Journal of Applied Mechanics and Technical Physics

, Volume 59, Issue 7, pp 1242–1250

# Free Vibrations of a Cylindrical Shell Partially Resting on Elastic Foundation

• S. A. Bochkarev
Article

## Abstract

The free vibrations of a circular cylindrical shell resting on a two-parameter Pasternak elastic foundation are investigated. The elastic medium is inhomogeneous along the shell length, and the inhomogeneity represents an alternation of areas with and without the medium. The behavior of the shell is considered in the framework of the classical shell theory based on the Kirchhoff–Love hypotheses. The corresponding geometric and physical relations, together with the equations of motion, are reduced to a system of eight ordinary differential equations for new unknowns. The problem is solved by applying the Godunov orthogonal sweep method, and the differential equations are integrated using the fourth-order Runge–Kutta method. The natural frequencies are calculated by applying a stepwise iterative procedure, followed by a further refinement based on the bisection method. The results are validated by comparing them with available numerical-analytical solutions. For simply supported, clamped-clamped, and clamped-free cylindrical shells, the numerical results reveal that the lowest vibration frequencies depend on the elastic medium characteristics and the type of the inhomogeneity. It is shown that a violation in the smoothness of the curves is caused by variations in the lowest frequency mode, the ratio of the size of the elastic foundation to the total length of the shell, and its stiffness, as well as by a combination of the boundary conditions specified at the ends of the shell.

## Keywords

classical shell theory cylindrical shell Godunov’s orthogonal sweep method free vibrations Pasternak elastic medium

