Approximation of beam support coefficient values at vibrations and buckling

The problem of calculating the first natural frequency of vibration and the first critical force for a beam with elastic supports is considered. An analytical review of the literature on solving such problems showed that in the theory of vibrations and the theory of stability of beams, consideration of the support conditions is based on the use of support coefficients, the values of which were obtained after solving the corresponding differential equation. The reviewed literature contains only a limited set of values of these coefficients, mainly for ideal supports of simple types: hinges, fixed, etc. Consideration of the stiffness of supports can only be found in individual editions and only for a limited number of values. In this work, the calculation of the support coefficients depending on the stiffness of the beam supports for the first natural frequency of vibrations and the first critical force is made. The obtained values were divided into three zones and approximated within each zone by quadratic functions. The use of quadratic approximation made it possible to obtain simple analytical dependencies suitable for engineering applied calculations, and the division of stiffness into zones provided an acceptable error of the obtained values. Also, quadratic dependencies made it possible to solve inverse problems for determining the stiffness of supports for a given value of the first natural frequency of vibrations or the first critical force. A detailed study of the error of the obtained approximating functions over the entire considered range of stiffness was carried out, which showed that the error in determining the coefficient of supports during fluctuations is not more than 2%, and in case of loss of stability - 6%. The error depends on the combination of stiffness of the supports, and can increase if the stiffnesses differ by more than an order of magnitude. The high sensitivity of the solution of the inverse problem to the input data was also established, which is the result of the high nonlinearity of the dependence of the coefficients of the supports on the stiffness. The obtained results can be used in engineering calculations of the first natural frequency of vibrations and the first critical force of a beam with elastic supports.