Analytical assessment of the frequency of natural vibrations of a truss with an arbitrary number of panels

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The aim of the work is to derive a formula for the dependence of the first frequency of the natural oscillations of a planar statically determinate beam truss with parallel belts on the number of panels, sizes and masses concentrated in the nodes of the lower truss belt. Truss has a triangular lattice with vertical racks. The solution uses Maple computer math system operators. Methods. The basis for the upper estimate of the desired oscillation frequency of a regular truss is the energy method. As a form of deflection of the truss taken deflection from the action of a uniformly distributed load. Only vertical mass movements are assumed. The amplitude values of the deflection of the truss is calculated by the Maxwell - Mohr’s formula. The forces in the rods are determined in symbolic form by the method of cutting nodes. The dependence of the solution on the number of panels is obtained by an inductive generalization of a series of solutions for trusses with a successively increasing number of panels. For sequences of coefficients of the desired formula, fourth-order homogeneous linear recurrence equations are compiled and solved. Results. The solution is compared with the numerical one, obtained from the analysis of the entire spectrum of natural frequencies of oscillations of the mass system located at the nodes of the truss. The frequency equation is compiled and solved using Eigenvalue search operators in the Maple system. It is shown that the obtained analytical estimate differs from the numerical solution by a fraction of a percent. Moreover, with an increase in the number of panels, the error of the energy method decreases monotonically. A simpler lower bound for the oscillation frequency according to the Dunkerley method is presented. The accuracy of the lower estimate is much lower than the upper estimate, depending on the size and number of panels.

About the authors

Mikhail N. Kirsanov

National Research University “Moscow Power Engineering Institute”

Author for correspondence.
14 Krasnokazarmennaya St, Moscow, 111250, Russian Federation

Doctor of Physical and Mathematical Sciences, Professor of the Department of Robotics, Mechatronics, Dynamics and Machine Strength of the Institute of Power Machinery and Mechanics


  1. Ufimtcev E. Dynamic Calculation of Nonlinear Oscillations of Flat Trusses. Part 2. Examples of Calculations. Procedia Engineering. 2017;206:850–856. doi: 10.1016/j.proeng.2017.10.561.
  2. Tejani G.G., Savsani V.J., Patel V.K., Mirjalili S. Truss optimization with natural frequency bounds using improved symbiotic organisms search. Knowledge-Based Systems. 2018;143:162–178. doi: 10.1016/j.knosys.2017.12.012.
  3. Ufimtsev E., Voronina M. Research of Total Mechanical Energy of Steel Roof Truss during Structurally Nonlinear Oscillations. Procedia Engineering. 2016;150:1891–1897. doi: 10.1016/j.proeng.2016.07.188.
  4. Jalbi S., Bhattacharya S. Closed form solution for the first natural frequency of offshore wind turbine jackets supported on multiple foundations incorporating soil-structure interaction. Soil Dynamics and Earthquake Engineering. 2018;113(May):593–613.
  5. Kilikevicius A., Fursenko A., Jurevicius M., Kilikeviciene K., Bureika G. Analysis of parameters of railway bridge vibration caused by moving rail vehicles. Measurement and Control (United Kingdom). 2019;52(9–10):1210–1219. doi: 10.1177/0020294019836123.
  6. Aldushkin R.V., Savin S.Yu. Investigation of the operation of triangular farms with static and dynamic effects. Construction and reconstruction. 2010;3(29):3–6. (In Russ.) Available from: 15503961_71987283.pdf (accessed: 01.05.2020).
  7. Bolotina T.D. The deflection of the flat arch truss with a triangular lattice depending on the number of panels. Bulletin of Scientific Conferences. 2016;(8)4–3:7–8.
  8. Timofeeva Т.А. Formulas for calculating the deflection of a flat lattice frame with an arbitrary number of panels. Structural mechanics and structures. 2019;4(23):26–33. (In Russ.)
  9. Boyko A.Y., Tkachuk G.N. Derivation of the formulas for the deflection of a flat hinged-rod frame in the of symbol mathematics Maple system. Structural mechanics and structures. 2019;4(23):15–25. (In Russ.)
  10. Belyankin N.A., Boyko A.Y. Formula for deflection of a girder with an arbitrary number of panels under the uniform load. Structural mechanics and structures. 2019;1(20):21–29. (In Russ.)
  11. Tkachuk G.N. The formula for the dependence of the deflection of an asymmetrically loaded flat truss with reinforced braces on the number of panels. Structural mechanics and structures. 2019;2(21):32–39. (In Russ.)
  12. Tinkov D.V. Comparative analysis of analytical solutions to the problem of deflection of truss structures. Magazine of Civil Engineering. 2015;5(57):66–73. (In Russ.)
  13. Hutchinson R.G., Fleck N.A. Microarchitectured cellular solids – the hunt for statically determinate periodic trusses. ZAMM Z. Angew. Math. Mech. 2005;9(85):607–617.
  14. Hutchinson R.G., Fleck N.A. The structural performance of the periodic truss. Journal of the Mechanics and Physics of Solids. 2006;54(4):756–782.
  15. Zok F.W., Latture R.M., Begley M.R. Periodic truss structures. Journal of the Mechanics and Physics of Solids. 2016;96:184–203.
  16. Rybakov L.S., Mishustin I.V. Natural vibrations of flat regular elastic trusses of orthogonal structure. Mechanics of composite materials and structures. 1999;2(5):3–16. (In Russ.)
  17. Mishustin I.V., Rybakov L.S. Oscillations of flat elastic trusses of orthogonal structure. News of the Academy of Sciences. Solid Mechanics. 2003;2:168–184. (In Russ.)
  18. Buka-Vaivade K., Kirsanov M.N., Serdjuks D.O. Calculation of deformations of a cantilever frame planar truss model with an arbitrary number of panels. Vestnik MGSU. 2020;15(4):510–517.
  19. Kirsanov M. Analytical Solution of a Spacer Beam Truss Deflection with an Arbitrary Number of Panels. Construction of Unique Buildings and Structures. 2020;88:8802.
  20. Kirsanov M.N., Tinkov D.V. Analytical expressions of the frequencies of small vibrations of a beam truss with an arbitrary number of panels. Structural Mechanics and Structures. 2019;20(1):14–20. (In Russ.)
  21. Tinkov D.V. Analytical solutions to problems on natural frequencies of oscillations of regular rod systems (Thesis of Candidate of Technical Sciences). Moscow; 2019. (In Russ.)
  22. WolframAlpha System. Available from: (accessed: 03.07.2020).



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Copyright (c) 2020 Kirsanov M.N.

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