Application of a Semi-Analytical Method for the Free Vibration and Damping Analysis of Sandwich Shell Structures

Application of a Semi-Analytical Method for the Free Vibration and Damping Analysis of Sandwich Shell Structures

Kwang-Hun Kim, Ju-Song Kim

Faculty of Mechanical Engineering No.1, Pyongyang University of Mechanical Engineering,

Democratic Peoples Republic of Korea

Song Ri

Faculty of Electronics and Automation, Kim Il Sung

University, Pyongyang, Democratic Peoples Republic of Korea

Hyon-Sik Hong

Institute of Mechanics, State Academy of Science, Democratic Peoples Republic of Korea

Jong-Hun Kye

Department of Physics, University of Sciences, Pyongyang, Democratic Peoples Republic of Korea

Abstarct – In this paper, the free vibration and damping characteristics of sandwich cylindrical and conical shells with viscoelastic core are investigated by using a semi-analytical method. Donnel's shell theory and the energy principle are applied to establish the theoretical formulations of the sandwich shell with viscoelastic core. The displacement components of the viscoelastic core are expressed as those of base and constraining layers by using continuity condition. The displacement field of the sandwich shell expanded by the Legendre polynomials in axial direction and Fourier series in circumferential direction. The presented method are verified to have enough reliability and accuracy for predicting the natural frequencies and modal loss factors of the sandwich cylindrical and conical shells with viscoelastic layer by comparing to the vibrational analysis results of published article.

Keywords: Sandwich shell, a semi-analytical method, Free vibration, Damping characteristics, Numerical analysis

1. INTRODUCTION

   Sandwich structures are widely used in various engineering applications duo to the high vibration and noise reduction ability caused by the large shear deformation of the viscoelastic material layer [1]. Therefore, the study on the dynamic characteristics of the sandwich structures have attracted a lot of interest of many researchers [2, 3]. Yang et al. [4] studied the vibration and damping characteristics of the sandwich conical shells and annular plates with arbitrary boundary conditions including classical and elastic ones by using a simple and efficient modified Fourier solution. Wang et al. [5] presented a semi-analytical method for the free vibration analysis of the functionally graded (FG) sandwich doubly-curved panels and shells of revolution with arbitrary boundary conditions. Bardell et al. [6] presented the vibration study of a general three-layer conical sandwich panel based on the h-p version of the finite element method. In their study, the h-p finite element formulation of sandwich panel was derived based on a set of trigonometric assumed displacement functions. Singha et al. [7] investigated the free vibration behavior of rotating pretwisted sandwich conical shell panels with functionally graded graphene-reinforced composite (FG-GRC) face sheets and homogenous core using finite element method in conjunction with HSDT. Sofiyev and Osmancelebioglu [8] demonstrated the effectiveness of functionally graded coatings in the vibration of sandwich truncated conical shells. In their study, the governing equations were established by using FSDT and Donnell kinematics assumptions. Jin et al. [9] developed an accurate solution for the vibration and damping characteristics of a three-layered passive constrained layer

   damping (PCLD) cylindrical shell with general elastically restrained boundaries by means of the modified FourierRitz method in conjunction with Donnell shell assumptions and linear viscoelastic theory. Sahu et al. [10] conducted the free vibration study of doubly curved sandwich shell panels having a core of viscoelastic material, constrained by a functionally graded material (FGM) layer by using finite element method (FEM) in framework of FSDT.

   Meanwhile, researchers proposed different numerical methods such as FEM [10, 11], differential quadrature method [12], pb-2 Ritz method [13], Non-Uniform Rational B-Splines (NURBS) method [14], spectral-Tchebychev solution technique [15], dynamic stiffness method [16-18], meshfree method [19, 20] for the dynamic analysis of composite shells and plates. The meshfree method is attracting attention from researchers due to their excellent behaviors [21-23].

