Nonlinear dynamics of axially moving plates

Journal of Sound and Vibration 332 (2013) 391–406

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Nonlinear dynamics of axially moving plates Mergen H. Ghayesh, Marco Amabili n, Michael P. Paı¨doussis Department of Mechanical Engineering, McGill University, 817 Sherbrooke St. West, Montreal, QC, Canada H3A 0C3

a r t i c l e i n f o

abstract

Article history: Received 4 March 2012 Received in revised form 1 August 2012 Accepted 17 August 2012 Handling Editor: A.V. Metrikine Available online 23 September 2012

The nonlinear dynamics for forced motions of an axially moving plate is numerically investigated using Von Ka´rma´n plate theory and retaining in-plane displacements and inertia. The equations of motion are obtained via an energy method based on Lagrange equations. This yields a set of second-order nonlinear ordinary differential equations with coupled terms. The equations are transformed into a set of first-order nonlinear ordinary differential equations and are solved via the pseudo-arclength continuation technique. The near-resonance nonlinear dynamics is examined via plotting the frequency–response curves of the system. Results are shown through frequency– response curves, time histories, and phase-plane diagrams. The effect of system parameters, such as the axial speed and the pretension, on the resonant responses is also highlighted. & 2012 Elsevier Ltd. All rights reserved.

1. Introduction Axially moving (or travelling) systems are widely found in engineering, e.g., in mechanical, automotive, industrial, and aerospace applications. Paper sheets, textile fibers, band saw blades, robotic manipulators, conveyor belts, and magnetic tapes are just a handful of specific examples. Due to their widespread application, vibration and stability analyses of this class of systems have been conducted for many years by many authors. An extensive review on the various aspects of the dynamics of axially moving systems has been provided by Wickert and Mote [1] and later by Chen [2]; specifically, the mathematical modeling, solution methods, parametrically or externally excited systems, the effect of elastic foundations, and different boundary conditions. The literature concerning the dynamics of one-dimensional axially moving systems (such as beams and strings) is quite abundant. In early studies, attention was focused on linear models, with the aim of determining the natural frequencies, complex mode functions (due to travelling waves), critical speeds, and the response near the first instability [3]. However, beyond the first instability, where the oscillation or buckling amplitude is large, the linear predictions are not reliable [4,5]. More complete models of the system as well as more sophisticated techniques started appearing in the 1990s by several investigators [6–15]. Pakdemirli and co-workers [8–11], for example, employed perturbation techniques, such as the method of multiple scales and matched asymptotic expansion, to investigate the vibrations and stability of the string- and beam-like systems. These studies were extended and pursued, for example, by: Chen and co-investigators [16–18], who conducted research on the viscoelastic models of this system, considered Timoshenko beam theory, and employed different analytical and numerical techniques; Marynowski and co-workers [19], who considered various energy dissipation mechanisms in the model; Suweken and Van Horssen [20], who investigated the dynamics of the system using a discretization perturbation technique; and Huang et al. [21], who considered a three-to-one internal resonance in the

n

Corresponding author. Tel.: þ1 514 398 3068; fax: þ1 514 398 7365. E-mail address: [email protected] (M. Amabili).

0022-460X/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jsv.2012.08.013

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system dynamics. Garziera and Amabili [22] investigated the damping effect of winding on the lateral vibrations of axially moving tapes. Recently, a systematic research on this topic was conducted in [23–25] by the first author and co-workers. These analyses involved a variety of system models, namely, linear (parametrically excited), nonlinear, partially supported, energy dissipative (e.g., based on the Kelvin–Voigt energy dissipative mechanism), and laminated composite models.

~ f cos(t) b

a

cv

Fig. 1. Schematic representation of an axially moving plate.

1

0.14 0.12

0.8

w2,1/h

w1,1/h

0.1 0.6

0.08 0.06

0.4

0.04 0.2 0.02 0 0.8

1

1.2

1.4

1.6

1.8

0 0.8

2

1

1.2

ω/ω1,1

1.4

1.6

1.8

2

ω/ω1,1

0.04

0.06 0.05

0.03

w4,1/h

w3,1/h

0.04 0.02

0.03 0.02

0.01 0.01 0 0.8

1

1.2

1.4 ω/ω1,1

1.6

1.8

2

0 0.8

1

1.2

1.4

1.6

1.8

2

ω/ω1,1

Fig. 2. The frequency–response curves of the system with cv ¼5 m/s and P ¼500 N/m: (a) maximum of the generalized coordinate w1,1; (b) maximum of the generalized coordinate w2,1; (c) maximum of the generalized coordinate w3,1; (d) maximum of the generalized coordinate w4,1. Bold lines and dotted lines represent the stable and unstable solutions, respectively.

