# Stability in parametric resonance of axially moving viscoelastic beams with time-dependent speed

**ABSTRACT** Stability in transverse parametric vibration of axially accelerating viscoelastic beams is investigated. The governing equation is derived from Newton's second law, the Kelvin constitution relation, and the geometrical relation. When the axial speed is a constant mean speed with small harmonic variations, the governing equation can be regarded as a continuous gyroscopic system under small periodically parametric excitations and a damping term. The method of multiple scales is applied directly to the governing equation without discretization. The stability conditions are obtained for combination and principal parametric resonance. Numerical examples are presented for beams with simple supports and fixed supports, respectively, to demonstrate the effect of viscoelasticity on the stability boundaries in both cases.

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**ABSTRACT:**This paper focuses on the bifurcation and chaos of an axially accelerating viscoelastic beam in the supercritical regime. For the first time, the nonlinear dynamics of the system under consideration are studied via the high-order Galerkin truncation as well as the differential and integral quadrature method (DQM & IQM). The speed of the axially moving beam is assumed to be comprised of a constant mean value along with harmonic fluctuations. The transverse vibrations of the beam are governed by a nonlinear integro-partial-differential equation, which includes the finite axial support rigidity and the longitudinally varying tension due to the axial acceleration. The Galerkin truncation and the DQM & IQM are, respectively, applied to reduce the equation into a set of ordinary differential equations. Furthermore, the time history of the axially moving beam is numerically solved based on the fourth-order Runge–Kutta time discretization. Based on the numerical solutions, the phase portrait, the bifurcation diagrams and the initial value sensitivity are presented to identify the dynamical behaviors. Based on the nonlinear dynamics, the effects of the truncation terms of the Galerkin method, such as 2-term, 4-term, and 6-term, are studied by comparison with DQM & IQM.International Journal of Bifurcation and Chaos 05/2014; 24(05). · 0.92 Impact Factor - SourceAvailable from: Mehmet Pakdemirli
- SourceAvailable from: Hu Ding

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JOURNAL OF

SOUND AND

VIBRATION

Journal of Sound and Vibration 284 (2005) 879–891

Stability in parametric resonance of axially moving viscoelastic

beams with time-dependent speed

Li-Qun Chena,?, Xiao-Dong Yangb

aDepartment of Engineering Mechanics, Shanghai Jiao Tong University, Shanghai 200030, China

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

Received 26 April 2004; received in revised form 5 July 2004; accepted 20 July 2004

Available online 15 December 2004

Abstract

Stability in transverse parametric vibration of axially accelerating viscoelastic beams is investigated. The

governing equation is derived from Newton’s second law, the Kelvin constitution relation, and the

geometrical relation. When the axial speed is a constant mean speed with small harmonic variations, the

governing equation can be regarded as a continuous gyroscopic system under small periodically parametric

excitations and a damping term. The method of multiple scales is applied directly to the governing equation

without discretization. The stability conditions are obtained for combination and principal parametric

resonance. Numerical examples are presented for beams with simple supports and fixed supports,

respectively, to demonstrate the effect of viscoelasticity on the stability boundaries in both cases.

r 2004 Elsevier Ltd. All rights reserved.

1. Introduction

Many engineering devices can be modeled as axially moving beams. One major problem is the

occurrence of large transverse vibrations due to tension or axial speed variation. Transverse

vibration of axially accelerating beams has been extensively analyzed. Although Pasin [1] first

studied the problem as early as in 1972, much progress was achieved recently. O¨z et al. [2] applied

the method of multiple scales to study dynamic stability of an axially accelerating beam with small

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www.elsevier.com/locate/jsvi

0022-460X/$-see front matter r 2004 Elsevier Ltd. All rights reserved.

doi:10.1016/j.jsv.2004.07.024

?Corresponding author.

