Two-dimensional nonlinear dynamics of an axially moving viscoelastic beam with time-dependent axial speed

Chaos, Solitons & Fractals 52 (2013) 8–29

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Chaos, Solitons & Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos

Two-dimensional nonlinear dynamics of an axially moving viscoelastic beam with time-dependent axial speed Mergen H. Ghayesh ⇑, Marco Amabili, Hamed Farokhi Department of Mechanical Engineering, McGill University, Montreal, Quebec, Canada H3A 0C3

a r t i c l e

i n f o

Article history: Received 6 September 2012 Accepted 16 March 2013

a b s t r a c t In the present study, the coupled nonlinear dynamics of an axially moving viscoelastic beam with time-dependent axial speed is investigated employing a numerical technique. The equations of motion for both the transverse and longitudinal motions are obtained using Newton’s second law of motion and the constitutive relations. A two-parameter rheological model of the Kelvin–Voigt energy dissipation mechanism is employed in the modelling of the viscoelastic beam material, in which the material time derivative is used in the viscoelastic constitutive relation. The Galerkin method is then applied to the coupled nonlinear equations, which are in the form of partial differential equations, resulting in a set of nonlinear ordinary differential equations (ODEs) with time-dependent coefficients due to the axial acceleration. A change of variables is then introduced to this set of ODEs to transform them into a set of first-order ordinary differential equations. A variable step-size modified Rosenbrock method is used to conduct direct time integration upon this new set of first-order nonlinear ODEs. The mean axial speed and the amplitude of the speed variations, which are taken as bifurcation parameters, are varied, resulting in the bifurcation diagrams of Poincaré maps of the system. The dynamical characteristics of the system are examined more precisely via plotting time histories, phase-plane portraits, Poincaré sections, and fast Fourier transforms (FFTs). Ó 2013 Elsevier Ltd. All rights reserved.

1. Introduction 1.1. Applications and categories Axially moving systems, (i.e. strings, beams, and plates), are present in a wide class of mechanical and industrial applications. They are extensively used in many engineering devices and machine components such as in conveyor belts, textile fibers, band saw blades, robotic manipulators, paper sheet processes, and aerial cable tramways. There has been a large amount of research on these systems because of their widespread applications. Due to the simultaneous presence of the axial speed and the rotation of the

⇑ Corresponding author. Tel.: +1 514 398 6290. E-mail address: [email protected] (M.H. Ghayesh). 0960-0779/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.chaos.2013.03.005

cross-section of the structure, a Coriolis force is generated which makes the system gyroscopic. The presence of the gyroscopic term makes the classical normal modes vanish and it does generate travelling wave components along the structure length, which is the most interesting characteristic of this class of systems. The literature on the axially moving beams and strings with either constant or time-dependent axial speed can be mainly categorized into two classes. In the first class, the equation of motion consists of only one partial differential equation which is obtained by either neglecting the longitudinal displacement or making an assumption on the longitudinal motion and reducing the two partial differential equations into one. In the second class of analysis, both longitudinal and transverse displacements are taken into account resulting in two coupled partial differential equations of motion.

M.H. Ghayesh et al. / Chaos, Solitons & Fractals 52 (2013) 8–29

9

⎯⎯⎯→

Fig. 1. Schematic representation of an axially moving viscoelastic beam with time-dependent axial speed.

(a)

(b)

(c)

(d)

Fig. 2. Bifurcation diagrams of Poincaré points for increasing mean axial speed of the system with c1 = 0.1; (a, b) the first two generalized coordinates of the transverse motion; (c, d) the first two generalized coordinates of the longitudinal motion; the critical value of the axial speed of this case is equal to 1.139 (i.e. (c0)cr = 1.139).

1.2. The literature review on the first class of analysis (i.e. the analyses concerned with only the transverse vibrations) The dynamics of this class of systems has been studied extensively in the literature. In these studies, either the classical structural models are employed or the viscoelastic ones such as the Kelvin–Voigt and Maxwell energy dissipation mechanisms are considered. The earlier studies, carried out by Mote [1] and Wickert [2], examined the

nonlinear vibrations of axially moving strings and beams. Huang et al. [3] analyzed the nonlinear vibrations of an axially moving beam subject to a periodic transverse force excitation. These investigations were continued by Marynowski and Kapitaniak [4], who employed the Zener internal damping in the modelling of an axially moving beam with time-dependent tension. Pellicano and Vestroni [5–8] investigated the nonlinear dynamics and bifurcations of an axially moving beam in the sub and supercriti-

10

M.H. Ghayesh et al. / Chaos, Solitons & Fractals 52 (2013) 8–29

(a)

(b) 0.002

q2

q1

0.01

0

0

-0.01 -0.002

2900

2910

2920

2930

t

2940

2900

2950

(c)

2910

2920

t

2930

2940

2950

(d)

0.01 0.001

dq2/dt

dq1/dt

0.005

0

0

-0.005 -0.001

-0.01

-0.015

-0.01

-0.005

0

q1

0.005

0.01

0.015

(e)

-0.002

-0.001

0

q2

0.001

0.002

(f) 0.0012

0.012

0.0008

FFT[q1]

FFT[q2]

0.008

0.0004

0.004

0

0

0.5

1

ω/Ω

1.5

2

0

0

0.5

1

ω/Ω

1.5

2

Fig. 3. Period-2 motion for the system of Fig. 2 at c0 = 1.2: (a, b) time traces of the q1 and q2 motions, respectively; (c, d) phase-plane diagrams of the q1 and q2 motions, respectively; (e, f) FFTs of the q1 and q2 motions, respectively; x and X are the Fourier and external frequencies, respectively.

cal axial speed regimes. Pakdemirli and co-workers [9–12] contributed to the field by conducting a series of research on the vibration of axially moving systems with timedependent velocities, using the method of multiple time-

scales and matched asymptotic expansion. Sze et al. [13] employed the incremental harmonic balance in order to examine the nonlinear vibrations of an axially moving beam. Xu and Zhu [14] investigated the nonlinear dynam-

11

M.H. Ghayesh et al. / Chaos, Solitons & Fractals 52 (2013) 8–29

(a)

(b)

0.008

0.02

q2

q1

0.004

0

0

-0.004

-0.02 -0.008

2940

(c)

2960

2980

t

2940

3000

(d)