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393

The literature related to dynamics of axially moving two-dimensional models (such as plates and membranes), on the other hand, is not large, even though the dynamics of stationary (not travelling) plates has received considerable attention by Amabili and co-workers [26–29], Ribeiro and Petyt [30], and Kim et al. [31], for instance. The early studies on the dynamics of axially moving two-dimensional systems considered linear models [32–38]. For instance, Lengoc and McCallion [32] analyzed the vibration response of an axially moving plate subjected to a tangential edge loading. The vibration and stability characteristics of an axially moving plate subjected to a uniform in-plane tension in the transport direction were examined by Lin [33], who determined the critical speed of the system. Dynamics of axially moving orthotropic thin plates were examined by Wang [34], who employed a mixed finite element formulation based on the Mindlin–Reissner plate theory. Kim et al. [35] developed a modal spectral element for a thin axially moving plate travelling at a constant speed and subjected to a uniform axial tension. Free vibrations of axially moving symmetrically laminated composite plates were investigated by Hatami et al. [36]. The Kelvin–Voigt viscoelastic model was used to model an axially moving plate with parabolically varying thickness in [37]. Stability analysis of an axially moving elastic thin plate was investigated analytically by Banichuk et al. [38]. Yang et al. [39] examined the linear dynamics of an axially moving composite plate. Further investigations on this topic focused on the nonlinear aspects of the problem; the literature in this case is not large [40–42]. For example, the natural frequencies and responses of an axially moving plate were determined by Luo and Hamidzadeh [43]. A perturbation approach was employed by the same authors [44] to determine the equilibrium solutions and buckling stability of an axially moving thin plate. The finite element method was employed by Hatami et al. [45] to investigate the nonlinear vibrations of an axially moving plate. Yang et al. [46] investigated the transverse vibrations of an axially moving plate using a finite difference method. In the present study, the geometrically nonlinear vibrations and stability of an axially moving plate subjected to an outof-plane excitation load are studied. The plate is modeled using Von Ka´rma´n plate theory as well as Kirchhoff’s hypothesis, retaining in-plane displacements and inertia. It is the first time that all in-plane and out-of-plane displacements are retained in the nonlinear analysis of an axially moving plate—the in-plane displacements store a portion of the energy in thin-walled structures and cannot be ignored. The Lagrange method is employed, which yields a set of discretized

0.5

w2,1/h

w1,1/h

0.05

0

0

-0.05 -0.5

0

0.001

0.002

0

t 0.005

w4,1/h

w3,1/h

0.002 t

0.004

0

-0.004

0.001

0

0.001

0.002 t

0

-0.005

0

0.001

0.002 t

Fig. 3. Time histories for the system of Fig. 2 at o ¼1.2652o1,1 for different generalized coordinates, as denoted in the sub-figures. The plots correspond to the left peak in Fig. 2, where the first out-of-plane mode, i.e., mode (1,1), is directly excited: (a) generalized coordinate w1,1; (b) generalized coordinate w2,1; (c) generalized coordinate w3,1; (d) generalized coordinate w4,1.

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equations of motion (nonlinear ordinary differential equations); 30 degrees of freedom are employed to investigate all possible modal interactions. Frequency–response curves of the system are obtained using the pseudo-arclength continuation method which is capable of following both stable and unstable solution branches as well as determining bifurcation points. The effects of pretension as well as the axial speed on the nonlinear dynamics of the system are highlighted. 2. Problem statement, equations of motion, and method of solution As shown in Fig. 1, a rectangular plate with in-plane dimensions a and b and thickness h is considered. Let (O; x, y, z) be a rectangular Cartesian coordinate system, with the x and y axes defining the mid-plane of the plate; z denotes the out-of-plane coordinate. The origin (O) of this Cartesian coordinate system is assumed to be located at one of plate corners. Let u¼u(x, y, t), v¼v(x, y, t), and w¼w(x, y, t) denote the displacement components of the mid-plane of the plate in the x, y, and z directions from the static equilibrium (u¼ v¼w¼0). The plate is assumed to be subject to a pretension per unit width P in the x direction. Moreover, the plate is considered to be travelling in the x direction at a constant axial speed cv. A concentrated harmonic force f~ cosðotÞ, orthogonal to the plate, is applied at the centre. The forcing amplitude is denoted by f~ , positive in the z direction. The following relations are given for the Von Ka´rma´n plate theory [28]:

exx ¼ ex,0 þzkx , eyy ¼ ey,0 þ zky , gxy ¼ gxy,0 þ zkxy ,

(1)

where   @u 1 @w 2 þ , @x 2 @x   @v 1 @w 2 ey,0 ¼ þ , @y 2 @y

ex,0 ¼

0.4

(dw2,1/dt)/hω1,1

(dw1,1/dt)/hω1,1

0.04

0

0

-0.04 -0.4 -0.4

0 w1,1/h

0.4

-0.04

0 w2,1/h

0.04

0.004

(dw4,1/dt)/hω1,1

(dw3,1/dt)/hω1,1

0.004

0

0

-0.004 -0.002

0 w3,1/h

0.002

-0.004

-0.004

0 w4,1/h

0.004

Fig. 4. Phase-plane portrait for the system of Fig. 2 at o ¼ 1.2652o1,1 for different generalized coordinates, as denoted in the sub-figures. The plots correspond to the left peak in Fig. 2, where the first out-of-plane mode, i.e., mode (1,1), is directly excited: (a) generalized coordinate w1,1; (b) generalized coordinate w2,1; (c) generalized coordinate w3,1; (d) generalized coordinate w4,1.

M.H. Ghayesh et al. / Journal of Sound and Vibration 332 (2013) 391–406

gxy,0 ¼

395

@u @v @w @w þ þ , @y @x @x @y @2 w , @x2 @2 w ky ¼  2 , @y kx ¼ 

kxy ¼ 2

@2 w : @x@y

(2)

In Eqs. (1) and (2), exx, eyy and gxy denote the strain components at an arbitrary point of the plate at a distance z from the mid-plane; ex,0, ey,0 and gx,0 are the strains of the mid-plane; kx, ky, and kxy denote the curvatures and torsion of the midplane. The above equations demonstrate the nonlinear relations between the strains and displacements; the source of nonlinearity in these relations and hence in the equations of motion is geometric, caused by large displacements. The kinetic energy of the system may be expressed as follows: Z a Z b " 2  2  2 # Z a Z b " 2  2  2 # 1 @u @v @w 1 @u @v @w T P ¼ rh þ þ þ þ dxdy þ rhcv 2 dxdy 2 @t @t @t 2 @x @x @x 0 0 0 0   Z aZ b Z aZ b @u @u @v @v @w @w 1 @u @u þ þ dxdy þ rh þ 2cv 2 dxdy, (3) cv 2 þ 2cv þ rhcv @t @x @t @x @t @x 2 @t @x 0 0 0 0 where r represents the mass density of the plate and t is time. As seen in this equation, there are three more terms on the right-hand side (compared to the case of a stationary plate) due to the axial speed. Considering Kirchhoff’s hypothesis, the elastic potential energy of the plate can be expressed as [28] Z Z Z 1 a b h=2 ðsxx exx þ syy eyy þ txy gxy Þdxdydz, (4) UP ¼ 2 0 0 h=2

0.2

w2,1/h

w1,1/h

0.04

0

-0.04

0

0

-0.2

0.001 t

0

0.001 t

0

0.001 t

0.04

w4,1/h

w3,1/h

0.004

0

0

-0.004 -0.04

0

0.001 t

Fig. 5. Time histories for the system of Fig. 2 at o ¼ 1.7051o1,1 for different generalized coordinates, as denoted in the sub-figures. The plots correspond to the right peak in Fig. 2: (a) generalized coordinate w1,1; (b) generalized coordinate w2,1; (c) generalized coordinate w3,1; (d) generalized coordinate w4,1.

