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Free Vibrations and Nonlinear Responses for a Cantilever Honeycomb Sandwich Plate.

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 low-velocity 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 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]. The impact behavior of honeycomb sandwich panels has been investigated by finite element method by Menna et al. [7]. The vibration-based spatial damage identification of the honeycomb-core 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 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. 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 in-plane 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 in-plane and transversal excitations, the model and the coordinate system are shown in Figure 1. The plate is of length a, width b, and thickness h. Let (u,v,w) and ([u.sub.0],[v.sub.0],[w.sub.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. The mid-plane rotations of a transverse normal about the x and y axes are denoted by [[empty set].sub.x] and [[empty set].sub.y], respectively. The plate is subjected to an in-plane excitation p = [p.sub.0] + [p.sub.1] cos [[OMEGA].sub.1]t and a transverse excitation F(x,y)cos [[OMEGA].sub.t], where [[OMEGA].sub.1] and [OMEGA] 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:

[mathematical expression not reproducible]. (1)

where [u.sub.0] = u (x, y, 0, t),[v.sub.0] = v(x, y, 0,f),[w.sub.0] = w(x, y, 0, t), [[empty set].sub.x] = [([partial derivative]u/[partial derivative]z).sub.z=0], and [[empty set].sub.y] = [([partial derivative]v/[partial derivative]z).sub.z=0].

The relations between strains and displacements according to von Karman nonlinear strains-displacements are given by

[mathematical expression not reproducible]. (2)

The constitutive relations of the honeycomb sandwich plate can be written as follows:

[mathematical expression not reproducible], (3)

where [Q.sub.11],[Q.sub.12],[Q.sub.22],[Q.sub.44],[Q.sub.55], and [Q.sub.66] are elastic constants which can be written as

[mathematical expression not reproducible]. (4)

The terms [E.sub.x] and [E.sub.y] are equivalent elastic modulus of the honeycomb core at x and y directions, respectively, and [v.sub.x] and [v.sub.y] 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]:

[mathematical expression not reproducible], (5)

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.sub.s] and [v.sub.s] are, respectively, Young's modulus and Poisson's ratio of the materials for the honeycomb core.

In (4), [G.sub.xz] and [G.sub.yz] are transverse shear moduli, which can be computed by using the following formula:

G = E/2(1+v). (6)

The relation between the equivalent density of the honeycomb core [[rho].sub.c] and the density of the materials [[rho].sub.s] is

[[rho].sub.c]/[[rho].sub.s] = d/l((t/l+2)/2 cos [theta](t/l + sin [theta]). (7)

The formula above for hexagon honeycomb core can be simplified as

[[rho].sub.c]/[[rho].sub.s] = 2/[square root of 3] d/l. (8)

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:

[mathematical expression not reproducible], (9)

where

[mathematical expression not reproducible]. (10)

The stiffness elements of the honeycomb sandwich plate are given by

[mathematical expression not reproducible]. (11)

3. Frequencies of Transverse Vibrations

In this section, the frequencies of transverse vibration for w direction are considered by using the Rayleigh-Ritz method. The deflections for w is as follows:

w(x,y,t) = W (x, y) sin [omega]t. (12)

The kinetic energy of the honeycomb sandwich plate is

T = 1/2[integral][integral]m [([partial derivative]w/[partial derivative]t).sup.2] dxdy. (13)

The maximum kinetic energy is of the following form:

[T.sub.max] = [[omega].sup.2]/2 [integral][integral] m[W.sup.2] dxdy. (14)

The potential energy of the honeycomb plate is of the following form:

[U.sub.max] = 1/2 [integral][integral][integral]([[sigma].sub.xx][[epsilon].sub.xx] + [[sigma].sub.yy][[epsilon].sub.yy] + [[sigma].sub.xy][[epsilon].sub.xy] + [[sigma].sub.yz][[gamma].sub.yz] + [[sigma].sub.xz][[gamma].sub.xz]) dv. (15)

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:

[mathematical expression not reproducible], (16)

where the equivalent shear forces are

[mathematical expression not reproducible]. (17)

The mode function [w.sub.0] can be written as the following form:

[w.sub.0] = [C.sub.11][X.sub.1][Y.sub.1] + [C.sub.12][X.sub.1][Y.sub.2] + [C.sub.21][X.sub.2][Y.sub.1] + [C.sub.22][X.sub.2][Y.sub.2], (18)

where [C.sub.11],[C.sub.12],[C.sub.21], and [C.sub.22] are modal parameters and

[mathematical expression not reproducible], (19)

where

[mathematical expression not reproducible]. (20)

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 matrix

[mathematical expression not reproducible], (21)

and 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 [[rho].sub.s] = 2.66 x [10.sup.3] kg/[m.sup.3], Young's modulus [E.sub.s] = 72 x [10.sup.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.sub.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. 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 2-truncated function for w0 is assumed as the following:

[w.sub.0] = [w.sub.1](t)[X.sub.1](x)[Y.sub.3](y) + [w.sub.2](t)[X.sub.2](x)[Y.sub.3](y), (22)

where

[Y.sub.3](x) = sin [[mu].sub.3]y + sinh [[mu].sub.3]y

- [[beta].sub.3] (cosh [[mu].sub.3]x + cos[[mu].sub.3]x),

cos [[mu].sub.3]b cosh [[mu].sub.3] b - 1 = 0, (23)

[[beta].sub.3] = sinh [[mu].sub.3]b - sin [[mu].sub.3]b/cosh [[mu].sub.3]b - cos [[mu].sub.3]b.

Truncated functions for other directions are

[mathematical expression not reproducible], (24)

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:

[F.sub.0] (t) = [F.sub.1](t)[X.sub.1](x)[Y.sub.3](y) + [F.sub.2](t)[X.sub.2](x)[Y.sub.3](y). (25)

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:

[mathematical expression not reproducible]. (26)

Equations (26) are typical nonlinear equations governing the vibrations of generalized coordinates undergoing external excitation with frequency [OMEGA] and parametric excitation with frequency [[OMEGA].sub.1].

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: [OMEGA] = [[OMEGA].sub.1] = 100 Hz, [gamma] = 150 N x s/m. The loads are chosen as [F.sub.1] = 25 Pa, [F.sub.2] = 10 Pa and [P.sub.0] = 25 Pa, [P.sub.1] = 24.9 Pa. Figure 3 illustrates chaotic motions for the honeycomb sandwich plate. Figures 3(a) and 3(b) present the phase portraits on the planes ([w.sub.1],[[??].sub.1]) and ([w.sub.2],[[??].sub.2]). Figures 3(c) and 3(d) give the time histories on the planes (t, [w.sub.1]) and (t,[w.sub.2]).

Figure 3(e) represents the phase portraits in three-dimensional space ([w.sub.1],[[??].sub.1],[w.sub.2]).

Chosen the transverse loads as [F.sub.1] = 5 Pa and [F.sub.2] = 7 Pa and the in-plane load [P.sub.0] = 80 Pa, Figure 4 presents the bifurcations of the honeycomb plate with variations of parametric excitation amplitude [P.sub.1]. From the bifurcation diagram, it can be seen that the honeycomb sandwich plate can have periodic and multiperiodic motions when parametric amplitude [P.sub.1] changes. It is periodic doubling bifurcation when the in-plane force [P.sub.1] is about 200 Pa.

Figure 5 shows periodic motions when the in-plane load [P.sub.1] = -260 Pa.

When the in-plane [P.sub.1] increases, multiperiodic motions are found for the honeycomb sandwich plate. Figure 6 represents 2-periodic motions as [P.sub.1] = 1300 Pa.

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.sub.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.sub.1] needs increase to bifurcate. In addition, if the force [P.sub.1] is larger than 6000 Pa, the amplitude of the plate increases obviously and then divergences.

When [P.sub.1] = 1000 Pa, Figure 8 shows periodic motion of the honeycomb sandwich plate.

When the force [P.sub.1] is chosen as [P.sub.1] = 5500 Pa, the multiperiodic motions are obtained as shown in Figure 9.

5. Conclusions

Dynamics of a cantilever honeycomb sandwich plate subjected to both in-plane and transverse excitations have been studied. The governing equations are derived by using Hamilton's principle and Reddy's third-order 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 in-plane excitations.

https://doi.org/10.1155/2018/8162873

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).

References

[1] L. J. Gibson and M. F. Ashby, Cellular Solids: Structure and Properties, Cambridge University Press, Cambridge, UK, 2nd edition, 1997.

[2] M. Meo, R. Vignjevic, and G. Marengo, "The response of honeycomb sandwich panels under low-velocity impact loading," International Journal of Mechanical Sciences, vol. 47, no. 9, pp. 1301-1325, 2005.

[3] B. L. Buitrago, C. Santiuste, S. Sanchez-Saez, E. Barbero, and C. Navarro, "Modelling of composite sandwich structures with honeycomb core subjected to high-velocity impact," Composite Structures, vol. 92, no. 9, pp. 2090-2096, 2010.

[4] Y. Q. Li, F. Li, and D. W. Zhu, "Geometrically nonlinear free vibrations of the symmetric rectangular honeycomb sandwich panels with simply supported boundaries," Composite Structures, vol. 92, no. 5, pp. 1110-1119, 2010.

[5] S. Feli and M. H. Namdari Pour, "An analytical model for composite sandwich panels with honeycomb core subjected to high-velocity impact," Composites PartB: Engineering, vol. 43, no. 5, pp. 2439-2447, 2012.

