Vibration analysis and design optimization of viscoelastic sandwich cylindrical shell

Journal of Sound and Vibration 331 (2012) 2729–2752

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Vibration analysis and design optimization of viscoelastic sandwich cylindrical shell Farough Mohammadi, Ramin Sedaghati n Department of Mechanical and Industrial Engineering, Concordia University, 1455 de Maisonneuve Boulevard West, Montreal, QC, Canada H3G 1M8

a r t i c l e i n f o

abstract

Article history: Received 15 July 2011 Received in revised form 3 February 2012 Accepted 3 February 2012 Handling Editor: H. Ouyang Available online 28 February 2012

Damping properties of viscoelastic sandwich structure can be improved by changing some parameters such as thickness of the layers, distribution of partial treatments, slippage between layers at the interfaces, cutting and its distribution at the top and core layers. Since the optimization problem may result in a thick core layer, for achieving more accuracy a new higher-order Taylor’s expansion of transverse and in-plane displacement fields is developed for the core layer of sandwich cylindrical shell in which the displacement fields at the core layer are compatibly described in terms of the displacement fields at the elastic faces. The presented model includes fewer parameters than the previously developed models and therefore decreases the number of degree of freedom in the finite element modeling. The transverse normal stress in the core layer is also considered. The formulations are developed to consider the slippage between layers at the interfaces. Finally, by combining the finite element method and the optimization algorithms based on the genetic algorithm and sequential quadratic programming technique, a design optimization methodology has been formulated to maximize the damping characteristics using the optimal number and location of cuts and partial treatments with optimal thicknesses of top and core layers. & 2012 Elsevier Ltd. All rights reserved.

1. Introduction Viscoelastic materials are frequently used in sandwich structures such as beam, plate and shell where damping is desired for a wide range of frequencies. The damping layer in sandwich structures can be used in two different configurations including constrained and unconstrained damping layer which have been studied by many researchers. Ross et al. [1] developed the analysis of constrained and unconstrained sandwich structures. They defined loss factor in terms of strain energy which can be evaluated using the ratio of the dissipated energy caused by viscoelastic layer to the maximum potential energy in harmonic motion. This definition was examined by Ungar and Kerwin [2]. They showed that the definition is only meaningful for massless structure and demonstrated that in order to consider the mass, the definition should be applied only at the natural frequencies. Three-layered sandwich beam was analyzed by Mead and Markus [3]. They derived a sixth-order differential equation of motion in terms of the deflection for the structure with different boundary conditions. Rao [4,5] analyzed the effects of rotational inertia and shear deformation for short sandwich beams. Higher-order expansion in displacement fields of the core layer considering slippage at the interfaces in sandwich beam structure was assumed by Bai and Sun [6]. They implemented the physical Lagrange multiplier in order to consider the higher-order displacement fields in the core of the beam structure. It was also shown that the slippage increases damping

n

Corresponding author. E-mail addresses: [email protected] (F. Mohammadi), [email protected], [email protected] (R. Sedaghati).

0022-460X/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.jsv.2012.02.004

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Nomenclature

Greek letters

a radius of the cylinder E Young’s modulus F the equivalent nodal force G shear modulus hi, i¼t,b, c thickness of the top (i¼t), bottom (i ¼b) and core (i¼c) layers Kn frequency dependent complex shear stiffness parameter of the adhesive layer K stiffness matrix k(o) real part of the frequency dependent complex shear stiffness parameter of the adhesive layer L length of the cylinder M mass matrix Nj, j¼1,2,3 Lagrangian shape functions q vector of nodal displacements s axial coordinate t,b,c indices for representing top, bottom and core, respectively u,v,w translational displacement field in axial, circumferential and thickness directions, through the thickness off each layer, respectively u0,v0,w0 translational displacement field of the middle plane in axial, circumferential and thickness directions, respectively

gxs,gxs,gxy

in-plane shear strain, transverse shear strain in axial direction, transverse shear strain in hoop direction, respectively ess,eyy,exx normal strains in axial, hoop and transverse directions, respectively Z(o) frequency dependent loss factor Zeff effective loss factor y circumferential coordinate u Poisson’s ratio x1,x2,x3 thickness coordinates located at the middle plane of the top, bottom and core layers, respectively r density sss,syy,sxx normal stresses in axial, hoop and transverse directions, respectively txs,txs,txy in-plane shear stress, transverse shear stress in axial direction, transverse shear stress in hoop direction, respectively u eigenvector Fc1 , Fc2 , Fc3 , Yc1 , Yc2 , Yc3 , Cc1 , Cc2 higher-order terms of displacement fields in Taylor’s series expansions in the core layer c1,c2 rotations of the normals to the middle plane in axial and circumferential directions, respectively o frequency

characteristics of the beam structure. The higher-order model of the displacement fields in the core layer of sandwich beam structure was also examined by Babert et al. [7]. Using the Green function, they presented an approximate solution by assuming perfect bonding at the interfaces to find the displacement fields at the core layer. By comparing the damping behavior resulted from their analysis to the experimental data, they showed that assuming the higher expansion of displacement field for sandwich beam structure provides more accurate model. Using the linear expansion of displacement field in the core layer, Sainsbury and Masti [8] developed a combination of two elements named as FEM_1 and FEM_2 in the finite element formulation of partial treatment in sandwich cylindrical shell in a way that compatibly bottom and top/ core layers were, respectively, presented by two finite element formulations namely FEM_1 (rectangular curved shape with four corner nodes) and FEM_2 (rectangular curved shape with four corner nodes and four mid-side nodes). Ramesh and Ganesan [9] compared the results from different theories including Wilkins Theory (WT), Khatua’s Theory (KT) and Discrete Layer Theory (DLT) in semi-analytical finite element modeling of the sandwich shell structure. Arau´jo et al [10] used the higher-order shear deformation theory to represent the displacement field at the viscoelastic core layer in vibration analysis of sandwich composite plate structure. Parametric studies and optimization to investigate the effect of the main parameters such as dimensions of the layers, material properties and partial treatment on the damping characteristics have been investigated by many researchers. Ramesh and Ganesan [11,12] presented the effect of geometry and material properties on loss factor using semi-analytical finite element modeling based on the first-order shear deformation theory (FSDT) for shells of revolution. Plunkett and Lee [13] optimized damping characteristics of a beam structure using partial treatment by cutting the constraining layer into optimal lengths. Mantena et al. [14], maximized loss factor in constrained sandwich beam structure by optimizing side length of the treatments. Optimal size and dimension of treatments in sandwich layered plate structure was also investigated by Huang et al. [15]. Trompette and Fatemi [16] obtained the best position of one cut considering lack of continuity in longitudinal displacement in cantilever beam to maximize loss factor. Lepik [17] optimized thickness distribution of axisymmetric viscoelastic cylindrical shell layer in order to minimize the deflection under impulsive loading. Zhenga et al. [18] minimized vibration response of cylindrical shell using the optimal size and location of treatments. Ajmi and Bourisli [19] maximized loss factor by considering the optimized number of treatments in partial treatment of beam structure and appropriate thickness ratio of top and core layers. Lepoittevin and Kress [20] optimized the distribution of cuts in constraining and constrained layers of sandwich beam structures. They showed that by embedding the cuts, transverse shear strain at core layer increases and consequently the damping increases.

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The optimization problem may result in thick core layer and consequently the lower-order expansion of the displacement fields in thickness coordinate of the core layer in sandwich cylindrical shell is not accurate enough. The main limitation in the previously developed higher-order displacement fields in the core layer such as the higher-order model considered by Arau´jo et al. [10] is lack of the compatibility between the displacement fields at the different layers in which implementing the boundary conditions does not compatibly give the displacement fields at the core layer in terms of the displacement fields at the top and bottom layers. In this study, based on the profile of displacement fields of the core layer in static deformation, a new higher-order expansion of transverse and in-plane displacement fields in the thickness direction of the core layer are developed to analyze the vibration damping of the sandwich cylindrical shell with thin or thick core layer. Using this way, not only a good approximation for the shape of the displacement profile in the thickness coordinate can be obtained, but also it is possible to drive the higher-order expansion in which the displacement field of the core layer can be represented by those of the elastic faces (i.e. the least number of variable). As explained further, the partial treatment and cut can be then modeled using the developed higher-order expansion. The results from the higherorder expansion are also compared to the lower-order expansion of the displacement field. By combining the finite element model with the optimization algorithms based on the genetic algorithm (GA) and sequential quadratic programming (SQP) technique, the optimal number, distribution and thicknesses of top and core layer of the partial treatments are simultaneously optimized to improve damping of the structure. The results show that damping can significantly improve while the total mass of the structure decreases. In order to improve damping, cuts can be embedded in the sandwich structure to increase transverse shear strains in the core layer at the cut locations. Considering this, cutting in this study has been modeled using the discontinuity at the nodal displacements in the elements at the top and core layers. Here, distribution and the total number of cuts are simultaneously optimized to improve the damping behavior in the sandwich cylindrical shell considering different boundary conditions. The objective function in this study is considered to be an effective loss factor defined based on combinations of loss factor in different circumferential and axial directions and also their contributions in the forced vibration.

