Mutual influence of the supports stiffness and the first natural frequency at bending vibrations of a spring-hinged beam

This paper discusses the analytical finding of the required stiffness values of the supports, which would provide the given first natural frequency of bending vibrations for a spring-hinged beam. The solution is obtained through the support coefficient, the value of which, according to the theory of beam vibrations, connects the stiffness of the support of the beam and the value of its first vibration frequency. For this purpose, preliminary approximation of dependence of support coefficient values on a support stiffness is performed using quadratic analytical functions. In this case, the problem of determining the stiffness of the supports becomes inverse and is reduced to finding combinations of the supports stiffness values, at which there is a solution of the quadratic analytical functions for the required support coefficient and, accordingly, the required first beam frequency. Normalization of the obtained support coefficients is proposed for a more convenient estimation of their influence on the first frequency of beam vibration.