Free vibration analysis of cylindrical sandwich panel with electro-rheological core and FG-GPLRC facing sheets based on First order shear deformation theory referred by Qatu

1. Introduction

The advanced sandwich structures are more and more applied because these structures have low weight, cost, and high performance. The use of soft materials such as viscoelastic materials and smart fluids (Electro/Magneto-rheological materials) as sandwich construction core suppresses vibration and noise. Electro-rheological materials change quickly from liquid to quasi-solid when subjected to electric fields. These materials perform as viscoelastic materials in low strain levels, but compared with viscoelastic materials, they have some advantages: controllability by applying electric fields and usability in a more significant temperature range. Numerous studies on vibration and dynamic analysis of these structures have been published.

Additionally, composite materials have a significant role in engineering structures and improving their operational characteristics. Composites are widely applied in aircraft, biomedical, aerospace, infrastructure, marine, energy, armor, recreational, and automotive applications. The high-strength, low-density, and high-stiffness characteristics make composites highly flexible in primary and secondary structures of both civilian aircraft and military.

Yalcintas and Coulter (1995) investigated vibration characteristics and variations of modal loss factors with different parameters of sandwich beam with ER core. Kang et al. (2001) investigated damping characteristics of electro-rheological composite beams using finite element method. Yeh et al. (2004) studied dynamic stability of sandwich beam with electro-rheological fluid core subjected to axial dynamic force by using finite element method and the harmonic balance method. Finite element formulation for vibration of sandwich rectangular plate with electro-rheological core was developed by Yeh and Chen (2004); they found that when an electric field is applied, the damping of the system is more effective. The effect of thickness of ER layer and electric field identity on the natural frequency and modal loss factor of sandwich orthotropic plate with ER core was study also by Yeh and Chen (2007) utilizing discrete finite element method. Yeh (2008)–(Yeh, 2011)used discrete finite element method to study vibration and damping characteristics of sandwich cylindrical shells with electro-rheological layer. Analytical solution for free and forced harmonic vibration of electro-rheological sandwich plate with simply supported edges studied by Hasheminejad and Maleki (2009). Mohammadi and Sedaghati (2012) developed a nonlinear finite element model for nonlinear vibration of sandwich shell with electro-rheological fluid for different boundary conditions. Tabassian and Rezaeepazhand (2013) investigated dynamic stability of smart sandwich beam with electro-rheological core resting on Winkler elastic foundation and subjected to external harmonic loads. Allahverdizadeh et al. (2013) used finite element method to present dynamic behavior of functionally graded electro-rheological sandwich beams with different boundary conditions, optimal parameters characterization, and vibration control of ER sandwich beams are investigation by Allahverdizadeh et al. (Allahverdizadeh et al., 2014). They used ASTM E756 standard for estimation of the storage and loss modules of the viscoelastic layer. Analytical solution for free vibration of three-layer sandwich plate with ER core was presented by Soleymani et al. (2015). Vibration analysis of electro rheological sandwich plate with CNT reinforced facing sheets rested on Winkler Pasternak foundation was carried out by Arani et al. (2017). Based on classical thin plate theory for face sheets and first order shear theory for the core layer, free and forced vibration of a sandwich plate with electro-rheological fluid core and functionally graded facing layers was investigated by Gholamzadeh Babaki and Shakouri (2019). Based on classical plate theory for face layer, Exact solution for SSSS and SSCC sandwich cylindrically curved panel with ER core and CNTRC facing sheets rested on Winkler Pasternak foundation was performed by Ghorbanpour Arani and Jamali (2020). Within the framework of modified couple stress theory, vibration characteristics of a micro cylindrical sandwich panel with foam core and graphene platelets reinforced composite (GPLRC) resting on elastic foundation is studied by Anvari et al. (2020). Vibration of GPLLRC cylindrical shell within the framework of FSDT was discussed by Arefi et al. (2021). Analytical and semi-analytical solution for static and free vibration analysis of porous GPLRC based on theory of elasticity is investigated by Rahimi et al. (2020). Vibration control of GPLRC cylindrical shell surrounded by piezoelectric layers was presented by Al-Furjan et al. (2021) using generalized differential quadrature (GDQ). Song et al. (2017) investigated free and forced vibration of functionally graded GPLRC plates with simply supported boundary condition using Navier’s method. Feng et al. (2017) used layer-wise approach to study nonlinear free vibration of GPLLRC beam within the frame work of Timoshenko beam theory along with von Kármán nonlinear strain displacement relationship. Mohseni and Shakouri (2020) studied free and forced vibration of simply supported sandwich plate with the viscoelastic core and FG-GPLRC face layers based on classical theory. They used rule of mixture to obtain effective material properties of GPLRC. The presence of the trapezoidal shape factor ( 1 + z R ) in the derivation of the fundamental equation of shells makes it challenging to integrate and consider the effects of depth ratio in shell theories. In most studies, the term z R was neglected that mean assumption of thin shell theories ( 1 + z R ≃ 1 ) . The equations with considering this term were presented by Qatu (1999) named the First-order shear deformation shell theory referred by Qatu (FSDTQ) and discussed in some research (Asadi and Qatu, 2012, 2013; Qatu et al., 2012; Qatu, 1995, 1999). Most of the existing works of sandwich structures with ER core restricted to beams and flat plates with isotropic facing sheets using finite element method and classical theories.