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

Page 2

bending stiffness. O¨zkaya and Pakdemirli [3] 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. O¨z and Pakdemirli [4] and O¨z [5]

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 [6] adopted a 1-term Galerkin discretization and the perturbation

method to study dynamic stability of an axially accelerating beam subjected to a tension

fluctuation. O¨zkaya and O¨z [7] applied artificial neural network algorithm to determine stability

of an axially accelerating beam.

All above-mentioned researchers considered elastic beams, and did not account for any

damping. The modeling of dissipative mechanisms is an important research topic of axially

moving material vibrations [8,9]. Viscoelasticity is an effective approach to model the damping

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. The literature that is specially related to axially accelerating viscoelastic beams is

relatively limited. Based on 3-term Galerkin truncation, Marynowski [10] and Marynowski and

Kapitaniak [11] compared the Kelvin model with the Maxwell model and the Bu ¨ gers model,

respectively, through numerical simulation of nonlinear vibration responses of an axially moving

beam at a constant speed, and found that all models yield similar results in the case of small

damping. Marynowski [12] further studied numerically nonlinear dynamical behavior of an

axially moving viscoelastic beam with time-dependent tension based on 4-term Galerkin

truncation. Based on 2-term Galerkin truncation, Yang and Chen [13] and Chen et al. [14]

applied the averaging method to analyze the stability of axially accelerating linear beams with

pinned or clamped ends, and Yang and Chen [15] studied numerically bifurcation and chaos of an

axially accelerating nonlinear beam.

In this paper, the stability is investigated for parametric vibration of axially accelerating

viscoelastic beams. The governing equation is derived from Newton’s second law, the constitution

relation, and the strain–displacement relation. The method of multiple scales is applied directly to

the governing equation. The stability boundaries for combination and principal resonance are

presented for beams with simple supports and fixed supports. The effects of viscoelasticity on the

boundaries are numerically demonstrated.

2. The governing equation

A uniform axially moving viscoelastic beam, with density r; cross-sectional area A; moment of

inertial I and initial tension P0; travels at the time-dependent axial transport speed vðTÞ between

two prismatic ends separated by distance L: Consider only the bending vibration described by the

transverse displacement VðX;TÞ; where T is the time and X is the axial coordinate. The Newton

second law of motion yields

rA

q2U

qT2þ 2v

q2U

qXqTþdv

dT

qU

qXþ v2q2U

qX2

??

¼ P0

q2UðX;TÞ

qX2

?q2MðX;TÞ

qX2

;

(1)

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where MðX;TÞ is the bending moment given by

MðX;TÞ ¼ ?

Z

A

ZsðX;Z;TÞdA;

(2)

where Z;X-plane is the principal plane of bending, and sðX;Z;TÞ is the disturbed normal stress.

The viscoelastic material of the beam obeys the Kelvin model, with the constitution relation

sðX;Z;TÞ ¼ EeðX;Z;TÞ þ ZqeðX;Z;TÞ

qT

;

(3)

where eðX;Z;TÞ is the axial strain, E is the stiffness constant, and Z is the viscosity coefficient. For

small deflections, the strain–displacement relation is

eðX;Z;TÞ ¼ ?Zq2UðX;TÞ

qX2

:

(4)

Substitution of Eqs. (3) and (4) into Eq. (2) and then substitution the resulting equation into Eq.

(1) lead to

rA

q2U

qT2þ 2v

q2U

qXqTþdv

dT

qU

qXþ v2q2U

qX2

??

