Exact Radial Natural Frequencies of Functionally Graded Hollow Long Cylinders

Abstract

In this work the exact free axisymmetric pure radial vibration of hollow infinite cylinders made of hypothetically functionally power-graded materials having identical inhomogeneity indexes for both Young’s modulus and the material density is addressed. The equation of motion is obtained as a linear second-order Bessel’s ordinary differential equation with constant coefficients based on the axisymmetric linear elasticity theory.  For traction free boundaries, a closed form frequency equation is offered. After verifying the present results for cylinders made of both isotropic and homogeneous materials, and isotropic functionally graded materials, an extensive parametric study is carried out to investigate the influences of both the thickness and inhomogeneity indexes on the natural frequencies. Results are presented in both graphical and tabular forms. It was revealed that the fundamental frequency in the radial mode is principally affected from the inhomogeneity parameters than the higher ones. However, the natural frequencies except the fundamental ones are dramatically affected from the thickness of the cylinder. As the thickness decreases, the natural frequencies considerably increase.  It is also revealed that, there is a linear relationship between the fundamental frequency and others in higher modes.

Keywords

• Free vibration,natural frequency,thick-walled hollow cylinder,functionally graded.

References

1. Chree, C., The equations of an isotropic elastic solid in polar and cylindrical coordinates, their solutions and applications. Transactions of the Cambridge Philosophical Society, 14, 250–309, 1889.
2. Greenspon, J. E., Flexural vibrations of a thick-walled circular cylinder according to the exact theory of elasticity. Journal of Aero/Space Sciences, 27, 1365–1373, 1957.
3. Gazis, D. C., Three dimensional investigation of the propagation of waves in hollow circular cylinder. Journal of the Acoustical Society of America, 31, 568–578, 1959.
4. Gladwell, G. M., Tahbildar, U. C., Finite element analysis of axisymmetric vibrations of cylinders. Journal of Sound and Vibration, 22, 143–157, 1972.
5. Gladwell, G. M., Vijay, D. K., Natural frequencies of free finite length circular cylinders. Journal of Sound and Vibration, 42, 387–397, 1975.
6. Hutchinson, J. R., Vibrations of solid cylinders. Journal of Applied Mechanics, 47, 901–907, 1980.
7. Hutchinson, J. R., El-azhari, S. A., Vibrations of free hollow circular cylinders. Journal of Applied Mechanics, 53, 641–646, 1986.
8. Singal, R. K., Williams, K. A., Theoretical and experimental study of vibrations of thick circular cylindrical shell and rings. Journal of Vibration, Acoustics, Stress and Reliability in Design, 110, 533–537, 1988.

