Research Article  Open Access
Free Vibrations and Nonlinear Responses for a Cantilever Honeycomb Sandwich Plate
Abstract
Dynamics of a cantilever honeycomb sandwich plate are studied in this paper. The governing equations of the composite plate subjected to both inplane and transverse excitations are derived by using Hamilton’s principle and Reddy’s thirdorder shear deformation theory. Based on the Rayleigh–Ritz method, some modes of natural frequencies for the cantilever honeycomb sandwich plate are obtained. The relations between the natural frequencies and the parameters of the plate are investigated. Further, the Galerkin method is used to transform the nonlinear partial differential equations into a set of nonlinear ordinary differential equations. Nonlinear dynamic responses of the cantilever honeycomb sandwich plate to such external and parametric excitations are discussed by using the numerical method. The results show that inplane and transverse excitations have an important influence on nonlinear dynamic characteristics. Rich dynamics, such as periodic, multiperiodic, quasiperiodic, and chaotic motions, are located and studied by the bifurcation diagram for some specific parameters.
1. Introduction
Honeycomb sandwich structures are used widely in aerospace field, vehicles, ships, transport packaging, building decorations, and so on. Honeycomb sandwich structures have many advantages such as light weight, high specific stiffness, and high specific strength, as well as good structural stability and energy absorption [1]. Therefore, many research works have focused on the honeycomb sandwich plates and shells and corresponding dynamic characters with different techniques. Meo et al. [2] studied the response of honeycomb sandwich panels on lowvelocity impact damage by using experimental investigation and numerical simulation. The finite element method is one of the useful methods to study dynamics of composite sandwich panels with honeycomb core [3]. Free vibrations of symmetric rectangular honeycomb sandwich panels with simply supported boundaries at the four edges were investigated by using the homotopy analysis method [4]. The analytical model of composite sandwich panels with honeycomb core subjected to highvelocity impact had been developed by Feli and Namdari Pour [5]. Buckling analysis of rectangular plate with hexagonal honeycomb core under combined axial compression and transverse shear loads is given in the reference [6]. The impact behavior of honeycomb sandwich panels has been investigated by finite element method by Menna et al. [7]. The vibrationbased spatial damage identification of the honeycombcore sandwich composite plates was given by using wavelet analysis method [8]. The dynamic behavior of a viscoelastic sandwich composite plate with honeycomb core subjected to the nonuniform blast load was investigated by both theoretical and experimental methods (see the details in Balkan and Mecitoglu [9]). The vibroacoustic bending properties of honeycomb sandwich panels with composite faces were studied by Laurent [10]. The stretch and bending of honeycomb sandwich plates were obtained with skin and height effects by using analytic homogenization method [11].
The abovementioned references mainly used numerical methods or experiments to study the mechanical properties or impact damage, and so on of the honeycomb sandwich structures. In this study, we focus on the case that the cantilever honeycomb sandwich plate is loaded by the joint external and parametric excitations. Free vibrations and nonlinear dynamics for the cantilever honeycomb sandwich plate are given.
This paper is organized as follows. In Section 2, based on Hamilton’s principle and Reddy’s thirdorder shear deformation theory, the formulas for the honeycomb sandwich plate subjected to inplane and transverse excitations are derived. Free transversal vibrations of the plate at cantilever boundary conditions are given by using Rayleigh–Ritz method in Section 3. The variations of natural frequencies with varied parameters of the honeycomb plate are obtained. In Section 4, the formulas for the honeycomb sandwich plate subjected to inplane and transverse excitations are transformed to ordinary differential equations by using the Galerkin method. The mode functions are selected under cantilever boundary conditions. Numerical simulations are derived to obtain nonlinear response of the honeycomb plate. Finally, conclusions are given.
2. Equations of Motion for the Plate
Consider a cantilever honeycomb sandwich plate subjected to both inplane and transversal excitations, the model and the coordinate system are shown in Figure 1. The plate is of length , width , and thickness . Let and be, respectively, the displacements of an arbitrary point and a point in the middle plane of the plate in the , and directions. The midplane rotations of a transverse normal about the and axes are denoted by and , respectively. The plate is subjected to an inplane excitation and a transverse excitation , where and are the excitation frequencies of the transverse and inplane excitations, respectively.