To the best knowledge of authors, vibration and damping behavior of a thick cylindrical sandwich panel with ER core and FG-GPLRC facing sheets has not yet been reported in the literature, and in the present work we discuss this matter. A layer-wise approach and Halpin–Tsai micromechanical model are used for facing sheets. The equations of motion within the framework of FSDTQ and applying Hamilton’s principle are derived. Fourier series expansion along the axial and circumferential coordinate is used to solve the equations of motion analytically. In the case of other edge boundary conditions, we used 2D-GDQ to numerically solve the governing equation of motion. As an essential property of the system, the natural frequency changes with the electric field change, and this event makes it possible to change the natural frequency at the right time and prevent the resonance phenomenon.

2. Problem formulation

Consider a cylindrical sandwich panel composed of an electro-rheological core and GPLRC facing sheets with lengths (L), span angle (θ₀), mid-surface radius (R_c), i-th layer thickness (h_i) where (i = t, b, and c), and global coordinate (r, θ , x) as shown in Figure 1. Generally, t, b, and c represent the top, bottom, and middle layers, respectively. The local axis of the i-th layer radial directions is denoted by z_i, and its origin is located on the i-th layer’s mid-surface. Where

r = z i + R i → r = R i ( 1 + z i R i )

(1)

OPEN IN VIEWER

Figure 1.

Geometry of cylindrical sandwich panel with electro-rheological core and graphene platelets reinforced composite (GPLRC) facing sheets.

The formulation of this study is based on the following assumption:

• A perfect bond without any Slippage between the layers is assumed. According to this assumption, the displacements at the interfaces are the same for the core and facing.

2.1. GPLRC properties

Each GPLRC facing sheet is composed of N_L layers of the same thickness. In each sub-layer, GPLs as effective rectangular fillers are perfectly bonded, and dispersed uniformly in polymer matrix while GPL weight fraction changes according to different GPL distribution patterns in the in the radial direction (UD, FG-X, FG-⋄, FG-Δ, and FG-∇) as depicted in Figure 2. The darker colors indicate a higher weight of GPLs. In the UD pattern, GPLs distributed unique, FG-⋄ and FG-X are symmetric patterns by having maximum and minimum weight fraction values in the mid-plane of the layer, in the FG-Δ and FG-∇weight fraction of GPLs decrease and increase linearly from the bottom surface to the top.OPEN IN VIEWER

Figure 2.

Different patterns of GPL distribution across thickness of facing sheets.

