Formulas for Calculating Deformations and Natural Frequency of Free Vibrations of a Hexagonal Tower

Abstract

Statement of the problem. A scheme of a six-sided prismatic statically determinate spatial girder is proposed. The task is set by induction to derive formulas for the dependence of the deflection of the structure and the lower limit of the main frequency of natural vibrations on the number of panels along the height of the prism. Materials and methods. The forces in the rods along with the reactions of the supports are found in an analytical form by means of the method of cutting nodes in the Maple symbolic mathematics system. One of the nodes at the base of the truss has a spherical support, one has a cylindrical support, the remaining four supports are racks. The top deflection is determined by the Maxwell-Mohr formula. From the analysis of the sequences of coefficients in the formulas for individual structures with a different number of panels, their common members are determined, which are included in the desired calculation formula. The Dunkerley method is used to estimate the first frequency of free oscillations. Results. For various types of loads, formulas for the dependence of girderf deflections on the number of panels are obtained. The coefficients in the solution are polynomial in the number of panels. The derived analytical dependence of the first frequency on the number of panels in comparison with the numerical solution has a small error, which decreases with increasing number of panels. Conclusions. A design of an axisymmetric statically determinate tower-type girder has been developed, which allows analytical solutions to the problem of deflection and the problem of the first natural frequency for an arbitrary number of panels. The resulting formulas can be used to assess the accuracy of numerical solutions and for preliminary calculations of models of structures of this type.