2. Lower-order expansion of displacement fields through the thickness of viscoelastic core layer First-order shear deformation theory (FSDT) can be used to define the displacement distribution through the thickness of elastic and viscoelastic layers. As shown in Fig. 1, displacement fields for a three-layered sandwich shell of revolution are defined in terms of the displacements of the middle plane including u0, v0 and w0 and the rotations of the normals to the middle plane in axial and circumferential directions denoted by c1 and c2, respectively. Considering lower-order expansion of displacement fields, the total displacement fields for each layer can be written as t

ut ðs, y, x1 ,tÞ ¼ uto ðs, y,tÞ þ x1 c1 ðs, y,tÞ t

vt ðs, y, x1 ,tÞ ¼ vto ðs, y,tÞ þ x1 c2 ðs, y,tÞ ,



ht ht o x1 o 2 2

(1)



hb h o x2 o b 2 2

(2)

wt ¼ wt ðs, y,tÞ b

ub ðs, y, x2 ,tÞ ¼ ubo ðs, y,tÞ þ x2 c1 ðs, y,tÞ b

vb ðs, y, x2 ,tÞ ¼ vbo ðs, y,tÞ þ x2 c2 ðs, y,tÞ , wb ¼ wb ðs, y,tÞ

Fig. 1. Sandwich cylindrical shell; displacements and rotations at each layer.

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uc ðs, y, x3 ,tÞ ¼ uco ðs, y,tÞ þ x3 c1 ðs, y,tÞ c

vc ðs, y, x3 ,tÞ ¼ vco ðs, y,tÞ þ x3 c2 ðs, y,tÞ ,



wb ¼ wb ðs, y,tÞ

hc hc o x3 o 2 2

(3)

Two different methods can be used to determine the displacement fields through the thickness of viscoelastic sandwich structures. In the first method, named method_1, the translational displacements at the top and bottom elastic layers are determined using the displacement fields at the core layer by implementing the boundary conditions at the interfaces. As shown in this study, this method facilitates determination of the displacement fields considering higher-order expansion of displacements through the thick core layer but fails to model the cuts and partial treatment in the sandwich structure. In the second method, named method_2, the displacement fields at the core layer are obtained in terms of the displacements at the bottom and top elastic layers. In spite of the fact that this method brings intricacy into the formulations particularly where the higher-order expansion of the core displacement fields is considered, it can effectively model the cuts and partial treatments in the sandwich structure. Both methods should give identical results for fully treated sandwich cylindrical shell. In this study slippage between layers at the interface has been considered in the formulations. As mentioned before, Bai and Sun [6] assumed an adhesive layer at the interface of the layers in sandwich beam structure and showed that slippage can increase loss factor in a certain range of viscoelastic property related to adhesive layer. The effect of slippage between fiber and matrix in composite structures on damping properties was investigated by McLean and Read [21] and also Nelson and Hancock [22]. They experimentally demonstrated that slippage reduces damping properties and therefore it should be avoided. Based on the methodology presented by Bai and Sun [6], for demonstrating the effect of slippage, we can assume that a very thin adhesive layer glues the top and bottom layers to the core layer. Considering this, at the interfaces the in-plane displacements are discontinuous. Using the displacement variation at the interfaces and the viscoelastic property of the adhesive layers, transverse shear stresses at interfaces are obtained. Therefore the boundary conditions at the interfaces can be expressed as follows: ht hc , x3 ¼ , tx3 s ¼ K n ðut uc Þ, tx3 y ¼ K n ðvt vc Þ 2 2 h h x2 ¼ b , x3 ¼  c , tx3 s ¼ K n ðuc ub Þ, tx3 y ¼ K n ðvc vb Þ 2 2

x1 ¼ 

(4)

in which tx3 s and tx3 y are, respectively, transverse shear stresses in axial and circumferential directions and Kn ¼k(o)(1 þiZ(o)) is a complex shear stiffness parameter of the adhesive layer representing the bonding at the interfaces which is assumed to be frequency dependent. The adhesive layer is assumed to be completely in contact with the core and faces. If a large value is assigned to k(o) then

tx3 s Kn

0

and

tx3 y Kn

0

and consequently the in-plane displacements are equal at the interfaces which means that the perfect bonding is provided. Substituting displacements from Eqs. (1)–(3) into Eq. (4) and shear stress–strain relationship yields:     @w ht t hc c c ¼ K n ut0  c1 uc0  c1 Gc c1 þ @s 2 2     @w hc c h b c n c b ¼ K u0  c1 u0  b c1 Gc c1 þ @s 2 2     @w h h t c c c t n Gc c2 þ ¼ K vt0  c2 vc0  c2 a@y 2 2     @w h h c c c b Gc c2 þ ¼ K n vc0  c2 vb0  b c2 (5) a@y 2 2 According to the method_1, by obtaining the translational displacement field at the top and bottom layers from Eq. (5) and then substituting into Eqs. (1)–(3), the total displacements at the elastic faces are obtained as  c  ðh ct þ h cc Þ t ut ¼ uc0 þ t 1 2 c 1 þ x1 c1 þ KGnc c1 þ @w ht ht @s (6) t c  c  ,  o x1 o t ðh c þ h c Þ 2 2 vt ¼ vc þ t 2 c 2 þ x c þ Gnc c þ @w 0

1

2

2

K

2

a@y

 c  b þ x2 c1  KGnc c1 þ @w @s b c  c @w  , b ðh c þ h c Þ vb ¼ vc0  b 2 2 c 2 þ x2 c2  KGnc c2 þ a@ y

ub ¼ uc0 

b

c

ðhb c1 þ hc c1 Þ 2



hb h o x2 o b 2 2

(7)

According to the method_2, the in-plane displacements (uc and vc) are related to the displacement fields at the top and c bottom layers. To implement this, the first two equations in Eq. (5) are simultaneously solved for uc0 and c1 , and the last c two equations in Eq. (5) are simultaneously solved for vc0 and c2 . These obtained parameters are subsequently substituted

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into Eq. (3) to derive uc ¼

    ub0 þ ut0 ðhb cb1 ht ct1 Þ Kn Kn 2Gc x3 @w b t þ x3 þ ðut0 ub0 Þ ðhb c1 þ ht c1 Þ  n n n 4 2 2Gc þK hc 4Gc þ 2K hc 2Gc þ K hc @s

(8)

vc ¼

    vb0 þvt0 ðhb cb2 ht ct2 Þ Kn Kn 2Gc x3 @w b t t b þ x3 þ ðv v Þ ðh c þ h c Þ  t b 2 2 4 2 2Gc þ K n hc 0 0 4Gc þ2K n hc 2Gc þK n hc a@y

(9)

where  hc/2o x3 ohc/2. For the lower-order model, deflection w is considered to be constant through thickness of the sandwich cylinder. 3. Higher-order expansion of displacement fields through the thickness of viscoelastic core layer The lower-order model can satisfy the compatibility in the displacement fields through the thickness of the sandwich structure in which according to the method_2 the in-plane displacements and the rotations are obtained in terms of the displacements at the top and bottom layers as shown in Section 2. However, in sandwich structures the lower-order expansion of displacement distribution through the thickness of the compliant core layer is not accurate enough particularly where the core layer is thick. In this study the following distribution of displacements is considered at the core layer of the sandwich cylindrical shell: 2

3

uc ¼ uc0 þ x3 Fc1 þ x3 Fc2 þ x3 Fc3 2

3

vc ¼ vc0 þ x3 Yc1 þ x3 Yc2 þ x3 Yc3 ,



2 wc ¼ wc0 þ x3 Cc1 þ x3 Cc2

hc hc o x3 o 2 2

(10)

In order to achieve the compatibility in the deformation of the layers in the sandwich structure, all displacement fields in the core layer including rotations and translations should be identified in terms of the displacements at the top and bottom layers. Since the boundary conditions at the two interfaces including the continuity in transverse deflection and slippage (or perfect bonding) give six equations for in-plane and transverse displacements, the following developed approach is considered to reduce the eleven parameters of the displacement fields in Eq. (10) to six parameters. The total displacement fields in the core layer for harmonic motion can be expressed as follows: 2 3 uc ðs, y, x3 Þ 6 v ðs, y, x Þ 7 qc ¼ 4 c (11) 3 5expðiotÞ wc ðs, y, x3 Þ The profile of displacements with respect to the thickness coordinate of the core layer can be defined based on the static deformation. However, the displacement functions are afterward adjusted to fulfill the dynamic motion of the core layer. Considering this, by neglecting the inertia forces the equilibrium equations are written as @sss @tsy @tx3 s þ þ ¼0 @s a@y @x3 @tsy @syy @tx3 y þ þ ¼0 @s a@y @x3 @tx3 s @tx3 y @sx3 x3 þ þ ¼0 @s a@y @x3

