Uniqueness of determination of the type of the boundary conditions at one end of a beam from three natural frequencies of its vibrations

The problem of identification of the type and coefficients of the Euler-Bernoulli boundary conditions at one end of a beam is considered. It is shown that three natural frequencies uniquely determine not only the coefficients of a boundary condition of definite type, but also the type of boundary condition that can be realized from the following set: clamping, free support, free end, floating clamping, elastic clamping, inertial element at the end. A method for solving this inverse problem is proposed and corresponding examples and counter-examples are given.