0.04

3000

0.01

dq2/dt

dq1/dt

2980

t

0.02

0.02

0

-0.02

-0.04

2960

0

-0.01

-0.02

0

-0.02

0.02

q1

(e)

-0.005

0

0.005

q2

(f) 0.0002

0

p2

p1

0.0001

0

-0.0001 -0.0001

-0.0002

-0.0002 2940

2960

t

2980

3000

2900

2910

t

2920

2930

Fig. 4. Chaotic motion for the system of Fig. 2 at c0 = 1.34: (a, b) time traces of the q1 and q2 motions, respectively; (c, d) phase-plane diagrams of the q1 and q2 motions, respectively; (e, f) time traces of the p1 and p2 motions, respectively.

ics of high-dimensional models of a translating beam. Suweken and Van Horssen [15,16] examined the sringand beam-like vibrations of a conveyor belt with a low and time-varying velocity, employing the method of multiple scales. Chakraborty and Mallik [17] employed the same analytical method so as to investigate the nonlinear dynamics of an axially moving beam with constant axial acceleration. Chen and co-workers [18–21] used different

analytical and numerical techniques along with different beam models to investigate the dynamics of axially moving beams. Ghayesh and co-investigators [22–29] pursued and extended these studies by investigating the nonlinear dynamics of an axially moving beam with constant and time-dependent axial speeds, employing both elastic and viscoelastic structural models, and employing both analytical and numerical techniques.

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M.H. Ghayesh et al. / Chaos, Solitons & Fractals 52 (2013) 8–29

(a)

(b) 0.002

q1

q2

0.025

0

0.02

-0.002

0.015 2900

(c)

2910

2920

t

2900

2930

2910

t

2920

2930

(d)

0.008

0.004

dq2/dt

dq1/dt

0.004

0

0

-0.004

-0.004

-0.008

0.018

0.02

0.022

q1

0.024

0.026

-0.002

(f)

(e)

0

q2

0.002

0.004

dq2/dt

dq1/dt

-0.004

0

-0.008

-0.024

q1

-0.022

-0.02

-0.004

0

0.002

q2

0.004

Fig. 5. Periodic motion for the system of Fig. 2 at c0 = 1.6: (a, b) time traces of the q1 and q2 motions, respectively; (c, d) phase-plane diagrams of the q1 and q2 motions, respectively; (e, f) Poincaré sections of the q1 and q2 motions, respectively; (g, h) time traces of the p1 and p2 motions, respectively; (i, j) Poincaré sections of the p1 and p2 motions, respectively.

1.3. The literature review on the second class of analysis (i.e. the analyses concerned with the coupled longitudinal– transverse dynamics) The literature regarding the coupled dynamics of axially moving systems (for both cases, with either a constant or

time-dependent axial speed), on the other hand, is rather limited. Riedel and Tan [30] investigated the coupled forced response of a nonlinear axially moving strip with a three-to-one internal resonance between the first two transverse modes. The coupled equations of motion for axially accelerated viscoelastic beams were derived in

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M.H. Ghayesh et al. / Chaos, Solitons & Fractals 52 (2013) 8–29

(g)

(h)

0.0002

-0.0001

0.0001

0

p2

p1

-0.00015

-0.0002

-0.0001

-0.00025

-0.0002 2900

2910

2920

2900

2930

2910

(i)

2920

2930

t

t

(j)

0.0002

0

dp2/dt

dp1/dt

0.0001

0

-0.0001

-0.0002

-0.0002

0.00012

0.00016

0.0002

-0.0002

-0.00018

-0.00016

p2

p1 Fig. 5. (continued)

[31,32]; however, the simplified version, in which only the transverse motion was considered, was used to investigate the nonlinear dynamics of the system. Ghayesh and coworkers [33–35] investigated the coupled nonlinear dynamics of an axially moving beam with constant and time-dependent axial speed.

structural models [34] and the viscoelastic one has not been investigated yet. This work continues and extends the work of the first author in [34] by introducing a viscoelastic structural model to the beam material.

1.4. The contribution of the current study to the field

The schematic representation of the system is depicted in Fig. 1. This figure shows an axially moving viscoelastic beam of length L, axial stiffness EA, and flexural rigidity EI. The beam is subject to a constant pretension p, moving at a time-dependent axial speed v(t), and hinged at both ends. The mass density is denoted by q and the viscosity coefficient is represented by g. The coupled nonlinear equations of motion for an axially accelerated Kelvin–Voigt viscoelastic beam are derived employing the force and moment balances under the following assumptions: (i) shear deformation and rotary inertia are neglected, i.e. the Euler–Bernoulli beam model is used; (ii) the material, not partial, time derivative is employed in the constitutive relation which produces axial speed-dependent energy dissipation terms in the govern-

In many applications, the axial speed is not constant due to various internal or external defects and geometrical or dynamical imperfections. As a result, an axial acceleration is generated which can cause unwanted vibrational resonances. On the other hand, the viscoelastic structural models of the system, which takes into account the internal energy dissipation mechanisms, can give more accurate results compared to the elastic structural models. It should be noted that the Kelvin–Voigt model provides a more complex model compared to the viscous damping model, since it changes with respect to both axial coordinate and axial speed. The literature on the coupled nonlinear dynamics of axially accelerated beams is limited only to the elastic

2. Equations of motion

14

M.H. Ghayesh et al. / Chaos, Solitons & Fractals 52 (2013) 8–29

(a)

(b)

(c)

(d)

Fig. 6. Bifurcation diagrams of Poincaré points for increasing mean axial speed of the system with c1 = 0.2; (a, b) the first two generalized coordinates of the transverse motion; (c, d) the first two generalized coordinates of longitudinal motion; the critical value of the axial speed of this case is equal to 1.139 (i.e. (c0)cr = 1.139).

ing equations; (iii) the type of nonlinearity is geometric, due to the stretching effect of the mid-plane of the beam; (iv) the boundary conditions are immovable. A two-parameter viscoelastic rheological model of the Kelvin–Voigt type is used in the modelling of the system as follows. As such the following constitutive relation holds:

c1 rx ¼ c2 ex þ c3



 @ ex @ ex þv ; @t @x

ð1Þ

in which rx and ex are defined as axial stress and strain at an arbitrary point of the cross-section of the beam and c1 = 1, c2 = E, and c3 = g. Moreover, the axial strain of the mid-plane of the beam is in the form

e¼

 2 @u 1 @w þ : @x 2 @x

ð2Þ

The following bending moment and axial force are obtained employing the force–stress and moment–stress relations along with taking into account Eqs. (1) and (2):