(12)

By neglecting the in-plane normal and in-plane shear stresses, the first and second equations in Eq. (12) can be rewritten as @tx3 s ¼ 0 ) tx3 s ¼ f ðs, yÞ, @x3

@tx3 y ¼ 0 ) tx3 y ¼ gðs, yÞ @x3

(13)

Eq. (13) indicates that the transverse shear stresses are constant through the viscoelastic core layer. The relation between the transverse shear stresses and strains can be written as

gx3 s ¼

tx3 s Gc

gx3 y ¼

¼

tx3 y Gc

2ð1 þ uc Þ tx3 s ¼ 2ð1 þuc Þb1 Ec

¼

2ð1 þuc Þ tx3 y ¼ 2ð1 þ uc Þb2 Ec

where

b1 ¼

tx3 s Ec

and

b2 ¼

tx3 y Ec

(14)

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Shear strains can be also obtained using the displacement fields in Eq. (10) as follows:

gx3 s ¼

c @wc0 @uc @wc @Cc1 2 2 @C2 ¼ Fc1 þ 2x3 Fc2 þ 3x3 Fc3 þ þ x3 þ x3 þ @x3 @s @s @s @s

gx3 y ¼ 0

c @wc0 @v @w @Cc1 2 @C2 2 þ ¼ Yc1 þ 2x3 Yc2 þ3x3 Yc3 þ þ x3 þ x3 @x3 a@y a@y a@y a@y

1

(15)

2

Equating the coefficient of x3, x3 and x3 from Eqs. (14) and (15) yields the following relations: @wc0 @Cc1 @Cc2 ¼ 2ð1 þ uc Þb1 , 2Fc2 þ ¼ 0, 3Fc3 þ ¼0 @s @s @s c c c @w @C1 @C2 Yc1 þ 0 ¼ 2ð1 þ uc Þb2 , 2Yc2 þ ¼ 0, 3Yc3 þ ¼0 a@y a@y a@y

Fc1 þ

(16)

Now, substituting Fc1 , Fc2 , Fc3 , Yc1 , Yc2 and Yc3 from Eq. (16) into the first and second equations in Eq. (10) yields the in-plane displacement fields through thickness of the core as   @wc0 x2 @Cc1 x33 @Cc2  3  uc ¼ uc0 þ x3 2ð1 þuc Þb1  @s 2 @s 3 @s  2 c c @w x @ C x3 @Cc2 1 0 vc ¼ vc0 þ x3 2ð1 þ uc Þb2   3 (17)  3 a@y 2 a@y 3 a@y By neglecting the in-plane normal stresses, the constitutive equations are written as u ess ¼ c ðeyy þ ex3 x3 Þ 1uc u eyy ¼  c ðex3 x3 þ ess Þ 1uc   EC ð1uc Þex3 x3 þ uc ðeyy þ ess Þ sx3 x3 ¼ ð1 þuc Þð12uc Þ

(18)

Substituting tx3 s , tx3 y from Eq. (14) and sx3 x3 from Eq. (18) into the last equation from Eq. (12) gives the following relation between b1 and b2: @=@x3 ½ð1uc Þex3 x3 þuc ðeyy þ ess Þ @b1 @b2 þ ¼ ð1 þ uc Þð12uc Þ @s a@y

(19)

which can be simplified by using the first and second equations from Eq. (18) and considering ex3 x3 ¼ ð@wc =@x3 Þ, as follows: @b1 @b2 @2 wc þ ¼ 2 @s a@y @x3

(20)

Now, by substituting wc from Eq. (10) into Eq. (20), Cc2 is obtained as

Cc2 ¼ 

ðð@b1 =@sÞ þ ð@b2 =a@yÞÞ 2

(21)

Finally, substituting Eq. (21) into Eq. (10) yields the transverse displacement through the thickness of the core layer as follows:   x2 @b1 @b2 þ wc ¼ w0c þ x3 Cc1  3 (22) 2 @s a@y Now, the total displacement fields in the core layer are obtained in terms of the three translational displacements uc0 , vc0 , wc0 and three rotational displacements Cc1 , b1, b2. According to the method_1, the boundary conditions from Eq. (5) and also the continuity in transverse deflection can be implemented to obtain translational displacements at the top and bottom layers in terms of displacement fields in the core layer. Therefore, the displacement fields through the thickness of top and bottom layers are derived as follows: !   2 3 @wc0 hc hc @Cc1 hc @2 b1 @ 2 b2 ht t Gc t c ut ¼ u0 þ 2ð1 þ uc Þb1   þ þ þ c1 þ x1 c1 þ 2ð1 þ uc Þb1 n 2 @s 8 @s 48 @s2 a@y@s 2 K !   2 3 @wc0 hc hc @Cc1 hc @2 b1 @ 2 b2 ht t Gc t c þ c2 þ x1 c2 þ2ð1 þ uc Þb2 n 2ð1 þuc Þb2  þ vt ¼ v0 þ þ  2 a@y 8 a@y 48 a@y@s a2 @y2 2 K  2 hc h @b1 @b2 þ wt ¼ wc0 þ Cc1  c (23) 2 8 @s a@y !   2 3 @wc0 hc @Cc1 hc @2 b1 @ 2 b2 h b Gc b 2ð1 þ uc Þb1    þ  b c1 þ x2 c1 2ð1 þuc Þb1 n 2 @s 8 @s 48 @s2 a@y@s 2 K

hc ub ¼ uc0 

F. Mohammadi, R. Sedaghati / Journal of Sound and Vibration 331 (2012) 2729–2752

vb ¼ vc0 

hc 2

!   2 3 @wc0 h @Cc1 hc @2 b1 @ 2 b2 h b Gc b 2ð1 þuc Þb2  þ   c  b c2 þ x2 c2 2ð1þ uc Þb2 n a@y 8 a@y 48 a@y@s a2 @y2 2 K  2 hc h @b1 @b2 þ wb ¼ wc0  Cc1  c 2 8 @s a@y

2735

(24)

According to the method_2, the displacements distribution through the thickness of the core layer should be obtained in terms of the displacement fields at the top and bottom layers. According to the continuity of transverse displacement either considering slippage or perfect bonding, the transverse displacements at the top and bottom of the core layer are, respectively, equal to those at the elastic top and bottom layers as 2

wt ¼ wc0 þ

hc c hc c C þ C , 2 1 4 2

2

wb ¼ wc0 

hc c hc c C þ C 2 1 4 2

Using Eq. (21) and solving the abovementioned equations for Cc1 and wc0 yields: w wb Cc1 ¼ t hc 2

wc0 ¼

wt þwb hc þ hc 8



 @b1 @b2 þ @s a @y

(25)

(26)

Considering the slippage condition at the interfaces, the in-plane translational displacements uc0 and vc0 at the core layer are directly obtained in terms of the displacement fields at the top and bottom layers. However, the rotations b1 and b2 are related to the displacement fields in the elastic layers by two coupled partial differential equations. Substituting Eqs. (1), (2) and (17) for in-plane displacement into Eq. (4), yields the following relations: !!   2 3 @wc0 ht t hc h @Cc1 hc @Cc2 2ð1 þ uC Þb1   c  ¼ Gnc 2ð1 þ uc Þb1 K n uto  c1  u0c þ 2 2 @s 8 @s 24 @s !!   2 3 @wc0 ht t hc h @Cc1 hc @Cc2 2ð1þ uC Þb2  K n vto  c2  v0c þ   c ¼ Gnc 2ð1 þuc Þb2 2 2 a@y 8 a@y 24 a@y !   2 3 @wc0 hc h @Cc1 hc @Cc2 h b 2ð1 þ uC Þb1   c þ ubo  b c1 ¼ Gnc 2ð1 þ uc Þb1 K n u0c  2 @s 8 @s 24 @s 2 !   2 3 @wc0 hc @Cc1 hc @Cc2 hb b n 0 hc b 2ð1 þ uC Þb2   K vc  þ vo  c2 ¼ Gnc 2ð1 þ uc Þb2 (27) 2 a@y 8 a@y 24 a@y 2 After substituting Cc2 from Eq. (21) into the abovementioned relations of Eq. (27) and solving for uc0 , vc0 , b1 and b2, one can obtain the following equations: uc0 ¼

ub0 þut0 hb cb1 ht ct1 h2c @Cc1 þ þ 2 4 8 @s

vb0 þvt0 hb cb2 ht ct2 h2c @Cc1 þ þ 2 4 8 a @y ! ! 3 b t K n hc @ðwt þ wb Þ K n hc @2 b1 @2 b2 hb c1 þ ht c1 n n b t þ ð2K ð1 þvc Þhc þ2Ec Þb1 þ þ þK u0 u0  ¼0 @s 2 12 a @y @s 2 @s2 vc0 ¼