Effective young module according to Halpin–Tsai micromechanical model is (Affdl and Kardos, 1976)

E = 3 8 ( 1 + ζ L η L V G P L 1 − η L V G P L ) E m + 5 8 ( 1 + ζ T η T V G P L 1 − η T V G P L ) E m

(2)

Mass density and Poisson’s ratio of the facing sheets sub-layer can be calculated by the rule of mixture

ρ ( i ) = ρ G P L V G P L ( i ) + ρ m ( 1 − V G P L ( i ) ) υ ( i ) = ρ G P L V G P L ( i ) + υ m ( 1 − V G P L ( i ) ) i = 1,2 , . . . , N L

(3)

In which E m , V G P L are Young module of polymer matric and GPL volume fraction, respectively, ζ and η are dimensional ratio of graphene plate and Young module ratio of GPL and polymer matrix, respectively, with the following relations

ζ L = 2 a G P L t G P L ζ T = 2 b G P L t G P L η L = E G P L E m − 1 E G P L E m + ζ L η T = E G P L E m − 1 E G P L E m + ζ T

(4)

where a_GPL, b_GPL, and t_GPL are length, width, and thickness of graphene platelet, respectively.

The relation of GPL volume fraction (V_GPL) for different patterns of distribution by dividing the layer to N_L sub-layer is (Feng et al., 2017)

U D : V G P L = V G P L * F G − ∇ : V G P L = V min + ( i − 1 ) ( V max − V min ) N L − 1 F G − Δ : V G P L = V max − ( i − 1 ) ( V max − V min ) N L − 1 F G − X : { V G P L = V max − ( i − 1 ) 2 ( V max − V min ) N L − 2 1 ≤ i ≤ N L 2 V G P L = V min + ( i − N L 2 − 1 ) 2 ( V max − V min ) N L − 2 N L 2 < i ≤ N L F G − ⋄ : { V G P L = V min + ( i − 1 ) 2 ( V max − V min ) N L − 2 1 ≤ i ≤ N L 2 V G P L = V max − ( i − N L 2 − 1 ) 2 ( V max − V min ) N L − 2 N L 2 < i ≤ N L i = 1,2 , . . . , N L

(5)

Here V G P L * = γ G P L γ G P L + ρ G P L ρ m ( 1 − γ G P L ) , V min , V max is total, minimum, and maximum GPL volume fraction, respectively, and γ_GPL is weight fraction of graphene platelet.

2.2. Electro-rheological properties

The electro-rheological material in the pre-yield region is modeled as a linear viscoelastic, which based on the Kelvin viscoelastic model ( τ x y = η γ ˙ + G γ ) and the simplifying (Yeh et al. (2004)), the relationship between its shear strain and shear stress with complex module is expressed as follows

G = G ′ + i G ″

(6)

where G′ and G″ called storage and loss module, respectively.

The complex module of different ER materials has been obtained by experiment and reported in literatures (Don, 1993; Yalcintas and Coulter, 1995). We used complex module introduced by Yalcintas and Coulter (1995) with relations as: G = G ′ + i G ″ , G ′ = 50000 E 2 , G ″ = 2600 E + 1700 which depends on electric field, E [kv.mm⁻¹].

2.3. Kinematic relations

Displacement fields for each layer of sandwich cylindrical panel based on FSDT are expressed as

u i ( x , θ , z , t ) = u 0 i ( x , θ , t ) + z i ψ x i v i ( x , θ , z , t ) = v 0 i ( x , θ , t ) + z i ψ θ i i = t , b , c w t ( x , θ , z , t ) = w 0 t ( x , θ , t ) w b ( x , θ , z , t ) = w 0 b ( x , θ , t ) w c ( x , θ , z , t ) = w 0 c ( x , θ , t ) + z c α c

(7)

where u₀, v₀, w₀ are displacement components at mid-surface, ψ x and ψ θ are angular displacements of the i-th layer in the θ and x directions, respectively. According to equation (7), the transverse displacement of each facing sheet is independent of the radial coordinate, whereas the core transvers displacement varies linearly along the radial coordinate.

