Formula for two-sided estimation of the fundamental frequency of oscillations of a lattice truss - M. N. Kirsanov (Vestnik MGSU)
Introduction. One of the main tasks of the theory of oscillations of building structures is the determination of the fundamental frequency of natural oscillations. Analytical solutions are rare here and, as a rule, are based on approximate estimates of the first frequency from above (Rayleigh’s method) or from below (Dunkerley’s estimate). Most often, the problem of natural oscillations is solved numerically by the finite element method using specialized packages. In this paper, the task is to derive analytical estimates of the dependence of the first oscillation frequency of a lattice truss on the number of panels, the geometric characteristics of the structure, and the parameters of the elastic properties of the material.Materials and methods. A flat statically determinable lattice is supported by its base on struts. The angular support is a fixed joint. Calculation of forces in structural elements is performed by cutting out nodes using standard operators of the Maple symbolic mathematics system. The rigidity of the truss is found by the Maxwell – Mohr formula. The mass of the truss is distributed uniformly over its nodes. Mass oscillations occur vertically. By generalizing a series of solutions for trusses with a successively increasing order to an arbitrary number of panels, the desired formulas are derived by induction.Results. A case of kinematic variability of the proposed truss scheme was noticed. Formulas for the first frequency are obtained by the Dunkerley and Rayleigh method. The two analytical solutions are compared with the numerical solution obtained for the entire frequency spectrum. Spectral constants and resonant safety regions were discovered in the spectra of a family of regular trusses.Conclusions. The two-sided method for estimating the first frequency is applicable to solving problems on regular constructions, where the final formula includes the order of regularity as a parameter. For the construction under consideration, the error of the Rayleigh method is comparable to the error of the Dunkerley method.