Deformations and spatial structure vibrations frequency of the rectangular contour type cover: analytical solutions

The object of research. A new scheme of a statically determined truss in the form of a closed rectangle with vertical support columns along the inner contour is proposed. The cell of the regularity of the construction is a quadrangular rod pyramid. All cells are united along the vertices by a rod square contour. Four additional horizontal rod supports are located at the corners of the structure. When determining the deflection and forces in the critical rods, the vertical load, evenly distributed over the truss nodes, was considered. The derivation of the formula for the dependence of the deflection of an arbitrary hinge on the cantilever part of the truss on the number of panels in the truss is given. Method. The derivation of formulas for deflections, forces, and frequencies of free vibrations is based on an inductive generalization of the solution sequence for structures with a different number of panels. Forces are found from the solution of a system of linear equations for the equilibrium of nodes. The deflection and the stiffness matrix of the structure are calculated in an analytical form using the Maxwell-Mohr formula. To find the oscillation frequency of nodes endowed with masses, the Dunkerley method is used. Results. The formulas for the deflection of nodes have a compact form and allow you to calculate the deflection of an arbitrary point on the outer (cantilever) contour of the truss. The lower estimate of the first oscillation frequency of nodes under the assumption of vertical displacements of points has a relative error compared to the numerical solution of the problem of the spectrum of all frequencies non-monotonically dependent on the number of panels. The absolute error decreases as the number of panels increases. Solutions of systems of equilibrium equations for nodes and all transformations are made in the system of symbolic mathematics Maple. Linear asymptotics of solutions is found for some forces.