Free Vibrations and Nonlinear Responses for a Cantilever Honeycomb Sandwich Plate

Dynamics of a cantilever honeycomb sandwich plate are studied in this paper. ,e governing equations of the composite plate subjected to both in-plane and transverse excitations are derived by using Hamilton’s principle and Reddy’s third-order shear deformation theory. Based on the Rayleigh–Ritz method, some modes of natural frequencies for the cantilever honeycomb sandwich plate are obtained. ,e 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. ,e results show that in-plane 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.


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].erefore, 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 low-velocity impact damage by using experimental investigation and numerical simulation.e 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].
e analytical model of composite sandwich panels with honeycomb core subjected to high-velocity 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].e impact behavior of honeycomb sandwich panels has been investigated by finite element method by Menna et al. [7].
e vibration-based spatial damage identification of the honeycomb-core sandwich composite plates was given by using wavelet analysis method [8].e 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]).e vibroacoustic bending properties of honeycomb sandwich panels with composite faces were studied by Laurent [10].e stretch and bending of honeycomb sandwich plates were obtained with skin and height effects by using analytic homogenization method [11].
e 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.
is paper is organized as follows.In Section 2, based on Hamilton's principle and Reddy's third-order shear deformation theory, the formulas for the honeycomb sandwich plate subjected to in-plane 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. e 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 in-plane and transverse excitations are transformed to ordinary di erential equations by using the Galerkin method.e mode functions are selected under cantilever boundary conditions.Numerical simulations are derived to obtain nonlinear response of the honeycomb plate.Finally, conclusions are given.

Equations of Motion for the Plate
Consider a cantilever honeycomb sandwich plate subjected to both in-plane and transversal excitations, the model and the coordinate system are shown in Figure 1. e plate is of length a, width b, and thickness h.Let (u, v, w) and (u 0 , v 0 , w 0 ) be, respectively, the displacements of an arbitrary point and a point in the middle plane of the plate in the x, y, and z directions.e mid-plane rotations of a transverse normal about the x and y axes are denoted by ϕ x and ϕ y , respectively.e plate is subjected to an in-plane excitation p p 0 + p 1 cos Ω 1 t and a transverse excitation F(x, y)cos Ωt, where Ω 1 and Ω are the excitation frequencies of the transverse and in-plane excitations, respectively.
Based on Reddy's third-order shear deformation theory in Reddy [12], the displacement components of the honeycomb sandwich plate can be represented as follows: w(x, y, z, t) w 0 (x, y, t). ( where u 0 u(x, y, 0, t), v 0 v(x, y, 0, t), w 0 w(x, y, 0, t), ϕ x (zu/zz) z 0 , and ϕ y (zv/zz) z 0 .e relations between strains and displacements according to von Karman nonlinear strains-displacements are given by ( e constitutive relations of the honeycomb sandwich plate can be written as follows: where Q 11 , Q 12 , Q 22 , Q 44 , Q 55 , and Q 66 are elastic constants which can be written as e terms E x and E y are equivalent elastic modulus of the honeycomb core at x and y directions, respectively, and v x and v y are equivalent Poisson's ratios.ese 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]: 2 Advances in Materials Science and Engineering where d is the thickness of the honeycomb core cellular cells and t and l are length of the straight and sloping wall of the honeycomb core cellular cells (Figure 2).E s and v s are, respectively, Young's modulus and Poisson's ratio of the materials for the honeycomb core.In (4), G xz and G yz are transverse shear moduli, which can be computed by using the following formula: e relation between the equivalent density of the honeycomb core ρ c and the density of the materials ρ s is e formula above for hexagon honeycomb core can be simpli ed 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:

Advances in Materials Science and Engineering
where e stiffness elements of the honeycomb sandwich plate are given by

Frequencies of Transverse Vibrations
In this section, the frequencies of transverse vibration for w direction are considered by using the Rayleigh-Ritz method.e deflections for w is as follows: w(x, y, t) � W(x, y)sin ωt.(12) e kinetic energy of the honeycomb sandwich plate is e maximum kinetic energy is of the following form: e potential energy of the honeycomb plate is of the following form: e stresses and strains in (15) are obtained from (2) and ( 3).
e boundary conditions for the cantilever honeycomb sandwich plate are given as follows: 4 Advances in Materials Science and Engineering where the equivalent shear forces are e mode function w 0 can be written as the following form: where C 11 , C 12 , C 21 , and C 22 are modal parameters and where e 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 matrix and let the determinant equal to 0; the first four frequencies for the honeycomb sandwich plate can be obtained numerically.e parameters of the honeycomb sandwich plate are as follows: density ρ s � 2.66 × 10 3 kg/m 3 , Young's modulus E s � 72 × 10 9 Pa, Poisson's ratio v � 0.33, the length a � 5 m and width b � 2 m, the thickness of the core is h c � 0.01 m, and the length of the straight and sloping wall of the honeycomb core cellular cells d � 0.0008 m and l � 0.01 m.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.e 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.