Based on Reddy’s thirdorder shear deformation theory in Reddy [12], the displacement components of the honeycomb sandwich plate can be represented as follows:where , , and .
The relations between strains and displacements according to von Karman nonlinear strainsdisplacements are given by
The constitutive relations of the honeycomb sandwich plate can be written as follows:where , and are elastic constants which can be written as
The terms and are equivalent elastic modulus of the honeycomb core at and directions, respectively, and and are equivalent Poisson’s ratios. These equivalent elastic parameters of the honeycomb core can be given by the following equations, and the details can be found in the study of Fu et al. [13]:where is the thickness of the honeycomb core cellular cells and and are length of the straight and sloping wall of the honeycomb core cellular cells (Figure 2). and are, respectively, Young’s modulus and Poisson’s ratio of the materials for the honeycomb core.
In (4), and are transverse shear moduli, which can be computed by using the following formula:
The relation between the equivalent density of the honeycomb core and the density of the materials is
The formula above for hexagon honeycomb core can be simplified as
Substituting (2) and (3) into the potential and kinetic functions, the equations of motion for the honeycomb sandwich plate are derived by using Hamilton’s principle:where
The stiffness elements of the honeycomb sandwich plate are given by
3. Frequencies of Transverse Vibrations
In this section, the frequencies of transverse vibration for direction are considered by using the Rayleigh–Ritz method. The deflections for is as follows:
The kinetic energy of the honeycomb sandwich plate is
The maximum kinetic energy is of the following form:
The potential energy of the honeycomb plate is of the following form:
The stresses and strains in (15) are obtained from (2) and (3).
The boundary conditions for the cantilever honeycomb sandwich plate are given as follows:where the equivalent shear forces are
The mode function can be written as the following form:where , and are modal parameters andwhere
The mode functions (19) satisfy the cantilever boundary conditions (16). Since the honeycomb plate system is a conservative system during free vibrations, the maximum kinetic energy is equal to the maximum potential energy. According to the Rayleigh–Ritz method, compute the Jacobian matrixand let the determinant equal to 0; the first four frequencies for the honeycomb sandwich plate can be obtained numerically.
The parameters of the honeycomb sandwich plate are as follows: density , Young’s modulus , Poisson’s ratio , the length and width , the thickness of the core is , and the length of the straight and sloping wall of the honeycomb core cellular cells and . Table 1 lists the first four orders of natural frequencies of the cantilever honeycomb sandwich plate with the changes of the honeycomb core thickness. From the data in Table 1, we can see that the natural frequency of the first four orders of the honeycomb sandwich panel increases slowly with the increase of the thickness of the honeycomb cores.

Table 2 shows the variation of the natural frequencies of the honeycomb sandwich panel with the density of the material of the honeycomb core under cantilever boundary conditions. From the data in Table 2, it can be concluded that the natural frequency of the honeycomb sandwich panel decreases with the increase of the density of the matrix material.

Table 3 shows the effect of the change in the wall length of the honeycomb cell of the honeycomb sandwich panel on the natural frequency of the first four bands of the honeycomb sandwich panel. From the data in Table 3, we can find that, with the increase of the cell length of the honeycomb core, the natural frequencies of the first four bands of the honeycomb sandwich panel showed a slight increase.

Table 4 shows the effect of the variation of the wall thickness of the honeycomb unit on the natural frequencies of the honeycomb sandwich panel. From the data in Table 4, it can be seen that, with the increase of the wall thickness of the honeycomb cell, the first four natural frequencies of the honeycomb sandwich panel show a decreasing tendency.

It can be concluded from Tables 1–4 that the thickness of the honeycomb core and the density of the materials have little effect on the natural frequencies. The frequencies increase with increase of cell length of the honeycomb core and with decrease of the wall thickness of the honeycomb core.
In this section, the influence of the honeycomb core size and the material density on the natural frequencies of the plate are given, which provides some guidance for the application of honeycomb plate in engineering to avoid resonance. In next section, nonlinear dynamics of the cantilever honeycomb sandwich plate jointed external and parametric excitations are studied.
