Abstract
The multipulse homoclinic orbits and chaotic dynamics of a reinforced composite plate with the carbon nanotubes (CNTs) under combined in-plane and transverse excitations are studied in the case of 1 : 1 internal resonance. The method of multiple scales is adopted to derive the averaged equations. From the averaged equations, the normal form theory is applied to reduce the equations to a simpler normal form associated with a double zero and a pair of pure imaginary eigenvalues. The energy-phase method proposed by Haller and Wiggins is utilized to examine the global bifurcations and chaotic dynamics of the CNT-reinforced composite plate. The analytical results demonstrate that the multipulse Shilnikov-type homoclinic orbits and chaotic motions exist in the system. Homoclinic trees are constructed to illustrate the repeated bifurcations of multipulse solutions. In order to verify the theoretical results, numerical simulations are given to show the multipulse Shilnikov-type chaotic motions in the CNT-reinforced composite plate. The results obtained here imply that the motion is chaotic in the sense of the Smale horseshoes for the CNT-reinforced composite plate.
1. Introduction
Carbon nanotubes (CNTs), as a new type of advanced materials, have attracted a lot of attention of researchers. This is because CNTs possess high strength and stiffness with high aspect ratio and low density. Due to these properties of CNTs, a number of studies on the nonlinear vibrations and dynamic responses of CNT-reinforced composite plates have been carried out by many researchers in recent years. By employing an equivalent continuum model, Formica et al. [1] investigated the vibration behaviors of CNT-reinforced composite plates. Zhu et al. [2] presented the bending and free vibration analysis of CNT-reinforced composite plates using the finite element method based on the first-order shear deformation plate theory. In their work, the authors showed the effects of the volume fractions of CNTs and the edge-to-thickness ratios on the bending responses, natural frequencies, and mode shapes of the plates. In addition, Wang and Shen [3] examined the nonlinear dynamic response of CNT-reinforced composite plates resting on elastic foundations in thermal environments. The motion equations were derived based on a higher-order shear deformation theory with a von Kármán-type of kinematic nonlinearity. In [4], a mixed Navier-layerwise differential quadrature method was employed by Malekzadeh and Heydarpour for the free vibration and static response analysis of functionally graded carbon nanotube- (FG-CNT-) reinforced composite laminated plates. Based on the conventional Ritz method accompanied with the Lagrangian multiplier technique, Kiani [5] analyzed the free vibration characteristics of FG-CNT-reinforced composite plates located on point supports. Rafiee et al. [6] applied Galerkin’s method to deal with the nonlinear dynamic stability of initially imperfect piezoelectric FG-CNT-reinforced composite plates under combined thermal and electrical loadings. Using the Fourier series expansion and state-space technique, Alibeigloo and Liew [7] investigated the bending behavior of FG-CNT-reinforced composite plates with simply supported edges subjected to thermomechanical loads. Subsequently, this work was extended by Alibeigloo and Emtehani [8] for various boundary conditions by employing the differential quadrature method. Recently, Zhang et al. [9] carried out the analysis of geometrically nonlinear large deformation of triangular FG-CNT-reinforced composite plates using the element-free improved moving least-squares Ritz (IMLS-Ritz) method.
In addition, with the increasing applications of functionally graded materials (FGM) in modern technology, the buckling and vibration analysis of FGM structures began to attract a widespread attention. Thus, many studies on the buckling and vibration analysis of FGM shells have been published in the literature. Zhang and Li [10] discussed the dynamic buckling of FGM truncated conical shells subjected to normal impact loads. Bagherizadeh et al. [11] investigated the mechanical buckling of simply-supported FGM cylindrical shells using third-order shear deformation shell theory. They found that system parameters have great influence on the buckling characteristics of FGM shells. Buckling behavior analysis based on Reddy’s high-order shear deformation theory has also been performed by Sun et al. [12] for FGM cylindrical shells subjected to an axial compression in the thermal environment. In recent years, there have been some numerical methods for analyzing the plates and shells. The multiquadric radial basis function (MQ) method was developed and applied by Ferreira [13] to discuss the effects of system parameters on the laminated composite plates. The method of discrete singular convolution (DSC) gives a fast and accurate solution of the mathematical physics and engineering problems. Then, Civalek and his coworkers made a number of remarkable studies using the DSC method; such investigations involved the analysis of composite conical shells and panels [14–16], plates on elastic foundations [17, 18], and so on.
