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

Authors: Junhua Zhang; Xiaodong Yang; Wei Zhang
Journal: Advances in Materials Science and Engineering
 (2018)

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## Abstract

Dynamics of a cantilever honeycomb sandwich plate are studied in this paper. The 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. 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 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.


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## Body

## 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 Section2, 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 Figure1. The plate is of length a, width b, and thickness h. Let u,v,w and u0,v0,w0 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 ϕx and ϕy, respectively. The plate is subjected to an in-plane excitation p=p0+p1cos⁡Ω1t and a transverse excitation Fx,ycos⁡Ωt, where Ω1 and Ω are the excitation frequencies of the transverse and in-plane excitations, respectively.Figure 1
The model of cantilever honeycomb sandwich plate.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:(1)ux,y,z,t=u0x,y,t+zϕxx,y,t−43h2z3ϕx+∂w0∂x,vx,y,z,t=v0x,y,t+zϕyx,y,t−43h2z3ϕy+∂w0∂y,wx,y,z,t=w0x,y,t.where u0=ux,y,0,t,v0=vx,y,0,t,w0=wx,y,0,t, ϕx=∂u/∂zz=0, and ϕy=∂v/∂zz=0.The relations between strains and displacements according to von Karman nonlinear strains-displacements are given by(2)εxx=∂u∂x+12∂w∂x2,εxz=12∂u∂z+∂w∂x,εxy=12∂u∂y+∂v∂x+∂w∂x∂w∂y,γyy=∂v∂y+12∂w∂y2,γyz=12∂v∂z+∂w∂y,γzz=∂w∂z.The constitutive relations of the honeycomb sandwich plate can be written as follows:(3)σxxσyyσxy=Q11Q120Q12Q22000Q66εxxεyyγxy,σyzσxz=Q4400Q55γyzγxz,where Q11,Q12,Q22,Q44,Q55, and Q66 are elastic constants which can be written as(4)Q11=Ex1−vxvy,Q12=vxEy1−vxvy,Q22=Ey1−vxvy.Q44=Gyz,Q55=Gxz,Q66=Gxy.The termsEx and Ey are equivalent elastic modulus of the honeycomb core at x and y directions, respectively, and vx and vy 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]:(5)Ex=Esd3l3cos⁡θ+d/lt/l+sin⁡θsin2⁡θ1+2.4+1.5vs+cot2⁡θd2/l2,vx=cos⁡θcos⁡θ+d/l1+1.4+1.5vsd2/l2t/l+sin⁡θsin⁡θ1+2.4+1.5vs+cot2θd2/l2,Ey=Esd3l3t/l+sin⁡θcos2⁡θcos⁡θ+d/l11+2.4+1.5vs+tan2⁡θ+2t/lsec2⁡θd2/l2,vy=t/l+sin⁡θsin⁡θcos⁡θcos⁡θ+d/l1+1.4+1.5vsd2/l21+2.4+1.5vs+tan2⁡θ+2t/lsec2⁡θd2/l2,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). Es and vs are, respectively, Young’s modulus and Poisson’s ratio of the materials for the honeycomb core.Figure 2
The unit cell of general hexagonal core layer.In (4), Gxz and Gyz are transverse shear moduli, which can be computed by using the following formula:(6)G=E21+v.The relation between the equivalent density of the honeycomb coreρc and the density of the materials ρs is(7)ρcρs=d/lt/l+22cos⁡θt/l+sin⁡θ.The formula above for hexagon honeycomb core can be simplified as(8)ρcρs=23dl.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:(9)∂Nxx∂x+∂Nxy∂y=I0u¨0+I1−c1I3ϕ¨x−c1I3∂w¨0∂x,∂Nxy∂x+∂Nyy∂y=I0v¨0+I1−c1I3ϕ¨y−c1I3∂w¨0∂y,∂Nxx∂x∂w0∂x+Nxx∂2w0∂x2+∂Nxy∂x∂w0∂y+2Nxy∂2w0∂x∂y+∂Nxy∂y∂w0∂x+∂Nyy∂y∂w0∂y+Nyy∂2w0∂y2+∂Qx∂x−c2∂Rx∂x+∂Qy∂y−c2∂Ry∂y+c1∂2Pxx∂x2+2∂2Pxy∂x∂y+∂2Pyy∂y2+F0cos⁡Ωt−γw˙0=I0w¨0−c12I6∂2w¨0∂x2+∂2w¨0∂y2+c1I3∂u¨0∂x+∂v¨0∂y+c1I4−c1I6∂ϕ¨x∂x+∂ϕ¨y∂y,∂Mxx∂x+∂Mxy∂y−c1∂Pxx∂x−c1∂Pxy∂y−Qx−c2Rx=I1−c1I3u¨0+I2−2c1I4+c12I6ϕ¨x−c1I4−c1I6∂w¨0∂x,∂Mxy∂x+∂Myy∂y−c1∂Pxy∂y−c1∂Pyy∂y−Qy−c2Ry=I1−c1I3v¨0+I2−2c1I4+c12I6ϕ¨y−c1I4−c1I6∂w¨0∂y,where(10)NxxNyyNxy=A11A120A12A22000A66∂u0∂x+12∂w0∂x2∂v0∂y+12∂w0∂y2∂u0∂y+∂v0∂x+∂w0∂x∂w0∂y,MxxMyyMxy=D11D120D21D22000D66∂ϕx∂x∂ϕy∂y∂ϕx∂y+∂ϕy∂x−c1F11F120F21F22000F66∂ϕx∂x+∂2w0∂x2∂ϕy∂y+∂2w0∂y2∂ϕx∂y+∂ϕy∂x+2∂2w0∂x∂y,PxxPyyPxy=F11F120F21F22000F66∂ϕx∂x∂ϕy∂y∂ϕx∂y+∂ϕy∂x−c1H11H120H21H22000H66∂ϕx∂x+∂2w0∂x2∂ϕy∂y+∂2w0∂y2∂ϕx∂y+∂ϕy∂x+2∂2w0∂x∂y,QxQy=A4400A55ϕy+∂w0∂yϕx+∂w0∂x−c2D4400D55ϕy+∂w0∂yϕx+∂w0∂x,RxRy=D4400D55ϕy+∂w0∂yϕx+∂w0∂x−c2F4400F55ϕy+∂w0∂yϕx+∂w0∂x.The stiffness elements of the honeycomb sandwich plate are given by(11)Aij,Bij,Dij,Eij,Fij,Hij=∫−h/2h/2Qij1,z,z2,z3,z4,z6dz,i,j=1,2,6,Aij,Dij,Fij=∫−h/2h/2Qij1,z,z2,z4dz,i,j=4,5.


