Forced response of a cylindrical waveguide with simulation of the wavenumber extraction problem

The steady state response of a cylindrical elastic waveguide of arbitrary cross section to a harmonic load is considered. The inverse problem of wavenumber extraction is simulated using finite element discretization of the cross section. The case of a concentrated lateral point load on a railroad rail is used for illustration, at a frequency of 202 Hz, corresponding to a frequency well below the first cutoff, but above the regime where simple strength of materials concepts are accurate. The work is conceived with a view towards applications in the nondestructive characterization of such beams by means of scanned laser vibrometry for measurement of lateral bending wavenumbers dependent on a contained static axial load. Simulations of such measurements are generated, and subjected to noise and to a variety of potential systematic errors. Linear and nonlinear least squares minimization of the residual between simulated measurements and fits show that the lateral bending wavenumber can be recovered accurately, with remarkable robustness, even in the presence of noise and systematic errors.