Nonlinear Responses of the Plate
In this section, the 2-truncated function for w 0 is assumed as the following: where Truncated functions for other directions are Advances in Materials Science and Engineering Hence, 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 in-plane 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 Next, numerical simulations are given to study the nonlinear responses of the honeycomb sandwich plate under transverse and in-plane 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 in-plane force and the transverse excitation, are chosen as follows: Ω Ω 1 100 Hz, c 150 N • s/m.e loads are chosen as F 1 25 Pa, F 2 10 Pa and P 0 25 Pa, P 1 24.9 Pa. Figure 3 illustrates chaotic motions for the honeycomb sandwich plate.Figures 3(a 4 presents the bifurcations of the honeycomb plate with variations of parametric excitation amplitude P 1 .From the bifurcation diagram, it can be seen that the honeycomb sandwich plate can have periodic and multiperiodic motions when parametric amplitude P 1 changes.It is periodic doubling bifurcation when the in-plane force P 1 is about 200 Pa.
Figure 5 shows periodic motions when the in-plane load P 1 −260 Pa.
When the in-plane P 1 increases, multiperiodic motions are found for the honeycomb sandwich plate.Figure 6 represents 2-periodic motions as P 1 1300 Pa.

Advances in Materials Science and Engineering
When the thickness of the honeycomb core cells changes to 0.0012 m and other parameters are kept the same, the bifurcation diagram with in-plane force changing has been presented in Figure 7.It can be seen from the bifurcation diagram that it is periodic doubling bifurcation when the in-plane force P 1 is about 5000 Pa.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 P 1 needs increase to bifurcate.In addition, if the force P 1 is larger than 6000 Pa, the amplitude of the plate increases obviously and then divergences.
When P 1 1000 Pa, Figure 8 shows periodic motion of the honeycomb sandwich plate.
When the force P 1 is chosen as P 1 5500 Pa, the multiperiodic motions are obtained as shown in Figure 9.

Conclusions
Dynamics of a cantilever honeycomb sandwich plate subjected to both in-plane and transverse excitations have been studied.e governing equations are derived by using Hamilton's principle and Reddy's third-order shear deformation theory.e natural frequencies of the cantilever honeycomb sandwich plate are obtained by Rayleigh-Ritz method.
en, the Galerkin truncation procedure is employed to transform the nonlinear partial di erential equations into a set of nonlinear ordinary   Advances in Materials Science and Engineering di erential 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 in-plane excitations.

Figure 1 :
Figure 1: e model of cantilever honeycomb sandwich plate.

Figure 2 :
Figure 2: e unit cell of general hexagonal core layer.

Figure 3 (
Figure3(e) represents the phase portraits in threedimensional space (w 1 , _ w 1 , w 2 ).Chosen the transverse loads as F 1 5 Pa and F 2 7 Pa and the in-plane load P 0 80 Pa, Figure4presents the bifurcations of the honeycomb plate with variations of parametric excitation amplitude P 1 .From the bifurcation diagram, it can be seen that the honeycomb sandwich plate can have periodic and multiperiodic motions when parametric amplitude P 1 changes.It is periodic doubling bifurcation when the in-plane force P 1 is about 200 Pa.Figure5shows periodic motions when the in-plane load P 1 −260 Pa.When the in-plane P 1 increases, multiperiodic motions are found for the honeycomb sandwich plate.Figure6represents 2-periodic motions as P 1 1300 Pa.

Figure 5 :
Figure 5: Periodic motions of the cantilever honeycomb sandwich plate.

Figure 8 :
Figure 8: Periodic motion of the cantilever honeycomb sandwich plate.

Table 2 :
Frequencies of different material densities of honeycomb sandwich plate.

Table 3 :
Frequencies of different lengths of the unit cell.

Table 4 :
Frequencies of different thicknesses of the unit cell.

Table 1 :
Frequencies of different core layer thicknesses of honeycomb sandwich plate.