Nonlinear vibrations of axially moving Timoshenko beams under weak and strong external excitations

JOURNAL OFSOUND ANDVIBRATION

Journal of Sound and Vibration 320 (2009) 10781099

frequencies and modes for simply supported boundary conditions. Wu and Mote [2] studied the linear

ARTICLE IN PRESS

www.elsevier.com/locate/jsvi0022-460X/$ - see front matter r 2008 Elsevier Ltd. All rights reserved.

doi:10.1016/j.jsv.2008.08.024

Corresponding author at: Department of Mechanics, Shanghai University, Shanghai 200444, China.E-mail address: [email protected] (L.-Q. Chen).1. Introduction

Many real-life engineering devices, such as band saws, power transmission chains, aerial cableways, andserpentine belts, involve the transverse vibration of axially moving beams. Despite its wide applications, thesedevices suffer from the occurrence of large transverse vibrations due to initial excitations. Transversevibrations of these devices have been investigated to avoid possible resulting fatigue, failure, and low quality.For example, the vibration of the blade of band saws causes poor cutting quality. The vibration of the beltleads to noise and accelerated wear of the belt in belt drive systems. Therefore, vibration analysis of axiallymoving beams is important for the design of the devices.There are many researches that have been carried out on axially moving systems in literatures. Mote [1] rst

investigated the effect of tension in an axially moving band and computed numerically the rst threeAbstract

In this paper, nonlinear vibrations under weak and strong external excitations of axially moving beams are analyzed

based on the Timoshenko model. The governing nonlinear partial-differential equation of motion is derived from

Newtons second law, accounting for the geometric nonlinearity caused by nite stretching of the beams. The complex

mode approach is applied to obtain the transverse vibration modes and the natural frequencies of the linear equation. The

method of multiple scales is employed to investigate primary resonances, nonsyntonic excitations, superharmonic

resonances, and subharmonic resonances. Some numerical examples are presented to demonstrate the effects of a varying

parameter, such as axial speed, external excitation amplitudes, and nonlinearity, on the response amplitudes for the rst

and second modes, when other parameters are xed. The stability of the response amplitudes is investigated and the

boundary of instability is located.

r 2008 Elsevier Ltd. All rights reserved.Nonlinear vibrations of axially moving Timoshenko beamsunder weak and strong external excitations

You-Qi Tanga, Li-Qun Chena,b,, Xiao-Dong Yangc

aShanghai Institute of Applied Mathematics and Mechanics, Shanghai 200072, ChinabDepartment of Mechanics, Shanghai University, Shanghai 200444, China

cDepartment of Engineering Mechanics, Shenyang Institute of Aeronautical Engineering, Shenyang 110034, China

Received 1 June 2008; received in revised form 17 August 2008; accepted 26 August 2008

Handling Editor: L.G. Tham

Available online 19 October 2008

ARTICLE IN PRESSY.-Q. Tang et al. / Journal of Sound and Vibration 320 (2009) 10781099 1079problem of parametric excitation of an axially moving band by periodic edge loading with differentassumptions. Wickert and Mote [3] presented a classical vibration theory, comprised of a modal analysis and aGreens function method, for the traveling string and the traveling beam, where natural frequencies and modesassociated with free vibration serve as a basis for analysis. Oz and Pakdemirli [4] and Oz [5] calculated the rsttwo natural frequency values for different exural stiffnesses in the cases of pinnedpinned ends andclampedclamped ends, respectively. Ghayesh and Khadem [6] investigated free nonlinear transverse vibrationof an axially moving beam in which rotary inertia and temperature variation effects have been considered andthey gave natural frequency versus the mean velocity and rotary inertia, critical speed versus the rotary inertia,and natural frequency versus the mean velocity and temperature for the rst two modes. Lee and Jang [7]investigated the effects of continuously incoming and outgoing semi-innite beam parts on dynamiccharacteristics and stability of an axially moving beam by using the spectral element method. In all of theabove literatures, either simple support or xed support is researched. Yang and Chen [8] studied axiallymoving elastic beams, and Chen and Yang [9] studied viscoelastic beams, on simple supports with torsionsprings and gave the rst two frequencies and modes.Lee et al. [10] used exact dynamic-stiffness matrix in structural dynamics to provide very accurate solutions,

while reducing the number of degrees of freedom to resolve the computational and cost problems. Chakrabortyand Mallik [11] used wave propagation in a simply supported traveling beam to derive forced responses. Zhangand Zu [12] investigated the nonlinear forced vibration of viscoelastic moving belts excited by the eccentricity ofpulleys. Pellicano and Vestroni [13] investigated the dynamic response of a simply supported traveling beamsubjected to a transverse load in the super-critical speed range. Yang and Chen [14] studied the nonlinear forcedvibration of axially moving viscoelastic beams excited by vibration of the supporting foundation.The axially moving beam has been traditionally represented by the EulerBernoulli beam theory by

assuming that the beam is relatively thin compared to its length. It appears that, to the authors knowledge,there have been very few studies on the axially moving beam for the Timoshenko model. Simpson [15] wasprobably the rst to derive equations of motion for the moving thick beam on the basis of the Timoshenkobeam theory, but no numerical results were given and he did not consider axial tension in his equations.Chonan [16] studied the steady-state response of a moving Timoshenko beam by applying the Laplacetransform method. Arboleda-Monsalve et al. [17] presented the dynamic-stiffness matrix and load vector of aTimoshenko beam column resting on a two-parameter elastic foundation with generalized end conditions. Meiet al. [18] presented wave vibration analysis of an axially loaded cracked Timoshenko beam.The present paper is organized as follows. Section 2 derives the governing nonlinear partial-differential

equation of motion. Section 3 employs the method of multiple scales to investigate nonlinear vibrations underweak and strong external excitations. Section 4 ends the paper with concluding remarks.

2. The governing equation

Uniform axially moving Timoshenko beams, with density r, cross-sectional area A, area moment of inertiaof the cross-section about the neutral axis J, initial tension P, shape factor k, modulus of elasticity E, axialtension N, beam shearing modulus G, travel at the constant axial transport speed G between two simplesupports separated by distance L under the distributed external excitation F in the transverse direction. Thebending vibration can be described by two variables dependent on axial coordinate X and time T, namely,transverse displacement Y(X,T) and angle of rotation of the beam cross-section C(X,T) due to the bendingmoment.Because shear deformations are considered, the angle of the beam y depends not only on the angle C but

also on the shear angle g:

yx; t cx; t gx; t (1)The bending moment M(x, t) and shear force Q(x, t) are related to the corresponding deformations:

M EJc;x,

Q M ;x AG c y AG c Y ;x (2)

k k

ARTICLE IN PRESSY.-Q. Tang et al. / Journal of Sound and Vibration 320 (2009) 107810991080Coupled governing equations are obtained according to Newtons second law:

