# Dynamic stability of an axially accelerating viscoelastic beam

**ABSTRACT** This work investigates dynamic stability in transverse parametric vibration of an axially accelerating viscoelastic tensioned beam. The material of the beam is described by the Kelvin model. The axial speed is characterized as a simple harmonic variation about the constant mean speed. The Galerkin method is applied to discretize the governing equation into a infinite set of ordinary-differential equations under the fixed–fixed boundary conditions. The method of averaging is employ to analyze the dynamic stability of the 2-term truncated system. The stability conditions are presented and confirmed by numerical simulations in the case of subharmonic and combination resonance. Numerical examples demonstrate the effects of the dynamic viscosity, the mean axial speed and the tension on the stability conditions.

**0**Bookmarks

**·**

**85**Views

- [Show abstract] [Hide abstract]

**ABSTRACT:**A mathematical model describing the nonlinear vibration of horizontal axis wind turbine (HAWT) blades is proposed in this paper. The system consists of a rotating blade and four components of deformation including longitudinal vibration (named axial extension), out-of-plane bend (named flap), in-plane/edgewise bend (named lead/lag) and torsion (named feather). It is assumed that the center of mass, shear center and aerodynamic center of a cross section all lie on the chord line, and don't coincide with each other. The structural damping of the blade, which is brought about by materials and fillers is taken into account based on the Kelvin–Voigt theory of composite materials approximately. The equivalent viscosity factor can be determined from empirical data, theoretical computation and experimental test. Gravitational loading and aerodynamic loading are considered as distributed forces and moments acting on blade sections. A set of partial differential equations governing the coupled, nonlinear vibration is established by applying the generalized Hamiltonian principle, and the current model is verified by previous models. The solution of equations is discussed, and examples concerning the static deformation, aeroelastic stability and dynamics of the blade are given.Applied Mathematical Modelling 10/2013; · 1.71 Impact Factor - SourceAvailable from: Hu Ding
##### Dataset: 《EJMSOL》2392

- SourceAvailable from: Hu Ding

Page 1

European Journal of Mechanics A/Solids 23 (2004) 659–666

Dynamic stability of an axially accelerating viscoelastic beam

Li-Qun Chena,b,∗, Xiao-Dong Yangb, Chang-Jun Chenga,b

aDepartment of Mechanics, Shanghai University, Shanghai 200436, China

bShanghai Institute of Applied Mathematics and Mechanics, Shanghai 200072, China

Received 13 October 2003; accepted 14 January 2004

Available online 25 March 2004

Abstract

This work investigates dynamic stability in transverse parametric vibration of an axially accelerating viscoelastic tensioned

beam. The material of the beam is described by the Kelvin model. The axial speed is characterized as a simple harmonic

variation about the constant mean speed. The Galerkin method is applied to discretize the governing equation into a infinite set

of ordinary-differential equations under the fixed–fixed boundary conditions. The method of averaging is employ to analyze the

dynamic stability of the 2-term truncated system. The stabilityconditions are presented and confirmed by numerical simulations

in the case of subharmonic and combination resonance. Numerical examples demonstrate the effects of the dynamic viscosity,

the mean axial speed and the tension on the stability conditions.

2004 Elsevier SAS. All rights reserved.

Keywords: Axially accelerating beam; Viscoelasticity; Method of averaging

1. Introduction

Axially moving beams can represent many engineering devices, such as band saws and serpentine belts. Despite many

advantages of these devices, vibrations associated with the devices have limited their applications. One major problem is the

occurrence of large transverse vibrations due to tension or axial speed variation termed as parametric vibrations.