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where exx, eyy and gxy represent the Green strains and sxx, syy and txy are the Kirchhoff stresses; these stresses and strains obey the following relations for homogeneous and isotropic materials:

sxx ¼

E ðexx þ neyy Þ, 1n2

syy ¼

E ðeyy þ nexx Þ, 1n2

txy ¼

E g , 2ð1 þ nÞ xy

(5)

where n is the Poisson ratio and E denotes Young’s modulus. In order to express the potential energy of the plate in terms of the mid-plane strains, curvatures and torsion, one may substitute Eqs. (1) and (5) into Eq. (4), which results in the following:  Z aZ b 1 Eh 1n 2 2 2 UP ¼ e þ e þ2 ne e þ g dxdy x,0 y,0 x,0 y,0 xy,0 2 1n2 0 0 2 3

þ

1 Eh 2 12ð1n2 Þ

Z

a

0

Z 0

b

  Z aZ b   1n 2 2 2 kxy dxdy þP kx þ ky þ 2nkx ky þ ex,0 dxdy, 2 0 0

(6)

where the first term on the right-hand side is due to the stretching effect of the mid-plane, the second one to the bending effect, and the third one to the pretension per unit width P. In Eq. (6), it is assumed that the normal stress in the x direction ð0Þ involves a uniform pre-stress sxx , while there are no pre-stress components in the shear stress and the normal stress in the y direction. Integration of this pre-stress over the thickness of the plate results in the pretension per unit width, i.e., R h=2 ð0Þ P ¼ h=2 s xx dz.

0.2 (dw2,1/dt)/hω1,1

(dw1,1/dt)/hω1,1

0.04

0

-0.04 -0.04

0

-0.2 0

0.04

-0.08

w1,1/h

0.08

0.005 (dw4,1/dt)/hω1,1

(dw3,1/dt)/hω1,1

0.04

0

0

-0.005

-0.04 -0.04

0 w2,1/h

0 w3,1/h

0.04

-0.004

0

0.004

w4,1/h

Fig. 6. Phase-plane portrait for the system of Fig. 2 at o ¼1.7051o1,1 for different generalized coordinates, as denoted in the sub-figures. The plots correspond to the right peak in Fig. 2: (a) generalized coordinate w1,1; (b) generalized coordinate w2,1; (c) generalized coordinate w3,1; (d) generalized coordinate w4,1.

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397

The virtual work due to the external forces on the plate may be expressed as Z aZ b ðqx u þqy v þqz wÞdxdy, W¼ 0

(7)

0

where qx, qy, and qz denote the distributed forces per unit area applied in the x, y and z direction, respectively. In this paper, it is assumed that the external forces in the x and y directions are zero (qx ¼qy ¼0); there is only a single concentrated harmonic force orthogonal to the plate applied at the centre. The external distributed force, resulting from this harmonic concentrated force, is given by qz ¼ f~ dðxa=2Þdðyb=2ÞcosðotÞ,

(8)

where f~ denotes the forcing amplitude, positive in the z direction, d is the Dirac delta function, and o is the excitation frequency. It should be noted that the results presented in this paper are obtained by directly exciting the first generalized coordinate for transverse motion. Inserting Eq. (8) into Eq. (7) gives the following virtual work done by the external force: W ¼ f~ cosðotÞðwÞx ¼ a=2,

(9)

y ¼ b=2 :

The Rayleigh dissipation function is employed to take into account the external nonconservative damping forces, which are assumed to be of the viscous type, as follows: Z a Z b " 2  2  2 # 1 @u @v @w þ þ F¼ c dxdy, (10) 2 0 0 @t @t @t where c is the damping coefficient.

0.6

1 0.8

w2,1/h

w1,1/h

0.4 0.6 0.4

0.2

0.2 0 0.8

1

1.2

1.4 1.6 ω/ω1,1

1.8

0 0.8

2

1

1.2

1.4 1.6 ω/ω1,1

1.8

1

1.2

1.4 1.6 ω/ω1,1

1.8

2

0.35 0.14 0.3

0.12 0.1

0.2

w4,1/h

w3,1/h

0.25

0.15

0.08 0.06

0.1

0.04

0.05

0.02

0 0.8

1

1.2

1.4 1.6 ω/ω1,1

1.8

2

0 0.8

2

Fig. 7. The frequency–response curves of the system with cv ¼15 m/s and P¼ 500 N/m: (a) maximum of the generalized coordinate w1,1; (b) maximum of the generalized coordinate w2,1; (c) maximum of the generalized coordinate w3,1; (d) maximum of the generalized coordinate w4,1. Bold lines and dotted lines represent the stable and unstable solutions, respectively.

