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Non-linear global dynamics of an axially moving plate
Author's Accepted Manuscript
Nonlinear global dynamics of an axially moving plate Mergen H. Ghayesh, Marco Amabili
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PII: DOI: Reference:
S0020-7462(13)00120-0 http://dx.doi.org/10.1016/j.ijnonlinmec.2013.06.005 NLM2183
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International Journal of Non-Linear Mechanics
Received date: 20 September 2012 Revised date: 27 May 2013 Accepted date: 8 June 2013 Cite this article as: Mergen H. Ghayesh, Marco Amabili, Nonlinear global dynamics of an axially moving plate, International Journal of Non-Linear Mechanics, http://dx.doi.org/10.1016/j.ijnonlinmec.2013.06.005 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Nonlinear global dynamics of an axially moving plate Mergen H. Ghayesh*, Marco Amabili Department of Mechanical Engineering, McGill University, 817 Sherbrooke St. West, Montreal, Quebec, Canada H3A 0C3 *Corresponding author: [email protected], Tel: (+1) 514 398‐6290
Abstract In the present study, the geometrically nonlinear dynamics of an axially moving plate is examined by constructing the bifurcation diagrams of Poincaré maps for the system in the sub and supercritical regimes. The von Kármán plate theory is employed to model the system by retaining in‐plane displacements and inertia. The governing equations of motion of this gyroscopic system are obtained based on an energy method by means of Lagrange equations which yields a set of second‐order nonlinear ordinary differential equations with coupled terms. A change of variables is employed to transform this set into a set of first‐order nonlinear ordinary differential equations. The resulting equations are solved using direct time integration, yielding time‐varying generalized coordinates for the in‐plane and out‐of‐plane motions. From these time histories, the bifurcation diagrams of Poincaré maps, phase‐plane portraits, and Poincaré sections are
1
constructed at points of interest in the parameter space for both the axial speed regimes. Keywords: Axially moving plates; Nonlinear dynamics; Bifurcations; Stability 1. Introduction The class of axially moving systems are widely used in many engineering devices and machine components; for example, they are simplified models of band saw blades, paper sheets, textile fibers, robotic manipulators, conveyor belts, and magnetic tapes. The axial speed plays an important role in the dynamical behaviour of this class of systems. The natural frequencies of the system decrease with increasing axial speed and at a certain value, which is called critical axial speed, the first natural frequency of a conservative system vanishes, which causes the system to lose stability via divergence, leading to buckling. The axial speed range before the critical axial speed is called subcritical regime and that after the critical speed is called supercritical regime. Studies concerning the dynamics of axially moving systems can be classified mainly into two general groups in terms of models being considered, i.e. one‐dimensional
2
such as beams and strings [1], and two‐dimensional such as membranes and plates. The linear and nonlinear dynamics of axially moving one‐dimensional systems (such as beams and strings) has received considerable attention in the literature. For example, Pakdemirli and co‐workers [2‐5] investigated the dynamics of axially moving beams and strings by means of the method of multiple timescales and matched asymptotic expansion. A systematic research on the dynamics of axially moving beams and strings were conducted by Chen and co‐workers [6, 7], who introduced viscoelastic models for the system, and employed different analytical and numerical techniques. Sub and supercritical nonlinear dynamics of an axially moving beam with special consideration to the case with an internal resonance was examined numerically by Ghayesh et al. [8]. The coupled longitudinal‐transverse dynamics of an axially accelerated beam was investigated by Ghayesh [9], who discovered period‐doubling bifurcations in the resonant dynamics of the system. The literature on the dynamics of axially moving two‐dimensional models (such as membranes and plates) is not large; the dynamics of stationary (not axially moving) plates has been studied extensively, for example, by Amabili and co‐workers [10‐12]. The linear models of axially moving two‐dimensional
3
systems were developed and analyzed, for instance, by: Lin [13], who examined the influence of a uniform in‐plane tension in the transport direction on the dynamics of an axially moving plate; Kim et al. [14], who developed a modal spectral element for a thin axially moving plate travelling at a constant speed and subjected to a uniform axial tension; Yin‐feng and Zhong‐min [15], who introduced the Kelvin‐Voigt viscoelastic model to the mathematical model of an axially moving plate with parabolically varying thickness; Banichuk et al. [16], who examined the stability of an axially moving elastic thin plate; and Yang et al. [17], who investigated the dynamics of an axially moving composite plate. These studies were extended and pursued for nonlinear models of axially moving two‐ dimensional models; the literature in this case is not large [18, 19]. For instance, Luo and Hamidzadeh [20] determined the natural frequencies and dynamic responses of an axially moving plate. Hatami et al. [21] employed the finite element method to examine the nonlinear vibrations of an axially moving plate. The three‐dimensional (i.e., by retaining all in‐plane and out‐of‐plane displacements and inertia) nonlinear global dynamics of axially moving plates has not been studied yet; in other words, bifurcation diagrams Poincaré maps for an axially moving plate has not been investigated in the literature yet. The current paper is the first to do so.
4
In this effort, the nonlinear global dynamics of an axially moving plate subjected to a distributed harmonic excitation load is investigated by constructing the bifurcation diagrams of Poincaré maps as either the axial speed or the forcing amplitude is varied. The plate is modeled via von Kármán plate theory taking into account Kirchhoff’s hypothesis, retaining in‐plane displacements and inertia; this paper is the first which retains all in‐plane and out‐of‐plane displacements in the nonlinear analysis of an axially moving plate‐‐ the computer codes should be well‐optimized for this operation. The potential and kinetic energies of the system are constructed and inserted in the Lagrange equations which yield a set of second‐order nonlinear coupled ordinary differential equations. These equations are solved via direct time integration by
means of Gear’s backward‐differentiation‐formula (BDF) dealing with the stiff nonlinear equations, yielding the amplitudes of oscillation as functions of time. Thus, bifurcation diagrams of Poincaré maps are constructed using either the axial speed or the forcing amplitude as the bifurcation parameter. The analyses also include the system in the supercritical regime. This paper is the first which investigates the nonlinear global dynamics of an axially moving plate. The nonlinear global dynamics of an axially moving plate retaining all in‐plane and
5
out‐of‐plane displacements and inertia has not been studied before in the literature; the bifurcation diagrams of Poincaré maps are constructed in this paper for the first time for an axially moving plate retaining all in‐plane and out‐ of‐plane displacements and inertia.
2. Model development and method of solution As shown in Fig. 1, a rectangular plate with Cartesian coordinate system (O; x, y, z), having the origin O at one corner is considered. In‐plane dimensions and thickness are denoted by a, b, and h, respectively; x and y axes define the mid‐plane of the plate and z denotes the out‐of‐plane coordinate. The displacements of each point of the mid‐plane of the plate are denoted by u=u(x,y,t), v=v(x,y,t), and w=w(x,y,t) in the x, y, and z directions from the static equilibrium (u=v=w=0), respectively. The plate is assumed to be subject to an in‐ plane pretension per unit width P in the x direction. The plate is also considered to be travelling in the x direction at a constant axial speed cv (in m/s). Moreover, a distributed harmonic force per unit area f1 sin (π x a ) sin (π y b ) co s(ω t ) , orthogonal to the plate, is applied; ω is the excitation frequency and the forcing
6
amplitude is denoted by f1 sin (π x a ) sin (π y b ) , positive in the z direction (f1 is in N/m2). In what follows, the kinetic and potential energies of the system are constructed as functions of the in‐plane and out‐of‐plane displacements; they are then inserted into the Lagrange equations so as to obtain the discretized equations of motion. Compared to the case of a stationary plate, due to presence of the axial speed, axial speed‐dependent terms emerge in the kinetic energy of the system; the kinetic energy of the system is constructed as 2 2 2 a b ⎡ ⎛ ∂u ⎞ 1 ⎛ ∂v ⎞ ⎛ ∂ w ⎞ ⎤ TP = ρ h ∫ ∫ ⎢⎜ ⎟ + ⎜ ⎟ + ⎜ ⎟ ⎥ dxdy 0 0 2 t t t ∂ ∂ ∂ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎦ ⎣ 2 2 2 a b ⎡ ⎛ ∂u ⎞ 1 ⎛ ∂ v ⎞ ⎛ ∂w ⎞ ⎤ + ρ hcv 2 ∫ ∫ ⎢⎜ ⎟ + ⎜ ⎟ + ⎜ ⎟ ⎥ dxdy 0 0 ∂ ∂ ∂ 2 x x x ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎦ ⎣
⎡ ∂u ∂u ∂v ∂ v ∂ w ∂ w ⎤ + + dxdy 0 ∫0 ⎢ ∂t ∂x ∂t ∂x ∂t ∂x ⎥⎦ ⎣ a b⎡ ∂u ∂u ⎤ 1 + 2cv 2 ⎥ dxdy , + ρ h ∫ ∫ ⎢ cv 2 + 2cv 0 0 ∂t ∂x ⎦ 2 ⎣ + ρ hcv ∫
a
b
(1)
where ρ denotes the mass density of the plate and t represents time. The following relations, between the strain of a generic point of the plate and the displacement field, are given for the von Kármán plate theory [12]:
7
∂u 1 ∂ w ⎞ ∂ 2w ε xx = + ⎛⎜ ⎟ −z 2 , ∂x 2 ⎝ ∂x ⎠ ∂x 2
2
∂v 1 ⎛ ∂ w ⎞ ∂ 2w (2) − ε yy = + ⎜ z , ∂y 2 ⎝ ∂y ⎟⎠ ∂y 2 ∂u ∂ v ∂ w ∂ w ∂ 2w γ xy = + + − 2z , ∂ y ∂x ∂ x ∂ y ∂ x ∂y
where ε xx , ε yy and γ xy represent the strain components at an arbitrary point of the plate at a distance z from the mid‐plane. As seen in Eq. (2), the source of nonlinearity is geometric due to large displacements. The relations between the Kirchhoff stresses ( σ xx , σ yy and τ xy ) and Green strains ( ε xx , ε yy and γ xy ) for homogeneous and isotropic materials are given by
E (ε xx +νε yy ) , 1 −ν 2 E σ yy = (ε yy +νε xx ) , (3) 1 −ν 2 E τ xy = γ xy , 2(1 +ν )
σ xx =
where ν and E are the Poisson ratio and Young’s modulus, respectively. Under Kirchhoff’s hypothesis, the elastic potential energy of the plate can be expressed as [12]
8
UP =
a b ⎛ ∂u 1 a b h /2 1 ∂w ⎞ σ xxε xx + σ yy ε yy + τ xy γ xy ) dxdydz + P ∫ ∫ ⎜⎜ + ⎜⎛ ( ⎟ ∫ ∫ ∫ 0 0 ∂x 2 0 0 − h /2 2 ⎝ ∂x ⎠ ⎝
2
⎞ ⎟⎟ dxdy. (4) ⎠
where it is assumed that there is a uniform pre‐stress σ xx(0) in the x direction, while there are no pre‐stress components in the normal stress in the y direction and the shear stress. Integrating this pre‐stress over the thickness of the plate gives the pretension per unit width, i.e. P = ∫
h /2
− h /2
σ (0) xx dz . Substitution of Eqs. (2) and
(3) into Eq. (4) results in the potential energy as a function of displacement field. The virtual work due to a distributed harmonic force per unit area f1 sin (π x a ) sin (π y b ) co s(ω t ) , orthogonal to the plate, may be expressed as
W =∫
a
0
∫
b
0
⎛ ⎞ ⎛πx ⎞ ⎛πy ⎞ ⎜ wf1 sin ⎜ a ⎟ sin ⎜ b ⎟ cos(ω t ) ⎟ dxdy , (5) ⎝ ⎠ ⎝ ⎠ ⎝ ⎠
where, it is assumed that the external forces in the x and y directions are zero. The nonconservative external damping forces, assumed to be of viscous type, are assumed using the Rayleigh dissipation function as follows: 2 2 2 1 a b ⎡ ⎛ ∂u ⎞ ⎛ ∂v ⎞ ⎛ ∂w ⎞ ⎤ F = c ∫ ∫ ⎢⎜ ⎟ + ⎜ ⎟ + ⎜ ⎟ ⎥ dxdy , (6) 2 0 0 ⎣ ⎝ ∂t ⎠ ⎝ ∂t ⎠ ⎝ ∂t ⎠ ⎦
where c is the damping coefficient.
