An Inverse Method to Measure the Flexural Wave Properties of a Beam

Abstract

This paper describes an inverse method to measure the complex flexural wavenumber and wave propagation coefficients of a beam undergoing transverse motion. The formulation begins with the Bernoulli-Euler partial differential beam equation solved in terms of spatial trigonometric functions multiplied by a temporal harmonic response term. The resulting model is a nonlinear expression, which has five unknowns that consist of four wave propagation coefficients and the flexural wavenumber, and is written so that it is equal to data from seven equally spaced measurement locations on the beam. These equations, which are functions of the unknown parameters, are then inverted so that the unknowns become functions of the data. This operation allows the inverse method to be linear with respect to the beam parameters, yielding closed-form estimations. The method is independent of boundary conditions as it depends only on the measurements. An experiment is conducted in which a rectangular aluminum beam is vibrated transversely on a shaker table, with accelerometers recording the beam response data at the seven locations. The data were combined to estimate the flexural wavenumber, which was compared to a modeled wavenumber using assumed material properties for the beam, resulting in a very favorable comparison. Next, the wave propagation coefficients were estimated, although no comparison was made to the model because these coefficients depend on boundary conditions that are difficult to determine. Finally, the experimental response at a beam location is compared to a modeled response using the estimated parameters.