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The in-plane and out-of-plane displacements, for the simply-supported plate with immovable edges, are approximated using the following series expansions: uðx,y,tÞ ¼

~ M N~ X X

um,n ðtÞ sinðmpx=aÞ sinðnpy=bÞ,

m¼1n¼1

vðx,y,tÞ ¼

wðx,y,tÞ ¼

M X N X m¼1n¼1 _ _ M N X X

vm,n ðtÞ sinðmpx=aÞ sinðnpy=bÞ, wm,n ðtÞ sinðmpx=aÞ sinðnpy=bÞ,

(11)

m¼1n¼1

1 0.5

w2,1/h

w1,1/h

0.5

0

0

-0.5 -0.5 -1

0

0.001

0

0.002

0.001

t 0.04

w4,1/h

w3,1/h

0.2

0

-0.2

0

0.001

0

-0.04

0.002

0

0.001

t 0.0005

w2,3/h

w1,3/h

0.002 t

0.001

0

-0.001

0.002 t

0

0.001

0.002 t

0

-0.0005

0

0.001

0.002 t

Fig. 8. Time histories for the system of Fig. 7 at o ¼ 1.4453o1,1 for different generalized coordinates, as denoted in the sub-figures. The plots correspond to the left peak in Fig. 7, where the first out-of-plane mode, i.e., mode (1,1), is directly excited: (a) generalized coordinate w1,1; (b) generalized coordinate w2,1; (c) generalized coordinate w3,1; (d) generalized coordinate w4,1; (e) generalized coordinate w1,3; (f) generalized coordinate w2,3.

M.H. Ghayesh et al. / Journal of Sound and Vibration 332 (2013) 391–406

399

where m and n are the numbers of half-waves in the x and y direction, respectively, and um,n(t), vm,n(t), and wm,n(t) are the generalized coordinates, which are unknown functions of time and should be determined through numerical solutions. In order to simplify the procedure for the derivation of the equations of motion, the following notation is introduced:  T q ¼ um,n ,vm,n ,wm,n :

(12)

The elements of the vector in Eq. (12), qi, are time-dependent generalized coordinates and the number of degrees of freedom determines the dimension of this vector. The Lagrange equations are employed to derive the equations of motion; the advantage of this method over Hamilton’s technique is that it bypasses setting up partial-differential equations, hence leading directly to set of approximate ordinary

1

(dw2,1/dt)/hω1,1

(dw1,1/dt)/hω1,1

1

0

0

-1 -1

0

-1

1

-0.5

w1,1/h

0

0

-0.04

-0.2 0

0.2

-0.03

0

w3,1/h

0.003 (dw2,3/dt)/hω1,1

(dw3,1/dt)/hω1,1

0.03

w4,1/h

0.002

0

-0.002 -0.001

0.5

0.04 (dw4,1/dt)/hω1,1

(dw3,1/dt)/hω1,1

0.2

-0.2

0 w2,1/h

0

-0.003 0 w1,3/h

0.001

-0.0005

0 w2,3/h

0.0005

Fig. 9. Phase-plane portraits for the system of Fig. 7 at o ¼ 1.4453o1,1 for different generalized coordinates, as denoted in the sub-figures. The plots correspond to the left peak in Fig. 7, where the first out-of-plane mode, i.e., mode (1,1), is directly excited: (a) generalized coordinate w1,1; (b) generalized coordinate w2,1; (c) generalized coordinate w3,1; (d) generalized coordinate w4,1; (e) generalized coordinate w1,3; (f) generalized coordinate w2,3.

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differential equations with coupled terms. The Lagrange equations of motion are given by d dt

! @T P @T P @U P  þ ¼ Q j, @q_ j @qj @qj

j ¼ 1,. . .,N,

(13)

where N is the number of degrees of freedom, and Qj are given by Qj ¼ 

@F @W þ , @q_ j @qj

0

-0.4

0

0.001

0

-0.4

0.002

0

t

w4,1/h

w3,1/h

0.002

0.2

0

0

0.001

0

-0.2

0.002

0

t

0.001

0.002

t

0.0004

w2,3/h

0.0004

w1,3/h

0.001 t

0.4

-0.4

(14)

0.4

w2,1/h

w1,1/h

0.4

j ¼ 1,. . .,N:

0

0

-0.0004

-0.0004 0

0.001 t

0.002

0

0.001 t

0.002

Fig. 10. Time histories for the system of Fig. 7 at o ¼1.8307o1,1 for different generalized coordinates, as denoted in the sub-figures. The plots correspond to the right peak in Fig. 7: (a) generalized coordinate w1,1; (b) generalized coordinate w2,1; (c) generalized coordinate w3,1; (d) generalized coordinate w4,1; (e) generalized coordinate w1,3; (f) generalized coordinate w2,3.

