 1/3, and (f) for (ωαx/ω0x)= 3.


0.4

0.2

0

−0.2

−0.4

T(t, α)

α

α = 1

α = 0

0

1020

t010

2030

40

(a)

0.4

0.2

0

−0.2

−0.40

T(t, α)

1020

30

010

2030

40

t

α

(b)

4030

2010

0

T(t, α)

0 10 20 30−0.4

0.4

t

α

(c)

0.5

0

−0.50

0 10 20 30 40

10

T(t, α)

t

α

(d)

Figure 4.1. Numerical simulations and graphical presentation of the results. Time functions T(t,α)surfaces for the different beam transversal vibrations kinetic and creep material parameters: (a)(ωαx/ω0x)= 1, (b) (ωαx/ω0x)= 1/4, (c) (ωαx/ω0x)= 1/3, and (d) (ωαx/ω0x)= 3.

5. The S. T. Ariaratnam idea applied to the stochastic stability of the creep beamtransversal vibrations dynamic shapes under axial bounded noise excitation

In the case

f (z)= 1 +mz

�(5.1)

equation (3.21) transforms to the form

∂2v(z, t)∂t2

+ c 20x∂2

∂z2

[∂2v(z, t)∂z2

f (1)e (z)

]+

1(ρ1A1 + ρ2A2

) ∂∂z

[FN

∂v(z, t)∂z

]

+ c 20xα

∂2

∂z2

{f (1)α (z)Dα

t

[∂2v(z, t)∂z2

]}− i 2

x∂4v(z, t)∂t2∂z2

= 0

(5.2)


0.5

0

−0.5

00

10 20 30 40

10

T(t, α)α

t

(a)

0.4

0.2

0

−0.2

−0.40

10

0 10 20 30 40

T(t, α) α

t

(b)

0.5

0

−0.5

0 10 20 30 40

T(t, α)

t

(c)

0.4

0.20

−0.2

−0.4

T(t, α)

0 1 2 3 4 5 6 7 8 9 10t

f (x)f 1(x)f 2(x)f 3(x)f 4(x)f 5(x)

f 6(x)f 7(x)f 8(x)f 9(x)f 10(x)

(d)

1

0.5

0

−0.5

−1

T(t, α)

0 5 10 15 20 25 30

t

f (x)f 1(x)f 2(x)f 3(x)f 4(x)f 5(x)

f 6(x)f 7(x)f 8(x)f 9(x)f 10(x)

(e)

43210

−1−2

T(t, α)

0 10 20 30 40 50 60

t

f (x)f 1(x)f 2(x)f 3(x)f 4(x)f 5(x)

f 6(x)f 7(x)f 8(x)f 9(x)f 10(x)

(f)

Figure 4.2. Numerical simulations and graphical presentation of the results. Time functions T(t,α)surfaces and curves families for the different beam transversal vibrations kinetic and discrete valuesof the creeping material parameters 0≤ α≤ 1: (a) and (c) (ωαx/ω0x)= 1, (b) and (d) (ωαx/ω0x)= 1/4,(e) (ωαx/ω0x)= 1/3, and (f) (ωαx/ω0x)= 3.


and (4.6) transforms to the form

d2

dz2

[Z′′(z)

(1 +m

z

�

)]+ i 2

x k4Z′′(z)− k4Z(z)= 0. (5.3)

In the case of hinge fixing of the ends of the bar, solution of (5.1) into first unperturbedform is found in a view

v(z, t)= T(t)sinnπz

�. (5.4)

By introducing (5.4) into (5.2) and applying the Bubnov-Galerkin method, we obtainthe following equations:

T(t)sinnπz

�+ c 2

0xT(t)d2

dz2

{−(nπ

�

)2(1 +m

z

�

)sin

nπz

�

}

− FN (t)(ρ1A1 +ρ2A2

)(nπ

�

)2

T(t)sinnπz

�+c 2

0xαd2

dz2

{−(nπ

�

)2(1+m

z

�

)sin

nπz

�

}Dαt

[T(t)

]

+ i 2x

(nπ

�

)2

T(t)sinnπz

�= 0,

T(t) +c 2

0x(nπ/�)4[1 +m/2][1 + i 2

x (nπ/�)2]{

1− FN (t)(nπ/�)2

c 20x(nπ/�)4[1 +m/2]

}T(t)

+c 2

0xα(nπ/�)4[1 +m/2][1 + i 2

x (nπ/�)2] Dα

t

[T(t)

]= 0.

