Consider a thin composite layered circular cylindrical shell with the length ℓ,the constant thickness h,and the radius r. Each layer is assumed to be homogeneous,linearly elastic,and isotropic or specially orthotropic. The x-coordinate of the shells is taken along the longitudinal direction,and the θ- and z-coordinates are in the circumferential direction and the radial direction,respectively. The equations of motion for the cylindrical shells are based on Love’s shell theory^{[16, 20]}. The equations of the cylindrical shells filled with a fluid are given as follows^[21]:

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where t is the time,p is the pressure,ρ is the density,and h is the thickness of the cylinder.

The stress resultants and the stress couples are

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where N_x,N_θ,and N_xθ are the stress resultants,M_x,M_θ,and M_xθ are the moment resultants,and σ_x,σ_θ,and τ_xθ are the stresses in the respective directions,respectively.

The strain-displacement relations of the circular cylindrical shells are

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where u,v,and w are the displacement functions of the midplane in the longitudinal,circumferential,and transverse directions,respectively.

For a thin laminated cylindrical shell,the stress-strain relations of the kth layer are defined by

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Substituting Eq. (3) into Eq. (4) and then substituting the obtained result into Eq. (2),we can obtain the force and moment resultants as follows:

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where A₁₁,A₁₂,A₂₂,A₆₆,B₁₁,B₁₂,B₂₂, B₆₆,and D₁₁,D₁₂,D₂₂,D₆₆ are the extensional rigidities,the bending-stretching coupling rigidities,and the bending rigidities defined by

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z_k is the distance from the midsurface to the surface of the kth layer. For a thin shell,Q₁₁^k ,Q₁₂^k,Q₁₂^k,and Q₆₆^k are reduced to the stiffnesses defined by

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Substituting Eq. (5) into Eq. (1) yields the equations of equilibrium in terms of the u,v,and w displacements as follows:

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where L_ij (i,j = 1,2,3) are differential operators given in Appendix A.

The displacement components u,v,and w are assumed as follows:

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where x and θ are the longitudinal coordinate and the rotational coordinate,respectively,ω is the angular frequency of the vibration,t is the time,and n is the circumferential node number.

The fluid is assumed to be incompressible. The irrotational flow of an inviscid fluid undergoing small oscillations is expressed as a wave equation. The equation of motion of the fluid can be written in the cylindrical coordinate system (x,θ,r) as follows^{[1, 2]}:

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where t is the time,p is the pressure,and c is the sound speed of the fluid. The x- and θ- coordinates are the same as those of the shell,while the r-coordinate is taken from the x-axis of the shell.

The associated form of the pressure field in the contained fluid,satisfying Eq. (10),is assumed to be

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where J_n is the Bessel function of the order n.

The fluid radial displacement and the shell radial displacement must be equal at the interface of the shell inner wall and the fluid in order to ensure that the fluid remains in contact with the shell wall. Then,the coupling condition is^{[1, 2]}

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Therefore,we have

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where ρ_f is the density of the contained fluid,and the prime on J_n denotes differentiation with respect to the argument r.

Introduce the following non-dimensional parameters:

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where L is the length,X is the distance coordinate,R is the radius,H is the thickness,λ is the frequency,and δ_k is the relative thickness of the kth layer.

Substituting Eq. (9) into Eq. (8) together with Eq. (11) and then using the non-dimensional parameters,we can obtain the equation of motion of the coupled system in the symmetric form as follows:

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where L_ij (i,j = 1,2,3) are the differential operators given in Appendix B.

Since the third column of Eq. (15) contains the derivatives of the third-order in U,the form of Eq. (15) is not convenient to the solution procedure we propose to adopt. Therefore,the equations are combined within themselves,and a modified set of equations are derived. To modify the equations,the first column of Eq. (15) is differentiated with respect to X and used to eliminate U^′′′(X) in the third equation. The modified set of equations are given by

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where L₃₁^* ,L₃₂^*,and L₃₃^* are the updated differential operators given in Appendix C.