   In this study, a semi-analytical method for the free vibration and damping analysis of sandwich cylindrical and conical shells with viscoelastic core with Legendre polynomials as displacement functions is presented. The theoretical formulations of the sandwich shell are established by using the energy principle, FSDT and Donnels shell thoery. Using the continuity condition in interface between the layers, the displacement components of the viscoelastic core are replaced by those of base and constraining layers. The displacement component of the sandwich shell are approximated by using Fourier series in conjunction with Legendre polynomials. The accuracy and reliability of the presented method are verified through the comparison with the results of published literature.

2. THEORETICAL FORMULATIONS

   Fig. 1 shows the geometry and coordinate system of a sandwich conical shell which is composed of the base layer, the viscoelastic core and the constraining layer. Orthogonal curvilinear coordinate systems (x, , z) are located on the middle surfaces of each layer of the conical shell. The symbols L and are the length and semi-vertex angle of the sandwich conical shell. Ri0 and hi (i=s,v,c) denote small edge radius and thickness of each layer, and the subscripts s, v, and c are indicated for base layer, viscoelastic core and constraining layer, respectively. The cylindrical shell is considered as a conical shell with semi-vertex angle =0.

   x Rv0 Rs0 Rc0 hs hv hc

   Fig. 1. Geometry and coordinate system of sandwich conical shell with viscoelastic core.

   Based on the FSDT and Donnels shell theory, the displacement components at any point of individual layer are expressed as [9]

   ui ui z xi v v z i s,v,c i i i w w i

   (1)

   where ui, vi and w represent the middle plane displacements of each layer in the x, and z directions, respectively. In the base and constraining layers, the shear rotations xi and i can be written as follows.

   are only allowed flexural and axial deformations. Considering Eq. (3), the strains of the core layer are expressed as:

   v 1 u u c w xz h c s x x v v 1 cos v 1 cos v c w z h 2R c h 2R s v v v v

   (4)

   where

   c hs hc 2hv x 2h v 1 h h c s c 2hv Rc Rs 1 h cos h cos s c 4 4Rv Rs Rc

   (5)

   The shear stresses of the core layer are expressed as:

   v G v v G v xz v xz z v z

   (6)

   where the shear modulus Gv of the viscoelastic material is composed of real and imaginary parts.

   Gv Gr iGi

   (7)

   where Gr and Gi denote the real part and imaginary part of the complex shear modulus of the viscoelastic aterial layer, respectively.

   Meanwhile, the stress-strain relationships of the base and constraining layers can be written as follows.

   i Q Q 0 i x 11 12 x i Q Q 0 i i s, c 12 11 i 0 0 Q i x 66 x

   (8)

   where the stiffness coefficients Qmn of the isotropic material are as

   Q Ei , Q i Ei , Q Ei 11 1 2 12 1 2 66 2(1 ) i i i

   (9)

   where Ei and i are the Youngs modulus and Poissons ratios of the base and constraining layers. The strain-displacement relationships of the base and constraining layers can be written as:

   w , xi x w i s, c i Ri

   (2)

   i ui w 2 x = x z x2 i sin u 1 vi cos w R i R R i i i sin w z 2w z R x R2 2 i i i 1 ui vi sin v x R x R i i i sin w 2z 2w 2z R x R2 i i

   (10)

   where R =R +xsin.

   i i0

   By using the displacement continuity between layers, the displacements of the viscoelastic layer can be expressed as:

   u 1 u u hs hc , v 2 s c 2 xs 2 xc 1 u u hs hc , xv h c s 2 xs 2 xc v v 1 v v hs hc , v 2 s c 2 s 2 c 1 v v hs hc v h c s 2 s 2 c v

   (3)

   Generally, the Youngs modulus of the viscoelastic layer is much smaller than that of the base and constraining layers. Therefore, it can be assumed that the viscoelastic material layer undergoes only shear strains while the other two layers

   The strain energies of the sandwich shell can be described as follows.

