Calculating of a spatial cantilever truss natural vibration frequency with an arbitrary number of panels: analytical solution
Автор: Kirsanov Mikhail Nikolaevich, Vorobev Oleg Vladimirovich
Статья в выпуске: 1 (94), 2021 года.
The research object was a spatial cantilever statically determinate truss composed of three planar trusses with a triangular lattice. The spectrum of natural frequencies of the structure was analyzed. The same concentrated masses model the inertial properties of the truss at the nodes. The task was to obtain an analytical dependence of the lowest vibration frequency of a truss on the number of panels, mass, linear dimensions of the structure, and material properties. Method. The induction method and the Maple computer mathematics system operators were used to determine the forces in the rods and generalize the result to an arbitrary number of panels. The problem was solved using the Dunkerley approach, which gives a lower frequency estimate. Maxwell-Mohr's formula determines the rigidity of the structure. Homogeneous linear recurrent equations were compiled and solved to find the common members of the sequences of coefficients in the formula for the frequency. Results. The accuracy of the formula obtained by the Dunkerley method was estimated from comparison with a numerical calculation of the entire spectrum of natural frequencies. The comparison shows the good accuracy of the derived formula. As the number of panels increases, the accuracy of the lower estimate increases too. The same frequency for all trusses was found in the spectra of trusses with a different number of panels. This frequency is the spectral constant and depends only on the size of the system, the stiffness of the members, and the mass. The existence of spectral isolines with the property of asymptotically tending to a certain constant value was shown.
Truss, vibration, dunkerley method, spectral constant, spectral isolines, symbolic solution, natural frequencies
Короткий адрес: https://readera.org/143175780
IDR: 143175780 | DOI: 10.4123/CUBS.94.2
Список литературы Calculating of a spatial cantilever truss natural vibration frequency with an arbitrary number of panels: analytical solution
- Cao, L., Liu, J., Chen, Y.F. Vibration performance of arch prestressed concrete truss girder under impulse excitation. Engineering Structures. 2018. 165. Pp. 386–395. DOI:10.1016/j.engstruct.2018.03.050.
- Zhang, X., Li, Q., Wang, Y., Wang, Q. Vibration of a U-shaped steel–concrete composite hollow waffle floor under human-induced excitations. Advances in Structural Engineering. 2020. 23(14). Pp. 2996–3008. DOI:10.1177/1369433220927278.
- Li, J., Zhang, R., Liu, J., Cao, L., Chen, Y.F. Determination of the natural frequencies of a prestressed cable RC truss floor system. Measurement: Journal of the International Measurement Confederation. 2018. 122. Pp. 582–590. DOI:10.1016/j.measurement.2017.08.048.
- Liu, M., Cao, D., Zhang, X., Wei, J., Zhu, D. Nonlinear dynamic responses of beamlike truss based on the equivalent nonlinear beam model. International Journal of Mechanical Sciences. 2021. 194. Pp. 106197. DOI:10.1016/j.ijmecsci.2020.106197.
- Kapuria, S., Ahmed, A. A coupled efficient layerwise finite element model for free vibration analysis of smart piezo-bonded laminated shells featuring delaminations and transducer debonding. International Journal of Mechanical Sciences. 2021. 194. DOI:10.1016/j.ijmecsci.2020.106195.
- Zotos, K. Performance comparison of Maple and Mathematica. Applied Mathematics and Computation. 2007. 188(2). Pp. 1426–1429. DOI:10.1016/j.amc.2006.11.008.
- Rapp, B.E. Introduction to Maple. Microfluidics: Modelling, Mechanics and Mathematics. Elsevier, 2017. Pp. 9–20.
- Rakhmatulina, A. R. , Smirnova, A.A. Analytical calculation and analysis of planar springel truss. Structural mechanics and structures. 2018. 17(2). Pp. 72–79. URL: http://vuz.exponenta.ru/PDF/NAUKA/Rahm-Smirn2018-2.pdf (date of application: 27.02.2021).
- Ilyushin, A.S. The formula for calculating the deflection of a compound externally statically indeterminate frame. Structural mechanics and structures. 2019. 22(3). Pp. 29–38. URL: https://elibrary.ru/item.asp?id=41201106 (date of application: 27.02.2021).
- Kirsanov M. Trussed Frames and Arches: Schemes and Formulas - Cambridge Scholars Publishing. URL: https://www.cambridgescholars.com/product/978-1-5275-5976-9 (date of application: 27.02.2021).
- Kirsanov M. Planar Trusses: Schemes and Formulas - Cambridge Scholars Publishing. URL: https://www.cambridgescholars.com/product/978-1-5275-3531-2 (date of application: 27.02.2021).
- Arutyunyan, V.B. Analytical calculation of the deflection street bracket for advertising. Postulat. 2019. 1. URL: http://vuz.exponenta.ru/PDF/NAUKA/Arut2019-1.pdf (date of application: 27.02.2021).
