Application of Haar wavelet method for free vibration of laminated composite conical–cylindrical coupled shells with elastic boundary condition

The laminated structure was shown in many previous works; therefore, there is no need of further explanation here. The geometric and coordinate systems of laminated composite conical-cylindrical coupled shell are described in figure 1.

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The conical shell is composed of the (x_c , θ_c , r_c ) coordinate system. Here, x_c is the coordinate along the generator of the cone starting at its small edge, θ_c is the circumferential coordinate and r_c is the coordinate perpendicular to middle of the conical shell. The displacements of the conical shell with respect to this coordinate system are expressed by u_c , v_c and w_c in the x_c , θ_c and r_c directions respectively. φ is the semi-vertex angle of the cone, R₁ and R₂ are the radii of the cone at its small and large edges respectively, L_c is the cone length along its generator and h_.c. is the uniform thickness of the conical shell. The radius of the cone at any point along its length is defined as ${R}_{c}({x}_{c})={R}_{1}+{x}_{c}\,\sin \,\varphi .$ The cylindrical coordinate system (x_l , θ_l , r_l ) is used for the cylindrical shell, where x_l , θ_l and r_l represent the axial, circumferential and radial directions respectively. The u_l , v_l and w_l represent the deformations of the cylindrical shell in the x_l , θ_l , and r_l directions respectively. In both the conical and cylindrical shells, thickness is supposed to be negligible comparing to length or radii of curvature of the shell and the normal to the middle surface is assumed to be always straight and vertical to the middle surface. Here, the subscripts c and l represent the conical and cylindrical shells respectively. The both ends of coupled structure are supported with artificial boundary springs at along the five displacement directions, and the boundary condition can be generalized according to the stiffness of boundary springs. In the similar way, the continuity condition can be generalized by setting the artificial connective spring at the connected part of the conical and cylindrical shells.

According to the FSDT, the displacements at any point of the shell can be expressed by the displacement at the middle surface and the rotating component [37].

$$\begin{eqnarray}&&\begin{array}{l}{U}_{\zeta }\left(\alpha ,\beta ,z,t\right)={u}_{\zeta }\left(\alpha ,\beta ,t\right)+z{\phi }_{\alpha ,\zeta }\left(\alpha ,\beta ,t\right)\\ {V}_{\zeta }\left(\alpha ,\beta ,z,t\right)={v}_{\zeta }\left(\alpha ,\beta ,t\right)+z{\phi }_{\beta ,\zeta }\left(\alpha ,\beta ,t\right)\\ {W}_{\zeta }\left(\alpha ,\beta ,z,t\right)={w}_{\zeta }\left(\alpha ,\beta ,t\right),\end{array}\end{eqnarray} \tag{ 1 }$$

where subscript ζ = (c, l), represent the conical and cylindrical shells respectively, u, v and w indicate the displacement components in α, β and z directions, and φ_α and φ_β are the rotations of transverse normal along the β and α axis.

The deformation at any point of shell can be defined by the strains and curvature changes at the middle surface.

$$\begin{eqnarray}&&{\varepsilon }_{\alpha }={\varepsilon }_{\alpha }^{0}+z{k}_{\alpha },\,{\varepsilon }_{\beta }={\varepsilon }_{\beta }^{0}+z{k}_{\beta },{\gamma }_{\alpha \beta }={\gamma }_{\alpha \beta }^{0}+z{k}_{\alpha \beta },{\gamma }_{\alpha z}={\gamma }_{\alpha z}^{0},{\gamma }_{\beta z}={\gamma }_{\beta z}^{0}\end{eqnarray} \tag{ 2 }$$

where ${\varepsilon }_{\alpha }^{0},\,{\varepsilon }_{\beta }^{0},\,{\gamma }_{\alpha \beta }^{0}$ indicate the normal and shear strains in the middle surface, k_α , k_β , k_αβ are the curvature and twist strains, and ${\gamma }_{\alpha z}^{0},{\gamma }_{\beta z}^{0}$ are transverse shear strains[37].

The strains and curvature changes at the middle surface can be expressed as follows.

$$\begin{eqnarray}&&\begin{array}{l}{\varepsilon }_{\alpha }^{0}=\displaystyle \frac{1}{A}\displaystyle \frac{\partial u}{\partial \alpha }+\displaystyle \frac{v}{AB}\displaystyle \frac{\partial A}{\partial \beta }+\displaystyle \frac{w}{{R}_{\alpha }},\,{k}_{\alpha }=\displaystyle \frac{1}{A}\displaystyle \frac{\partial {\phi }_{\alpha }}{\partial \alpha }+\displaystyle \frac{{\phi }_{\beta }}{AB}\displaystyle \frac{\partial A}{\partial \beta }\\ {\varepsilon }_{\beta }^{0}=\displaystyle \frac{1}{B}\displaystyle \frac{\partial v}{\partial \beta }+\displaystyle \frac{u}{AB}\displaystyle \frac{\partial B}{\partial \alpha }+\displaystyle \frac{w}{{R}_{\beta }},\,{k}_{\beta }=\displaystyle \frac{1}{B}\displaystyle \frac{\partial {\phi }_{\beta }}{\partial \beta }+\displaystyle \frac{{\phi }_{\alpha }}{AB}\displaystyle \frac{\partial B}{\partial \alpha }\\ {\gamma }_{\alpha \beta }^{0}=\displaystyle \frac{A}{B}\displaystyle \frac{\partial }{\partial \beta }\left(\displaystyle \frac{u}{A}\right)+\displaystyle \frac{B}{A}\displaystyle \frac{\partial }{\partial \alpha }\left(\displaystyle \frac{v}{A}\right),\,{k}_{\alpha \beta }=\displaystyle \frac{A}{B}\displaystyle \frac{\partial }{\partial \beta }\left(\displaystyle \frac{{\phi }_{\alpha }}{A}\right)+\displaystyle \frac{B}{A}\displaystyle \frac{\partial }{\partial \alpha }\left(\displaystyle \frac{{\phi }_{\beta }}{B}\right)\\ {\gamma }_{\alpha z}^{0}=\displaystyle \frac{1}{A}\displaystyle \frac{\partial w}{\partial \alpha }-\displaystyle \frac{u}{{R}_{\alpha }}+{\phi }_{\alpha },\,{\gamma }_{\beta z}^{0}=\displaystyle \frac{1}{B}\displaystyle \frac{\partial w}{\partial \beta }-\displaystyle \frac{v}{{R}_{\beta }}+{\phi }_{\beta }\end{array}\end{eqnarray} \tag{ 3 }$$

The coordinates, characteristics of the Lamé parameters and radii of curvatures are as follows;

$$\begin{eqnarray}&&\begin{array}{l}{\rm{Cylindrical}}\,\mathrm{shell}:\alpha =x,\beta =\theta ,{R}_{\alpha }=\infty ,{R}_{\beta }=R,A=1,B=R\\ {\rm{Conical}}\,{\rm{shell}}:\alpha =x,\beta =\theta :{R}_{\alpha }=\infty ,{R}_{\beta }=x\,\tan \,\varphi :A=1,B=x\,\sin \,\varphi \end{array}\end{eqnarray} \tag{ 4 }$$

According to the generalized Hooke's law, at the kth layer of laminated shells, corresponding stress-strain relationship can be expressed as follows;

$$\begin{eqnarray}&&\left\{\begin{array}{l}{\sigma }_{\alpha ,\zeta }\\ {\sigma }_{\beta ,\zeta }\\ {\tau }_{\alpha \beta ,\zeta }\\ {\tau }_{\alpha z,\zeta }\\ {\tau }_{\beta z,\zeta }\end{array}\right\}=\left[\begin{array}{ccccc}{\bar{Q}}_{11}^{k} & {\bar{Q}}_{12,\zeta }^{k} & {\bar{Q}}_{16}^{k} & 0 & 0\\ {\bar{Q}}_{12}^{k} & {\bar{Q}}_{22}^{(k)} & {\bar{Q}}_{26}^{k} & 0 & 0\\ {\bar{Q}}_{16}^{k} & {\bar{Q}}_{26}^{k} & {\bar{Q}}_{66}^{k} & 0 & 0\\ 0 & 0 & 0 & {\bar{Q}}_{44}^{k} & {\bar{Q}}_{45}^{k}\\ 0 & 0 & 0 & {\bar{Q}}_{45}^{k} & {\bar{Q}}_{55}^{k}\end{array}\right]\left\{\begin{array}{l}{\varepsilon }_{\alpha ,\zeta }\\ {\varepsilon }_{\beta ,\zeta }\\ {\gamma }_{\alpha \beta ,\zeta }\\ {\gamma }_{\alpha z,\zeta }\\ {\gamma }_{\beta z,\zeta }\end{array}\right\}\end{eqnarray} \tag{ 5 }$$

where σ_α and σ_β are normal stresses, τ_αβ , τ_αz and τ_βz are shear stresses. ${\bar{Q}}_{ij}^{k}\left(i,j=1,2,4,5,6\right)$ is the stiffness coefficients of laminated layers and obtained from the following equation.

$$\begin{eqnarray}&&\begin{array}{l}{\bar{Q}}_{11}^{k}={Q}_{11}^{k}{m}^{4}+2\left({Q}_{12}^{k}+2{Q}_{66}^{k}\right){m}^{2}{n}^{2}+{Q}_{22}^{k}{n}^{4}\\ {\bar{Q}}_{12}^{k}=\left({Q}_{11}^{k}+{Q}_{22}^{k}-4{Q}_{66}^{k}\right){m}^{2}{n}^{2}+{Q}_{12}^{k}\left({m}^{4}+{n}^{4}\right),\,{\bar{Q}}_{13}^{k}={Q}_{13}^{k}{m}^{2}+{Q}_{23}^{k}{n}^{2}\\ {\bar{Q}}_{22}^{k}={Q}_{11}^{k}{n}^{4}+2\left({Q}_{12}^{k}+2{Q}_{66}^{k}\right){m}^{2}{n}^{2}+{Q}_{22}^{k}{m}^{4},\,{\bar{Q}}_{23}^{k}={Q}_{23}^{k}{m}^{2}+{Q}_{13}^{k}{n}^{2},{\bar{Q}}_{33}^{k}={Q}_{33}^{k}\\ {\bar{Q}}_{16}^{k}=\left({Q}_{11}^{k}-{Q}_{12}^{k}-2{Q}_{66}^{k}\right){m}^{3}n+\left({Q}_{12}^{k}-{Q}_{22}^{k}+2{Q}_{66}^{k}\right)m{n}^{3}\\ {\bar{Q}}_{26}^{k}=\left({Q}_{11}^{k}-{Q}_{12}^{k}-2{Q}_{66}^{k}\right)m{n}^{3}+\left({Q}_{12}^{k}-{Q}_{22}^{k}+2{Q}_{66}^{k}\right){m}^{3}n,\,{\bar{Q}}_{36}^{k}={Q}_{13}^{k}mn-{Q}_{23}^{k}mn\\ {\bar{Q}}_{66}^{k}=\left({Q}_{11}^{k}+{Q}_{22}^{k}-2{Q}_{12}^{k}-2{Q}_{66}^{k}\right){m}^{2}{n}^{2}+{Q}_{66}^{k}\left({m}^{4}+{n}^{4}\right)\\ {\bar{Q}}_{44}^{k}={Q}_{44}^{k}{m}^{2}+{Q}_{55}^{k}{n}^{2},{\bar{Q}}_{45}^{k}=\left({Q}_{55}^{k}-{Q}_{44}^{k}\right)mn,\,{\bar{Q}}_{55}^{k}={Q}_{55}^{k}{m}^{2}+{Q}_{44}^{k}{n}^{2}\\ m=\,\cos \,{\alpha }_{fiber}^{k},\,n=\,\sin \,{\alpha }_{fiber}^{k}\end{array}\end{eqnarray} \tag{ 6 }$$

where, ${\alpha }_{fiber}^{k}$ is the fiber direction angle and indicates the angle between the principle and radius directions of the kth layer. ${Q}_{ij}^{k}$ is the reduced elasticity coefficient at the kth layer and expressed as follows [37].