@2w @3w @3w þ gI 2 þ gv I 3 ; @x2 @x @t @x !  2 ! @u 1 @w @ 2 u @w @ 2 w þ þ þ gA N ¼ p þ EA @x 2 @x @x@t @x @x@t ! @ 2 u @w @ 2 w þ gv A : þ @x2 @x @x2

M ¼ EI

ð3Þ

ð4Þ

As Newton’s second law of motion is applied to an infinitesimal element of the beam, the coupled nonlinear equations of motion for the longitudinal and transverse motions are obtained as !   2 @ 2 u dv @u @2u 2@ u þ 2v þv qA þ 1þ @x @x@t @x2 @t2 dt ! ! @ 2 u @w @ 2 w @3u @ 2 w @ 2 w @w @ 3 w þ þ þ gA ¼ EA þ @x2 @x @x2 @x2 @t @x2 @x@t @x @x2 @t 0 1 !2 @3u @2w @w @ 3 wA þ þ gv A@ 3 þ ; ð5Þ 2 @x @x @x @x3

15

M.H. Ghayesh et al. / Chaos, Solitons & Fractals 52 (2013) 8–29

(a)

(b) 0.02

q2

q1

0.002

0

0

-0.002

-0.02

2960

(c)

2980

t

2960

3000

(d)

0.03

2980

t

3000

0.008

0.02 0.004

dq2/dt

dq1/dt

0.01

0

0

-0.01 -0.004

-0.02

-0.03

-0.01

0

q1

-0.008

0.01

(f)

0

-0.0001

2960

t

2980

3000

-0.001

0

0.001

q2

0.002

0.0001

p2

p1

(e) 0.0001

-0.002

0

-0.0001

2960

t

2980

3000

Fig. 7. Period-2 oscillation for the system of Fig. 6 at c0 = 1.0: (a, b) time traces of the q1 and q2 motions, respectively; (c, d) phase-plane diagrams of the q1 and q2 motions, respectively; (e, f) time traces of the p1 and p2 motions, respectively.

16

M.H. Ghayesh et al. / Chaos, Solitons & Fractals 52 (2013) 8–29

(a)

(b)

0.01

0.02

q2

q1

0.005

0

0

-0.005

-0.02

2960

2980

-0.01

3000

2960

2980

t

3000

t

(c)

(d) 0.02

dq2/dt

dq1/dt

0.05

0

-0.05

0

-0.02

-0.04

-0.02

0

0.02

0.04

-0.01

-0.005

q1

0

0.005

0.01

0.005

0.01

q2

(e)

(f) 0.04

0.02

0.01

dq2/dt

dq1/dt

0.02

0

0

-0.01

-0.02

-0.02 -0.04 -0.02

-0.01

0

q1

0.01

0.02

-0.01

-0.005

0

q2

Fig. 8. Chaotic motion for the system of Fig. 6 at c0 = 1.35: (a, b) time traces of the q1 and q2 motions, respectively; (c, d) phase-plane diagrams of the q1 and q2 motions, respectively; (e, f) Poincaré sections of the q1 and q2 motions, respectively.

M.H. Ghayesh et al. / Chaos, Solitons & Fractals 52 (2013) 8–29

(a)

(b)

(c)

(d)

17

Fig. 9. Bifurcation diagrams of Poincaré points for increasing mean axial speed of the system with c1 = 0.3; (a, b) the first two generalized coordinates of the transverse motion; (c, d) the first two generalized coordinates of the longitudinal motion; the critical value of the axial speed of this case is equal to 1.139 (i.e. (c0)cr = 1.139).

! 2 @ 2 w dv @w @2w @4w @5w 2@ w þ 2v þv qA þ þ EI 4 þ gI 4 2 2 @x@t @x @x @x @t dt @x @t !  2 @5w @2w @u @ 2 w @ 2 u @w 3 @w @ 2 w þ þ þ gv I 5  p 2 ¼ EA @x @x @x @x2 @x2 @x 2 @x @x2  2 3 ! 2 2 3 2 2 @ u @ w @ u @w @w @ w @ w @w @ w þ2 þ þ þ gA @x@t @x2 @x2 @t @x @x @x2 @x@t @x @x2 @t 0 1 !2   2 3 2 2 3 2 @ u @ w @ u @w @w @ w @w @ w A: þ2 þ þ þ gv A@ 2 @x @x2 @x3 @x @x @x2 @x @x3 ð6Þ

Introducing the following dimensionless parameters:

sffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffi u w x p qA    ; u ¼ ; w ¼ ; x ¼ ; s¼t ; v ¼v L L L p qAL2 sffiffiffiffiffiffi sffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffi rffiffiffiffiffiffiffi EI EA gI p gA p ; v1 ¼ ; f¼ ; vf ¼ ; c¼ 3 2 p pL qA qA pL pL ð7Þ 

substituting them into Eqs. (5) and (6), and neglecting the asterisk notation for brevity, the following dimensionless

coupled nonlinear equations are obtained for the longitudinal and transverse motions, respectively:   @ 2 u dv @u @2u @2u þ 2v þ 1 þ þ v2 2 @ s2 ds @x @x@ s @x ! ! 2 2 3 @w @ w @ u @ 2 w @ 2 w @w @ 3 w 2 @ u ¼ v1 þ þ þ þf @x2 @x @x2 @x2 @ s @x2 @x@ s @x @x2 @ s 0 1 !2 @3u @2w @w @ 3 wA þ þ fv @ 3 þ ; ð8Þ 2 @x @x @x @x3  @2 w @ 2 w dv @w @2 w  2 @4 w @5 w @5w þ 2v þ þ v 1 þ v 2f þc þv 5 @ s2 ds @x @x@ s @x2 @x4 @x4 @ s @x  2 2 ! 2 2 @u @ w @ u @w 3 @w @ w þ þ ¼ v 21 @x @x2 @x2 @x 2 @x @x2  2 3 ! 2 2 3 @ u @ w @ u @w @w @ 2 w @ 2 w @w @ w þ 2 þ þ þf @x@ s @x2 @x2 @ s @x @x @x2 @x@ s @x @x2 @ s 0 1 !2   2 @ 2 u @ 2 w @ 3 u @w @w @ 2 w @w @ 3 wA @ : þ2 þ þ þ fv @x2 @x2 @x3 @x @x @x2 @x @x3

!

