Dependence of the first natural frequency of the trussed frame on the number of panels: Analytical solution

Abstract

The inertial properties of a planar model of a hinged supported frame-type truss are modeled by point masses at the nodes. Each mass has two degrees of freedom. The stiffness matrix of a statically determinate structure is determined by the Maxwell-Mohr formula. To find the forces in the truss rods in an analytical form, the method of cutting nodes is used, which is implemented in the Maple computer mathematics system. The lower bound of the first natural frequency is sought by the Dunkerley method. Individual solutions for trusses with a sequentially increasing number of panels are generalized to an arbitrary case using the double induction method according to the number of panels in the girder and the number of panels in the supporting side parts of the truss. The analytical dependence is compared with the numerical solution of the problem on the spectrum of natural frequencies of the structure. It is shown that with an increase in the number of panels, both in the girder and in the side racks, the accuracy of the analytical assessment increases.