Analytical Estimation of the Natural Oscillation Frequency of a Planar Lattice

Abstract

   Introduction. A new scheme of a flat statically determinate regular lattice is proposed. The lattice rods are hinged.   The study aims at deriving a formula for the dependence on the number of panels of the first natural oscillation frequency of nodes endowed with masses, each of which has two degrees of freedom in the lattice plane. The rigidity of all rods is assumed to be the same, the supports (movable and fixed hinges) — nondeformable.   Another objective of the study is to find the dependence of the stresses in the most compressed and stretched rods on the number of panels in an analytical form.   Materials and Methods. An approximate Dunkerley’s method was used to determine the lower bound for the lattice natural frequency. The lattice rigidity was found in analytical form according to Maxwell-Mohr formula. The rod stresses and the reactions of the supports were determined from the equilibrium equations compiled for all lattice nodes. Generalization of the result to an arbitrary number of panels was performed by induction using Maple symbolic math operators for analytical solutions to a number of problems for lattices with different number of panels.   Results. The lower analytical estimate of the first oscillation frequency was in good agreement with the numerical solution for the minimum frequency of the oscillation spectrum of the structure. Formulas were found for the stresses in four most compressed and stretched rods and their linear asymptotics. All required transformations were made in the system of Maple symbolic math.   Discussion and Conclusions. The obtained dependence of the first frequency of lattice oscillations on the number of panels, mass and dimensions of the structure has a compact form and can be used as a test problem for numerical solutions and optimization of the structure.