? P0

q2U

qX2þ EIq4U

qX4þ ZI

q5U

qTqX4¼ 0:

(5)

Introduce the dimensionless variables and parameters:

u ¼U

L;

x ¼X

L;

t ¼ T

ffiffiffiffiffiffiffiffiffiffiffiffi

rAL2

P0

s

;

g ¼ v

ffiffiffiffiffiffiffi

P0

rA

s

;

v2

f¼

EI

P0L2;?a ¼

IZ

ffiffiffiffiffiffiffiffiffiffiffiffi

L3

rAP0

p

;

ð6Þ

where bookkeeping device ? is a small dimensionless parameter accounting for the fact that the

viscosity coefficient is very small. Eq. (5) can be cast into the dimensionless form

q2u

qt2þ 2gq2u

qxqtþdg

dt

qu

qxþ ðg2? 1Þq2u

qx2þ v2

f

q4u

qx4þ ?a

q5u

qx4qt¼ 0:

(7)

3. Stability condition via the method of multiple scales

In the present investigation, the axial speed is assumed to be a small simple harmonic variation,

with the amplitude ?g1and the frequency o; about the constant mean speed g0;

gðtÞ ¼ g0þ ?g1sin ot:

Here the bookkeeping device ? is used to indicate the fact that the fluctuation amplitude is small,

with the some order as the dimensionless viscosity coefficient. In spite of the apparent connection

between the dimensionless viscosity coefficient and the amplitude of the variation through the

bookkeeping device ?; they are actually independent because each of them includes, respectively,

an arbitrary parameter a or g1of order one. Substitution of Eq. (8) into Eq. (7) and neglecting

(8)

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higher order ? terms in the resulting equation yield

Mq2u

qt2þ Gqu

¼ ?2?g1sin otq2u

qtþ Ku

qxqt? 2?g0g1sin otq2u

qx2? ?og1cos otqu

qx? ?a

q5u

qx4qt;

ð9Þ

where the mass, gyroscopic, and linear stiffness operators are, respectively, defined as

M ¼ I;

G ¼ 2g0

q

qx;

K ¼ ðg2

0? 1Þq2

qx2þ v2

f

q4

qx4:

(10)

The method of multiple scales will be employed to solve Eq. (9) directly. A first-order uniform

approximation is sought in the form

uðx;t;?Þ ¼ u0ðx;T0;T1Þ þ ?u1ðx;T0;T1Þ þ ???;

(11)

where T0¼ t is a fast scale characterizing motions occurring at ok(one of the natural frequencies

of the corresponding unperturbed linear system), and T1¼ ?t is a slow scale characterizing the

modulation of the amplitudes and phases due to viscoelasticity and possible resonance.

Substitution of Eq. (11) and the following relationship

q

qt¼

q

qT0þ ?

q

qT1þ ???;

q2

qt2¼

q2

qT2

0

þ 2?

q2

qT0qT1þ ???

(12)

into Eq. (9) and then equalization of coefficients of ?0and ? in the resulting equation lead to

Mq2u0

qT2

0

þ Gqu0

qT0þ Ku0¼ 0(13)

and

Mq2u1

qT2

0

þ Gqu1

q2u0

qT0qT1? 2g0

q5u0

qx4qT0:

qT0þ Ku1

¼ ?2

q2u0

qxqT1? 2g1sinot

q2u0

qxqT0þ g0

q2u0

qx2

??

? g1o cos otqu0

qx

? a

ð14Þ

Wickert and Mote [16] have obtained the solution to Eq. (13)

u0ðx;T0;T1Þ ¼

X

k¼0;1;...

bfkðxÞAkðT1ÞeiokT0þ¯fkðxÞ¯AkðT1Þe?iokT0c;

(15)

where the over bar denotes complex conjugation, and the kth natural frequency and the kth

complex eigenfunction can be determined by the boundary conditions.

If the variation frequency o approaches the sum of any two natural frequencies of system (13),

summation parametric resonance may occur. A detuning parameter s is introduced to quantify

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the deviation of o from onþ om; and o is described by

o ¼ onþ omþ ?s;

(16)

where onand omare, respectively, the nth and mth natural frequencies of system (13).

To investigate the summation parametric response, Eq. (15) can be expressed as

u0ðx;T0;T1Þ ¼ fnðxÞAnðT1ÞeionT0þ fmðxÞAmðT1ÞeiomT0þ cc;

where cc stands for the complex conjugate of all preceding terms on the right hand of an equation.