$$\begin{eqnarray}&&\begin{array}{l}{Q}_{11}^{k}=\displaystyle \frac{{E}_{11}^{k}}{1-{\mu }_{12}^{k}{\mu }_{21}^{k}},\,{Q}_{22}^{k}=\displaystyle \frac{{E}_{22}^{k}}{1-{\mu }_{12}^{k}{\mu }_{21}^{k}},\,{Q}_{12}^{k}=\displaystyle \frac{{\mu }_{12}^{k}{E}_{22}^{k}}{1-{\mu }_{12}^{k}{\mu }_{21}^{k}}\\ \,{Q}_{44}^{k}={G}_{23}^{k},\,{Q}_{55}^{k}={G}_{13}^{k},\,{Q}_{66}^{k}={G}_{12}^{k}\end{array}\end{eqnarray} \tag{ 7 }$$

where G₁₁, G₁₃ and G₂₃ indicate the shear moduli. E₁₁ and E₂₂ represent the effective Young's moduli of laminated composite shells in the principle material coordinates. μ₁₂ and μ₂₁ are Poisson's ratios. If the stresses are integrated with respect to shear, the force and moment results are obtained as follows.

$$\begin{eqnarray}&&\begin{array}{l}\left({N}_{\alpha ,\zeta },{N}_{\beta ,\zeta },{N}_{\alpha \beta ,\zeta },{M}_{\alpha ,\zeta },{M}_{\beta ,\zeta },{M}_{\alpha \beta ,\zeta }\right)=\displaystyle {\int }_{-h/2}^{h/2}\left({\sigma }_{\alpha ,\zeta },{\sigma }_{\beta ,\zeta },{\tau }_{\alpha \beta ,\zeta },z{\sigma }_{\alpha ,\zeta },z{\sigma }_{\beta ,\zeta },z{\tau }_{\alpha \beta ,\zeta }\right)dz\\ \,\left({Q}_{\alpha ,\zeta },{Q}_{\beta ,\zeta }\right)=\kappa \displaystyle {\int }_{-h/2}^{h/2}\left({\tau }_{\alpha z,\zeta },{\tau }_{\beta z,\zeta }\right)dz\end{array}\end{eqnarray} \tag{ 8 }$$

Also, the stress-strain relationship is written in the matrix form as follows.

$$\begin{eqnarray}&&\begin{array}{l}\left\{\begin{array}{l}{N}_{\alpha ,\zeta }\\ {N}_{\beta ,\zeta }\\ {N}_{\alpha \beta ,\zeta }\\ {M}_{\alpha ,\zeta }\\ {M}_{\beta ,\zeta }\\ {M}_{\alpha \beta ,\zeta }\end{array}\right\}\,=\,\left[\begin{array}{cc}\begin{array}{ccc}{A}_{11} & {A}_{12} & {A}_{16}\\ {A}_{12} & {A}_{22} & {A}_{26}\\ {A}_{16} & {A}_{26} & {A}_{66}\end{array} & \begin{array}{ccc}{B}_{11} & {B}_{12} & {B}_{16}\\ {B}_{12} & {B}_{22} & {B}_{26}\\ {B}_{16} & {B}_{26} & {B}_{66}\end{array}\\ \begin{array}{ccc}{B}_{11} & {B}_{12} & {B}_{16}\\ {B}_{12} & {B}_{22} & {B}_{26}\\ {B}_{16} & {B}_{26} & {B}_{66}\end{array} & \begin{array}{ccc}{D}_{11} & {D}_{12} & {D}_{16}\\ {D}_{12} & {D}_{22} & {D}_{26}\\ {D}_{16} & {D}_{26} & {D}_{66}\end{array}\end{array}\right]\left\{\begin{array}{l}{\varepsilon }_{\alpha ,\zeta }^{0}\\ {\varepsilon }_{\beta ,\zeta }^{0}\\ {\gamma }_{\alpha \beta ,\zeta }^{0}\\ {k}_{\alpha ,\zeta }\\ {k}_{\beta ,\zeta }\\ {k}_{\alpha \beta ,\zeta }\end{array}\right\},\\ \,\left\{\begin{array}{l}{Q}_{\alpha ,\zeta }\\ {Q}_{\beta ,\zeta }\end{array}\right\}\,=\,\kappa \left[\begin{array}{cc}{A}_{55} & {A}_{45}\\ {A}_{45} & {A}_{44}\end{array}\right]\left\{\begin{array}{l}{\gamma }_{\alpha z,\zeta }^{0}\\ {\gamma }_{\beta z,\zeta }^{0}\end{array}\right\}\end{array}\end{eqnarray} \tag{ 9 }$$

where, κ is the modified shear coefficient and set as 5/6 in this paper. A_ij , B_ij and D_ij are stretching, stretching -bending and bending coefficients respectively and expressed as follows.

$$\begin{eqnarray}&&\begin{array}{l}{A}_{ij}=\displaystyle \sum _{k=1}^{N}{\bar{Q}}_{ij}^{k}\left({Z}_{k+1}-{Z}_{k}\right),\,\,\,\,\left(i,j=1,2,4,5,6\right)\\ {B}_{ij}=\displaystyle \frac{1}{2}\displaystyle \sum _{k=1}^{N}{\bar{Q}}_{ij}^{k}\left({Z}_{k+1}^{2}-{Z}_{k}^{2}\right),\,\,\,\,\left(i,j=1,2,6\right)\\ {D}_{ij}=\displaystyle \frac{1}{3}\displaystyle \sum _{k=1}^{N}{\bar{Q}}_{ij}^{k}\left({Z}_{k+1}^{3}-{Z}_{k}^{3}\right),\,\,\,\,\left(i,j=1,2,6\right)\end{array}\end{eqnarray} \tag{ 10 }$$

where, N indicates the total number of laminated layers, and the strain energy occurred by the vibration of laminated composite rotating shell is expressed as follows.

$$\begin{eqnarray}&&\begin{array}{l}{U}_{V,\zeta }=\displaystyle \frac{1}{2}\displaystyle {\int }_{\alpha }\displaystyle {\int }_{\beta }\left\{\begin{array}{l}{N}_{\alpha ,\zeta }{\varepsilon }_{\alpha ,\zeta }^{0}+{N}_{\beta ,\zeta }{\varepsilon }_{\beta ,\zeta }^{0}+{N}_{\alpha \beta ,\zeta }{\gamma }_{\alpha \beta ,\zeta }^{0}+{M}_{\alpha ,\zeta }{k}_{\alpha ,\zeta }\\ +{M}_{\beta ,\zeta }{k}_{\beta ,\zeta }+{M}_{\alpha \beta ,\zeta }{k}_{\alpha \beta ,\zeta }+{Q}_{\alpha ,\zeta }{\gamma }_{\alpha z,\zeta }^{0}+{Q}_{\beta ,\zeta }{\gamma }_{\beta z,\zeta }^{0}\end{array}\right\}ABd\beta d\alpha \\ \,=\,\displaystyle \frac{1}{2}\displaystyle {\int }_{\alpha }\displaystyle {\int }_{\beta }\left\{\begin{array}{l}{N}_{\alpha ,\zeta }\left(\displaystyle \frac{1}{A}\displaystyle \frac{\partial {u}_{\zeta }}{\partial \alpha }+\displaystyle \frac{{v}_{\zeta }}{AB}\displaystyle \frac{\partial A}{\partial \beta }+\displaystyle \frac{{w}_{\zeta }}{{R}_{\alpha }}\right)+{N}_{\beta ,\zeta }\left(\displaystyle \frac{1}{B}\displaystyle \frac{\partial {v}_{\zeta }}{\partial \beta }+\displaystyle \frac{{u}_{\zeta }}{AB}\displaystyle \frac{\partial B}{\partial \alpha }+\displaystyle \frac{{w}_{\zeta }}{{R}_{\beta }}\right)\\ \,+\,{N}_{\alpha \beta ,\zeta }\left(\displaystyle \frac{A}{B}\displaystyle \frac{\partial }{\partial \beta }\left(\displaystyle \frac{{u}_{\zeta }}{A}\right)+\displaystyle \frac{B}{A}\displaystyle \frac{\partial }{\partial \alpha }\left(\displaystyle \frac{{v}_{\zeta }}{A}\right)\right)+{M}_{\alpha ,\zeta }\left(\displaystyle \frac{1}{A}\displaystyle \frac{\partial {\phi }_{\alpha ,\zeta }}{\partial \alpha }+\displaystyle \frac{{\phi }_{\beta ,\zeta }}{AB}\displaystyle \frac{\partial A}{\partial \beta }\right)\\ \,+\,{M}_{\beta ,\zeta }\left(\displaystyle \frac{1}{B}\displaystyle \frac{\partial {\phi }_{\beta ,\zeta }}{\partial \beta }+\displaystyle \frac{{\phi }_{\alpha ,\zeta }}{AB}\displaystyle \frac{\partial B}{\partial \alpha }\right)+{M}_{\alpha \beta ,\zeta }\left(\displaystyle \frac{A}{B}\displaystyle \frac{\partial }{\partial \beta }\left(\displaystyle \frac{{\phi }_{\alpha ,\zeta }}{A}\right)+\displaystyle \frac{B}{A}\displaystyle \frac{\partial }{\partial \alpha }\left(\displaystyle \frac{{\phi }_{\beta ,\zeta }}{B}\right)\right)\\ \,+\,{Q}_{\alpha ,\zeta }\left(\displaystyle \frac{1}{A}\displaystyle \frac{\partial {w}_{\zeta }}{\partial \alpha }-\displaystyle \frac{{u}_{\zeta }}{{R}_{\alpha }}+{\phi }_{\alpha ,\zeta }\right)+{Q}_{\beta ,\zeta }\left(\displaystyle \frac{1}{B}\displaystyle \frac{\partial {w}_{\zeta }}{\partial \beta }-\displaystyle \frac{{v}_{\zeta }}{{R}_{\beta }}+{\phi }_{\beta ,\zeta }\right)\end{array}\right\}d\beta d\alpha \end{array}\end{eqnarray} \tag{ 11 }$$

The corresponding kinetic energy function is given as follows.