However, in the course of our study, we found that there are only few studies on the global bifurcations and multipulse chaotic dynamics for the CNT-reinforced composite plate. In order to eliminate or suppress large nonlinear vibrations and chaotic motions of the CNT-reinforced composite plate, we should deepen and complete the theoretical analysis on the CNT-reinforced composite plate model, discuss the complex dynamic behaviors, explore the existence conditions of the multipulse Shilnikov-type orbits, and analyze the impact of parameters on the system, so as to ensure the stability and controllability of the CNT-reinforced composite plate. The present work is therefore motivated to examine the global bifurcations and multipulse chaotic dynamics of the CNT-reinforced composite plate considered by Guo and Zhang [19]. The energy-phase method developed by Haller and Wiggins [20] is applied to study the multipulse homoclinic bifurcations and chaotic dynamics for the CNT-reinforced composite plate. In recent decades, two theories that are usually employed to detect the global bifurcations and multipulse Shilnikov-type chaotic dynamics in high-dimensional systems are the energy-phase method [20] and the extended Melnikov method [21]. Also, some researchers have utilized the developed theories to engineering applications, see, for example, [22–29].
Based on the research by Guo and Zhang [19], the averaged equations are obtained for the case of 1 : 1 internal resonance, principal parametric resonance, and 1/2 subharmonic resonance of the CNT-reinforced composite plate. From the averaged equations, the explicit expressions of the 3-order normal form are derived employing the normal form theory. The energy-phase method is utilized to detect the existence of the multipulse Shilnikov-type homoclinic orbits in the CNT-reinforced composite plate. Homoclinic trees are constructed to illustrate the repeated bifurcations of multipulse solutions. In addition, numerical simulations are given to verify the analytical predictions.
The organization of this paper is as follows. In Section 2, we derive the averaged equations and normal form for the CNT-reinforced composite plate. According to the normal form derived in Section 2, the dynamics of the unperturbed system is discussed in Section 3. In Section 4, a detailed dynamic analysis of the perturbed system is presented, and the existence of multipulse homoclinic orbits is confirmed. Numerical simulations are given in Section 5 to show that the chaotic motions can occur in the CNT-reinforced composite plate. A short conclusion is reported in Section 6.
2. Formulation of the Problem
2.1. Equations of Motion and Perturbation Analysis
This paper focuses on studies of the multipulse homoclinic orbits and chaotic dynamics of a four-edge simply-supported CNT-reinforced composite rectangular plate subjected to the in-plane and transverse excitations. The model is shown in Figure 1 [19]. The edge length and width of the plate in the and directions are, respectively, and , and the thickness is . A Cartesian coordinate system is located in the middle surface of the CNT-reinforced composite rectangular plate. The displacements of an arbitrary point within the plate are , , and in the , , and directions, respectively. The in-plane excitation along the direction at is given by . The transverse excitation subjecting to the CNT-reinforced composite rectangular plate is of the form .
According to the research by Guo and Zhang [19], the axial and transverse displacement fields at any point for the CNT-reinforced composite plate are given aswhere , , and represent the displacements at the midplane of the plate in the , , and directions. and denote the rotations of the transverse normal at the midplane about the and axes. Employing the von Kármán-type plate theory and in terms of the displacements, the strains and the curvatures in the midplane can be expressed as
The force and moment resultants associated with the strains and curvatures of the plate constitutive equations arewhere the coupling matrix and is equal to zero in the stiffness matrix. Then, the nonlinear governing equations of motion for the CNT-reinforced composite plate are obtained as follows [19]:
Substituting the stress resultants of equations (1a)–(1c), (2), and (3a)–(3e) into equations (4a)–(4e), the governing equations of motion for the CNT-reinforced composite plate in terms of generalized displacements are derived as follows [19]:where is the damping coefficient, and all the other coefficients can be found in [19]. The associated boundary conditions for the simply-supported CNT composite plate can be written as
Also, the following nondimensional parameters are introduced:
In this paper, our research is focused on the multipulse global bifurcations and chaotic dynamics of the CNT composite plate in its first two modes. Hence, we express in the following form:where and are the amplitudes of two modes. The other variables and transverse excitation are given as
Then, substituting these expressions into equations (5a)–(5e) and applying the Galerkin integration procedure, we can obtain the displacements , , , and with respect to . Consequently, the nonlinear ordinary differential equations of this system in terms of transverse displacements are derived. According to the results of convergence studies given by Hao et al. [30], the dimensionless equations of motion for the CNT-reinforced composite plate in the first two modes are derived as [19]where all the coefficients can be found in [19]. We consider the case of 1 : 1 internal resonance, principal parametric resonance, and 1/2 subharmonic resonance of the CNT-reinforced composite plate, for which the resonant relations are listed as follows:where and are two detuning parameters. For convenience, we let in the following analysis. The uniform solutions of equations (10a) and (10b) take the following form:where and . We employ the method of multiple scales to equations (10a) and (10b) and obtain the averaged equations as follows:where .