ð9Þ

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M.H. Ghayesh et al. / Chaos, Solitons & Fractals 52 (2013) 8–29

(a)

(b)

(c)

(d)

Fig. 10. Bifurcation diagrams of Poincaré points for increasing mean axial speed of the system with c1 = 0.4; (a, b) the first two generalized coordinates of the transverse motion; (c, d) the first two generalized coordinates of the longitudinal motion; the critical value of the axial speed of this case is equal to 1.139 (i.e. (c0)cr = 1.139).

The following boundary conditions for a hinged–hinged beam with immovable edges are used

ujx¼0 ¼ ujx¼1 ¼ 0; wjx¼0 ¼ wjx¼1 ¼ 0;

ð10Þ  @ 2 w  @x2 

¼ x¼0

 @ 2 w  @x2 

¼ 0:

ð11Þ

numerically, the Galerkin method is applied to the equations of motion to discretize them into a set of nonlinear ordinary differential equations. Hence, the eigenfunctions of a hinged–hinged linear stationary beam are chosen as the basis functions for the following approximate series expansion:

x¼1

M X /r ðxÞqr ðsÞ;

The underlined terms in Eqs. (8) and (9) are related to the steady viscous effect, which are due to the simultaneous presence of the axial speed and material viscosity and were usually neglected in the literature. If the underlined terms are omitted, the equations of [31] are recovered.

wðx; sÞ ¼

3. Method of solution

where /r(x) is the rth dimensionless eigenfunction for the transverse motion of a hinged–hinged linear stationary beam, and qr(s) and pr(s) denote the rth generalized coordinates in the transverse and longitudinal directions, respectively. The axial speed is also assumed to be composed of a constant mean value along with harmonic fluctuations as follows:

The dimensionless equations of the motion of the system (Eqs. (8) and (9)), which are in the form of partial differential equations, represent a continuous system with infinite number of degrees of freedom. In order to be able to examine the dynamical characteristics of the system

ð12Þ

r¼1

uðx; sÞ ¼

N X

/r ðxÞpr ðsÞ;

ð13Þ

r¼1

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M.H. Ghayesh et al. / Chaos, Solitons & Fractals 52 (2013) 8–29

(a)

(b) 0.005

q2

q1

0.02

0

0

-0.005

-0.02

2970

2980

2990

3000

2970

2980

t

(c)

(d)

3000

0.04

0.02

dq2/dt

dq1/dt

0.05

0

0

-0.02

-0.05

-0.02

-0.01

0

0.01

0.02

-0.04

-0.005

0

q1

0.005

q2

(f)-0.012

0.032

-0.0125

dq2/dt

dq1/dt

(e) 0.036

0.028

0.024

2990

t

-0.013

0

0.002

0.004

0.006

0.008

-0.0135

-0.005

-0.004

-0.003

-0.002

q2

q1

Fig. 11. Period-2 oscillation for the system of Fig. 10 at c0 = 1.0: (a, b) time traces of the q1 and q2 motions, respectively; (c, d) phase-plane diagrams of the q1 and q2 motions, respectively; (e, f) Poincaré sections of the q1 and q2 motions, respectively.

v ðsÞ ¼ c0 þ c1 sinðXsÞ;

ð14Þ

in which c0, c1, and X denote the mean value of the axial speed, the amplitude of the speed variations, and the frequency of the speed variations, respectively.

Substituting Eqs. (12)–(14) into Eqs. (8) and (9), multiplying the resultant equations by the appropriate eigenfunctions, and integrating with respect to x from 0 to 1, yields the following M + N coupled second-order nonlinear ordinary differential equations with time-dependent coefficients:

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M.H. Ghayesh et al. / Chaos, Solitons & Fractals 52 (2013) 8–29

(a)

(b) 0.04

q2

q1

0.01

0

0

-0.01

-0.04

2960

(c)

2970

2980

t

2990

(d)

0.2

0.1

2970

2980

2990

3000

0

0.01

0.02

t

0.1

0.05

dq2/dt

dq1/dt

2960

3000

0

0

-0.05

-0.1

-0.1 -0.2

-0.02

0

q1

0.02

0.04

-0.02

(f) 0.006

0.005

0.003

dp2/dt

0.01

dp1/dt

(e)

-0.04

0

q2

0

-0.003

-0.005

-0.01

-0.01

-0.001

0

p1

0.001

-0.006

-0.0008

-0.0004

p2

0

Fig. 12. Chaotic motion for the system of Fig. 10 at c0 = 1.75: (a, b) time traces of the q1 and q2 motions, respectively; (c, d) phase-plane diagrams of the q1 and q2 motions, respectively; (e, f) phase-plane diagrams of the p1 and p2 motions, respectively.

21

M.H. Ghayesh et al. / Chaos, Solitons & Fractals 52 (2013) 8–29

(a)

(b)

(c)

(d)

Fig. 13. Bifurcation diagrams of Poincaré points for increasing amplitude of the axial speed variations of the system with c0 = 1.0; (a, b) the first two generalized coordinates of the transverse motion; (c, d) the first two generalized coordinates of the longitudinal motion; the critical value of the axial speed of this case is equal to 1.139 (i.e. (c0)cr = 1.139). N Z X

1

 Z €j þ c1 X cosðXsÞ /i /j dx p

0

j¼1

j¼1

" f

N Z X 0

j¼1

N Z X

1

0

j¼1

M X M Z X þ

"