9
The following expressions are used to expand the in‐plane and out‐of‐ plane displacements, satisfying identically the geometric boundary conditions of the simply‐supported plate with immovable edges: M
N
u(x , y ,t ) = ∑∑ um ,n (t )sin(mπ x / a)sin(nπ y / b), m =1 n =1 M
N
v(x , y ,t ) = ∑∑ vm ,n (t )sin(mπ x / a)sin(nπ y / b), (7) m =1 n =1 M N
w(x , y ,t ) = ∑∑ wm ,n (t )sin(mπ x / a)sin(nπ y / b), m =1 n =1
where m and n are the numbers of half‐waves in the x and y direction, respectively; um ,n (t ) , v m ,n (t ) , and w m ,n (t ) denote the generalized coordinates— these generalized coordinates are unknown functions of time which will be determined through numerical solutions. The exact values of M and N in Eq. (7) are not determined at this step, in order not to lose generality. Instead, the number of generalized coordinates employed is defined by subscripts in Section 3, last paragraph (above Section 3.1). All modes in series are not used one after the other; for more information, please see Ref. [10]. In what follows, the following notation is introduced for brevity: q = {um ,n , vm ,n , wm ,n } , (8) T
10
where the elements of the vector qi, are time‐dependent generalized coordinates and the dimension of this vector is equal to the number of degrees of freedom. The generalized forces Q j are obtained by differentiating the Rayleigh dissipation function and of the virtual work done by external forces
Qj = −
∂F ∂W , j = 1,...., N , (9) + ∂q j ∂q j
where N is the number of degrees of freedom. The Lagrange equations, which bypass setting up partial‐differential equations, and hence result in a set of nonlinear ordinary differential equations of motions are given as follows: d ⎛ ∂TP ⎜ dt ⎜⎝ ∂q j
⎞ ∂TP ∂UP + = Q j , j = 1,...., N , (10) ⎟⎟ − ∂ ∂ q q j j ⎠
Substitution of Eq. (7) into Eq. (2) and inserting the resulting expression along with Eq. (3) into Eq. (4) gives the potential energy of the system in terms of the generalized coordinates. Furthermore, inserting Eq.(7) into Eq. (1) results in the kinetic energy in terms of generalized coordinates. Moreover, substituting Eq. (7) into Eqs. (5) and (6), and inserting the resultant equations into Eq. (9) gives the
11
generalized forces in terms of generalized coordinates. Substituting these functions into Eq. (10) leads to a set of N coupled second‐order nonlinear ordinary differential equations. Transforming this set into 2N first‐order nonlinear ordinary differential equations via a change of variables by means of the transformation yi =qi with i = 1,2,..., N allows for application of standard numerical techniques. This set is solved via direct time integration by means of Gear’s backward‐differentiation‐formula (BDF), yielding time histories of all generalized coordinates. 3. Bifurcation diagrams of Poincaré maps The nonlinear global dynamics of the system is investigated in this section by means of direct time integration via Gear’s backward‐differentiation‐ formula (BDF). The forcing amplitude and axial speed are taken as bifurcation parameters in the first and second parts, respectively. Poincaré section at each bifurcation parameter is plotted, yielding bifurcation diagrams of Poincaré maps, which helps us in identifying different motions such as periodic, period‐n, quasiperiodic, and chaotic ones. At each bifurcation parameter, the time response is obtained for a sufficient time interval in order to exclude any possible transient effects. The phase space is sectioned in every period of the
12
exciting force in order to construct the bifurcation diagrams of Poincaré maps. In order to follow a certain attractor by increasing the given bifurcation parameter, the final value of generalized coordinates at each step is taken as the initial condition for the next step. Moreover, it is implied response and amplitude are with respect to the w1,1 motion and the amplitude of the w1,1 motion where it is sectioned, respectively. A rectangular aluminium plate with the following dimensions and mechanical properties are considered: a=0.515 m, b=0.184 m, h=0.0003 m, E=69 GPa, ρ=2700 kg/m3, and ν=0.33. The same modal damping ratio of ζ=0.0117 is utilized for all generalized coordinates, according to Ref. [10] and [12]; the pretension P=10 N/m is used for all cases. A thirty‐degree‐of‐freedom system involving 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; specifically, 60 first‐order nonlinear ordinary differential equations with coupled terms are solved via direct time integration.
13
3.1. The forcing amplitude as the bifurcation parameter The bifurcation diagrams of Poincaré maps for the first four generalized coordinates corresponding to the out‐of‐plane motion versus the forcing amplitude are shown in Fig. 2, 5, 7, 8, and 10 for the axial speeds of cv= 5.0000, 10.0000, 13.0000, 15.0000 , and 28.0000. The excitation frequency is set to the linear fundamental natural frequency for the out‐of‐plane motion and π/12 phase‐shift is induced in the excitation force. Figure 2 (a‐d) presents the bifurcation diagram of the system with cv= 5.0000. As seen in Fig. 2(a), for zero forcing amplitude it was found that the response is zero; this is as expected as it is physically meaningful for the system in the subcritical regime. As the forcing amplitude is increased gradually from zero, the periodic response amplitude increases accordingly until f1=4.0268; typical characteristics of the periodic motion at f1=2.0134 are shown in Fig. 3 through time histories, phase‐plane portraits, and Poincaré sections of the generalized coordinates w1,1, w2,1, w3,1, and w4,1, respectively. As the forcing amplitude is increased slightly, the motion becomes quasiperiodic in the interval [4.1074 4.5906]; some typical characteristics of the quasiperiodic motion in this interval are shown in Fig. 4 (a‐l) for f1=4.2685—closed circuits in sub‐figures (i‐l) illustrate the quasiperiodicity. At f1=4.6711, the system returns to its original
14
period and maintains that period until f1=7.0067, where a jump occurs. As the forcing amplitude is increased further, another jump occurs at f1=8.4564. At f1=9.1007, the motion becomes quasiperiodic and the system maintains that quasiperiodicity until the next event. This quasiperiodic motion leads the system to have dominantly chaotic motion in the forcing amplitude range of [9.5034 12.0000]. Increasing the axial speed to 10.0000, from 5.0000 in Fig. 2, Fig. 5 is generated. As seen in Fig. 5 (a), as the forcing amplitude is increased the response amplitude increases until a jump occurs at f1=1.6913, which is accompanied by a sudden reduction in the amplitude. By further incrementing the forcing amplitude, the response amplitude increases until f1=5.6376. The motion is dominantly quasiperiodic in the interval of [5.7181 7.3289], except at few points where it becomes chaotic. At f1=7.4094, the system returns to its original period and maintains that period until f1=9.1812, where a small jump occurs. Increasing the forcing amplitude even further, the motion becomes quasiperiodic in the interval [10.3087 12.0000]; the quasiperiodic motion characteristics are illustrated in Fig.6 for f1=12.000. Figure 7 shows the bifurcation diagrams obtained for the case with cv= 13.0000; this axial speed is higher than that of the previous cases (Figs. 2 and 5).
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For this case, due to the effect of increased axial speed, the number of complex regions in the cascade of bifurcations decreases. Investigating the dynamics of the system with higher axial speeds, specifically cv= 15.0000 (Fig. 8 (a‐d)) and 28.0000 (Fig. 10 (a‐d)) reveals that, compared to the case of Fig. 7 (cv= 13.0000), due to increased axial speed, the cascade of strange attractors increases and becomes stronger; typical characteristics of a chaotic motion is shown in Fig. 9 (a‐l). Moreover, for the system in the supercritical axial speed regime (see Fig. 10 (d), for example), when the external excitation amplitude is zero, the response amplitude for some generalized coordinates is a non‐zero value; this implies that the system has lost its stability via a pitchfork bifurcation leading to buckling, even though f1=0.0000. 3.2. The axial speed as the bifurcation parameter In this section, the axial speed is chosen as the bifurcation parameter in the bifurcation diagrams of Poincaré maps for six constant forcing amplitudes. These bifurcation diagrams are plotted in Figs. 11, 12, 13, 14, 16, and 17 for the system with f1=1.0000, 2.0000, 3.0000, 4.0000, 6.0000, and 8.0000, respectively.