2.2. Computation of Normal Form
In order to study the multipulse chaotic dynamics of the CNT-reinforced composite plate, the first is to reduce equations (13a)–(13d) to a simpler normal form. It is clear that equations (13a)–(13d) possess a trivial zero solution at which the Jacobian matrix can be written as
The characteristic polynomial of Jacobian matrix (14) is
It is easy to see that when , , and are satisfied simultaneously, systems (13a)–(13d) have a pair of pure imaginary eigenvalues and one non-semisimple double zero:
Setting and and considering , , and as the perturbation parameters, then equations (13a)–(13d) without the perturbation parameters become
For the trivial zero solution , the Jacobian matrix of equations (17a)–(17d) is evaluated as
Executing the Maple program designed by Zhang et al. [31] leads to the following 3-order normal form of equations (17a)–(17d):
Accordingly, the normal form of equations (13a)–(13d) with the perturbation parameters is obtained as
Lettingwe have
We then construct the following transformation to find the unfolding of equations (22a)–(22d):
Substituting equation (23) into equations (22a)–(22d) and canceling the nonlinear terms including parameter , the unfolding of equations (22a)–(22d) can be derived aswhere and . Introduce the scale transformations
Then, equation (24a)–(24d) can be expressed as the Hamilton form with perturbations:
The Hamiltonian function takes the form
3. Dynamics of the Unperturbed System
Setting in equations (26a)–(26d) results in the completely integrable equations defined as the unperturbed system. Hence, we will now study the nonlinear dynamics of the following unperturbed system:
Consider the first two equations of systems (28a)–(28d):
Note that homoclinic bifurcations occur in systems (29a) and (29b) for . Also, when , we can obtain that systems (29a) and (29b) have only the trivial zero solution being a center. On a curve expressed by , that is,three solutions bifurcate from the trivial zero solution through a pitchfork bifurcation denoted by and , respectively, where
For all , systems (29a) and (29b) possess one hyperbolic saddle point connected to itself by a pair of homoclinic orbits, , that is, . Therefore, the set defined byis a two-dimensional invariant manifold in the full four-dimensional phase space. Based on the analysis given by Haller and Wiggins [20], we know that the two-dimensional invariant manifold is normally hyperbolic and contains three-dimensional stable and unstable manifolds. Let and , respectively, denote the stable and unstable manifolds. The existence of the homoclinic orbit in systems (29a) and (29b) to means that and intersect nontransversally along a three-dimensional homoclinic manifold described as follows:where . Letting , , systems (29a) and (29b) can be rewritten as
The Hamiltonian for Hamilton systems (34a) and (34b) is
When , there is a homoclinic loop formed by one saddle point and a pair of homoclinic orbits . Using equations (34a) and (34b) and (35), the analytical expressions for the homoclinic orbits of (34a) and (34b) are then obtained as
We now consider the dynamics of the unperturbed system of (26a) and (26d) restricted to given by
If the condition holds, constant is a periodic orbit, and if , constant is a circle of fixed points. A value of for which is referred to as a resonant value , and these fixed points are identified as resonant fixed points. A resonant value is denoted by such that
Thus, the resonant value is derived as
For , the phase shift can be defined aswhich will be used in subsequent studies to determine the condition under which the multipulse Shilnikov-type homoclinic orbits may exist. Substituting the first equations of (36a) and (36b) into the fourth equations of (28a)–(28d) leads to
After integration, we obtain
Accordingly, the phase shift is also obtained as
4. Dynamics of the Perturbed System
After obtaining the detailed properties of the subspace for systems (28a)–(28d), the next step is to examine the effects of small perturbation terms on systems (28a)–(28d). The energy-phase method developed by Haller and Wiggins [20] is utilized to determine the existence of multipulse orbits and chaotic dynamics in the CNT-reinforced composite rectangular plate.