(5.5)

We pointed out the following notations:

ω20xn =

c 20x(nπ/�)4[1 +m/2][

1 + i 2x (nπ/�)2

] ,

hxnξ(t)= FN (t)c 2

0x(nπ/�)2[1 +m/2],

ω20xαn =

c 20xα(nπ/�)4[1 +m/2][

1 + i 2x (nπ/�)2

]

(5.6)

and we obtain the following fractional differential equation with respect to the time func-tion:

T(t) + ω20xn

{1−hxnξ(t)

}T(t) + ω2

0xαnDαt

[T(t)

]= 0. (5.7)

To solve the previous equation we can apply Ariaratnam’s idea [1]. The randombounded noise axial excitation ξ(t) is taken in the following form:

F(t)= F0ξ(t)= F0 sin[Ωt+ σB(t) + γ

], (5.8)


where B(t) is the standard Wiener process, and γ is a random uniformly distributed vari-able in interval [0,2π], then ξ(t) is a stationary process having autocorrelation functionand spectral density function:

R(τ,Ω)= 12h2one

−σ2τ/2 cosΩτ,

S(ω,Ω)=∫ +∞

−∞R(τ,Ω)eiωτdτ = 1

2h0nσ

2 ω2 +Ω2 + σ2/4[(ω2−Ω2− σ2/4

)2+ σ2ω2

] .(5.9)

Stochastic process |ξ(t)| ≤ 1 is bounded for all values of time t.The next idea of Ariaratnam is to apply the averaging method, and for that reason

we must introduce the amplitude an(t) and the phase Φn(t), which are time unknownfunctions, by means of the transformation relation of Tn(t):

Tn(t)= an(t)cosΦn(t), Tn(t)=−an(t)ω0n sinΦn(t). (5.10)

Substituting these relations in (5.7) and using (5.8) as well as Φn(t)= (Ω/2)t+ φn(t) andΔn = ω0xn−Ω/2, we can write the following system fractional differential equation withrespect to the amplitude an(t) and the phase Φn(t), exactly equivalent to (5.7):

an(t)=−12ω0xnan(t)h0n sin(Ωt+ψ)sin

(Ωt+ 2φn

)+ω2αxn

ω0xnsinΦn(t)Dα

t

[an(t)cosΦn(t)

],

(5.11)

˙φn(t)= Δn(t)− 1

2ω0xnh0n sin(Ωt+ψ)

[1 + cos

(Ωt+ 2φn

)]

+ω2αxn

an(t)ωoxncosΦn(t)Dα

t

[an(t)cosΦn(t)

], ψ(t)= σB(t),

(5.12)

where

Dαt

[an(t)cosΦn(t)

]= 1Γ(1−α)

d

dt

∫ t0

an(τ)cosΦn(τ)(t− τ)α

dτ. (5.13)

By applying the averaging method, we assume that excitation and beam kinetic pa-rameters values h0n, ω2

αxn, Δn(t) are small depending on small parameter ε (see [12, 16])as βn =O(ε), Δn =O(ε) and μ=O(ε), where 0 < ε ≤ 1. The assumption or the conditionΔn = O(ε) shows that frequencies of external random bounded excitation Ω are in thevicinity of the frequency 2ω0xn of fundamental parametric resonance in the nth form ofperturbed parametric resonance state.

We introduce the following notations:

∫ +∞

0R(τ)eiωτdτ =Hc(ω) + iHs(ω), where the kernel is in the form R(τ)= τ−α.

(5.14)


Having in consideration that (see [5])

Dαt

[f (t)

]= 1Γ(1−α)

d

dt

∫ t0

f (τ)(t− τ)α

dτ = 1Γ(1−α)

∫ t0

f ′(τ)(t− τ)α

dτ + f(0+) t−α

Γ(1−α),

Dαt

[an(t)cosΦn(t)

]= 1Γ(1−α)

∫ t0

−an(τ)ω0xn sinΦn(τ)(t− τ)α

dτ + an(0+)cosΦn

(0+) t−α

Γ(1−α)(5.15)

and after averaging the right-hand side of (5.12) with respect to total phase Φn and takingthat Ωt = 2Φn− 2φn, we obtain the averaged equations:

an(t)=−14ω0xnan(t)h0n cos

(ψ− 2φn

)− ω2αxn

2Γ(1−α)an(t)Hen

(Ω

2

),

˙φn(t)= Δn(t)− 1

4ω0xnhon sin

(ψ− 2φn

)+

ω2αxn

2Γ(1−α)Hen

(Ω

2

),

ψ(t)= σB(t),

(5.16)

where limT→∞(1/T)∫ T

0 eiΦn(t)t−α dt = 0.

