Bilateral Analytical Estimation of the First Frequency of a Plane Truss

TLDR: In this paper, the vertical displacement of nodes is derived from the Maxwell-Mohr method with the premise of linear elasticity, and Dunkerley's and Rayleigh's estimations of primary truss frequency on the number of panels are deduced from the inductive analysis of the set of particular trusses with an increasing number of columns.

Abstract: The object of research is the statically determinate cantilever truss. The trass consists of rectangular panels with downward diagonal beams. The truss has two supports, one of which is fixed hinged, and another one is roller support. Masses are located in the nodes of top and bottom chords. Forces in the bars and reactions at supports are determined using the method of joint isolation. The vertical displacement of nodes is derived from the Maxwell-Mohr method with the premise of linear elasticity. Dependence of vertical displacement, Dunkerley’s and Rayleigh’s estimations of primary truss frequency on the number of panels is deduced from the inductive analysis of the set of particular trusses with an increasing number of panels. Recurrence equations that meet particular coefficients are derived using special functions of the computer algebra system Maple. Obtained solutions are polynomial, with the number of panels as variables. Rayleigh’s quotient is calculated with the assumption that the first mode of vibration is equal to truss deflection under the uniformly distributed load. Graphs of the dependencies of obtained estimations on nodes masses, the number of panels, stiffness, and size of the truss are plotted.