Uniqueness in the determination of vibration sources in rectangular Germain–Lagrange plates using displacement measurements over line segments with arbitrary small length

Abstract

The theme of this work is related to the field of vibration and source detection, which is important in naval, aerospace and civil engineering industries. The detection of unexpected vibration sources, in general, signals malfunctioning, or even an undesired presence in the case of defense systems. The focus will be on thin plates, which are among the basic building blocks of large complex structures. Here, we consider loads acting on a rectangular plate R of the product form g(t)Q(x), where the function of time g has a continuous first derivative and the spatial load distribution Q is a square-integrable function over R. We prove that the observation of the displacement of a line segment with arbitrary length parallel to one of the sides of the plate is enough for the determination of Q, provided that the interval of time is long enough. We also prove that the normal derivative along a side of the rectangle measured for an arbitrarily small interval of time is sufficient to determine the spatial load distribution Q. The method used to obtain the results is based on the series decomposition of the dynamic response and an analysis of the almost periodic distribution that arises from it.