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A meshfree approach for free vibration analysis of ply drop-off laminated conical, cylindrical shells and annular plates

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Abstract

In this paper, a novel meshfree approach is proposed for the free vibration analysis of ply drop-off laminated conical and cylindrical shells and annular plates. The theoretical formulation for free vibration analysis is based on the first-order shear deformation theory, and the field functions are approximated by a novel meshfree Tchebychev-radial point interpolation method shape function using Tchebychev polynomials and Gaussian radial functions as the basis. The governing equations and boundary conditions for the substructures of ply drop-off laminated composite shell are derived, and the equations of the whole system are obtained by combining them using a continuous condition. The boundary and continuous conditions are generalized by the introduction of an artificial spring technique, and the type of boundary conditions is selected according to the spring stiffness values. The accuracy and reliability of the proposed method are verified by comparing the results in the literature and of the finite element program ABAQUS. The free vibration characteristics including natural frequencies and mode shapes of ply drop-off laminated composite shells with various geometrical dimensions and boundary conditions are presented through numerical examples.

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References

  1. Nallim, L.Z., Oller, S.: An analytical–numerical approach to simulate the dynamic behavior of arbitrarily laminated composite plates. Compos Struct. 85, 311–325 (2008)

    Google Scholar 

  2. Khov, H., Li, W.L., Gibson, R.F.: An accurate solution method for the static and dynamic deflections of orthotropic plates with general boundary conditions. Compos Struct 90, 474–481 (2009)

    Google Scholar 

  3. Mohammadi, M., Mohseni, E., Moeinfar, M.: Bending, buckling and free vibration analysis of incompressible functionally graded plates using higher order shear and normal deformable plate theory. Appl. Math. Model. 69, 47–62 (2019)

    MathSciNet  MATH  Google Scholar 

  4. Merdacia, S., Belghoul, H.: High-order shear theory for static analysis of functionally graded plates with porosities. C.R. Mec. 347, 207–217 (2019)

    Google Scholar 

  5. Amabili, M.: Nonlinear vibrations and stability of laminated shells using a modified first-order shear deformation theory. Eur. J. Mech. A Solids 68, 75–87 (2018)

    MathSciNet  MATH  Google Scholar 

  6. Amabili, M.: A new third-order shear deformation theory with non-linearities in shear for static and dynamic analysis of laminated doubly curved shells. Compos. Struct. 128, 260–273 (2015)

    Google Scholar 

  7. Amabili, M., Reddy, J.N.: The nonlinear, third-order thickness and shear deformation theory for statics and dynamics of laminated composite shells. Comp. Struct. 244, 112265 (2020)

    Google Scholar 

  8. Amabili, M., Balasubramanian, P.: Nonlinear forced vibrations of laminated composite conical shells by using a refined shear deformation theory. Comp. Struct. 249, 112522 (2020)

    Google Scholar 

  9. Amabili, M., Reddy, J.N.: A new non-linear higher-order shear deformation theory for large-amplitude vibrations of laminated doubly curved shells. Int. J. Non Linear Mech. 45, 409–418 (2010)

    Google Scholar 

  10. Amabili, M., Balasubramanian, P.: Nonlinear vibrations of truncated conical shells considering multiple internal resonances. Nonlinear Dyn. 100, 77–93 (2020)

    Google Scholar 

  11. Amabili, M.: Non-linearities in rotation and thickness deformation in a new third-order thickness deformation theory for static and dynamic analysis of isotropic and laminated doubly curved shells. Int. J. Non Linear Mech. 69, 109–128 (2015)

    Google Scholar 

  12. Chen, W.Q., Wang, Y.F., Cai, J.B., Ye, G.R.: Three-dimensional analysis of cross-ply laminated cylindrical panels with weak interfaces. Int. J. Solids Struct. 41, 2429–2446 (2004)

    MATH  Google Scholar 

  13. Malekzadeh, P.: Three-dimensional free vibration analysis of thick laminated annular sector plates using a hybrid method. Compos. Struct. 90, 428–437 (2009)

