The formula for the first natural frequency and the frequency spectrum of a spatial regular truss

Abstract

A scheme of a statically determinate spatial truss is proposed. The gable cover of the structure is formed by isosceles rod triangles with supports in the form of racks on the sides. A formula is derived for the lower boundary of the structure’s first natural frequency under the assumption that its mass is concentrated in the nodes. To calculate the stiffness of the truss according to the Maxwell - Mohr formula, the forces in the rods are found by cutting out the nodes in an analytical form. The lower limit of the fundamental frequency is calculated using the Dunkerley partial frequency method. A series of solutions obtained for trusses with a different number of panels is generalized to an arbitrary order of a regular truss by induction using Maple symbolic mathematics operators. Comparison of the analytical solution with the numerical value of the first frequency of the spectrum shows good agreement between the results. The spectra of a series of regular trusses of various orders are analyzed. Two spectral constants of the problem are found, one of which is the highest frequency of truss vibrations, which does not depend on their order.

Keywords

• spatial truss
• dunkerley method
• fundamental frequency
• analytical solution
• natural oscillations
• regular construction
• spectrum
• spectral constant
• induction method
• maxwell-mohr formula