$$\begin{eqnarray}&&\begin{array}{l}{T}_{\zeta }=\displaystyle \frac{1}{2}\displaystyle {\int }_{\alpha }\displaystyle {\int }_{\beta }\displaystyle {\int }_{-h/2}^{h/2}\left[\rho \left(z\right)\left({{\dot{U}}_{\zeta }}^{2}+{{\dot{V}}_{\zeta }}^{2}+{{\dot{W}}_{\zeta }}^{2}\right)\right]ABdzd\beta d\alpha \\ \,=\displaystyle \frac{1}{2}\displaystyle {\int }_{\alpha }\displaystyle {\int }_{\beta }\left[\begin{array}{l}{\bar{I}}_{0}{\left(\displaystyle \frac{\partial {u}_{\zeta }}{\partial t}\right)}^{2}+{\bar{I}}_{0}{\left(\displaystyle \frac{\partial {v}_{\zeta }}{\partial t}\right)}^{2}+{\bar{I}}_{0}{\left(\displaystyle \frac{\partial {w}_{\zeta }}{\partial t}\right)}^{2}+2{\bar{I}}_{1}\left(\displaystyle \frac{\partial {u}_{\zeta }}{\partial t}\right)\left(\displaystyle \frac{\partial {\phi }_{\alpha ,\zeta }}{\partial t}\right)\\ +2{\bar{I}}_{1}\left(\displaystyle \frac{\partial {v}_{\zeta }}{\partial t}\right)\left(\displaystyle \frac{\partial {\phi }_{\beta ,\zeta }}{\partial t}\right)+{\bar{I}}_{2}{\left(\displaystyle \frac{\partial {\phi }_{\alpha ,\zeta }}{\partial t}\right)}^{2}+{\bar{I}}_{2}{\left(\displaystyle \frac{\partial {\phi }_{\beta ,\zeta }}{\partial t}\right)}^{2}\end{array}\right]ABd\beta d\alpha \end{array}\end{eqnarray} \tag{ 12 }$$

In equation (12), I₀, I₁, I₂ are the inertia item and defined as follows.

$$\begin{eqnarray}&&\begin{array}{l}{\bar{I}}_{0}={I}_{0}+\displaystyle \frac{{I}_{1}}{{R}_{\alpha }}+\displaystyle \frac{{I}_{1}}{{R}_{\beta }}+\displaystyle \frac{{I}_{2}}{{R}_{\alpha }{R}_{\beta }}\\ {\bar{I}}_{1}={I}_{1}+\displaystyle \frac{{I}_{2}}{{R}_{\alpha }}+\displaystyle \frac{{I}_{2}}{{R}_{\beta }}+\displaystyle \frac{{I}_{3}}{{R}_{\alpha }{R}_{\beta }}\\ {\bar{I}}_{2}={I}_{2}+\displaystyle \frac{{I}_{3}}{{R}_{\alpha }}+\displaystyle \frac{{I}_{3}}{{R}_{\beta }}+\displaystyle \frac{{I}_{4}}{{R}_{\alpha }{R}_{\beta }}\\ \left({I}_{0},{I}_{1},{I}_{2},{I}_{3},{I}_{4}\right)=\displaystyle \sum _{k=1}^{N}\displaystyle {\int }_{{z}_{k}}^{{z}_{k+1}}{\rho }^{k}\left(1,z,{z}^{2},{z}^{3},{z}^{4}\right)dz\end{array}\end{eqnarray} \tag{ 13 }$$

The potential energy stored in the boundary springs can be written as

$$\begin{eqnarray}&&\begin{array}{l}{U}_{c}=\displaystyle \frac{1}{2}\displaystyle {\int }_{0}^{2\pi }\displaystyle {\int }_{-h/2}^{h/2}{\left\{{k}_{ub,0}{u}_{c}^{2}+{k}_{vb,0}{v}_{c}^{2}+{k}_{wb,0}{w}_{c}^{2}+{k}_{\varphi b,0}{\phi }_{{a}_{c}}^{2}+{k}_{\theta b,0}{\phi }_{{\beta }_{c}}^{2}\right\}}_{x=0}Bd{\beta }_{c}dz\\ \,+\displaystyle \frac{1}{2}\displaystyle {\int }_{0}^{2\pi }\displaystyle {\int }_{-h/2}^{h/2}{\left\{{k}_{ub,1}{u}_{l}^{2}+{k}_{vb,1}{v}_{c}^{2}+{k}_{wb,1}{w}_{l}^{2}+{k}_{\varphi b,1}{\phi }_{{a}_{l}}^{2}+{k}_{\theta b,1}{\phi }_{{\beta }_{l}}^{2}\right\}}_{x=L}Bd{\beta }_{l}dz\end{array}\end{eqnarray} \tag{ 14 }$$

where, k_ub , k_vb , k_wb , k_φb and k_θb are stiffness of the boundary springs at two ends of the coupled structure. The strain energy stored in the connective spring at the combined part of conical and cylindrical shells are expressed as follows.

$$\begin{eqnarray}&&{U}_{c}=\displaystyle \frac{1}{2}\displaystyle {\int }_{0}^{2\pi }\displaystyle {\int }_{-h/2}^{h/2}\left\{\begin{array}{l}{k}_{uc}{\left[{u}_{l}-\left({u}_{c}\sin {\varphi }_{c}-{w}_{c}\cos {\varphi }_{c}\right)\right]}^{2}+{k}_{vc}{\left({v}_{l}-{v}_{c}\right)}^{2}\\ +{k}_{wc}{\left[{w}_{l}-\left({u}_{c}\cos {\varphi }_{c}+{w}_{c}\sin {\varphi }_{c}\right)\right]}^{2}+{k}_{\varphi c}{\left({\phi }_{{\alpha }_{l}}-{\phi }_{{\alpha }_{c}}\right)}^{2}+{k}_{\theta c}{\left({\phi }_{{\beta }_{l}}-{\phi }_{{\beta }_{c}}\right)}^{2}\end{array}\right\}Bd\beta dz\end{eqnarray} \tag{ 15 }$$

where, k_uc , k_vc , k_wc , k_φc and k_θc are stiffness of the connective springs at the combined plane of the coupled structure.

In this article, it is assumed that there exists no energy by the external force as the main purpose is to calculate the natural frequency of combinational structure.

The total Lagrangian functional (L) of the laminated composite conical-cylindrical coupled shell is written as follows.

$$\begin{eqnarray}&&L=T-{U}_{V}-{U}_{b}-{U}_{c}\end{eqnarray} \tag{ 16 }$$

Applying the Hamilton's principle, the constitution equation of the shell is obtained as follows.

$$\begin{eqnarray}&&\delta \displaystyle {\int }_{{t}_{0}}^{{t}_{1}}\left(T-{U}_{V}-{U}_{b}-{U}_{c}\right)dt=0\end{eqnarray} \tag{ 17 }$$

$$\begin{eqnarray}&&{N}_{\alpha }\displaystyle \frac{\partial B}{\partial \alpha }+B\displaystyle \frac{\partial {N}_{\alpha }}{\partial \alpha }+A\displaystyle \frac{\partial {N}_{\alpha \beta }}{\partial \beta }-{N}_{\beta }\displaystyle \frac{\partial B}{\partial \alpha }+\displaystyle \frac{AB}{{R}_{\alpha }}{Q}_{\alpha }=AB\left({\bar{I}}_{0}\displaystyle \frac{{\partial }^{2}u}{\partial {t}^{2}}+{\bar{I}}_{1}\displaystyle \frac{{\partial }^{2}{\phi }_{\alpha }}{\partial {t}^{2}}\right)\end{eqnarray} \tag{ 18.a }$$

$$\begin{eqnarray}&&A\displaystyle \frac{\partial {N}_{\beta }}{\partial \beta }+{N}_{\alpha \beta }\displaystyle \frac{\partial B}{\partial \alpha }+B\displaystyle \frac{\partial {N}_{\alpha \beta }}{\partial \alpha }+{N}_{\alpha \beta }\displaystyle \frac{\partial B}{\partial \alpha }+\displaystyle \frac{AB}{{R}_{\beta }}{Q}_{\beta }=AB\left({\bar{I}}_{0}\displaystyle \frac{{\partial }^{2}v}{\partial {t}^{2}}+{\bar{I}}_{1}\displaystyle \frac{{\partial }^{2}{\phi }_{\beta }}{\partial {t}^{2}}\right)\end{eqnarray} \tag{ 18.b }$$

$$\begin{eqnarray}&&-AB\left(\displaystyle \frac{{N}_{\alpha }}{{R}_{\alpha }}+\displaystyle \frac{{N}_{\beta }}{{R}_{\beta }}\right)+{Q}_{\alpha }\displaystyle \frac{\partial B}{\partial \alpha }+B\displaystyle \frac{\partial {Q}_{\alpha }}{\partial \alpha }+A\displaystyle \frac{\partial {Q}_{\beta }}{\partial \beta }=AB\left({\bar{I}}_{0}\displaystyle \frac{{\partial }^{2}w}{\partial {t}^{2}}\right)\end{eqnarray} \tag{ 18.c }$$

$$\begin{eqnarray}&&{M}_{\alpha }\displaystyle \frac{\partial B}{\partial \alpha }+B\displaystyle \frac{\partial {M}_{\alpha }}{\partial \alpha }+A\displaystyle \frac{\partial {M}_{\alpha \beta }}{\partial \beta }-{M}_{\beta }\displaystyle \frac{\partial B}{\partial \alpha }+AB{Q}_{\alpha }=AB\left({\bar{I}}_{1}\displaystyle \frac{{\partial }^{2}u}{\partial {t}^{2}}+{\bar{I}}_{2}\displaystyle \frac{{\partial }^{2}{\phi }_{\alpha }}{\partial {t}^{2}}\right)\end{eqnarray} \tag{ 18.d }$$

$$\begin{eqnarray}&&A\displaystyle \frac{\partial {M}_{\beta }}{\partial \beta }+{M}_{\alpha \beta }\displaystyle \frac{\partial B}{\partial \alpha }+B\displaystyle \frac{\partial {M}_{\alpha \beta }}{\partial \alpha }+{M}_{\alpha \beta }\displaystyle \frac{\partial B}{\partial \alpha }+AB{Q}_{\beta }=AB\left({\bar{I}}_{0}\displaystyle \frac{{\partial }^{2}v}{\partial {t}^{2}}+{\bar{I}}_{1}\displaystyle \frac{{\partial }^{2}{\phi }_{\beta }}{\partial {t}^{2}}\right)\end{eqnarray} \tag{ 18.e }$$

The constitution equations of individual rotating shells are expressed in matrix form as follows.

$$\begin{eqnarray}&&\left\{\left[\begin{array}{l}{L}_{11}\,{L}_{12}\,{L}_{13}\,{L}_{14}\,{L}_{15}\\ {L}_{21}\,{L}_{22}\,{L}_{23}\,{L}_{24}\,{L}_{25}\\ {L}_{31}\,{L}_{32}\,{L}_{33}\,{L}_{34}\,{L}_{35}\\ {L}_{41}\,{L}_{42}\,{L}_{43}\,{L}_{44}\,{L}_{45}\\ {L}_{51}\,{L}_{52}\,{L}_{53}\,{L}_{54}\,{L}_{55}\end{array}\right]-\left[\begin{array}{l}{G}_{11}\,{G}_{12}\,{G}_{13}\,{G}_{14}\,{G}_{15}\\ {G}_{21}\,{G}_{22}\,{G}_{23}\,{G}_{24}\,{G}_{25}\\ {G}_{31}\,{G}_{32}\,{G}_{33}\,{G}_{34}\,{G}_{35}\\ {G}_{41}\,{G}_{42}\,{G}_{43}\,{G}_{44}\,{G}_{45}\\ {G}_{51}\,{G}_{52}\,{G}_{53}\,{G}_{54}\,{G}_{55}\end{array}\right]\right\}\left[\begin{array}{l}u\\ v\\ w\\ {\phi }_{\alpha }\\ {\phi }_{\beta }\end{array}\right]=0\end{eqnarray} \tag{ 19 }$$

where, the detailed expression formula of coefficients L_ij , and G_ij can be found in appendix A.