9. Gosh, A. K., Axisymmetric vibration of a long cylinder. Journal of Sound and Vibration, 186(5), 711–721, 1995.
10. Leissa, A. W., So, J., Accurate vibration frequencies of circular cylinders from three-dimensional analysis. Journal of the Acoustical Society of America, 98, 2136–2141, 1995.
11. So, J., Leissa, A. W., Free vibration of thick hollow circular cylinders from three-dimensional analysis. Journal of Vibration and Acoustics, 119, 89–95, 1997.
12. Liew, K. M., Hung, K. C., Lim, M. K., Vibration of stress free hollow cylinders of arbitrary cross section. Journal of Applied Mechanics, ASME, 62, 718–724, 1995.
13. Hung, K. C., Liew, K. M., Lim, M. K., Free vibration of cantilevered cylinders: effects of cross-sections and cavities. Acta Mechanica, 113, 37–52, 1995.
14. Wang, H., Williams, K. Vibrational modes of thick cylinders of finite length. Journal of Sound and Vibration, 191, 955–971, 1996.
15. Zhou, D., Cheung, Y. K., Lo, S. H., Au, F. T. K., 3D vibration analysis of solid and hollow circular cylinders via Chebyshev–Ritz method. Computer Methods in Applied Mechanics and Engineering, 192, 1575–1589, 2003.
16. Mofakhami, M. R., Toudeshky, H. H., Hashemi, S. H. H., Finite cylinder vibrations with different end boundary conditions. Journal of Sound and Vibration, 297, 293–314, 2006.
17. Abbas, İ., Natural frequencies of a poroelastic hollow cylinder. Acta Mechanica, 186(1–4), 229–237, 2006.
18. Yahya, G. A., Abd-Alla, A. M., Radial vibrations in an isotropic elastic hollow cylinder with rotation. Journal of Vibration and Control, 22(13), 3123–3131, 2016.
19. Nelson, R. B., Dong, S. B., Kalra, R. D., Vibrations and waves in laminated orthotropic circular cylinders. Journal of Sound and Vibration, 18, 429–444, 1971.
20. Huang, K. H., Dong, S. B., Propagating waves and edge vibrations in anisotropic composite cylinders. Journal of Sound and Vibration, 96, 363–379, 1984.
21. Yuan, F. G., Hsieh, C. C., Three-dimensional wave propagation in composite cylindrical shell. Composite Structures, 42, 153–167, 1988.
22. Kharouf, N., Heyliger, P. R., Axisymmetric free vibrations of homogeneous and laminated piezoelectric cylinders. Journal of Sound and Vibration, 174(4), 539-561, 1994.
23. Markus, S., Mead, D. J., Axisymmetric and asymmetric wave motion in orthotropic cylinders. Journal Sound Vibration, 181, 127–147, 1995.
24. Markus, S., Mead, D. J., Wave motion in a 3-layered, orthotropic isotropic orthotropic composite shell. Journal Sound Vibration, 181, 149–167, 1995.
25. Ding, H. J., Wang, H. M., Chen, W. Q., Elastodynamic solution of a non-homogeneous orthotropic hollow cylinder. Acta Mechanica Sinica, 18, 621–628, 2002.
26. Ding, H. J., Wang, H. M., Chen, W. Q., A solution of a non-homogeneous orthotropic cylindrical shell for axisymmetric plane strain dynamic thermo elastic problems. Journal of Sound and Vibration, 263, 815–829, 2003.
27. Heyliger, P. R., Jilani, A., The free vibrations of inhomogeneous elastic cylinders and spheres. International Journal of Solids and Structures, 29(22), 2689-2708, 1992.
28. Loy, C. T., Lam, J. N., Reddy, J. N., Vibration of functionally graded cylindrical shells. Int. J. Mech. Sci. 41, 309–324, 1999.
29. Han, X., Liu, G. R., Xi, Z. C., Lam, K. Y., Characteristics of waves in a functionally graded cylinder. Int. Journal for Numerical Methods in Engineering, 53(3), 653–676, 2002.
30. Patel, B. P., Gupta, S. S., Loknath, M. S., Kadu, C. P., Free vibration analysis of functionally graded elliptical cylindrical shells using higher-order theory. Compos. Struct., 69, 259–270, 2005.
31. Pelletier, J. L., Vel, S. S., An exact solution for the steady-state thermoelastic response of functionally graded orthotropic cylindrical shells. Int. J. Solids Struct., 43, 1131–1158, 2006.
32. Arciniega, R. A., Reddy, J. N., Large deformation analysis of functionally graded shells. Int. J. Solids Struct., 44, 2036–2052, 2007.
33. Yang, J., Shen, H. S., Free vibration and parametric resonance of shear deformable functionally graded cylindrical panels. J. Sound Vib., 261, 871–893, 2007.
34. Jianqiao, Y., Qiujuan, M., Shan, S., Wave propagation in non-homogeneous magneto-electro-elastic hollow cylinders. J. Ultrasonic, 48, 664–677, 2008.
35. Abd-Alla, A. M., Nofal, T. A., Farhan, A. M., Effect of the non-homogenity on the composite infinite cylinder of isotropic material. Physics Letters A, 372, 4861–4864, 2008.
36. Tornabene, F., Viola, E., Inman, D. J., 2-D differential quadrature solution for vibration analysis of functionally graded conical, cylindrical shell and annular plate structures. J. Sound Vib., 328, 259-290, 2009.
37. Keleş, İ., Tütüncü, N., Exact analysis of axisymmetric dynamic response of functionally graded cylinders (or disks) and spheres. Journal of Applied Mechanics, 78(6), 061014 (7 pages), 2011.
38. Shen, H. S., Xiang, Y., Nonlinear vibration of nanotube-reinforced composite cylindrical shells in thermal environments. Comput. Methods Appl. Mech. Engrg., 213, 196-205, 2012.
39. Shen, H. S., Xiang, Y., Nonlinear vibration of nanotube-reinforced composite cylindrical panels resting on elastic foundations in thermal environments. Compos. Struct., 111, 291-300, 2014.
40. Moradi-Dastjerdi, R., Foroutan, M., Pourasghar, A., Dynamic analysis of functionally graded nanocomposite cylinders reinforced by carbon nanotube by a mesh-free method. Mater. Des., 44, 256-266, 2013.
41. Bowman, F., Introduction to Bessel Functions, New York, Dover, 1958.
42. Hildebrand, F. B., Advanced Calculus for Applications, Prentice-Hall. Inc. Englewood Cliffs, New Jersey, 1962.
43. Abramowitz, M., Stegun, I. A. (Eds.), Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, 9th printing, New York, Dover, 1972.