    Google Scholar 

  14. Tong, B., Li, Y., Zhu, X., Zhang, Y.: Three-dimensional vibration analysis of arbitrary angle-ply laminated cylindrical shells using differential quadrature method. Appl. Acoust. 146, 390–397 (2019)

    Google Scholar 

  15. Rastogi, V., Urmaliya, P.K., Verma, A.K., Kumhar, V.: Free vibration of isotropic and laminated composite plate using three-dimensional finite element analysis. Mater. Today Proc. 18, 2823–2831 (2019)

    Google Scholar 

  16. Jin, G., Su, Z., Shi, S., Ye, T., Gao, S.: Three-dimensional exact solution for the free vibration of arbitrarily thick functionally graded rectangular plates with general boundary conditions. Compos. Struct. 108, 565–577 (2014)

    Google Scholar 

  17. Ye, T., Jin, G., Zhu, Su., Jia, X.: A unified Chebyshev–Ritz formulation for vibration analysis of composite laminated deep open shells with arbitrary boundary conditions. Arch. Appl. Mech. 84, 441–471 (2014)

    MATH  Google Scholar 

  18. Xie, X., Zheng, H., Jin, G.: Free vibration of four-parameter functionally graded spherical and parabolic shells of revolution with arbitrary boundary conditions. Compos. B 77, 59–73 (2015)

    Google Scholar 

  19. Liu, T., Wang, A., Wang, Q., Qin, B.: Wave based method for free vibration characteristics of functionally graded cylindrical shells with arbitrary boundary conditions. Thin Walled Struct. 148, 106580 (2020)

    Google Scholar 

  20. Tornabene, F., Fantuzzi, N., Viola, E., Reddy, J.N.: Winkler-Pasternak foundation effect on the static and dynamic analyses of laminated doubly-curved and degenerate shells and panels. Comp. Part B 57, 269–296 (2014)

    Google Scholar 

  21. Kim, K., Kwak, S., Jang, P., Sok, M., Jon, S., Ri, K.: Free vibration analysis of elastically connected composite laminated double-plate system with arbitrary boundary conditions by using meshfree method. AIP Adv. 11, 035119 (2021)

    Google Scholar 

  22. Talebitooti, R., Shenaei Anbardan, V.: Haar wavelet discretization approach for frequency analysis of the functionally graded generally doubly-curved shells of revolution. Appl. Math. Model. 67, 645–675 (2019)

    MathSciNet  MATH  Google Scholar 

  23. Qin, B., Zhong, R., Wu, Q., Wang, T., Wang, Q.: A unified formulation for free vibration of laminated plate through Jacobi–Ritz method. Thin Walled Struct. 144, 106354 (2019)

    Google Scholar 

  24. Jin, G., Ye, T., Su, Z.: Structural Vibration: A Uniform Accurate Solution for Laminated Beams, Plates and Shells with General Boundary Conditions. Springer, New York (2015)

    MATH  Google Scholar 

  25. Xiang, X., Guoyong, J., Wanyou, Li., Zhigang, L.: A numerical solution for vibration analysis of composite laminated conical, cylindrical shell and annular plate structures. Compos. Struct. 111, 20–30 (2014)

    Google Scholar 

  26. Tornabene, F.: Free vibrations of anisotropic doubly-curved shells and panels of revolution with a free-form meridian resting on Winkler–Pasternak elastic foundations. Compos. Struct. 94, 186–206 (2011)

    Google Scholar 

  27. Ye, T., Jin, G., Zhang, Y.: Vibrations of composite laminated doubly-curved shells of revolution with elastic restraints including shear deformation, rotary inertia and initial curvature. Compos. Struct. 133, 202–225 (2015)

    Google Scholar 

  28. Li, H., Pang, F., Wang, X., Yuan, Du., Chen, H.: Free vibration analysis for composite laminated doubly-curved shells of revolution by a semi analytical method. Compos. Struct. 201, 86–111 (2018)

    Google Scholar 

  29. Barbero, E.J.: Finite Element Analysis of Composite Materials with Abaqus. CRC Press, London (2013)

    Google Scholar 

  30. Varughese, B., Mukherjee, A.: A ply drop-off element for analysis of tapered laminated composites. Compos. Struct. 39, 123–144 (1997)