The boundary condition equations at two ends of the conical-cylindrical coupled shell are obtained as follows:

$$\begin{eqnarray}&&{\rm{Left}}\,\mathrm{boundary}:\left\{\begin{array}{l}{N}_{\alpha ,c}-{k}_{ub,0}{u}_{c}=0\\ {N}_{\alpha \beta ,c}-{k}_{vb,0}{v}_{c}=0\\ {Q}_{\alpha ,c}-{k}_{wb,0}{w}_{c}=0\\ {M}_{\alpha ,c}-{k}_{\varphi b,0}{\phi }_{\alpha ,c}=0\\ {M}_{\alpha \beta ,c}-{k}_{\theta b,0}{\phi }_{\theta ,c}=0\end{array}\right.,{\rm{Right}}\,{\rm{boundary}}:\left\{\begin{array}{l}{N}_{\alpha .l}+{k}_{ub,1}{u}_{l}=0\\ {N}_{\alpha \beta ,l}+{k}_{vb,1}{v}_{l}=0\\ {Q}_{\alpha ,l}+{k}_{wb,1}{w}_{l}=0\\ {M}_{\alpha ,l}+{k}_{\varphi {b},1}{\phi }_{\alpha ,l}=0\\ {M}_{\alpha \beta ,l}+{k}_{\theta b,1}{\phi }_{\beta ,l}=0\end{array}\right.\end{eqnarray} \tag{ 20 }$$

Also, the continuity condition equations at the combined part of conical-cylindrical coupled shell are expressed as follows.

$$\begin{eqnarray}&&\begin{array}{l}{N}_{\alpha ,c}+{k}_{uc}\left[{u}_{c}-\left({u}_{l}\,\cos \,\varphi +{w}_{l}\,\sin \,\varphi \right)-\right]=0,\\ {N}_{\alpha \beta ,c}+{k}_{vc}\left({v}_{c}-{v}_{l}\right)=0,\\ {Q}_{\alpha ,c}+{k}_{wc}\left[{w}_{c}-\left({w}_{l}\,\cos \,\varphi -{u}_{l}\,\sin \,\varphi \right)\right]=0,:\,{\rm{at}}\,{\rm{conical}}\,{\rm{shell}}\\ {M}_{\alpha ,c}+{k}_{\varphi c}\left({\phi }_{\alpha ,c}-{\phi }_{\alpha ,l}\right)=0,\\ {M}_{\alpha \beta ,c}+{k}_{\theta c}\left({\phi }_{\beta ,c}-{\phi }_{\beta ,l}\right)=0\end{array}\end{eqnarray} \tag{ 21.a }$$

$$\begin{eqnarray}&&\begin{array}{l}{N}_{\alpha ,l}-{k}_{uc}\left[{u}_{l}-\left({u}_{c}\,\cos \,\varphi -{w}_{c}\,\sin \,\varphi \right)\right]=0,\\ {N}_{\alpha \beta ,l}-{k}_{vc}\left({v}_{l}-{v}_{c}\right)=0,\\ {Q}_{\alpha ,l}-{k}_{wc}\left[{w}_{l}-\left({u}_{c}\,\sin \,\varphi +{w}_{c}\,\cos \,\varphi \right)\right]=0,:\,{\rm{at}}\,{\rm{cylindrical}}\,{\rm{shell}}\\ {M}_{\alpha ,l}-{k}_{\varphi c}\left({\phi }_{\alpha ,l}-{\phi }_{\alpha ,c}\right)=0,\\ {M}_{\alpha \beta ,l}-{k}_{\theta c}\left({\phi }_{\beta ,l}-{\phi }_{\beta ,c}\right)=0\end{array}\end{eqnarray} \tag{ 21.b }$$

According to the refernece [68], the displacement and rotation components in the circumferential direction can be extended into the Fourier series by applying the separation of variables.

$$\begin{eqnarray}&&\begin{array}{l}{u}_{\zeta }\left(\alpha ,\beta ,t\right)={U}_{\zeta }\left(\alpha \right)\cos (n\theta ){e}^{i\omega t}\\ {v}_{\zeta }\left(\alpha ,\beta ,t\right)={V}_{\zeta }\left(\alpha \right)\sin \left(n\theta \right){e}^{i\omega t}\\ {w}_{\zeta }\left(\alpha ,\beta ,t\right)={W}_{\zeta }\left(\alpha \right)\cos \left(n\theta \right){e}^{i\omega t}\\ {\phi }_{\alpha ,\zeta }\left(\alpha ,\beta ,t\right)={{\rm{\Phi }}}_{\zeta }\left(\alpha \right)\cos \left(n\theta \right){e}^{i\omega t}\\ {\phi }_{\beta ,\zeta }\left(\alpha ,\beta ,t\right)={{\rm{\Theta }}}_{\zeta }\left(\alpha \right)\sin \left(n\theta \right){e}^{i\omega t}\end{array}\end{eqnarray} \tag{ 22 }$$

where, ω is angular frequency and non-negative integer n is number of waves of the corresponding mode. U(α), V(α), W(α), Φ(α)and Θ(α) are unknown variable function to determine.

Substituting equations (22) into (19) and performing some algebraic operations, the construction equation can be written in following form.

$$\begin{eqnarray}&&\begin{array}{l}{{\rm{\Psi }}}_{11,\zeta }^{0}{U}_{\zeta }+{{\rm{\Psi }}}_{11,\zeta }^{1}\displaystyle \frac{d{U}_{\zeta }}{d\alpha }+{{\rm{\Psi }}}_{11,\zeta }^{2}\displaystyle \frac{{d}^{2}{U}_{\zeta }}{d{\alpha }^{2}}+{{\rm{\Psi }}}_{12,\zeta }^{0}{V}_{\zeta }+{{\rm{\Psi }}}_{12,\zeta }^{1}\displaystyle \frac{d{V}_{\zeta }}{d\alpha }+{{\rm{\Psi }}}_{13,\zeta }^{0}{W}_{\zeta }+{{\rm{\Psi }}}_{13,\zeta }^{1}\displaystyle \frac{d{W}_{\zeta }}{d\alpha }\\ \,+\,{{\rm{\Psi }}}_{14,\zeta }^{0}{{\rm{\Phi }}}_{\zeta }+{{\rm{\Psi }}}_{14,\zeta }^{1}\displaystyle \frac{d{{\rm{\Phi }}}_{\zeta }}{d\alpha }+{{\rm{\Psi }}}_{14,\zeta }^{2}\displaystyle \frac{{d}^{2}{{\rm{\Phi }}}_{\zeta }}{d{\alpha }^{2}}+{{\rm{\Psi }}}_{15,\zeta }^{0}{{\rm{\Theta }}}_{\zeta }+{{\rm{\Psi }}}_{15,\zeta }^{1}\displaystyle \frac{d{{\rm{\Theta }}}_{\zeta }}{d\alpha }\\ \,+\,{\bar{I}}_{0}{\omega }^{2}{R}^{2}{U}_{\zeta }+{\bar{I}}_{1}{\omega }^{2}{R}^{2}{{\rm{\Phi }}}_{\zeta }=0\end{array}\end{eqnarray} \tag{ 23.a }$$

$$\begin{eqnarray}&&\begin{array}{l}{{\rm{\Psi }}}_{21,\zeta }^{0}{U}_{\zeta }+{{\rm{\Psi }}}_{21,\zeta }^{1}\displaystyle \frac{d{U}_{\zeta }}{d\alpha }+{{\rm{\Psi }}}_{22,\zeta }^{0}{V}_{\zeta }+{{\rm{\Psi }}}_{22,\zeta }^{1}\displaystyle \frac{d{V}_{\zeta }}{d\alpha }+{{\rm{\Psi }}}_{22,\zeta }^{2}\displaystyle \frac{{d}^{2}{V}_{\zeta }}{d{\alpha }^{2}}+{{\rm{\Psi }}}_{23,\zeta }^{0}{W}_{\zeta }\\ +\,{{\rm{\Psi }}}_{24,\zeta }^{0}{{\rm{\Phi }}}_{\zeta }+{{\rm{\Psi }}}_{24,\zeta }^{1}\displaystyle \frac{d{{\rm{\Phi }}}_{\zeta }}{d\alpha }+{{\rm{\Psi }}}_{25,\zeta }^{0}{{\rm{\Theta }}}_{\zeta }+{{\rm{\Psi }}}_{25,\zeta }^{1}\displaystyle \frac{d{{\rm{\Theta }}}_{\zeta }}{d\alpha }+{{\rm{\Psi }}}_{25,\zeta }^{2}\displaystyle \frac{{d}^{2}{{\rm{\Theta }}}_{\zeta }}{d{\alpha }^{2}}+{\bar{I}}_{0}{\omega }^{2}{R}^{2}{V}_{\zeta }+{\bar{I}}_{1}{\omega }^{2}{R}^{2}{{\rm{\Theta }}}_{\zeta }=0\end{array}\end{eqnarray} \tag{ 23.b }$$

$$\begin{eqnarray}&&\begin{array}{l}{{\rm{\Psi }}}_{31,\zeta }^{0}{U}_{\zeta }+{{\rm{\Psi }}}_{31,\zeta }^{1}\displaystyle \frac{d{U}_{\zeta }}{d\alpha }+{{\rm{\Psi }}}_{32,\zeta }^{0}{V}_{\zeta }+{{\rm{\Psi }}}_{33,\zeta }^{0}{W}_{\zeta }+{{\rm{\Psi }}}_{33,\zeta }^{1}\displaystyle \frac{d{W}_{\zeta }}{d\alpha }+{{\rm{\Psi }}}_{33,\zeta }^{2}\displaystyle \frac{{d}^{2}{W}_{\zeta }}{d{\alpha }^{2}}\\ \,+\,{{\rm{\Psi }}}_{34,\zeta }^{0}{{\rm{\Phi }}}_{\zeta }+{{\rm{\Psi }}}_{34,\zeta }^{1}\displaystyle \frac{d{{\rm{\Phi }}}_{\zeta }}{d\alpha }+{{\rm{\Psi }}}_{35,\zeta }^{0}{{\rm{\Theta }}}_{\zeta }+{I}_{0}{\omega }^{2}{R}^{2}{W}_{\zeta }=0\end{array}\end{eqnarray} \tag{ 23.c }$$