Relation of displacements continuity at the interfaces of facing and core layers are

u c ( x , θ , h c 2 , t ) = u t ( x , θ , − h t 2 , t ) v c ( x , θ , h c 2 , t ) = v t ( x , θ , − h t 2 , t ) w c ( x , θ , h c 2 , t ) = w t ( x , θ , − h t 2 , t ) u c ( x , θ , − h c 2 , t ) = u b ( x , θ , h b 2 , t ) v c ( x , θ , − h c 2 , t ) = v b ( x , θ , h b 2 , t ) w c ( x , θ , − h c 2 , t ) = w b ( x , θ , h b 2 , t )

(8)

From equation (8), displacements of the core layer can be determined in term of facing layer displacement as follows

u 0 c = u 0 t 2 + u 0 b 2 − h t 4 ψ x t + h b 4 ψ x b v 0 c = v 0 t 2 + v 0 b 2 − h t 4 ψ θ t + h b 4 ψ θ b w 0 c = 1 2 ( w 0 t + w 0 b ) ψ x c = 1 h c ( u 0 t − u 0 b − h t 2 ψ x t − h b 2 ψ x b ) ψ θ c = 1 h c ( v 0 t − v 0 b − h t 2 ψ θ t − h b 2 ψ θ b ) α c = 1 h c ( w 0 t − w 0 b )

(9)

Strain–displacement relations in cylindrical coordinate in linear elastic continuum are (Sadd, 2009)

ε x x i = ∂ u i ∂ x ε θ θ i = 1 r ( ∂ v i ∂ θ + w i ) ε r r i = ∂ w i ∂ r γ θ r i = 1 r ∂ w i ∂ θ − v i r + ∂ v i ∂ r γ x r i = ∂ w i ∂ x + ∂ u i ∂ r γ x θ i = 1 r ∂ u i ∂ θ + ∂ v i ∂ x i = t , b , c

(10)

Substitution of equation (7) into equation (10) and using equation (1) results in strain components in term of mid-plain displacements and curvatures

ε x x i = ε 0 x x i + z i κ x x i ε θ θ i = 1 1 + z i R i ( ε 0 θ θ i + z i κ θ θ i ) γ x z i = γ 0 x z i γ x θ i = 1 1 + z i R i ( ε 0 θ x i + z i κ θ x i ) + ε 0 x θ i + z i κ x θ i γ θ z i = 1 1 + z i R i γ 0 r θ i i = t , b γ x z c = ∂ w 0 c ∂ x + z c ∂ α c ∂ x + ψ x c γ θ z c = 1 1 + z c R c ( ∂ w 0 c R c ∂ θ + z c ∂ α c R c ∂ θ − v 0 c R c + ψ θ c )

(11)

In which

ε 0 x x i = ∂ u 0 i ∂ x κ x x i = ∂ ψ x i ∂ x ε 0 θ θ i = ( ∂ v 0 i R i ∂ θ + w 0 i R i ) κ θ θ i = ∂ ψ θ i R i ∂ θ γ 0 x z i = ∂ w 0 i ∂ x + ψ x i ε 0 θ x i = ∂ u 0 i R i ∂ θ κ θ x i = ∂ ψ x i R i ∂ θ ε 0 x θ i ∂ v 0 i ∂ x κ x θ i = ∂ ψ θ i ∂ x γ 0 θ z i = ∂ w 0 i R i ∂ θ − v 0 i R i + ψ θ i i = t , b

(12)

2.4. Constitutive relations

The stress-strain relation for facing sheets is

{ σ x x σ θ θ τ x θ τ r θ τ r x } ( k ) = [ Q 11 Q 12 0 0 0 Q 21 Q 22 0 0 0 0 0 Q 44 0 0 0 0 0 Q 55 0 0 0 0 0 Q 66 ] ( k ) { ε x x ε θ θ γ x θ γ r θ γ r x } ( k ) k = 1,2 , . . . , N L

(13)

where Q i j ( k ) is elastic constant for each sub layer (See appendix A).