Q;X dX cos y rAY ;TT 2GY ;XT cos y G2Y ;XX cos ydXN sin yN;X dX sin y F dX 0,

rJc;TT M;X Q 0 (3)where F is the external transverse excitation. The axial tension N is composed of the initial tension and thetension due to the transverse displacement:

N P 12EAY ;X 2 (4)

Substituting Eqs. (2) and (4) into Eq. (3) and using the approximate expressions cos y 1 and sin y y,x yieldthe governing equation for the transverse vibration of axially moving Timoshenko beams:

rJ 1 kPAG

kEG

Y ;XXTT EJ 1

kP

AG

Y ;XXXX PY ;XX

3

2EAY ;2XY ;XX

r2Jk

GY ;TTTT G2Y ;XXTT 2GY ;XTTT

3rEJk2G

Y ;2XY ;XXTT 4Y ;XY ;XTY ;XXT

2Y ;XY ;XXY ;XTT 2Y ;2XTY ;XX rEJkG

2GY ;XXXT G2Y ;XXXX 3E2Jk

2GY ;2XY ;XXXX

6Y ;XY ;XXY ;XXX 2Y ;3XX rAY ;TT 2GY ;XT G2Y ;XX F 0 (5)The boundary conditions for the simple supports at both ends are

Y jX0 0; Y jXL 0; EJc;XX0 0; EJc;X

XL 0 (6)

It is assumed that the external transverse excitation is a spatially uniformly distributed periodic force asF F1 cos(OT), where F1 is the excitation amplitude and O the excitation frequency.Introduce the dimensionless variables and parameters

y Y0:5L

; x XL

; t TP

rAL2;

sv G

rAP

;

rk0

kJP

GA2L2; k1

kJE

GAL2,

k2 EA

P; k3

J

AL2; k4

EJ

PL2; o O

rAL2

P;

sb LF 1

0:5rP(7)

where e, a dimensionless small number, denotes the small but nite transverse deformation of beams.Dimensionless parameter k0 associated with k1 accounts for the effects of shear distortion, parameter k2represents the effect of nonlinearity, parameter k3 represents the effects of rotary inertia, and parameter k4denotes the stiffness of the beam. Integer r indicates the order of the external excitation. For weak externalexcitations, F1 is of order ey, and r 1; for strong external excitations, F1 is of order y, and r 0.Substituting Eq. (7) into Eqs. (5) and (6) yields the dimensionless form:

y;tt 2vy;xt v2 1y;xx k0 k1 k3 k0v2y;xxtt k1 k4 k1v2y;xxxx k0y;tttt 2vy;xttt 2k1vy;xxxt

32

k2y;2xy;xx k1y;2xy;xxtt

4y;xy;xty;xxt 2y;xy;xxy;xtt 2y;2xty;xx k1k2y;2xy;xxxx 6y;xy;xxy;xxx 2y;3xx

rb cosoT0 (8)yx0 0; y

x1 0; y;xx

x0 0; y;xx

x1 0 (9)

Lee and Jang [7] proposed a set of governing equations for axially moving EulerBernoulli beams toaccount for the momentum transport through both of the end boundaries. For the transverse vibration, theirequation differs from the traditional one with an additional y,xxtt term. However, the present investigation stilladopts the traditional formulation. It should be noticed that Eq. (8) contains term y,xxtt. Therefore, the

following analysis may be valid for models developed by Lee and Jangs approach.

qT2 k0

qT4 2v

qx qT0 2k0v

qx qT3 v 1

qx2 2k1vqx3 qT0

ARTICLE IN PRESSY.-Q. Tang et al. / Journal of Sound and Vibration 320 (2009) 10781099 10811 :

0 0 0

k0 k1 k3 k0v2q4y1

qx2 qT20 k1 k4 k1v2

q4y1qx4

2 q2y0

qT0 qT1 4k0

q4y0qT30 qT1

2v q2y0

qx qT1 6k0v

q4y0qx qT20 qT1

32k2

qy0qx

2q2y0qx2

3k1q2y0qx qT0

2q2y0qx2

3k1qy0qx

q3y0qx qT20

q2y0qx2

3k1k2q2y0qx2

3

6k1qy0qx

q2y0qx qT0

q3y0qx2 qT0

2k0 k1 k3 k0v2q4y0

qx2 qT0 qT1

32k1

qy0qx

2 q4y0qx2 qT20

9k1k2qy0qx

q2y0qx2

q3y0qx3

2k1vq4y0

qx3 qT1

32k1k2

qy0qx

2q4y0qx4

b cosot

(13)

The solution to Eq. (12) can be assumed as

y f xA T ;T eionT0 f xA T ;T eionT0 (14)3. The multi-scale analysis

The method of multiple scales will be employed to Eq. (8). One assumes an expansion of dimensionlessdisplacement and its time derivatives

yx; t; y0x;T0;T1;T2 y1x;T0;T1;T2 2y2x;T0;T1;T2 (10)

d

dt qqT0

qqT1

2 qqT2

,

d2

dt2 q

2

qT20 2 q

2

qT0 qT1 2 q

2

qT21 2 q

2

qT0 qT2

,

d3

dt3 q

3

qT30 3 q

3

qT20 qT1 32 q

3

qT0 qT21 q

3

qT20 qT2

! ,

d4

dt4 q

4

qT40 4 q

4

qT30 qT1 2 6 q

4

qT20 qT21

4 q4

qT30 qT2

! (11)

3.1. Nonlinear vibrations under external excitations

First, we consider nonlinear vibrations under weak external excitations of axially moving beams on simplesupports based on the Timoshenko beam model. At r 1, substituting Eq. (10) into Eq. (8), and thenequalizing the coefcients e0 and e1 in the resulting equation, one obtains

0 :

q2y0qT20

k0q4y0qT40

2v q2y0

qx qT0 2k0v

q4y0qx qT30

v2 1q2y0qx2

2k1vq4y0

qx3 qT0

k0 k1 k3 k0v2q4y0

qx2 qT20 k1 k4 k1v2

q4y0qx4

0(12)

q2y1 q4y1 q

2y1 q4y1 2 q

2y1 q4y10 n n 1 2 n n 1 2

ARTICLE IN PRESSY.-Q. Tang et al. / Journal of Sound and Vibration 320 (2009) 107810991082where fn and on denote the nth mode function and natural frequency and overbar represents complexconjugate. Substituting Eq. (14) into Eqs. (12) and (9) yields

k0fno4n 2ik0vf0no3n k0 k1 k3 k0v2f00n fn

o2n 2ivf0n k1f000n on

v2 1f00n k1 k4 k1v2f0000n 0 (15)

fn0 0; fn1 0; f00n0 0; f00n1 0 (16)The solution to ordinary differential Eq. (15) can be expressed by

fnx C1neib1nx C2neib2nx C3neib3nx C4neib4nx (17)Substituting Eq. (17) into Eqs. (15) and (16) yields

k1 k4 k1v2b4in 2vk1onb3in k0 k1 k3 k0v2o2n 1 v2b2in 2k0vo3n 2vonbin k0o4n o2n 0 (18)