16
The excitation frequency is set to 200.0000 rad/s, and π/12 phase‐shift is induced in the excitation force. In Fig. 11 (a‐d) with f1=1.0000, the axial speed is increased and shows gradual increase in the amplitude until cv=7.6510. The motion becomes dominantly quasiperiodic in the interval of [7.8523 8.8591]; at cv=9.0604, the system returns to its original period and maintains that period until cv=12.6846. There are two attractors in the interval [12.8859 24.7651], namely quasiperiodic and chaotic, except at cv=22.5503 and 22.7517, where there are periodic limit cycles; a jump occurs at cv=22.3490 which is accompanied by a quasiperiodic motion. The system regains the original period at cv=24.9664 and maintains it until cv=30.0000. The bifurcation diagram of the same system of Fig. 11(a, b), but with a higher forcing amplitude (i.e. f1=2.0000) is shown in Fig. 12(a‐d). As seen in Fig. 12(a), the motion is periodic in the interval [0.0000 12.6846]; there is a small jump in the vicinity of cv=8.4564. A dominant quasiperiodicity is observed in the interval [12.8859 14.6980] with few exceptional chaotic attractors. Further increasing the axial speed, the system regains the original period at cv=14.8993 and maintains that period until cv=17.9195. Increasing the axial speed slightly, the motion becomes dominantly chaotic in the interval [18.1208 25.7718],
17
except at few point where it is either periodic or quasiperiodic. Increasing the axial speed even further, the motion becomes periodic at cv=25.9732 which continues until cv=30.0000. The bifurcation diagrams of the system with higher forcing amplitudes are shown in Figs. 13, 14, 16, and 17 for the system with f1=3.0000, 4.0000, 6.0000, and 8.0000, respectively; for all of these cases, the system shows very interesting dynamics including periodic, period‐doubling, quasiperiodic, and chaotic motions—see Fig.14 and 15 for a period‐2 motion. From comparison of the system dynamics in these figures, one may conclude that it is hard to make any general conclusion regarding the effect of the forcing amplitude on the occurrence and strength of different attractors. Nevertheless, varying the forcing amplitude changes the dynamical behaviour of the system not only quantitatively, but also qualitatively. Moreover, it is shown that higher axial speed does not necessarily lead the system to have a chaotic motion. Furthermore, as is expected from physical perspective, at zero axial speed, there is a non‐trivial periodic attractor.
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4. Conclusions The sub and supercritical nonlinear global dynamics of an axially moving plate subjected to a distributed harmonic excitation load has been investigated numerically. The bifurcation diagrams of Poincaré maps were constructed by varying either the axial speed or the forcing amplitude as the bifurcation parameter. The system was modelled as a von Kármán plate, taking into account Kirchhoff’s hypothesis and retaining in‐plane displacements and inertia. The equations of motion were obtained via an energy method based on Lagrange equations yielding a set of second‐order nonlinear coupled ordinary differential equations. These equations were integrated numerically via Gear’s backward‐ differentiation‐formula (BDF) and the bifurcation diagrams of Poincaré maps were constructed. The results showed that, depending on different system parameters, very interesting and rich dynamical behaviour involving periodic, quasiperiodic, period‐2 and chaotic motions is displayed by the system. In connection with the cases where the forcing amplitude is varied as the bifurcation parameter, it was found that the system with supercritical axial speed loses stability even when the excitation amplitude is zero. Regarding the cases where the axial speed is the control parameter, it was shown that the bifurcation diagrams for different
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forcing amplitudes differ from each other both qualitatively and quantitatively. However, it is impossible to make any general conclusion on the trend of these changes. Moreover, analyzing the bifurcation diagrams, varying either the axial speed or the forcing amplitude, revealed that, the system always starts from simple periodic motion, and there is no sudden strange attractor emerging immediately as the bifurcation parameter is increased.
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References: [1] E. Carrera, G. Giunta, M. Petrolo, Beam Structures: Classical and Advanced Theories, John Wiley & Sons Ltd, 2011. [2] H.R. Öz, M. Pakdemirli, E. Özkaya, Transition behaviour from string to beam for an axially accelerating material, Journal of Sound and Vibration, 215 (1998) 571‐576. [3] M. Pakdemirli, E. Özkaya, Approximate boundary layer solution of a moving beam problem, Mathematical and Computational Applications, 3 (1998) 93‐100. [4] M. Pakdemirli, A.G. Ulsoy, Stability analysis of an axially accelerating string, Journal of Sound and Vibration, 203 (1997) 815‐832. [5] M. Pakdemirli, A.G. Ulsoy, A. Ceranoglu, Transverse vibration of an axially accelerating string, Journal of Sound and Vibration, 169 (1994) 179‐196. [6] 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) 996‐1008. [7] B. Wang, L.‐Q. Chen, Asymptotic stability analysis with numerical confirmation of an axially accelerating beam constituted by the standard linear solid model, Journal of Sound and Vibration, 328 (2009) 456‐466. [8] M.H. Ghayesh, H.A. Kafiabad, T. Reid, Sub‐ and super‐critical nonlinear dynamics of a harmonically excited axially moving beam, International Journal of Solids and Structures, 49 (2012) 227‐243. [9] M.H. Ghayesh, Coupled longitudinal–transverse dynamics of an axially accelerating beam, Journal of Sound and Vibration, 331 (2012) 5107‐5124. [10] M. Amabili, Nonlinear vibrations of rectangular plates with different boundary conditions: theory and experiments, Computers & Structures, 82 (2004) 2587‐2605. [11] M. Amabili, Theory and experiments for large‐amplitude vibrations of rectangular plates with geometric imperfections, Journal of Sound and Vibration, 291 (2006) 539‐ 565. [12] M. Amabili, Nonlinear Vibrations and Stability of Shells and Plates, Cambridge University Press, New York, 2008. [13] C.C. Lin, Stability and vibration characteristics of axially moving plates, International Journal of Solids and Structures, 34 (1997) 3179‐3190. [14] J. Kim, J. Cho, U. Lee, S. Park, Modal spectral element formulation for axially moving plates subjected to in‐plane axial tension, Computers & Structures, 81 (2003) 2011‐ 2020. [15] Z. Yin‐feng, W. Zhong‐min, Vibrations of axially moving viscoelastic plate with parabolically varying thickness, Journal of Sound and Vibration, 316 (2008) 198‐210. [16] N. Banichuk, J. Jeronen, P. Neittaanmäki, T. Tuovinen, On the instability of an axially moving elastic plate, International Journal of Solids and Structures, 47 (2010) 91‐99. [17] X.‐D. Yang, L.‐Q. Chen, J.W. Zu, Vibrations and stability of an axially moving rectangular composite plate, Journal of Applied Mechanics, 78 (2011) 011018.
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[18] Y.‐Q. Tang, L.‐Q. Chen, Nonlinear free transverse vibrations of in‐plane moving plates: Without and with internal resonances, Journal of Sound and Vibration, 330 (2011) 110‐126. [19] Y.‐Q. Tang, L.‐Q. Chen, Primary resonance in forced vibrations of in‐plane translating viscoelastic plates with 3:1 internal resonance, Nonlinear Dynamics, 1‐14. [20] A.C.J. Luo, H.R. Hamidzadeh, The nonlinear vibration and stability of axially travelling thin plates, in: Proceedings of the ASME Design Engineering Technical Confrence, 2003, pp. 1135‐1143. [21] S. Hatami, M. Azhari, M.M. Saadatpour, Nonlinear analysis of axially of axially moving plates using FEM, International Journal of Structural Stability and Dynamics, 7 (2007) 589‐607.
22
b
cv
a
Fig.1. Schematic representation of an axially moving plate
23
(a)
(b)
(c)
(d)
Fig.2. Bifurcation diagrams of Poincaré points for increasing forcing amplitude on the system with cv=5.0000; (a) the generalized coordinate w1,1; (b) the generalized coordinate w2,1; (c) the generalized coordinate w3,1; (d) the generalized coordinate w4,1.
24
(a)
(b) 0.3
1
w2,1/h
w1,1/h
0.5
0
0
-0.5
-1 2940
2960
2980
-0.3 2940
3000
Time (dimensionless seconds)
2960
2980
3000
Time (dimensionless seconds)
(c)
(d) 0.03
w4,1/h
w3,1/h
0.04
0
-0.04 2940
2960
2980
0
-0.03 2940
3000
2960
2980
3000
Time (dimensionless seconds)
Time (dimensionless seconds)
(e)
(f) 0.2
0.5
(dw2,1/dt)/hω1,1
(dw1,1/dt)/hω1,1
0.1
0
0
-0.1 -0.5
-0.5
0
w1,1/h
0.5
-0.2
-0.1
0
0.1
0.2
w2,1/h
25
(g)
(h) 0.02
0.04
(dw4,1/dt)/hω1,1
(dw3,1/dt)/hω1,1
0.02
0
0
-0.02
-0.04
-0.01
0
0.01
-0.02
-0.01
w3,1/h
0
0.01
0.02
w4,1/h
(i)
(j) 1
0.15
(dw2,1/dt)/hω1,1
(dw1,1/dt)/hω1,1
0.5
0
-0.5
-1 -1
-0.5
0
0.5
0
1
w1,1/h
0
0.02
0.04
0.06
w2,1/h
(k)
(l)
0.01
(dw4,1/dt)/hω1,1
(dw3,1/dt)/hω1,1
0.02
0 -0.006
-0.004
-0.002
w3,1/h
0
0
0
0.0005
0.001
0.0015
0.002
w4,1/h Fig.3. Periodic oscillation for the system of Fig.2 at f1=2.0134 (ω1,1= 140.3774 rad/s): (a‐d) time traces of the the generalized coordinates w1,1, w2,1, w3,1, and w4,1, respectively; (e‐h) phase‐plane portraits of the the generalized coordinates w1,1, w2,1, w3,1, and w4,1, respectively; (i‐l) Poincaré sections of the the generalized coordinates w1,1, w2,1, w3,1, and w4,1, respectively.
