Estimation of structural wave numbers from spatially sparse response measurements

A method is presented for estimating the complex wave numbers and amplitudes of waves that propagate in damped structures, such as beams, plates, and shells. The analytical basis of the method is a wave field that approximates response measurements in an aperture where no excitations are applied. At each frequency, the method iteratively adjusts wave numbers to best approximate response measurements, using wave numbers at neighboring frequencies as initial estimates in the search. In comparison to existing methods, the method generally requires far fewer measurement locations and does not require evenly spaced locations. The number of locations required by the method scales with the number of waves that propagate in the structure, whereas the number of locations required by existing methods scales with the minimum wavelength. In addition, the method allows convenient inclusion of the analytic relationships between wave numbers that exist for flexural vibrations of beams and plates. Advantages of the method are illustrated by an example in which a beam is excited by a transverse force at one end. Using analytic data and experimental measurements, the method produces a wave field that matches response measurements to within 1 percent. One interesting feature of the new method is that, when applied to analytic data, it supplies more robust wave number estimates using responses at unevenly spaced locations.