N Z X j¼1

1

0

/i /0j dx



j¼1

1

0

j¼1 k¼1

0

j¼1 1

0

j¼1 k¼1



1

0

j¼1 k¼1

#

 ! pj

j¼1 k¼1

0

1

M Z X

1

 M Z X €j þ c1 X cosðXsÞ /i /j dx q

0

j¼1

p_ j

 M X M Z X /i /00j dx pj þ

 M X M Z X /i /00j dx p_ j þ

0

/i /0j dx

0

1

/i /00j dx



j¼1

pj

þ 2c0 c1 sinðXsÞ  1Þ

#  /i /0j /00k dx qj qk

þ v 2f

 /i /00j /0k dx qj q_ k

M Z 1 X j¼1

0

j¼1 k¼1

ð15Þ

þ

0

 /i /00j dx qj

" Z  M X 0000 /i /j dx qj þ c j¼1

1

0

 0000 /i /j dx q_ j

"  # N X M Z X 00000 2 /i /j dx qj  v 1 j¼1 k¼1

N X M Z X þ

 /i /00j /00k dx qj qk

#  0 000 /i /j /k dx qj qk ; i ¼ 1; 2; . . . ; N;

0

j¼1

þv

1

0

 /i /0j dx q_ j þ ðc20 þ c21 ðsinðXsÞÞ2

M Z 1 X j¼1

M Z X

1

 /i /0j dx qj

1

0

j¼1

M Z X þ 2ðc0 þ c1 sinðXsÞÞ

2

 M X M Z X /i /000 dx p þ j j

M X M Z 1 X

þ

1

1

/i /0j /00k dx qj q_ k  fðc0 þ c1 sinðXsÞÞ

0

j¼1 k¼1



1

N Z X XsÞÞ þ 2c0 c1 sinðXsÞÞ

þ c21 ðsinð

"  v 21

/i dx þ

N Z X

0

N Z X þ 2ðc0 þ c1 sinðXsÞÞ

þ ðc20

1

0

 1 /i /00j /0k dx pj qk

M X M X M Z 3X

2 j¼1 k¼1 l¼1

0

1

#  /i /0j /0k /00l dx qj qk ql

0

1

 /i /0j /00k dx pj qk

22

M.H. Ghayesh et al. / Chaos, Solitons & Fractals 52 (2013) 8–29

(a)

(b) 0.02

q2

q1

0.002

0

0

-0.002 -0.02 2970

2980

2990

t

3000

(c)

2970

2980

2990

t

3000

(d) 0.005

dq2/dt

dq1/dt

0.02

0

0

-0.005

-0.02

-0.01

0

q1

-0.002

0.01

(f) 0.0018

0.008

0.0012

0.002

FFT[q2]

FFT[q1]

(e) 0.012

0

q2

0.004

0

0.0006

0

0.5

1

ω/Ω

1.5

2

0

0

0.5

1

ω/Ω

1.5

2

Fig. 14. Period-2 oscillation for the system of Fig. 13 at c1 = 0.24: (a, b) time traces of the q1 and q2 motions, respectively; (c, d) phase-plane diagrams of the q1 and q2 motions, respectively; (e, f) FFTs of the q1 and q2 motions, respectively; x and X are the Fourier and external frequencies, respectively.

M.H. Ghayesh et al. / Chaos, Solitons & Fractals 52 (2013) 8–29

(a)

23

(b)

(c)

(d)

Fig. 15. Bifurcation diagrams of Poincaré points for increasing amplitude of the axial speed variations of the system with c0 = 1.2; (a, b) the first two generalized coordinates of the transverse motion; (c, d) the first two generalized coordinates of the longitudinal motion; the critical value of the axial speed of this case is equal to 1.139 (i.e. (c0)cr = 1.139).

" N X M Z X f 0

j¼1 k¼1

þ2

1

 N X M Z X /i /0j /00k dx p_ j qk þ j¼1 k¼1

M X M X M Z X 0

j¼1 k¼1 l¼1 M X M X M Z X þ 0

j¼1 k¼1 l¼1

1

#  1 /i /0j /0k /00l dx qj qk q_ l " N X M Z X j¼1 k¼1

j¼1 k¼1

qj qk ql þ

0

1

0

1

 /i /00j /00k dx pj qk

 M X M X M Z X 0 /i /000 / dx p q þ 2 j k j k j¼1 k¼1 l¼1

M X M X M Z 1 X j¼1 k¼1 l¼1

i ¼ 1;2; ...;M;

0

 /i /0j /0k /00l dx qj q_ k ql

 fðc0 þ c1 sinðXsÞÞ N X M Z X þ

 /i /00j /0k dx p_ j qk

1

0

0

1

 /i /0j /00k /00l dx

#  0 0 000 /i /j /k /l dx qj qk ql ;

ð16Þ

where the dot and prime superscripts denote differentiations with respect to the dimensionless time and axial coordinate, respectively.

Introducing the following change of variables

yi ¼ q_ i

ði ¼ 1; 2; . . . ; MÞ;

ð17Þ

xi ¼ p_ i

ði ¼ 1; 2; . . . ; NÞ;

ð18Þ

and substituting them into Eqs. (15) and (16) results in a set of first-order nonlinear ordinary differential equations with coupled terms and time-dependent coefficients. This set of 2(M + N) first-order nonlinear ODEs is solved numerically, employing direct time integration via the variable step-size modified Rosenbrock method. In the present study, M = N = 6 are considered, resulting in 24 coupled first-order nonlinear ordinary differential equations with time-dependent coefficients. 4. The nonlinear global dynamics of the system In this section, the nonlinear global dynamics of the system is investigated numerically by choosing either the mean axial speed or the amplitude of the speed variations

24

M.H. Ghayesh et al. / Chaos, Solitons & Fractals 52 (2013) 8–29

(b)

(a)

0.005

q2

q1

0.02

0

0

-0.005

-0.02

2920

2940

2960

t

2980

2920

3000

2940

2960

2980

t

3000

(d)

(c)

0.02

0.05

dq2/dt

dq1/dt

0.01

0

0

-0.01

-0.05

-0.02

-0.02

0.02

-0.005

(f)0.0006

0.0012

0.0004

FFT[q2]

0.0018

FFT[q1]

(e)

0

q1

0.0006

0.0002

0

0

0.5

1

1.5

ω/Ω

2

2.5

3

0

0

0.5

0

0.005

q2

1

1.5

ω/Ω

2

2.5

3

Fig. 16. Chaotic motion for the system of Fig. 15 at c1 = 0.228: (a, b) time traces of the q1 and q2 motions, respectively; (c, d) phase-plane diagrams of the q1 and q2 motions, respectively; (e, f) FFTs of the q1 and q2 motions, respectively; x and X are the Fourier and external frequencies, respectively.

as the bifurcation parameter. The discretized equations are then directly integrated via a variable step-size modified Rosenbrock technique, resulting in generalized coordinates of the system as functions of time. The computer codes were run for a time interval of [0, 3000] dimensionless sec-

onds and only the last 30% of time traces were retained so as to avoid any possible transient effects. The bifurcation diagrams of the system are constructed by sectioning the phase-space in every period of the speed variations. In order to continue a certain attractor as the bifurcation