1 1 1 1

b21n b22n b

23n b

24n

eib1n eib2n eib3n eib4n

b21n eib1n b22n e

ib2n b23n eib3n b24n e

ib4n

0BBBB@

1CCCCA

1

C2n

C3n

C4n

0BBB@

1CCCAC1n 0 (19)

For the nontrivial solution of Eq. (19), the determinant of the coefcient matrix must be zero:

eib1nb2n eib3nb4n b21n b22nb23n b24n eib1nb3n eib2nb4n b23n b21nb22n b24n eib1nb3n eib1nb4n b22n b23nb21n b24n 0 (20)

Using Eqs. (19) and (20), one can obtain the modal function of the simply supported beam as follows:

fx c1 eib1nx b24n b21neib3n eib1n b24n b22neib3n eib2n

eib2nx b24n b21neib2n eib1n

b24n b23neib2n eib3neib3x

(

1 b24n b21neib3n eib1n

b24n b22neib3n eib2n b

24n b21neib2n eib1n

b24n b23neib2n eib3n

!eib4nx

)(21)

The nth eigenvalues bjn (j 1, 2, 3, 4) and the corresponding natural frequency can be calculatednumerically considering boundary conditions (9). Fig. 1 presents the natural frequencies for the rst andsecond modes. It has been found that the natural frequencies decrease with increasing axial speed. The exactvalue at which the rst natural frequency vanishes is called the critical speed and afterwards the system isunstable about the zero equilibrium.If the axial speed variation frequency approaches any natural frequency of the system (12), primary

response may occur. A detuning parameter s is introduced to quantify the deviation of o from on and o isdescribed by

o on s (22)where on denote the nth natural frequency of free vibration described by Eq. (12). To investigate the primaryresponse with the possible contributions of modes not involved in the resonance, the solution to Eq. (13) canbe expressed as

y f xA T ;T eionT0 f xA T ;T eiomT0 cc (23)0 n n 1 2 m m 1 2

ARTICLE IN PRESSY.-Q. Tang et al. / Journal of Sound and Vibration 320 (2009) 10781099 1083where man. Substituting Eqs. (22) and (23) into Eq. (13) yields

q2y1qT20

k0q4y1qT40

2v q2y1

qx qT0 2k0v

q4y1qx qT30

v2 1 q2y1qx2

2k1vq4y1

qx3 qT0

k0 k1 k3 k0v2q4y1

qx2 qT20 k1 k4 k1v2

q4y1qx4

3 0 2 0 2 00 000 qAn

10

8

6

4

2

1

2

0.5 1 1.5 2 2.5 3 3.5

k4 = 1k4 = 1

k4 = 0.36 k4 = 0.36

k4 = 0.64k4 = 0.64

k4 = 0.16 k4 = 0.16

k4 = 0.04 k4 = 0.04

35

30

25

20

15

10

5

0.5 1 1.5 2 2.5 3

Fig. 1. Natural frequency diagram: (a) the rst mode and (b) the second mode. 2ionfn 2ik0onfn vfn 3k0vonfn ik0 k1 k3 k0v onfn k1vfn qT1 322k2f0nf

0nf

00n 2k1o2nf0nf

0nf

00n k2f02nf

00n k1o2nf02nf

00n 6k1k2f002nf

00n 6k1k2f

0nf

00nf

000n

6k1k2f0nf00nf

000n 6k1k2f0nf00nf

000n 2k1k2f0nf

0nf

0000n k1k2f02nf

0000n A2nAn 3k2f0nf

0mf

00m k1o2nf0nf

0mf

00m

k2f0mf0mf

00n k1o2nf0mf

0mf

00n k2f0mf0nf

00m k1o2nf0mf0nf

00m 6k1k2f00mf00nf

00m 3k1k2f

0mf

00nf

000m

3k1k2f0nf00mf

000m 3k1k2f

0mf

00mf

000n 3k1k2f0mf

00mf

000n 3k1k2f0nf00mf

000m 3k1k2f0mf00nf

000m k1k2f0nf

0mf

0000m

k1k2f0mf0mf

0000n k1k2f0mf0nf

0000m AmAnAm

1

2b eisT1

eionT0

2iomfm 2ik0o3mfm vf0m 3k0vo2mf0m ik0 k1 k3 k0v2omf00m k1vf000mqAmqT1

322k2f0mf

0mf

00m 2k1o2mf0mf

0mf

00m k2f02mf

00m k1o2mf02mf

00m 6k1k2f002mf

00n 6k1k2f0mf00mf000m

6k1k2f0mf00mf

000m 6k1k2f0mf00mf

000m 2k1k2f0mf

0mf

0000m k1k2f02mf

0000m A2mAm 3k2f0mf

0nf

00n k1o2mf0mf

0nf

00n

k2f0nf0nf

00m k1o2mf0nf

0nf

00m k2f0nf0mf

00n k1o2mf0nf0mf

00n 6k1k2f00nf00mf

00n 3k1k2f

0nf

00mf

000n

3k1k2f0mf00nf

000n 3k1k2f

0nf

00nf

000m 3k1k2f0nf

00nf

000m 3k1k2f0mf00nf

000n 3k1k2f0nf00mf

000n k1k2f0mf

0nf

0000n

k1k2f0nf0nf

0000m k1k2f0nf0mf

0000n AnAmAn eiomT0 ccNST (24)

where the prime denotes derivation with respect to the dimensionless spatial variable x, cc stands for complexconjugate of the proceeding terms, NST for non-secular terms, and h.o.t. for high orders of e.

ARTICLE IN PRESSY.-Q. Tang et al. / Journal of Sound and Vibration 320 (2009) 107810991084The solvability condition, which demands orthogonal relationships, has been presented by Nayfeh [19]:

2ionfn 2ik0o3nfn vf0n 3k0vo2nf0n ik0 k1 k3 k0v2onf00n k1vf000n qAnqT1

322k2f0nf

0nf

00n 2k1o2nf0nf

0nf

00n k2f02nf

00n k1o2nf02nf

00n 6k1k2f002nf

00n 6k1k2f

0nf

00nf

000n

6k1k2f0nf00nf000n 6k1k2f0nf00nf000n 2k1k2f0nf0nf

0000n k1k2f02nf

0000n A2nAn

3k2f0nf0mf

00m k1o2nf0nf

0mf

00m k2f0mf

0mf

00n k1o2nf0mf

0mf

00n k2f0mf0nf

00m k1o2nf0mf0nf

00m

6k1k2f00mf00nf00m 3k1k2f

0mf

00nf

000m 3k1k2f0nf

00mf

000m 3k1k2f

0mf

00mf

000n 3k1k2f0mf

00mf

000n

3k1k2f0nf00mf000m 3k1k2f0mf00nf000m k1k2f0nf0mf

0000m k1k2f0mf

0mf

0000n k1k2f0mf0nf

0000m jAmj2An

12b eisT1

;fn

0 (25)