The spline approximation is a lower order approximation,which yields a better accuracy than a global higher order approximation. Bickley^[19] successfully tested the spline collocation method over a two point boundary value problem with cubic splines. The displacement functions U(X),V (X),and W(X) are approximated by the cubic and quintic spline functions U^*(X),V^*(X),and W^*(X) as follows:

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where H(X −X_j) is the Heaviside step function,and N is the number of intervals in the range of X ∈ ^{[0, 1]}. The division points

are chosen as the knots of the splines as well as the collocation points. Imposing the condition that the differential equations given by Eq. (16) are satisfied by these splines at the knots,we can obtain a set of 3N +3 homogeneous equations into 3N +15 unknown spline coefficients a_i,b_j,c_i,d_j , e_i,and f_j (i = 0,1,2,3,4; j = 0,1,2,· · · ,N − 1).

The following boundary conditions are used to analyze the problem:

(i) C-C

(ii) S-S

From any of the boundary conditions,we can obtain eight more equations on the spline coefficients. Combining these eight equations with the earlier 3N + 3 homogeneous equations,we can get 3N + 15 homogeneous equations with the same number unknowns. Thus,these equations can be reduced to a system,which can be written as follows:

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where M and P are square matrices,q is a column matrix of the spline coefficients,and λ is the frequency parameter. This is a generalized eigenvalue problem.

The convergence study for the frequency parameter is carried out under the C-C boundary condition. The values of N are started from

onwards. From Table 1,we can see that the choice of

Table 1 Convergence study for two-layered cylindrical shells filled with fluid under C-C boundary condition )

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is sufficient,since the computed values of λ improve with the increase in N,but the improvement come down steadily when

Comparisons of the results are made in order to verify the efficiency of the present method. Table 2 shows a non-dimensional frequency for the isotropic cylindrical shells filled with a fluid under the C-C boundary condition. The results are compared with those in Refs. ^[1] and ^[2]. From Table 2,we can see that the present results agree well with those obtained in Refs. ^[1] and ^[2]. Some values in our case are a little higher. This may be due to the different material properties and the different adopted method.

Table 2 Comparison of frequency (Hz) for isotropic cylindrical shells filled with fluid under C-C boundary condition, where L = 20, R = 1, and h = 0.01 )

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In Table 3,the used parametric values are

Table 3 Comparison of frequency (Hz) for cylindrical shells under C-C boundary condition )

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The natural frequencies under the C-C boundary condition for

are compared with those obtained in Refs. ^[22],^[23],and ^[24] (see Table 3).

In this work,the frequency parameter is analyzed for two layered cylindrical shells filled with a fluid. The considered materials are HSG,SGE,and PRD. The variations of the frequency parameters with respect to the thickness ratio (δ),the length of cylinder (L),the thickness of cylinder (H),and the circumferential node number (n) under the C-C and S-S boundary conditions are studied.

The variations of the frequency parameters λ_m (m = 1,2,3) with respect to the ratio of the thickness of the inner layer to the entire thickness δ under the C-C boundary condition and HSG-PRD are depicted in Fig. 1. From Fig. 1,we can see that,when δ = 0,the shell becomes a homogeneous PRD shell; when δ = 1,the shell becomes a homogenous HSG shell; when 0 < δ < 1,both layers exist. From Fig. 1,we can see that λ₁,λ₂,and λ₃ increase with the increase in δ. However,when the value of δ is between 0 and 0.2,λ₂ and λ₃ increase rapidly,while λ₁ increases slowly.