3

ð2K n ð1 þ vc Þhc þ 2Ec Þb2 þ

K n hc @ðwt þ wb Þ K n hc þ a @y 2 12

! ! b t @ 2 b1 @ 2 b2 hb c2 þht c2 n t b þ v v  þ K ¼0 0 0 a @y @s a2 @y2 2

(28)

(29)

(30)

(31)

Now, the only unknown parameters are the rotations b1 and b2 which can be obtained by solving Eqs. (30) and (31). The solution is given in the next section. 4. Semi-analytical finite element method Lagrange equation is used to establish the equations of motion. The kinetic and potential energies are obtained in each layer using the following equations: Z h=2 Z 2p Z Le =2 1 ri ðu_ 2i þ v_ 2i þ w_ 2i ÞdV i (32) Ti ¼ ðh=2Þ 0 ðLe =2Þ 2 Ui ¼

Z

h=2

ðh=2Þ

Z 2p Z 0

ðLe =2Þ ðLe =2Þ

1 i i ðs e þ siyy eiyy þ sixi xi eixi xi þ tisy gisy þ tixi s gixi s þ tixi y gixi y ÞdV i 2 ss ss

(33)

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where i¼t, b and c, dVi ¼(aþ xi) dy ds dxi and xi is the coordinate in thickness direction at the middle of each layer and (xt,xb,xc)¼ (x1,x2,x3). In Eq. (33) the energy terms related to the in-plane and normal stresses of the core layer, i.e. tcsy , scyy and scss are neglected due to the much small modulus in the core layer compared to the elastic faces as the normal and in-plane shear stresses are mainly carried by the elastic faces. This approximation is also reported in many literatures such as Refs. [6,7]. Strain–displacement relation using linear Green’s Strain for the cylindrical shell element can be expressed as follows [23]:       @u 1 @vi w @wc @vi 1 @ui þ eiss ¼ i eiyy ¼ þ i ecx3 x3 ¼ , gisy ¼ 1 þ ðxi =aÞ a@y 1 þ ðxi =aÞ a@y @s a @xi @s   @vi 1 @wi vi @ui @wi i i , gxi s ¼ (34) gxi y ¼ þ  þ @xi 1 þðxi =aÞ a@y a @xi @s The ratio xi/a is negligible if the thickness of each layer is small compared to the radius. Using the constitutive equations and substituting displacement fields for each layer into Eqs. (32) and (33), the kinetic and potential energies are obtained in terms of the displacements. Semi-analytical finite element modeling can be used to define the translational and rotational displacement fields at the top, bottom and core layers in which the displacements in the circumferential direction are expressed analytically using the Fourier series and also are discretized in the axial direction using the Lagrangian shape functions [11,24,25]. Considering this, the possible displacement fields at the top, bottom and core layers for the lower- or higher-order model of displacement distribution based on the method_1 or method_2 can be defined as wi0 ¼

m X 1 X

N j wij cosðnyjÞ,

j¼1n¼0

ui0 ¼

m X 1 X

m X 1 X

vi0 ¼

m X 1 X

Cc1 ¼

N j uij cosðnyjÞ,

j¼1n¼0

b1 ¼

N j Cc1j cosðnyjÞ

j¼1n¼0

m X 1 X

Nj bj1 cosðnyjÞ,

b2 ¼

j¼1n¼0

ci1 ¼

N j vij sinðnyjÞ

j¼1n¼0

m X 1 X

m X 1 X

Nj bj2 sinðnyjÞ

i¼1n¼0 i

ci2 ¼

Nj cj1 cosðnyjÞ,

j¼1n¼0

m X 1 X

i

Nj cj2 sinðnyjÞ

(35)

j¼1n¼0

in which m is the number of nodes for each cylindrical shell element in axial direction and i¼b, t and c. Since there is no preference for the orientation of circumferential modes, an arbitrary phase angle j must be included [24]. Generally, j ¼ j0 and j ¼ j0 þ p/2 in which j0 is an arbitrary constant. For the free vibration j0 ¼0 and for the forced vibration j0 depends on the distribution of the forces. For example in the case of point loading, if the coordinates are located at the same point where the load is exerted, then j0 ¼0 [25]. Using the configurations of the mode shapes, the physical intuition of the displacement functions are well explained by Soedel [25]. If three nodes are considered in axial direction, the shape functions N1, N2 and N3 are given as N1 ¼

2s2 L2e



s , Le

N2 ¼ 1

4s2 L2e

,

N3 ¼

2s2 L2e

þ

s Le

(36)

As mentioned before, the rotations b1 and b2 are two known rotations in method_1 however, according to the method_2, they should be determined by solving partial differential equations given in Eqs. (30) and (31). Here, we briefly discuss the solution of theses equations for b1 and b2. The solution consists of homogeneous and particular parts. The general solution of the coupled differential Eqs. (30) and (31) can be obtained by solving the following differential equations which are obtained after eliminating the nonhomogeneous terms in Eqs. (30) and (31) as ! 3 K n hc @2 b1 @2 b2 ð2K n ð1þ vc Þhc þ 2Ec Þb1 þ þ ¼0 (37) 12 a@y@s @s2 3

ð2K n ð1 þvc Þhc þ 2Ec Þb2 þ

K n hc 12

@ 2 b1 @ 2 b2 þ a@y@s a2 @y2

! ¼0

(38)

Eqs. (37) and (38) can be rewritten in the following form: ðC 1 þC 2 Dss Þb1 þ C 2 Dsy b2 ¼ 0

(39)

ðC 1 þ C 2 Dyy Þb2 þC 2 Dsy b1 ¼ 0

(40)

where 3

C 1 ¼ ð2K n ð1 þvc Þhc þ2Ec Þ,

C2 ¼

K n hc , 12

Dss ¼

@2 , @s2

Dyy ¼

@2 2

a2 @y

,

Dsy ¼

@2 a@y@s

F. Mohammadi, R. Sedaghati / Journal of Sound and Vibration 331 (2012) 2729–2752

2737

Multiplying Eq. (39) by (  C1 þC2Dyy) and Eq. (40) by (C2DSy) and subtracting from each other give ðC 21 C 1 C 2 ðDss þ Dyy Þ þ C 22 Dss Dyy C 22 Dsy Dsy Þb1 ¼ 0

(41)

According to Clairaut’s theorem, for a function which has continuous second derivative with respect to y and s: @2 @2 @2 @2 ¼ a@y@s a@y@s @s2 a2 @y2

or

Dss Dyy ¼ Dsy Dsy

After simplifying Eq. (41), using Clairaut’s theorem and then dividing Eq. (41) by C1C2, one can obtain the following partial differential equation: ! @2 b @2 b kb1 þ a2 21 þ 21 ¼ 0, 0 o s o s0 , 0 o y o y0 (42) @s @y where

"

k ¼ 24a

2

Similarly, one can obtain the following equation: kb2 þ a2

@2 b2 @2 b2 þ 2 @s2 @y

1þ uc 2

hc

þ

#

Ec K n hc

3

! ¼ 0,

0 o s o s0 ,

0 o y o y0

(43)

It can be realized that for a moderate thick core layer, k is a large non-dimensional value. The solution is now presented for b1 and the similar results can be concluded for b2. The boundary conditions may be expressed as follows:

b1 ðs, y0 Þ ¼ EðsÞ, b1 ðs,0Þ ¼ HðsÞ, b1 ðs0 , yÞ ¼ GðyÞ, b1 ð0, yÞ ¼ FðyÞ

(44)

According to the superposition principle, the homogeneous solution of b1 is expressed as (45)

b1 ¼ b11 þ b12 þ b13 þ b14 in which the boundary conditions are rewritten as follows:

b11 ðs, y0 Þ ¼ EðsÞ, b11 ðs,0Þ ¼ 0, b11 ðs0 , yÞ ¼ 0, b11 ð0, yÞ ¼ 0,

b12 ðs, y0 Þ ¼ 0, b12 ðs,0Þ ¼ HðsÞ b12 ðs0 , yÞ ¼ 0, b12 ð0, yÞ ¼ 0

b13 ðs, y0 Þ ¼ 0, b13 ðs,0Þ ¼ 0, b13 ðs0 , yÞ ¼ GðyÞ, b13 ð0, yÞ ¼ 0,

b14 ðs, y0 Þ ¼ 0, b14 ðs,0Þ ¼ 0 b14 ðs0 , yÞ ¼ 0, b14 ð0, yÞ ¼ FðyÞ

The partial differential equation for b11 can be written as kb11 þ a2

@2 b11 @2 b11 þ 2 @s2 @y

(46)

! ¼ 0,

0 os o s0 ,

0 o y o y0

(47)

Using the separation of variables method, b11 is expressed as

b11 ¼ XðsÞYðyÞ where by separating the equation and implementing the boundary conditions X and Y are obtained as   np XðsÞ ¼ A1n sin s s0 0sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! n2 p2 a2 A n2 p2 a2 þA3 y exp  kþ y YðyÞ ¼ A2n exp@ k þ n s0 2 s20