Axial transport acceleration frequently appears in engineering systems. For example, if an axially moving beam models a

belt on a pair of rotating pulleys, the rotation vibration of the pulleys will result in a small fluctuation in the axial speed of the

belt. Although vibration analysis of parametrically excited, axially moving beams has been studied extensively, the literature

that is specially related to the axially accelerating beams is relatively limited. Pasin (1972) studied the stability of transverse

vibrations of beams with periodically reciprocating motion in axial direction. Öz, Pakdemirli and Özkaya (1998) applied the

method of multiple scales to study dynamic stability of an axially accelerating beam with small bending stiffness. Özkaya

and Pakdemirli (2000) applied the method of multiple scales and the method of matched asymptotic expansions to construct

non-resonant boundary layer solutions for an axially accelerating beam with small bending stiffness. Öz and Pakdemirli (2000)

and Öz (2001) used the method of multiple scales to calculate analytically the stability boundaries of an axially accelerating

tensioned beam under simply-supported conditions and fixed–fixed conditions respectively. Parker and Lin (2001) adopted a

1-term Galerkin discretization and the perturbation method to study dynamic stability of an axially accelerating beam subjected

to a tension fluctuation. Özkaya and Öz (2002) applied artificial neural network algorithm to determine stability of an axially

accelerating beam.

All above-mentioned researchers assumed the beam under their consideration is elastic, and did not account for

any dissipative mechanisms. However, the modeling of dissipative mechanisms is an important research topic of axially

*Corresponding author.

E-mail address: lqchen@online.sh.cn (L.-Q. Chen).

0997-7538/$ – see front matter 2004 Elsevier SAS. All rights reserved.

doi:10.1016/j.euromechsol.2004.01.002

Page 2

660

L.-Q. Chen et al. / European Journal of Mechanics A/Solids 23 (2004) 659–666

moving material vibrations (Wickert and Mote, 1988; Abrate, 1992). Viscoelasticity is an effective approach to model the

dissipative mechanism because some beam-like engineering devices are composed of some viscoelastic metallic or ceramic

reinforcement materials like glass-cord and viscoelastic polymeric materials such as rubber. Based on 3-term Galerkin

truncation, Marynowski (2002) and Marynowski and Kapitaniak (2002) compared the Kelvin model with the Maxwell model

and the Bügers model respectively through numerical simulation of nonlinear vibration responses of an axially moving beam at

a constant speed. They found that in the case of small damping, all models yield similar results. To authors’ knowledge, there

are no researches on transverse vibration of an axially accelerating viscoelastic beam. To address the lack of research in this

aspect, this paper investigates dynamic stability of an axially accelerating viscoelastic beam.

2. The governing equation and its Galerkin discretization

A uniform axially moving viscoelastic beam, with linear density ρ, cross-sectional area moment of inertial I and initial

tension P, travelsatthetime-dependent axial transport speed c(T) between twoprismaticendsseparated bydistance L.Because

only small damping is considered here, according to the results of Marynowski (2002) and Marynowski and Kapitaniak (2002),

one can assume the viscoelasticity of the beam material to be defined by the Kelvin model

?

where σ is the axial stress, ε is the axial strain, E0is the stiffness constant, and E0α is the viscosity coefficient. Consider only

the bending vibration described by the transverse displacement V(X,T), where T is the time, and X is the axial coordinate.

The Newton second law of motion yields

?∂2V

The prismatic joints between which the beam travelscan be modeled as fixed–fixed ends. Hence the boundary conditions are

????(0,T)

c(T) = c0(1+hcosΩT).

The assumption has its the physical meaning. For example, if the axially moving beam models a belt on a pair of rotating

pulleys, the rotation vibration of the pulleys will result in a small fluctuation in the axial speed of the belt.

Substituting Eq. (4) into Eq. (2) and transforming the resulting equation into the dimensionless form yield the dimensionless

governing equation of transverse motion

∂2v

∂t2+2γ(1+hcosωt)∂2v

where

?

?

σ = E0

1+α∂

∂T

?

ε,

(1)

ρ

∂T2+2c∂2V

∂X∂T+dc

dt

∂V

∂X+c2∂2V

∂X2

?

−P∂2V

∂X2+E0I

?

1+α∂

∂T

?∂4V

∂X4= 0.

(2)

V(0,t) = V(L,t) =0,

∂V

∂X

=∂V

∂X

????(L,T)

=0.

(3)

It is assumed that the transport speed is characterized as a simple harmonic variation about the constant mean speed, i.e.,

(4)

∂x∂t−γhω∂v

∂xsinωt +?γ2−µ2+2γ2hcosωt?∂2v

?