2iomfm 2ik0o3mfm vf0m 3k0vo2mf0m ik0 k1 k3 k0v2omf00m k1vf000mqAmqT1

322k2f0mf

0mf

00m 2k1o2mf0mf

0mf

00m k2f02mf

00m k1o2mf02mf

00m 6k1k2f002mf

00n 6k1k2f

0mf

00mf

000m

6k1k2f0mf00mf

000m 6k1k2f0mf00mf000m 2k1k2f0mf

0mf

0000m k1k2f02mf

0000m A2mAm 3k2f0mf

0nf

00n

k1o2mf0mf0nf

00n k2f0nf

0nf

00m k1o2mf0nf

0nf

00m k2f0nf0mf

00n k1o2mf0nf0mf

00n 6k1k2f00nf00mf

00n

3k1k2f0nf00mf000n 3k1k2f0mf00nf

000n 3k1k2f

0nf

00nf

000m 3k1k2f0nf

00nf

000m 3k1k2f0mf00nf

000n

3k1k2f0nf00mf000n k1k2f0mf0nf

0000n k1k2f0nf

0nf

0000m k1k2f0nf0mf

0000n jAnj2Am

;fm

0 (26)

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

hf ; gi Z 10

f gdx (27)

Application of the distributive law of the inner product to Eqs. (25) and (26) leads to

qAnqT1

knA2nAn mnmjAmj2An bwn eisT1 0 (28)

qAmqT1

kmA2mAm mmnjAnj2Am 0 (29)

where

kn 3

22k2

Z 10

f0nf0nf

00nfn dx 2k1o2n

Z 10

f0nf0nf

00nfn dx k2

Z 10

f02nf00nfn dx k1o2n

Z 10

f02nf00nfn dx

6k1k2Z 10

f002nf00nfn dx6k1k2

Z 10

f0nf00nf

000n fn dx6k1k2

Z 10

f0nf00nf

000n fn dx6k1k2

Z 10

f0nf00nf

000nfn dx

2k1k2Z 10

f0nf0nf

0000n fn dx k1k2

Z 10

f02nf0000n fn dx

2 ion

Z 10

fnfn dx 2ik0o3nZ 10

fnfn dx

vZ 1

f0nfn dx 3k0vo2nZ 1

f0nfn dx ik0 k1 k3 k0v2onZ 1

f00nfn dx k1vZ 1

f000n fn dx

(30a)

0 0 0 0

ARTICLE IN PRESSY.-Q. Tang et al. / Journal of Sound and Vibration 320 (2009) 10781099 1085mnm 3 k2Z 10

f0nf0mf

00mfn dx k1o2n

Z 10

f0nf0mf

00mfn dx k2

Z 10

f0mf0mf

00nfn dx

k1o2nZ 10

f0mf0mf

00nfn dx k2

Z 10

f0mf0nf

00mfn dx k1o2n

Z 10

f0mf0nf

00mfn dx

6k1k2Z 10

f00mf00nf

00mfn dx 3k1k2

Z 10

f0mf00nf

000mfn dx 3k1k2

Z 10

f0nf00mf

000mfn dx

3k1k2Z 10

f0mf00mf

000n fn dx 3k1k2

Z 10

f0mf00mf

000n fn dx 3k1k2

Z 10

f0nf00mf

000mfn dx

3k1k2Z 10

f0mf00nf

000mfn dx k1k2

Z 10

f0nf0mf

0000m fn dx k1k2

Z 10

f0mf0mf

0000n fn dx

k1k2Z 10

f0mf0nf

0000m fn dx

2 ion

Z 10

fnfn dx

2ik0o3nZ 10

fnfn dx vZ 10

f0nfn dx

3k0vo2nZ 10

f0nfn dx ik0 k1 k3 k0v2onZ 10

f00nfn dx k1vZ 10

f000n fn dx

(30b)

wn Z 10

fn dx4 ion

Z 10

fnfn dx

2ik0o3nZ 10

fnfn dx vZ 10

f0nfn dx 3k0vo2nZ 10

f0nfn dx

ik0 k1 k3 k0v2onZ 10

f00nfn dx k1vZ 10

f000nfn dx

(30c)

Express the solution to Eqs. (28) and (29) in polar form

An anT1;T2 eibnT1;T2,Am amT1;T2 eibmT1;T2 (31)

Substituting Eq. (31) into Eq. (29) yields

qamqT1

Rekmb3m RemmnjAnj2bm,

amqbmqT1

Imkmb3m ImmmnjAnj2bm (32)

For steady-state solutions, the amplitude am and the new phase bm angle in Eq. (32) should be constant.Setting a0m 0 and b0m 0 gives

0 Rekmb3m RemmnjAnj2bm,0 Imkmb3m ImmmnjAnj2bm (33)

Eq. (33) obviously has a zero solution. If we assume that there is a non-zero solution, Eq. (33) yields

0 Rekmb2m RemmnjAnj2,0 Imkmb2m ImmmnjAnj2 (34)

The two equations in Eq. (34) cannot come into existence at the same time. Thus, the assumption is wrong.The solution to Eq. (29) has only zero stationary solution and decays to zero exponentially. Therefore, the mthmode has actually no effect on the stability.Substituting Eq. (31) into Eq. (28) and eliminating the mth mode yields

qanqT1

bRewn cos yn b Imwn sin yn,

anqynqT

ans Imkna3n bRewn sin yn b Imwn cos yn (35)

1

ARTICLE IN PRESSY.-Q. Tang et al. / Journal of Sound and Vibration 320 (2009) 107810991086where

yn sT1 bn (36)For steady-state solutions, the amplitude an and the new phase yn angle in Eq. (35) should be constant. Settinga0n 0 and y0n 0 leads to

s Imkna2n b

anjwnj (37)

ana0 in Eq. (35) yields

qanqT1

bRewn cos yn b Imwn sin yn,qynqT1

s Imkna2n b

anRewn sin yn

b

anImwn cos yn (38)

The stability of the nontrivial state can be obtained by perturbing these polar modulation equations andchecking the eigenvalues of the resulting Jacobian matrix.To determine the stability of the nontrivial state, these equations are perturbed to obtain

DqanqT1

;DqynqT1

T JfDan;DyngT (39)

where T denotes transpose.The Jacobian matrix whose eigenvalues determine the stability and bifurcations of the system is

J 0 san Imkna3n

san

3 Imknan 0

0@

1A (40)

The J-matrix characteristic function

l2 s Imkna2n

s 3Imkna2n 0 (41)

where l is an eigenvalue of the system. Based on RouthHurwitz theorem, the boundary of instability yields

s Imkna2n

s 3 Imkna2n 0 (42)

