Identification of boundary conditions using natural frequencies

The present investigation concerns a disc of varying thickness of whose flexural stiffness D varies with the radius r according to the law D = D-0 r(m), where D-0 and m are constants. The problem of finding boundary conditions for fastening this disc, which are inaccessible to direct observation, from the natural frequencies of its axisymmetric flexural oscillations is considered. The problem in question belongs to the class of inverse problems and is a completely natural problem of identification of boundary conditions. The search for the unknown conditions for fastening the disc is equivalent to finding the span of the vectors of unknown conditions coefficients. It is shown that this inverse problem is well posed. Two theorems on the uniqueness and a theorem on stability of the solution of this problem are proved, and a method for establishing the unknown conditions for fastening the disc to the walls is indicated. An approximate formula for determining the unknown conditions is obtained using the first three natural frequencies. The method of approximate calculation of unknown boundary conditions is explained with the help of three examples of different cases for fastening the disc ( rigid clamping, free support, elastic fixing).