$$\begin{eqnarray}&&\begin{array}{l}{{\rm{\Psi }}}_{41,\zeta }^{0}{U}_{\zeta }+{{\rm{\Psi }}}_{41,\zeta }^{1}\displaystyle \frac{d{U}_{\zeta }}{d\alpha }+{{\rm{\Psi }}}_{41,\zeta }^{2}\displaystyle \frac{{d}^{2}{U}_{\zeta }}{d{\alpha }^{2}}+{{\rm{\Psi }}}_{42,\zeta }^{0}{V}_{\zeta }+{{\rm{\Psi }}}_{42,\zeta }^{1}\displaystyle \frac{d{V}_{\zeta }}{d\alpha }+{{\rm{\Psi }}}_{43,\zeta }^{0}{W}_{\zeta }+{{\rm{\Psi }}}_{43,\zeta }^{1}\displaystyle \frac{d{W}_{\zeta }}{d\alpha }\\ \,+\,{{\rm{\Psi }}}_{44,\zeta }^{0}{{\rm{\Phi }}}_{\zeta }+{{\rm{\Psi }}}_{44,\zeta }^{1}\displaystyle \frac{d{{\rm{\Phi }}}_{\zeta }}{d\alpha }+{{\rm{\Psi }}}_{44,\zeta }^{2}\displaystyle \frac{{d}^{2}{{\rm{\Phi }}}_{\zeta }}{d{\alpha }^{2}}+{{\rm{\Psi }}}_{45,\zeta }^{0}{{\rm{\Theta }}}_{\zeta }+{{\rm{\Psi }}}_{45,\zeta }^{1}\displaystyle \frac{d{{\rm{\Theta }}}_{\zeta }}{d\alpha }\\ \,+\,{\bar{I}}_{1}{\omega }^{2}{R}^{2}{U}_{\zeta }+{\bar{I}}_{2}{\omega }^{2}{R}^{2}{{\rm{\Phi }}}_{\zeta }=0\end{array}\end{eqnarray} \tag{ 23.d }$$

$$\begin{eqnarray}&&\begin{array}{l}{{\rm{\Psi }}}_{51,\zeta }^{0}{U}_{\zeta }+{{\rm{\Psi }}}_{51,\zeta }^{1}\displaystyle \frac{d{U}_{\zeta }}{d\alpha }+{{\rm{\Psi }}}_{52,\zeta }^{0}{V}_{\zeta }+{{\rm{\Psi }}}_{52,\zeta }^{1}\displaystyle \frac{d{V}_{\zeta }}{d\alpha }+{{\rm{\Psi }}}_{52,\zeta }^{2}\displaystyle \frac{{d}^{2}{V}_{\zeta }}{d{\alpha }^{2}}+{{\rm{\Psi }}}_{53,\zeta }^{0}{W}_{\zeta }\\ \,+\,{{\rm{\Psi }}}_{54,\zeta }^{0}{{\rm{\Phi }}}_{\zeta }+{{\rm{\Psi }}}_{54,\zeta }^{1}\displaystyle \frac{d{{\rm{\Phi }}}_{\zeta }}{d\alpha }+{{\rm{\Psi }}}_{55,\zeta }^{0}{{\rm{\Theta }}}_{\zeta }+{{\rm{\Psi }}}_{55,\zeta }^{1}\displaystyle \frac{d{{\rm{\Theta }}}_{\zeta }}{d\alpha }+{{\rm{\Psi }}}_{55,\zeta }^{2}\displaystyle \frac{{d}^{2}{{\rm{\Theta }}}_{\zeta }}{d{\alpha }^{2}}\\ \,+\,{\bar{I}}_{1}{\omega }^{2}{R}^{2}{V}_{\zeta }+{\bar{I}}_{2}{\omega }^{2}{R}^{2}{{\rm{\Theta }}}_{\zeta }=0\end{array}\end{eqnarray} \tag{ 23.e }$$

where, the detailed expression formula of coefficients ${{\rm{\Psi }}}_{ij,\zeta }^{k}$ can be found in appendix B.

In the similar way, the same forms of equation can be obtained with respect to boundary and continuity conditions.

For boundary conditions;

Left boundary:

$$\begin{eqnarray}&&\left\{\begin{array}{l}{A}_{11}{R}_{c}\displaystyle \frac{d{U}_{c}}{d\alpha }+\left({A}_{12}\,\sin \,\varphi -{k}_{ub,0}{R}_{c}\right){U}_{c}+{A}_{12}n{V}_{c}+{A}_{12}\,\cos \,\varphi {W}_{c}+{B}_{11}{R}_{c}\displaystyle \frac{d{{\rm{\Phi }}}_{c}}{d\alpha }+{B}_{12}\,\sin \,\varphi {{\rm{\Phi }}}_{c}+{B}_{12}{{\rm{\Theta }}}_{c}=0\\ -{A}_{66}n{U}_{c}+{A}_{66}{R}_{c}\displaystyle \frac{d{V}_{c}}{d\alpha }-\left({A}_{66}\,\sin \,\varphi +{k}_{vb,0}{R}_{c}\right){V}_{c}-{B}_{66}n{{\rm{\Phi }}}_{c}+{B}_{66}{R}_{c}\displaystyle \frac{d{{\rm{\Theta }}}_{c}}{d\alpha }-{B}_{66}\,\sin \,\varphi {{\rm{\Theta }}}_{c}=0\\ \kappa {A}_{55}{R}_{c}\displaystyle \frac{d{W}_{c}}{d\alpha }-{k}_{wb,0}{R}_{c}{W}_{c}+\kappa {A}_{55}{R}_{c}{{\rm{\Phi }}}_{c}=0\\ {B}_{11}{R}_{c}\displaystyle \frac{d{U}_{c}}{d\alpha }+{B}_{12}\,\sin \,\varphi {U}_{c}+{B}_{12}n{V}_{c}+{B}_{12}\,\cos \,\varphi {W}_{c}+{D}_{11}{R}_{c}\displaystyle \frac{d{{\rm{\Phi }}}_{c}}{d\alpha }+\left({D}_{12}\,\sin \,\varphi -{k}_{\varphi b,0}{R}_{c}\right){\rm{\Phi }}+{D}_{12}n{{\rm{\Theta }}}_{c}=0\\ -{B}_{66}n{U}_{c}+{B}_{66}{R}_{c}\displaystyle \frac{d{V}_{c}}{d\alpha }-{B}_{66}\,\sin \,\varphi {V}_{c}-{D}_{66}n{{\rm{\Phi }}}_{c}+{D}_{66}{R}_{c}\displaystyle \frac{d{{\rm{\Theta }}}_{c}}{d\alpha }-\left({D}_{66}\,\sin \,\varphi +{k}_{\theta b,0}{R}_{c}\right){{\rm{\Theta }}}_{c}=0\end{array}\right.\end{eqnarray} \tag{ 24.a }$$

Right boundary:

$$\begin{eqnarray}&&\left\{\begin{array}{l}{A}_{11}{R}_{l}\displaystyle \frac{d{U}_{l}}{d\alpha }+{k}_{ub,1}{R}_{l}{U}_{l}+{A}_{12}n{V}_{l}+{A}_{12}{W}_{l}+{B}_{11}{R}_{l}\displaystyle \frac{d{{\rm{\Phi }}}_{l}}{d\alpha }+{B}_{12}n{{\rm{\Theta }}}_{l}=0\\ -{A}_{66}n{U}_{l}+{A}_{66}{R}_{l}\displaystyle \frac{d{V}_{l}}{d\alpha }+{k}_{vb,1}{R}_{l}{V}_{l}-{B}_{66}n{{\rm{\Phi }}}_{l}+{B}_{66}{R}_{l}{{\rm{\Theta }}}_{l}=0\\ \kappa {A}_{55}{R}_{l}\displaystyle \frac{d{W}_{l}}{d\alpha }+{k}_{wb,1}{R}_{l}{W}_{l}+\kappa {A}_{55}{R}_{l}{{\rm{\Phi }}}_{l}=0\\ {B}_{11}{R}_{l}\displaystyle \frac{d{U}_{l}}{d\alpha }+{B}_{12}n{V}_{l}+{B}_{12}{W}_{l}+{D}_{11}{R}_{l}\displaystyle \frac{d{{\rm{\Phi }}}_{l}}{d\alpha }+{k}_{\varphi b,1}{R}_{l}{{\rm{\Phi }}}_{l}+{D}_{12}n{{\rm{\Theta }}}_{l}=0\\ -{B}_{66}n{U}_{l}+{B}_{66}{R}_{l}\displaystyle \frac{d{V}_{l}}{d\alpha }-{D}_{66}n{{\rm{\Phi }}}_{l}+{D}_{66}{R}_{l}\displaystyle \frac{d{{\rm{\Theta }}}_{l}}{d\alpha }+{k}_{\theta b,1}{R}_{l}{{\rm{\Theta }}}_{l}=0\end{array}\right.\end{eqnarray} \tag{ 24.b }$$

For continuity conditions

Conical shell:

$$\begin{eqnarray}&&\left\{\begin{array}{l}{A}_{11}{R}_{l}\displaystyle \frac{d{U}_{c}}{d\alpha }+{A}_{12}\,\sin \,\varphi {U}_{c}+{k}_{uc}{R}_{l}\left({U}_{c}-{U}_{l}\,\cos \,\varphi \right)+{A}_{12}n{V}_{c}\\ +{A}_{12}\,\cos \,\varphi {W}_{c}-{k}_{uc}{R}_{l}\,\sin \,\varphi {W}_{l}+{B}_{11}{R}_{l}\displaystyle \frac{d{{\rm{\Phi }}}_{c}}{d\alpha }+{B}_{12}\,\sin \,\varphi {{\rm{\Phi }}}_{c}+{B}_{12}{{\rm{\Theta }}}_{c}=0\\ -{A}_{66}n{U}_{c}+{A}_{66}{R}_{l}\displaystyle \frac{d{V}_{c}}{d\alpha }-{A}_{66}\,\sin \,\varphi {V}_{c}+{k}_{vc}{R}_{l}\left({V}_{c}-{V}_{l}\right)-{B}_{66}n{{\rm{\Phi }}}_{c}+{B}_{66}{R}_{l}\displaystyle \frac{d{{\rm{\Theta }}}_{c}}{d\alpha }-{B}_{66}\,\sin \,\varphi {{\rm{\Theta }}}_{c}=0\\ {k}_{wc}{R}_{l}\,\sin \,\varphi {U}_{l}+\kappa {A}_{55}{R}_{l}\displaystyle \frac{d{W}_{c}}{d\alpha }+{k}_{wc}{R}_{l}\left({W}_{c}-{W}_{l}\,\cos \,\varphi \right)+\kappa {A}_{55}{R}_{l}{{\rm{\Phi }}}_{c}=0\\ {B}_{11}{R}_{l}\displaystyle \frac{d{U}_{c}}{d\alpha }+{B}_{12}\,\sin \,\varphi {U}_{c}+{B}_{12}n{V}_{c}+{B}_{12}\,\cos \,\varphi {W}_{c}\\ +{D}_{11}{R}_{l}\displaystyle \frac{d{{\rm{\Phi }}}_{c}}{d\alpha }+{D}_{12}\,\sin \,\varphi {{\rm{\Phi }}}_{c}+{k}_{\varphi c}{R}_{l}\left({{\rm{\Phi }}}_{c}-{{\rm{\Phi }}}_{l}\right)+{D}_{12}n{{\rm{\Theta }}}_{c}=0\\ -{B}_{66}n{U}_{c}+{B}_{66}{R}_{l}\displaystyle \frac{d{V}_{c}}{d\alpha }-{B}_{66}\,\sin \,\varphi {V}_{c}-{D}_{66}n{{\rm{\Phi }}}_{c}+{D}_{66}{R}_{l}\displaystyle \frac{d{{\rm{\Theta }}}_{c}}{d\alpha }-{D}_{66}\,\sin \,\varphi +{k}_{\theta c}{R}_{l}\left({{\rm{\Theta }}}_{c}-{{\rm{\Theta }}}_{l}\right)=0\end{array}\right.\end{eqnarray} \tag{ 25.a }$$