   Re 2

   U Us Uv Uc 1 2 L hs 2 s s s s s s R dzdxd 2 x x x x s 0 0 hs 2 1 2 L hv 2 v v v v R dzdxd 2 xz xz z z v 0 0 hv 2 1 2 L hc 2 c c c c c c R dzdxd 2 x x x x c 0 0 hc 2

   (11)

   where Us is the nodal displacement vector, and K and M are the stiffness and mass matrices. From the above Eq.(17), the complex eigenvalues are obtained. The real part of the natural frequency of the sandwich shell and the modal loss factor are defined as follows.

   (18)

   (19)

   f

   2

   2

   The kinetic energies of the sandwich shell can be given

   as

3. Numerical Results

   Im

   Re 2

   h 2 L T T T T s s u2 v2 w 2 R dxd s v c 2 s s s 0 0 h 2 L h 2 L v v w 2R dxd c c u2 v2 w 2 R dxd 2 v 2 c c c 0 0 0 0

   (12)

   where i (i=s,v,c) denotes the density of each layer.

   The elastic energies stored in distributed springs of base and constraining layers can be given as

   1 2 U k u2 k v2 k w2 K w x R d BC 2 u 0 s v0 s w0 w0 x 0 s 0 1 2 k u2 k v2 k w2 K w x R d 2 u1 s v1 s w1 w1 x L s 0 1 2 k u2 k v2 k w2 K w x R d 2 u 0 c v0 c w0 w0 x 0 c 0 1 2 k u2 k v2 k w2 K w x R d 2 u1 c v1 c w1 w1 x L c 0

   (13)

   The total Lagrangian energy function of the sandwich conical shell can be written as follows.

   L T U U BC

   (14)

   By introducing the Legendre polynomial, the displacement components can be expanded as follows.

   N M u (x) cos(n )U s sin(n )U s eit s m mn mn m 1 n 0 N M v (x) sin(n )V s cos(n )V s eit s m mn mn m 1 n 0 N M c m mn mn u (x) cos(n )U c sin(n )U c eit m 1 n 0 N M v (x) sin(n )V c cos(n )V c eit c m mn mn m 1 n 0 N M w (x) cos(n )W sin(n )W eit m mn mn m 1 n 0

   (15)

   where m(x) is the m-order Legendre polynomial for the

   displacement function; is an angular frequency, t denotes

   It is very important to determine the proper degree of polynomial that can simultaneously guarantee the accuracy of the solution and the computational efficiency.

   Because, increasing the degree of the polynomial will reduce the computational efficiency: the computation time for the solution process will be long, and the increase of the excessive polynomial degree will result in the case that the solution does not converge and diverges. For the determination of the proper degree of a polynomial, the natural frequency convergence characteristics with increasing polynomial degree are investigated and the results are shown in Fig. 2.

   Fig.2. Convergence of modal loss factor for a sandwich conical shell.

   As shown in Fig. 2, the present method can be ensured stable convergence for predicting the natural frequencies and modal loss factors of the sandwich conical shell. The boundary conditions are generalized by the introduction of an artificial spring technique, and the type of boundary conditions is selected according to the spring stiffness. The effects of spring stiffness values of elastic boundary on the natural frequencies and modal loss factors of sandwich conical and cylindrical shells are investigated in Fig. 3.

   time. The symbols

   U s , V s

   , U c , V c , W , U s

   , V s ,

   mn mn

   mn mn mn

   mn mn

   c c

   U

   V

   ,

   mn mn

   and

   Wmn unknown coefficients of the Legendre

   polynomials that you want to obtain;

   Minimizing the above Lagrangian energy function with respect to the unknown nodal displacement components,

   L 0, q U s ,V s ,U c ,V c ,W ,U s ,V s ,U c ,V c ,W q mn mn mn mn mn mn mn mn mn mn

   (16)

   K 2 M U 0 s

   (17)

   Substituting Eq. (15) into Eq.(14), following governing equations are obtained.

   Fig.3. Convergence of natural frequency for a sandwich shells with different boundary spring stiffness;

   1. conical, (b) cylindrical.