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401

Substituting Eq. (11) into Eqs. (9) and (10), and inserting the resulting equations into Eq. (14) gives the generalized forces in terms of generalized coordinates. Furthermore, inserting Eq. (11) into Eqs. (2), (6) and (3), results in the potential and kinetic energies in terms of the generalized coordinates. Substituting these functions, which are functions of generalized coordinates, into Eq. (13) results in a set of N coupled nonlinear ordinary differential equations. In order to use available numerical methods, this set is transformed into a set of first-order nonlinear ordinary differential equations via the transformation yi ¼ q_ i with i¼1,2,y,N; this operation gives a set of 2N first-order nonlinear ordinary coupled differential equations. This set is solved via pseudo-arclength continuation and collocation techniques which is a suitable numerical method to obtain the frequency–response curves; the AUTO code [47] – which is capable of calculating continuation of the solutions, bifurcations and branches, as well as determining their stability – is employed.

0.5 (dw2,1/dt)/hω1,1

(dw1,1/dt)/hω1,1

0.5

0

0

-0.5

-0.5

-0.4

0 w1,1/h

0.4

-0.4

0 w2,1/h

0.4

0.2 (dw4,1/dt)/hω1,1

(dw3,1/dt)/hω1,1

0.5

0

0

-0.2 -0.5 -0.4

0 w3,1/h

0.4

-0.2

0

-0.002

0.2

0.004

(dw2,3/dt)/hω1,1

(dw1,3/dt)/hω1,1

0.002

0 w4,1/h

-0.0005

0 w1,3/h

0.0005

0

-0.004 -0.0004

0 w2,3/h

0.0004

Fig. 11. Phase-plane portraits for the system of Fig. 7 at o ¼ 1.8307o1,1 for different generalized coordinates, as denoted in the sub-figures. The plots correspond to the right peak in Fig. 7: (a) generalized coordinate w1,1; (b) generalized coordinate w2,1; (c) generalized coordinate w3,1; (d) generalized coordinate w4,1; (e) generalized coordinate w1,3; (f) generalized coordinate w2,3.

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3. Numerical results The frequency–response curves of the system are obtained in this section by means of the pseudo-arclength continuation technique. The effect of the system parameters on these curves is also highlighted. The analysis is performed in two steps. In the first step, the excitation frequency is set far from resonance and the forcing amplitude is used as the control parameter; then, starting with zero initial conditions, the first control parameter is increased, until the desired forcing amplitude is reached. The second step performs the continuation and obtains the resonant response using the excitation frequency as the control parameter. Calculations have been performed for a rectangular aluminium plate with the following dimensions and mechanical properties: a ¼0.515 m, b ¼0.184 m, h¼0.0003 m, E¼69 GPa, r ¼2700 kg/m3, and n ¼0.33. The forcing amplitude, f~ , is set to 0.007 for all cases studied here; only generalized coordinate w1,1 is excited. The same modal damping ratio of z1,1 ¼0.0117 is used for all generalized coordinates. Also, thirty degrees of freedom are utilized; i.e., 60 first-order nonlinear ordinary differential equations with coupled terms are solved numerically. Particularly, the following generalized coordinates are used: w1,1, w2,1, w3,1, w4,1, w1,3, w2,3, w3,3, w4,3, w1,2, u1,1, u2,1, u3,1, u4,1, u5,1, u1,3, u2,3, u3,3, u4,3, u5,3, u1,2, u2,2, v1,1, v2,1, v1,2, v2,2, v3,2, v4,2, v5,2, v1,3, and v2,3. In order to validate the numerical approach used in this paper, the frequency–response curve of the stationary (not travelling) plate, free of pretension (i.e., cv ¼P¼0), has been obtained and compared to that given by Amabili in Ref. [26]; the results are given in Appendix A. In what follows, three cases – the first two with different axial speeds (i.e., cv ¼5 m/s and 15 m/s) and a constant pretension P¼500 N/m, and the third one with cv ¼6 m/s and P¼0 N/m – are studied in the first part of this section. Then, the effects of the axial speed and pretension on the resonant response of the system are examined. The frequency–response curve of the system with cv ¼ 5 m/s is shown in Fig. 2(a–d) in the frequency neighbourhood of the fundamental transverse mode, for different out-of-plane generalized coordinates; sub-figure (a) shows the maximum

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Fig. 12. The frequency–response curves of the system with cv ¼6 m/s and P ¼0: (a) maximum of the generalized coordinate w1,1; (b) maximum of the generalized coordinate w2,1; (c) maximum of the generalized coordinate w3,1; (d) maximum of the generalized coordinate w4,1. Bold lines and dotted lines represent the stable and unstable solutions, respectively.