Cylindrical shell:

$$\begin{eqnarray}&&\left\{\begin{array}{l}{A}_{11}{R}_{l}\displaystyle \frac{d{U}_{l}}{d\alpha }-{k}_{uc}{R}_{l}\left({U}_{l}-{U}_{c}\right)+{A}_{12}n{V}_{l}+{A}_{12}{W}_{l}-{k}_{uc}{R}_{l}\,\sin \,\varphi {W}_{c}+{B}_{11}{R}_{l}\displaystyle \frac{d{{\rm{\Phi }}}_{l}}{d\alpha }+{B}_{12}n{{\rm{\Theta }}}_{l}=0\\ -{A}_{66}n{U}_{l}+{A}_{66}{R}_{l}\displaystyle \frac{d{V}_{l}}{d\alpha }-{k}_{vc}{R}_{l}\left({V}_{l}-{V}_{c}\right)-{B}_{66}n{{\rm{\Phi }}}_{l}+{B}_{66}{R}_{l}{{\rm{\Theta }}}_{l}=0\\ {k}_{wc}{R}_{l}\,\sin \,\varphi {U}_{c}+\kappa {A}_{55}{R}_{l}\displaystyle \frac{d{W}_{l}}{d\alpha }-{k}_{wc}{R}_{l}\left({W}_{l}-{W}_{c}\,\cos \,\varphi \right)+\kappa {A}_{55}{R}_{l}{{\rm{\Phi }}}_{l}=0\\ {B}_{11}{R}_{l}\displaystyle \frac{d{U}_{l}}{d\alpha }+{B}_{12}n{V}_{l}+{B}_{12}{W}_{l}+{D}_{11}{R}_{l}\displaystyle \frac{d{{\rm{\Phi }}}_{l}}{d\alpha }-{k}_{\varphi c}{R}_{l}\left({{\rm{\Phi }}}_{l}-{{\rm{\Phi }}}_{c}\right)+{D}_{12}n{{\rm{\Theta }}}_{l}=0\\ -{B}_{66}n{U}_{l}+{B}_{66}{R}_{l}\displaystyle \frac{d{V}_{l}}{d\alpha }-{D}_{66}n{{\rm{\Phi }}}_{l}+{D}_{66}{R}_{l}\displaystyle \frac{d{{\rm{\Theta }}}_{l}}{d\alpha }+{k}_{\theta c}{R}_{l}\left({{\rm{\Theta }}}_{l}-{{\rm{\Theta }}}_{c}\right)=0\end{array}\right.\end{eqnarray} \tag{ 25.b }$$

Haar wavelet consisting of a group of square waves has size of +1and −1 in some intervals and has zeros in elsewhere, and it is known one of the simplest compactly supported orthogonal wavelet among the wavelet families. The details of Haar wavelet series and their integrals can be found in the following references [67–69, 72–74].

Haar wavelet series are used to discretize the derivatives in entire governing equations including boundary conditions. The Haar wavelet series is defined in the interval [0, 1], therefore, to apply the HWDM, the linear transformation statute is used for the coordinate conversion from length interval [0, L] of the rectangular plate to the interval [0, 1] of the Haar wavelet series, that is,

$$\begin{eqnarray}&&\xi =\displaystyle \frac{x}{{L}_{x}}\end{eqnarray} \tag{ 26 }$$

In the HWDM, the highest order derivatives of the displacements are expressed using the Haar wavelet series and the lower order derivatives can be obtained by integrating Haar wavelet series. The highest order derivative of the displacements in the governing equations of motion and boundary condition expressions is second order, which can be obtained by means of the Haar wavelet series as follows;

$$\begin{eqnarray}&&\begin{array}{l}\displaystyle \frac{{d}^{2}U\left(\xi \right)}{d{\xi }^{2}}=\displaystyle \sum _{i=1}^{2M}{a}_{i}{h}_{i}\left(\xi \right),\,\displaystyle \frac{{d}^{2}V\left(\xi \right)}{d{\xi }^{2}}=\displaystyle \sum _{i=1}^{2M}{b}_{i}{h}_{i}\left(\xi \right),\,\displaystyle \frac{{d}^{2}W\left(\xi \right)}{d{\xi }^{2}}=\displaystyle \sum _{i=1}^{2M}{c}_{i}{h}_{i}\left(\xi \right)\\ \,\,\,\,\,\displaystyle \frac{{d}^{2}{\rm{\Phi }}(\xi )}{d{\xi }^{2}}=\displaystyle \sum _{i=1}^{2M}{d}_{i}{h}_{i}\left(\xi \right),\,\displaystyle \frac{{d}^{2}{\rm{\Theta }}\left(\xi \right)}{d{\xi }^{2}}=\displaystyle \sum _{i=1}^{2M}{e}_{i}{h}_{i}\left(\xi \right),\end{array}\end{eqnarray} \tag{ 27 }$$

where a_i , b_i , c_i and d_i are unknown coefficients of the Haar wavelets. The first order derivatives of displacements are calculated by integrating equation (27) and the the displacement functions can be obtained by integrating the above result again. The first order derivatives of displacements and displacement functions are written as follows.

$$\begin{eqnarray}&&\begin{array}{l}\displaystyle \frac{dU\left(\xi \right)}{d\xi }=\displaystyle \sum _{i=1}^{2M}{a}_{i}{P}_{1,i}\left(\xi \right)+\displaystyle \frac{dU\left(0\right)}{d\xi },\,U\left(\xi \right)=\displaystyle \sum _{i=1}^{2M}{a}_{i}{P}_{2,i}\left(\xi \right)+\xi \displaystyle \frac{dU\left(0\right)}{d\xi }+U\left(0\right)\\ \displaystyle \frac{dV\left(\xi \right)}{d\xi }=\displaystyle \sum _{i=1}^{2M}{a}_{i}{P}_{1,i}\left(\xi \right)+\displaystyle \frac{dV\left(0\right)}{d\xi },\,V\left(\xi \right)=\displaystyle \sum _{i=1}^{2M}{b}_{i}{P}_{2,i}\left(\xi \right)+\xi \displaystyle \frac{dV\left(0\right)}{d\xi }+V\left(0\right)\\ \displaystyle \frac{dW\left(\xi \right)}{d\xi }=\displaystyle \sum _{i=1}^{2M}{c}_{i}{P}_{1,i}\left(\xi \right)+\displaystyle \frac{dW\left(0\right)}{d\xi },\,W\left(\xi \right)=\displaystyle \sum _{i=1}^{2M}{c}_{i}{P}_{2,i}\left(\xi \right)+\xi \displaystyle \frac{dW\left(0\right)}{d\xi }+W\left(0\right)\\ \displaystyle \frac{d{\rm{\Phi }}(\xi )}{d\xi }=\displaystyle \sum _{i=1}^{2M}{d}_{i}{P}_{1,i}\left(\xi \right)+\displaystyle \frac{d{\rm{\Phi }}\left(0\right)}{d\xi },\,{\rm{\Phi }}\left(\xi \right)=\displaystyle \sum _{i=1}^{2M}{d}_{i}{P}_{2,i}\left(\xi \right)+\xi \displaystyle \frac{d{\rm{\Phi }}\left(0\right)}{d\xi }+{\rm{\Phi }}\left(0\right)\\ \displaystyle \frac{d{\rm{\Theta }}\left(\xi \right)}{d\xi }=\displaystyle \sum _{i=1}^{2M}{e}_{i}{P}_{1,i}\left(\xi \right)+\displaystyle \frac{d{\rm{\Theta }}\left(0\right)}{d\xi },\,{\rm{\Theta }}\left(\xi \right)=\displaystyle \sum _{i=1}^{2M}{e}_{i}{P}_{2,i}\left(\xi \right)+\xi \displaystyle \frac{d{\rm{\Theta }}\left(0\right)}{d\xi }+{\rm{\Theta }}\left(0\right)\end{array}\end{eqnarray} \tag{ 28 }$$

Equations (27) and (28) can be expressed in the discretized matrix form as follows:

$$\begin{eqnarray}&&\begin{array}{l}\displaystyle \frac{{d}^{2}{\boldsymbol{U}}}{d{\xi }^{2}}={\boldsymbol{Ha}}+{{\boldsymbol{H}}}_{11}{\boldsymbol{f}},\,\displaystyle \frac{d{\boldsymbol{U}}}{d\xi }={{\boldsymbol{P}}}_{1}{\boldsymbol{a}}+{{\boldsymbol{P}}}_{11}{\boldsymbol{f}},\,{\boldsymbol{U}}={{\boldsymbol{P}}}_{2}{\boldsymbol{a}}+{{\boldsymbol{P}}}_{22}f,\\ \displaystyle \frac{{d}^{2}{\boldsymbol{V}}}{d{\xi }^{2}}={\boldsymbol{Hb}}+{{\boldsymbol{H}}}_{11}{\boldsymbol{g}},\,\displaystyle \frac{d{\boldsymbol{V}}}{d\xi }={{\boldsymbol{P}}}_{1}{\boldsymbol{b}}+{{\boldsymbol{P}}}_{11}{\boldsymbol{g}},\,{\boldsymbol{V}}={{\boldsymbol{P}}}_{2}{\boldsymbol{b}}+{{\boldsymbol{P}}}_{22}{\boldsymbol{g}},\\ \displaystyle \frac{{d}^{2}{\boldsymbol{W}}}{d{\xi }^{2}}={\boldsymbol{Hc}}+{{\boldsymbol{H}}}_{11}{\boldsymbol{h}},\,\displaystyle \frac{d{\boldsymbol{W}}}{d\xi }={{\boldsymbol{P}}}_{1}{\boldsymbol{c}}+{{\boldsymbol{P}}}_{11}{\boldsymbol{h}},\,{\boldsymbol{W}}={{\boldsymbol{P}}}_{2}{\boldsymbol{c}}+{{\boldsymbol{P}}}_{22}{\boldsymbol{h}},\\ \displaystyle \frac{{d}^{2}{\boldsymbol{\Phi }}}{d{\xi }^{2}}={\boldsymbol{Hd}}+{{\boldsymbol{H}}}_{11}{\boldsymbol{k}},\,\displaystyle \frac{d{\boldsymbol{\Phi }}}{d\xi }={{\boldsymbol{P}}}_{1}{\boldsymbol{d}}+{{\boldsymbol{P}}}_{11}{\boldsymbol{k}},\,{\boldsymbol{\Phi }}={{\boldsymbol{P}}}_{2}{\boldsymbol{d}}+{{\boldsymbol{P}}}_{22}{\boldsymbol{k}},\\ \displaystyle \frac{{d}^{2}{\boldsymbol{\Theta }}}{d{\xi }^{2}}={\boldsymbol{He}}+{{\boldsymbol{H}}}_{11}{\boldsymbol{l}},\,\displaystyle \frac{d{\boldsymbol{\Theta }}}{d\xi }={{\boldsymbol{P}}}_{1}{\boldsymbol{e}}+{{\boldsymbol{P}}}_{11}{\boldsymbol{l}},\,{\boldsymbol{\Theta }}={{\boldsymbol{P}}}_{2}{\boldsymbol{e}}+{{\boldsymbol{P}}}_{22}{\boldsymbol{l}},\end{array}\end{eqnarray} \tag{ 29 }$$