- 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). Pp. 1210–1219. DOI:10.1177/0020294019836123.
- Pekcan, G., Itani, A.M., Linke, C. Enhancing seismic resilience using truss girder frame systems with supplemental devices. Journal of Constructional Steel Research. 2014. 94. Pp. 23–32.
- Martins, A.M.B., Simões, L.M.C., Negrão, J.H.J.O. Optimization of extradosed concrete bridges subjected to seismic action. Computers and Structures. 2021. 245. DOI:10.1016/j.compstruc.2020.106460.
- Resatalab, S., Ahmadi, M.T., Alembagheri, M. Seismic response sensitivity analysis of intake towers interacting with dam, reservoir and foundation. Magazine of Civil Engineering. 2020. 98(7). Pp. 9901–9901. DOI:10.18720/MCE.99.1. URL: https://engstroy.spbstu.ru/article/2020.99.1 (date of application: 27.02.2021).
- Tarasov, V.A. Double Seismic Insulation System of Turbine Unit Foundation. Construction of Unique Buildings and Structures. 2020. 91(6). Pp. 9101–9101. DOI:10.18720/CUBS.91.1. URL: https://unistroy.spbstu.ru/article/2020.91.1 (date of application: 27.02.2021).
- Lardeur, P., Arnoult, É., Martini, L., Knopf-Lenoir, C. The Certain Generalized Stresses Method for the static finite element analysis of bar and beam trusses with variability. Finite Elements in Analysis and Design. 2012. 50. Pp. 231–242. DOI:10.1016/j.finel.2011.09.013.
- Liu, M., Cao, D., Zhu, D. Coupled vibration analysis for equivalent dynamic model of the space antenna truss. Applied Mathematical Modelling. 2021. 89. Pp. 285–298. DOI:10.1016/j.apm.2020.07.013.
- Kirsanov, M.N., Buka-Vaivade, K. Analytical expressions of frequencies of small oscillations of a beam truss with an arbitrary number of panels. Structural mechanics and structures. 2019. 23(4). Pp. 7–14. URL: http://vuz.exponenta.ru/PDF/NAUKA/Karina23.pdf (date of application: 27.02.2021).
- Hutchinson, R.G., Fleck, N.A. The structural performance of the periodic truss. Journal of the Mechanics and Physics of Solids. 2006. 54(4). Pp. 756–782. DOI:10.1016/j.jmps.2005.10.008.
- Zok, F.W., Latture, R.M., Begley, M.R. Periodic truss structures. Journal of the Mechanics and Physics of Solids. 2016. 96. Pp. 184–203. DOI:10.1016/j.jmps.2016.07.007.
- Serpik, I.N., Alekseytsev, A. V. Optimization of flat steel frame and foundation posts system. Magazine of Civil Engineering. 2016. 61(1). Pp. 14–24. DOI:10.5862/MCE.61.2. URL: https://engstroy.spbstu.ru/article/2016.61.2 (date of application: 27.02.2021).
- Degertekin, S.O., Yalcin Bayar, G., Lamberti, L. Parameter free Jaya algorithm for truss sizing-layout optimization under natural frequency constraints. Computers and Structures. 2021. 245. Pp. 106461. DOI:10.1016/j.compstruc.2020.106461.
- Lamberti, L. An efficient simulated annealing algorithm for design optimization of truss structures. Computers and Structures. 2008. 86(19–20). Pp. 1936–1953. DOI:10.1016/j.compstruc.2008.02.004.
- Serpik, I.N., Mironenko, I. V., Averchenkov, V.I. Algorithm for evolutionary optimization of reinforced concrete frames subject to nonlinear material deformation. Procedia Engineering. 2016. 150. Pp. 1311–1316. DOI:10.1016/j.proeng.2016.07.304.
- Low, K.H. Modified Dunkerley formula for eigenfrequencies of beams carrying concentrated masses. International Journal of Mechanical Sciences. 2000. 42(7). Pp. 1287–1305. DOI:10.1016/S0020-7403(99)00049-1.
- Levy, C. An iterative technique based on the Dunkerley method for determining the natural frequencies of vibrating systems. Journal of Sound and Vibration. 1991. 150(1). Pp. 111–118. DOI:10.1016/0022-460X(91)90405-9.
- Trainor, P.G.S., Shah, A.H., Popplewell, N. Estimating the fundamental natural frequency of towers by Dunkerley’s method. Journal of Sound and Vibration. 1986. 109(2). Pp. 285–292. DOI:10.1016/S0022-460X(86)80009-8.
- Low, K.H. Frequencies of beams carrying multiple masses: Rayleigh estimation versus eigenanalysis solutions. Journal of Sound and Vibration. 2003. 268(4). Pp. 843–853. DOI:10.1016/S0022-460X(03)00282-7.