(48)

(49)

(50)

Considering this fact that k is a large value, the second term in Eq. (50) is eliminated. Using Eqs. (49) and (50), b11 is now obtained as follows: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !   1 X n2 p2 a2 np b11 ¼ A3n exp kþ y s (51) sin s0 s20 n¼1 where A3n ¼A1nA2n. Using the boundary condition and the Fourier series, A3n is obtained as follows: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !Z    s0 2 n2 p2 a2 np A3n ¼ exp  k þ y EðsÞsin s ds 0 s0 s0 s20 0 Substituting A3n into Eq. (51) yields

b11 ¼

1 X n¼1

"

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !# n2 p2 a2 ðyy0 Þ An ðsÞexp kþ s20

(52)

(53)

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F. Mohammadi, R. Sedaghati / Journal of Sound and Vibration 331 (2012) 2729–2752

where 2 s0

An ðsÞ ¼

Z

s0

EðsÞsin

0

     np np s ds sin s s0 s0

If we assume that A is the maximum value of An(s) in the interval 0os os0, b11 satisfies the following inequality: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi " !# 1 X n2 p2 a2 ð b11 r A exp kþ y  y Þ 0 s20 n¼1 For 0 o y o y0 all exponential terms are negative values. Thus, one can write the following inequalities: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! pffiffiffiffi n2 p2 a2 ðyy0 Þ r expð kðyy0 ÞÞ kþ exp s20 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !   n2 p2 a2 np exp ð kþ y  y Þ rexp að y  y Þ 0 0 s0 s20

(54)

(55)

Using Eq. (55), Eq. (54) is splitted into two segments as

b11 rA

m1 X   pffiffiffiffi expð kðyy0 ÞÞ þ n¼1

1 X n ¼ m1 þ 1

exp

 ! np aðyy0 Þ s0

(56)

where m1 is an arbitrary finite integer value. By implementing the geometric progression formula, the inequality in Eq. (56) can be rewritten as

pffiffiffiffi expðpaðm1 þ 1Þðyy0 Þ=s0 Þ 0 (57) b11 r A m1 expð kðyy0 ÞÞ þ 1expðpaðyy0 Þ=s0 Þ Since the argument of exponential function is a negative value within the interval 0 o y o y0 and k is a large nondimensional value, the first and second terms in Eq. (57) are negligible. Eq. (57) is only violated when y is sufficiently close to y0 in which b11 approaches to a finite value within a small interval. Considering this, the kinetic and strain energies associated with b11 are negligible after the integrations. Using the same approach, it can be proved that b12 and b14 are zero and one can obtain an inequality for b13 similar to Eq. (57).Therefore the homogeneous solution can be neglected. The particular solution can be written in the following forms:

b1 ¼

1 X

ða0 þa1 s þ a2 s2 þ   ÞcosðnyjÞ

n¼0

b2 ¼

1 X

ðb0 þ b1 s þ b2 s2 þ   ÞsinðnyjÞ

(58)

n¼0

Three terms of the polynomial in axial direction can be considered for the three nodes along the axial direction of the cylinder. By substituting Eq. (58) into Eqs. (30) and (31) and also equating the coefficient of s0, s1 and s2 in both differential equations, six linear equations are obtained which can be solved for a0, a1, a2, b0, b1 and b2. These equations are given in the Appendix. The coefficients are then substituted into Eq. (58) to find the displacement fields at the damping layer. Degrees of freedom at each node considering lower- and higher-order Taylor’s expansion in the thickness coordinate based on the method_1 and method _2 are summarized in Table 1. The kinetic and strain energy must be calculated for each layer using the described displacement distributions. The total kinetic and strain energies are then obtained by summing of the energies in top, bottom and core layers as U ¼ U t þ U c þU b ,

T ¼ Tt þ Tc þ Tb

(59)

Lagrange equation can be employed to establish the equations of motion which can be written as follows:       d @T @T @U  þ ¼ Fi dt @q_ i @qi @qi Table 1 DOF required in linear and nonlinear displacement distribution models based on method_1 and method_2. Displacement expansion through the core layer

Approach

DOF

FEM (a)

Lower order

method_1

½uc0 vc0 wc0 c1 c2 c1 c2 c1 c2 

FEM (b)

Lower order

method_2

FEM (c)

Higher order

method_1

FEM (d)

Higher order

method_2

½ub0 ½uc0 ½ub0

vb0 vc0 vb0

w0 wc0 wb0

c

c

cb1 bc1 cb1

cb2 bc2 cb2

t

ut0 Cc1

t

b

b

vt0 ct1

ct1 ct2

ct2  cb1 cb2  t

t

ut0 vt0 wt0 c1 c2 

(60)

F. Mohammadi, R. Sedaghati / Journal of Sound and Vibration 331 (2012) 2729–2752

2739

Substituting Eq. (59) into Eq. (60) and assembling for all elements gives the equations of motion in the finite element form as Mq€ þKq ¼ F

(61)

where K ¼K0 (o) þK00 (o) is a complex frequency dependent stiffness matrix. Due to the frequency dependency of viscoelastic properties of the damping layer, stiffness matrix is nonlinear with respect to frequency. In the case of free vibration, Eq. (61) can be written in the following form: ðo2 Mþ KðoÞÞq ¼ 0

(62)

Eq. (62) cannot directly be solved due to the nonlinearity in stiffness matrix. Natural frequencies can be found directly using forced harmonic response method for a certain range of frequency. Under harmonic load F¼F0eiot, Eq. (61) can be expressed as ðo2 Mþ KðoÞÞq ¼ F0

(63)

Therefore, the natural frequencies and consequently the corresponding loss factors can be calculated using the frequency response by solving a set of linear equations in Eq. (63). This method can be expensive since the solution strongly depends on the resolution of frequency axis. The alternative method is to solve eigenvalue problem of Eq. (62) iteratively using the evaluated stiffness matrix at the natural frequencies. The evaluated stiffness matrix at the natural frequency of the bare cylinder could be the best starting point for each mode in the iteration process. As mentioned before, the modal loss factor is defined as the ratio of dissipated energy per radian and maximum potential energy at natural frequency which can be described as [26]:

Zi ¼

uTi K00 ui uTi K0 ui

(64)

in which ui is the normalized eigenvector in free vibration and Zi is the loss factor related to the ith mode. 4.1. Unconstrained treatment Unconstrained treatment can be modeled by eliminating the constraining elastic layer. Therefore the three-layered structure is substituted by two-layered structure which is simpler but inefficient considering the weight point of view [27]. Due to small shear strain at viscoelastic layer in unconstrained treatment, this configuration is useful when a light structure exhibiting small damping behavior is desired. However, as shown in this study, in order to achieve considerable damping in unconstrained sandwich structure, thick viscoelastic layer should be considered which leads to increase in the structural weight. The method_1 can effectively be used to formulate the unconstrained treatment. In order to contribute the strain energy related to the in-plane stresses which are predominant in the damping layer for unconstrained treatment, the lower-order model is employed which is more reasonable since both shear stresses and in-plane normal stresses are considered in the model. According to the method_1, the displacement distributions for the bottom and viscoelastic layers given in Eqs. (3) and (7) are used in the finite element modeling. 4.2. Partial treatment Partial treatment leads to reduction of the total weight in sandwich structure and may provide better vibration suppression performance. This can be efficient for the sandwich structures where the objective is reducing the weight while maximizing the damping. Partial treatment in the developed finite element method can be modeled by eliminating the top and core layers for the untreated elements. Considering this, the stiffness and mass matrices of the bare and treated elements are assembled at the base layer during the assembly process. This can be possible by utilizing method_2 that includes the FEMs (b) and (d) shown in Table 1 which are describing, respectively, lower- and higher-order Taylor’s expansion of displacement fields in the thickness coordinate of the core layer. The configuration of treatments (8 configurations) on the bare cylinder for parametric studies presented in this study is shown in Fig. 2 in which fifteen cylindrical shell elements are considered in finite element modeling and the symbol  shows the location of the treatments for each configuration. 4.3. Cut modeling The configuration of one cut is shown in Fig. 3(a). Lepoittevin and Kress [20] showed that the cut leads to increase in the transverse shear strain at the core layer and consequently the damping increases. Cuts could be considered as untreated elements in which the size of the elements is very small. Therefore after assembling, the size of stiffness and mass matrices would be too large in which the numerical calculations become too expensive. The cuts here are modeled by considering this fact that the nodal displacement at the elastic top and viscoelastic core layers are discontinuous at the location of the cuts [16] (Fig. 3(a)). Considering this, the nodal displacements and the entries of the corresponding mass and stiffness matrices at the cut locations are taken apart in the system stiffness and mass matrices. Although, the effect of damping due

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F. Mohammadi, R. Sedaghati / Journal of Sound and Vibration 331 (2012) 2729–2752

Fig. 2. Configuration of treatments on the bare cylindrical shell.