## 3. Frequencies of Transverse Vibrations

In this section, the frequencies of transverse vibration forw direction are considered by using the Rayleigh–Ritz method. The deflections for w is as follows:(12)wx,y,t=Wx,ysin⁡ωt.The kinetic energy of the honeycomb sandwich plate is(13)T=12∫∫m∂w∂t2dxdy.The maximum kinetic energy is of the following form:(14)Tmax=ω22∫∫mW2dxdy.The potential energy of the honeycomb plate is of the following form:(15)Umax=12∫∫∫σxxεxx+σyyεyy+σxyγxy+σyzγyz+σxzγxzdV.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:(16)x=0:w=v=u=ϕx=ϕy=0,x=a:Nxy=Mxx=Mxy−c1Pxy=Q¯x=0,y=0:Nyy=Nxy=Myy=Mxy−c1Pxy=Q¯y=0,y=b:Nyy=Nxy=Myy=Mxy−c1Pxy=Q¯y=0,∫0aNxxx=0,ady=∫0ap0+p1cos⁡Ω1tdy,where the equivalent shear forces are(17)Q¯x=Qx+∂Mxy∂y−c2Rx+c1∂Pxx∂x+∂Pxy∂y,Q¯y=Qy+∂Mxy∂x−c2Ry+c1∂Pyy∂y+∂Pxy∂y.The mode functionw0 can be written as the following form:(18)w0=C11X1Y1+C12X1Y2+C21X2Y1+C22X2Y2,where C11,C12,C21, and C22 are modal parameters and(19)X1=sinλ1x−sinhλ1x+α1coshλ1x−cosλ1x,X2=sinλ2x−sinhλ2x+α2coshλ2x−cosλ2x,Y1=1,Y2=32yb−1,where(20)cos⁡λia⁡coshλia+1=0,i=1,2,α1=sinhλ1a+sinλ1acoshλ1a+cosλ1a,α2=sinhλ2a+sinλ2acoshλ2a+cosλ2a.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(21)∂Tmax−Umax∂C11,∂Tmax−Umax∂C12,∂Tmax−Umax∂C21,∂Tmax−Umax∂C22,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: densityρs=2.66×103kg/m3, Young’s modulus Es=72×109Pa, Poisson’s ratio v=0.33, the length a=5m and width b=2m, the thickness of the core is hc=0.01m, and the length of the straight and sloping wall of the honeycomb core cellular cells d=0.0008m and l=0.01m. 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 1
Frequencies of different core layer thicknesses of honeycomb sandwich plate.
Modal
h
c
m
0.01
0.02
0.03
0.04
0.05
1
360.28803
360.28803
360.28805
360.28807
360.28809
2
1243.29964
1243.29970
1243.29979
1243.29992
1243.30009
3
2207.55777
2207.55929
2207.56181
2207.56540
2207.56995
4
2507.85212
2507.86810
2507.89476
2507.93202
2507.97997Table2 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 2
Frequencies of different material densities of honeycomb sandwich plate.
Modal
ρ
s
10
3
kg
/
m
3
2.66
2.67
2.68
2.69
2.70
1
360.28803
359.61270
358.94115
358.27335
357.60927
2
1243.29964
1240.96918
1238.65177
1236.34730
1234.05564
3
2207.55777
2203.41989
2199.30518
2195.21344
2191.14447
4
2507.85212
2503.15136
2498.47694
2493.82861
2489.20612Table3 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 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
2747.21435Table4 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.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