Fig. 1 Variations of frequency parameters with respect to relative layer thickness under C-C bound- ary condition and HSG-PRD, where n = 2, and H = 0.02

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Figure 2 describes the variations of the frequency parameters λ₁,λ₂,and λ₃ with respect to the ratio of the thickness of the inner layer to the entire thickness δ (0 < δ < 1) under HSG-PRD and the C-C boundary condition. From Fig. 2,we can see that,when δ = 0,the shell becomes a homogeneous PRD shell; when δ = 1,the shell becomes a homogenous HSG shell; when 0 < δ < 1,both layers exist. It can be seen that,when δ increases from 0 to 0.2,λ₁ increases gradually,while λ₂ and λ₃ increase rapidly. From Fig. 2(a),we can see that λ₁,λ₂,and λ₃ increase slowly when 0.2 < δ < 1. The nature of the curves in Fig. 2(b) is almost identical to those in Fig. 2(a). It can be observed that the values of λ₁,λ₂,and λ₃ in Fig. 2(b) are smaller than those obtained in Fig. 2(a) when δ = 0.

Fig. 2 Variations of frequency parameters with respect to relative layer thickness under C-C bound- ary condition and HSG-PRD, where n = 2, and H = 0.05

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Figure 3 shows the variations of the frequency parameters λ₁,λ₂,and λ₃ with respect to the ratio of the thickness of the inner layer to the entire thickness δ under HSG-SGE and the C-C boundary condition. From Fig. 3,we can see that ,when δ = 0,the shell becomes a homogeneous SGE shell; when δ = 1,the shell becomes a homogenous HSG shell; when 0 < δ < 1,both layers exist. From Fig. 3(a),we can see that λ₁,λ₂,and λ₃ change with the increase in δ. λ₁ decreases slowly when 0 ≤ δ ≤ 0.6,and then increases gradually afterwards. λ₂ and λ₃ increase rapidly when 0 ≤ δ ≤ 0.2,decrease slowly when 0.2 ≤ δ ≤ 0.6,and increase gradually afterwards. From Fig. 3(b),we can see that λ₁,λ₂,and λ₃ change with the increases in δ. The pattern variations for λ₁ and λ₃ are similar to those obtained in Fig. 3(a). λ₂ slowly decreases when 0 ≤ δ ≤ 0.6,and gradually increases when δ > 0.6.

Fig. 3 Variations of frequency parameters with respect to relative layer thickness under C-C bound- ary condition and HSG-SGE, where n = 4, and H = 0.02

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Figure 4 corresponds to the effects of the length parameter L on ω₁,ω₂,and ω₃ of the cylindrical shells under HSG-SGE and the C-C boundary condition. Since λ_m is explicitly a function of the length of the cylinder,when studying the effect of the length of the cylinder on its vibrational behavior,the angular frequency ω_m (×10³ Hz) is considered instead of λ_m. From Fig. 4(a),we can see that ω₁,ω₂,and ω₃ decrease smoothly when L increases,and the decreases are fast when 0 < L < 0.75 while slow when 0.75 < L < 2. From Fig. 4(b),we can see that the pattern variations of ω₁,ω₂,and ω₃ are the same as those obtained in Fig. 4(a).

Fig. 4 Effects of length on angular frequency under C-C boundary condition and HSG-SGE, where H = 0.02, and δ = 0.4

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Figure 5 describes the effects of the thickness parameter H on λ₁,λ₂,and λ₃ under HSGSGE and the C-C boundary condition. From Fig. 5(a),we can see that λ₁,λ₂,and λ₃ increase slowly when 0.01 ≤ H ≤ 0.02. When 0.02 < H < 0.06,λ₃ increases rapidly,while λ₁ and λ₂ increase slowly. The values of λ_m (_m = 1,2,3) are higher for higher modes. From Fig. 5(b),we can see that the patterns of the variations of λ₁,λ₂,and λ₃ are similar to those obtained in Fig. 5(a).