Consider an axially moving beam with k4 0.64 and v 2.0. The response amplitudes at exact resonancefor the rst two natural frequencies are shown in Fig. 2. The rst two natural frequencies of the unperturbedsystem are o1 4.7393 and o2 23.7017. In the rst-mode response, the coefcients are b 0.04,k2 100,000, and in the second-mode response, the coefcients are b 0.1, k2 100,000.Fig. 3 shows the effects of a different parameter k2. With an increase of k2, response under the same

conditions decreases. In the rst-mode response, the coefcient is b 0.04, and in the second-mode response,the coefcient is b 0.1. The solid lines are for coefcient k2 50,000, the dashed for coefcientk2 100,000, and the dotted lines for coefcient k2 150,000.The effects of the foundation vibration amplitude on the response amplitudes are illustrated in Fig. 4. From

the response diagrams, it is clear that the excitation amplitudes increase the amplitude of the excited system.The coefcient is k2 100,000. In the rst-mode response, the solid lines are for coefcient b 0.02, thedashed for coefcient b 0.04, and the dotted lines for coefcient b 0.06. In the second-mode response, thesolid lines are for coefcient b 0.05, the dashed are for coefcient b 0.1, and the dotted lines for coefcientb 0.15.The stability of the response amplitudes is illustrated in Fig. 5. In the rst-mode response, the coefcients

are b 0.04 and k2 100,000. In the second-mode response, the coefcients are b 0.1 and k2 100,000.

The solid lines denote the response amplitudes and the dashed denote the boundary of instability.

ARTICLE IN PRESSY.-Q. Tang et al. / Journal of Sound and Vibration 320 (2009) 10781099 10871

0.8

10-3 10-47

6

5The inner region of the boundary of instability is unstable and the outer is stable. Thus a jump phenomenonappears.

3.2. Nonlinear vibrations under strong external excitations

Next, we consider nonlinear vibrations under strong external excitations of axially moving beams on simplesupports based on the Timoshenko beam model. At r 0, substituting Eq. (10) into Eq. (8), and then

1

0.8

0.6

0.4

0.2

0

a n a n

10-3 10-4

-10 -5 0 5 10

7

6

5

4

3

2

1

0-10 -5 0 5 10

Fig. 3. Comparison of response amplitudes of a different parameter k2: (a) the rst mode and (b) the second mode.

0.6

0.4

0.2

0

a n a n-10 -5 0 5 10

4

3

2

1

0-10 -5 0 5 10

Fig. 2. Response amplitudes diagram: (a) the rst mode and (b) the second mode.

ARTICLE IN PRESSY.-Q. Tang et al. / Journal of Sound and Vibration 320 (2009) 1078109910881

0.8

10-3 10-47

6

5equalizing the coefcients e0 and e1 in the resulting equation, one obtains

0 :

q2y0qT20

k0q4y0qT40

2v q2y0

qx qT0 2k0v

q4y0qx qT30

v2 1q2y0qx2

2k1vq4y0

qx3 qT0

k0 k1 k3 k0v2q4y0

qx2 qT20 k1 k4 k1v2

q4y0qx4

b cosoT0(43)

1

0.8

0.6

0.4

0.2

0

a n a n

10-3 10-4

-10 -5 0 5 10

7

6

5

4

3

2

1

0-10 -5 0 5 10

Fig. 5. Stability of the response amplitudes: (a) the rst mode and (b) the second mode.

0.6

0.4

0.2

0

a n a n

-10 -5 0 5 10

4

3

2

1

0-10 -5 0 5 10

Fig. 4. Comparison of response amplitudes of different excitation amplitudes b: (a) the rst mode and (b) the second mode.

ARTICLE IN PRESSY.-Q. Tang et al. / Journal of Sound and Vibration 320 (2009) 10781099 10891 :

q2y1qT20

k0q4y1

qT40 2v q

2y1qx qT0

2k0v q4y1

qx qT30 v2 1q

2y1qx2

2k1v q4y1

qx3 qT0

k0 k1 k3 k0v2q4y1

qx2 qT20 k1 k4 k1v2

q4y1qx4

2 q2y0

qT0 qT1 4k0

q4y0qT30 qT1

2v q2y0

qx qT1 6k0v

q4y0qx qT20 qT1

32k2

qy0qx

2q2y0qx2

3k1q2y0qx qT0

2q2y0qx2

3k1qy0qx

q3y0qx qT20

q2y0qx2

3k1k2q2y0qx2

3

6k1qy0qx

q2y0qx qT0

q3y0qx2 qT0

2k0 k1 k3 k0v2q4y0

qx2 qT0 qT1

32k1

qy0qx

2 q4y0qx2 qT20

9k1k2qy0qx

q2y0qx2

q3y0qx3

2k1vq4y0

qx3 qT1 32k1k2

qy0qx

2q4y0qx4

(44)

Express the solution to Eq. (43) in the following form:

y0x;T0;T1;T2 fxqT0;T1;T2 cc (45)where f(x) is the complex mode function satisfying the boundary condition. The mode function has beencalculated above in Eq. (21).Substituting Eq. (45) into Eq. (43), multiplying both sides by f(x), and integrating on [0,1] gives

q4 a1 _ _ _q a2 q a3 _q a4q h cosoT0 (46)where

a1 2vR 10 ff

0 dxR 10 f

2 dx; a2

R 10 f

2 dx k0 k1 k3 k0v2R 10 ff

00 dx

k0R 10 f

2 dx; h b

R 10 fdx

k0R 10 f

2 dx,

a3 2vR 10 ff

0 dx k1R 10 ff

000 dx

k0R 10f2 dx

; a4 v2 1 R 10 ff00 dx k1 k4 k1v2 R 10 ff0000 dx

k0R 10f2 dx

(47)

Express the solution to Eq. (46) in polar form

q AT1;T2 eio0T0 B eioT0 (48)Substituting Eq. (48) into (46) yields

B h2o4 ia1o3 a2o2 ia3o a4

(49)

The solution of Eq. (43) is

y0x;T0;T1;T2 fxAT1;T2 eio0T0 B eioT0 cc (50)Substituting Eq. (50) into Eq. (44) yields

q2y1qT20

k0 q4y1

qT40 2v q

2y1qx qT0

2k0v q4y1

qx qT30 v2 1 q

2y1qx2

2k1v q4y1

qx3 qT0

k0 k1 k3 k0v2q4y1

qx2 qT20 k1 k4 k1v2

q4y1qx4

3 9k1o20 k2f02f00 2k1k2f003 6k1k2f0f00f000 k1k2f02f0000A3 e3io0T02

2ARTICLE IN PRESSY.-Q. Tang et al. / Journal of Sound and Vibration 320 (2009) 107810991090 6k1k2f0f00f000 f0f00f000 f0f00f000 k1k2f02f0000 2f0f0f0000AB2 eio02oT0

f3k1o2 k2f02f00 2f0f0f00 6k1k2f002f00 f0f00f000 f0f00f000 f0f00f000 2k1k2f0f0f0000 k1k2f02f0000ABA

32k1o2 k2f02f00 2f0f0f00 6k1k2f002f00 f0f00f000 f0f00f000 f0f00f000

2k1k2f0f0f0000 k1k2f02f0000B2Bg eioT0


2k1k2f0f0f0000 k1k2f02f0000ABB


2k1k2f0f0f0000 k1k2f02f0000A2A 2io01 2k0o20f v1 3k0o20f0 io0k0 k1 k3 k0v2f00

k1vf000qAqT1

eio0T0 ccNST h:o:t: (51)

where the prime denotes derivation with respect to dimensionless spatial variable x, cc stands for complexconjugate of the proceeding terms, NST for non-secular terms, and h.o.t. for high orders of e.