?

∂x2+∂4v

∂x4+η

∂5v

∂x4∂t= 0,

(5)

v =V

L,

x =X

L,

t = T

?

E0I

ρL4,γ = c0

ρL2

E0I,

µ=

PL2

E0I,

η =α

E0I

ρL4,ω = Ω

ρL4

EI.

(6)

The corresponding dimensionless boundary conditions are

v(0,t) = v(1,t) = 0,

∂v

∂x

????(0,T)

=∂v

∂x

????(1,T)

= 0.

(7)

The Galerkin method is employed to simplify Eq. (5). Under given boundary conditions, the solution of Eq. (5) may be

expanded into the series of eigenfunctions of the stationary beam with two fixed ends

v(x,t) = q(t)Tϕ(x),

wheretheinfinitecolumnmatrixes q(t) and ϕ(x) arerespectively assembledby thegeneralizeddisplacements and thestationary

eigenfunctions, namely

q(t) =?q1(t),q2(t),...,qn(t),...?T,

(8)

ϕ(x) =?ϕ1(x),ϕ2(x),...,ϕn(x),...?T.

(9)

Page 3

L.-Q. Chen et al. / European Journal of Mechanics A/Solids 23 (2004) 659–666

661

For the beam under the fixed–fixed boundary conditions (7), the stationary eigenfunctions are (Bishop and Johnson, 1979)

ϕi(x) = cosβix −coshβix −cosβi−coshβi

sinβi−sinhβi

where βiis the ith root of the frequency equation

cosβicoshβi−1 =0.

Taking the appropriate derivatives and substituting them into Eq. (5), one obtains the residual

R(x,t) = ¨ qTϕ +?2γ(1+hcosωt)˙ qT−γωhsinωtqT?ϕ?+?γ2−µ2+2γ2hcosωt?qTϕ??+?qT+η˙ qT?ϕ????.

If the weighting functions are also chosen as the stationary beam eigenfunctions, then application of the Galerkin method

requires that the residual (12) should satisfy

(sinβix −sinhβix),

(10)

(11)

(12)

1

?

0

R(x,t)ϕ(x)Tdx = 0T

∞,

(13)

where 0∞is an infinite zero column matrix. Substituting Eq. (12) into Eq. (13) and transposing the resulting equation, one

obtains the Galerkin discretization of the governing equation

I∞¨ qT+B?2γ(1+hcosωt)˙ qT−γωhsinωtqT?+?γ2−µ2+2γ2hcosωt?CqT+Λ?qT+η˙ qT?= 0∞,

where

1

?

00

1

?

00

The orthonormality of the stationary beam eigenfunctions leads to

Λ= diag?β4

In the case that h = 0 and η = 0, Eq. (5) is reduced to the generating system

∂2v

∂t2+2γ∂2v

which governs freetransverse vibration of an elasticbeam axiallymoving at a constant speed. Followingtheprocedure proposed

by Wickert and Mote (1990), one can determine numerically the natural frequencies ωi(i = 1,2,...) for given parameters γ

and µ.

(14)

I∞=

ϕ(x)ϕ(x)Tdx,

B =

1

?

?

ϕ(x)ϕ?(x)Tdx,

C =

ϕ(x)ϕ??(x)Tdx,

Λ =

1

ϕ(x)ϕ????(x)Tdx.

(15)

I∞= diag{11

...},

1

β4

2

...β4

n

...?.

(16)

∂x∂t+?γ2−µ2?∂2v

∂x2+∂4v

∂x4= 0,

(17)

3. The averaged equation and stability analysis

In present investigation, the authors will consider subharmonic resonance and combination resonance near the first two

eigenfrequencies of the generating system (17). Therefore only first two terms are retained in the Galerkin expansion (8). In this

case, both B and C are 2×2 matrices, and direct calculations yield

B = bJ2,

where

4β2

2

β4

2

ci= −βi

+(cos2βi+4cosβi−3)sinh2βi

Introduce a new set of state variables

ζ = (ζ1,ζ2,ζ3,ζ4)T=(q1,q2, ˙ q1, ˙ q2)T.