26
(a)
(b) 1
0.5
w2,1/h
w1,1/h
0.5
0
0
-0.5
-1 2940
2960
2980
3000
Time (dimensionless seconds)
-0.5 2940
(c)
2960
2980
3000
Time (dimensionless seconds)
(d) 0.5
w4,1/h
w3,1/h
0.2
0
0
-0.2
-0.5 2940
2960
2980
3000
Time (dimensionless seconds)
(e)
2940
2960
2980
3000
Time (dimensionless seconds)
(f) 1 0.4
(dw2,1/dt)/hω1,1
(dw1,1/dt)/hω1,1
0.5
0
0
-0.5 -0.4
-1 -1
-0.2
-0.5
0
w1,1/h
0.5
1
0
w2,1/h
0.2
27
(g)
(h) 0.4
0.4
(dw4,1/dt)/hω1,1
(dw3,1/dt)/hω1,1
0.2
0
0
-0.2
-0.4
-0.4 -0.2
0
-0.2
0.2
0
0.2
w4,1/h
w3,1/h
(i)
(j) 0.3
(dw1,1/dt)/hω1,1
(dw2,1/dt)/hω1,1
0
-0.4
-0.8
0
-0.2
0.6
0.8
-0.1
0
w1,1/h
(k)
0.1
0.2
w2,1/h
1
(l) 0.2
(dw4,1/dt)/hω1,1
(dw3,1/dt)/hω1,1
0.3
0
0
-0.2
-0.3
-0.1
0
w3,1/h
-0.05
0
0.05
0.1
w4,1/h
Fig.4. Quasiperiodic oscillation for the system of Fig.2 at f1=4.2685 (ω1,1= 140.3774 rad/s): (a‐d) time traces of the the generalized coordinates w1,1, w2,1, w3,1, and w4,1, respectively; (e‐h) phase‐ plane portraits of the the generalized coordinates w1,1, w2,1, w3,1, and w4,1, respectively; (i‐l) Poincaré sections of the the generalized coordinates w1,1, w2,1, w3,1, and w4,1, respectively.
28
(a)
(b)
(c)
(d)
Fig.5. Bifurcation diagrams of Poincaré points for increasing forcing amplitude on the system with cv=10.0000; (a) the generalized coordinate w1,1; (b) the generalized coordinate w2,1; (c) the generalized coordinate w3,1; (d) the generalized coordinate w4,1.
29
(a)
(b) 0.5
w2,1/h
w1,1/h
1
0
0
-1
2940
2960
2980
-0.5 2940
3000
(c)
2960
2980
3000
Time (dimensionless seconds)
Time (dimensionless seconds)
(d) 0.2
w4,1/h
w3,1/h
0.2
0
0
-0.2
-0.2
2940
2960
2980
2940
3000
(e)
2960
2980
3000
Time (dimensionless seconds)
Time (dimensionless seconds)
(f) 1
(dw2,1/dt)/hω1,1
(dw1,1/dt)/hω1,1
1
0
0
-1
-1
-1
0
w1,1/h
1
-0.4
0
w2,1/h
0.4
30
(g)
(h) 0.4
(dw4,1/dt)/hω1,1
(dw3,1/dt)/hω1,1
0.5
0
0
-0.5
-0.4 -0.1
0
-0.1
0.1
0
0.1
w4,1/h
w3,1/h
(i)
(j)
-0.4
(dw1,1/dt)/hω1,1
(dw2,1/dt)/hω1,1
0.8
-0.6
0.6
1
1.04
1.08
1.12
0.22
w1,1/h
0.24
0.26
0.28
w2,1/h
(k)
(l) 0.4
(dw4,1/dt)/hω1,1
(dw3,1/dt)/hω1,1
-0.1
0.3
-0.2
0.2
-0.3
0.04
0.06
w3,1/h
0.08
0.08
0.1
0.12
0.14
w4,1/h Fig.6. Quasiperiodic oscillation for the system of Fig.5 at f1=12.000 (ω1,1= 99.6403 rad/s): (a‐d) time traces of the the generalized coordinates w1,1, w2,1, w3,1, and w4,1, respectively; (e‐h) phase‐ plane portraits of the the generalized coordinates w1,1, w2,1, w3,1, and w4,1, respectively; (i‐l) Poincaré sections of the the generalized coordinates w1,1, w2,1, w3,1, and w4,1, respectively.
31
(a)
(b)
(c)
(d)
Fig.7. Bifurcation diagrams of Poincaré points for increasing forcing amplitude on the system with cv=13.0000; (a) the generalized coordinate w1,1; (b) the generalized coordinate w2,1; (c) the generalized coordinate w3,1; (d) the generalized coordinate w4,1.
32
(a)
(b)
(c)
(d)
Fig.8. Bifurcation diagrams of Poincaré points for increasing forcing amplitude on the system with cv=15.0000; (a) the generalized coordinate w1,1; (b) the generalized coordinate w2,1; (c) the generalized coordinate w3,1; (d) the generalized coordinate w4,1.
33
(a)
(b) 0.5
w2,1/h
w1,1/h
1
0
0
-1
2940
2960
2980
3000
Time (dimensionless seconds)
-0.5 2940
(c)
2960
2980
3000
Time (dimensionless seconds)
(d) 0.5
w4,1/h
w3,1/h
0.3
0
0
-0.3 -0.5 2940
2960
2980
2940
3000
2960
2980
3000
Time (dimensionless seconds)
Time (dimensionless seconds)
(e)
(f) 4
3
(dw2,1/dt)/hω1,1
(dw1,1/dt)/hω1,1
2
0
-2
0
-3 -4 -1
-0.5
0
w1,1/h
0.5
1
-0.5
0
0.5
w2,1/h
34
(g)
(h)
3
(dw4,1/dt)/hω1,1
(dw3,1/dt)/hω1,1
3
0
0
-3
-3
-0.4
0
0.4
-0.4
-0.2
w3,1/h
0
0.2
0.4
w4,1/h
(i)
(j) 6 4
2
(dw2,1/dt)/hω1,1
(dw1,1/dt)/hω1,1
3
0
0
-2
-4
-3 -6 -0.5
0
0.5
1
0
1.5
0.5
w2,1/h
w1,1/h
(k)
(l) 6
3
(dw4,1/dt)/hω1,1
(dw3,1/dt)/hω1,1
3
0
0
-3
-3 -6
-0.6
-0.4
-0.2
0
w3,1/h
0.2
0.4
-0.4
0
0.4
w4,1/h
Fig.9. Chaotic oscillation for the system of Fig.8 at f1=11.1946 (ω1,1= 29.9911 rad/s): (a‐d) time traces of the the generalized coordinates w1,1, w2,1, w3,1, and w4,1, respectively; (e‐h) phase‐plane portraits of the the generalized coordinates w1,1, w2,1, w3,1, and w4,1, respectively; (i‐l) Poincaré sections of the the generalized coordinates w1,1, w2,1, w3,1, and w4,1, respectively.
35
(a)
(b)
(c)
(d)
Fig.10. Bifurcation diagrams of Poincaré points for increasing forcing amplitude on the system with cv=28.0000; (a) the generalized coordinate w1,1; (b) the generalized coordinate w2,1; (c) the generalized coordinate w3,1; (d) the generalized coordinate w4,1.
36
(a)
(b)
(c)
(d)
Fig.11. Bifurcation diagrams of Poincaré points for increasing axial speed of the system with f1=1.0000; (a) the generalized coordinate w1,1; (b) the generalized coordinate w2,1; (c) the generalized coordinate w3,1; (d) the generalized coordinate w4,1.
37
(a)
(b)
(c)
(d)
Fig.12. Bifurcation diagrams of Poincaré points for increasing axial speed of the system with f1=2.0000; (a) the generalized coordinate w1,1; (b) the generalized coordinate w2,1; (c) the generalized coordinate w3,1; (d) the generalized coordinate w4,1.
38
(a)
(b)
(c)
(d)
Fig.13. Bifurcation diagrams of Poincaré points for increasing axial speed of the system with f1=3.0000; (a) the generalized coordinate w1,1; (b) the generalized coordinate w2,1; (c) the generalized coordinate w3,1; (d) the generalized coordinate w4,1.
39
(a)
(b)
(c)
(d)
Fig.14. Bifurcation diagrams of Poincaré points for increasing axial speed of the system with f1=4.0000; (a) the generalized coordinate w1,1; (b) the generalized coordinate w2,1; (c) the generalized coordinate w3,1; (d) the generalized coordinate w4,1.
40
(a)
(b)
0.3
w2,1/h
w1,1/h
0.1
0
0
-0.3
-0.1
3550
3560
3570
3580
3590
3550
3600
3560
3570
3580
3590
3600
Time (dimensionless seconds)
Time (dimensionless seconds)
(c)
(d) -0.9
w4,1/h
w3,1/h
0.2
0
-1
-0.2
-1.1 3550
3560
3570
3580
3590
3600
3550
Time (dimensionless seconds)
3560
3570
3580
3590
3600
Time (dimensionless seconds)
(e)
(f) 0.1
(dw2,1/dt)/hω
(dw1,1/dt)/hω
0.2
0
0
-0.1
-0.2
-0.2
-0.05
0
w1,1/h
0.05
0.1
-0.4
-0.2
0
0.2
w2,1/h
41
(g)
(h) 0.2
(dw4,1/dt)/hω
(dw3,1/dt)/hω
0.1
0
0
-0.2
-0.1
-0.2
0
0.2
-1.05
w3,1/h
-1
-0.95
w4,1/h
(i)
(j) 0.2
(dw2,1/dt)/hω
(dw1,1/dt)/hω
0.1
0
-0.1
0
-0.1
-0.2
0
0.1
0.2
0.3
w1,1/h
-0.06
-0.04
-0.02
(k)
0
0.02
0.04
w2,1/h
(l)
(dw4,1/dt)/hω
(dw3,1/dt)/hω
0.1 0
-0.02
1.1
0
1.2
1.3
w3,1/h
1.4
-0.02
-0.01
0
0.01
0.02
0.03
0.04
w4,1/h
Fig.15. Period‐2 oscillation for the system of Fig.14 at cv=30.0000: (a‐d) time traces of the the generalized coordinates w1,1, w2,1, w3,1, and w4,1, respectively; (e‐h) phase‐plane portraits of the the generalized coordinates w1,1, w2,1, w3,1, and w4,1, respectively; (i‐l) Poincaré sections of the the generalized coordinates w1,1, w2,1, w3,1, and w4,1, respectively.
42
(a)
(b)
(c)
(d)
Fig.16. Bifurcation diagrams of Poincaré points for increasing axial speed of the system with f1=6.0000; (a) the generalized coordinate w1,1; (b) the generalized coordinate w2,1; (c) the generalized coordinate w3,1; (d) the generalized coordinate w4,1.
43
(a)
(b)
(c)
(d)
Fig.17. Bifurcation diagrams of Poincaré points for increasing axial speed of the system with f1=8.0000; (a) the generalized coordinate w1,1; (b) the generalized coordinate w2,1; (c) the generalized coordinate w3,1; (d) the generalized coordinate w4,1.
44
Highlights: ‐ ‐ ‐ ‐
The three‐dimensional nonlinear global dynamics of axially moving plates is investigated numerically. The governing equations of motion are obtained by means of Lagrange equations. These equations are solved via direct time integration by means of Gear’s backward‐differentiation‐formula (BDF). Bifurcation diagrams of Poincaré maps are constructed.