(u 0t i v0t i w0t i ψ 1ti ψ 2t i )

(u0t j v0t j w0t j ψ 1t j ψ 2t j )

2 cuts 4 cuts

i j

6 cuts 8 cuts 10 cuts 12 cuts 14 cuts

(u 0b v0b w0b ψ 1b ψ 2b ) Fig. 3. (a) Discontinuity of nodal displacements at the cut location and (b) configuration of cuts on the sandwich cylindrical shell.

to the friction between the elements is not considered in this model but it can show the effect of additional shear strain in the core layer on the damping behavior. As it has been mentioned before, the FEM (b) and (d) which are based on the method_2 can be used to model the cuts since considering the discontinuity in the nodal displacements at the top layer bring about discontinuity of displacement at the core layer so that the displacements at the base layer are still continuous. Discontinuity in the nodal displacements of the core in the method_1, leads to discontinuity of displacement at top and bottom layers which does not exhibit the behavior and the effect of the cuts in the finite element modeling. Numerical studies show that cutting influences damping properties. The location of cuts is important for different boundary conditions. The best position of the cuts seems to be between the elements of the sandwich structure with higher modal strain energy since the cuts leads to additional shear deformation and consequently more dissipation of energy. In order to observe the effect of the cuts and their distribution, seven configurations of cuts as shown in Fig. 3(b) have been used for the parametric studies presented in this study in which the discontinuity are considered between different elements according to the number of cuts. 5. Optimization problem The parametric studies presented in this study show that the damping characteristics are affected by the ratio of the top and core thicknesses to the base layer thickness, slippage between layers, total number and distribution of untreated elements in the partial treatment process and also total number of cuts and their distribution along the axis of the sandwich cylindrical shell. These parameters can be optimized simultaneously or separately in the optimization problem. The objective function can also be the loss factor for each mode or combinations (linear or logarithmic) of all loss factors at different modes. In this study, an effective loss factor is introduced according to combinations of all circumferential and axial modes. As mentioned before, loss factor can be determined at the corresponding natural frequencies in circumferential and axial modes. In practice, the contribution of each mode is different for a particular displacement vector X0. Therefore, a weighted combination of loss factor is required to be defined as an objective function. The participation value of each mode can be evaluated in an undamped structure. This value is then considered as a weighting factor for the corresponding loss factor and consequently the weighted combinations of all loss factors related to all modes is considered to be the objective function. For the undamped structure under harmonic loading, the amplitude vector for

F. Mohammadi, R. Sedaghati / Journal of Sound and Vibration 331 (2012) 2729–2752

2741

each circumferential mode can be written as [28] m X

Xn0 ¼

cni uni

(65)

i¼1

where uni is the normalized eigenvector (:9uni :¼1) for nth circumferential and ith axial mode, cni indicates how much of each axial mode is present in the nth circumferential mode and Xn0 represents displacement vector through the cylindrical shell due to a particular loading. For each circumferential mode, uni are orthonormal with respect to stiffness and mass matrices as follows: n n unT iaj i M uj ¼ 0, n n unT iaj i K uj ¼ 0,

(66)

According to this property, cni can be obtained as cni ¼

n n unT i M X0 n n nT ui M ui

(67)

In order to realize the effect of each particular axial mode on the displacement vector Xn0 , all the eigenvectors should be projected on the Xn0 . Therefore a meaningful relation can be defined to determine the participation of each mode as follows: 9cni Xn0 Uuni 9 n n n i ¼ 1 9ci X0 Uui 9

ani ¼ Pm

(68)

The percentage of each circumferential mode can be directly calculated from the total displacement at each point P in the axial direction: 9Xn0 ðPÞ9 n n ¼ 1 9X0 ðPÞ9

ln ¼ P1

(69)

Therefore the effective loss factor is determined by combining the percentage of all modes in circumferential and axial direction which are participating to form the displacement vector X0 as follows:

Zeff ¼

1 X m X

ln ani Zni

(70)

n¼1i¼1

where Zni is the loss factor at circumferential mode n and axial mode i and Zeff is the effective loss factor. The effective loss factor Zeff is now considered as the objective function to be maximized. The constraint of the optimization problem is the total mass of the sandwich structure. The goal here is to suppress the vibration of the base layer. The core and top layers should not have significant impact on the total mass of the structure (mass constraint). In this study, first the optimized number of cuts and their distributions are obtained for fully treated sandwich cylinder using the genetic algorithm available in MATLAB. Binary numbers [0 1] are employed to define the location of the cuts in the sandwich cylinder. Therefore the initial population are created using random binary vectors in a way that 0 is assigned to a place where there is no cut and 1 is assigned to the cut location. For the individuals which include only binary numbers, the useful ‘‘Scattered’’ function is employed for creating the Crossover Children which merges the parents by choosing randomly from the variables of the first and second parent. The Mutation function has been customized in a way that with probability less than 10% random numbers of the variables are switched from 0 to 1 and vice versa. Another optimization has been conducted to optimize simultaneously total number of treatments, the distribution of the treatment and thickness of the top and core layers. In this case, the total mass of the structure is the constraint of the optimization problem. The Genetic Algorithm (GA) combined with Sequential Quadratic Programming (SQP) available in MATLAB are used in the optimization process. First all parameters are optimized by employing GA in which the binary vectors, 0 and 1 are assigned to the treated and untreated elements, respectively. Since the thicknesses of the top and bottom layers at the treatments are real value, therefore a crossover function is customized to create the children so that the ‘‘Scattered’’ function is used to create binary vector for the location of the treatments and the ‘‘Intermediate’’ function is used to create the thicknesses of the top and bottom layers in which weighted average of the thicknesses from the parents are evaluated. The same Mutation function mentioned before is used to create mutation children except that for the elements of an individual representing the top and core thicknesses, the Gaussian function is used to add random number to these elements. The optimum thickness can be further improved using SQP subjected to the constraint on total mass. The result obtained by GA is used as the starting point for SQP solution. 6. Results and discussion First, the developed finite element models have been validated by comparing the results to those reported in the literatures. The validation is shown using numerical, analytical and experimental data available in the literatures for the bare cylindrical shell. However due to the lack of experimental studies on damping effect of the cuts in sandwich

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F. Mohammadi, R. Sedaghati / Journal of Sound and Vibration 331 (2012) 2729–2752

cylindrical shell, the results for damping behavior are validated using the available data in the literature for sandwich cylindrical shell without the cuts. Different finite element models based on the lower and higher order of Taylor’s expansion of displacement fields in the thickness coordinate of the core layer using method_1 and method_2 are also compared with each other. Parametric studies are also presented to observe the effect of the partial treatment and the distribution of treated elements on the loss factor of constrained sandwich cylindrical shell. Similar results are also presented for the effects of cuts and their distribution on damping behavior. Moreover, the effect of imperfect bonding is shown for different fundamental circumferential modes. The optimization problem is first solved to find the optimum number and distribution of cuts in a fully treated sandwich cylinder and, finally, the optimization problem is solved to decrease the weight of the sandwich structure using partial treatments in a way that a light structure with maximum effective loss factor is achieved. The results from the parametric study are presented for clamped–free boundary conditions and the optimization results are presented for clamped–free, simply supported and clamped–clamped boundary conditions. 6.1. FEM validation The free vibration characteristics for bare cylindrical shell and sandwich cylindrical shell at different circumferential and axial modes are compared with other studies and some available experimental data. These studies include the work presented by Petyt [29] in which natural frequencies of a curved rectangular bare plate are obtained using theoretical methods and experiments. Au and Cheung [30] analyzed free vibration and stability of shells using spline finite strip method. Djoudi and Bahai [31] developed finite element model of bare cylindrical shallow shell based on the assumed strain fields. Chung [32] developed analytical method to evaluate natural frequencies in the bare cylindrical shell. Using Eq. (62), in Table 2 natural frequencies for simply supported bare cylinder are obtained and compared with those in literature. In Table 3, another comparison is presented for clamped–free boundary condition. The dimensions and material properties are mentioned in the caption of each table. As it can be realized, very good agreement exists between the presented results and those reported in the references. For the viscoelastivc sandwich cylindrical shell, the obtained natural frequencies and corresponding loss factors at different fundamental circumferential modes using lower-order model based on the method_1, FEM (a), are shown in Fig. 4. The material properties and the dimensions of the three layers are given in Table 4. The results are compared to those reported in Refs. [8,9] and as it can be realized very good agreement exists. Further results are presented for sandwich structure in which the viscoelastic properties of the damping core layer depend on the frequency. The properties extracted from Ref. [33] and the dimensions are given in Table 5. Using method_1, the frequency response under point load at the end of the clamped–free cylinder is obtained at the point where the load is exerted. Fig. 5 shows that for a thin damping layer the FRF for the lower- and higher-order models are nearly similar. However, at the natural frequencies the damping behavior of the higher-order model is slightly more than the lower-order model. The ratio of damping layer and top elastic layers to the base layer is assumed to be Rc and Rt, respectively. In Fig. 6. for Rt ¼0.2 and different values of Rc, the effective loss factors resulted from different finite element models (mentioned in Table 1) are compared for the clamped–free boundary condition. It should be noted that in this study the effective loss factor is calculated based on considering 10 circumferential modes and their first 3 axial modes for sandwich cylindrical shell under point loading. As expected, the results from the lower- and higher-order models deviate as Rc becomes larger (core viscoelastic becomes thicker). However, one can realize that the difference between lower- and higher-order models Table 2 Natural frequencies (Hz) of the cylindrical shell for simply support boundary condition, a ¼148.234 mm; L ¼298.2 mm; hb ¼0.508 mm; E ¼2.035  1011 N m  2; u ¼0.285; r ¼ 7846 kg m  3. m

n

Present study

Ref. [29]

