EExact Solutions for Forced Vibration of Non-Uniform Rods by Laplace Transformation

Year 2011, Volume: 24 Issue: 2, 347 - 353, 05.04.2011

Kerimcan Celebı İbrahim Keles Naki Tutuncu

Abstract

Longitudinal forced vibration behavior of non-uniform rods subjected to dynamic axial load is studied. Exact displacement solutions are obtained using the Laplace transformation method. Free vibration behavior is readily obtained in the analysis. Natural frequencies available in the literature for the cases considered are fully recovered. Inverse transformation into the time domain is performed using calculus of residues. Closed-form displacement expressions are tractable and efficiently implemented. Their efficiency is demonstrated by comparing the results with those obtained using Mode Superposition Method.

Key Words: Longitudinal vibrations; Forced vibrations;

Natural  frequency; Non-uniform rod; Laplace    transformation;

Residue theorem.

 

 

 

Keywords

Longitudinal vibrations; Forced vibrations; Natural frequency; Non-uniform rod; Laplace

References

• Kumar, B.M., Sujith, R.I., “Exact solutions for the longitudinal vibration of non-uniform rods”, J. Sound. Vib., 207: 721-729 (1997).
• Eisenberger, M., “Exact longitudinal vibration frequencies of a variable cross-section rod”, Appl. Acoust., 34: 123-130 (1991).
• Abrate, S., “Vibration of non-uniform rods and beams”, J. Sound. Vib., 185: 703-716(1995).
• Li, Q.S., Wu, J.R., Xu, J., “Longitudinal vibration of multi-step non-uniform structures with lumped masses and spring supports”, Appl. Acoust., 63: 333-350 (2002).
• Li, Q.S., “Exact solutions for free longitudinal vibrations of non-uniform rods”, J. Sound. Vib., 234: 1-19(2000).
• Qiusheng, L., Hong, C., Guiqing, L., “Static and dynamic analysis of straight bars with variable cross-section”, Comput. Struct., 59: 1185-1191 (1996).
• Raj, A., Sujith, R.I., “Closed-form solutions for the free longitudinal vibration of inhomogeneous rods”, J. Sound. Vib., 283: 1015-1030 (2005).
• Nachum, S., Altus, E., “Natural frequencies and mode shapes of deterministic and stochastic non- homogeneous rods and beams”, J. Sound. Vib., 302: 903-924 (2007).
• Horgan, C.O., Chan, A.M., “Vibration of inhomogeneous strings, rods and membranes”, J. Sound. Vib., 225: 503-513 (1999).
• Manolis, G.D., Beskos, D.E., “Dynamic stress concentration studies by boundary integrals and Laplace transform”, Int. J. Numer. Meth. Eng., 17: 573-599(1981).
• Clough, R.W., Penzien, J., “Dynamics of Structures”, McGraw-Hill, New York, (1993).