    Google Scholar 

  31. He, K., Ganesan, R., Hosa, S.V.: Modified shear-lag model for analysis of a composite laminate with drop-off plies. Compos. Sci. Technol. 63, 1453–1462 (2003)

    Google Scholar 

  32. Liu, G.R., Gu, Y.T.: An Introduction to Meshfree Methods and Their Programming. Springer, Dordrecht (2005)

    Google Scholar 

  33. Chinnaboon, B., Chucheepsakul, S., Katsikadelis, J.T.: A BEM-based domain meshless method for the analysis of Mindlin plates with general boundary conditions. Comput. Methods Appl. Mech. Eng. 200(13–16), 1379–1388 (2011)

    MathSciNet  MATH  Google Scholar 

  34. Sorić, J., Jarak, T.: Mixed meshless formulation for analysis of shell-like structures. Comput. Methods Appl. Mech. Eng. 199(17–20), 1153–1164 (2010)

    MathSciNet  MATH  Google Scholar 

  35. Moghaddam, M.R., Baradaran, G.H.: Three-dimensional free vibrations analysis of functionally graded rectangular plates by the meshless local Petrov–Galerkin (MLPG) method. Appl. Math. Comput. 304, 153–163 (2017)

    MathSciNet  MATH  Google Scholar 

  36. Shivanian, E.: Meshless local Petrov–Galerkin (MLPG) method for three-dimensional nonlinear wave equations via moving least squares approximation. Eng. Anal. Bound. Elem. 50, 249–257 (2015)

    MathSciNet  MATH  Google Scholar 

  37. Wen, P.H.: Meshless local Petrov–Galerkin (MLPG) method for wave propagation in 3D poroelastic solids. Eng. Anal. Bound. Elem. 34(4), 315–323 (2010)

    MathSciNet  MATH  Google Scholar 

  38. Liu, G.R., Zhao, X., Dai, K.Y., Zhong, Z.H., Li, G.Y., Han, X.: Static and free vibration analysis of laminated composite plates using the conforming radial point interpolation method. Compos. Sci. Technol. 68, 354–366 (2008)

    Google Scholar 

  39. Kwak, S., Kim, K., Ri, Y., Jong, G., Ri, H.: Natural frequency calculation of open laminated conical and cylindrical shells by a meshless method. Eur. Phys. J. Plus 135(434), 1–33 (2020)

    Google Scholar 

  40. Kim, K., Kwak, S., Jang, P., Sok, M., Jon, S., Ri, K.: Free vibration analysis of elastically connected composite laminated double-plate system with arbitrary boundary conditions by using meshfree method. AIP Adv. 11, 035119 (2021). https://doi.org/10.1063/5.0040270

    Article  Google Scholar 

  41. Bediz, B.: A spectral-Tchebychev solution technique for determining vibrational behavior of thick plates having arbitrary geometry. J. Sound Vib. 432, 272–289 (2018)

    Google Scholar 

  42. Gottlieb, D., Orszag, S.A.: Numerical Analysis of Spectral Methods, Theory and Applications. SIAM-CBMS, Philadelphia (1977)

    MATH  Google Scholar 

  43. Zhou, D., Au, F., Cheung, Y., Lo, S.: Three-dimensional vibration analysis of circular and annular plates via the Chebyshev Ritz method. Int. J. Solid Struct. 40(12), 3089–3105 (2003)

    MATH  Google Scholar 

  44. Filiz, S., Bediz, B., Romero, L.A., Ozdoganlar, O.B.: A spectral-Tchebychev solution for three-dimensional vibrations of parailelepipeds under mixed boundary conditions. J. Appl. Mech. 79, 051012 (2012)

    Google Scholar 

  45. Filiz, S., Bediz, B., Romero, L.A., Ozdoganlar, O.B.: Three dimensional dynamics of pretwisted beams: A spectral-Tchebychev solution. J. Sound Vib. 333, 2823–2839 (2014)