45
Nonlinear global dynamics of an axially moving plate Mergen H. Ghayesh, Marco Amabili
www.elsevier.com/locate/nlm
PII: DOI: Reference:
S0020-7462(13)00120-0 http://dx.doi.org/10.1016/j.ijnonlinmec.2013.06.005 NLM2183
To appear in:
International Journal of Non-Linear Mechanics
Received date: 20 September 2012 Revised date: 27 May 2013 Accepted date: 8 June 2013 Cite this article as: Mergen H. Ghayesh, Marco Amabili, Nonlinear global dynamics of an axially moving plate, International Journal of Non-Linear Mechanics, http://dx.doi.org/10.1016/j.ijnonlinmec.2013.06.005 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Nonlinear global dynamics of an axially moving plate Mergen H. Ghayesh*, Marco Amabili Department of Mechanical Engineering, McGill University, 817 Sherbrooke St. West, Montreal, Quebec, Canada H3A 0C3 *Corresponding author: [email protected], Tel: (+1) 514 398‐6290
Abstract In the present study, the geometrically nonlinear dynamics of an axially moving plate is examined by constructing the bifurcation diagrams of Poincaré maps for the system in the sub and supercritical regimes. The von Kármán plate theory is employed to model the system by retaining in‐plane displacements and inertia. The governing equations of motion of this gyroscopic system are obtained based on an energy method by means of Lagrange equations which yields a set of second‐order nonlinear ordinary differential equations with coupled terms. A change of variables is employed to transform this set into a set of first‐order nonlinear ordinary differential equations. The resulting equations are solved using direct time integration, yielding time‐varying generalized coordinates for the in‐plane and out‐of‐plane motions. From these time histories, the bifurcation diagrams of Poincaré maps, phase‐plane portraits, and Poincaré sections are
1
constructed at points of interest in the parameter space for both the axial speed regimes. Keywords: Axially moving plates; Nonlinear dynamics; Bifurcations; Stability 1. Introduction The class of axially moving systems are widely used in many engineering devices and machine components; for example, they are simplified models of band saw blades, paper sheets, textile fibers, robotic manipulators, conveyor belts, and magnetic tapes. The axial speed plays an important role in the dynamical behaviour of this class of systems. The natural frequencies of the system decrease with increasing axial speed and at a certain value, which is called critical axial speed, the first natural frequency of a conservative system vanishes, which causes the system to lose stability via divergence, leading to buckling. The axial speed range before the critical axial speed is called subcritical regime and that after the critical speed is called supercritical regime. Studies concerning the dynamics of axially moving systems can be classified mainly into two general groups in terms of models being considered, i.e. one‐dimensional
2
such as beams and strings [1], and two‐dimensional such as membranes and plates. The linear and nonlinear dynamics of axially moving one‐dimensional systems (such as beams and strings) has received considerable attention in the literature. For example, Pakdemirli and co‐workers [2‐5] investigated the dynamics of axially moving beams and strings by means of the method of multiple timescales and matched asymptotic expansion. A systematic research on the dynamics of axially moving beams and strings were conducted by Chen and co‐workers [6, 7], who introduced viscoelastic models for the system, and employed different analytical and numerical techniques. Sub and supercritical nonlinear dynamics of an axially moving beam with special consideration to the case with an internal resonance was examined numerically by Ghayesh et al. [8]. The coupled longitudinal‐transverse dynamics of an axially accelerated beam was investigated by Ghayesh [9], who discovered period‐doubling bifurcations in the resonant dynamics of the system. The literature on the dynamics of axially moving two‐dimensional models (such as membranes and plates) is not large; the dynamics of stationary (not axially moving) plates has been studied extensively, for example, by Amabili and co‐workers [10‐12]. The linear models of axially moving two‐dimensional
3
systems were developed and analyzed, for instance, by: Lin [13], who examined the influence of a uniform in‐plane tension in the transport direction on the dynamics of an axially moving plate; Kim et al. [14], who developed a modal spectral element for a thin axially moving plate travelling at a constant speed and subjected to a uniform axial tension; Yin‐feng and Zhong‐min [15], who introduced the Kelvin‐Voigt viscoelastic model to the mathematical model of an axially moving plate with parabolically varying thickness; Banichuk et al. [16], who examined the stability of an axially moving elastic thin plate; and Yang et al. [17], who investigated the dynamics of an axially moving composite plate. These studies were extended and pursued for nonlinear models of axially moving two‐ dimensional models; the literature in this case is not large [18, 19]. For instance, Luo and Hamidzadeh [20] determined the natural frequencies and dynamic responses of an axially moving plate. Hatami et al. [21] employed the finite element method to examine the nonlinear vibrations of an axially moving plate. The three‐dimensional (i.e., by retaining all in‐plane and out‐of‐plane displacements and inertia) nonlinear global dynamics of axially moving plates has not been studied yet; in other words, bifurcation diagrams Poincaré maps for an axially moving plate has not been investigated in the literature yet. The current paper is the first to do so.
4
In this effort, the nonlinear global dynamics of an axially moving plate subjected to a distributed harmonic excitation load is investigated by constructing the bifurcation diagrams of Poincaré maps as either the axial speed or the forcing amplitude is varied. The plate is modeled via von Kármán plate theory taking into account Kirchhoff’s hypothesis, retaining in‐plane displacements and inertia; this paper is the first which retains all in‐plane and out‐of‐plane displacements in the nonlinear analysis of an axially moving plate‐‐ the computer codes should be well‐optimized for this operation. The potential and kinetic energies of the system are constructed and inserted in the Lagrange equations which yield a set of second‐order nonlinear coupled ordinary differential equations. These equations are solved via direct time integration by
means of Gear’s backward‐differentiation‐formula (BDF) dealing with the stiff nonlinear equations, yielding the amplitudes of oscillation as functions of time. Thus, bifurcation diagrams of Poincaré maps are constructed using either the axial speed or the forcing amplitude as the bifurcation parameter. The analyses also include the system in the supercritical regime. This paper is the first which investigates the nonlinear global dynamics of an axially moving plate. The nonlinear global dynamics of an axially moving plate retaining all in‐plane and
5
out‐of‐plane displacements and inertia has not been studied before in the literature; the bifurcation diagrams of Poincaré maps are constructed in this paper for the first time for an axially moving plate retaining all in‐plane and out‐ of‐plane displacements and inertia.
2. Model development and method of solution As shown in Fig. 1, a rectangular plate with Cartesian coordinate system (O; x, y, z), having the origin O at one corner is considered. In‐plane dimensions and thickness are denoted by a, b, and h, respectively; x and y axes define the mid‐plane of the plate and z denotes the out‐of‐plane coordinate. The displacements of each point of the mid‐plane of the plate are denoted by u=u(x,y,t), v=v(x,y,t), and w=w(x,y,t) in the x, y, and z directions from the static equilibrium (u=v=w=0), respectively. The plate is assumed to be subject to an in‐ plane pretension per unit width P in the x direction. The plate is also considered to be travelling in the x direction at a constant axial speed cv (in m/s). Moreover, a distributed harmonic force per unit area f1 sin (π x a ) sin (π y b ) co s(ω t ) , orthogonal to the plate, is applied; ω is the excitation frequency and the forcing
6
amplitude is denoted by f1 sin (π x a ) sin (π y b ) , positive in the z direction (f1 is in N/m2). In what follows, the kinetic and potential energies of the system are constructed as functions of the in‐plane and out‐of‐plane displacements; they are then inserted into the Lagrange equations so as to obtain the discretized equations of motion. Compared to the case of a stationary plate, due to presence of the axial speed, axial speed‐dependent terms emerge in the kinetic energy of the system; the kinetic energy of the system is constructed as 2 2 2 a b ⎡ ⎛ ∂u ⎞ 1 ⎛ ∂v ⎞ ⎛ ∂ w ⎞ ⎤ TP = ρ h ∫ ∫ ⎢⎜ ⎟ + ⎜ ⎟ + ⎜ ⎟ ⎥ dxdy 0 0 2 t t t ∂ ∂ ∂ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎦ ⎣ 2 2 2 a b ⎡ ⎛ ∂u ⎞ 1 ⎛ ∂ v ⎞ ⎛ ∂w ⎞ ⎤ + ρ hcv 2 ∫ ∫ ⎢⎜ ⎟ + ⎜ ⎟ + ⎜ ⎟ ⎥ dxdy 0 0 ∂ ∂ ∂ 2 x x x ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎦ ⎣
⎡ ∂u ∂u ∂v ∂ v ∂ w ∂ w ⎤ + + dxdy 0 ∫0 ⎢ ∂t ∂x ∂t ∂x ∂t ∂x ⎥⎦ ⎣ a b⎡ ∂u ∂u ⎤ 1 + 2cv 2 ⎥ dxdy , + ρ h ∫ ∫ ⎢ cv 2 + 2cv 0 0 ∂t ∂x ⎦ 2 ⎣ + ρ hcv ∫
a
b
(1)
where ρ denotes the mass density of the plate and t represents time. The following relations, between the strain of a generic point of the plate and the displacement field, are given for the von Kármán plate theory [12]:
7
∂u 1 ∂ w ⎞ ∂ 2w ε xx = + ⎛⎜ ⎟ −z 2 , ∂x 2 ⎝ ∂x ⎠ ∂x 2
2
∂v 1 ⎛ ∂ w ⎞ ∂ 2w (2) − ε yy = + ⎜ z , ∂y 2 ⎝ ∂y ⎟⎠ ∂y 2 ∂u ∂ v ∂ w ∂ w ∂ 2w γ xy = + + − 2z , ∂ y ∂x ∂ x ∂ y ∂ x ∂y
where ε xx , ε yy and γ xy represent the strain components at an arbitrary point of the plate at a distance z from the mid‐plane. As seen in Eq. (2), the source of nonlinearity is geometric due to large displacements. The relations between the Kirchhoff stresses ( σ xx , σ yy and τ xy ) and Green strains ( ε xx , ε yy and γ xy ) for homogeneous and isotropic materials are given by
E (ε xx +νε yy ) , 1 −ν 2 E σ yy = (ε yy +νε xx ) , (3) 1 −ν 2 E τ xy = γ xy , 2(1 +ν )
σ xx =
where ν and E are the Poisson ratio and Young’s modulus, respectively. Under Kirchhoff’s hypothesis, the elastic potential energy of the plate can be expressed as [12]
8
UP =
a b ⎛ ∂u 1 a b h /2 1 ∂w ⎞ σ xxε xx + σ yy ε yy + τ xy γ xy ) dxdydz + P ∫ ∫ ⎜⎜ + ⎜⎛ ( ⎟ ∫ ∫ ∫ 0 0 ∂x 2 0 0 − h /2 2 ⎝ ∂x ⎠ ⎝
2
⎞ ⎟⎟ dxdy. (4) ⎠
where it is assumed that there is a uniform pre‐stress σ xx(0) in the x direction, while there are no pre‐stress components in the normal stress in the y direction and the shear stress. Integrating this pre‐stress over the thickness of the plate gives the pretension per unit width, i.e. P = ∫
h /2
− h /2
σ (0) xx dz . Substitution of Eqs. (2) and
(3) into Eq. (4) results in the potential energy as a function of displacement field. The virtual work due to a distributed harmonic force per unit area f1 sin (π x a ) sin (π y b ) co s(ω t ) , orthogonal to the plate, may be expressed as
W =∫
a
0
∫
b
0
⎛ ⎞ ⎛πx ⎞ ⎛πy ⎞ ⎜ wf1 sin ⎜ a ⎟ sin ⎜ b ⎟ cos(ω t ) ⎟ dxdy , (5) ⎝ ⎠ ⎝ ⎠ ⎝ ⎠
where, it is assumed that the external forces in the x and y directions are zero. The nonconservative external damping forces, assumed to be of viscous type, are assumed using the Rayleigh dissipation function as follows: 2 2 2 1 a b ⎡ ⎛ ∂u ⎞ ⎛ ∂v ⎞ ⎛ ∂w ⎞ ⎤ F = c ∫ ∫ ⎢⎜ ⎟ + ⎜ ⎟ + ⎜ ⎟ ⎥ dxdy , (6) 2 0 0 ⎣ ⎝ ∂t ⎠ ⎝ ∂t ⎠ ⎝ ∂t ⎠ ⎦
where c is the damping coefficient.