And for the core layer we have

{ τ θ z c τ x z c } = [ G 0 0 G ] { γ θ z c γ x z c }

(14)

Stress resultants relation is

{ N x x N x θ Q x z M x x M x θ } ( i ) = ∫ − h i / 2 h i / 2 { σ x x σ x θ τ x z z σ x x z σ x θ } ( i ) ( 1 + z R ) d z { N θ θ N θ x Q θ z M θ θ M θ x } ( i ) = ∫ − h i / 2 h i / 2 { σ θ θ σ θ x τ θ z z σ θ θ z σ θ x } ( i ) d z i = t , b

(15a)

{ Q x z c Q θ z c S x z c S θ z c } = ∫ − h c / 2 h c / 2 { τ x z c ( 1 + z R ) τ θ z c τ x z c z ( 1 + z R ) τ θ z c z } d z

(15b)

The integral relations in the equations (15a) and (15b) should be computed in terms of unknown displacements; however, this is a bit difficult due to the lack of z R . The integration for the core is done directly but the integration for the facing is done with another simplifier assumption which we will discuss below.

By substituting equations (11), (12), and (13) into equation (15a), the expression 1 1 + z R appears in the integral which causes to the difficulty of the integration. One of the ways to integrate easier is geometrically expanding this expression

( 1 1 + z R ) = 1 − z R + z 2 R 2 − z 3 R 3 + . . . ⏟ n e g l i g i b l e

By doing so, we still do not omit z R , but we can omit higher-order terms

It results in the following force and moment relationship with strain and curvature at the mid surface for each facing layer (Qatu, 2004)

( N x x i N θ θ i N x θ i N θ x i M x x i M θ θ i M x θ i M θ x i ) = ( A ¯ 11 i A 12 i A ¯ 16 i A 16 i B ¯ 11 i B 12 i B ¯ 16 i B 16 i A 12 i A ^ 22 i A 26 i A ^ 26 i B 12 i B ^ 22 i B 26 i B ^ 26 i A ¯ 16 i A 26 i A ¯ 66 i A 66 i B ¯ 16 i B 26 i B ¯ 66 i B 66 i A 16 i A ^ 26 i A 66 i A ^ 66 i B 16 i B ^ 26 i B 66 i B ^ 66 i B ¯ 11 i B 12 i B ¯ 16 i B 16 i D ¯ 11 i D 12 i D ¯ 16 i D 16 i B 12 i B ^ 22 i B 26 i B ^ 26 i D 12 i D ^ 22 i D 26 i D ^ 26 i B ¯ 16 i B 26 i B ¯ 66 i B 66 i D ¯ 16 i D 26 i D ¯ 66 i D 66 i B 16 i B ^ 26 i B 66 i B ^ 66 i D 16 i D ^ 26 i D 66 i D ^ 66 i ) { ε 0 x x i ε 0 θ θ i ε 0 x θ i ε 0 θ x i κ x x i κ θ θ i κ x θ i κ θ x i } i = t , b

(16)

where A i j , B i j , D i j , E i j , . . . are the stiffness components (See appendix A).

By direct integrating from equation (15b) the following equations is derived

{ Q x z c Q θ z c S x z c S θ z c } = { G [ h c ( ∂ w 0 c ∂ x + ψ x c ) + h c 3 12 1 R c ∂ α c ∂ x ] G [ ( ∂ w 0 c ∂ θ − v 0 c + R c ψ θ c − R c ∂ α c ∂ θ ) ln ( R c + h c 2 R c − h c 2 ) + h c ∂ α c ∂ θ ] G h c 3 12 ( 1 R c ∂ w 0 c ∂ x + 1 R c ψ x c + ∂ α c ∂ x ) G [ ( R c 2 ⁡ ln ( R c + h c 2 R c − h c 2 ) − R c h c ) ∂ α c ∂ θ + ( h c − R c ln ( R c + h c 2 R c − h c 2 ) ) ( ∂ w 0 c ∂ θ − v 0 c + R c ψ θ c ) ] }

(17)

2.5. Equations of motion

Hamilton’s principle with the following relation is used to derive equations of motion

δ Π = δ ∫ t 1 t 2 ( U − T ) d t = 0

(18)

where U = U t + U c + U b T = T t + T c + T b and δ denotes first variation, U and T are strain and kinetic energy with the following relations