2364.42566It can be concluded from Tables1–4 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 forw0 is assumed as the following:(22)w0=w1tX1xY3y+w2tX2xY3y,where(23)Y3x=sin⁡μ3y+sinh⁡μ3y−β3cosh⁡μ3x+cos⁡μ3x,cos⁡μ3b⁡cosh⁡μ3b−1=0,β3=sinh⁡μ3b−sin⁡μ3bcosh⁡μ3b−cos⁡μ3b.Truncated functions for other directions are(24)u0=u1t∂X1x∂xY3y+u2t∂X2x∂xY3y,v0=v1tX1x∂Y3y∂y+v2tX2x∂Y3y∂y,ϕx=ϕx1t∂X1x∂xY3y+ϕx2t∂X2x∂xY3y,ϕy=ϕy1tX1x∂Y3y∂y+ϕy2tX2x∂Y3y∂y.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:(25)F0t=F1tX1xY3y+F2tX2xY3y.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:(26)w¨1+f17γw˙1+f15w1+f16w2+f19w1P0+P1cos⁡Ω1t+f11w13+f12w12w2+f13w1w22+f14w23=f18F1cos⁡Ωt,w¨2+f27γw˙2+f25w1+f26w2+f29w2P0+P1cos⁡Ω1t+f21w13+f22w12w2+f23w1w22+f24w23=f28F2cos⁡Ωt.Equations (26) are typical nonlinear equations governing the vibrations of generalized coordinates undergoing external excitation with frequency Ω and parametric excitation with frequency Ω1.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=Ω1=100Hz,γ=150N⋅s/m. The loads are chosen as F1=25Pa, F2=10Pa and P0=25Pa, P1=24.9Pa. Figure 3 illustrates chaotic motions for the honeycomb sandwich plate. Figures 3(a) and 3(b) present the phase portraits on the planes w1,w˙1 and w2,w˙2. Figures 3(c) and 3(d) give the time histories on the planes t,w1 and t,w2. Figure 3(e) represents the phase portraits in three-dimensional space w1,w˙1,w2.Figure 3
Chaotic motion of the cantilever honeycomb sandwich plate.
(a)
(b)
(c)
(d)
(e)Chosen the transverse loads asF1=5Pa and F2=7Pa and the in-plane load P0=80Pa, Figure 4 presents the bifurcations of the honeycomb plate with variations of parametric excitation amplitude P1. From the bifurcation diagram, it can be seen that the honeycomb sandwich plate can have periodic and multiperiodic motions when parametric amplitude P1 changes. It is periodic doubling bifurcation when the in-plane force P1 is about 200Pa.Figure 4
The bifurcation diagram of the system with change ofP1.Figure5 shows periodic motions when the in-plane load P1=−260Pa.Figure 5
Periodic motions of the cantilever honeycomb sandwich plate.
(a)
(b)
(c)
(d)
(e)When the in-planeP1 increases, multiperiodic motions are found for the honeycomb sandwich plate. Figure 6 represents 2-periodic motions as P1=1300Pa.Figure 6
Period-2 motion of the cantilever honeycomb sandwich plate.
(a)
(b)
(c)
(d)
(e)When the thickness of the honeycomb core cells changes to0.0012m 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 P1 is about 5000Pa. 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 P1 needs increase to bifurcate. In addition, if the force P1 is larger than 6000Pa, the amplitude of the plate increases obviously and then divergences.Figure 7
The bifurcation diagram of the system with change ofP1.WhenP1=1000Pa, Figure 8 shows periodic motion of the honeycomb sandwich plate.Figure 8
Periodic motion of the cantilever honeycomb sandwich plate.
(a)
(b)
(c)
(d)
(e)When the forceP1 is chosen as P1=5500Pa, the multiperiodic motions are obtained as shown in Figure 9.Figure 9
Multiperiodic motions of the cantilever honeycomb sandwich plate.
(a)
(b)
(c)
(d)
(e)


## 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.



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