4.1. The Case of Dissipative Perturbations
We start by considering the influence of such small perturbations on manifold . Based on the research by Haller and Wiggins [20], we know that the manifold is invariant for small nonzero . Therefore, the perturbed annulus is taken to be the same as . That is,
In order to study the dynamic behaviors of the perturbed vector field restricted to near the resonance , we now introduce the following scale transformations:
The last two equations of (26a)–(26d) then becomewhere the prime represents the differentiation with respect to . When , systems (46a) and (46b) become
Systems (47a) and (47b) are a Hamiltonian system with the Hamiltonian
A simple calculation indicates that systems (47a) and (47b) have two singular points expressed by
It is known that if , the singular point is a center point. And if , the singular point is a saddle connected to itself by a homoclinic orbit. Figure 2(a) shows the phase portrait of systems (47a) and (47b). It is observed that, for sufficiently small parameter , the singular point still remains a hyperbolic saddle point , while for small perturbations, the singular point changes to a hyperbolic sink , see Figure 2(b) for the phase portrait of perturbed systems (46a) and (46b).
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On the basis of equation (48), at , the basin of the attractor for can be estimated as follows:
Substituting in (49a) and (49b) into (50) yields
The neighborhood of is defined aswhere the constant is sufficiently large so that the unperturbed homoclinic orbit is enclosed within the annulus.
4.2. The Existence of Multipulse Homoclinic Orbits
Based on the results obtained by Haller and Wiggins [20], the dissipative energy-difference function is of the formwhereand represents the region contained between a pair of homoclinic orbits in the plane, is the boundary of , and denotes the phase difference given in (43). The third term in (53) can be calculated as follows:
Thus, the dissipative energy-difference function becomeswhere . A dissipation factor is defined as . Equation (56) can then be rewritten in the following form:
The zeros of can be achieved by solving the following equation:
The upper bound of the dissipation factor is derived from equation (58):
According to (59), it is found that, for small dissipative effects , the following upper bound on the pulse numbers is obtained:which means that the homoclinic tree is finite even for arbitrarily small dissipation. The result in equation (60) also indicates that the upper bound is inversely proportional to the dissipation factor . We now identify zeros of the energy-difference function in the interval . From (57), we find that, for any satisfyingthe zeros of in arewhere . Define a set that contains all transverse zeros of as
It is easy to know that both zeros are transverse under condition (61). We now need to introduce the two additional anglesand utilize them to construct the two sets of transverse zeros of as
Consider a domain enclosed inside the homoclinic orbit of (47a) and (47b) located in the interval . The periodic orbits in are classified according to their pulse numbers, see Figure 3. Note that, under condition (61), all internal orbits outside intersect transversally. Thus, for any periodic or homoclinic orbit outside , the pulse number is . Define the energy sequencesuch that provides the energy level related to an orbit closer to the center. Define the open set of internal orbits in as
The pulse sequence is then defined as
Since the energy of the periodic orbits in decreases monotonically with the orbits shrinking to the center, we necessarily have
The layer sequence is defined aswhere Int() denotes the interior of a set. And the construction of the layer sequence is shown in Figure 3. We can easily see that all these sequences defined above are finite by (60). For any periodic orbit , the pulse number is . As illustrated in Figure 3, the layer radii are defined as
A recursive algorithm for the calculation of the distribution of pulse numbers and the layer radii is implemented on a computer. For different values of the dissipation factor , we obtain the corresponding pulse diagrams and layer radius diagrams as a function of the phase shift for , illustrated in Figures 4 and 5. As can be seen in Figure 4, the horizontal line segments at each level identify that an infinity of -pulse orbits exist for all values of the phase shift in the interval below that line. Besides, the diagrams in Figure 5 exhibit a gradual breakup of the homoclinic tree as is increased. This observation means that the system parameters have a vital effect on the distribution of the pulse numbers and the layer radii.
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We now detect the existence of the multipulse Shilnikov-type orbits based on Theorem 4.5 in [20]. Note that the sinks of (46a) and (46b) become the centers of (47a) and (47b). Therefore, systems (47a) and (47b) have a nondegenerate equilibrium
The values of for which the center falls in the zero set given in (63) must be determined, that is,which yields
Solving (74), the dissipation factor is obtained as
The result is only valid when
It is also necessary to prove the nondegeneracy conditionwhenever (74) and (76) hold. Carrying out the differentiation in expression (74), we find that (77) fails to be satisfied when
It is easy to see that equations (74) and (78) cannot hold simultaneously under condition (76). Thus, nondegeneracy condition (77) holds. We now have to guarantee that the landing point of any -pulse orbit taking off from a slow sink lies in the domain of attraction of a sink on . To verify it, we consider the point in the interval and is apart from the approximate landing point . The -coordinate of this point is calculated asin which we recall that
If , we redefine by subtracting . The main purpose here is to achieve translation of the landing point which is the closest to the saddle point in the interval . We require thator, equivalently, thatto make sure that the projection of landing points falls in the domain of attraction of one of the sinks.