δ U t = ∫ V t ( σ x x t δ ε x x t + σ θ θ t δ ε θ θ t + τ x θ t δ γ x θ t + τ x z t δ γ x z t + τ θ z t δ γ θ z t ) d V t δ U b = ∫ V b ( σ x x b δ ε x x b + σ θ θ b δ ε θ θ b + τ x θ b δ γ x θ b + τ x z b δ γ x z b + τ θ z b δ γ θ z b ) d V b δ U c = ∫ V c τ x z c δ γ x z c + τ θ z c δ γ θ z c d V c δ T i = ∫ V i ρ i ( ∂ u i ∂ t ∂ δ u i ∂ t + ∂ v i ∂ t ∂ δ v i ∂ t + ∂ w i ∂ t ∂ δ w i ∂ t ) d v i → i = t , b , c

(19)

Using equations (9), (11), and (15a) in (19) and next substitution in equation (18) leads to the following governing equation of motions

δ u 0 i : − R i ∂ N x x i ∂ x − ∂ N θ x i ∂ θ + ζ R c h c Q x z c + R i I 0 i ∂ 2 u 0 i ∂ t 2 + R t I 1 i ∂ 2 ψ x i ∂ t 2 + ( R c I 0 c 2 + ζ R c I 1 c h c ) ∂ 2 u 0 c ∂ t 2 + ( R c I 1 c 2 + ζ R c I 2 c h c ) ∂ ψ x c ∂ t 2 = 0 δ v 0 i : − ∂ N θ θ i ∂ θ − R i ∂ N x θ i ∂ x − Q θ z i + ( − 1 2 + ζ R c h c ) Q θ z c + R i I 0 i ∂ 2 v 0 i ∂ t 2 + R i I 1 i ∂ 2 ψ θ i ∂ t 2 + ( R c I 0 c 2 + ζ R c I 1 c h c ) ∂ 2 v 0 c ∂ t 2 + ( R c I 1 c 2 + ζ R c I 2 c h c ) ∂ 2 ψ θ c ∂ t 2 = 0 δ w 0 i : N θ θ i − R i ∂ Q x z i ∂ x − ∂ Q θ z i ∂ θ − R c 2 ∂ Q x z c ∂ x − ζ R c h c ∂ S x z c ∂ x − 1 2 ∂ Q θ z c ∂ θ − ζ 1 h c ∂ S θ z c ∂ θ + R i I 0 i ∂ 2 w 0 i ∂ t 2 + ( R c I 0 c 2 + ζ R c I 1 c h c ) ∂ 2 w 0 c ∂ t 2 + ( R c I 1 c 2 + ζ R c I 2 c h c ) ∂ 2 α c ∂ t 2 = 0 δ ψ x i : − R i ∂ M x x i ∂ x − ∂ M θ x i ∂ θ + R i Q x z i − R c h i 2 h c Q x z c + R i I 1 i ∂ 2 u 0 i ∂ t 2 + R i I 2 i ∂ 2 ψ x i ∂ t 2 − ( ζ R c I 0 c h i 4 + R c I 1 c h i 2 h c ) ∂ 2 u 0 c ∂ t 2 − ( ζ R c I 1 c h i 4 + R c I 2 c h i 2 h c ) ∂ 2 ψ x c ∂ t 2 = 0 δ ψ θ i : − ∂ M θ θ i ∂ θ − R i ∂ M x θ i ∂ x + R i Q θ z i + ( ζ h i 4 − h i R c 2 h c ) Q θ z c + R i I 1 i ∂ 2 v 0 i ∂ t 2 + R i I 2 i ∂ 2 ψ θ i ∂ t 2 − ( ζ R c I 0 c h i 4 + R c I 1 c h i 2 h c ) ∂ 2 v 0 c ∂ t 2 − ( ζ R c I 1 c h i 4 + R c I 2 c h i 2 h c ) ∂ 2 ψ θ c ∂ t 2 = 0

(20)

where i = t , b , i = t → ζ = + 1 , i = b → ζ = − 1 and inertia terms are

I j i = ∑ k = 1 N L ∫ z k z k + 1 ρ k ( 1 + z i R i ) z i j d z → i = t , b j = 0,1,2 I j c = ∫ − h c 2 h c 2 ρ c ( 1 + z c R c ) z c j d z → j = 0,1,2

Additionally, following boundary conditions are derived:

At x = 0 , L :

N x x i o r δ u 0 i = 0 N x θ i o r δ v 0 i = 0 ( Q x z i R i + ζ R c h c S x z c + R c 2 Q x z c ) o r δ w 0 i = 0 M x x i o r δ ψ x i = 0 M x θ i o r δ ψ θ i = 0

(21)

And at θ = 0 , θ 0 :

N θ x i o r δ u 0 i = 0 N θ θ i o r δ v 0 i = 0 ( Q θ z i + ζ 1 h c S x z c + 1 2 Q θ z c ) o r δ w 0 i = 0 M θ x i o r δ ψ x i = 0 M θ θ i o r δ ψ θ i = 0

(22)

From equations (21) and (22) relation for different edges boundary conditions are expressed as:

Simply-Supported

x = 0 , L N x x i = v 0 i = w 0 i = M x x i = ψ θ i = 0 θ = 0 , θ 0 N θ θ i = u 0 i = w 0 i = M θ θ i = ψ x i = 0

(23)

Clamped

x = 0 , L u 0 i = v 0 i = w 0 i = ψ x i = ψ θ i = 0 θ = 0 , θ 0 u 0 i = v 0 i = w 0 i = ψ x i = ψ θ i = 0

(24)

Free

x = 0 , L N x x i = N x θ i = ( Q x z i R i + ζ R c h c S x z c + R c 2 Q x z c ) = M x x i = M x θ i = 0 θ = 0 , θ 0 N θ θ i = N θ x i = ( Q θ z i + ζ 1 h c S x z c + 1 2 Q θ z c ) = M θ θ i = M θ x i = 0

(25)

Finally, by using equations (9), (16) and (17) in Equation (20), equations of motion are obtained in terms of unknown displacement of facing sheets as follow:

[ D ] { δ } = 0

(26)

where D is a 10 × 10 coefficient matrix and δ is displacements vector (see appendix B).

3. Solution procedure

Due to the nature of harmonic vibration of sandwich cylindrical panel, displacement components are expressed as follows via using separation of variable technique

δ ( x , θ , t ) = δ ( x , θ ) e i λ t

(27)

3.1. Analytical solution

Following Fourier series expansion of displacement components along the axial and circumferential coordinates satisfy relation for simply supported edges, equation (23)

u 0 i ( x , θ ) = ∑ m = 1 ∞ ∑ n = 1 ∞ U m n i cos ( α m x ) sin ( β n θ ) v 0 i ( x , θ ) = ∑ m = 1 ∞ ∑ n = 1 ∞ V m n i sin ( α m x ) cos ( β n θ ) w 0 i ( x , θ ) = ∑ m = 1 ∞ ∑ n = 1 ∞ W m n i sin ( α m x ) sin ( β n θ ) ψ x i ( x , θ ) = ∑ m = 1 ∞ ∑ n = 1 ∞ ψ x m n i cos ( α m x ) sin ( β n θ ) ψ θ i ( x , θ ) = ∑ m = 1 ∞ ∑ n = 1 ∞ ψ θ m n i sin ( α m x ) cos ( β n θ )

(28)

In which α m = m π L a n d β n = n π θ 0 . By substituting equation (28) into equation (26) results in the following system of homogeneous equation

( [ K ] − λ 2 [ M ] ) { δ } = 0

(29)

where [K] and [M] are stiffness and mass matrices, respectively, (See appendix C). Nontrivial solution to equation (27) leads to the following system of eigenvalue equations

| [ K ] − λ 2 [ M ] | = 0

(30)

Solving equation (30) results in the natural frequencies and modal loss factor as follows

N a t u r a l f r e q u e n c y ( ω ) = r e a l ( λ 2 ) M L F = i m a g e ( λ 2 ) r e a l ( λ 2 )

(31)

3.2. Generalized differential quadrature

In the case of other edges boundary conditions, equations of motions are solved numerically by using GDQ.