3.2.1. Nonsyntonic excitations

When the frequency o is far from 0, o0, o0/3, and 3o0, the solvability condition demands the orthogonalrelationship


2k1k2f0f0f0000 k1k2f02f0000ABB


2k1k2f0f0f0000 k1k2f02f0000A2A 2io01 2k0o20f v1 3k0o20f0 io0k0 k1 k3 k0v2f00

k1vf000qAqT

;f

0 (52) 329k1o2 k2f02f00 2k1k2f003 6k1k2f0f00f000 k1k2f02f0000B3 e3ioT0

92k2 k1o2 4k1oo0 4k1o20f02f00 2k1k2f003 6k1k2f0f00f000

k1k2f02f0000A2B eio2o0T0

32k2 k1o2 4k1oo0 4k1o20f0

2f00 2f0f0f00 6k1k2f00f00

2

6k1k2f0f00f000 f0f00f000 f0f00f000 k1k2f02f0000 2f0f0f0000A2B eio2o0T0

92k2 k1o20 4k1oo0 4k1o2f02f00 2k1k2f003 6k1k2f0f00f000

k1k2f02f0000AB2 eio02oT0

3 k2 k1o20 4k1oo0 4k1o2f02f00 2f0f0f00 6k1k2f00f00

21

ARTICLE IN PRESSY.-Q. Tang et al. / Journal of Sound and Vibration 320 (2009) 10781099 1091where the inner product is dened for complex functions on [0,1] as

hf ; gi Z 10

f gdx (53)

Application of the distributive law of inner product to Eq. (52) leads to

wABB kA2A qAqT1

0 (54)

where

w 3 k1o20 k2Z 10

ff02f00 dx 2Z 10

ff0f0f00 dx

6k1k2Z 10

ff002f00 dx

Z 10

ff0f00f000 dxZ 10

ff0f00f000 dxZ 10

ff0f00f000 dx

2k1k2

Z 10

ff0f0f0000 dx k1k2Z 10

ff02f0000dx

2 io01 2k0o20

Z 10

ffdx v1 3k0o20Z 10

ff0 dx io0k0 k1 k3 k0v2

Z 10

ff00 dx k1vZ 10

ff000 dx

(55a)

k 12w (55b)

These coefcients in Eq. (55) can be determined by the natural frequencies and the modal functions calculatedfrom Eq. (12) with the boundary condition. Consider the transformation

A a1T1;T2 eia2T1;T2,B b1 eib2 (56)


qa1qT1

a31kR a1b21wR (57a)

qa2qT1

a21kI b21wI (57b)

The rst approximate solution is

y0 a1 coso0t a2 B cosot O (58)These coefcients a1 and a2 can be determined from Eqs. (57a) and (57b). The amplitude of the free-vibrationterm will be attenuated. Steady-state response will be only forced vibration term.

3.2.2. Superharmonic resonances and response amplitudes

To study the superharmonic resonances, a detuning parameter s is introduced to quantify the deviation ando is described by

3o o0 s (59)Solvability requires

wABB kA2A zB3 eisT1 qAqT

0 (60)

1

ARTICLE IN PRESSwhere

w 3 k1o20 k2Z 10

ff02f00 dx 2Z 10

ff0f0f00 dx

6k1k2Z 10

ff002f00 dx

Z 10

ff0f00f000 dxZ 10

ff0f00f000 dxZ 10

ff0f00f000 dx

2k1k2

Z 10

ff0f0f0000 dx k1k2Z 10

ff02f0000dx

2 io01 2k0o20

Z 10

ffdx v1 3k0o20Z 10


Z 10

ff00 dxk1vZ 10

ff000 dx

(61a)

k 12w (61b)

z 32

k1o20 k2Z 10

ff02f00 dx 2k1k2Z 10

ff003 dx 6k1k2Z 10

ff0f00f000 dx

k1k2Z 10

ff02f0000 dx

2 io01 2k0o20Z 10

ffdx v1 3k0o20Z 10

ff0 dx

io0k0 k1 k3 k0v2Z 10

ff00 dx k1vZ 10

ff000 dx

(61c)

Consider the transformation

A a1T1;T2 eia2T1;T2; B b1 eib2 (62)Substituting Eq. (62) into Eq. (60) yields

qa1qT1

a31kR a1b21wR b31zR cos3b2 sT1 a2 zI sin3b2 sT1 a2 (63a)

a1qa2qT1

a31kI a1b21wI b31zR sin3b2 sT1 a2 zI cos3b2 sT1 a2 (63b)

Then, we change Eq. (63) into an autonomous system

qa1qT1

a31kR a1b21wR b31zR cosg zI sing (64a)

qgqT1

s a21kI b21wI b31a1

zR sing zI cosg (64b)

where

g 3b2 sT1 a2 (65)For steady-state solutions, the amplitude a1 and the new phase g angle in Eq. (64) should be constant. Settinga01 0 and g0 0 and then eliminating from Eq. (64) leads to

s a21kI b21wI

b61zI 2 zR2 a21a21kR b21wR2

qa

(66)

Y.-Q. Tang et al. / Journal of Sound and Vibration 320 (2009) 1078109910921

When a1a0 in Eq. (64), the Jacobian matrix characteristic function of the balanced solution on the righthand,

l2 22a21kR b21wRl 3a21kR b21wRa21kR b21wR s a21kI b21wI s 3a21kI b21wI 0 (67)


3a21kR b21wRa21kR b21wR s a21kI b21wI s 3a21kI b21wI 0 (68)

ARTICLE IN PRESS

1

0.8

0.6

a n a n

10-3 10-44

3

2

1

0.8

0.6

0.4

0.2

0

a n a n

10-3 10-4

-10 -5 0 5 10

4

3

2

1

0-5 0 5

Fig. 6. Response amplitude diagram: (a) the rst mode and (b) the second mode.