M.H. Ghayesh et al. / Chaos, Solitons & Fractals 52 (2013) 8–29

(a)

(b)

(c)

(d)

25

Fig. 17. Bifurcation diagrams of Poincaré points for increasing amplitude of the axial speed variations of the system with c0 = 1.5; (a, b) the first two generalized coordinates of the transverse motion; (c, d) the first two generalized coordinates of the longitudinal motion; the critical value of the axial speed of this case is equal to 1.139 (i.e. (c0)cr = 1.139).

parameter, i.e. the mean axial speed or the amplitude of the speed variations, is varied, the final condition of each step is taken as the initial condition of the next step. In this section, it is meant that response and amplitude are with respect to the q1 motion and the amplitude of the q1 motion where it is sectioned, respectively. 4.1. The mean axial speed as the bifurcation parameter In this section, the mean value of the axial speed, c0, is chosen as the bifurcation parameter. The bifurcation diagrams of the system are plotted for four different values of the amplitude of the axial speed variations, namely c1 = 0.1, 0.2, 0.3, and 0.4, which are shown in Figs. 2, 6, 9 and 10, respectively. The following system parameters are selected for these figures: v1 = 33.5261, vf = 0.1732, X = 1.2, c = 0.00000749, and f = 10. In Fig. 2(a) with c1 = 0.1, corresponding to the first generalized coordinate of the transverse motion, as the mean

axial speed is increased, the system stays at the trivial equilibrium configuration until c0 = 1.02 is reached, where the first bifurcation occurs and the system jumps to a nontrivial attractor. In the mean axial speed range of [1.02, 1.14], the system displays a simple periodic motion, continued by a period-3 motion at c0 = 1.14 and 1.15. The system displays a period-2 motion in the interval [1.18, 1.25]. Typical characteristics of a period-2 motion at c0 = 1.20 are illustrated in Fig. 3 through (a, b) the time histories of the q1 and q2 motions, (c, d) the phase-plane portraits of the q1 and q2 motions, (e, f) the fast Fourier transforms of q1 and q2 motions, respectively. By further incrementing the mean axial speed, a chaotic motion occurs at c0 = 1.26; the system exhibits this type of motion until c0 = 1.37 is reached, where the motion becomes periodic once again. This type of motion is displayed by the system thereafter until c0 = 2.00 is hit. Figs. 4 and 5 illustrate the characteristics of chaotic and periodic motions at c0 = 1.34 and 1.60, respectively.

26

M.H. Ghayesh et al. / Chaos, Solitons & Fractals 52 (2013) 8–29

(a)

(c)

(b)

(d)

Fig. 18. Bifurcation diagrams of Poincaré points for decreasing mean axial speed of the system with c1 = 0.2; (a, b) the first two generalized coordinates of the transverse motion; (c, d) the first two generalized coordinates of the longitudinal motion; the critical value of the axial speed of this case is equal to 1.139 (i.e. (c0)cr = 1.139).

Fig. 6 shows the bifurcation diagrams of Poincaré maps for the system with c0 = 0.2. The amplitude of the speed variations is a little higher than that of the system of Fig. 2 where c0 = 0.1. Due to increased amplitude of the speed variations, the first bifurcation occurs at a lower mean axial speed, namely at c0 = 0.89 (compared to the previous case). A period-2 motion is observed afterwards in the mean axial speed range of [0.90, 1.13] followed by a period6 motion at c0 = 1.14, 1.15, and 1.16. Typical characteristics of a period-2 motion are illustrated in Fig. 7 for c0 = 1.0 through (a, b) the time histories of the q1 and q2 motions, (c, d) the phase-plane portraits of the q1 and q2 motions, and (e, f) the time histories of the p1 and p2 motions, showing a symmetry breaking motion for the p2 generalized coordinate. As the mean axial speed is increased further, a period-4 motion dominates until c0 = 1.24 is hit, where the motion becomes chaotic. The system displays a chaotic motion in the interval [1.24, 1.47]; it is worthwhile noting that, compared to the previous case (Fig. 2), the onset of chaotic motion occurs earlier, namely at c0 = 1.24. The system exhibits a simple periodic motion thereafter which

lasts until c0 = 2.00. The dynamical characteristics of a chaotic attractor at c0 = 1.35 is shown in Fig. 8. As the amplitude of the axial speed variations is increased to c1 = 0.3, from c1 = 0.2 in Fig. 6, a new bifurcation diagram is generated (Fig. 9). In this case, the first bifurcation occurs at c0 = 0.84, which is less than that of the system of Fig. 6. A dominant period-2 motion is observed in the mean axial speed range of [0.84, 1.01]. Onset of chaos is predicted at c0 = 1.02; this mean axial speed is less than that of the system of Fig. 6. The system displays a chaotic motion dominantly in the mean axial speed range of [1.02, 1.19]. Two periodic attractor coexist in the interval [1.20, 1.42], where the system jumps repeatedly between one to the other. After a range of period-2 and chaotic motions, the system regains the original period at c0 = 2.00 and maintains that period until c0 = 2.00 is reached. The final set of bifurcation diagrams in this section corresponds to the case with c1 = 0.4 (Fig. 10), which is a little higher than the amplitude of the speed variations of the system of Fig. 9. Due to increased amplitude of the axial speed variations, the first bifurcations occurs at a lower va-

M.H. Ghayesh et al. / Chaos, Solitons & Fractals 52 (2013) 8–29

(a)

(c)

27

(b)

(d)

Fig. 19. Bifurcation diagrams of Poincaré points for decreasing amplitude of the axial speed variations of the system with c0 = 1.2; (a, b) the first two generalized coordinates of the transverse motion; (c, d) the first two generalized coordinates of the longitudinal motion; the critical value of the axial speed of this case is equal to 1.139 (i.e. (c0)cr = 1.139).