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of the generalized coordinate w1,1, (b) the maximum of the generalized coordinate w2,1, (c) the maximum of the generalized coordinate w3,1, and (d) the maximum of the generalized coordinate w4,1. The fundamental linear natural frequency for the out-of-plane motion is obtained as o1,1 ¼209.4185 rad/s. The linear natural frequency corresponding to the (2, 1) mode is obtained as o2,1 ¼355.8621 rad/s. The ratio between these two frequencies is o2,1/o1,1 ¼ 1.6993. As seen in Fig. 2(a), the first peak in the response appears for the excitation frequency around the fundamental one; while the second peak (which is very small in Fig. 2(a)) arises due to the above-mentioned relation between the (1, 1) and (2, 1) natural frequencies for the out-of-plane motion. There are two limit-point bifurcations present (corresponding to the first peak), namely at o ¼1.2670o1,1 and 1.0716o1,1; the first one is responsible for the transition from a stable solution to an unstable one and the second one for the regaining of stability. Comparing sub-figures (a) and (b) reveals that the amplitude of the second peak for the (2, 1) generalized coordinate is larger than that of the (1, 1) generalized coordinate. It is also seen that the first peak shows a hardening-type behaviour, while the second one shows a weak nonlinearity. Typical periodic motion characteristics of the system of Fig. 2 at o ¼1.2652o1,1 are illustrated in Fig. 3(a–d) and Fig. 4(a–d) through time histories and phase-plane portraits for different generalized coordinates; these plots correspond to the left (first) curve in Fig. 2. — the time axis in all time histories is normalized to the fundamental linear natural frequency for the out-of-plane motion. Time traces reveal three pieces of useful information: the first is the phase difference between different generalized coordinates, the second is the contribution of higher harmonics in the response corresponding to each generalized coordinate (e.g., sub-figure (c)), the third is that the displacement during an oscillation period is symmetric on the two sides of the plate; in some models, a geometric imperfection can cause asymmetry in the response. Typical characteristics of a periodic motion at o ¼1.7051o1,1 – which corresponds to the second (right) curve – are shown in Figs. 5 and 6, illustrating the absence of contributions by higher harmonics. Increasing the axial speed from 5 m/s (in Fig. 2(a–d)) to 15 m/s, Fig. 7(a–d) is generated. The linear natural frequencies for the out-of-plane motion corresponding to the modes (1, 1) and (2, 1) are as follows: o1,1 ¼164.8394 rad/s and o2,1 ¼277.7017 rad/s with the ratio of o2,1/o1,1 ¼1.6847. There are two peaks in the frequency–response curves, both showing a hardening-type behaviour. Comparing Fig. 2(a–d) and Fig. 7(a–d) reveals that, as opposed to Fig. 2(a–d), there is

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Fig. 13. The frequency–response curves of the system for several axial speeds. The values of cv are denoted on the curves: (a) maximum of the generalized coordinate w1,1; (b) maximum of the generalized coordinate w2,1; (c) maximum of the generalized coordinate w3,1; (d) maximum of the generalized coordinate w4,1.

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Fig. 14. The frequency–response curves of the system for several pretensions. The values of P are denoted on the curves: (a) maximum of the generalized coordinate w1,1; (b) maximum of the generalized coordinate w2,1; (c) maximum of the generalized coordinate w3,1; (d) maximum of the generalized coordinate w4,1.