Y.-Q. Tang et al. / Journal of Sound and Vibration 320 (2009) 10781099 10930.4

0.2

0-10 -5 0 5 10

1

0-5 0 5


ARTICLE IN PRESSY.-Q. Tang et al. / Journal of Sound and Vibration 320 (2009) 107810991094Consider an axially moving beam with k4 0.64 and v 2.0. The response amplitudes at exact resonancefor the rst two natural frequencies are shown in Fig. 6. The rst two natural frequencies of the unperturbedsystem are o1 4.7393 and o2 23.7017. In the rst-mode response, the coefcients are b 0.03,k2 100,000, and in the second-mode response, the coefcients are b 0.5, k2 100,000.Fig. 7 indicates the effects of a different parameter k2. With an increase of k2, response under the same

1

0.8

0.6

0.4

0.2

0

a n a n

10-3 10-4

-10 -5 0 5 10

4

3

2

1

0-5 0 5

Fig. 8. Comparison of response amplitudes of different excitation amplitudes b: (a) the rst mode and (b) the second mode.conditions decreases. In the rst-mode response, the coefcient is b 0.03, and in the second-mode response,the coefcient is b 0.5. The dotted lines are for coefcient k2 50,000, the dashed for coefcientk2 100,000, and the solid lines for coefcient k2 150,000.The effects of the external excitation amplitudes on the response amplitudes are illustrated in Fig. 8. From

the response diagrams, it is clear that the external excitation amplitudes increase the amplitude of the excitedsystem. The coefcient here is k2 100,000. In the rst-mode response, the solid lines are for coefcientb 0.05, the dotted lines for coefcient b 0.04, and the dashed for coefcient b 0.03. In the second-moderesponse, the solid lines are for coefcient b 0.7, the dotted lines for coefcient b 0.5, and the dashed forcoefcient b 0.3.The stability of the response amplitudes is illustrated in Fig. 9. In the rst-mode response, the coefcients

are b 0.03 and k2 100,000. In the second-mode response, the coefcients are b 0.7 and k2 100,000.The solid lines denote the response amplitudes and the dashed denote the boundary of instability.The inner region of the boundary of instability is unstable and the outer is stable. Thus a jump phenomenonappears.

3.2.3. Subharmonic resonances and responses amplitudes

To study the subharmonic resonances, a detuning parameter s is introduced to quantify the deviation and ois described by

o 3o0 s (69)Solvability requires

wABB kA2A zA2B eisT1 qAqT1

0 (70)

ARTICLE IN PRESSY.-Q. Tang et al. / Journal of Sound and Vibration 320 (2009) 10781099 1095where

w 3 k1o20 k2Z 10

ff02f00 dx 2Z 10

ff0f0f00 dx

6k1k2Z 10

ff002f00 dx

Z 10

ff0f00f000 dxZ 10

ff0f00f000 dxZ 10

ff0f00f000 dx

2k1k2Z 1 0 0 0000

Z 1 02 0000

2

1

0.8

0.6

0.4

0.2

0

a n a n

10-3 10-4

-5 0 5 10

4

3

2

1

0-5 0 5

Fig. 9. Stability of the response amplitudes: (a) the rst mode and (b) the second mode.0

ff f f dx k1k20

ff f dx 2 io01 2k0o0

Z 10

ffdx v1 3k0o20Z 10


Z 10

ff00 dxk1vZ 10

ff000 dx

(71a)

k 12w (71b)

z 32

k1o20 k2Z 10

ff02f00 dx 2

Z 10

ff0f0f00 dx

6k1k2Z 10

ff00f002dx

6k1k2Z 10

ff0f00f000 dxZ 10

ff0f00f000 dxZ 10

ff0f00f000 dx

k1k2Z 10

ff02f0000 dx 2

Z 10

ff0f0f 0000dx

2 io01 2k0o20Z 10

ffdx v1 3k0o20

Z 10

ff0 dxio0k0 k1 k3 k0v2Z 10

ff00 dx k1vZ 10

ff000 dx

(71c)

Consider the transformation

A a1T1;T2 eia2T1;T2; B b1 eib2 (72)


qa1qT1

a31kR a1b21wR a21b1 zR cosb2 sT1 3a2 zI sinb2 sT1 3a2

(73a)

qa2qT1

a21kI b21wI a1b1 zR sinb2 sT1 3a2 zI cosb2 sT1 3a2

(73b)

Then, we change Eq. (73) into an autonomous system

qa1qT1

a31kR a1b21wR a21b1zR cos g zI sin g (74a)

qgqT1

s 3a21kI 3b21wI 3a1b1zR sing zI cos g (74b)

where

g b2 sT1 3a2 (75)For steady-state solutions, the amplitude a1 and the new phase g angle in Eq. (74) should be constant. Settinga01 0 and g0 0 and then eliminating from Eq. (74) leads toq

ARTICLE IN PRESSY.-Q. Tang et al. / Journal of Sound and Vibration 320 (2009) 107810991096s 3a21kI b21wI 3 a21b21zI 2 zR2 a21kR b21wR2 (76)When a1a0 in Eqs. (74), the J-matrix characteristic function of balanced solution on the right hand,

l2 22a21kR b21wRl 3a21kR b21wRa21kR b21wR 3 s

3 a21kI b21wI

s3 a21kI b21wI

0 (77)


a21kR b21wRa21kR b21wR s3 a21kI b21wI

s3 a21kI b21wI

0 (78)

Consider an axially moving beam with k4 0.64 and v 2.0. The response amplitudes at frequencies threetimes those of exact resonance for the rst two natural frequencies are shown in Fig. 10. The rst two natural

1

0.8

0.6

0.4

0.2

0

1

0.8

0.6

0.4

0.2

0

a n a n

10-3 10-3

0 5 10 15 20 25

0 20 40 60 80Fig. 10. Response amplitude diagram: (a) the rst mode and (b) the second mode.

frequencies of the unperturbed system are o1 4.7393 and o2 23.7017. In the rst-mode response, thecoefcients are b 0.03, k2 100,000, and in the second-mode response, the coefcients are b 5,k2 100,000.Fig. 11 shows the effects of a different parameter k2. In the rst-mode response, the coefcient is b 0.03,

and in the second-mode response, the coefcient is b 5. The dotted lines are for coefcient k2 50,000, thedashed for coefcient k2 100,000, and the solid lines for coefcient k2 150,000.The effects of the external excitation amplitudes on the response amplitudes are illustrated in Fig. 12. From

the response diagrams, it is clear that the excitation amplitudes increase the amplitude of the excited system.The coefcient is k2 100,000. In the rst-mode response, the solid lines are for coefcient b 0.1, the dotted

ARTICLE IN PRESS

10-3 10-31

0.8

0.6

a n

1

0.8

0.6

a n

1

0.8

0.6

0.4

0.2

0

a n

1

0.8

0.6

0.4

0.2

0

a n

0 10 20 30 40

0 20 40 60 80 100

10-3 10-3


Y.-Q. Tang et al. / Journal of Sound and Vibration 320 (2009) 10781099 10970.4

0.2

0

0.4

0.2

00 5 10 15 20 25

0 20 40 60 80

Fig. 12. Comparison of response amplitudes of different excitation amplitudes b: (a) the rst mode and (b) the second mode.

