Formulas for calculating the natural frequency of a flat truss with additional supports
https://doi.org/10.22227/2305-5502.2026.1.9
Abstract
Introduction. A diagram of a statically determinate truss with parallel chords and an algorithm for deriving an analytical dependence of the magnitude of the structure deflection under the action of a uniformly distributed nodal load and the first frequency of natural oscillations on the number of panels are proposed.
Materials and methods. The material of the truss rods is elastic, the hinges are ideal, the load is nodal. The truss is externally statically indeterminate. All calculations of forces in symbolic form and transformations are performed in the computer mathematics system. The Maxwell – Mohr formula is used to calculate the deflection. The formula for the first frequency is derived based on a version of the approximate Dunkerley method under the assumption that the truss mass is uniformly distributed over its nodes. The nodes perform vertical oscillations. To calculate the coefficients in the formulas for the dependence of deflection and frequency, the induction method is used with respect to the number of panels. The solution of the obtained recurrent equations is performed in the Maple computer mathematics system.
Results. The coefficients of the formula for calculating the deflection have the form of polynomials with respect to the number of panels of a degree not higher than fourth. It was found that for a certain number of panels the truss allows kinematic variability. An example of a kinematically consistent distribution of virtual velocities of truss nodes is given. The method used to estimate the oscillation frequency showed good accuracy in comparison with the numerical method, which takes into account all degrees of freedom of the adopted truss model. The calculation was performed for kinematically admissible numbers of panels. The general term of the sequence of such numbers is given.
Conclusions. The methods and algorithm used to estimate deformations and natural frequency have proven their efficiency and can be applied to similar calculations of regular structures. The found cases of kinematic variability indicate the need for kinematic verification of the rod structures used in practice.
About the Author
M. N. Kirsanov
National Research University “Moscow Power Engineering Institute” (MPEI)
Russian Federation
Russian Federation
Mikhail N. Kirsanov — Doctor of Physical and Mathematical Sciences, Professor of the Department of Robotics, Mechatronics, Dynamics and Strength of Machines
14 Krasnokazarmennaya st., Moscow, 111250
Scopus: 16412815600, ResearcherID: H-9967-2013, Google Scholar: FfoNGFwAAAAJ, IstinaResearcherID: 2939132
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Review
For citations:
Kirsanov M.N. Formulas for calculating the natural frequency of a flat truss with additional supports. Construction: Science and Education. 2026;16(1):138-151. (In Russ.) https://doi.org/10.22227/2305-5502.2026.1.9
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