where H and P ₁ , P ₂ are the Haar wavelet and its integrals defined as

$$\begin{eqnarray}{{\boldsymbol{H}}}_{1}=\left[\begin{array}{cccc}{h}_{1}\left({\xi }_{1}\right) & {h}_{2}\left({\xi }_{1}\right) & \cdots & {h}_{n}\left({\xi }_{1}\right)\\ {h}_{1}\left({\xi }_{2}\right) & {h}_{2}\left({\xi }_{2}\right) & \cdots & {h}_{n}\left({\xi }_{2}\right)\\ \vdots & \vdots & \ddots & \vdots \\ {h}_{1}\left({\xi }_{n}\right) & {h}_{2}\left({\xi }_{n}\right) & \cdots & {h}_{n}\left({\xi }_{n}\right)\end{array}\right]{{\boldsymbol{H}}}_{11}=\left[\begin{array}{cc}0 & 0\\ 0 & 0\\ \vdots & \vdots \\ 0 & 0\end{array}\right]\end{eqnarray} \tag{ 30 }$$

$$\begin{eqnarray}{{\boldsymbol{P}}}_{1}=\left[\begin{array}{cccc}{P}_{1,1}\left({\xi }_{1}\right) & {P}_{1,2}\left({\xi }_{1}\right) & \cdots & {P}_{1,n}\left({\xi }_{1}\right)\\ {P}_{1,1}({\xi }_{2}) & {P}_{1,2}\left({\xi }_{2}\right) & \cdots & {P}_{1,n}\left({\xi }_{2}\right)\\ \vdots & \vdots & \ddots & \vdots \\ {P}_{1,1}\left({\xi }_{n}\right) & {P}_{1,2}\left({\xi }_{n}\right) & \cdots & {P}_{1,n}\left({\xi }_{n}\right)\end{array}\right]{{\boldsymbol{P}}}_{11}=\left[\begin{array}{cc}1 & 0\\ 1 & 0\\ \vdots & \vdots \\ 1 & 0\end{array}\right]\end{eqnarray} \tag{ 31 }$$

$$\begin{eqnarray}{{\boldsymbol{P}}}_{2}=\left[\begin{array}{cccc}{P}_{2,1}\left({\xi }_{1}\right) & {P}_{2,2}\left({\xi }_{1}\right) & \cdots & {P}_{2,n}\left({\xi }_{1}\right)\\ {P}_{2,1}\left({\xi }_{2}\right) & {P}_{2,2}\left({\xi }_{2}\right) & \cdots & {P}_{2,n}\left({\xi }_{2}\right)\\ \vdots & \vdots & \ddots & \vdots \\ {P}_{2,1}\left({\xi }_{n}\right) & {P}_{2,2}\left({\xi }_{n}\right) & \cdots & {P}_{2,n}\left({\xi }_{n}\right)\end{array}\right]{{\boldsymbol{P}}}_{22}=\left[\begin{array}{cc}{\xi }_{1} & 1\\ {\xi }_{2} & 1\\ \vdots & \vdots \\ {\xi }_{n} & 1\end{array}\right]\end{eqnarray} \tag{ 32 }$$

and notations are defined as follows.

$$\begin{eqnarray}&&\begin{array}{l}{\boldsymbol{U}}={\left[U\left({\xi }_{1}\right),U\left({\xi }_{2}\right),\cdots ,U\left({\xi }_{n}\right)\right]}^{T},{\boldsymbol{a}}={\left[{{a}}_{{1}},{{a}}_{{2}},\cdots ,{{a}}_{{n}}\right]}^{T},\,{\boldsymbol{f}}={\left[\displaystyle \frac{dU\left({\xi }_{0}\right)}{d\xi },U\left({\xi }_{0}\right)\right]}^{T}\\ {\boldsymbol{V}}={\left[V\left({\xi }_{1}\right),V\left({\xi }_{2}\right),\cdots ,V\left({\xi }_{n}\right)\right]}^{T},\,{\boldsymbol{b}}={\left[{{b}}_{{1}},{{b}}_{{2}},\cdots ,{{b}}_{{n}}\right]}^{T},\,{\boldsymbol{g}}={\left[\displaystyle \frac{dV\left({\xi }_{0}\right)}{d\xi },V\left({\xi }_{0}\right)\right]}^{T}\\ {\boldsymbol{W}}={\left[W\left({\xi }_{1}\right),W\left({\xi }_{2}\right),\cdots ,W\left({\xi }_{n}\right)\right]}^{T},\,{\boldsymbol{c}}={\left[{{c}}_{{1}},{{c}}_{{2}},\cdots ,{{c}}_{{n}}\right]}^{T},\,{\boldsymbol{h}}={\left[\displaystyle \frac{dW\left({\xi }_{0}\right)}{d\xi },W\left({\xi }_{0}\right)\right]}^{T}\\ {\boldsymbol{\Phi }}={\left[{\rm{\Phi }}\left({\xi }_{1}\right),{\rm{\Phi }}\left({\xi }_{2}\right),\cdots ,{\rm{\Phi }}\left({\xi }_{n}\right)\right]}^{T},\,{\boldsymbol{d}}={\left[{{d}}_{{1}},{{d}}_{{2}},\cdots ,{{d}}_{{n}}\right]}^{T},\,{\boldsymbol{k}}={\left[\displaystyle \frac{d{\rm{\Phi }}\left({\xi }_{0}\right)}{d\xi },{\rm{\Phi }}\left({\xi }_{0}\right)\right]}^{T}\\ {\boldsymbol{\Theta }}={\left[{\rm{\Theta }}\left({\xi }_{1}\right),{\rm{\Theta }}\left({\xi }_{2}\right),\cdots ,{\rm{\Theta }}\left({\xi }_{n}\right)\right]}^{T},\,{\boldsymbol{e}}={\left[{{e}}_{{1}},{{e}}_{{2}},\cdots ,{{e}}_{{n}}\right]}^{T},\,{\boldsymbol{l}}={\left[\displaystyle \frac{d{\rm{\Theta }}\left({\xi }_{0}\right)}{d\xi },{\rm{\Theta }}\left({\xi }_{0}\right)\right]}^{T}\end{array}\end{eqnarray} \tag{ 33 }$$

where, f , g , h , k and l indicate the integral constants, which can be obtained by applying the boundary condition. The highest order of the displacements of boundary condition equations is first order, and the first order derivatives and displacements in equation (28) are calculated when ξ = 0 and ξ = 1. The discretization of the boundary condition equation can be manipulated in the same way as that of the displacement, and it can be written in the matrix form as follows.

$$\begin{eqnarray}&&\begin{array}{l}\displaystyle \frac{d{{\boldsymbol{U}}}_{{\bf{b}}}}{d\xi }={{\bf{P}}}_{{\bf{bc}}1}{\boldsymbol{a}}+{{\bf{P}}}_{{\bf{bc}}11}{\boldsymbol{f}},\,\displaystyle \frac{d{{\boldsymbol{V}}}_{{\bf{b}}}}{d\xi }={{\bf{P}}}_{{\bf{bc}}1}{\boldsymbol{b}}+{{\bf{P}}}_{{\bf{bc}}11}{\boldsymbol{g}},\,\displaystyle \frac{d{{\boldsymbol{W}}}_{{\bf{b}}}}{d\xi }={{\bf{P}}}_{{\bf{bc}}1}{\boldsymbol{c}}+{{\bf{P}}}_{{\bf{bc}}11}{\boldsymbol{h}},\\ \,\displaystyle \frac{d{{\boldsymbol{\Phi }}}_{{\bf{b}}}}{d\xi }={{\bf{P}}}_{{\bf{bc}}1}{\boldsymbol{d}}+{{\bf{P}}}_{{\bf{bc}}11}{\boldsymbol{k}},\displaystyle \frac{d{{\boldsymbol{\Theta }}}_{{\bf{b}}}}{d\xi }={{\boldsymbol{P}}}_{{\bf{bc}}1}{\boldsymbol{e}}+{{\boldsymbol{P}}}_{{\bf{bc}}11}{\boldsymbol{l}}\end{array}\end{eqnarray} \tag{ 34 }$$

$$\begin{eqnarray}&&\begin{array}{l}{{\boldsymbol{U}}}_{{\bf{b}}}={{\boldsymbol{P}}}_{{\bf{bc}}2}{\boldsymbol{a}}+{{\boldsymbol{P}}}_{{\bf{bc}}22}{\boldsymbol{f}},\,{{\boldsymbol{V}}}_{{\bf{b}}}={{\boldsymbol{P}}}_{{\bf{bc}}2}{\boldsymbol{b}}+{{\boldsymbol{P}}}_{{\bf{bc}}22}{\boldsymbol{g}},\,{{\boldsymbol{W}}}_{{\bf{b}}}={{\boldsymbol{P}}}_{{\bf{bc}}2}{\boldsymbol{c}}+{{\boldsymbol{P}}}_{{\bf{bc}}22}{\boldsymbol{h}},\\ \,{{\boldsymbol{\Phi }}}_{{\bf{b}}}={{\boldsymbol{P}}}_{{\bf{bc}}2}{\boldsymbol{d}}+{{\boldsymbol{P}}}_{{\bf{bc}}22}{\boldsymbol{k}},\,{{\boldsymbol{\Theta }}}_{{\bf{b}}}={{\boldsymbol{P}}}_{{\bf{bc}}2}{\boldsymbol{e}}+{{\boldsymbol{P}}}_{{\bf{bc}}22}{\boldsymbol{l}}\end{array}\end{eqnarray} \tag{ 35 }$$

where

Left boundary:

$$\begin{eqnarray}\begin{array}{l}{{\bf{P}}}_{{\bf{bc}}1}=\left[\begin{array}{cccc}{p}_{1,1}\left(0\right) & {p}_{1,2}\left(0\right) & \cdots & {p}_{1,n}\left(0\right)\end{array}\right]\,{{\bf{P}}}_{{\bf{bc}}11}=\left[\begin{array}{cc}1 & 1\end{array}\right]\\ {{\bf{P}}}_{{\bf{bc}}2}=\left[\begin{array}{cccc}{p}_{2,1}\left(0\right) & {p}_{2,2}\left(0\right) & \cdots & {p}_{2,n}\left(0\right)\end{array}\right]\,{{\bf{P}}}_{{\bf{bc}}22}=\left[\begin{array}{cc}\xi \left(0\right) & 1\end{array}\right]\end{array}\end{eqnarray} \tag{ 36 }$$

Right boundary:

$$\begin{eqnarray}\begin{array}{l}{{\bf{P}}}_{{\bf{bc}}1}=\left[\begin{array}{cccc}{p}_{1,1}\left(1\right) & {p}_{1,2}\left(1\right) & \cdots & {p}_{1,n}\left(1\right)\end{array}\right]\,{{\bf{P}}}_{{\bf{bc}}11}=\left[\begin{array}{cc}1 & 1\end{array}\right]\\ {{\bf{P}}}_{{\bf{bc}}2}=\left[\begin{array}{cccc}{p}_{2,1}\left(1\right) & {p}_{2,2}\left(1\right) & \cdots & {p}_{2,n}\left(1\right)\end{array}\right]\,{{\bf{P}}}_{{\bf{bc}}22}=\left[\begin{array}{cc}\xi \left(1\right) & 1\end{array}\right]\end{array}\end{eqnarray} \tag{ 37 }$$

With respect to the continuity condition, the equations can be expressed in the similar way as the boundary condition.

