Beam support stiffness analytic solution for the first eigen-frequency and critical force

The work discusses the problem of providing the required first natural frequency of bending vibrations of the beam under the action of a longitudinal force by introducing the necessary stiffness of the supports. Considering and combining the equations of free vibrations of the beam and the equations describing the loss of its stability, the operability condition was obtained because of providing a minimum given value of the first natural frequency of vibrations considering the action of the axial force. In this case, the achievement of the zero frequency of natural vibration corresponds to the loss of stability, which allows solving both problems. This problem is mathematically complex, and in the known scientific literature its solution is usually given only in graphical or tabular form. The problem lies in the nonlinear dependence of the coefficients of supports on the stiffness during vibrations and loss of stability. To solve this problem, the approximation of the nonlinear coefficients of the supports by the least squares method and the obtaining of quadratic approximating functions was used. As a result, the problem of determining the required stiffness of the supports brought to a fourth-degree resolving algebraic equation, for which an analytic solution exists. The obtained solution allows the stiffness of the beam supports, which provides the required value of the first natural frequency of vibrations of the beam and its first critical load in the form of external compressive force or temperature effects. Replacing the nonlinear dependencies of the support coefficients with the stiffness of the supports with simpler quadratic functions led to relatively simple analytic dependencies that allow the resolution equation to be transformed according to the particular problem being solved. At the same time, quadratic functions influenced the calculation error, to reduce which, the range of the support stiffness under consideration was limited and divided into three zones. The results of calculations using the proposed analytical solution were compared with numerical calculations using finite element method. The comparison of the calculation results showed an error of not more than 5 % for the considered range of stiffness of the supports, which is quite enough for engineering calculations of beam structures. To limit the error of the result, it is recommended that the stiffnesses of both supports be equal or of the same order.