Formulas for the first two frequencies of natural oscillations of a flat truss

Introduction. Calculation of the natural oscillation frequency is the basis of the study of structural dynamics and is usually based on numerical methods. In cases where the structure is statically determinate and has a periodic structure, analytical solutions are also possible for estimating the first natural frequency. The most well-known here are the Rayleigh method for estimating the frequency from above and the Dunkerley method, which gives an approximate estimate from below. In this paper, simple analytical estimates are derived for the dependences of the first oscillation frequencies of a flat truss on the number of panels and the parameters of the structure.

Materials and methods. A statically determinate beam truss has a rise in the middle part. A simplified version of the Dunkerley method is used for the analytical calculation of the first natural oscillation frequency. The forces in the rods included in the formula are calculated in symbolic form by cutting out nodes using standard operators of the Maple computer mathematics system. The Maxwell – Mohr formula is used to determine the rigidity of the structure. It is assumed that the truss mass is uniformly distributed over all its nodes. The induction method is used to generalize the sequence of individual solutions for trusses of different orders to an arbitrary number of panels. The formula for the second natural frequency is obtained by the three-point collocation method based on the condition of similarity of the curve of the first frequency dependence on the number of panels.

Results. Formulas for the first two frequencies of natural oscillations of the truss are derived. Analytical solutions are compared with numerical ones obtained for the entire frequency spectrum. It is shown that with an increase in the number of panels, the accuracy of the analytical solution increases.

Conclusions. The analytical method for estimating the first and second frequencies is applicable to solving problems on regular structures. The advantage of the method is that its accuracy is independent of the order of regularity of the structure. The simple form of the result allows it to be used to select the optimal parameters of the object without the use of labor-intensive computer calculations.

1. Kovalenko G.V., Makeev V.B., Dementyeva V.V. Study of natural vibration frequencies of trusses based on the finite element method (FEM). Young Thought: Science, Technology, Innovation : Proceedings of the VII (XIII) All-Russian Scientific and Technical Conference of Students, Master’s Students, Postgraduates and Young Scientists. 2015; 44-48. EDN WWZURN. (rus.).

2. Seventekidis P., Giagopoulos D. Generalization gap estimation for damage classification tasks when finite element simulated data is used for training: A numerical study and experimental validation on a steel truss. Engineering Structures. 2024; 319:118855. DOI: 10.1016/j.engstruct.2024.118855

3. Ignatiev V.A., Ignatiev A.V. Finite element method in the form of the classical mixed method of structural mechanics (theory, mathematical models and algorithms). Moscow, ASV Publishing House, 2022; 306. EDN CSOLBF. (rus.).

4. Rapp B.E. Introduction to Maple. Microfluidics: Modelling, Mechanics and Mathematics. 2017; 9-20. DOI: 10.1016/b978-1-4557-3141-1.50002-2

5. Zotos K. Performance comparison of Maple and Mathematica. Applied Mathematics and Computation. 2007; 188(2):1426-1429. DOI: 10.1016/j.amc.2006.11.008

6. Goloskokov D.P. Analyzing simply supported plates using maple system. 2014 International Conference on Computer Technologies in Physical and Engineering Applications (ICCTPEA). 2014; 55-56. DOI: 10.1109/ICCTPEA.2014.6893273

7. Goloskokov D.P., Matrosov A.V. Approximate analytical solutions in the analysis of thin elastic plates. AIP Conference Proceedings. 2018; 1959:070012. DOI: 10.1063/1.5034687

8. Matrosov A.V., Suratov V.A. Stress-strain state in the corner points of a clamped plate under uniformly distributed normal load. Materials Physics and Mechanics. 2018; 36(1):124-146. DOI: 10.18720/MPM.3612018_16. EDN XOBJZJ.

9. Kirsanov M.N. Planar trusses. Deflections and oscillation frequencies : reference book. Moscow, INFRA-M, 2025; 196. DOI: 10.12737/2173835. EDN FRTTXN. (rus.).

10. Sviridenko O.V., Komerzan E.V. The dependence of the natural oscillation frequency of the console truss on the number of panels. Construction of Unique Buildings and Structures. 2022; 3(101):10101. DOI: 10.4123/CUBS.101.1. EDN CKQDPU.

11. Luong Cong L. Dependence of the region of resonance-safe frequencies on the sizes of statically determinable flat truss. Structural Mechanics and Structures. 2024; 2(41):16-26. DOI: 10.36622/2219-1038.2024.41.2.002. EDN AKAVDU. (rus.).

12. Luan L.C. Estimates of deflection and natural frequency of vibrations of a hinged-rod truss with an arbitrary number of panels. Structural Mechanics and Structures. 2024; 4(43):42-53. DOI: 10.36622/2219-1038.2024.43.4.004. EDN MGQUXN. (rus.).

13. Luong C.L. Resonance safety zones of a truss structure with an arbitrary number of panels. Construction of Unique Buildings and Structures. 2024; 4(113):11304. DOI: 10.4123/CUBS.113.4. EDN FLBVJY.

14. Astakhov S.V. Analytical assessment of the deflection of the rod model of a four-slope roof frame. Structural Mechanics and Structures. 2024; 4(43):34-41. DOI: 10.36622/2219-1038.2024.43.4.003. EDN GITKYF. (rus.).

15. Maslov A.N. The first natural frequency of a planar regular truss. Analytical solution. Construction of Unique Buildings and Structures. 2023; 4(109):10912. DOI: 10.4123/CUBS.109.12. EDN QNZNZW.

16. Komerzan E.V., Maslov A.N. Estimation of the L-shaped spatial truss fundamental frequency oscillations. Structural Mechanics and Structures. 2023; 2(37):35-45. DOI: 10.36622/VSTU.2023.37.2.004. EDN UGWBIP. (rus.).

17. Vychuzhina Z.K. Calculation of the flat truss natural oscillations first frequency. Structural Mechanics and Structures. 2023; 3(38):136-142. DOI: 10.36622/VSTU.2023.38.3.013. EDN FQOHWL. (rus.).

18. Komerzan E.V., Ninalalov I.G., Sviridenko O.V. Calculation of a planar model composite truss fundamental frequency. Structural Mechanics and Structures. 2023; 4(39):27-34. DOI: 10.36622/VSTU.2023.39.4.003. EDN DLVBCD. (rus.).

19. 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 [Monthly Journal on Construction and Architecture]. 2020; 15(4):510-517. DOI: 10.22227/1997-0935.2020.4.510-517. EDN GGKIPC. (rus.).

20. Galishnikova V.V., Ignatiev V.A. Regular rod systems. Theory and calculation methods. Volgograd, VolgGASU, 2006. EDN QNMMHT. (rus.).

21. Petrichenko E.A. Lower bound of the natural oscillation frequency of the fink truss. Structural Mechanics and Structures. 2020; 3(26):21-29. EDN PINHFN. (rus.).

22. Low K.H. A modified Dunkerley formula for eigenfrequencies of beams carrying concentrated masses. International Journal of Mechanical Sciences. 2000; 42(7):1287-1305. DOI: 10.1016/s0020-7403(99)-00049-1