Stability in parametric resonance of axially moving viscoelastic beams with timedependent 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.

Article: Periodic responses and chaotic behaviors of an axially accelerating viscoelastic Timoshenko beam
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ABSTRACT: This paper investigates the steadystate periodic response and the chaos and bifurcation of an axially accelerating viscoelastic Timoshenko beam. For the first time, the nonlinear dynamic behaviors in the transverse parametric vibration of an axially moving Timoshenko beam are studied. The axial speed of the system is assumed as a harmonic variation over a constant mean speed. The transverse motion of the beam is governed by nonlinear integropartialdifferential equations, including the finite axial support rigidity and the longitudinally varying tension due to the axial acceleration. The Galerkin truncation is applied to discretize the governing equations into a set of nonlinear ordinary differential equations. Based on the solutions obtained by the fourthorder Runge–Kutta algorithm, the stable steadystate periodic response is examined. Besides, the bifurcation diagrams of different bifurcation parameters are presented in the subcritical and supercritical regime. Furthermore, the nonlinear dynamical behaviors are identified in the forms of time histories, phase portraits, Poincaré maps, amplitude spectra, and sensitivity to initial conditions. Moreover, numerical examples reveal the effects of various terms Galerkin truncation on the amplitude–frequency responses, as well as bifurcation diagrams.Nonlinear Dynamics 01/2014; · 2.42 Impact Factor  SourceAvailable from: Süleyman M. Bağdatli[Show abstract] [Hide abstract]
ABSTRACT: The transverse vibrations of an axially accelerating Euler–Bernoulli beam resting on simple supports are investigated. The supports are at the ends, and there is a support in between. The axial velocity is a sinusoidal function of time varying about a constant mean speed. Since the supports are immovable, the beam neutral axis is stretched during the motion, and hence, nonlinear terms are introduced to the equations of motion. Approximate analytical solutions are obtained using the method of multiple scales. Natural frequencies are obtained for different locations of the support other than end supports. The effect of nonlinear terms on natural frequency is calculated for different parameters. Principal parametric resonance occurs when the velocity fluctuation frequency is equal to approximately twice of natural frequency. By performing stability analysis of solutions, approximate stable and unstable regions were identified. Effects of axial velocity and location of intermediate support on the stability regions have been investigated.Journal of Vibration and Acoustics. 01/2011; 133:031013. 
Article: Vibration and Stability Analysis of Axially Moving Beams with Variable Speed and Axial Force
International Journal of Structural Stability and Dynamics 08/2014; 14(06):1450015. · 1.06 Impact Factor
Page 1
JOURNAL OF
SOUND AND
VIBRATION
Journal of Sound and Vibration 284 (2005) 879–891
Stability in parametric resonance of axially moving viscoelastic
beams with timedependent speed
LiQun Chena,?, XiaoDong 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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0022460X/$see front matter r 2004 Elsevier Ltd. All rights reserved.
doi:10.1016/j.jsv.2004.07.024
?Corresponding author.
Email 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 nonresonant 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 fixedfixed conditions,
respectively. Parker and Lin [6] adopted a 1term 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 abovementioned 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 beamlike engineering devices are composed of some viscoelastic
metallic or ceramic reinforcement materials like glasscord and viscoelastic polymeric materials
such as rubber. The literature that is specially related to axially accelerating viscoelastic beams is
relatively limited. Based on 3term 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 timedependent tension based on 4term Galerkin
truncation. Based on 2term 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; crosssectional area A; moment of
inertial I and initial tension P0; travels at the timedependent 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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Page 3
where MðX;TÞ is the bending moment given by
MðX;TÞ ¼ ?
Z
A
ZsðX;Z;TÞdA;
(2)
where Z;Xplane 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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Page 4
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 firstorder 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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Page 5
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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Page 6
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 nonzero 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 nonzero 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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Page 7
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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Page 8
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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886
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
Page 11
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
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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 partialdifferential 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
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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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