Substitution of Eqs. (16) and (17) into Eq. (14) and expression of the trigonometric functions in

exponential form yield

(17)

Mq2u1

qT2

0

þ Gqu1

qT0þ Ku1

¼?2_Anðionfnþ g0f0

nÞ þ g1

1

2ðom? onÞ¯f0

1

2ðon? omÞ¯f0

mþ ig0¯f00

m

??

?

¯fmeisT1? iaonAnf0000

n

?

?

þ cc þ NST;

?

?

eionT0

?2_Amðiomfmþ g0f0

mÞ þ g1

nþ ig0¯f00

n

?

¯fneisT1? iaomAmf0000

m

eiomT0

ð18Þ

where the dot and the prime denote derivation with respect to the slow time variable T1and the

dimensionless spatial variable x; respectively, and NST stands for the terms that will not bring

secular terms into the solution. Eq. (18) has a bounded solution only if a solvability condition

holds. The solvability condition demands the orthogonal relationships

?2_Anðionfnþ g0f0

nÞ þ g1

1

2ðom? onÞ¯f0

1

2ðon? omÞ¯f0

mþ ig0¯f00

m

??

¯fmeisT1? iaonAnf0000

?

n;fn

?

?

?

¼ 0;

?

?2_Amðiomfmþ g0f0

mÞ þ g1

nþ ig0¯f00

n

?

¯fneisT1? iaomAmf0000

m;fm

¼ 0;

ð19Þ

where the inner product is defined for complex functions on [0,1] as

hf;gi ¼

Z1

0

f ¯ gdx:

(20)

Application of the distributive law of the inner product to Eq. (19) leads to

_Anþ acnnAnþ g1dnm¯AmeisT1¼ 0;

_Amþ acmmAmþ g1dmn¯AneisT1¼ 0;

ð21Þ

where

ckk¼

iok

R1

0¯f0

0fk¯fkdx þ g0

0f0000

k¯fkdx

2ðiok

R1

0fk¯fkdx þ g0

R1

0f0

k¯fkdxÞ

R1

ðk ¼ n;mÞ;

dkj¼ ?ðoj? okÞR1

j¯fkdx þ 2ig0

0¯f00

k¯fkdxÞ

j¯fkdx

4ðiok

R1

R1

0f0

ðk ¼ n;m; j ¼ m;nÞ:

ð22Þ

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These coefficients can be determined by the modal parameters calculated from Eq. (13), and are

independent of parametric excitation due to the variation of axial speed.

The transformation

AnðT1Þ ¼ BnðT1ÞeisT1=2;

AmðT1Þ ¼ BmðT1ÞeisT1=2

(23)

changes Eq. (21) into an autonomous system

_Bnþ is

_Bmþ is

2Bnþ acnnBnþ g1dnm¯Bm¼ 0;

2Bmþ acmmBmþ g1dmn¯Bn¼ 0:

ð24Þ

Obviously, Eq. (24) (and thus Eq. (21) has a zero solution. Suppose that the non-zero solutions of

Eq. (24) take the form

Bn¼ bnelT1;

Bm¼ bme¯lT1;

(25)

where bnand bmare real coefficients, and l is a complex to be determined. Substituting Eq. (25)

into Eq. (24) and taking the complex conjugate of the second resulting equation yield

?l ?s

g1¯dmnbnþ ?l þs

2i ? acnn

?

??

bnþ g1dnmbm¼ 0;

2i ? a¯ cmm

?

bm¼ 0:

ð26Þ

Eq. (26), a set of homogeneous linear algebraic equations of bnand bm; has non-zero solutions if

and only if its determinant of coefficient vanishes. Therefore,

s

2i þ acnn

l2þ aðcnnþ cmmÞl þ

??

?s

2i þ a¯ cmm

??

? g2

1dnm¯dmn¼ 0:

(27)

When l has positive real part, the system is unstable.