Fig. 5 Effects of thickness parameter on frequency parameters under C-C boundary condition and HSG-SGE, where L = 1.25, and δ = 0.4

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The effects of the circumferential node number n on λ₁,λ₂,and λ₃ under the C-C boundary condition and HSG-SGE and HSG-PRD are shown in Figs. 6(a)-6(d). From Fig. 6(a),we can see that λ₁,λ₂,and λ₃ increase rapidly when 1 ≤ n ≤ 2. After that,λ₂ slowly decreases until n = 5,λ₁ slowly decreases when 1 ≤ n ≤ 5,while λ₃ increases gradually. When 5 < n < 6,λ₁,λ₂,and λ₃ increase rapidly. From Fig. 6(b),we can see that there are rapid increases in λ₂ and λ₃ when 1 ≤ n ≤ 2. After that,λ₂ increases slowly until n = 5. When 5 < n < 6,λ₁,λ₂,and λ₃ increase rapidly. From Fig. 6(c),we can see that the patterns of the variations of λ₁,λ₂,and λ₃ are similar to those obtained in Fig. 6(a). However,when n = 1,the values of λ₁,λ₂,and λ₃ are smaller than those in Fig. 6(a). This is because of the applied different combination of materials. The variation nature of λ₁,λ₂,and λ₃ in Fig. 6(d) are the same as those in Fig. 6(b). However,the values of λ₁,λ₂,and λ₃ when n = 1 are smaller than those in Fig. 6(b). This is due to the different combination of the materials.

Fig. 6 Variations of frequency parameters with respect to circumferential node number under C-C boundary condition and HSG-SGE or HSG-PRD

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Figure 7 shows the variations of the frequency parameters λ₁,λ₂,and λ₃ with respect to the ratio of the thickness of the inner layer to the entire thickness δ under the S-S boundary condition and HSG-PRD. From Fig. 7,we can see that,when δ = 0,the shell becomes a homogeneous PRD shell; when δ = 1,the shell becomes a homogenous HSG shell; when 0 < δ < 1,both layers exist. λ₁,λ₂,and λ₃ increase with the increase in δ. However,λ₂ and λ₃ increase faster than λ₁. It can also be observed that the frequencies for all modes in Fig. 7(b) are higher than those in Fig. 7(a).

Fig. 7 Variations of frequency parameters with respect to relative layer thickness under S-S boundary condition and HSG-PRD, where n = 2, and H = 0.02

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Figure 8 depicts the variations of λ₁,λ₂,and λ₃ with respect to the ratio of the thickness of the inner layer to the entire thickness δ under HSG-PRD and the S-S boundary condition. From Fig. 8,we can see that,when δ = 0,the shell becomes a homogeneous PRD shell; when δ = 1,the shell becomes a homogenous HSG shell; when 0 < δ < 1,both layers exist. It can be observed that,when 0 ≤ δ ≤ 0.2,λ₁ increases gradually,while λ₂ and λ₃ increase rapidly. After that,λ₁,λ₂,and λ₃ increase slowly when δ increases.

Fig. 8 Variations of frequency parameters with respect to relative layer thickness under C-C bound- ary condition and HSG-PRD material, where n = 2, and H = 0.05

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Figure 9 exhibits the variations of the frequency parameters λ₁,λ₂,and λ₃ with respect to the ratio of the thickness of the inner layer to the entire thickness δ under HSG-SGE and the S-S boundary condition. From Fig. 9,we can see that,when δ = 0,the shell becomes a homogeneous SGE shell; when δ = 1,the shell becomes a homogenous HSG shell; when 0 < δ < 1,both layers exist. From Fig. 9(a),we can see that λ₁,λ₂,and λ₃ change with the increase in δ. λ₁ increases gradually when 0 < δ < 0.2,and decreases slowly afterwards. λ₂ and λ₃ rapidly increase when 0 6 δ 6 0.2. λ₂ decreases slowly when 0.2 < δ < 0.8,and after that increases gradually. λ₃ rapidly decreases when 0.2 ≤ δ ≤ 0.8,while increases significantly afterwards. From Fig. 9(b),we can see that,λ₁,λ₂,and λ₃ change with the increase in δ. λ₁ decreases when 0 < δ < 1. λ₂ and λ₃ rapidly increase when 0 ≤ δ ≤ 0.2,then decrease slowly when 0.2 ≤ δ ≤ 0.8,and increase gradually when δ > 0.8.