lue of the mean axial speed, namely at c0 = 0.67. A simple periodic motion is observed in the mean axial speed range of [0.68, 0.88], continued by a period-3 motion in the interval [0.89, 0.91]. The system displays a dominant period-2 motion afterwards in the range of [0.92, 1.07]. Typical characteristics of a period-2 motion at c0 = 1.00 is illustrated in Fig. 11 through (a, b) the time histories of the q1 and q2 motions, (c, d) the phase-plane portraits of the q1 and q2 motions, (e, f) the Poincaré sections of the q1 and q2 motions, respectively. A dominant chaotic attractor is observed afterwards until c0 = 1.79 is hit. The dynamical characteristics of this type of motion for the first two generalized coordinates of the longitudinal and transverse motions are depicted in Fig. 12 for c0 = 1.75. The system exhibits a simple periodic motion thereafter which continues until c0 = 2.00. From Figs. 2, 6, 9 and 10, it can be concluded that, in general, due to increased amplitude of the axial speed variation, the number of complex attractors as well as the cascade of higher order bifurcations increase in different

ranges of the mean axial speed. Moreover, the first bifurcation and the onset of quasiperiodic and chaotic motions occur at lower values of the mean axial speed for the system with higher amplitudes of the axial speed variations. 4.2. The amplitude of the axial speed variations as the bifurcation parameter The amplitude of the axial speed variations, c1, is chosen as the bifurcation parameter in this section. The bifurcation diagrams of Poincaré maps are constructed for three cases, namely c0 = 1.0, 1.2, and 1.5, which are plotted in Figs. 13, 15 and 17, respectively. The other system parameters in these figures are chosen as follows: v1 = 33.5261, vf = 0.1732, X = 1.2, c = 0.00000749, and f = 10. As shown in Fig. 13, corresponding to the case with c0 = 1.0, the first bifurcation occurs at c1 = 0.114, leading to a periodic oscillations. The periodic motion is displayed by the system thereafter until c1 = 0.190 is reached, where a period-2 motion occurs. This type of motion prevails in

28

M.H. Ghayesh et al. / Chaos, Solitons & Fractals 52 (2013) 8–29

the range of [0.190, 0.310], with windows of period-4 and period-6 motions in between. Typical characteristics of a period-2 motion at c1 = 0.240 are depicted in Fig. 14. A chaotic motion is observed in the interval [0.312, 0.322]. The system displays a dominant periodic motion thereafter, with a window of chaotic motion, until c1 = 0.392 is hit, where the motion becomes period-2; this type of motion is observed thereafter until c1 = 0.400. Increasing the mean axial speed to c0 = 1.2, from c0 = 1.0 in Fig. 13, a new set of bifurcation diagrams is constructed and plotted in Fig. 15. As the amplitude of the axial speed variations is increased, the amplitude of the transverse motion decreases until c1 = 0.018 is reached, where the first chaotic motion is observed; as seen in Fig. 15(a), the q1 amplitude starts from a nonzero value, implying that the system is in the supercritical regime. Period-2, period-3, and chaotic motions are observed in the range of [0.020, 0.060]. The system displays different types of motions including periodic, period-2, period-3, period-4, and period-6, in the interval [0.061, 0.126]. A dominant chaotic motion is observed afterwards which continues until c1 = 0.156. The system exhibits very rich dynamics with various types of attractors thereafter until c1 = 0.400 is hit. Typical characteristics of a chaotic motion for the first two generalized coordinates of the transverse motion are illustrated in Fig. 16. The last figure is related to the case with c0 = 1.5, which is a little higher than that of the previous case. As shown in Fig. 17, the system exhibits a periodic motion in the range of [0.000, 0.191], continued by a dominant chaotic motion until c1 = 0.400. Analyzing the results of this section, i.e. Figs. 13, 15 and 17, reveals that due to increased mean axial speed, the chaotic motion occurs over a wider ranges of the amplitude of the speed variations. For higher values of the mean axial speed, the trivial equilibrium configuration vanishes (due to transition from the sub to supercritical values) and even for small values of the amplitude of the speed variations, the system is attracted by nontrivial periodic attractors.

5. Conclusion In the present study, the coupled nonlinear dynamics of an axially accelerated viscoelastic beam has been investigated numerically. The equations of motion were derived employing the force and moment balance, resulting in two nonlinear partial differential equations describing the motion of the system. These equations were then discretized employing the Galerkin scheme and transformed into a set of nonlinear second-order ODEs. A change of variables was then introduced to these second-order ODEs, which resulted in a new set of first-order nonlinear ODEs with coupled terms and time-dependent coefficients. The equations were solved numerically employing a variable step-size modified Rosenbrock method, which resulted in the bifurcation diagrams of Poincaré maps of the system vs. the mean axial speed or the amplitude of the axial speed variations. Examining the global nonlinear dynamics of the system showed that for the first case, in which the mean axial

speed was the bifurcation parameter, due to increased amplitude of the axial speed variations, complexity as well as the range of chaotic regions increase. Moreover, the first bifurcation and onset of quasiperiodic and chaotic motions occur at lower mean axial speeds. The results for the second case, where the amplitude of the axial speed variations is the bifurcation parameter, revealed that for higher values of the mean axial speed, the trivial equilibrium state vanishes and the range of chaotic motions increase. Appendix A. The bifurcation diagrams of Poincaré maps of the system were obtained as either the mean axial speed or the amplitude of the axial speed variations was increased as the bifurcation parameter; i.e. forward bifurcation diagrams were constructed. It is worthwhile noting that the bifurcation diagrams of the same system with the same parameters may be different for the case when the bifurcation parameter is decreased from the final value; i.e. when a backward bifurcation diagram is constructed. In order to examine the differences between the forward and backward bifurcation diagrams, the backward bifurcation diagrams of two cases are constructed; in particular, the bifurcation diagrams of the system of Figs. 6 and 15 are reconstructed as the bifurcation parameter is decreased. The backward counterpart of Fig. 6 is plotted in Fig. 18, in which the mean value of the axial speed is decreased as the bifurcation parameter, and that of Fig. 15 is depicted in Fig. 19, where the amplitude of the axial speed variations is decreased as the bifurcation parameter. Comparing Figs. 6 and 18, it can be observed that the bifurcation diagrams are almost the same for both cases; however, for the case with decreasing mean axial speed (Fig. 18), two period-4 attractors appear at c0 = 1.04 and 1.05 and new period-4 and period-6 attractors emerge in the range of [1.13, 1.24]. Moreover, the amplitude of the final region of periodic motion decreases for the system of Fig. 18. The difference for the case when the amplitude of the axial speed variations is varied as the bifurcation parameter (i.e. Figs. 15 and 19) is more visible. Comparing Figs. 15 and 19, it is seen that the early chaotic regions (before c1 = 0.20) becomes much narrower for the system of Fig. 19 (with decreasing bifurcation parameter); the chaotic regions of Fig. 19 that occur after c1 = 0.20, become slightly narrower and the attractors between theses chaotic regions become periodic. References [1] Mote Jr CD. On the nonlinear oscillation of an axially moving string. J Appl Mech 1966;33:463–4. [2] Wickert JA. Non-linear vibration of a traveling tensioned beam. Int J Non-Linear Mech 1992;27:503–17. [3] Huang JL, Su RKL, Li WH, Chen SH. Stability and bifurcation of an axially moving beam tuned to three-to-one internal resonances. J Sound Vib 2011;330:471–85. [4] Marynowski K, Kapitaniak T. Zener internal damping in modelling of axially moving viscoelastic beam with time-dependent tension. Int J Non-Linear Mech 2007;42:118–31. [5] Pellicano F, Vestroni F. Nonlinear dynamics and bifurcations of an axially moving beam. ASME J Vib Acoust 2000;122:21–30. [6] Pellicano F, Zirilli F. Boundary layers and non-linear vibrations in an axially moving beam. Int J Non-Linear Mech 1998;33:691–711.

