Natural Frequencies and Mode Shapes of Deterministic and Stochastic Non-Homogeneous Rods


Article Preview

In this study the natural frequencies and mode shapes of the kth order of nonhomogeneous (deterministic and stochastic) rods are found. The solution is based on the functional perturbation method (FPM). The natural frequency and mode shape of the kth order is found analytically to any desired degree of accuracy. In the deterministic it is shown that the FPM accuracy range for the frequency ω and the mode shape is less then 1%. The stochastic case demonstrates the power of this method. The material and geometrical properties will be considered as statistically homogeneous random field with exponential two-point correlation. It is shown that the accuracy depends on the stochastic information used, the correlation distance (roughly the “grain size”), and whether we are interested in the properties of ω or ω2.



Edited by:

Patrick Sean Keogh




S. Nachum and E. Altus, "Natural Frequencies and Mode Shapes of Deterministic and Stochastic Non-Homogeneous Rods", Applied Mechanics and Materials, Vols. 5-6, pp. 207-216, 2006

Online since:

October 2006





[1] S. Abrate: Vibration of non-uniform rods and beams, Journal of Sound and Vibration, Vol. 185 (4) (1995), 703-716.


[2] B. M. Kumar and R. I. Sujith: Exact solutions for the longitudinal vibrations of non-uniform rods, Journal of Sound and Vibration, Vol. 207 (5) (1997), 721-729.


[3] C. O. Horgan and A. M. Chan: Vibration of inhomogeneous strings rods and membranes, Journal of Sound and Vibration, Vol. 225 (3) (1999), 503-513.


[4] S. Candan and I. Elishakoff: Constructing the axial stiffness of longitudinally vibrating rod from fundamental mode shape, International Journal of Solids and Structures, Vol. 38 (2001), 3443-3452.


[5] I. Elishakoff: Eigenvalues of Inhomogeneous Structures: Unusual Closed-Form Solutions, Boca Raton, CRC Press, (2005).

[6] M. Hoshiya and H. Shah: Free vibration of stochastic beam-column, Journal of the Engineering Mechanics Division, Vol. 4 (1971), 1239-1255.

[7] R. Ganesan and V.K. Kowda: Free vibration of composite beam-columns with stochastic material and geometrical properties subjected to axial loads, Journal of Reinforced Plastics and Composites, Vol. 24 (2005), 69-91.


[8] R. Vaicatis: Free vibrations of beam with random characteristics, Journal of Sound and Vibration, Vol. 35 (1) (1974), 13-21.

[9] C. S. Manohar and A.J. Keane: Axial vibrations of stochastic rod, Journal of Sound and Vibration, Vol. 165 (2) (1993), 341-359.


[10] M. Beran: Statistical Continuum Mechanics, in Inter Publisher. New York, McGraw-Hill Book Company, 1968, 1-141.

[11] S. Timoshenko, D.H. Young and W. Weaver Jr: Vibration Problems in Engineering, (Ed. 4), New York, McGraw-Hill, 1974, 363-459.

[12] V.I. Smirnov: A Course of Higher Mathematics, Vol. 1, New York, Pergamon Press, (1964).