9
The following expressions are used to expand the in‐plane and out‐of‐ plane displacements, satisfying identically the geometric boundary conditions of the simply‐supported plate with immovable edges: M
N
u(x , y ,t ) = ∑∑ um ,n (t )sin(mπ x / a)sin(nπ y / b), m =1 n =1 M
N
v(x , y ,t ) = ∑∑ vm ,n (t )sin(mπ x / a)sin(nπ y / b), (7) m =1 n =1 M N
w(x , y ,t ) = ∑∑ wm ,n (t )sin(mπ x / a)sin(nπ y / b), m =1 n =1
where m and n are the numbers of half‐waves in the x and y direction, respectively; um ,n (t ) , v m ,n (t ) , and w m ,n (t ) denote the generalized coordinates— these generalized coordinates are unknown functions of time which will be determined through numerical solutions. The exact values of M and N in Eq. (7) are not determined at this step, in order not to lose generality. Instead, the number of generalized coordinates employed is defined by subscripts in Section 3, last paragraph (above Section 3.1). All modes in series are not used one after the other; for more information, please see Ref. [10]. In what follows, the following notation is introduced for brevity: q = {um ,n , vm ,n , wm ,n } , (8) T
10
where the elements of the vector qi, are time‐dependent generalized coordinates and the dimension of this vector is equal to the number of degrees of freedom. The generalized forces Q j are obtained by differentiating the Rayleigh dissipation function and of the virtual work done by external forces
Qj = −
∂F ∂W , j = 1,...., N , (9) + ∂q j ∂q j
where N is the number of degrees of freedom. The Lagrange equations, which bypass setting up partial‐differential equations, and hence result in a set of nonlinear ordinary differential equations of motions are given as follows: d ⎛ ∂TP ⎜ dt ⎜⎝ ∂q j
⎞ ∂TP ∂UP + = Q j , j = 1,...., N , (10) ⎟⎟ − ∂ ∂ q q j j ⎠
Substitution of Eq. (7) into Eq. (2) and inserting the resulting expression along with Eq. (3) into Eq. (4) gives the potential energy of the system in terms of the generalized coordinates. Furthermore, inserting Eq.(7) into Eq. (1) results in the kinetic energy in terms of generalized coordinates. Moreover, substituting Eq. (7) into Eqs. (5) and (6), and inserting the resultant equations into Eq. (9) gives the
11
generalized forces in terms of generalized coordinates. Substituting these functions into Eq. (10) leads to a set of N coupled second‐order nonlinear ordinary differential equations. Transforming this set into 2N first‐order nonlinear ordinary differential equations via a change of variables by means of the transformation yi =qi with i = 1,2,..., N allows for application of standard numerical techniques. This set is solved via direct time integration by means of Gear’s backward‐differentiation‐formula (BDF), yielding time histories of all generalized coordinates. 3. Bifurcation diagrams of Poincaré maps The nonlinear global dynamics of the system is investigated in this section by means of direct time integration via Gear’s backward‐differentiation‐ formula (BDF). The forcing amplitude and axial speed are taken as bifurcation parameters in the first and second parts, respectively. Poincaré section at each bifurcation parameter is plotted, yielding bifurcation diagrams of Poincaré maps, which helps us in identifying different motions such as periodic, period‐n, quasiperiodic, and chaotic ones. At each bifurcation parameter, the time response is obtained for a sufficient time interval in order to exclude any possible transient effects. The phase space is sectioned in every period of the
12
exciting force in order to construct the bifurcation diagrams of Poincaré maps. In order to follow a certain attractor by increasing the given bifurcation parameter, the final value of generalized coordinates at each step is taken as the initial condition for the next step. Moreover, it is implied response and amplitude are with respect to the w1,1 motion and the amplitude of the w1,1 motion where it is sectioned, respectively. A rectangular aluminium plate with the following dimensions and mechanical properties are considered: a=0.515 m, b=0.184 m, h=0.0003 m, E=69 GPa, ρ=2700 kg/m3, and ν=0.33. The same modal damping ratio of ζ=0.0117 is utilized for all generalized coordinates, according to Ref. [10] and [12]; the pretension P=10 N/m is used for all cases. A thirty‐degree‐of‐freedom system involving 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; specifically, 60 first‐order nonlinear ordinary differential equations with coupled terms are solved via direct time integration.
13
3.1. The forcing amplitude as the bifurcation parameter The bifurcation diagrams of Poincaré maps for the first four generalized coordinates corresponding to the out‐of‐plane motion versus the forcing amplitude are shown in Fig. 2, 5, 7, 8, and 10 for the axial speeds of cv= 5.0000, 10.0000, 13.0000, 15.0000 , and 28.0000. The excitation frequency is set to the linear fundamental natural frequency for the out‐of‐plane motion and π/12 phase‐shift is induced in the excitation force. Figure 2 (a‐d) presents the bifurcation diagram of the system with cv= 5.0000. As seen in Fig. 2(a), for zero forcing amplitude it was found that the response is zero; this is as expected as it is physically meaningful for the system in the subcritical regime. As the forcing amplitude is increased gradually from zero, the periodic response amplitude increases accordingly until f1=4.0268; typical characteristics of the periodic motion at f1=2.0134 are shown in Fig. 3 through time histories, phase‐plane portraits, and Poincaré sections of the generalized coordinates w1,1, w2,1, w3,1, and w4,1, respectively. As the forcing amplitude is increased slightly, the motion becomes quasiperiodic in the interval [4.1074 4.5906]; some typical characteristics of the quasiperiodic motion in this interval are shown in Fig. 4 (a‐l) for f1=4.2685—closed circuits in sub‐figures (i‐l) illustrate the quasiperiodicity. At f1=4.6711, the system returns to its original
14
period and maintains that period until f1=7.0067, where a jump occurs. As the forcing amplitude is increased further, another jump occurs at f1=8.4564. At f1=9.1007, the motion becomes quasiperiodic and the system maintains that quasiperiodicity until the next event. This quasiperiodic motion leads the system to have dominantly chaotic motion in the forcing amplitude range of [9.5034 12.0000]. Increasing the axial speed to 10.0000, from 5.0000 in Fig. 2, Fig. 5 is generated. As seen in Fig. 5 (a), as the forcing amplitude is increased the response amplitude increases until a jump occurs at f1=1.6913, which is accompanied by a sudden reduction in the amplitude. By further incrementing the forcing amplitude, the response amplitude increases until f1=5.6376. The motion is dominantly quasiperiodic in the interval of [5.7181 7.3289], except at few points where it becomes chaotic. At f1=7.4094, the system returns to its original period and maintains that period until f1=9.1812, where a small jump occurs. Increasing the forcing amplitude even further, the motion becomes quasiperiodic in the interval [10.3087 12.0000]; the quasiperiodic motion characteristics are illustrated in Fig.6 for f1=12.000. Figure 7 shows the bifurcation diagrams obtained for the case with cv= 13.0000; this axial speed is higher than that of the previous cases (Figs. 2 and 5).
15
For this case, due to the effect of increased axial speed, the number of complex regions in the cascade of bifurcations decreases. Investigating the dynamics of the system with higher axial speeds, specifically cv= 15.0000 (Fig. 8 (a‐d)) and 28.0000 (Fig. 10 (a‐d)) reveals that, compared to the case of Fig. 7 (cv= 13.0000), due to increased axial speed, the cascade of strange attractors increases and becomes stronger; typical characteristics of a chaotic motion is shown in Fig. 9 (a‐l). Moreover, for the system in the supercritical axial speed regime (see Fig. 10 (d), for example), when the external excitation amplitude is zero, the response amplitude for some generalized coordinates is a non‐zero value; this implies that the system has lost its stability via a pitchfork bifurcation leading to buckling, even though f1=0.0000. 3.2. The axial speed as the bifurcation parameter In this section, the axial speed is chosen as the bifurcation parameter in the bifurcation diagrams of Poincaré maps for six constant forcing amplitudes. These bifurcation diagrams are plotted in Figs. 11, 12, 13, 14, 16, and 17 for the system with f1=1.0000, 2.0000, 3.0000, 4.0000, 6.0000, and 8.0000, respectively.