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an unstable solution branch in the second peak of Fig. 7(a–d). The stable branches are shown with bold lines and the unstable ones with dotted lines. Theoretically, as seen in Fig. 7(a), as the excitation frequency is increased from o ¼0.8o1,1, the response amplitude increases accordingly until it reaches the first limit point bifurcation at o ¼1.4506o1,1, where it becomes unstable. The amplitude of this now unstable solution branch decreases until o ¼1.0824o1,1 is reached, where the system regains stability by means of the second limit-point bifurcation. As the excitation frequency is increased further, the response amplitude initially decreases and then increases until stability is lost again at o ¼1.8328o1,1 via the third limit-point bifurcation. Stability is regained via the fourth limit-point bifurcation at o ¼1.7291o1,1. As seen in subfigures (a) and (b), the amplitude of the second peak is almost the same for the (1, 1) and (2, 1) generalized coordinates; sub-figures (c and d) show that the amplitude of the second peak for the (3, 1) and (4, 1) generalized coordinates is larger than that of the first one. Two excitation frequencies, o ¼ 1.4453o1,1 and o ¼ 1.8307o1,1, corresponding to the left and right peaks (curves), are chosen and the vibration responses are studied in detail via Fig. 8(a–f) and Fig. 9(a–f) for the first case and by means of Fig. 10(a–f) and Fig. 11(a–f) for the second case. As seen in these figures, the responses in different generalized coordinates are again symmetric, with the contribution of higher harmonics for generalized coordinates w3,1, w4,1, w1,3, w2,3 for the first case (Fig. 8(c–f)), and w1,3 and w2,3 for the second case (Fig. 10(e and f)). The frequency–response curve of the system with no pretension is plotted in Fig. 12(a–d) for the axial speed of 6 m/s. The linear natural frequencies corresponding to the (1, 1) and (2, 1) modes of the out-of-plane motion are obtained as: o1,1 ¼131.9597 rad/s and o2,1 ¼ 181.9923 rad/s. The ratio between these two frequencies is o2,1/o1,1 ¼1.3792; due to this fact, a second peak arises in the vicinity of o ¼1.4o1,1, as seen in Fig. 12(a–d). Sub-figures (c and d) show that the modal interactions are more prominent for the w3,1 and w4,1 generalized coordinates. The effect of the axial speed on the resonant response of the system is highlighted in Fig. 13(a–d); the pretension is set to 500 N/m. Again, due to the ratio between the natural frequencies corresponding to the (1, 1) and (2, 1) modes, there are two peaks in the plot. As seen in sub-figure (a), as the axial speed is increased, the hardening behavior of the first curve (on the left-hand side) for the fundamental generalized coordinate increases and the multi-valued region of the response become wider; this behaviour is observed for the second peak in all generalized coordinates shown in Fig. 13. Fig. 14(a–d) shows the frequency–response of the system for several values of the pretension, P, while the axial speed is set to cv ¼5 m/s. Again, there are two peaks in these curves. As the pretension is set to larger values, the hardening-type behaviour of the first curve (on the left-hand side) for the fundamental generalized coordinate decreases. Another interesting feature observed is that the peaks of the second curves (on the right-hand side) are shifted by the pretension, while this effect is not that prominent for the first curves. 4. Conclusions The nonlinear vibrations and stability of an axially moving (travelling) plate have been investigated in this paper via the pseudo-arclength continuation technique, with special consideration to the effect of the axial speed and pretension. Considering all in-plane and out-of-plane displacements, Von Ka´rma´n plate theory as well as Kirchhoff’s hypothesis was employed to construct the potential and kinetic energies of the system. The energy expressions were inserted in the Lagrange equation which yields a set of second-order nonlinear coupled ordinary differential equations. The equations were transformed into a set of first-order nonlinear ordinary differential equations via change of variables and solved numerically using the pseudo-arclength continuation technique. The results were presented in the form of time histories, phase-plane portraits, and frequency–response curves. A large number of degrees-of-freedom in the discretization was employed to discover almost all modal interactions; this number results in satisfactory convergence. Results for the resonant response showed that, depending on the system parameters, there are two peaks in the frequency–response diagram of the system, due to the relation between the (1, 1) and (2,1) natural frequencies for the out-of-plane motion. Moreover, the only type of bifurcation present is the limit point, followed by either stable or unstable periodic solutions. As the axial speed is increased, the hardening behaviour of the left curve for the fundamental generalized coordinate as well as the right curve for the (1,1), (2,1), (3,1), and (4,1) generalized coordinates for the out-of-plane motion increase and the multi-valued region of the response becomes wider. The hardening-type behaviour of the left curve for the fundamental generalized coordinate decreases, as the pretension is increased. Moreover, the peaks of the right curve are shifted by the pretension. Appendix A. Validation The frequency–response of a stationary plate which is free of pretension (i.e., with cv ¼P ¼0) is obtained and compared to that given by Amabili in Ref. [26], displaying excellent agreement, as seen in Fig. 15. References [1] [2] [3] [4]

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