By introducing the change of the variables by the relations ρn(t) = lnan(t) and θn =φn − ψ/2 into the previous pair of the stochastic differential equations, we obtain thesystem

dρn(t)=[− 1

4ω0xnh0n cos2θn− ω2

αxn

2Γ(1−α)(t)Hen

(Ω

2

)]dt,

dθn(t)=[Δn(t) +

14ω0xnh0n sin2θn +

ω2αxn

2Γ(1−α)Hsn

(Ω

2

)]dt− 1

2σdB(t).

(5.17)

6. The Lyapunov exponent and stochastic stability

The Lyapunov exponent (see [1]) of the creeping beam stochastic transversal vibrationsin the nth form of perturbed parametric resonance state given by the averaged stochasticequations system (5.17) may be defined by the following expression:

λn = limt→∞

12t

ln{[Tn(t)

]2+

1ω2on

[Tn(t)

]2}=⇒ λn = lim

t→∞1t

ln[an(t)

]= limt→∞

1tρn(t). (6.1)

Now, the Lyapunov exponent is a measure of the average exponential growth of theamplitude process an(t) of the creep beam transversal vibrations in the nth form of per-turbed parametric resonance process. λn is a deterministic number with probability one(w.p.1) for the system given by (5.17). Solutions of the averaged differential equationsdepending on initial values Tn(t0) and Tn(t0), in general, are two values of the Lyapunovexponent λn in the corresponding nth form of perturbed parametric resonance process.If both Lyapunov exponents are negative, the trivial solution in the corresponding nthform of perturbed parametric resonance process are stable processes.


In order to calculate λn, we must integrate both sides of (5.17) and we obtain thefollowing expression:

λn = limt→∞

1tρ(t)=−1

4ω0xnh0nE

[cos2θn

]− 12

ω2αxn

Γ(1−α)Hen

(Ω

2

). (6.2)

By using corresponding results obtained by Stratonovich [35] and Ariaratnam [1] forthe Lyapunov exponent we obtain the following asymptotic result:

λn =−14ωoxnhonF

(honωoxnσ2

,4Δnσ2

)− 1

2ω2αxn

Γ(1−α)Hen

(Ω

2

). (6.3)

In the previous expressions and calculations we used invariant (stationary) probabilitydensity function satisfying the periodicity condition when

Δ0n(t)= ωon− Ω

2+

12

ω2αxn

Γ(1−α)Hsn

(Ω

2

)= 0. (6.4)

For the Lyapunov exponent we obtain

λn =−14h0nω0xn

I1(h0nω0xn/σ2

)I0(h0nω0xn/σ2

) − ω2αxnn

2Γ(1−α)Hcn

(Ω

2

), (6.5)

where I0, I1 are Bessel functions of real argument, and F(v · q) is a function of Besselfunctions of imaginary argument.

7. Concluding remarks

From the obtained analytical and numerical results for natural transversal creeping vi-brations of a fractional order derivative hereditary rod with two layers, it can be seenthat fractional order derivative hereditary properties are convenient for changing timefunction depending on material creep parameters, and that fundamental eigenfunctiondepending on space coordinate is dependent only on boundary conditions and geomet-rical properties of layers.

Acknowledgments

Parts of this research were supported by the Ministry of Sciences, Technologies and De-velopment of Republic Serbia through Mathematical Institute SANU Belgrade Grantsno. 1616 Real Problems in Mechanics and no. ON144002 Theoretical and Applied Me-chanics of Rigid and Solid Body, Mechanics of Materials, and the Faculty of MechanicalEngineering University of Nis Grant no. 1828 Dynamics and Control of Active Structure.This support is greatly acknowledged. The author wishes to express her gratitude to thereviewers for the useful suggestions which have contributed to the improvement of thepaper.


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Katica (Stevanovic) Hedrih: Faculty of Mechanical Engineering, University of Nis, 18000 Nis, Serbiaand Montenegro; Mathematical Institute SANU, 11001 Belgrade, Serbia and MontenegroE-mail addresses: [email protected]; [email protected]

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