The natural frequency of a mast with an arbitrary number of panels

Abstract

Introduction. The article addresses a spatial model of a statically definable mast truss consisting of four identical planar trusses with a crosswise grid system and a base with four supports at the corners. The authors solve the problem of deriving the analytical dependence between the bottom vibration frequency of the mast truss and the number of panels, mass, linear dimensions of its construction and properties of the material. Materials and methods. To calculate the values of forces, arising in the rods of a mast truss with an arbitrary number of panels, and analyze the obtained results, the induction method and operators of the Maple computer system for mathematics were used. The problem of deriving the analytical dependence between the bottom frequency of vibrations of the mast truss and its parameters is solved using the Dunkerley method, which generates the bottom estimate of the natural frequency. The rigidity of the truss structure is calculated according to the Maxwell – Mohr formula. To calculate the common members of sequences of coefficients, homogeneous linear recurrent equations are derived and solved in the frequency formula. Results. A formula is obtained for estimating the first frequency of natural vibrations of a truss. The formula coefficients have the form of polynomials of no higher than the fourth order. The accuracy of the calculation formula, obtained using the Dunkerley method, is estimated by the comparison with the first frequency, obtained through the numerical calculation of the entire spectrum of natural frequencies. Conclusions. The analysis of the analytical results and their comparison with the numerical ones shows high accuracy of the derived formula. The authors have identified a dependence, whereby an increase in the number of mast truss panels boosts the accuracy of the bottom estimate of the natural frequency.