Formula for estimating the fundamental frequency of a frame-type planar truss

An inductive algorithm is given for deriving the dependence of a planar truss's fundamental frequency of natural vibrations on the number of panels. The truss is statically determinate; the rods are elastic, and the joints of the rods in the nodes are articulated. The mass of the truss is evenly distributed over its nodes. Vertical vibrations of nodes are considered. The approximate Dunkerley method is used to calculate the lower bound of the fundamental frequency. The forces in the rods are determined by the method of cutting nodes. The stiffness of the structure is calculated using the Maxwell - Mohr formula. The sequence of solutions for trusses with a different number of panels is generalized to an arbitrary case by induction. The equilibrium conditions for nodes are reduced to solving a system of linear algebraic ones in the Maple computer mathematics system. The found analytical solution is compared with the numerical solution obtained in the Maple system as the lowest frequency of the entire spectrum and with an independent numerical solution using the finite element method in the SOLIDWORKS system. The error of the analytical solution compared to the numerical one does not exceed a few percent and decreases with an increase in the number of truss panels. Spectral constants, isolines, and a resonant safety region were discovered in the spectra of a family of regular trusses of various orders.