C = diag{c1

c2},

(18)

b =

1β2

1−β4

?4coshβisinβi+(cosh2βi−3)sin2βi+4(cosβi−coshβi)2

?

1+(tanβ1−tanhβ1)(tanβ2−tanhβ2)

(sinβ1−sinhβ1)(sinβ2−sinhβ2)

?

,

J2=

?0

−1

01

?

,

???4(sinβi−sinhβi)2?

(i = 1,2).

(19)

(20)

Page 4

662

L.-Q. Chen et al. / European Journal of Mechanics A/Solids 23 (2004) 659–666

Then Eq. (14) can be truncated and cast into the form

˙ζ = Aζ +hA1ζ cosωt +ωhA2ζ sinωt +ηA3ζ,

(21)

where

A = −

?

02

I2

2γB

(γ2−µ2)C +Λ

?02

?

,

A1= −2γ

?02

?1

02

B

?

γC

?

,

A2= γ

?02

02

02

B

?

,

A3= −η

02

Λ

02

?

,

02=

?00

00

?

,

I2=

0

10

.

(22)

The matrix A describes the dynamics of transverse vibration of an elastic beam axially moving at a constant speed. When

the speed is lower than the critical speed, matrix A has four pure imaginary eigenvalues, ±iω1and ±iω2. Thus there exists

canonical transformation matrix T such that

?−ω1J2

Introduce the state variable transformation

TTAT = Ω =

02

02

−ω2J2

?

.

(23)

ξ =Tζ.

(24)

Then Eq. (21) can be expressed in terms of new state variables as

˙ξ =Ωξ +hD1ξ cosωt +ωhD3xsinωt +ηD3ξ,

(25)

where

D1= TTA1T,

If the speed variation frequency ω approaches ω0= 2ω1,2ω2, or ω1± ω2, instability may occur in subharmonic or

combination parametric resonance. A small parameter δ is introduced to quantify the deviation of ω from ω0, and ω is described

by ω =ω0(1+δ). Introduce a new time τ = ωt. Eq. (25) can be rewritten as

˙ξ =Ω0ξ −δΩ0ξ +hD1

ω0

ω0

where

?−k1J2

In Eq. (27), δ,h and η are all small parameters.

To apply the averaging method, rewrite ξ in a amplitude-phase form

D2= TTA2T,

TD3= TTA3T.

(26)

ξ cosτ +hD2ξ sinτ +ηD3

ξ,

(27)

Ω0=Ω

ω0

=

02

02

−k2J2

?

,k1=ω1

ω0

, k2=ω2

ω0

.

(28)

ξ1= a1cosϕ1,ξ2= a1sinϕ1,ξ3=a2cosϕ2,ξ4= a2sinϕ2,

(29)

where

ϕj= kjτ +θj

Substitution of Eqs. (29) and (30) into Eq. (27) leads to

(j = 1,2).

(30)

˙ a1= g1cosϕ1+g2sinϕ1,

˙ a2= g3cosϕ2+g4sinϕ2,

a1˙θ1= g2cosϕ1−g1sinϕ1,

a2˙θ2= g4cosϕ2−g3sinϕ2,

(31)

where

gi=?D1,i

1a1cosϕ1+D2,i

+?D1,i

1a1sinϕ1+D3,i

2a1sinϕ1+D3,i

3a1sinϕ1+D3,i

1a2cosϕ2+D4,i

2a2cosϕ2+D4,i

3a2cosϕ2+D4,i

1a1sinϕ2

? h

ω0

cosτ

2a1cosϕ1+D2,i

+?D1,i

In Eq. (32), the superscript of an element denotes its row and column in the matrix, and

2a1sinϕ2

?hsinτ

ω0

3a1cosϕ1+D2,i

3a1sinϕ2

? η

+δri.

(32)

r1= k1a1sinϕ1,r2= −k1a1cosϕ1,r3= k2a2sinϕ2,r4= −k2a2cosϕ2.

