Kinematic analysis and estimation of the frequency of natural oscillations of a planar lattice

Abstract

Introduction. A statically determined regular symmetrical rectangular lattice is fixed on two supports. The mass of the lattice is distributed over its nodes. A kinematic analysis is presented that reveals the variability of the lattice for a certain number of panels. An algorithm for searching for the distribution of virtual velocities of lattice nodes in this case is described. A formula is derived for the dependence of the lower estimate of the first frequency of natural oscillations of the structure on the number of panels from the range of permissible values. Materials and methods. Calculation of forces in the bars of a statically determinate lattice is performed by the method of cutting out nodes using Maple computer mathematics system operators. To determine the rigidity of the structure, the Maxwel – Mohr formula is used on the assumption that the elastic moduli and cross-sectional areas of all lattice rods are the same. Results. It is shown that when the number of panels is a multiple of three, the determinant of the matrix of the system of equilibrium equations degenerates, and the system becomes an instantly variable mechanism. A corresponding picture of the distribution of knot velocities is given. Using the approximate Dunkerley estimate, a formula is derived for the dependence of the lower limit of the first oscillation frequency of the truss on the number of panels. The generalization of a series of particular solutions to an arbitrary number of panels is performed in the Maple system by induction. Conclusions. The proposed lattice model admits an analytical solution for the lower estimate of the first natural frequency. Comparison of the result with the numerical solution obtained for the lowest frequency of the entire frequency spectrum of natural lattice vibrations shows its high accuracy. It is shown that the error of the found analytical estimate decreases with an increase in the number of panels. The solution can be used in optimization problems and for a preliminary assessment of the designed structure.