Ref. [30]

Ref. [8]

Experiment [29]

1

1 2 3 4 5 6 7 8

3271 1862.2 1102 705.9 497.9 400.7 381.6 417.7

3270.5 1862 1101.8 705.7 497.5 400.1 380.7 416.7

3270.6 1862 1101.8 705.9 497.5 400.1 380.7 416.7

3266.8 1861.1 1101.8 706.5 499.2 402.9 384.7 422

– – 1101 715 534 410 393 426

2

1 2 3 4 5 6 7 8

4838.5 3725.8 2743.4 2018.7 1515.7 1175.6 954.4 825.2

4837.7 3725 2742.6 2018 1515 1174.9 953.6 824.2

4837.8 3725.1 2742.8 2018.2 1515.2 1175.1 953.7 824.3

4837.3 3724 2741.3 2016.8 1514.7 1175.9 956.3 829

– – – – – – – –

F. Mohammadi, R. Sedaghati / Journal of Sound and Vibration 331 (2012) 2729–2752

2743

Table 3 Natural frequencies (Hz) of the cylindrical shell for clamped–free boundary condition, a¼ 216.2mm; L¼ 512.2 mm; hb ¼1.5 mm; E ¼ 1.83  1011 N m  2; u¼ 0.3; r ¼ 7492 kg m  3. m

n

Present study

Ref. [31]

Nastran [31]

Analytical [32]

1

1 2 3 4 5 6 7 8 9 10

857.5 405.6 225.1 174.6 203.6 274.9 369.8 482.5 611.1 755.4

– 403.91 224.79 172.4 203.37 274.04 369.8 482.96 611.42 755.14

– 410.1 232.2 180.5 206.2 275.5 370.1 483.5 614 –

– 403.72 223.34 171.77 199.16 268.86 361.92 472.54 599.03 740

2

2 3 4 5 6 7 8 9 10

1440.5 932.9 649.4 500.0 448.6 473.1 550.0 661.2 796.5

1446.56 940.2 647.25 509.11 442.16 477.32 542.69 656.06 788.39

– 943.2 671.4 529.3 478 496.9 567.2 673.2 –

1437.11 928.28 644.48 494.69 442 464.59 539.45 648.34 781.15

3

2 3 4 5 6 7 8 9 10

2491.3 1841.5 1376.1 1066.3 874.7 795.97 763.3 812.9 910.8

2530.79 1876.8 1412.72 1103.56 875.29 789.5 748.35 809.48 884.6

– – – – – – – – –

2487.6 1834.82 1367.64 1057.12 864.82 767.65 750.67 798.18 893.6

10−1

10−2

Present Study WT [9] Present Study

KT [9] FEM_1+FEM_2[8]

Loss Factor

Non-dimensional Frequency

DLT [9]

10−3

10−2

DLT [9] WT [9] KT [9] FEM_1+FEM_2[8]

10−3

10−4

10−4 1

2

3

6 7 4 5 8 9 Circumferential mode number

10

11

1

2

3

4 5 8 9 6 7 Circumferential mode number

10

11

Fig. 4. (a) Non-dimensional frequency (O ¼ rbhbao2/Eb) and (b) loss factor for clamped–clamped cylinder. Table 4 Properties and dimension of the three-layered sandwich cylinder—properties of damping layer is independent from frequency [9].

Base layer Viscoelastic layer Constraining layer

Radius (mm)

Length (mm)

Thickness (mm)

Young modulus (N m  2)

Density (kg m  3)

Poisson’s ratio

100 100 100

100 100 100

1 1 1

2.1  1011 (2.3þ 0.8i)  107 2.1  1011

7850 1340 7850

0.3 0.4 0.3

are insignificant (l%) for value of Rc up to 3. The difference of 2.5% is observed for the thick core layer (3 oRc o5). The results also show that the method_1 and method_2 for the lower- and higher-order models generate similar results. The small difference (0.8%) between FEM (c) and (d) arises from neglecting the general solution in Eqs. (30) and (31).

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Table 5 Properties and dimension of the three-layered sandwich cylinder—properties of damping layer is frequency dependent [33]. Radius (mm) Base layer Viscoelastic layer Constraining layer

216.2 216.2 216.2

Length (mm) 511.2 511.2 511.2

Thickness (mm) 1.5 hc ht

Young modulus (N m  2) 11

1.83  10 380,560 (o/2p)0.475(1þ 1.46i) 1.83  1011

Density (kg m  3)

Poisson’s ratio

7492 1140 7492

0.3 0.34 0.3

Fig. 5. Frequency response for clamped–free constrained sandwich cylindrical shell; hb ¼hc ¼1.5 mm, ht ¼0.3 mm, comparison between lower- and higher-order models.

Fig. 6. Comparison between the results of the lower- and higher-order models for effective loss factor in clamped–free boundary condition using, Rt ¼0.2.

6.2. Effect of thickness ratios of the core and constrained layers Based on the higher-order model, the parametric study is performed to investigate the effect of Rc and Rt on the loss factor associated with first four fundamental circumferential modes (n ¼1,2,3,4 and m¼1). It is assumed that perfect bonding exists at the interfaces. According to the results shown in Fig. 7, in order to achieve higher damping in the case of the thin core, thickness of the constraining layer should increases. However the effect is more considerable up to a certain ranges of Rt. After these ranges, increase in Rt slightly increases the loss factor in the expense of increasing the mass of the structure. In the case of the thick viscoelastic core layer, increase in Rt leads to decrease in loss factor. Also for the thin constraining layer, thick core layer shows higher damping property than thin core. On the other hand, for thick constraining layer, by increasing the thickness of the core layer, loss factors at the predominant modes first decreases and then increases.

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Fig. 7. Variation of loss factor versus Rt and Rc for constrained clamped–free cylindrical shell—Higher-order model; FEM (c); (a) n¼ 1, (b) n¼ 2, (c) n¼ 3, (d) n¼ 4.

Fig. 8. Loss factor versus shear rigidity of adhesive layer for clamped–free boundary condition at first axial mode and different circumferential mode number; Rc ¼1, Rt ¼ 0.2—Higher-order model, FEM (c).

6.3. Effect of the slippage between layers In order to investigate the effect of bonding at the interfaces, the shear stiffness parameter K* of the adhesive layer can be related to the shear modulus of the core layer as K*hc ¼CGc in which C is a non-dimensional parameter which determine the bonding at the interfaces [6]. Large value of C should be assigned if perfect bonding is desired at the interfaces. Fig. 8 shows the loss factor at first five circumferential and first axial modes versus parameter C using the higher-order model. It is interesting to note that initially by increasing factor C (especially for lower modes) loss factors slightly decreases and then rapidly increase and converge to the value in the case of perfect bonding (for C 4100). It can be concluded that to achieve more conservative design, perfect bonding between layers should be provided. It should be noted that to introduce the true slippage, the damping due to the contact at the adhesive layer should be taken into the accounts. However, the most part of the damping is due to the transverse shear strains in the damping layer which will be affected if the in-plane displacement fields are discontinuous at the interface. The discontinuities result in small force at the interfaces and accordingly the transverse shear strains in the core layer reduce.

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6.4. Unconstrained sandwich cylinder Fig. 9 shows frequency response of the unconstrained sandwich cylindrical shell. According to the previous discussions in Section 4.1, the lower-order model has been employed for the unconstrained treatment. The main disadvantage of

Fig. 9. Frequency response for clamped–free unconstrained sandwich cylindrical shell; comparison between thin and thick core layers.

Fig. 10. Effect of core layer thickness on the loss factor of unconstraint sandwich cylindrical shell at different fundamental circumferential mode for clamped–free boundary condition.

Fig. 11. Damping properties of fully treated sandwich cylindrical shell without cutting; Rc ¼ 1, Rt ¼0.2, comparison between lower- and higher-order models using FEM (b) and (d).

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unconstrained sandwich structure is poor damping property due to small shear stress at the damping layer. Due to increase in the total mass of the structure while small change in the stiffness, the natural frequency considerably decreases as the thickness of the core layer increases. The effect of the core thickness on the loss factor of the clamped–free uncontained sandwich cylindrical shell is also demonstrated in Fig. 10. As expected, the loss factor increases as the core layers become thicker. This is especially more pronounced for the lower modes.