16
The excitation frequency is set to 200.0000 rad/s, and π/12 phase‐shift is induced in the excitation force. In Fig. 11 (a‐d) with f1=1.0000, the axial speed is increased and shows gradual increase in the amplitude until cv=7.6510. The motion becomes dominantly quasiperiodic in the interval of [7.8523 8.8591]; at cv=9.0604, the system returns to its original period and maintains that period until cv=12.6846. There are two attractors in the interval [12.8859 24.7651], namely quasiperiodic and chaotic, except at cv=22.5503 and 22.7517, where there are periodic limit cycles; a jump occurs at cv=22.3490 which is accompanied by a quasiperiodic motion. The system regains the original period at cv=24.9664 and maintains it until cv=30.0000. The bifurcation diagram of the same system of Fig. 11(a, b), but with a higher forcing amplitude (i.e. f1=2.0000) is shown in Fig. 12(a‐d). As seen in Fig. 12(a), the motion is periodic in the interval [0.0000 12.6846]; there is a small jump in the vicinity of cv=8.4564. A dominant quasiperiodicity is observed in the interval [12.8859 14.6980] with few exceptional chaotic attractors. Further increasing the axial speed, the system regains the original period at cv=14.8993 and maintains that period until cv=17.9195. Increasing the axial speed slightly, the motion becomes dominantly chaotic in the interval [18.1208 25.7718],
17
except at few point where it is either periodic or quasiperiodic. Increasing the axial speed even further, the motion becomes periodic at cv=25.9732 which continues until cv=30.0000. The bifurcation diagrams of the system with higher forcing amplitudes are shown in Figs. 13, 14, 16, and 17 for the system with f1=3.0000, 4.0000, 6.0000, and 8.0000, respectively; for all of these cases, the system shows very interesting dynamics including periodic, period‐doubling, quasiperiodic, and chaotic motions—see Fig.14 and 15 for a period‐2 motion. From comparison of the system dynamics in these figures, one may conclude that it is hard to make any general conclusion regarding the effect of the forcing amplitude on the occurrence and strength of different attractors. Nevertheless, varying the forcing amplitude changes the dynamical behaviour of the system not only quantitatively, but also qualitatively. Moreover, it is shown that higher axial speed does not necessarily lead the system to have a chaotic motion. Furthermore, as is expected from physical perspective, at zero axial speed, there is a non‐trivial periodic attractor.
18
4. Conclusions The sub and supercritical nonlinear global dynamics of an axially moving plate subjected to a distributed harmonic excitation load has been investigated numerically. The bifurcation diagrams of Poincaré maps were constructed by varying either the axial speed or the forcing amplitude as the bifurcation parameter. The system was modelled as a von Kármán plate, taking into account Kirchhoff’s hypothesis and retaining in‐plane displacements and inertia. The equations of motion were obtained via an energy method based on Lagrange equations yielding a set of second‐order nonlinear coupled ordinary differential equations. These equations were integrated numerically via Gear’s backward‐ differentiation‐formula (BDF) and the bifurcation diagrams of Poincaré maps were constructed. The results showed that, depending on different system parameters, very interesting and rich dynamical behaviour involving periodic, quasiperiodic, period‐2 and chaotic motions is displayed by the system. In connection with the cases where the forcing amplitude is varied as the bifurcation parameter, it was found that the system with supercritical axial speed loses stability even when the excitation amplitude is zero. Regarding the cases where the axial speed is the control parameter, it was shown that the bifurcation diagrams for different
19
forcing amplitudes differ from each other both qualitatively and quantitatively. However, it is impossible to make any general conclusion on the trend of these changes. Moreover, analyzing the bifurcation diagrams, varying either the axial speed or the forcing amplitude, revealed that, the system always starts from simple periodic motion, and there is no sudden strange attractor emerging immediately as the bifurcation parameter is increased.
20
References: [1] E. Carrera, G. Giunta, M. Petrolo, Beam Structures: Classical and Advanced Theories, John Wiley & Sons Ltd, 2011. [2] H.R. Öz, M. Pakdemirli, E. Özkaya, Transition behaviour from string to beam for an axially accelerating material, Journal of Sound and Vibration, 215 (1998) 571‐576. [3] M. Pakdemirli, E. Özkaya, Approximate boundary layer solution of a moving beam problem, Mathematical and Computational Applications, 3 (1998) 93‐100. [4] M. Pakdemirli, A.G. Ulsoy, Stability analysis of an axially accelerating string, Journal of Sound and Vibration, 203 (1997) 815‐832. [5] M. Pakdemirli, A.G. Ulsoy, A. Ceranoglu, Transverse vibration of an axially accelerating string, Journal of Sound and Vibration, 169 (1994) 179‐196. [6] 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) 996‐1008. [7] B. Wang, L.‐Q. Chen, Asymptotic stability analysis with numerical confirmation of an axially accelerating beam constituted by the standard linear solid model, Journal of Sound and Vibration, 328 (2009) 456‐466. [8] M.H. Ghayesh, H.A. Kafiabad, T. Reid, Sub‐ and super‐critical nonlinear dynamics of a harmonically excited axially moving beam, International Journal of Solids and Structures, 49 (2012) 227‐243. [9] M.H. Ghayesh, Coupled longitudinal–transverse dynamics of an axially accelerating beam, Journal of Sound and Vibration, 331 (2012) 5107‐5124. [10] M. Amabili, Nonlinear vibrations of rectangular plates with different boundary conditions: theory and experiments, Computers & Structures, 82 (2004) 2587‐2605. [11] M. Amabili, Theory and experiments for large‐amplitude vibrations of rectangular plates with geometric imperfections, Journal of Sound and Vibration, 291 (2006) 539‐ 565. [12] M. Amabili, Nonlinear Vibrations and Stability of Shells and Plates, Cambridge University Press, New York, 2008. [13] C.C. Lin, Stability and vibration characteristics of axially moving plates, International Journal of Solids and Structures, 34 (1997) 3179‐3190. [14] J. Kim, J. Cho, U. Lee, S. Park, Modal spectral element formulation for axially moving plates subjected to in‐plane axial tension, Computers & Structures, 81 (2003) 2011‐ 2020. [15] Z. Yin‐feng, W. Zhong‐min, Vibrations of axially moving viscoelastic plate with parabolically varying thickness, Journal of Sound and Vibration, 316 (2008) 198‐210. [16] N. Banichuk, J. Jeronen, P. Neittaanmäki, T. Tuovinen, On the instability of an axially moving elastic plate, International Journal of Solids and Structures, 47 (2010) 91‐99. [17] X.‐D. Yang, L.‐Q. Chen, J.W. Zu, Vibrations and stability of an axially moving rectangular composite plate, Journal of Applied Mechanics, 78 (2011) 011018.
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[18] Y.‐Q. Tang, L.‐Q. Chen, Nonlinear free transverse vibrations of in‐plane moving plates: Without and with internal resonances, Journal of Sound and Vibration, 330 (2011) 110‐126. [19] Y.‐Q. Tang, L.‐Q. Chen, Primary resonance in forced vibrations of in‐plane translating viscoelastic plates with 3:1 internal resonance, Nonlinear Dynamics, 1‐14. [20] A.C.J. Luo, H.R. Hamidzadeh, The nonlinear vibration and stability of axially travelling thin plates, in: Proceedings of the ASME Design Engineering Technical Confrence, 2003, pp. 1135‐1143. [21] S. Hatami, M. Azhari, M.M. Saadatpour, Nonlinear analysis of axially of axially moving plates using FEM, International Journal of Structural Stability and Dynamics, 7 (2007) 589‐607.
22
b
cv
a
Fig.1. Schematic representation of an axially moving plate
23
(a)
(b)
(c)
(d)
Fig.2. Bifurcation diagrams of Poincaré points for increasing forcing amplitude on the system with cv=5.0000; (a) the generalized coordinate w1,1; (b) the generalized coordinate w2,1; (c) the generalized coordinate w3,1; (d) the generalized coordinate w4,1.
24
(a)
(b) 0.3
1
w2,1/h
w1,1/h
0.5
0
0
-0.5
-1 2940
2960
2980
-0.3 2940
3000
Time (dimensionless seconds)
2960
2980
3000
Time (dimensionless seconds)
(c)
(d) 0.03
w4,1/h
w3,1/h
0.04
0
-0.04 2940
2960
2980
0
-0.03 2940
3000
2960
2980
3000
Time (dimensionless seconds)
Time (dimensionless seconds)
(e)
(f) 0.2
0.5
(dw2,1/dt)/hω1,1
(dw1,1/dt)/hω1,1
0.1
0
0
-0.1 -0.5
-0.5
0
w1,1/h
0.5
-0.2
-0.1
0
0.1
0.2
w2,1/h
25
(g)
(h) 0.02
0.04
(dw4,1/dt)/hω1,1
(dw3,1/dt)/hω1,1
0.02
0
0
-0.02
-0.04
-0.01
0
0.01
-0.02
-0.01
w3,1/h
0
0.01
0.02
w4,1/h
(i)
(j) 1
0.15
(dw2,1/dt)/hω1,1
(dw1,1/dt)/hω1,1
0.5
0
-0.5
-1 -1
-0.5
0
0.5
0
1
w1,1/h
0
0.02
0.04
0.06
w2,1/h
(k)
(l)
0.01
(dw4,1/dt)/hω1,1
(dw3,1/dt)/hω1,1
0.02
0 -0.006
-0.004
-0.002
w3,1/h
0
0
0
0.0005
0.001
0.0015
0.002
w4,1/h Fig.3. Periodic oscillation for the system of Fig.2 at f1=2.0134 (ω1,1= 140.3774 rad/s): (a‐d) time traces of the the generalized coordinates w1,1, w2,1, w3,1, and w4,1, respectively; (e‐h) phase‐plane portraits of the the generalized coordinates w1,1, w2,1, w3,1, and w4,1, respectively; (i‐l) Poincaré sections of the the generalized coordinates w1,1, w2,1, w3,1, and w4,1, respectively.
