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Natural frequency calculation of open laminated conical and cylindrical shells by a meshless method

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Abstract

In this paper, a new and an efficient solution method based on local gradient smoothing method has been applied to free vibration problem of open composite laminated cylindrical and conical shells with elastic boundary conditions. The theoretical model is formulated by the first-order shear deformation theory, and the motion equation is obtained by the Hamilton’s principle. The motion equation is discretized by meshless shape function; in this process, the derivatives of the shape function are approximated by local gradient smoothing method. The accuracy, applicability and efficiency of this method are demonstrated for free vibrations of open composite laminated cylindrical and conical shells with different geometric, material parameters and boundary condition. The numerical results show good convergence characteristics and good agreement between the present method and the existing literature. And through several numerical examples, some useful results for free vibration results of open composite laminated cylindrical and conical shells are obtained, which may serve as a benchmark solutions for researchers to check their analytical and numerical methods.

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Data Availability Statement

This manuscript has associated data in a data repository. [Authors’ comment: All data included in this manuscript are available upon request by contacting with the corresponding author.]

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Authors and Affiliations

  1. College of Mechanical Science and Technology, Kim Chaek University of Technology, Pyongyang, 999093, Democratic People’s Republic of Korea

    Songhun Kwak & Gwangil Jong

  2. Department of Engineering Machine, Pyongyang University of Mechanical Engineering, Pyongyang, 999093, Democratic People’s Republic of Korea

    Kwanghun Kim, Yonguk Ri & Hyonil Ri

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  1. Songhun Kwak
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  2. Kwanghun Kim
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  3. Yonguk Ri
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  4. Gwangil Jong
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  5. Hyonil Ri
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Correspondence to Kwanghun Kim.

Appendix

Appendix

$$ \varvec{K}_{Ii} = \left[ {\begin{array}{l@{\quad}l@{\quad}l@{\quad}l@{\quad}l} {L_{11} } & {L_{12} } & {L_{13} } & {L_{14} } & {L_{15} } \\ {L_{21} } & {L_{22} } & {L_{23} } & {L_{24} } & {L_{25} } \\ {L_{31} } & {L_{32} } & {L_{33} } & {L_{34} } & {L_{35} } \\ {L_{41} } & {L_{42} } & {L_{43} } & {L_{44} } & {L_{45} } \\ {L_{51} } & {L_{52} } & {L_{53} } & {L_{54} } & {L_{55} } \\ \end{array} } \right],\,\varvec{M}_{Ii} = \left[ {\begin{array}{l@{\quad}l@{\quad}l@{\quad}l@{\quad}l} { - I_{0} \phi_{i} } & 0 & 0 & { - I_{1} \phi_{i} } & 0 \\ 0 & { - I_{0} \phi_{i} } & 0 & 0 & 0 \\ 0 & 0 & { - I_{0} \phi_{i} } & 0 & 0 \\ { - I_{1} \phi_{i} } & 0 & 0 & { - I_{2} \phi_{i} } & 0 \\ 0 & { - I_{1} \phi_{i} } & 0 & 0 & { - I_{2} \phi_{i} } \\ \end{array} } \right] $$

Detailed expressions of differential operators \( L_{ij} \):

