Hexagonal rod pyramid: deformations and natural oscillation frequency
A new scheme of a statically determinate dome truss is proposed. The purpose of the study is to obtain exact formulas for structural deflections under a uniform load and to find upper and lower analytical estimates of the first frequency of natural oscillations depending on the number of panels, sizes, and masses concentrated in the truss nodes. Calculation of forces in the truss rods is performed by cutting nodes. The system of equations in projections on the coordinate axes, compiled in the Maple software, includes the forces in the rods and the reactions of vertical supports located along two contours of the structure at the base. The amount of deflection and stiffness of the entire truss is calculated using the Mohr integral. To determine the lower estimate of the first frequency an approximate Dunkerley method is used. The formula for the upper limit of the first frequency is derived by the Rayleigh energy method. In the Rayleigh method, the shape of the deflection from the action of a uniformly distributed load is taken as the deflection of the truss. Displacements of loads are assumed to be only vertical. The overall dependence of the solution on the number of panels is obtained by induction on a series of solutions for trusses with a successively increasing number of panels. The operators of the Maple system of symbolic mathematics are used. Based on the calculation results, it was concluded that the distribution of forces over the structure rods does not depend on the number of panels. Asymptotes were found on the graphs of the obtained analytical dependences of the deflection on the number of panels for different truss heights. The estimates of the first natural frequency are compared with the numerical solution obtained from the analysis of the natural frequency spectrum. The coefficients of the frequency equation are found 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 54 %, and the upper estimate by the Rayleigh method has an error of about 2 %. The formula for the lower Dunkerley frequency estimate is simpler than the Rayleigh estimate.