(33)

Page 5

L.-Q. Chen et al. / European Journal of Mechanics A/Solids 23 (2004) 659–666

663

For small parameters δ, h and η, both the amplitudes and the phases are slowly varying with time. Therefore, the right-

hand side of Eq. (31) can be replaced by their time averages over a period, and the solutions of the averaged system are the

same as the original one (Bogoliubov and Mitropolsky, 1961). Ariaratnam and Namchchivaya (1986) and Asokanthan and

Ariaratnam (1994) applied the method of averaging to study axially moving materials. The averaged equations take different

forms depending on the value of the speed variation frequency.

For kj= 1/2 (j = 1,2), the variation frequency lies in the neighborhoods of 2ωj. The non-trivial components of the

averaged equation take the form

˙ aj= (Ujhcos2θj+Vjhsin2θj+Mjη)aj,

aj˙θj=

?

Vjhcos2θj−Ujhsin2θj+Njη −1

2δ

?

aj

(j = 1,2),

(34)

where

U1=1

M1=D1,1

4

?D1,1

1

−D2,2

ω0

+D2,2

2ω0

?D3,4

1

+D1,2

N1=D2,1

2

+D2,1

2

?

,V1=1

4

?D1,2

U2=1

1

+D2,1

ω0

?D3,3

+D4,4

2ω0

1

−D1,1

−D4,4

ω0

2

+D2,2

2

?

,

33

,

3

−D1,2

2ω0

?

3

,

4

11

+D3,4

N2=D4,3

2

+D4,3

−D3,4

2ω0

2

?

,

(35)

V2=1

4

1

+D4,3

ω0

1

−D3,3

2

+D4,4

2

,M2=D3,3

33

,

33

.

Introducing two new variables

χj= ajcos(θj+ϕj),ςj= ajsin(θj+ϕj) (j = 1,2),

(36)

where

ϕj=1

2tan−1

?Uj

Vj

?

.

(37)

Then Eq. (34) can be expressed in the new variables as

??

˙ ςi=

˙ χi=Miηξi+

??

The characteristic equation of the linear ordinary-differential equation (38) is

?

with the roots

?

U2

i+V2

ih−Niη +1

ih−Niη +1

?

2δ

?

ςi,

U2

i+V2

2δξ1+Miηςi.

(38)

λ2−2Miηλ +M2

iη2+

Niη −1

2δ

?2

−?U2

i+V2

i

?h2= 0(39)

λ1,2= Mjη ±

?U2

j+V2

j

?h2−

?

Njη −2ωj−ω

4ωj

?2

.

(40)

If the roots have negative real parts, the solutions are stable, and if the real part of at least one of the roots is positive, then the

solution is unstable. Therefore stability boundaries in the neighborhoods of 2ωjcan be located in the (ω,h) plane. In this case,

the parametric excitation frequency is near twice the first or second natural frequency. Hence the first or second subharmonic

resonance results in instability.

In the case of summation resonance, k1+k2=1. One can cast the averaged equation in the form

˙ χ1= (M11η +iN11η −iδk1)χ1+(U11+iV11)χ2,

˙ χ2= (U12+iV12)χ1+(M12η +iN12η −iδk2)χ2,

where

(41)

χ1= a1cosθ1+ia1sinθ1,χ2= a2cosθ2−ia2sinθ2,

(42)

Page 6

664

L.-Q. Chen et al. / European Journal of Mechanics A/Solids 23 (2004) 659–666

U11=1

4

?D3,1

?D1,3

1

−D4,2

ω0

−D2,4

ω0

+D2,2

2ω0

1

+D4,1

2

+D3,2

2

?

?

−D2,1

2ω0

,V11=1

4

?D4,1

?D2,3

1

+D3,2

ω0

+D1,4

ω0

+D4,4

2ω0

1

−D3,1

2

+D4,2

2

?

?

,

U12=1

4

11

+D2,3

2

+D1,4

2

,V12=1

4

11

−D1,3

2

+D2,4

2

,

M11=D1,1

33

,N11=D1,2

33

,M12=D3,3

33

,N12=D3,4

3

−D4,3

2ω0

3

.

(43)

The analysis of the characteristic roots leads to stability boundaries in the neighborhoods of ω1+ω2can be located in the (ω,h)

plane. Similarly, one can obtain the averaged equation in the case of difference resonance.

