Bilateral Analytical Estimation of the Natural Oscillation Frequency of a Planar Triangular Truss

Building Materials

The object of research is a statically determinate planar truss. The scheme of a lattice console structure in the form of a triangle is considered. Also, a structural feature is the presence of upper and lower belts, as well as braces. Method. The calculation of forces in the rods is carried out by cutting nodes from the solution of the system of equilibrium equations for all nodes in the projection on the coordinate axis. To derive formulas for the dependence of 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 is calculated using the Maxwell-Mohr formula in analytical form. Dunkerley and Rayleigh methods are used to find estimates of the lowest oscillation frequency of nodes endowed with masses. All mathematical transformations are performed in the Maple symbolic mathematics system. Results. Under the assumption that vibrations of loads concentrated in the nodes of the structure occur only along the vertical, and the rigidities of all rods are the same, compact formulas are obtained for upper and lower estimates of the first (main) frequency of natural oscillations of the system for an arbitrary number of panels. The upper estimate of the first oscillation frequency of the nodes  has a rather higher than the lower estimate. The analytical solution is compared with the lowest oscillation frequency obtained numerically. The accuracy of the upper estimate of the frequency is very high and almost independent of the construction order.