$$ L_{11} = A_{11} \frac{{\partial^{2} \phi_{i} }}{{\partial x^{2} }} + \frac{{2A_{16} }}{R}\frac{{\partial^{2} \phi_{i} }}{\partial x\partial \theta } + \frac{{A_{66} }}{{R^{2} }}\frac{{\partial^{2} \phi_{i} }}{{\partial \theta^{2} }} + A_{11} \frac{\sin \varphi }{R}\frac{{\partial \phi_{i} }}{\partial x} - A_{22} \frac{{\sin^{2} \varphi }}{{R^{2} }}\phi_{i} $$
$$ \begin{aligned} L_{12} & = A_{16} \frac{{\partial^{2} \phi_{i} }}{{\partial x^{2} }} + \frac{{(A_{12} + A_{66} )}}{R}\frac{{\partial^{2} \phi_{i} }}{\partial x\partial \theta } + \frac{{A_{26} }}{{R^{2} }}\frac{{\partial^{2} \phi_{i} }}{{\partial \theta^{2} }} + (A_{16} - A_{26} )\frac{\sin \varphi }{R}\frac{{\partial \phi_{i} }}{\partial x}\\ & - (A_{22} + A_{66} )\frac{\sin \varphi }{{R^{2} }}\frac{{\partial \phi_{i} }}{\partial \theta } + A_{26} \frac{{\sin^{2} \varphi }}{{R^{2} }}\phi_{i} \\ \end{aligned} $$
$$ L_{13} = A_{12} \frac{\cos \varphi }{R}\frac{{\partial \phi_{i} }}{\partial x} + A_{26} \frac{\cos \varphi }{{R^{2} }}\frac{{\partial \phi_{i} }}{\partial \theta } - A_{22} \frac{\sin \varphi \cos \varphi }{{R^{2} }}\phi_{i} $$
$$ L_{14} = B_{11} \frac{{\partial^{2} \phi_{i} }}{{\partial x^{2} }} + \frac{{2B_{16} }}{R}\frac{{\partial^{2} \phi_{i} }}{\partial x\partial \theta } + \frac{{B_{66} }}{{R^{2} }}\frac{{\partial^{2} \phi_{i} }}{{\partial \theta^{2} }} + B_{11} \frac{\sin \varphi }{R}\frac{{\partial \phi_{i} }}{\partial x} - B_{22} \frac{{\sin^{2} \varphi }}{{R^{2} }}\phi_{i} $$
$$ \begin{aligned} L_{15} & = B_{16} \frac{{\partial^{2} \phi_{i} }}{{\partial x^{2} }} + \frac{{(B_{12} + B_{66} )}}{R}\frac{{\partial^{2} \phi_{i} }}{\partial x\partial \theta } + \frac{{B_{26} }}{{R^{2} }}\frac{{\partial^{2} \phi_{i} }}{{\partial \theta^{2} }} + (B_{16} - B_{26} )\frac{\sin \varphi }{R}\frac{{\partial \phi_{i} }}{\partial x}\\ & - (B_{22} + B_{66} )\frac{\sin \varphi }{{R^{2} }}\frac{{\partial \phi_{i} }}{\partial \theta } + B_{26} \frac{{\sin^{2} \varphi }}{{R^{2} }}\phi_{i} \\ \end{aligned} $$
$$ \begin{aligned} L_{21} & = A_{16} \frac{{\partial^{2} \phi_{i} }}{{\partial x^{2} }} + \frac{{(A_{12} + A_{66} )}}{R}\frac{{\partial^{2} \phi_{i} }}{\partial x\partial \theta } + \frac{{A_{26} }}{{R^{2} }}\frac{{\partial^{2} \phi_{i} }}{{\partial \theta^{2} }} + (2A_{16} + A_{26} )\frac{\sin \varphi }{R}\frac{{\partial \phi_{i} }}{\partial x}\\ & + (A_{22} + A_{66} )\frac{\sin \varphi }{{R^{2} }}\frac{{\partial \phi_{i} }}{\partial \theta } + A_{26} \frac{{\sin^{2} \varphi }}{{R^{2} }}\phi_{i} \\ \end{aligned} $$
$$ \begin{aligned} L_{22} & = A_{66} \frac{{\partial^{2} \phi_{i} }}{{\partial x^{2} }} + \frac{{2A_{26} }}{R}\frac{{\partial^{2} \phi_{i} }}{\partial x\partial \theta } + \frac{{A_{22} }}{{R^{2} }}\frac{{\partial^{2} \phi_{i} }}{{\partial \theta^{2} }} + A_{66} \frac{\sin \varphi }{R}\frac{{\partial \phi_{i} }}{\partial x}\\ & - (A_{66} \frac{{\sin^{2} \varphi }}{{R^{2} }} + A_{44} \frac{{\cos^{2} \varphi }}{{R^{2} }})\phi_{i} \\ \end{aligned} $$
$$ L_{23} = (A_{26} + A_{45} )\frac{\cos \varphi }{R}\frac{{\partial \phi_{i} }}{\partial x} + (A_{22} + A_{44} )\frac{\cos \varphi }{{R^{2} }}\frac{{\partial \phi_{i} }}{\partial \theta } + A_{26} \frac{\sin \varphi \cos \varphi }{{R^{2} }}\phi_{i} $$
