Inverse Problem for the Vibrating Beam in the Free--Clamped Configuration

• Published in: Philosophical transactions of the Royal Society of London. Series A: Mathematical, physical, and engineering sciences Vol. 304; no. 1483; p. 211
• Main Author: V. Barcilon
• Format: Journal Article
• Language: English
• Published: The Royal Society 18.02.1982
• ISSN: 1364-503X; 1471-2962
• DOI: 10.1098/rsta.1982.0012

We consider the problem of reconstructing the flexural rigidity r(x) and the density $\rho $(x) of a beam. The unknown beam is assumed to have a free left end and a clamped right end. The data consist of the displacement and angle of the centre line of the free left end after an initial impulse. The information content of this seismogram-like impulse response is equivalent to three spectra {$\omega _{n}$}, {$\nu _{n}$}, {$\mu _{n}$} and two gross constants F$_{1}$, F$_{2}$. These data do not specify the structure of the vibrating beam uniquely, but rather a class of beams. All the beams in this class share the same structure over that portion of their length which is actually set in motion; they can differ over the portion that is stationary. A method for constructing r(x) and $\rho $(x) is presented. It consists of two steps. First $\rho $(x) and r(x) are determined over a small interval (0,x) adjacent to the free left end. Next, this known portion of the beam is stripped off and the response of the resulting truncated beam is computed via the initial data. The procedure is then repeated. Finally, the question of the existence of a solution is discussed. More specifically, conditions on {$\omega _{n}$}, {$\nu _{n}$} and {$\mu _{n}$} are given that ensure that r(x) and $\rho $(x) are physically meaningful.