{"references": ["Arnold, R.N., Warburton, G.B., 1948. Flexural vibrations of the walls of\nthin cylindrical shells. Proceedings of the Royal Society of London A;\n197:238-256.", "Ludwig, A., Krieg, R., 1981.An analysis quasi-exact method for\ncalculating eigen vibrations of thin circular shells. J. Sound vibration;\n74,155-174.", "Chung, H., 1981. Free vibration analysis of circular cylindrical shells. J.\nSound vibration; 74, 331-359.", "Soedel, W., 1980.A new frequency formula for closed circular\ncylindrical shells for a large variety of boundary conditions. J. Sound\nvibration; 70,309-317.", "Forsberg, K., 1964. Influence of boundary conditions on modal\ncharacteristics of cylindrical shells. AIAA J; 2, 182- 189.", "Lam, K.L., Loy, C.T., 1995. Effects of boundary conditions on\nfrequencies characteristics for a multi- layered cylindrical shell. J. Sound\nvibration; 188, 363-384", "Loy, C.T., Lam, K.Y., 1996.Vibration of cylindrical shells with ring\nsupport. I.Joumal of Impact Engineering; 1996; 35:455.", "Koizumi, M., 1993. The concept of FGM Ceramic Transactions,\nFunctionally Gradient Materials.", "Makino A, Araki N, Kitajima H, Ohashi K. Transient temperature\nresponse of functionally gradient material subjected to partial, stepwise\nheating. Transactions of the Japan Society of Mechanical Engineers, Part\nB 1994; 60:4200-6(1994).\n[10] Anon, 1996.FGM components: PM meets the challenge. Metal powder\nReport. 51:28-32.\n[11] Zhang, X.D., Liu, D.Q., Ge, C.C., 1994. Thermal stress analysis of axial\nsymmetry functionally gradient materials under steady temperature field.\nJournal of Functional Materials; 25:452-5.\n[12] Wetherhold, R.C., Seelman, S., Wang, J.Z., 1996. Use of functionally\ngraded materials to eliminate or control thermal deformation.\nComposites Science and Technology; 56:1099-104.\n[13] Najafizadeh, M.M., Hedayati, B. Refined Theory for Thermoelastic\nStability of Functionally Graded Circular Plates. Journal of thermal\nstresses; 27:857-880.\n[14] Soedel, W., 1981. Vibration of shells and plates. MARCEL DEKKER,\nINC, New York.\n[15] Loy, C.T., Lam, K.Y., Reddy, J.N., 1999.Vibration of functionally\ngraded cylindrical shells; 41:309-324.\n[16] Najafizadeh, M.M., Isvandzibaei, M.R., 2007. Vibration of functionally\ngraded cylindrical shells based on higher order shear deformation plate\ntheory with ring support. Acta Mechanica; 191:75-91."]}

In this paper a study on the vibration of thin cylindrical shells with ring supports and made of functionally graded materials (FGMs) composed of stainless steel and nickel is presented. Material properties vary along the thickness direction of the shell according to volume fraction power law. The cylindrical shells have ring supports which are arbitrarily placed along the shell and impose zero lateral deflections. The study is carried out based on third order shear deformation shell theory (T.S.D.T). The analysis is carried out using Hamilton-s principle. The governing equations of motion of FGM cylindrical shells are derived based on shear deformation theory. Results are presented on the frequency characteristics, influence of ring support position and the influence of boundary conditions. The present analysis is validated by comparing results with those available in the literature.