Therefore, the entire systems of the laminated composite conical-cylindrical coupled shell including boundary condition are discretized using the HWDM and the entire motion equations can be expressed in the matrix form as follows.

$$\begin{eqnarray}\left[\begin{array}{cc}{{\bf{K}}}_{dd} & {{\bf{K}}}_{db}\\ {{\bf{K}}}_{bb} & {{\bf{K}}}_{bd}\end{array}\right]\left[\begin{array}{l}{{\boldsymbol{A}}}_{d}\\ {{\boldsymbol{A}}}_{b}\end{array}\right]-{\omega }^{2}\left[\begin{array}{cc}{{\bf{M}}}_{dd} & {{\bf{M}}}_{db}\\ {\bf{0}} & {\bf{0}}\end{array}\right]\left[\begin{array}{l}{{\boldsymbol{A}}}_{d}\\ {{\boldsymbol{A}}}_{b}\end{array}\right]=0\end{eqnarray} \tag{ 38 }$$

$$\begin{eqnarray}&&{{\bf{A}}}_{d}={\left[{\boldsymbol{a}},{\boldsymbol{b}},{\boldsymbol{c}},{\boldsymbol{d}},{\boldsymbol{e}},\right]}^{T},{{\bf{A}}}_{b}={\left[{\boldsymbol{f}},{\boldsymbol{g}},{\boldsymbol{h}},{\boldsymbol{k}},{\boldsymbol{l}},\right]}^{T}\end{eqnarray} \tag{ 39 }$$

where A _d and A _b represent the Haar wavelet coefficients and the integral constants, respectively, and subscripts d and b indicate the discrete equilibrium equations of motion, the boundary conditions and connecting conditions. The general discrete motion equations (Eq. (23)) are represented by K_dd , K_db and M_dd , M_db . In addition, the discrete boundary and connecting condition equations (Eq. (24, 25)) are represented by K_bb and K_bd . Their concrete expressions are shown in appendix C.

Since the maximal level of resolution J of Haar wavelet is set as 8, maximal dimension of Haar wavelet is 2^J+1 = 512. Therefore, the matrix degrees of K_dd , K_db , K_bb and K_bd are 2560 × 2560, 2560 × 20, 20 × 2560 and 20 × 20 respectively, that is, in equation (38), the degree of stiffness matrix is 2580 × 2580.

A _b at both sides of equation (38) can be eliminated by performing some algebraic manipulations, and the standard characteristic equation is expressed as follows.

$$\begin{eqnarray}&&\left[{{\bf{K}}}_{dd}-{{\bf{K}}}_{db}{{\bf{K}}}_{bd}^{-1}{{\bf{K}}}_{bb}\right]{{\boldsymbol{{\rm A}}}}_{d}={\omega }^{2}\left[{{\bf{M}}}_{dd}-{{\bf{M}}}_{db}{{\bf{K}}}_{bd}^{-1}{{\bf{K}}}_{bb}\right]{{\boldsymbol{{\rm A}}}}_{d}\end{eqnarray} \tag{ 40 }$$

Through the further simplification of the above equation, the following matrix expressions can be obtained.

$$\begin{eqnarray}&&\left({\bf{K}}-{\omega }^{2}{\bf{M}}\right){{\boldsymbol{{\rm A}}}}_{d}=0\end{eqnarray} \tag{ 41 }$$

where, ${\bf{K}}=\left[{{\bf{K}}}_{dd}-{{\bf{K}}}_{db}{{\bf{K}}}_{bd}^{-1}{{\bf{K}}}_{bb}\right]$ and ${\bf{M}}=\left[{{\bf{M}}}_{dd}-{{\bf{M}}}_{db}{{\bf{M}}}_{bd}^{-1}{{\bf{M}}}_{bb}\right]$ are stiffness and mass matrixes of the structure, respectively and their matrix dimension are 2560 × 2560.

In the numerical calculation, the number of terms of series must be truncated at an appropriate finite number by considering the computational cost of the numerical solution. Therefore, convergence studies are needed to establish the number of terms that should be used to obtain accurate results. The convergence criterion of the present method for a three-layered, cross-ply [0°/90°/0°] laminated composite conical-cylindrical coupled shell with respect to different the maximal level of resolution J is examined in table 1.

As can be seen from table 1, as J increases, the frequency paramater of the coupled structure reaches a certain value. It is obvious that as J increases, the accracy of solution increases, but the calculation time and cost also increases, and if J exceeds a certain value, the error becomes very small. Therefore, in this paper, the maximal level of resolution J of Haar wavelet is set as 8 to calculate the natural frequency of laminated composite conical—cylindrical coupled structure.

In the theoretical formula, the boundary and continuity conditions of coupled shell structure are simulated by the artificial spring technique composed of three linear springs and two rotating springs. As a result, the boundary and continuity conditions are changed according to the stiffness of springs. Therefore, it is necessary to conduct the convergence study on the stiffness of elastic springs to determine the boundary and continuity conditions. In order to observe the variation characteristics of frequency according to the stiffness of individual springs, the stiffness of the spring considered increases from 10³ to 10¹⁵ and the rest four elastic springs are set as 0. Figure 2 shows the frequency variation characteristics of the laminated composite conical-cylindrical coupled shell with the increase of stiffness of the boundary elastic springs. The geometric and material parameters are same as above. As can be seen in figure 2, the frequencies are almost unchanged with the increase of stiffness of the elastic spring up to a certain value (10⁵). However, out of this value, the frequency starts to increase rapidly and keeps level again after the stiffness reaches 10¹². In order to simplify the presentation, the clamped, free, simply-supported boundary conditions are represented as C, F and S, respectively. Also, three kinds of elastic boundary conditions are selected as E_I , E_II and E_III. Table 2 shows the stiffness of elastic springs corresponding to individual boundary conditions.

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In the previous section, the characteristic parameters are determined through the convergence study for the calculation of natural frequency of laminated composite conical-cyclindrical coupled shell. Based on these results, in this section, the accuracy of the proposed method is examined by compairng with the results of prevous works and that obtaind by the fininte element method. Table 3 shows the comparison results of the frequency paramaters obtained by the proposed method with that of the prevous works in the isotropic conical-cyclindrical coupled shell. As can be seen in the table, the calculation results of isotropic conical-cylindrical coupled shell obtained by the proposed method agree well with that of previous works, which indicate that the proposed method is suitable for the analysis of free vibration analysis of the coupled structure.

Then, the accuray of the proposed method for the free vibration analysis of the laminated composite conical-cylindrical coupled shell is validated. However, due to the lack of the references on the laminated composite conical-cylindrical coupled shell, the accurary of the proposed method is validated by the comparison with the results from the finite element method. The software ABAQUS is used in the finite element analysis. For the finite element analysis, the element type is set as SR4 and number of element is set as 8432. The geometirc parameters are same as Tbale 1, but only the thickness(h) is set as 0.15m differently.

Table 4 shows the comparison of the first ten basic natural frequencies obtained by the proposed method with those from the finite element analysis in the laminated composite conical-cylindrical coupled shell with several boundary conditions. As can be seen in table 4, the results in two methods agrees well each other, which shows that the proposed method is very applicable not only for the isotropic material but aslo for the free vibration analysis of the laminated composite conical-cylindrical coupled shell.

Based on the validity of convergence and accuracy of the proposed method, in this section, through the analysis on the several parameters influencing the natural frequency of the laminated composite conical-cylindrical coupled shell, new frequency results are proposed.

Table 5 shows the variation of frequency parameters of the laminated composite conical-cylindrical coupled shell with different thickness under several classical boundary conditions. The calculation is performed under the conditions that the fiber direction angles of the laminate material are [30°/60°/30°], the semi-vertex angle φ = 30°, R_c = 0.5 m, R_l = 1 m, L_c = 1 m, L_l  = 2 m and circumferential wave numbers n = 2. Table 5 clearly shows that as the thickness of coupled structure increases, the frequency parameters also increase no matter what the boundary condition is. Also, it can be seen that if the boundary condition is changed (for example, C–F boundary and F–C boundary), the frequency is also changed.

To help readers to understand the mode shape of coupled structure, figure 3 shows the first four types of modes under the C–C and F–C boundary conditions.

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To consider the effect of elastic boundary conditions on the natural frequency of the conical-cylindrical coupled shell, table 6 shows the frequency parameters under the several classical-elastic and elastic-elastic boundary conditions.

The laminated material is composed of four layers and the calculation is performed when fiber direction angles are [45°/−45°/45°/−45°], thickness(h) of conical and cylindrical shells are both 0.1 m and n = 0 ∼ 3. Other geometrics of coupled shell are same as table 5. Table 6 shows that the frequency is obviously different according to the change of boundary condition. Also, according to the boundary conditions, the basic frequency appears at different n values.

Next, the frequency parameters of the conical-cylindrical coupled shell according to the number of layers and forms of the laminated materials under the several classical boundary conditions and elastic boundary conditions are shown in table 7. The calculation is performed when the length (L_l ) of conical shell is 1 m, thickness of conical and cylindrical are both 0.1 m and n = 2. Table 7 shows that as the number of layer increase, the natural frequency of the laminated composite conical-cylindrical coupled shell also increases. Also, the frequency is changed in the different laminated structures even if the number of layers is same.

Tables 8–10 shows the frequency parameters of conical-cylindrical coupled shell with different laminated structures under the classical boundary, classical-elastic boundary and elastic boundary conditions, providing researchers the reference and benchmark data on the laminated composite conical-cylindrical coupled shell. The material and geometric parameters used in this research is set as table 5. The thickness of combinational shell is set as 0.1 m in both conical and cylindrical shells.

Tables 8–10 shows that the frequency parameters are changed according to the laminated structures under the same boundary condition, while they are varied according to the different boundary conditions under the same laminated structure.

Next, the characteristics of frequency variation of the laminated composite coupled shell are considered when the fiber direction angle is changed. The material and geometric parameters are same as table 7 and all the individual shells are laminated with three layers. The investigation is conducted when the fiber direction angles are [0°/α°/0°]. As can be clearly seen in figure 4, when the fiber direction angle increases, the frequency increases first, and then decreases again from a certain value to 90°. Also, in case of 90°, the frequency is changed symmetrically.

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Next, the variation characteristics of frequency of the laminated composite conical-cylindrical coupled shell are considered according to the increases of the elastic modulus ratios when the fiber direction angles are [0°/90°/0°]. As can be seen in figure 5, same as individual shells [52], as the elastic modulus ratio increases, the frequency of conical-cylindrical coupled structure also increases.

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Lastly, the variation characteristic of frequency is considered according to the change of geometric parameters of individual shells. First, under the several classical boundary conditions, the variation characteristics of frequency of the coupled structure are studied increasing the length of conical shell when the geometric parameters of conical shell are constant. The material and geometric parameters used are same as above. Figure 6 shows the variation characteristics of frequency of the coupled structure according to the increase of length of the conical shell. As can be seen in figure 6, when other geometric parameters are constant, the frequency of coupled structure decreases as the length of conical shell increase. Especially, it can be known that when the length increases, the frequency decreases rapidly at the first time and then slowly again beyond a certain limitation.

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Next, when the semi-vertex angle of conical shell increases, the variation characteristics of frequency of the coupled structure are investigated under the several elastic boundary conditions. As can be seen in figure 7, as the semi-vertex angle of conical shell increases, the frequency of coupled structure also increases.

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