$$ \begin{aligned} L_{24} & = B_{16} \frac{{\partial^{2} \phi_{i} }}{{\partial x^{2} }} + \frac{{(B_{12} + B_{66} )}}{R}\frac{{\partial^{2} \phi_{i} }}{\partial x\partial \theta } + \frac{{B_{26} }}{{R^{2} }}\frac{{\partial^{2} \phi_{i} }}{{\partial \theta^{2} }} + (2B_{16} + B_{26} )\frac{\sin \varphi }{R}\frac{{\partial \phi_{i} }}{\partial x}\\ & + (B_{22} + B_{66} )\frac{\sin \varphi }{{R^{2} }}\frac{{\partial \phi_{i} }}{\partial \theta } + \left( {B_{26} \frac{{\sin^{2} \varphi }}{{R^{2} }} + A_{45} \frac{\cos \varphi }{R}} \right)\phi_{i} \\ \end{aligned} $$
$$ \begin{aligned} L_{25} & = B_{66} \frac{{\partial^{2} \phi_{i} }}{{\partial x^{2} }} + \frac{{2B_{26} }}{R}\frac{{\partial^{2} \phi_{i} }}{\partial x\partial \theta } + \frac{{B_{22} }}{{R^{2} }}\frac{{\partial^{2} \phi_{i} }}{{\partial \theta^{2} }} + B_{66} \frac{\sin \varphi }{R}\frac{{\partial \phi_{i} }}{\partial x}\\ & - \left( {B_{66} \frac{{\sin^{2} \varphi }}{{R^{2} }} - A_{44} \frac{\cos \varphi }{R}} \right)\phi_{i} \\ \end{aligned} $$
$$ L_{31} = - A_{12} \frac{\cos \varphi }{R}\frac{{\partial \phi_{i} }}{\partial x} - A_{26} \frac{\cos \varphi }{{R^{2} }}\frac{{\partial \phi_{i} }}{\partial \theta } - A_{22} \frac{\sin \varphi \cos \varphi }{{R^{2} }}\phi_{i} $$
$$ L_{32} = - (A_{26} + A_{45} )\frac{\cos \varphi }{R}\frac{{\partial \phi_{i} }}{\partial x} - (A_{22} + A_{44} )\frac{\cos \varphi }{{R^{2} }}\frac{{\partial \phi_{i} }}{\partial \theta } + A_{26} \frac{\sin \varphi \cos \varphi }{{R^{2} }}\phi_{i} $$
$$ L_{33} = A_{55} \frac{{\partial^{2} \phi_{i} }}{{\partial x^{2} }} + \frac{{2A_{45} }}{R}\frac{{\partial^{2} \phi_{i} }}{\partial x\partial \theta } + \frac{{A_{44} }}{{R^{2} }}\frac{{\partial^{2} \phi_{i} }}{{\partial \theta^{2} }} + A_{55} \frac{\sin \varphi }{R}\frac{{\partial \phi_{i} }}{\partial x} - A_{22} \frac{{\cos^{2} \varphi }}{{R^{2} }}\phi_{i} $$
$$ L_{34} = \left( {A_{55} - B_{12} \frac{\cos \varphi }{R}} \right)\frac{{\partial \phi_{i} }}{\partial x} + \left( {\frac{{A_{45} }}{R} - B_{26} \frac{\cos \varphi }{{R^{2} }}} \right)\frac{{\partial \phi_{i} }}{\partial \theta } + \left( {A_{55} \frac{\sin \varphi }{R} - B_{22} \frac{\sin \varphi \cos \varphi }{{R^{2} }}} \right)\phi_{i} $$
$$ L_{35} = \left( {A_{45} - B_{26} \frac{\cos \varphi }{R}} \right)\frac{{\partial \phi_{i} }}{\partial x} + \left( {\frac{{A_{44} }}{R} - B_{22} \frac{\cos \varphi }{{R^{2} }}} \right)\frac{{\partial \phi_{i} }}{\partial \theta } + \left( {A_{45} \frac{\sin \varphi }{R} + B_{26} \frac{\sin \varphi \cos \varphi }{{R^{2} }}} \right)\phi_{i} $$
$$ L_{41} = L_{14} $$
$$ \begin{aligned} L_{42} & = B_{16} \frac{{\partial^{2} \phi_{i} }}{{\partial x^{2} }} + \frac{{(B_{12} + B_{66} )}}{R}\frac{{\partial^{2} \phi_{i} }}{\partial x\partial \theta } + \frac{{B_{26} }}{{R^{2} }}\frac{{\partial^{2} \phi_{i} }}{{\partial \theta^{2} }} + (B_{16} - B_{26} )\frac{\sin \varphi }{R}\frac{{\partial \phi_{i} }}{\partial x}\\ & - (B_{22} + B_{66} )\frac{\sin \varphi }{{R^{2} }}\frac{{\partial \phi_{i} }}{\partial \theta } + \left( {B_{26} \frac{{\sin^{2} \varphi }}{{R^{2} }} + A_{45} \frac{\cos \varphi }{R}} \right)\phi_{i} \\ \end{aligned} $$