ARTICLE IN PRESSY.-Q. Tang et al. / Journal of Sound and Vibration 320 (2009) 107810991098lines for coefcient b 0.06, and the dashed for coefcient b 0.03. In the second-mode response, the solidlines are for coefcient b 5.5, the dotted lines for coefcient b 5, and the dashed for coefcient b 4.5.The stability of the response amplitudes is illustrated in Fig. 13. In the rst-mode response, the coefcients

are b 0.03 and k2 100,000. In the second-mode response, the coefcients are b 5 and k2 100,000. Thesolid lines denote the response amplitudes and the dashed denote the boundary of instability. The inner regionof the boundary of instability is unstable and the outer is stable. Thus a jump phenomenon appears.

4. Conclusions

In this investigation, nonlinear vibrations under weak and strong external excitations of axially moving

10-3 10-31

0.8

0.6

0.4

0.2

0

a n a n

0 5 10 15 20 25

1

0.8

0.6

0.4

0.2

00 20 40 60 80

Fig. 13. Stability of the response amplitudes: (a) the rst mode and (b) the second mode.beam on simple supports are studied based on the Timoshenko beam model. A partial-differential nonlinearequation is derived from Newtons second law. The multiple-scale method is used to discuss nonsyntonicexcitations, superharmonic resonances, and subharmonic resonances. The nontrivial steady-state response andits existence conditions are presented. The system shows a typical muti-valued nonlinear phenomenon. Thenumerical examples investigated reveal the following.The natural frequencies decrease with an increase in the axial speed for simple supports. The rst natural

frequency vanishes at the critical speed and afterwards the system is unstable about the zero equilibrium.The assumption that the primary response with the possible contributions of modes does not involve

resonance is wrong. The modes have only zero stationary solution, which decays to zero exponentially.Therefore, the modes have actually no effect on the stability.For the efforts of different parameters about nonlinear vibrations under weak and strong external

excitations, the response amplitudes increase with decreasing nonlinear effects, and with increasing externalexcitation amplitudes.

Acknowledgements

This work was supported by the National Outstanding Young Scientists Fund of China (Project no.10725209), the National Natural Science Foundation of China (Project no. 10672092), Shanghai MunicipalEducation Commission Scientic Research Project (no. 07ZZ07), Innovation Foundation for Graduates ofShanghai University (Project no. A.16-0101-07-011), and Shanghai Leading Academic Discipline Project(no. Y0103).

References

[1] C.D. Mote Jr., Dynamic stability of an axially moving band, Journal of the Franklin Institute 285 (1968) 329346.

[2] W.Z. Wu, C.D. Mote Jr., Parametric excitation of an axially moving band by periodic edge loading, Journal of Sound Vibration 110

(1986) 2739.

[3] J.A. Wickert, C.D. Mote Jr., Classical vibration analysis of axially moving continua, ASME Journal of Applied Mechanics 57 (1990)

738744.

[4] H.R. Oz, M. Pakdemirli, Vibrations of an axially moving beam with time dependent velocity, Journal of Sound and Vibration 227

(1999) 239257.

[5] H.R. Oz, On the vibrations of an axially traveling beam on xed supports with variable velocity, Journal of Sound and Vibration 239

(2001) 556564.

[6] M.H. Ghayesh, S.E. Khadem, Rotary inertia and temperature effects on non-linear vibration, steady-state response and stability of

an axially moving beam with time-dependent velocity, International Journal of Mechanical Sciences 50 (2008) 389404.

[7] U. Lee, I. Jang, On the boundary conditions for axially moving beams, Journal of Sound and Vibration 306 (2007) 675690.

[8] X.D. Yang, L.Q. Chen, The stability of an axially acceleration beam on simple supports with torsion springs, Acta Mechanica Solida

Sinica 18 (2005) 18.

[9] L.Q. Chen, X.D. Yang, Vibration and stability of an axially moving viscoelastic beam with hybrid supports, European Journal of

Mechanics A/Solids 25 (2006) 9961008.

[10] U. Lee, J. Kim, H. Oh, Spectral analysis for the transverse vibration of an axially moving Timoshenko beam, Journal of Sound and

Vibration 271 (2004) 685703.

[11] G. Chakraborty, A.K. Mallik, Wave propagation in an vibration of a traveling beam with and without non-linear effects, part II:

ARTICLE IN PRESSY.-Q. Tang et al. / Journal of Sound and Vibration 320 (2009) 10781099 1099forced vibration, Journal of Sound and Vibration 236 (2) (2000) 291305.

[12] L. Zhang, J.W. Zu, Non-linear vibrations of viscoelastic moving belts, part II: forced vibration analysis, Journal of Sound and

Vibration 216 (1) (1998) 93105.

[13] F. Pellicano, F. Vestroni, Complex dynamics of high-speed axially moving systems, Journal of Sound and Vibration 258 (1) (2002)

3144.

[14] X.D. Yang, L.Q. Chen, Non-linear forced vibration of axially moving viscoelastic beams, Acta Mechanica Solida Sinica 19 (2006) 19.

[15] A. Simpson, Transverse modes and frequencies of beams translating between xed end supports, Journal of Mechanical Engineering

Science 15 (1973) 159164.

[16] S. Chonan, Steady state response of an axially moving strip subjected to a stationary lateral load, Journal of Sound and Vibration 107

(1) (1986) 155165.

[17] L.G. Arboleda-Monsalve, D.G. Zapata-Medina, J.D. Aristizabal-Ochoa, Timoshenko beam-column with generalized end conditions

on elastic foundation: dynamic-stiffness matrix and load vector, Journal of Sound and Vibration 310 (2008) 10571079.

[18] C. Mei, Y. Karpenko, S. Moody, D. Allen, Analytical approach to free and forced vibrations of axially loaded cracked Timoshenko

beams, Journal of Sound and Vibration 291 (2006) 10411060.

[19] A.H. Nayfeh, Introduction to Perturbation Techniques, Wiley, New York, 1982.

Nonlinear vibrations of axially moving Timoshenko beams under weak and strong external excitationsIntroductionThe governing equationThe multi-scale analysisNonlinear vibrations under external excitationsNonlinear vibrations under strong external excitationsNonsyntonic excitationsSuperharmonic resonances and response amplitudesSubharmonic resonances and responses amplitudes

ConclusionsAcknowledgementsReferences

Documents

Nonlinear vibrations of axially moving Timoshenko beams under weak and strong external excitations