4. Numerical results

Both the amplitude h and the frequency ω of axial speed fluctuation play an important part in the stability of responses. The

procedure in the previous section can be used to present the stability boundaries in the (ω,h) plane for a set of given parameters.

Numerical resultsindicatethereexist instabilityregions inthesubharmonic resonance and thesummationresonance, whilethere

is no instability region in the difference resonance.

First of all, the stability boundaries in the (ω,h) plane obtained through the averaging method are checked by numerical

simulations. The Runge–Kutta algorithm is employed to solve numerically the original equation (27). For fixed η,γ and µ,

numerical solution to Eq. (27) can be computed for every point in the (ω,h) plane. Therefore, a stability boundary can be

numerical obtained. Fig. 1 shows the case that η = 0.0002, γ = 8.0 and µ = 10.0. The comparison indicates that, for enough

small h, the analytical results agree excellently with the numerical ones in the first and second subharmonic resonance and the

summation resonance, while there is no instability region in the difference resonance. Hence the analytical results are confirmed

by numerical calculations.

The effect of the viscosity coefficient on the stability boundaries is examined in Fig. 2, in which γ = 8.0 and µ = 10.0. In

the subharmonic resonance and the summation resonance, the stability boundaries for η = 0.0,0.0001,0.0002 are respectively

depicted in the dashdot line, the dashed line and the solid line. In all cases, the larger dynamic viscosity leads to the lager

instability threshold of h for given ω, and the smaller instability range of ω for given h. Besides, the stability boundary in

the summation resonance is most sensitive to the change of the dynamic viscosity, while the stability boundary in the first

subharmonic resonance is most insensitive.

The effect of the mean axial speed on the stability boundaries is illustrated in Fig. 3, in which η = 0.0002 and µ = 10.0.

In the first and second subharmonic resonance and the summation resonance, the stability boundary for γ = 7.8,8.0,8.2 are

respectively depicted in the dashdot line, the dashed line and the solid line. In the first subharmonic resonance, with the decrease

of the mean axial speed, the instability regions drift towards the direction of the increasing ω. In the second subharmonic

resonance, the decreasing mean axial speed results in a slight shrinkage of instability region both in the directions of ω and h.

In the summation resonance, with the decrease mean axial speed, the instability regions drift towards the directions of the

increasing ω and h. That is, the smaller mean axial speed results in the larger instability threshold of h for given ω, and the

(a)(b)(c)

Fig. 1. Comparison between analytical results (solid line) and numerical results (crosses). (a) ω =2ω1, (b) ω = 2ω2, (c) ω =ω1+ω2.

Page 7

L.-Q. Chen et al. / European Journal of Mechanics A/Solids 23 (2004) 659–666

665

(a) (b)(c)

Fig. 2. Effect of the viscosity coefficient on the stability boundaries. (a) ω = 2ω1, (b) ω = 2ω2, (c) ω = ω1+ω2.

(a)(b) (c)

Fig. 3. Effect of the mean axial speed on the stability boundaries. (a) ω = 2ω1, (b) ω = 2ω2, (c) ω = ω1+ω2.

(a)(b)(c)

Fig. 4. Effect of the tension on the stability boundaries. (a) ω = 2ω1, (b) ω =2ω2, (c) ω = ω1+ω2.

larger instability threshold and the smaller instability range of ω for given h. Besides, the stability boundaries in the second

subharmonic resonance is not so sensitive to the variations of the mean speed as those in the first subharmonic resonance.

The effect of the tension on the stability boundaries is shown in Fig. 4, in which η = 0.0002 and γ = 8.0. In the first and

second subharmonic resonance and the summation resonance, the stability boundary for µ = 9.8,10.0,10.2, are respectively

depicted in the dashdot line, the dashed line and the solid line. Comparing Fig. 4 with Fig. 3, one finds that the effect of the

decreasing tension is very similar to that of the increasing mean axial speed.