In GDQ, partial derivative of any arbitrary function F (x, θ ) at a sample point (x, θ ) = (x_i, θ _j) is substitute by summation of weighting coefficients multiplied by the function values at sampling points in the domain of function variables

( ∂ n + m F ∂ x n ∂ θ m ) ( x , θ ) = ( x i , θ i ) = ∑ p = 1 N x ∑ q = 1 N θ A i p ( n ) B j q ( m ) F ( x n , θ m ) { i = 1,2 , . . . , N x j = 1,2 , . . . , N θ }

(32)

In which, N x and N θ are the number of sample points along the axial and circumferential direction, also A⁽¹⁾ and B⁽¹⁾ are the first weighting coefficients and written as

A i j ( 1 ) = { ∏ k = 1 k ≠ i , j N x ( x i , x k ) / ∏ k = 1 k ≠ j N x ( x i , x k ) i , j = 1,2 , . . . , N x ; i ≠ j ∑ k = 1 k ≠ j N x 1 ( x i − x k ) i = j = 1,2 , . . . , N x B i j ( 1 ) = { ∏ k = 1 k ≠ i , j N θ ( θ i − θ k ) / ∏ k = 1 k ≠ j N θ ( θ i − θ k ) i , j = 1,2 , . . . , N θ ; i ≠ j ∑ k = 1 k ≠ j N θ 1 ( θ i − θ k ) i , j = 1,2 , . . . , N θ

(33)

And the weighting coefficients for higher derivative computed as

A i j ( m ) = { m ( A i i ( m − 1 ) A i j ( 1 ) − A i j ( m ) x i − x j ) i , j = 1,2 , . . . , N x ; i ≠ j − ∑ k = 1 k ≠ j N x A i k ( m ) i = j = 1,2 , . . . , N x B i j ( m ) = { m ( B i i ( m − 1 ) B i j ( 1 ) − B i j ( m ) θ i − θ j ) i , j = 1,2 , . . . , N x ; i ≠ j − ∑ k = 1 k ≠ j N x B i k ( m ) i = j = 1,2 , . . . , N x

(34)

In this study, Chebyshev–Gauss–Lobatto grid distribution is selected to discretize the domain in x and θ directions as follows

x i = L 2 ( 1 − Cos ( i − 1 ) π N x − 1 ) θ i = θ 0 2 ( 1 − Cos ( i − 1 ) π N θ − 1 )

(35)

By applying GDQ to the equations of motion, following homogeneous matrix equation are derived

( [ K ¯ ] − λ 2 [ M ¯ ] ) { δ ¯ } = 0

(36)

where δ ¯ = { u ¯ 0 t , u ¯ 0 b , v ¯ 0 t , v ¯ 0 b , w ¯ 0 t , w ¯ 0 b , ψ ¯ x t , ψ ¯ x b , ψ ¯ θ t , ψ ¯ θ b } T u ¯ 0 t = { u 0 t ( x 1 , y 1 ) , . . . . , u 0 t ( x N x , y 1 ) , . . . . , u 0 t ( x N x , y N θ ) } T and K ¯ and M ¯ are stiffness and mass matrix, respectively, which can be derived by applying Equation (32) to Equation (26). It is noted that the other sub-vector in δ ¯ are similar to u ¯ 0 t .

Next, applying equation (32) to the boundary conditions gives the following matrix equation

[ T ] { δ } = 0

(37)

Here T is coefficient matrix which depends on the case of boundary conditions.

Assembling equations (34) and (35) leads to following homogeneous matrix equation

( [ K d d K d b T b d T b b ] − ω 2 [ M d d 0 0 0 ] ) { δ ¯ d δ ¯ b } = 0

(38)

In above relation, the subscripts ‘b’ and ‘d’ denote degrees of freedom corresponding to the boundary and domain, respectively. By elimination of the boundary nodes in above equation the standard eigenvalue problem can be obtain as

( [ Z ] + λ 2 [ M d d ] ) { δ ¯ d } = 0

(39)

In which, [ Z ] = [ K d d ] − [ K d b ] [ K b b ] − 1 [ K b d ] .

Nontrivial solution to equation (37) leads to the following eigenvalue matrix equations

| [ Z ] + λ 2 [ M d d ] | = 0

(40)

Solving equation (38) results in the natural frequencies and modal loss factor.