Simplified method for estimating the first natural frequency of a symmetric arch truss

A planar model of a symmetric statically determinate arch truss is considered. The mass of the truss is evenly distributed over its nodes. The Maxwell–Mohr formula, assuming that the truss rods have the same cross-section, allows one to determine the stiffness matrix of a given structure. The Dunkerley method and a variant of the Rayleigh method are proposed to be used to estimate the first natural frequency of the truss. The mass of the structure is conditionally concentrated in its nodes. Only small vertical oscillations are considered. The generalization of a number of solutions for trusses with different number of panels to the general case is carried out by induction. A simplified method for calculating the first frequency based on the Rayleigh method is proposed. To simplify the sums of partial frequencies and squared frequencies included in the Rayleigh solution, the area of the curve limiting the frequency values is replaced by its approximate value, which is calculated by the triangle area formula. The solution includes the value of the maximum deflection of the truss from the action of a distributed load. The results obtained by different analytical methods are compared with the results obtained by the numerical method. All transformations were carried out analytically using the Maple computer mathematical system. The results showed that with an increase in the number of panels, the accuracy of the Dunkerley analytical estimate increases, and the proposed method changes insignificantly. Spectral constants and frequency safety regions were found in the spectra of a family of trusses of different orders.