Model of a spatial dome cover. Deformations and oscillation frequency

The object of research. A new scheme of a statically determinate spatial truss is considered. The design has a hexagonal dome resting on two belts. The belts are supported by vertical racks. Two corner supports have spherical and cylindrical hinges. The outer support contra consists of 6n  horizontal rods, the inner one consists of 6(n-1) rods. The contours are connected by skews. Formulas are derived for the deflection of the vertex and the angular hinge depending on n. The upper and lower analytical estimates of the first frequency of natural oscillations of the structure are found. Method. Calculation of the forces in the rods is carried out by cutting out the nodes from the solution of the system of equilibrium equations for all nodes in the projection on the coordinate axes. To derive formulas for the dependence of breakdowns, forces, and the frequency of free oscillations, an inductive generalization of the sequence of solutions for structures with a different number of panels is used. The structural stiffness matrix and deflection are calculated using the Maxwell - Mohr formula in analytical form. To find estimates of the lowest frequency of vibrations of nodes endowed with masses, the Dunkerley and Rayleigh methods are used. Results. The vertical load distributed over the nodes and the concentrated load applied to the top are considered. Formulas for the forces in the characteristic bars of the structure are derived. A picture of the distribution of forces throughout the structure is presented. The resulting formulas for the deflection and frequency estimates have a compact form. The upper estimate of the first oscillation frequency of nodes under the assumption of vertical displacements of points has fairly high accuracy. The analytical solution is compared with the lowest oscillation frequency obtained numerically. All analytical transformations are performed in the Maple symbolic mathematics system. Some asymptotics of solutions is found.