      Based on the above study, the spring stiffness values for different boundary conditions considered in this study are set as shown in Table 1, in which the symbols F, C, S and SD mean free, clamped, simply supported and shear diaphragm boundary conditions, respectively.

      Table 1. Stiffness values for different boundary springs.

      B.C.	Boundary spring stiffness value
      	ku,	kv	kw	Kw
      F	0	0	0	0
      C	1014	1014	1014	1014
      <>S	1014	1014	1014	0
      SD	0	1014	1014	0

      Next, the free vibration results of a sandwich cylindrical shell are compared with those of literature to verify the accuracy of the proposed method. In Table 2, the natural frequencies for a sandwich cylindrical shell with various boundary conditions are compared with those of literature.

      Table. 2. Comparison of natural frequencies for a sandwich cylindrical shell with various boundary conditions.

      B.Cs	n	Ref.[9]	Present	Diff,%
      C-C	1	873.13	873.128	0.0002
      	2	821.85	821.847	0.0004
      	3	764.95	764.945	0.0007
      C-S	1	820.19	820.185	0.0006
      	2	765.25	765.244	0.0008
      	3	702.95	702.942	0.0011
      C-SD	1	799.83	799.824	0.0008
      	2	757.92	757.918	0.0003
      	3	702.45	702.442	0.0011
      C-F	1	669.29	669.283	0.0010
      	2	519.66	519.647	0.0025
      	3	406.55	406.536	0.0034

      As observed from Table 2, the frequency results obtained by the proposed method agree well with those of the literature.

      Based on the verification study of the proposed method, the effect of some parameters on the natural frequency and the modal loss factor of the sandwich cylindrical and conical shells are investigated. First, the effect of semi-vertex angle of a sandwich conical shell on the frequency parameters is considered in Table 3.

      Table. 3. Frequency parameters for a sandwich conical shell with various semi-vertex angles (m=1).

      	n	B.C.
      		C-C	C-S	C-SD	S-S
      /6	1	12.915	12.549	11.745	12.544
      	2	10.968	10.685	10.595	10.608
      	3	9.205	8.905	8.869	8.749
      	4	7.958	7.620	7.410	7.403
      /4	1	10.043	9.601	8.804	9.596
      	2	8.781	8.418	8.234	8.351
      	3	7.572	7.207	7.206	7.062
      	4	6.708	6.312	6.215	6.105
      /3	1	7.107	6.634	6.024	6.606
      	2	6.384	5.940	5.756	5.847
      	3	5.703	5.248	5.244	5.081
      	4	5.265	4.778	4.745	4.559

      The variations of the dimensionless frequencies and the modal loss factors of a sandwich cylindrical shell with various thickness ratio hv/h are shown in Fig. 4.

      Fig.4. Variation of frequency parameters and modal loss factor for a sandwich cylindrical shell with various thickness ratios; (a) C-C, (b) C-SD.

      It is clear that the frequency parameters decrease as the thickness of viscoelastic layer increases. But modal loss factors have no such change law.

4. CONCLUSION

In this paper, a semi-analytical method is proposed to analyze the vibration and damping characteristics of sandwich cylindrical and conical shells with viscoelastic material core. The theoretical formulations of the structural model are established by using the energy principle in framework of FSDT and Donnels shell theory. The energy of the sandwich shell is composed of that of base layer, viscoelastic core and constraining layer. The displacement components at any point of the sandwich shell are expanded by the Legendre polynomials in the meridional direction and Fourier series in the circumferential direction. Numerical examples for free vibration and damping analyses of sandwich cylindrical and conical shells with viscoelastic core are presented to verify the reliability and accuracy of the presented method. First, the free vibration analysis results of the sandwich shells obtained by the proposed method are

compared with those of published literature. Finally, the effects of several parameters such as geometric dimension, material properties and boundary condition on the frequency parameter and modal loss factor of the shell are investigated.

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