$$ L_{43} = \left( {B_{12} \frac{\cos \varphi }{R} - A_{55} } \right)\frac{{\partial \phi_{i} }}{\partial x} + \left( {B_{26} \frac{\cos \varphi }{{R^{2} }} - \frac{{A_{45} }}{R}} \right)\frac{{\partial \phi_{i} }}{\partial \theta } + (B_{12} - B_{22} )\frac{\sin \varphi \cos \varphi }{{R^{2} }}\phi_{i} $$
$$ L_{44} = D_{11} \frac{{\partial^{2} \phi_{i} }}{{\partial x^{2} }} + \frac{{2D_{16} }}{R}\frac{{\partial^{2} \phi_{i} }}{\partial x\partial \theta } + \frac{{D_{66} }}{{R^{2} }}\frac{{\partial^{2} \phi_{i} }}{{\partial \theta^{2} }} + D_{11} \frac{\sin \varphi }{R}\frac{{\partial \phi_{i} }}{\partial x} - \left( {D_{22} \frac{{\sin^{2} \varphi }}{{R^{2} }} + A_{55} } \right)\phi_{i} $$
$$ \begin{aligned} L_{45} & = D_{16} \frac{{\partial^{2} \phi_{i} }}{{\partial x^{2} }} + \frac{{(D_{12} + D_{66} )}}{R}\frac{{\partial^{2} \phi_{i} }}{\partial x\partial \theta } + \frac{{D_{26} }}{{R^{2} }}\frac{{\partial^{2} \phi_{i} }}{{\partial \theta^{2} }} + (D_{16} - D_{26} )\frac{\sin \varphi }{R}\frac{{\partial \phi_{i} }}{\partial x}\\ & - (D_{22} + D_{66} )\frac{\sin \varphi }{{R^{2} }}\frac{{\partial \phi_{i} }}{\partial \theta } + \left( {D_{26} \frac{{\sin^{2} \varphi }}{{R^{2} }} - A_{45} } \right)\phi_{i} \\ \end{aligned} $$
$$ \begin{aligned} L_{51} & = B_{16} \frac{{\partial^{2} \phi_{i} }}{{\partial x^{2} }} + \frac{{(B_{12} + B_{66} )}}{R}\frac{{\partial^{2} \phi_{i} }}{\partial x\partial \theta } + \frac{{B_{26} }}{{R^{2} }}\frac{{\partial^{2} \phi_{i} }}{{\partial \theta^{2} }} + (2B_{16} + B_{26} )\frac{\sin \varphi }{R}\frac{{\partial \phi_{i} }}{\partial x}\\ & + (B_{22} + B_{66} )\frac{\sin \varphi }{{R^{2} }}\frac{{\partial \phi_{i} }}{\partial \theta } + B_{26} \frac{{\sin^{2} \varphi }}{{R^{2} }}\phi_{i} \\ \end{aligned} $$
$$ L_{52} = L_{25} $$
$$ L_{53} = \left( {B_{26} \frac{\cos \varphi }{R} - A_{45} } \right)\frac{{\partial \phi_{i} }}{\partial x} + \left( {B_{22} \frac{\cos \varphi }{{R^{2} }} - \frac{{A_{44} }}{R}} \right)\frac{{\partial \phi_{i} }}{\partial \theta } + B_{26} \frac{\sin \varphi \cos \varphi }{{R^{2} }}\phi_{i} $$
$$ \begin{aligned} L_{54} & = D_{16} \frac{{\partial^{2} \phi_{i} }}{{\partial x^{2} }} + \frac{{(D_{12} + D_{66} )}}{R}\frac{{\partial^{2} \phi_{i} }}{\partial x\partial \theta } + \frac{{D_{26} }}{{R^{2} }}\frac{{\partial^{2} \phi_{i} }}{{\partial \theta^{2} }} + (2D_{16} + D_{26} )\frac{\sin \varphi }{R}\frac{{\partial \phi_{i} }}{\partial x}\\ & + (D_{22} + D_{66} )\frac{\sin \varphi }{{R^{2} }}\frac{{\partial \phi_{i} }}{\partial \theta } + \left( {D_{26} \frac{{\sin^{2} \varphi }}{{R^{2} }} - A_{45} } \right)\phi_{i} \\ \end{aligned} $$
$$ L_{55} = D_{66} \frac{{\partial^{2} \phi_{i} }}{{\partial x^{2} }} + \frac{{2D_{26} }}{R}\frac{{\partial^{2} \phi_{i} }}{\partial x\partial \theta } + \frac{{D_{22} }}{{R^{2} }}\frac{{\partial^{2} \phi_{i} }}{{\partial \theta^{2} }} + D_{66} \frac{\sin \varphi }{R}\frac{{\partial \phi_{i} }}{\partial x} - \left( {D_{66} \frac{{\sin^{2} \varphi }}{{R^{2} }} + A_{44} } \right)\phi_{i} $$

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Kwak, S., Kim, K., Ri, Y. et al. Natural frequency calculation of open laminated conical and cylindrical shells by a meshless method. Eur. Phys. J. Plus 135, 434 (2020). https://doi.org/10.1140/epjp/s13360-020-00438-0

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