4. Nonlinear Responses of the Plate
In this section, the 2truncated function for is assumed as the following:where
Truncated functions for other directions areHence, the displacements have been transferred into the generalized coordinates, and the truncated functions satisfy the cantilever boundary conditions (16). Similarly, the transversal force is also truncated into the generalized coordinates:
Substituting the mode functions (22), (24), (25) into (9), neglecting the inertia terms for inplane and rotary since they are small compared with that of transverse, one can obtain the nonlinear ordinary differential equations as follows:
Equations (26) are typical nonlinear equations governing the vibrations of generalized coordinates undergoing external excitation with frequency and parametric excitation with frequency .
Next, numerical simulations are given to study the nonlinear responses of the honeycomb sandwich plate under transverse and inplane loads of (26). It is assumed that the plate is made of aluminum, and the physical and geometric parameters are the same as that of Section 3. Other parameters, that is, the inplane force and the transverse excitation, are chosen as follows: . The loads are chosen as , and , . Figure 3 illustrates chaotic motions for the honeycomb sandwich plate. Figures 3(a) and 3(b) present the phase portraits on the planes and . Figures 3(c) and 3(d) give the time histories on the planes and . Figure 3(e) represents the phase portraits in threedimensional space .
(a)
(b)
(c)
(d)
(e)
Chosen the transverse loads as and and the inplane load , Figure 4 presents the bifurcations of the honeycomb plate with variations of parametric excitation amplitude . From the bifurcation diagram, it can be seen that the honeycomb sandwich plate can have periodic and multiperiodic motions when parametric amplitude changes. It is periodic doubling bifurcation when the inplane force is about .
Figure 5 shows periodic motions when the inplane load .
(a)
(b)
(c)
(d)
(e)
When the inplane increases, multiperiodic motions are found for the honeycomb sandwich plate. Figure 6 represents 2periodic motions as .
(a)
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When the thickness of the honeycomb core cells changes to and other parameters are kept the same, the bifurcation diagram with inplane force changing has been presented in Figure 7. It can be seen from the bifurcation diagram that it is periodic doubling bifurcation when the inplane force is about . Compared with bifurcation diagram (Figure 4), the bifurcation point moves to the right; that is, with the increase of the thickness of the honeycomb core, the force needs increase to bifurcate. In addition, if the force is larger than , the amplitude of the plate increases obviously and then divergences.
When , Figure 8 shows periodic motion of the honeycomb sandwich plate.
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(e)
When the force is chosen as , the multiperiodic motions are obtained as shown in Figure 9.
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5. Conclusions
Dynamics of a cantilever honeycomb sandwich plate subjected to both inplane and transverse excitations have been studied. The governing equations are derived by using Hamilton’s principle and Reddy’s thirdorder shear deformation theory. The natural frequencies of the cantilever honeycomb sandwich plate are obtained by Rayleigh–Ritz method. Then, the Galerkin truncation procedure is employed to transform the nonlinear partial differential equations into a set of nonlinear ordinary differential equations. Nonlinear dynamic responses of the cantilever honeycomb sandwich plate to such external and parametric excitations are studied using numerical method. From numerical simulations, it can be concluded that there exist chaotic motions as well as periodic and multiperiodic motions in the cantilever honeycomb sandwich plate subjected to the transversal and inplane excitations.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Authors’ Contributions
All the authors contributed equally to this work. Junhua Zhang and Wei Zhang conceived and designed the project. Junhua Zhang did the theoretical deduction and analyzed the results. Xiaodong Yang performed the numerical simulations. Junhua Zhang and Xiaodong Yang wrote the manuscript. Wei Zhang provided useful suggestions to this work.
Acknowledgments
The authors gratefully acknowledge the support of the National Natural Science Foundation of China (11472057, 11732005), the Qinxin Talents Cultivation Program of Beijing Information Science and Technology University (QXTCP B201701), and the Science and Technology Project of Beijing Municipal Education Commission (KM201711232002).
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Copyright
Copyright © 2018 Junhua Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.