Fig. 9 Variations of frequency parameters with respect to relative layer thickness under S-S boundary condition and HSG-SGE, where n = 4, and H = 0.02

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Figure 10 describes the variations of ω₁ (×10³ Hz),ω₂ (×10³ Hz),and ω₃ (×10³ Hz) of the cylindrical shells with respect to the length parameter under HSG-SGE and the S-S boundary condition. From Fig. 10,we can see that,when L increases,ω₁,ω₂,and ω₃ decrease smoothly. However,the decreases are fast when 0 < L < 0.75 while slow when 0.75 > L > 2. Besides,the values of the frequencies obtained under the S-S boundary condition are lower than those under the C-C boundary condition.

Fig. 10 Effects of length on angular frequency under S-S boundary condition and HSG-SGE, where H = 0.02, and δ = 0.4

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Figure 11 shows the variations of λ₁,λ₂,and λ₃ in the thickness parameter H under HSGSGE and the S-S boundary condition. From Fig. 11,we can see that λ₁,λ₂,and λ₃ increase smoothly when H increases. From Fig. 11(a),we can see that,λ₁ is almost constant for all values of H. λ₂ and λ₃ increase slowly when 0.01 ≤ H ≤ 0.02. After that,λ₃ increases rapidly when 0.02 < H < 0.06,while λ₂ increases slowly. From Fig. 11(b),we can see that,λ₁,λ₂,and λ₃ increase when H increases.

Fig. 11 Effects of thickness parameter on frequency parameters under S-S boundary condition and HSG-SGE, where n = 2, and δ = 0.4

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Figure 12 shows the effects of n on λ₁,λ₂,and λ₃ under HSG-SGE and the S-S boundary condition. From Fig. 12(a),we can see that λ₁ decreases while λ₂ and λ₃ increase rapidly when 1 ≤ n ≤ 2. After that,λ₁ and λ₂ slowly decrease until n = 5,while λ₃ increases gradually. When 5 < n < 6,λ₁,λ₂,and λ₃ increase rapidly. From Fig. 12(b),we can see that,λ₁ decreases while λ₂ and λ₃ increase when 1 ≤ n ≤ 2. After that,λ₁ and λ₂ slowly decrease until n = 5,λ₂ and λ₃ decrease first,then increase slowly,while λ₃ increases gradually. When 5 < n < 6,λ₁,λ₂,and λ₃ increase rapidly.

Fig. 12 Variations of frequency parameters with respect to circumferential node number under S-S boundary condition and HSG-SGE, where L = 1.25, and δ = 0.4

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The vibrations of the laminated cylindrical shells filled with a quiescent,incompressible,and inviscid fluid based on Love’s shell theory are studied. The Bickley-type spline is used to approximate the displacements,and the eigenvalue problem is solved for finding the frequency parameters. The spline method has very attractive characteristics for the computational work in terms of the convergence and accuracy,comparing with the finite element method and the wave propagation method^{[1, 2]}. Three types of materials are used in the analysis. The effects of the relative layer thickness,the length to thickness ratio,the length to radius ratio,and the circumferential node numbers on the frequency parameters of two layered cylindrical shells filled with a fluid under C-C and S-S boundary conditions are analyzed.

It is concluded that the fluid in the cylindrical shells strongly affects the frequency parameters. It is observed that the frequency parameters of the cylindrical shells filled with the fluid give lower values than the cylindrical shells without the fluid^[15]. The frequency parameter decreases when L increases. The effects of the thickness on the frequency parameter show that the frequency parameter increases when the thickness parameter increases. Besides,the frequency parameter increases when the circumferential node number increases. The values for all the frequencies obtained under the C-C boundary condition are higher than those under the S-S boundary condition. Therefore,it depends on the chosen material and parameters. The analysis can be extended to study other aspects such as shell structures with flowing fluids.