A Wavelet Approach for the Analysis of Bending Waves in a Beam on Viscoelastic Random Foundation


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This paper deals with the mathematical model of dynamic behaviour of a beam resting on viscoelastic random foundation for which the modulus of subgrade reaction is assumed to be a homogeneous random function of the space variable. An approximate analytical solution for the fourth-order differential equation with random parameters is obtained in the case of a ∞ C -class correlation function. This higher order regularity of correlation function implies the regularity of associated stochastic function [1] in the sense of the mean-square analysis [2]. The numerical results for the average displacement have been obtained by using Bourret’s approximation method. A special method of finding inverse Laplace transform based on the wavelet theory is adopted and used in the numerical examples.



Edited by:

Patrick Sean Keogh




P. Koziol et al., "A Wavelet Approach for the Analysis of Bending Waves in a Beam on Viscoelastic Random Foundation", Applied Mechanics and Materials, Vols. 5-6, pp. 239-246, 2006

Online since:

October 2006




[1] K. Sobczyk: Stochastic Differential Equations with Applications to Physics and Engineering (Kluwer Academic Publ., Dordrecht 1991).

[2] T.T. Soong: Random Differential Equations in Science and Engineering (Academic Press, New York 1973).

[3] I. Elishakoff: Probabilistic Methods in the Theory of Structures (John Wiley & Sons Inc., New York 1983).

[4] O.R. Jaiswal, R.N. Iyengar: Acta Mechanica 96 (1993) p.67.

[5] P. Koziol, Z. Hryniewicz: Proc. of the 14 th Polish Conf. on Comp. Meth. in Mech., Rzeszow (1999) p.119.

[6] P. Koziol, Z. Hryniewicz: in press, doi: 10. 1016/j. ijsolstr. 2006. 02. 018, International Journal of Solids and Structures (2006).

[7] G. Adomian: Stochastic Systems (Academic Press, New York 1983).

[8] I. Daubechies: SIAM Journal on Mathematical Analysis 24(2) (1993) p.499.

[9] J. Wang, Y. Zhou, H. Gao: Communications in Numerical Methods in Engineering 19 (2003) p.959.

[10] L. Brekhovskikh, V. Goncharov: Mechanics of Continua and Wave Dynamics (Heidelberg: Springer-Verlag, Berlin 1985).

[11] Z. Hryniewicz: Applied Mathematical Modelling, Vol. 21, May (1997) p.247.

[12] Z. Hryniewicz: Journal of Sound and Vibration 278 (2004) p.1013.

[13] G. Heckman, H. Schlichtkrull: Harmonic Analysis and Special Functions on Symmetric Spaces (Academic Press, San Diego 1994).

[14] D. Hong, J. Wang, R. Gardner: Real Analysis with an Introduction to Wavelets and Applications (Elsevier Inc., Oxford 2005).

[15] Y. Meyer: Wavelets and Operators (Cambridge University Press, Cambridge 1992).

[16] G. Beylkin, R. Coifman, V. Rokhlin: Communications on Pure and Applied Mathematics 44 (1991) p.141.