26
(a)
(b) 1
0.5
w2,1/h
w1,1/h
0.5
0
0
-0.5
-1 2940
2960
2980
3000
Time (dimensionless seconds)
-0.5 2940
(c)
2960
2980
3000
Time (dimensionless seconds)
(d) 0.5
w4,1/h
w3,1/h
0.2
0
0
-0.2
-0.5 2940
2960
2980
3000
Time (dimensionless seconds)
(e)
2940
2960
2980
3000
Time (dimensionless seconds)
(f) 1 0.4
(dw2,1/dt)/hω1,1
(dw1,1/dt)/hω1,1
0.5
0
0
-0.5 -0.4
-1 -1
-0.2
-0.5
0
w1,1/h
0.5
1
0
w2,1/h
0.2
27
(g)
(h) 0.4
0.4
(dw4,1/dt)/hω1,1
(dw3,1/dt)/hω1,1
0.2
0
0
-0.2
-0.4
-0.4 -0.2
0
-0.2
0.2
0
0.2
w4,1/h
w3,1/h
(i)
(j) 0.3
(dw1,1/dt)/hω1,1
(dw2,1/dt)/hω1,1
0
-0.4
-0.8
0
-0.2
0.6
0.8
-0.1
0
w1,1/h
(k)
0.1
0.2
w2,1/h
1
(l) 0.2
(dw4,1/dt)/hω1,1
(dw3,1/dt)/hω1,1
0.3
0
0
-0.2
-0.3
-0.1
0
w3,1/h
-0.05
0
0.05
0.1
w4,1/h
Fig.4. Quasiperiodic oscillation for the system of Fig.2 at f1=4.2685 (ω1,1= 140.3774 rad/s): (a‐d) time traces of the the generalized coordinates w1,1, w2,1, w3,1, and w4,1, respectively; (e‐h) phase‐ plane portraits of the the generalized coordinates w1,1, w2,1, w3,1, and w4,1, respectively; (i‐l) Poincaré sections of the the generalized coordinates w1,1, w2,1, w3,1, and w4,1, respectively.
28
(a)
(b)
(c)
(d)
Fig.5. Bifurcation diagrams of Poincaré points for increasing forcing amplitude on the system with cv=10.0000; (a) the generalized coordinate w1,1; (b) the generalized coordinate w2,1; (c) the generalized coordinate w3,1; (d) the generalized coordinate w4,1.
29
(a)
(b) 0.5
w2,1/h
w1,1/h
1
0
0
-1
2940
2960
2980
-0.5 2940
3000
(c)
2960
2980
3000
Time (dimensionless seconds)
Time (dimensionless seconds)
(d) 0.2
w4,1/h
w3,1/h
0.2
0
0
-0.2
-0.2
2940
2960
2980
2940
3000
(e)
2960
2980
3000
Time (dimensionless seconds)
Time (dimensionless seconds)
(f) 1
(dw2,1/dt)/hω1,1
(dw1,1/dt)/hω1,1
1
0
0
-1
-1
-1
0
w1,1/h
1
-0.4
0
w2,1/h
0.4
30
(g)
(h) 0.4
(dw4,1/dt)/hω1,1
(dw3,1/dt)/hω1,1
0.5
0
0
-0.5
-0.4 -0.1
0
-0.1
0.1
0
0.1
w4,1/h
w3,1/h
(i)
(j)
-0.4
(dw1,1/dt)/hω1,1
(dw2,1/dt)/hω1,1
0.8
-0.6
0.6
1
1.04
1.08
1.12
0.22
w1,1/h
0.24
0.26
0.28
w2,1/h
(k)
(l) 0.4
(dw4,1/dt)/hω1,1
(dw3,1/dt)/hω1,1
-0.1
0.3
-0.2
0.2
-0.3
0.04
0.06
w3,1/h
0.08
0.08
0.1
0.12
0.14
w4,1/h Fig.6. Quasiperiodic oscillation for the system of Fig.5 at f1=12.000 (ω1,1= 99.6403 rad/s): (a‐d) time traces of the the generalized coordinates w1,1, w2,1, w3,1, and w4,1, respectively; (e‐h) phase‐ plane portraits of the the generalized coordinates w1,1, w2,1, w3,1, and w4,1, respectively; (i‐l) Poincaré sections of the the generalized coordinates w1,1, w2,1, w3,1, and w4,1, respectively.
31
(a)
(b)
(c)
(d)
Fig.7. Bifurcation diagrams of Poincaré points for increasing forcing amplitude on the system with cv=13.0000; (a) the generalized coordinate w1,1; (b) the generalized coordinate w2,1; (c) the generalized coordinate w3,1; (d) the generalized coordinate w4,1.
32
(a)
(b)
(c)
(d)
Fig.8. Bifurcation diagrams of Poincaré points for increasing forcing amplitude on the system with cv=15.0000; (a) the generalized coordinate w1,1; (b) the generalized coordinate w2,1; (c) the generalized coordinate w3,1; (d) the generalized coordinate w4,1.
33
(a)
(b) 0.5
w2,1/h
w1,1/h
1
0
0
-1
2940
2960
2980
3000
Time (dimensionless seconds)
-0.5 2940
(c)
2960
2980
3000
Time (dimensionless seconds)
(d) 0.5
w4,1/h
w3,1/h
0.3
0
0
-0.3 -0.5 2940
2960
2980
2940
3000
2960
2980
3000
Time (dimensionless seconds)
Time (dimensionless seconds)
(e)
(f) 4
3
(dw2,1/dt)/hω1,1
(dw1,1/dt)/hω1,1
2
0
-2
0
-3 -4 -1
-0.5
0
w1,1/h
0.5
1
-0.5
0
0.5
w2,1/h
34
(g)
(h)
3
(dw4,1/dt)/hω1,1
(dw3,1/dt)/hω1,1
3
0
0
-3
-3
-0.4
0
0.4
-0.4
-0.2
w3,1/h
0
0.2
0.4
w4,1/h
(i)
(j) 6 4
2
(dw2,1/dt)/hω1,1
(dw1,1/dt)/hω1,1
3
0
0
-2
-4
-3 -6 -0.5
0
0.5
1
0
1.5
0.5
w2,1/h
w1,1/h
(k)
(l) 6
3
(dw4,1/dt)/hω1,1
(dw3,1/dt)/hω1,1
3
0
0
-3
-3 -6
-0.6
-0.4
-0.2
0
w3,1/h
0.2
0.4
-0.4
0
0.4
w4,1/h
Fig.9. Chaotic oscillation for the system of Fig.8 at f1=11.1946 (ω1,1= 29.9911 rad/s): (a‐d) time traces of the the generalized coordinates w1,1, w2,1, w3,1, and w4,1, respectively; (e‐h) phase‐plane portraits of the the generalized coordinates w1,1, w2,1, w3,1, and w4,1, respectively; (i‐l) Poincaré sections of the the generalized coordinates w1,1, w2,1, w3,1, and w4,1, respectively.
35
(a)
(b)
(c)
(d)
Fig.10. Bifurcation diagrams of Poincaré points for increasing forcing amplitude on the system with cv=28.0000; (a) the generalized coordinate w1,1; (b) the generalized coordinate w2,1; (c) the generalized coordinate w3,1; (d) the generalized coordinate w4,1.
36
(a)
(b)
(c)
(d)
Fig.11. Bifurcation diagrams of Poincaré points for increasing axial speed of the system with f1=1.0000; (a) the generalized coordinate w1,1; (b) the generalized coordinate w2,1; (c) the generalized coordinate w3,1; (d) the generalized coordinate w4,1.
37
(a)
(b)
(c)
(d)
Fig.12. Bifurcation diagrams of Poincaré points for increasing axial speed of the system with f1=2.0000; (a) the generalized coordinate w1,1; (b) the generalized coordinate w2,1; (c) the generalized coordinate w3,1; (d) the generalized coordinate w4,1.
38
(a)
(b)
(c)
(d)
Fig.13. Bifurcation diagrams of Poincaré points for increasing axial speed of the system with f1=3.0000; (a) the generalized coordinate w1,1; (b) the generalized coordinate w2,1; (c) the generalized coordinate w3,1; (d) the generalized coordinate w4,1.
39
(a)
(b)
(c)
(d)
Fig.14. Bifurcation diagrams of Poincaré points for increasing axial speed of the system with f1=4.0000; (a) the generalized coordinate w1,1; (b) the generalized coordinate w2,1; (c) the generalized coordinate w3,1; (d) the generalized coordinate w4,1.
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(a)
(b)
0.3
w2,1/h
w1,1/h
0.1
0
0
-0.3
-0.1
3550
3560
3570
3580
3590
3550
3600
3560
3570
3580
3590
3600
Time (dimensionless seconds)
Time (dimensionless seconds)
(c)
(d) -0.9
w4,1/h
w3,1/h
0.2
0
-1
-0.2
-1.1 3550
3560
3570
3580
3590
3600
3550
Time (dimensionless seconds)
3560
3570
3580
3590
3600
Time (dimensionless seconds)
(e)
(f) 0.1
(dw2,1/dt)/hω
(dw1,1/dt)/hω
0.2
0
0
-0.1
-0.2
-0.2
-0.05
0
w1,1/h
0.05
0.1
-0.4
-0.2
0
0.2
w2,1/h
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(g)
(h) 0.2
(dw4,1/dt)/hω
(dw3,1/dt)/hω
0.1
0
0
-0.2
-0.1
-0.2
0
0.2
-1.05
w3,1/h
-1
-0.95
w4,1/h
(i)
(j) 0.2
(dw2,1/dt)/hω
(dw1,1/dt)/hω
0.1
0
-0.1
0
-0.1
-0.2
0
0.1
0.2
0.3
w1,1/h
-0.06
-0.04
-0.02
(k)
0
0.02
0.04
w2,1/h
(l)
(dw4,1/dt)/hω
(dw3,1/dt)/hω
0.1 0
-0.02
1.1
0
1.2
1.3
w3,1/h
1.4
-0.02
-0.01
0
0.01
0.02
0.03
0.04
w4,1/h
Fig.15. Period‐2 oscillation for the system of Fig.14 at cv=30.0000: (a‐d) time traces of the the generalized coordinates w1,1, w2,1, w3,1, and w4,1, respectively; (e‐h) phase‐plane portraits of the the generalized coordinates w1,1, w2,1, w3,1, and w4,1, respectively; (i‐l) Poincaré sections of the the generalized coordinates w1,1, w2,1, w3,1, and w4,1, respectively.
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(a)
(b)
(c)
(d)
Fig.16. Bifurcation diagrams of Poincaré points for increasing axial speed of the system with f1=6.0000; (a) the generalized coordinate w1,1; (b) the generalized coordinate w2,1; (c) the generalized coordinate w3,1; (d) the generalized coordinate w4,1.
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(a)
(b)
(c)
(d)
Fig.17. Bifurcation diagrams of Poincaré points for increasing axial speed of the system with f1=8.0000; (a) the generalized coordinate w1,1; (b) the generalized coordinate w2,1; (c) the generalized coordinate w3,1; (d) the generalized coordinate w4,1.
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Highlights: ‐ ‐ ‐ ‐
The three‐dimensional nonlinear global dynamics of axially moving plates is investigated numerically. The governing equations of motion are obtained by means of Lagrange equations. These equations are solved via direct time integration by means of Gear’s backward‐differentiation‐formula (BDF). Bifurcation diagrams of Poincaré maps are constructed.
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