The wave propagation in a beam on a random elastic foundation

• Published in: Probabilistic engineering mechanics 2007-04, Vol.22 (2), p.150-158
• Main Authors: Schevenels, M., Lombaert, G., Degrande, G., Clouteau, D.
• Format: Article
• Language: English
• Subjects: Bethe–Salpeter equation; Dyson equation; Exact sciences and technology; Fundamental areas of phenomenology (including applications); Green’s functions; Monte Carlo simulation; Neumann expansion; Physics; Random wave propagation; Solid mechanics; Static elasticity (thermoelasticity...); Structural and continuum mechanics; Vibration, mechanical wave, dynamic stability (aeroelasticity, vibration control...)
• ISSN: 0266-8920; 1878-4275
• DOI: 10.1016/j.probengmech.2006.09.003

This paper presents a study of wave propagation in an infinite beam on a random Winkler foundation. The spatial variation of the foundation spring constant is modelled as a random field and the influence of the correlation length is studied. As it is impossible to determine the general stochastic Green’s function, the configurational average of the Green’s function and its correlation function are considered. These functions are found through the transformation of the stochastic equation of motion into the Dyson equation for the mean or coherent field and the Bethe–Salpeter equation for the field correlation, as used in the study of wave propagation in random media. The approximate solutions of the Dyson and the Bethe–Salpeter equations are validated by means of a Monte Carlo simulation and compared with the results of a classical Neumann expansion method. Although both methods only involve the second order statistics of the random field, the approximation of the Dyson and the Bethe–Salpeter equations gives better results than the Neumann expansion, at the expense of a larger computational effort. Furthermore, the results show that a small spatial variation of the spring constant has an influence on the response if the correlation length and the wavelength have a similar order of magnitude, while the waves in the beam cannot resolve the spatial variation in the case where the correlation length is much smaller than the wavelength.