Separate l; cnn; and cmminto real and imaginary parts,

l ¼ lRþ ilI;

cnn¼ cR

nnþ icI

nn;

cmm¼ cR

mmþ icI

mm:

(28)

Substituting Eq. (28) into Eq. (27) and separating the resulting equation into real and imaginary

parts lead to

lR2? lI2þ aðcR

þ

2lRlIþ aðcI

þ a cR

nnþ cR

?s

nnþ cI

s

2þ acI

mmÞlR? aðcI

2þ acI

mmÞlRþ aðcR

?

nnþ cI

1Reðdnm¯dmnÞ ¼ 0;

nnþ cR

s

2þ acI

mmÞlIþ a2cR

nncR

mm

s

2þ acI

nn

?

mm

??

? g2

mmÞlI

mm nn

?

? cR

nnmm

??hi

? g2

1Imðdnm¯dmnÞ ¼ 0:

ð29Þ

For aa0; Eq. (29) has the solution lR¼ 0 on the condition

Imðdnm¯dmnÞ ¼ 0;

Reðdnm¯dmnÞ40;

s ¼ ?g1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Reðdnm¯dmnÞ

q

:

(30)

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For aa0; substituting lR¼ 0 into Eq. (29) and eliminating lIin the resulting equation give

s

2ðcR

?

þ ðcR

nn? cR

mmÞ þ aðcR

nncI

mm? cR

mmcI

nnÞ þg2

1

aImðdnm¯dmnÞ

??2

þ ðcR

nnþ cR

mmÞðcI

mmÞ2s2

nnþ cI

4þsa

mmÞs

2ðcR

nn? cR

mmÞ þ aðcR

nncI

mm? cR

mmcI

nnÞ þg2

1

aImðdnm¯dmnÞ

?

nnþ cR

2ðcI

nnþ cI

mmÞ þ a2ðcR

nncR

mmþ cI

nncI

mmÞ þ g2

1Reðdnm¯dmnÞ

??

¼ 0:

ð31Þ

Eq. (31) is the analytical expression of the stability boundary in summation parametric resonance.

If the variation frequency o approaches two times of a natural frequency of system (13),

principal parametric resonance may occur. Denote

o ¼ 2onþ ?s:

(32)

Let m ¼ n in Eq. (31), then the resulting equation gives the stability boundary in nth principal

parametric resonance. For a ¼ 0; the stability boundary is expressed by

s ¼ ?g1jdnnj:

For aa0; the stability boundary is expressed by

g4

1

a2jdnnj4þ 4g2

(33)

1

acR

nncI

nnjdnnj2þ 4cR2

nn

s2

4þ sacI

nnþ a2ðcR2

nnþ cI2

nnÞ þ g2

1jdnnj2

??

¼ 0;

(34)

where

cnn¼

ion

R1

0f0000

n¯fndx

2ðion

R1

0fn¯fndx þ g0

R1

0f0

n¯fndxÞ

;

dnn¼ ?

2ig0

R1

0¯f00

n¯fndx

4ðion

R1

0fn¯fndx þ g0

R1

0f0

n¯fndxÞ

:

(35)

If the variation frequency o approaches the difference of any two natural frequencies of system

(13), difference parametric resonance may occur. The stability in difference parametric resonance

can be treated similarly. Denote

o ¼ on? omþ ?s:

(36)

The stability boundaries are expressed by Eqs. (30) and (31), respectively, for a ¼ 0 and aa0;

while the coefficients in them are given by

cnn¼

ion

R1

0¯f0

0fn¯fndx þ g0

0¯f0

R1

0f0000

n¯fndx

2ðion

R1

0fn¯fndx þ g0

R1

0f0

n¯fndxÞ

R1

R1

;

cmm¼

iom

R1

0f0000

m¯fmdx

2ðiom

R1

0fm¯fmdx ? g0

R1

0f0

m¯fmdxÞ

;

dnm¼ðomþ onÞR1

dmn¼ðonþ omÞR1

4ðiom

m¯fndx ? 2ig0

0¯f00

n¯fndxÞ

0¯f00

0f0

m¯fndx

4ðion

R1

0fm¯fmdx ? g0

R1

R1

0f0

;

n¯fmdx þ 2ig0

n¯fmdx

m¯fmdxÞ

:

ð37Þ

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4. Stability boundaries of beams with simple supports

For an axially moving beam with simple supports, the boundary conditions in dimensionless

form are

uð0;tÞ ¼ uð1;tÞ ¼ 0;

q2u

qx2

????

x¼0

¼q2u

qx2

????

x¼1

¼ 0:

(38)

Under the boundary conditions (38), the eigenfunction corresponding the kth natural frequency

okis [4]

fkðxÞ ¼ eib1kx?ðb2

4k? b2

ðb2

4k? b2

ðb2

1kÞðeib3k? eib1kÞ

4k? b2

? 1 ?ðb2

2kÞðeib3k? eib2kÞeib2kx?ðb2

1kÞðeib3n? eib1kÞ

4k? b2

4k? b2

ðb2

4k? b2

ðb2

1kÞðeib3k? eib1kÞ

4n? b2

1kÞðeib3k? eib1kÞ

4k? b2

3nÞðeib3k? eib3kÞeib2kx

2kÞðeib3k? eib2kÞ?ðb2

3kÞðeib3k? eib3kÞ

"#

eib4kx;

ð39Þ

where bjkðj ¼ 1;2;3;4Þ and okcan be solved from the following algebraic equations:

v4

fðb1kþ b2kþ b3kþ b4kÞ ¼ g2

b1kb2kþ b1kb3kþ b1kb4kþ b2kb3kþ b2kb4kþ b3kb4k¼ 0;

v4

0? 1;

fðb1kb2kb3kþ b1kb2kb4kþ b1kb3kb4kþ b2kb3kb4kÞ ¼ 2g0ok;

v4

kfb1kb2kb3kb4k¼ ?o2

ð40Þ

and the transcendental equation

ðb2

1k? b2

þ eiðb2kþb4kÞc þ ðb2

2kÞðb2

3k? b2

4kÞbeiðb1kþb2kÞþ eiðb3kþb4kÞc þ ðb2

1k? b2

2k? b2

4kÞðb2

3k? b2

1kÞbeiðb1kþb3kÞ

4kÞðb2

2k? b2

3kÞ½eiðb2kþb3kÞ? eiðb1kþb4kÞ? ¼ 0:

ð41Þ

Consider an axially moving beam with nf¼ 0:8 and g ¼ 2:0: The first two natural frequencies

and coefficients in corresponding eigenfunctions (39), numerically solved from Eqs. (40) and (41),

are o1¼ 5:3692; b11¼ 3:9906; b21¼ ?1:2424 þ 2:4397i; b31¼ ?1:2424 ? 2:4397i; b41¼ ?1:5058

and

o2¼ 30:1200;

?4:9503:

In summation parametric resonance, Eq. (22) gives c11¼ 45:8597; c22¼ 709:7023; d12¼

1:2427 þ 0:7843i; and d21¼ 0:2948 þ 0:1860i: In the case that ckkis real, Eq. (22) reduces to

s

2ðcR

b12¼ 7:4497;

b22¼ ?1:2497 þ 6:0726i;

b32¼ ?1:2497 ? 6:0726i;

b42¼

nn? cR

mmÞ

hi2

þ ðcR

nnþ cR

mmÞ2s2

4þ a2ðcR

nncR

mmÞ ? g2

1Reðdnm¯dmnÞ

??