    Google Scholar 

  46. Bediz, B., Romero, L.A., Ozdoganlar, O.B.: Three dimensional dynamics of rotating structures under mixed boundary conditions. J. Sound Vib. 358, 176–191 (2015)

    Google Scholar 

  47. Kurylov, Y., Amabili, M.: Polynomial versus trigonometric expansions for nonlinear vibrations of circular cylindrical shells with different boundary conditions. J. Sound Vib. 329, 1435–1449 (2010)

    Google Scholar 

  48. Kurylov, Y., Amabili, M.: Nonlinear vibrations of clamped-free circular cylindrical shells. J. Sound Vib. 330, 5363–5381 (2011)

    Google Scholar 

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Acknowledgements

We would like to take the opportunity to express my hearted gratitude to all those who made a contribution to the completion of our article.

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Authors and Affiliations

  1. Department of Mechanical Science and Technology, Kim Chaek University of Technology, Pyongyang, 950003, Democratic People’s Republic of Korea

    Songhun Kwak, Gwangil Jong & Jaeliong Cha

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

    Kwanghun Kim

  3. Digital Library, Kim Chaek University of Technology, Pyongyang, 950003, Democratic People’s Republic of Korea

    U. Juhyok

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  1. Songhun Kwak
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Appendices

Appendix 1

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

Appendix 2

$$C_{11}^{i} = A_{11} \frac{{\partial \phi_{i} }}{\partial x} + A_{12} \frac{\sin \alpha }{R}\phi_{i} - k_{u0} \phi_{i}$$
$$C_{12}^{i} = A_{12} \frac{n}{R}\phi_{i}$$
$$C_{13}^{i} = A_{12} \frac{\cos \alpha }{R}\phi_{i}$$
$$C_{14}^{i} = C_{41}^{i} = B_{11} \frac{{\partial \phi_{i} }}{\partial x} + B_{12} \frac{\sin \alpha }{R}\phi_{i}$$
$$C_{15}^{i} = B_{12} \frac{n}{R}\phi_{i}$$
$$C_{21}^{i} = - A_{66} \frac{n}{R}\phi_{i}$$
$$C_{22}^{i} = A_{66} \frac{{\partial \phi_{i} }}{\partial x} - A_{66} \frac{\sin \alpha }{R}\phi_{i} - k_{v0} \phi_{i}$$
$$C_{23}^{i} = C_{31}^{i} = C_{32}^{i} = C_{35}^{i} = C_{53}^{i} = 0$$
$$C_{24}^{i} = C_{51}^{i} = - B_{66} \frac{n}{R}\phi_{i}$$
$$C_{25}^{i} = C_{52}^{i} = B_{66} \frac{{\partial \phi_{i} }}{\partial x} - B_{66} \frac{\sin \alpha }{R}\phi_{i}$$
$$C_{33}^{i} = A_{55} \frac{{\partial \phi_{i} }}{\partial x} - k_{w0} \phi_{i}$$
$$C_{34}^{i} = A_{55} \phi_{i}$$
$$C_{42}^{i} = B_{12} \frac{n}{R}\phi_{i}$$
$$C_{43}^{i} = B_{12} \frac{\cos \alpha }{R}\phi_{i}$$
$$C_{44}^{i} = D_{11} \frac{{\partial \phi_{i} }}{\partial x} + D_{12} \frac{\sin \alpha }{R}\phi_{i} - k_{x0} \phi_{i}$$
$$C_{45}^{i} = D_{12} \frac{n}{R}\phi_{i}$$
$$C_{54}^{i} = - D_{66} \frac{n}{R}\phi_{i}$$
$$C_{55}^{i} = D_{66} \frac{{\partial \phi_{i} }}{\partial x} - D_{66} \frac{\sin \alpha }{R}\phi_{i} - k_{\theta 0} \phi_{i}$$

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Kwak, S., Kim, K., Jong, G. et al. A meshfree approach for free vibration analysis of ply drop-off laminated conical, cylindrical shells and annular plates. Acta Mech 232, 4775–4800 (2021). https://doi.org/10.1007/s00707-021-03084-4

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