Model of a hexagonal prismatic truss. Oscillation frequency spectrum

The object of research is a spatial truss. A scheme of a statically determinate tower-type truss is proposed. The purpose of the study is to derive formulas for the dependences of the first frequency of natural oscillations on the dimensions of the structure and to numerically analyze the spectra of a family of regular trusses of various orders. It is assumed that the mass of the structure is concentrated in the truss nodes. Method. To determine the analytical expressions for the forces in the rods, the equilibrium equations of the nodes are compiled in matrix form using the operators of the Maple computer mathematics system. The rigidity of the structure required to calculate the vibration frequencies is calculated using the Maxwell - Mohr formula. The lower analytical estimate of the first frequency is obtained in the Dunkerley approximation by calculating the partial frequencies. Only horizontal oscillations of the weights are assumed, each weight has two degrees of freedom. Generalizing a series of solutions for trusses with a successively increasing number of panels, we obtain the dependence of the lower frequency estimate on the number of panels. Results. A formula is obtained for the first oscillation frequency as a function of the number of panels. A good agreement between the found analytical solution and the numerical solution obtained with allowance for all degrees of freedom of the structure is shown. Graphs of the dependence of the first frequency and the relative error of the analytical solution on the number of panels are plotted. With an increase in the number of panels, the error of the found solution decreases from 7% to 3%. The natural frequency spectrum of the truss is analyzed. Spectral isolines and constants are found in the frequency set of a series of regular trusses