¼ 0:

(42)

Therefore, the instability region is given as

?2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 þ k2

g2

1Reðdnm¯dmnÞ ? a2cR

nncR

mm

s

oso2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 þ k2

g2

1Reðdnm¯dmnÞ ? a2cR

nncR

mm

s

;

(43)

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Page 9

where

k ¼cR

nn? cR

cR

mm

nnþ cR

mm

:

(44)

The instability region exists on the condition that cR

the axial speed variation amplitude is large enough, namely,

nncR

mmand Reðdnm¯dmnÞ have the same sign and

g14a

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Reðdnm¯dmnÞ

cR

nncR

mm

s

:

(45)

The stability boundaries for the summation resonance of first two modes in plane s ? g1are

shown in Fig. 1 for a ¼ 0; 0.0005, 0.001. The increasing viscosity coefficient makes the stability

boundaries move towards the increasing direction of g1in plane ðo;g1Þ and the instability regions

become narrow. That is, the larger viscosity coefficient leads to the larger instability threshold of

g1for given s; and the smaller instability range of s for given g1:

In principal parametric resonance, Eq. (35) gives d11¼ ?1:0456 þ 1:1879i; d22¼ ?0:4182 þ

0:9776i: The instability region is

?2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

g2

1jdnnj2? a2cR2

nn

q

oso2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

g2

1jdnnj2? a2cR2

nn

q

:

(46)

The instability region exists on the condition that the axial speed variation amplitude is beyond a

critical value,

g14ajcR

nnj

jdnnj:

(47)

The stability boundaries for the first and second principal resonance in plane s ? g1are shown,

respectively, in Fig. 2 for a ¼ 0; 0.02, 0.05 and Fig. 3 for a ¼ 0; 0.001, 0.002. In both cases, the

increasing viscosity coefficient makes the stability boundaries move towards the increasing

direction of g1in plane ðo;g1Þ and the instability regions become narrow.

In difference parametric resonance, Eq. (37) gives c11¼ 45:8597; c22¼ 741:7379 d12¼

?3:6139 ? 2:2809i; d21¼ 0:5997 þ 0:6081i: For real c11and c22; the stability boundary is given

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Fig. 1. The stability boundaries for the summation resonance of beams with simple supports.

L.-Q. Chen, X.-D. Yang / Journal of Sound and Vibration 284 (2005) 879–891

887

Page 10

by Eq. (42). In this example, Reðdnm¯dmnÞ is negative. Thus there is no instability region in the

difference resonance.

To depict the stability boundaries in the same scale, the different viscosity coefficients are

chosen in Figs. 1–3. These figures indicate that the stability boundary for the summation

resonance is most sensitive to the change of the viscosity coefficient, while the stability boundary

in the first principal resonance is most insensitive.

5. Stability boundaries of beams with fixed supports

For an axially moving beam with simple supports, the boundary conditions in dimensionless

form are

uð0;tÞ ¼ uð1;tÞ ¼ 0;

qu

qx

????

x¼0

¼qu

qx

????

x¼1

¼ 0:

(48)

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Fig. 2. The stability boundaries for the first principal resonance with simple supports.

Fig. 3. The stability boundaries for the second principal resonance with simple supports.

L.-Q. Chen, X.-D. Yang / Journal of Sound and Vibration 284 (2005) 879–891

888

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Under the boundary conditions (45), the eigenfunction corresponding the kth natural frequency

okis [5]

fkðxÞ ¼ eib1kx?ðb4k? b1kÞðeib3k? eib1kÞ

? 1 ?ðb4k? b1kÞðeib3n? eib1kÞ

ðb4k? b2kÞðeib3k? eib2kÞeib2kx?ðb4k? b1kÞðeib3k? eib1kÞ

ðb4k? b2kÞðeib3k? eib2kÞ?ðb4k? b1kÞðeib3k? eib1kÞ

ðb4n? b3nÞðeib3k? eib3kÞeib2kx

ðb4k? b3kÞðeib3k? eib3kÞ

??

eib4kx;

ð49Þ

where bjkðj ¼ 1;2;3;4Þ and ok can be solved from Eq. (40) and the following transcendental

equation:

ðb1k? b2kÞðb3k? b4kÞbeiðb1kþb2kÞþ eiðb3kþb4kÞc þ ðb2k? b4kÞðb3k? b1kÞbeiðb1kþb3kÞ

þ eiðb2kþb4kÞ? þ ðb1k? b4kÞðb2k? b3kÞ½eiðb2kþb3kÞ? eiðb1kþb4kÞ? ¼ 0

Consider an axially moving beam with nf¼ 0:8 and g ¼ 4:0: The first two natural frequencies

and coefficients in corresponding eigenfunctions (49), numerically solved from Eqs. (40) and (50),

are o1¼ 6:8647; b11¼ 6:6676; b21¼ ?2:4953 þ 2:5344i; b31¼ ?2:4953 ? 2:5344i; b41¼ ?1:6771

and o2¼ 43:3456; b12¼ 10:2236; b22¼ ?2:4997 þ 6:9798i; b32¼ ?2:4997 ? 6:9798i; b42¼

?5:2241:

In summation parametric resonance, Eq. (22) gives c11¼ 203:4929; c22¼ 1893:0621; d12¼

?0:1772 ? 0:2642i; and d21¼ ?0:0601 ? 0:0895i: The stability boundaries in the summation

resonance of first two modes in plane s ? g1are illustrated in Fig. 4 for a ¼ 0; 0.0005, 0.001. In

principal parametric resonance, Eq. (35) gives d11¼ 1:5272 ? 0:6178i; d22¼ 0:7776 ? 0:7987i: The

stability boundaries for the first and second principal resonance in plane s ? g1are illustrated,

respectively, in Fig. 5 for a ¼ 0; 0.005, 0.01 and Fig. 6 for a ¼ 0; 0.0005, 0.001. In all figures, the

instability regions draft towards the increasing direction of the amplitude with the increase of the

viscosity coefficient. The stability boundary in the first principal resonance is less sensitive to the

change of the viscosity coefficient. In difference parametric resonance, Eq. (37) gives c11¼

203:4929; c22¼ 483:0170; d12¼ 2:1967 þ 3:2696i; d21¼ ?4:0192 þ 0:3636i: There is no instability

region in the difference resonance.

ð50Þ

ARTICLE IN PRESS

Fig. 4. The stability boundaries for the summation resonance of beams with fixed supports.

L.-Q. Chen, X.-D. Yang / Journal of Sound and Vibration 284 (2005) 879–891

889

Page 12

6. Conclusions

Transverse stability is studied for axially moving viscoelastic beams with the speed that is

harmonically fluctuating about a constant mean value. Such a parametric vibration system can be

cast into an autonomous continuous gyroscopic system under a small time dependent

perturbation. The method of multiple scales is applied to a partial-differential equation governing

the transverse parametric vibration. The stability boundary is derived from the solvability

condition. Axially accelerating beams with simple supports and fixed supports are numerically

investigated. Numerical results demonstrate that instability occurs if the axial speed fluctuation

frequency is close to the sum of any two natural frequencies (summation parametric resonance) or

two times of a natural frequency (principal parametric resonance) of the unperturbed system. A

detuning parameter is used to quantify the deviation between the speed fluctuation frequency and

the sum of two natural frequencies or the multiple of a natural frequency. The stability boundaries

are numerically determined in the axial speed fluctuation detuning parameter–amplitude plane for

varying viscosity coefficient. With the increase of the viscosity coefficient, the lager instability

threshold of speed fluctuation amplitude becomes large for given detuning parameter, and the

ARTICLE IN PRESS

Fig. 5. The stability boundaries for the first principal resonance with fixed supports.

Fig. 6. The stability boundaries for the second principal resonance with fixed supports.

L.-Q. Chen, X.-D. Yang / Journal of Sound and Vibration 284 (2005) 879–891

890

Page 13

instability range of the detuning parameter becomes small for given speed fluctuation amplitude.

In addition, the viscosity coefficient influents more on the stability boundary in higher order

principal parametric resonance.

Acknowledgements

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

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