## References

1. 1.
Pasternak, P.L., Osnovy novogo metoda rascheta fundamentov na uprugom osnovanii pri pomoshchi dvukh koeffitsientov posteli (On a New Method of Analysis of an Elastic Foundation by Means of Two Foundation Constants), Moscow: Gosstroyizdat, 1954.Google Scholar
2. 2.
Vlasov, V.Z. and Leontev, N.N., Balki, plity i obolochki na uprugom osnovanii (Beams, Plates and Shells on Elastic Foundation), Moscow: Fizmatgiz, 1960.Google Scholar
3. 3.
Gorbunov-Posadov, M.I. and Malikova, T.A., Raschet konstruktsij na uprugom osnovanii (Calculation of Structures on Elastic Foundation), Moscow: Stroyizdat, 1973.Google Scholar
4. 4.
Paliwal, D.N., Pandey, R.K., and Nath, T., Free vibrations of circular cylindrical shell on Winkler and Pasternak foundations, Int. J. Pres. Ves. Pip., 1996, vol. 69, no. 1, pp. 79–89.
5. 5.
Malekzadeh, P., Farid, M., Zahedinejad, P., and Karami, G., Three-dimensional free vibration analysis of thick cylindrical shells resting on two-parameter elastic supports, J. Sound Vibrat., 2008, vol. 313, nos. 3–5, pp. 655–675.
6. 6.
Gheisari, M., Molatefi, H., and Ahmadi, S.S., Third order formulation for vibrating non-homogeneous cylindrical shells in elastic medium, J. Solid Mech., 2011, vol. 3, no. 4, pp. 346–352.Google Scholar
7. 7.
Leonenko, D.V., Natural vibrations of the three-layered cylindrical shells in the elastic pasternak’s medium, Mekh. Mashin, Mekhanizm. Mater., 2013, no. 4 (25), pp. 57–59.Google Scholar
8. 8.
Ye, T., Jin, G., Shi, S., and Ma, X., Three-dimensional free vibration analysis of thick cylindrical shells with general end conditions and resting on elastic foundations, Int. J. Mech. Sci., 2014, vol. 84, pp. 120–137.
9. 9.
Kuznetsova, E.L., Leonenko, D.V., and Starovoitov, E.I., Natural vibrations of three-layer circular cylindrical shells in an elastic medium, Mech. Solids, 2015, vol. 50, no. 3, pp. 359–366.
10. 10.
Khalifa, M.A., Natural frequencies and mode shapes of variable thickness elastic cylindrical shells resting on a Pasternak foundation, J. Vibrat. Control., 2016, vol. 22, no. 1, pp. 37–50.
11. 11.
Sofiyev, A.H., Karaca, Z., and Zerin, Z., Non-linear vibration of composite orthotropic cylindrical shells on the non-linear elastic foundations within the shear deformation theory, Compos. Struct., 2017, vol. 159, pp. 53–62.
12. 12.
Sheng, G.G. and Wang, X., Thermal vibration, buckling and dynamic stability of functionally graded cylindrical shells embedded in an elastic medium, J. Reinf. Plast. Comp., 2008, vol. 27, no. 2, pp. 117–134.
13. 13.
Shah, A.G., Mahmood, T., Naeem, M.N., Iqbal, Z., and Arshad, S.H., Vibrations of functionally graded cylindrical shells based on elastic foundations, Acta Mech., 2010, vol. 211, nos. 3–4, pp. 293–307.
14. 14.
Najafov, A.M., Sofiyev, A.H., and Kuruoglu, N., Torsional vibration and stability of functionally graded orthotropic cylindrical shells on elastic foundations, Meccanica, 2013, vol. 48, no. 4, pp. 829–840.
15. 15.
Mohammadimehr, M., Moradi, M., and Loghman, A., Influence of the elastic foundation on the free vibration and buckling of thin-walled piezoelectric-based FGM cylindrical shells under combined loading, J. Solid Mech., 2014, vol. 6, no. 4, pp. 347–365.Google Scholar
16. 16.
Kamarian, S., Sadighi, M., Shakeri, M., and Yas, M.H., Free vibration response of sandwich cylindrical shells with functionally graded material face sheets resting on Pasternak foundation, J. Sandwich Struct. Mater., 2014, vol. 16, no. 5, pp. 511–533.
17. 17.
Bahadori, R. and Najafizadeh, M.M., Free vibration analysis of two-dimensional functionally graded axisymmetric cylindrical shell on Winkler-Pasternak elastic foundation by first-order shear deformation theory and using Navier-differential quadrature solution methods, Appl. Math. Model., 2015, vol. 39, no. 16, pp. 4877–4894.
18. 18.
Sofiyev, A.H., Keskin, S.N., and Sofiyev, Ali H., Effects of elastic foundation on the vibration of laminated non-homogeneous orthotropic circular cylindrical shells, Shock Vibrat., 2004, vol. 11, no. 2, pp. 89–101.
19. 19.
Sheng, G.G., Wang, X., Fu, G., and Hu, H., The nonlinear vibrations of functionally graded cylindrical shells surrounded by an elastic foundation, Nonlin. Dyn., 2014, vol. 78, no. 2, pp. 1421–1434.
20. 20.
Sofiyev, A.H., Large amplitude vibration of FGM orthotropic cylindrical shells interacting with the nonlinear Winkler elastic foundation, Composites, Part B, 2016, vol. 98, pp. 141–150.
21. 21.
Sofiyev, A.H., Hui, D., Haciyev, V.C., Erdem, H., Yuan, G.Q., Schnack, E., and Guldal, V., The nonlinear vibration of orthotropic functionally graded cylindrical shells surrounded by an elastic foundation within first order shear deformation theory, Composites, Part B, 2017, vol. 116, pp. 170–185.
22. 22.
Amabili, M. and Dalpiaz, G., Free vibrations of cylindrical shells with non-axisymmetric mass distribution on elastic bed, Meccanica, 1997, vol. 32, no. 1, pp. 71–84.
23. 23.
Amabili, M. and Garziera, R., Vibrations of circular cylindrical shells with non-uniform constraints, elastic bed and added mass, Part I: Empty and fluid-filled shells, J. Fluids Struct., 2000, vol. 14, no. 5, pp. 669–690.
24. 24.
Gunawan, H., Mikami, T., Kanie, S., and Sato, M., Finite element analysis of cylindrical shells partially buried in elastic foundations, Comput. Struct., 2005, vol. 83, nos. 21–22, pp. 1730–1741.
25. 25.
Gunawan, H., Mikami, T., Kanie, S., and Sato, M., Free vibration characteristics of cylindrical shells partially buried in elastic foundations, J. Sound Vibrat., 2006, vol. 290, nos. 3–5, pp. 785–793.
26. 26.
Bakhtiari-Nejad, F. and Mousavi Bideleh, S.M., Nonlinear free vibration analysis of prestressed circular cylindrical shells on the Winkler/Pasternak foundation, Thin Wall. Struct., 2012, vol. 53, pp. 26–39.
27. 27.
Kim, Y.-W., Free vibration analysis of FGM cylindrical shell partially resting on Pasternak elastic foundation with an oblique edge, Composites, Part B, 2015, vol. 70, pp. 263–276.
28. 28.
Tan, B.H., Lucey, A.D., and Howell, R.M., Aero-hydro-elastic stability of flexible panels: prediction and control using localised spring support, J. Sound Vibrat., 2013, vol. 332, no. 26, pp. 7033–7054.
29. 29.
Karmishin, A.V., Lyaskovets, V.A., Myachenkov, V.I., and Frolov, A.N., Statika i dinamika tonkostennykh obolochechnykh konstruktsii (The Statics and Dynamics of Thin-Walled Shell Structures), Moscow: Mashinostroyeniye, 1975.Google Scholar
30. 30.
Godunov, S.K., Numerical solution of boundary-value problems for systems of linear ordinary differential equations, Usp. Mat. Nauk, 1961, vol. 16, no. 3, pp. 171–174.