Fig. 12. Partial treatment and its distribution effect on damping properties; Rc ¼ 1, Rt ¼0.2, (a) 1 treatment, (b) 2 treatments, (c) 3 treatments, (d) 4 treatments, (e) 5 treatments, (f) 6 treatments, (g) 7 treatments, (h) 8 treatments.

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6.5. Effect of cuts, partial treatments and their distribution on loss factor and frequency response function First, the sandwich structure layer is assumed to be fully treated without cut at the core and top layers. The damping property using the lower- and higher-order models in terms of the loss factor at each fundamental circumferential mode and also in terms of the effective loss factor for Rc ¼1 and Rt ¼0.2 is shown in Fig. 11. According to the results, the higherorder model exhibits more damping properties than the lower-order model. Effects of partial treatment on the loss

Fig. 13. Cut and its distribution effect on damping properties; Rc ¼ 1, Rt ¼ 0.2, (a) 2 cut, (b) 4 cuts, (c) 6 cuts, (d) 8 cuts, (e) 10 cuts, (f) 12 cuts, (g) 14 cuts.

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factor considering the treatment configurations shown in Fig. 2 are presented in Fig. 12. According to these configurations, although the partial treatment process can reduce significantly total weight of the structure, an inappropriate distribution may decrease the damping properties. It will be shown further that one can optimize the partial treatment locations and thickness ratio to achieve higher loss factor compared with that of fully treated structure. The effect of cuts considering the configurations given in Fig. 3(b) is also shown in Fig. 13. One can realize that cutting in the core and top layers can increase the damping property compared with the fully treated sandwich cylinder and also increase in the total number of cuts does not guarantee increase in damping properties. The FRFs for the sandwich cylindrical shell including 14 cuts and also for the same cylinder with 8 treatments (no cutting) are compared with FRF of the fully treated cylindrical shell in Fig. 14. Increase in damping due to the cuts leads to smaller amplitude in the frequency response. However, less damping due to the partial treatment using the configuration of 8 elements leads to higher amplitudes at the natural frequencies. 6.6. Optimization results As mentioned before, in this study displacement vector X0 resulted from harmonic point loading condition is considered to find the effective loss factor. The load is applied at the tip of the clamped–free and at the middle of the simply supported and clamped–clamped sandwich cylinders. Optimization results for number of cuts and their distribution along the axial direction considering different boundary conditions are shown in Table 6. According to the results, the effective loss factor is improved for clamped–free, simply support and clamped–clamped, respectively, by 5.2%, 6.5% and 6.6% with respect to that of fully treated sandwich cylinder with no cut. Symmetric distribution of cuts is observed for the clamped–clamped and simply supported boundary conditions. Since X0 resulted from point loading at the middle of the cylinder has similar configuration for both simply supported and clamped–clamped boundary conditions, the optimization results are similar in both cases. To observe the damping effect of the optimized cutting on the clamped– free sandwich cylinder, its frequency response is compared to the frequency response of the fully treated sandwich cylinder without cutting in Fig. 15. One can realize that the optimum cutting can considerably decrease the amplitude in the frequency response function especially at higher frequencies. Another optimization is performed to find optimum treatment in which a light structure exhibiting maximum damping property is achieved. The total number of treatments, distribution of treatments, and thickness of top and core layer are simultaneously optimized. It has been assumed that the total mass of the structure does not exceed 25% of the bare structure mass. The optimum treatment distribution with the optimum thickness ratio of the core and top layers for different boundary conditions are presented in Table 7. The optimum number of treatment for clamped–free boundary

Fig. 14. Effect of cuts and partial treatments on frequency response function; Rc ¼1, Rt ¼ 0.2.

Table 6 Optimum cutting distribution on fully treated cylinder, Rt ¼ 0.2, Rc ¼ 1.

Cut distribution for clamped-free sandwich cylinder (eff = 0.1481)

No cutting eff = 0.1408

Cut distribution for simply supported sandwich cylinder (eff = 0.1003) eff = 0.0942 Cut distribution for clamped-clamped sandwich cylinder (eff = 0.0997) eff = 0.0935

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Fig. 15. Frequency response of fully treated clamped–free sandwich cylinder; Rc ¼ 1, Rt ¼0.2, comparison between optimum cutting and no cutting.

Table 7 Optimum partial treating and thickness ratio of top and core layers.

Treatment distribution for clamped-free sandwich cylinder (eff = 0.1716)

Rt

Rc

0.163

2.083

0.098

1.781

0.079

1.881

Treatment distribution for simply supported sandwich cylinder (eff = 0.1207) Treatment distribution for clamped-clamped sandwich cylinder (eff = 0.1106)

Fig. 16. Comparison between frequency responses of optimum treating and full treating in clamped–free sandwich cylinder.

condition is found to be 7 and they are clustered at the free end of the shell. However for the simply-supported and clamped–clamped boundary conditions the optimum number of partial treatments is found to be 9 and they are clustered at the middle of the shell due to symmetry condition. It can be realized that for achieving maximum shear strain, the treated elements are mainly placed at the locations in which the deflection is high. The percentages of improvement of effective loss factor for the clamped–free, simply-supported and clamped–clamped are, respectively, 25.1%, 28.1% and 18.3%. Therefore the damping characteristics can considerably improved whilst the total weight of the structure is decreased. In order to understand the effect of the optimum treatment on damping, the frequency responses for the optimum design and the primary design in the clamped–free sandwich cylinder are shown in Fig. 16. It can be concluded that the optimum treatment provides better damping comparing to the initial design especially at the lower modes.

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7. Conclusion Considering slippage between layers at the interfaces, a new higher-order Taylor’s expansions of displacement fields in the core layer were developed to investigate damping characteristics of sandwich cylindrical shell. The developed higherorder model can compatibly describe the displacement field in the core layer in terms of displacements at the elastic faces. The parametric study showed that the slippage should be prevented. The higher-order model was considered in the optimization problems due to the feasible displacement profile achieved by this model especially for thick constrained viscoelastic core layer. Optimization results showed that by embedding an optimum distribution of the cuts, the loss factors particularly at the higher modes are considerably increased. It was also shown that a light structure with high damping characteristics can be achieved by optimizing simultaneously partial treatment distribution and the thicknesses of treating layers. Appendix Six linear equations to be solved for finding the coefficients of the polynomials in the particular solution 1 n 3 2 K hc Lc ð2aa2 þnb1 Þð2aEc L2e þ 2K n hc aL2e ð1 þ vC ÞÞa0 þ K n aL2e ðut2 ub2 Þ 12 1 1 t b  K n hc aLe ðwb1 þ wt1 wb3 wt3 Þ K n aL2 ðht c21 þ hb c21 Þ ¼ 0 2 2 1 n 3 2 K hc Le nb2 ð2aEc L2e þ 2K n hc aL2e ð1 þvC ÞÞa1 þ K n hc að2wt1 þ 2wt3 þ 2wb1 þ 2wb3 4wt2 4wb2 Þ 6 ! t t b b ðc11 c31 Þ ðc c31 Þ ht þ 11 hb þ ut3 ub3 þub1 ut1 ¼ 0 þ K n aLe 2 2 ð2aEc L2e þ2K n hc aLe 2 ð1 þ vC ÞÞa2 t t t b b b t b t þ K að2u3 þ4u2 þ 2u1 4ut2 2ub1 2ub3 þ ð2c21 c11 c31 Þht þ ð2c21 c11 c31 Þhb Þ ¼ 0 n 2 1 n 3 2 K hc aLe n t K hc Le nðnb0 þ aa1 Þð2a2 Ec L2e þ 2K n hc a2 L2e ð1 þ vC ÞÞb0  ðw2 þ wb2 Þ  n

12

t

n 2 2

K a Le

b

ht c22 þ hb c22 vt2 þ vb2 2

2

!

¼0

1 K n hc anLe 3 ð2wb1 wb3 þ wt1 wt3 Þ  K n hc L2e nðaa2 þ 2nb1 Þð2a2 Ec L2e þ 2K n hc a2 L2e ð1 þvC ÞÞb1 þ 6 2 ! t t b b ðc12 c32 Þ ðc c32 Þ ht þ 12 hb þ vt3 vb3 þ vb1 vt1 ¼ 0 þK n a2 Le 2 2 1 n 3 2 2 K hc Le n b2 þð2a2 Ec L2e þ 2K n hc a2 L2e ð1 þ vC ÞÞb2 þ K n hc anðwt1 2wt2 þ wt3 þ wb1 2wb2 þwb3 Þ 12 t t t b b b þK n a2 ððc12 2c22 þ c32 Þht þðc12 2c22 þ c32 Þhb 2vt3 þ4vt2 2vt1 þ 2vb3 4vb2 þ 2vb1 Þ ¼ 0

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