[6] F. Lopez Jimenez and N. Triantafyllidis, "Buckling of rectangular and hexagonal honeycomb under combined axial compression and transverse shear," International Journal of Solids and Structures, vol. 50, no. 24, pp. 3934-3946, 2013.

[7] C. Menna, A. Zinno, D. Asprone, and A. Prota, "Numerical assessment of the impact behavior of honeycomb sandwich structures," Composite Structures, vol. 106, pp. 326-339, 2013.

[8] A. Katunin, "Vibration-based spatial damage identification in honeycomb-core sandwich composite structures using wavelet analysis," Composite Structures, vol. 118, pp. 385-391, 2014.

[9] D. Balkan and Z. Mecitoglu, "Nonlinear dynamic behavior of viscoelastic sandwich composite plates under non-uniform blast load: theory and experiment," International Journal of Impact Engineering, vol. 72, pp. 85-104, 2014.

[10] G. Laurent, "Vibroacoustic flexural properties of symmetric honeycomb sandwich panels with composite faces," Journal of Sound and Vibration, vol. 343, pp. 71-103, 2015.

[11] Y. M. Li, M. P. Hoang, B. Abbes, F. Abbes, and Y. Q. Guo, "Analytical homogenization for stretch and bending of honeycomb sandwich plates with skin and height effects," Composite Structures, vol. 120, pp. 406-416, 2015.

[12] J. N. Reddy, Mechanics of Laminated Composite Plates and Shells: Theory and Analysis, CRC Press, New York, USA, 2004.

[13] M. Fu, O. Xu, and Y. Chen, "An overview of equivalent parameters of honeycomb core," Materials Review, vol. 29, no. 3, pp. 127-134, 2015.

Junhua Zhang (iD), (1) Xiaodong Yang (iD), (2) and Wei Zhang (iD) (2)

(1) College of Mechanical Engineering, Beijing Information Science and Technology University, Beijing 100192, China

(2) College of Mechanical Engineering, Beijing University of Technology, Beijing 100124, China

Correspondence should be addressed to Junhua Zhang; zjhuar@163.com

Received 11 July 2017; Accepted 22 November 2017; Published 6 March 2018

Academic Editor: Michael J. Schutze

Caption: Figure 1: The model of cantilever honeycomb sandwich plate.

Caption: Figure 2: The unit cell of general hexagonal core layer.

Caption: Figure 3: Chaotic motion of the cantilever honeycomb sandwich plate.

Caption: Figure 4: The bifurcation diagram of the system with change of [P.sub.1].

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

Caption: Figure 6: Period-2 motion of the cantilever honeycomb sandwich plate.

Caption: Figure 7: The bifurcation diagram of the system with change of [P.sub.1].

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

Caption: Figure 9: Multiperiodic motions of the cantilever honeycomb sandwich plate.
Table 1: Frequencies of different core layer thicknesses of
honeycomb sandwich plate.

Modal                             [h.sub.c](m)
           0.01         0.02          0.03          0.04

1       360.28803    360.28803     360.28805     360.28807
2       1243.29964   1243.29970    1243.29979    1243.29992
3       2207.55777   2207.55929    2207.56181    2207.56540
4       2507.85212   2507.86810    2507.89476    2507.93202

Modal
           0.05

1       360.28809
2       1243.30009
3       2207.56995
4       2507.97997

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

Modal                [[rho].sub.s] ([10.sup.3]kg/[m.sup.3])

           2.66         2.67           2.68           2.69

1       360.28803    359.61270      358.94115      358.27335
2       1243.29964   1240.96918     1238.65177     1236.34730
3       2207.55777   2203.41989     2199.30518     2195.21344
4       2507.85212   2503.15136     2498.47694     2493.82861

Modal

           2.70

1        357.60927
2       1234.05564
3       2191.14447
4       2489.20612

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

Modal                               l (m)

          0.008        0.009        0.010        0.011        0.012

1       322.25141    341.79923    360.28803    377.87327    394.67576
2       1112.04101   1179.49760   1243.29964   1303.98366   1361.96651
3       1974.49969   2094.27319   2207.55777   2315.30612   2418.25837
4       2243.09113   2379.15742   2507.85212   2630.25750   2747.21435

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

Modal                               t (m)
          0.0005       0.0006       0.0007       0.0008       0.0009

1       455.73231    416.02478    385.16410    360.28803    339.68281
2       1572.66346   1435.63876   1329.14322   1243.29964   1172.19414
3       2792.36425   2549.06814   2359.97852   2207.55777   2081.30542
4       3172.20989   2895.81819   2681.00668   2507.85212   2364.42566
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Title Annotation:Research Article
Author:Zhang, Junhua; Yang, Xiaodong; Zhang, Wei
Publication:Advances in Materials Science and Engineering
Date:Jan 1, 2018
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