On special generalized coordinates application for studying joint flexural vibrations of the blades of the main rotor, which is fixed on the elastic-damping support

DOI: 10.34759/trd-2019-108-4


Zagordan A. A.*, Zagordan N. L.**,

Federal Research Center “Computer Science and Control” of Russian Academy of Sciences, 44-2, Vavilova str., Moscow, 119333, Russia

*e-mail: a.zagordan@gmail.com
**e-mail: zagordann@gmail.com


The article considers bending vibrations of the main rotor blades, linked through the “pylon–main rotor” support, as naturally stranded rods, in both traction plane and rotation plane. The general system of equations of vibrations for the “main rotor–support” dynamic system, and assumption resonant diagrams of the main rotor with corresponding amplitude-frequency characteristics in the form of dependencies of changes of coefficients dynamic by frequency for the loads, passing to the main rotor shaft, are presented.

To transform the system of linear integro-differential equations with partial derivatives and periodic coefficients, governing the joint bending oscillations of the main rotor blades and the main rotor hub on an elastic damping support, into the system of ordinary differential equations with periodic coefficients, Bubnov-Galerkin method was used. Transformation of the initial generalized coordinates governing each individual blade oscillations in rotating coordinate system into special generalized coordinates governing the joint oscillations of all rotor blades is demonstrated as well.

To analyze the computational results on forced vibrations being obtained, and collation with the experimental data on vibrations recording at the rotating parts of the hub, the motion pattern and vibratory processes character of the hub and blades, rotating in the rotating and non-rotating coordinate systems, are considered. The graphs, demonstrating the trajectory change of the point on the blade, and vibrational process character while transition from the rotating system to non-rotating one, are presented.

Computing of resonant diagrams and amplitude-frequency characteristics for MI-38 helicopter with six- and five-blade rotor with flexible plastic-composite blades. Comparative analysis of dynamic response of the “main rotor–support” system with various number of blades was performed together with corresponding amplitude-frequency characteristics plotting.

Based on the comparative analysis of the “main rotor–support” system dynamic performance for the two variants of the main rotor, and inference on the six-blade main rotor preference over the five-blade one was drawn for the MI-38 helicopter.


vibrations, composite blades, joint vibrations, bearing blade, Bubnov-Galerkin method, generalized coordinate system


  1. Biderman V.L. Teoriya mekhanicheskikh kolebanii (Theory of mechanical oscillations), Moscow, Vysshaya shkola, 1980, 408 p.

  2. Baskin V.E., Vil’dgrube JI.C., Vozhdaev B.C., Maikapar G.N. Teoriya nesushchego vinta (Theory of main rotor), Moscow, Mashinostroenie, 1973, 364 p.

  3. Mil’ M.L. et al. Vertolety: raschet i proektirovanie (Helicopters: calculation and design), Moscow, Mashinostroenie, 1967, vol. 2, 424 p.

  4. Dzhonson U. Teoriya vertoleta (Helicopter theory), Moscow, Mir, 1983, vol. 1. – 502 p, vol. 2. – 529 p.

  5. Wnuk M.P. Nonlinear Fracture Mechanics, Vienna, Springer Vienna, 1990, 451 p.

  6. Vibratsii v tekhnike: spravochnik (Vibrations in Engineering), Moscow, Mashinostroenie, 1978, vol. 1, 358 p.

  7. Coleman R.P., Feingold A.M. Theory of self-excited mechanical oscillations of helicopter rotors with hinged blades. NACA Technical Report 1351, 1958, available at: https://digital.library.unt.edu/ark:/67531/metadc60767

  8. Ignatkin Yu.M., Makeev P.V, Shomov A.I. Trudy MAI, 2010, no. 38, available at: http://trudymai.ru/eng/published.php?ID=14148

  9. Chernous’ko F.L., Akulenko L.D., Sokolov B.N. Upravlenie kolebaniyami (Oscillations control), Moscow, Nauka, 1980, 384 p.

  10. Rikards R.B. Metod konechnykh elementov v teorii obolochek i plastin (Finite element method for shell and beam theory), Riga, Zinatne, 1988, 284 p.

  11. Zagordan A.A. Tekhnika vozdushnogo flota, 2007, no. 3 – 4, pp. 686 – 687.

  12. Mikheev S.V. Prikladnaya mekhanika v vertoletostroenii (Applied mechanics in helicopter engineering), Moscow, Al’teks, 2003, 264 p.

  13. Bramvell A.P.C. Dinamika vertoletov (Helicopter dynamics), Moscow, Mashinostroenie, 1982, 368 p.

  14. Bakhov O.P. Aerouprugost’ i dinamika konstruktsii vertoleta (Aeroelasticity and dynamics of helicopter design), Moscow, Mashinostroenie, 1985, 172 p.

  15. Chen P.C., Chopra I. Wind tunnel test of a smart rotor with individual blade twist control, Proceedings of the SPIE, 1997, vol. 3041, pp. 217 – 230.

  16. Sethi V., Song G. Pole. Placement Vibration Control of a Flexible Composite beam using Piezoceramic Sensors and Actuators, Journal of Thermoplastic Composite Materials, 2006, no. 19, pp. 293 – 308.

  17. Momterrubio L. and Sharfe I. Influence of landing gear on helicopter ground resonance, Canadian Aeronautics and Space Journal, vol. 48, issue 2, June 2002.

  18. Elizabeth M. Lee-Rausch, Robert T. Biedron, FUN3D Airloads Predictions for the Full-Scale UH-60A Airloads Rotor in a Wind Tunnel, Journal of the American Helicopter Society, 2014, vol. 59, no. 3, pp. 133 – 144.

  19. Ignatkin Yu.M., Makeev P.V., Shomov A.I. Trudy MAI, 2016, no. 87, available at: http://trudymai.ru/eng/published.php?ID=65636

  20. Golovkin M.A., Kochish S.I., Kritskii B.S. Trudy MAI, 2012, no. 55, available at: http://trudymai.ru/eng/published.php?ID=30023


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