M.H. Ghayesh et al. / Chaos, Solitons & Fractals 52 (2013) 8–29 [7] Pellicano F, Vestroni F. Complex dynamics of high-speed axially moving systems. J Sound Vib 2002;258:31–44. [8] Pellicano F. On the dynamic properties of axially moving systems. J Sound Vib 2005;281:593–609. [9] Pakdemirli M, Ulsoy AG, Ceranoglu A. Transverse vibration of an axially accelerating string. J Sound Vib 1994;169:179–96. [10] Öz HR, Pakdemirli M. Vibrations of an axially moving beam with time-dependent velocity. J Sound Vib 1999;227:239–57. [11] ÖZkaya E, Pakdemirli M. Vibrations of an axially accelerating beam with small flexural stiffness. J Sound Vib 2000;234:521–35. [12] Öz HR, Pakdemirli M, Boyaci H. Non-linear vibrations and stability of an axially moving beam with time-dependent velocity. Int J NonLinear Mech 2001;36:107–15. [13] Sze KY, Chen SH, Huang JL. The incremental harmonic balance method for nonlinear vibration of axially moving beams. J Sound Vib 2005;281:611–26. [14] Xu GY, Zhu WD. Nonlinear and time-varying dynamics of highdimensional models of a translating beam with a stationary load subsystem. J Vib Acoust 2010;132:061012. [15] Suweken G, Van Horssen WT. On the transversal vibrations of a conveyor belt with a low and time-varying velocity. Part II: the beam-like case. J Sound Vib 2003;267:1007–27. [16] Suweken G, Van Horssen WT. On the transversal vibrations of a conveyor belt with a low and time-varying velocity. Part I: the string-like case. J Sound Vib 2003;264:117–33. [17] Chakraborty G, Mallik AK. Stability of an accelerating beam. J Sound Vib 1999;227:309–20. [18] Chen L-Q, Wu J, Zu JW. The chaotic response of the viscoelastic traveling string: an integral constitutive law. Chaos Solitons Fractals 2004;21:349–57. [19] Chen L-Q, Yang X-D. Stability in parametric resonance of axially moving viscoelastic beams with time-dependent speed. J Sound Vib 2005;284:879–91. [20] Zhang N-H, Chen L-Q. Nonlinear dynamical analysis of axially moving viscoelastic strings. Chaos Solitons Fractals 2005;24:1065–74. [21] Xiaodong Y, Li-Qun C. Non-linear forced vibration of axially moving viscoelastic beams. Acta Mech Solida Sin 2006;19:365–73. [22] Ghayesh MH. Nonlinear forced dynamics of an axially moving viscoelastic beam with an internal resonance. Int J Mech Sci 2011;53:1022–37.

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[23] Ghayesh MH. On the natural frequencies, complex mode functions, and critical speeds of axially traveling laminated beams: parametric study. Acta Mech Solida Sin 2011;24:373–82. [24] Ghayesh MH, Païdoussis MP, Amabili M. Subcritical parametric response of an axially accelerating beam. Thin-Walled Struct 2012;60:185–93. [25] Ghayesh MH, Kafiabad HA, Reid T. Sub- and super-critical nonlinear dynamics of a harmonically excited axially moving beam. Int J Solids Struct 2012;49:227–43. [26] Kazemirad S, Ghayesh M, Amabili M. Thermo-mechanical nonlinear dynamics of a buckled axially moving beam. Arch Appl Mech 2013;83:25–42. [27] Ghayesh M. Stability and bifurcations of an axially moving beam with an intermediate spring support. Nonlinear Dyn 2012;69:193–210. [28] Ghayesh MH, Amabili M. Nonlinear dynamics of axially moving viscoelastic beams over the buckled state. Comput Struct 2012;112– 113:406–21. [29] Ghayesh MH, Amabili M. Steady-state transverse response of an axially moving beam with time-dependent axial speed. Int J NonLinear Mech 2013;49:40–9. [30] Riedel CH, Tan CA. Coupled, forced response of an axially moving strip with internal resonance. Int J Non-Linear Mech 2002;37:101–16. [31] Chen LH, Zhang W, Yang FH. Nonlinear dynamics of higherdimensional system for an axially accelerating viscoelastic beam with in-plane and out-of-plane vibrations. J Sound Vib 2010;329:5321–45. [32] Chen L-Q, Tang Y-Q. Combination and principal parametric resonances of axially accelerating viscoelastic beams: recognition of longitudinally varying tensions. J Sound Vib 2011;330:5598–614. [33] Ghayesh MH, Kazemirad S, Amabili M. Coupled longitudinal– transverse dynamics of an axially moving beam with an internal resonance. Mech Mach Theory 2012;52:18–34. [34] Ghayesh MH. Coupled longitudinal–transverse dynamics of an axially accelerating beam. J Sound Vib 2012;331:5107–24. [35] Ghayesh M, Amabili M, Païdoussis M. Nonlinear vibrations and stability of an axially moving beam with an intermediate spring support: two-dimensional analysis. Nonlinear Dyn 2012;70:335–54.