Trihedral Rod Pyramid: Deformations and Natural Vibration Frequencies

Строительные конструкции, здания и сооружения

The object of research is a new truss scheme of a statically determinate dome structure. The purpose of the study is to derive formulas for the dependences of the deflection under the action of a uniform load and the first frequency of natural vibrations on the number of panels, sizes, and masses concentrated in the truss nodes. Method. The forces in the truss rods are found from the equilibrium equations of the nodes. The system of equations also includes the reactions of vertical supports located along the contour of the structure. It is shown that the distribution of forces over the structure rods does not depend on the number of panels. The deflection values ​​and stiffness of the truss structure are calculated using the Maxwell – Mohr formula. The lower analytical estimate of the first frequency is obtained by the Dunkerley method, the upper one by the Rayleigh energy method. As a form of truss deflection in the Rayleigh method, the deflection from the action of a uniformly distributed load is taken. Only vertical oscillations of the weights are assumed. Results. The dependence of the solution on the number of panels is obtained by generalizing a series of solutions for trusses with a successively increasing number of panels. The solution uses operators of the Maple computer mathematics system. Graphs of the dependence of the deflection on the number of panels for different truss heights are plotted. The horizontal asymptote of the solution of the deflection problem is found. The value for the first natural frequency is compared with the numerical solution obtained from the analysis of the entire spectrum of natural frequencies of the vertical oscillations of the mass system located in the truss nodes. The frequency equation is compiled and solved using the eigenvalue search operators in the Maple system. It is shown that the lower analytical estimate based on the calculation of partial frequencies differs from the numerical solution by no more than 37%, while the upper estimate has an error of 7%. In this case, the formula for the lower Dunkerley frequency estimate turns out to be more compact. The natural frequency spectrum of the truss is analyzed. Isolines were found in the set of frequencies for a series of regular trusses.