Analytical Solutions for The Torsional Vibrations of Variable Cross-Section Rods

Abstract

The objective of this paper is to present exact analytical solutions for the torsional vibration of rods with nonuniform cross-section. Using appropriate transformations the equation of motion of torsional vibration of a rod with varying cross-section is reduced to analytically solvable standard differential equations whose form depends upon the specific area variation. Solutions are obtained for a rod with for a polynomial area variation. The solutions are obtained in terms of special functions such as Bessel and Neumann functions. Simple formulas to predict the natural frequencies of non-uniform rods with various end conditions are presented. The natural frequencies of variable cross-section rods for these end conditions are calculated and their dependence on taper is discussed

Keywords

• Torsional Vibration, Isotropic, Rod, Variable Cross-Section, Analytical Solution

References

1. Nagaraj, V. T., and Sahu, N., Torsional Vibrations of non-uniform rotating blades with attachment flexibility. Journal of Sound and Vibration, 80 (1982) 401-411.
2. Rezeka, S. F., Torsional vibrations of a nonprismatic hollow shaft. Journal of Vibration, Acoustics, Stress and Reliability in Design, ASME, 111 (1989) 486-489.
3. Eisenberger, M., Torsional Vibrations of open and variable cross-section bars. Thin- Walled Structures, 28 (1997) 269-278.
4. Li, Q.S., Torsional vibration of multi-step non-uniform rods with various concentrated elements. Journal of Sound and Vibration, 260 (2003) 637-651.
5. Gorman, D.J., Free vibration analysis of beams and shafts. New York: Wiley; 1975.
6. Belvins, R.D., Formulas for natural frequency and mode shape. New York: D. Van Nostrand; 1979.
7. Kameswara Rao, C., Torsional frequencies and mode shapes of generally constrained shafts and piping. Journal of Sound and Vibration, 125 (1988) 115–21.
8. Gere, JM., Torsional vibrations of beams of thin walled open cross section. J Appl Mech, 21 (1954) 381–7.

9. Carr, JB., The torsional vibrations of uniform thin walled beams of open section. Aeronaut J R Aeronaut Soc, 73 (1969):672–4.
10. Christiano, P., and Salmela, L., Frequencies of beams with elastic warping restraint. J Struct Div ASCE, 97 (1971) 1835–40.
11. Wekezer, J.W., Vibrational analysis of thin-walled bass with open cross sections. J Struct Eng ASCE, 115 (1989) 2965–78.
12. Abdel-Ghaffar, A.M., Free torsional vibrations of suspension bridges. J Struct Div ASCE, 1979;105:767–88. [13] Krajcinovic, D., A consistent discrete elements technique for thin-walled assemblages. Int J Solids Struct, 1969;5:639–62.
13. Mallick DV, Dungar R. Dynamic characteristics of core wall structures subjected to torsion and bending. Struct Eng 1977;55:251–61.
14. Banerjee JR, Guo S, Howson WP. Exact dynamic stiffness matrix of a bending–torsion coupled beam including warping. Int J Comput Struct 1996;59:613–21.
15. Matsui Y, Hayashikawa C. Dynamic stiffness analysis for torsional vibration of continuous beams with thin-walled crosssection. J Sound Vib 2001;243(2):301–16.
16. Kameswara Rao C, Appala Saytam A. Torsional vibrations and stability of thin-walled beams on continuous elastic foundation. AIAA J 1975;13:232–4.
17. Kameswara Rao C, Mirza S. Torsional vibrations and buckling of thin walled beams on elastic foundation. Thin Wall Struct 1989;7:73–82.
18. Zhang Z, Chen S. A new method for the vibration of thin-walled beams. Int J Comput Struct 1991;39:597–601. [20] Lee J, Kim SE. Flexural–torsional coupled vibration of thinwalled composite beams with channel sections. Comput Struct 2002;80:133–44.
19. Kollar LP. Flexural–torsional vibration of open section composite beams with shear deformation. Int J Solids Struct 2001;38: 7543–58.
20. Ganapathi M, Patel BP, Sentilkumar T. Torsional vibrations and damping analysis of sandwich beams. J Reinf Plast Compos 1999;18:96–117.
21. Sapountzakis EJ, Mokos VG. Warping shear stresses in nonuniform torsion of composite bars by BEM. Comput Meth Appl Mech Eng 2003;192:4337–53.
22. Sapountzakis EJ. Nonuniform torsion of multi-material composite bars by the boundary element method. Int J Comput Struct 2001;79:2805–16.
23. Sapountzakis EJ, Mokos VG. Nonuniform torsion of composite bars by boundary element method. J Eng Mech ASCE 2001;127(9):945–53.
24. Patel BP, Ganapathi M. Non-linear torsional vibration and damping analysis of sandwich beams. J Sound Vib 2001;240(2):385–93.
25. Sujith, R.I., Waldherr, G.A., and Zinn, B.T., Exact solution for one-dimensional acoustic fields in ducts with axial temperature gradient. AIAA Paper 94-0359, Proceedings of the 21st Aerospace Sciences Meeting, Reno, Nevada, 10-13 January (1994).
26. Sujith, R.I., Waldherr, G.A., and Zinn, B.T., Exact solution for one-dimensional acoustic fields in ducts with axial temperature gradient. Journal of Sound and Vibration, 184 (1995) 389-402.