Based on the above analysis, the results in this section give rise to the following theorem.
Theorem 1. For any integer , there exists a positive number and a finite union of codimension-one surfaces in the parameter space near the set satisfying , (75), and (82) such that, for any and , the following conclusions hold:(1)If the integer is even, then each of the saddle-focus-type equilibria contained in the slow manifold admits two generalized Shilnikov-type homoclinic orbits. If is odd, then there exist two cycles of Shilnikov-type heteroclinic orbits connecting the two saddle-foci to each other. In both cases, the -pulse orbits form pairs which are symmetric with respect to the subspace .(2)There exists an open set of parameters containing for which system (26) admits Smale horseshoes in its dynamics.
5. Numerical Simulations
In order to verify the analytical predictions, we choose systems (13a)–(13d) to perform numerical simulations. The fourth-order Runge–Kutta algorithm is employed to indicate the existence of the multipulse Shilnikov-type homoclinic orbits and chaotic motions in the CNT-reinforced composite plate. Figure 6 is obtained to show that there exist multipulse chaotic motions for the CNT-reinforced composite plate. In Figure 6, the parameters and initial conditions are selected as , , , , , , , , , , , , , , and the phase shift at is . We choose the pulse number ; then, (75) gives . Using equations (79) and (80a) and (80b), we have , , , and . Obviously, condition (82) holds, which implies that the chaotic motion presented in Figure 6 is multipulse Shilnikov-type chaotic motion.
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6. Conclusions
In this paper, the multipulse homoclinic orbits and chaotic dynamics of a CNT-reinforced composite plate under combined in-plane and transverse excitations are studied. The method of multiple scales is adopted to acquire the averaged equations in the case of 1 : 1 internal resonance, principal parametric resonance, and 1/2 subharmonic resonance. On the basis of the averaged equations obtained, the normal form theory is employed to derive the expressions of normal form associated with a double zero and a pair of pure imaginary eigenvalues. The energy-phase method proposed by Haller and Wiggins [20] is utilized to detect the presence of the multipulse homoclinic orbits and chaotic dynamics for the resonant case. As the trajectory of motion comes close to the sink point , every Shilnikov-type orbit takes off again and repeats this similar motion in the full four-dimensional phase space and eventually leads to the multipulse Shilnikov-type orbits. Our analysis in Section 4 also indicates that the multipulse Shilnikov-type orbits depend on the system parameters and dissipative perturbations. It is known that the existence of multipulse Shilnikov-type orbits implies the existence of chaos in the sense of Smale horseshoes. Homoclinic trees are presented to describe the repeated bifurcations of multipulse solutions. From the diagrams, we can see a gradual breakup of the homoclinic tree with the increase of the dissipation factor. This observation denotes that the damping coefficient and system parameters affect the distribution of the pulse numbers and the layer radii. The analytical results obtained here are extensions of those appearing in [19].
To confirm the theoretical results, numerical simulations are applied to examine the chaotic dynamics of the CNT-reinforced composite plate. The numerical results demonstrate that there exist multipulse homoclinic orbits and chaotic dynamics in systems (13a)–(13d) when the conditions are satisfied. As we all know, under certain conditions, the multipulse chaotic motions in the averaged equations can result in the multipulse amplitude-modulated chaotic vibrations in the original system, which indicates the existence of multipulse chaotic dynamics in the CNT-reinforced composite plate. The research in this paper provides a detailed explanation of the multipulse jumping behaviors for the CNT-reinforced composite plate under combined in-plane and transverse excitations. It is established simultaneously that the damping coefficient and system parameters have a significant influence on the nonlinear dynamics of the CNT-reinforced composite plate. Therefore, as mentioned above, the nonlinear dynamic behaviors of the CNT-reinforced composite plate can be controlled by varying the system parameters.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare no conflicts of interest.
Acknowledgments
This research was supported by the National Natural Science Foundation of China (11872201, 11572148, and 11772148), the National Research Foundation for the Doctoral Program of Higher Education of China (20133218110025), the Fundamental Research Funds for the Central Universities (NS2017046), and the Natural Science Research Project for College and University of Anhui Province (KJ2018A0048).