Identification of the fixedness and loadedness of an end of an Euler–Bernoulli beam from its natural vibration frequencies

We consider a uniform Euler–Bernoulli beam whose left end is fixed and whose right end has a load fixed by two springs. Being hit, the beam starts vibrating. The aim of the paper is to determine the fixedness parameters (the spring stiffness coefficients) and the loadedness parameters (the mass and moment of inertia of the load) of the right end of the beam from the natural frequencies of its bending vibrations. We show that the four unknown parameters of the boundary conditions on the right end of the beam are uniquely determined by five natural frequencies of bending vibrations. A counterexample is exhibited demonstrating that four natural frequencies are insufficient for the unique identification of these four nonnegative parameters.