Page 8

666

L.-Q. Chen et al. / European Journal of Mechanics A/Solids 23 (2004) 659–666

In all figures, the instability range of ω increases with the growth of h. The smallest instability threshold appears in the first

subharmonic resonance, and the largest instability threshold occurs in the summation resonance.

5. Conclusions

This paper treats dynamic stability of an axially accelerating viscoelastic beam. The method of averaging is applied to the

2-term Galerkin truncation of the governing equation. Numerical simulations are presented to confirm and demonstrate the

analytical results. From the analytical and numerical work, the following conclusions can be drawn.

(1) There exist instability regions in the subharmonic resonance and the summation resonance, while there is no instability

region in the difference resonance. In all cases that the instability occurs, the instability range of the axial speed fluctuation

frequency increases with the growth the axial speed fluctuation amplitude.

(2) In the first subharmonic resonance, the increase of the mean axial speed and the decrease of the tension lead to the increase

of instability threshold of the axial speed fluctuation frequency, and the increase of the viscosity coefficient leads to a slight

increase of instability threshold of the axial speed fluctuation amplitude.

(3) In the second subharmonic resonance, the increasing viscosity coefficient results in the increasing instability threshold of

the axial speed fluctuation amplitude, and either the increase of the mean axial speed or the decrease of the tension makes

the instability regions reduce slightly.

(4) In the summation resonance, the increase of the viscosity coefficient brings about a dramatic increase of the instability

threshold of the axial speed fluctuation amplitude, and either the increase of the mean axial speed or the decrease of the

tension leads to the increasing instability thresholds of the axial speed fluctuation amplitude and frequency.

Acknowledgement

The research is supported by the National Natural Science Foundation of China (Project No. 10172056).

References

Abrate, A.S., 1992. Vibration of belts and belt drives. Mech. Mach. Theory 27 (6), 645–659.

Ariaratnam, S.T., Namchchivaya, N.S., 1986. Dynamic stability of pipes conveying pulsating fluid. J. Sound Vib. 107, 215–230.

Asokanthan, S.F., Ariaratnam, S.T., 1994. Flexural instabilities in axially moving bands. J. Vib. Acoust. 116, 275–279.

Bishop, R.E.D., Johnson, D.C., 1979. The Mechanics of Vibration. Cambridge University Press, Cambridge.

Bogoliubov, N., Mitropolsky, Y.A., 1961. Asymptotic Methods in the Theory of Nonlinear Oscillations. Gordon & Breach, New York.

Marynowski, K., 2002. Non-linear dynamic analysis of an axially moving viscoelastic beam. J. Theoret. Appl. Mech. 40, 465–482.

Marynowski, K., Kapitaniak, T., 2002. Kelvin–Voigt versus Bügers internal damping in modeling of axially moving viscoelastic web. Inter. J.

Non-Linear Mech. 37, 1147–1161.

Öz, H.R., 2001. On the vibrations of an axially traveling beam on fixed supports with variable velocity. J. Sound Vib. 239, 556–564.

Öz, H.R., Pakdemirli, M., Özkaya, E., 1998. Transition behaviour from string to beam for an axially accelerating material. J. Sound Vib. 215,

571–576.

Özkaya, E., Pakdemirli, M., 2000. Vibrations of an axially accelerating beam with small flexural stiffness. J. Sound Vib. 234, 521–535.

Özkaya, E., Öz, H.R., 2002. Determination of natural frequencies and stability regions of axially moving beams using artificial neural networks

method. J. Sound Vib. 254, 782–789.

Parker, R.G., Lin, Y., 2001. Parametric instability of axially moving media subjected to multifrequency tension and speed fluctuations. J. Appl.

Mech. 68, 49–57.

Pasin, F., 1972. Über Die Stabilität der Beigeschwingungen von in Längsrichtung Periodisch Hin und Herbewegten Stäben. Ingenieur-Arch. 41,

387–393.

Wickert, J.A., Mote, C.D. Jr., 1988. Current research on the vibration and stability of axially moving materials. Shock Vib. Dig. 20, 3–13.

Wickert, J.A., Mote, C.D. Jr., 1990. Classical vibration analysis of axially moving continua. J. Appl. Mech. 57, 738–744.