Method for calculating elastic vibrations of a cyclically symmetric structure

DOI: 10.34759/trd-2021-121-05

Аuthors

Grishanina T. V.^*, Guseva E. E.^**

Moscow Aviation Institute (National Research University), 4, Volokolamskoe shosse, Moscow, А-80, GSP-3, 125993, Russia

*e-mail: t.grishanina@mai.ru
**e-mail: evgenia-kesha@mail.ru

Abstract

The article presents a new approach to the oscillation equations composing of elastic structures with cyclically symmetric structure. A cyclically symmetric elastic system is under consideration. At the k = 0, 1, 2, ..., N — 1 nodes of the system, located on a circle at equidistant points with angular coordinates of θ_k, the identic elastic rods of constant cross-section are being connected, working in tension-compression, torsion and bending-shear in and out of the plane of the system. To compose the oscillations equations of the system, both displacement and rotation angles components, symmetrical with respect to the radial plane passing through the k-th node, are being represented as cosine expansions in the circumferential direction, while skew symmetric ones are represented as sine expansions with wave numbers of n = 0, 1, 2, ..., N/2.

Expressions of potential and kinetic energies for all elements of the system are being composed. With account for the cos nθ_k and sin nθ_k functions orthogonality conditions for different n on a system of equidistant points, the terms of these expressions for different n from the set of n = 0, 1, 2, ..., are being obtained uncoupled among themselves. As the result, the Lagrange equations in generalized coordinates for a cyclically symmetric system with 6N degrees of freedom disintegrate into separate groups of six equations for each number n being accounted for.

Thus, the solution of the problem of a cyclically symmetric system oscillations of еру 6N order is being reduced to solving a number of problems for uncoupled subsystems of equations of the sixth order, representing separately harmonics n = 0, 1, 2, ..., ≤ N/2.

Keywords:

cyclically symmetric systems, rod systems, antenna, oscillation equations, uncoupled equations

References

1. Usyukin V.I. Stroitel’naya mekhanika konstruktsii kosmicheskoi tekhniki (Construction mechanics of space technology structures), Moscow, Mashinostroenie, 1988, 392 p.
2. Lopatin A.V., Rutkovskaya M.A. Vestnik Sibirskogo gosudarstvennogo universiteta nauki i tekhnologi imeni Akademika M.F. Reshetneva, 2007, no. 2, pp. 51–57.
3. Lopatin A.V., Rutkovskaya M.A. Vestnik Sibirskogo gosudarstvennogo universiteta nauki i tekhnologi imeni Akademika M.F. Reshetneva, 2007, no. 3, pp. 78–81.
4. Sgadova N.A., Strulev I.M. Trudy MAI, 2010, no. 38. URL: http://trudymai.ru/eng/published.php?ID=14168
5. Russkikh S.V., Shklyarchuk F.N. Kosmonavtika i raketostroenie, 2020, no. 2 (113), pp. 86-98.
6. Russkikh S.V., Shklyarchuk F.N. Izvestiya Rossiiskoi akademii nauk. Mekhanika tverdogo tela, 2021, no. 5, pp. 99-112. DOI: 10.31857/S0572329921050093
7. Kui M.Kh. Trudy MAI, 2015, no. 82. URL: http://trudymai.ru/eng/published.php?ID=58836
8. Russkikh S.V., Nagornov A.Yu., Shavnya R.A. 17-ya Mezhdunarodnaya konferentsiya «Aviatsiya i kosmonavtika — 2018», Moscow, Lyuksor, 2018, pp. 341-342.
9. Gryanik M.V., Loman V.I. Razvertyvaemye zerkal’nye antenny zontichnogo tipa (Deployable umbrella-type mirror antennas), Moscow, Radio i svyaz’, 1987, 72 p.
10. Obraztsov I.F., Savel’ev L.M., Khazanov Kh.S. Metod konechnykh elementov v zadachakh stroitel’noi mekhaniki letatel’nykh apparatov (The finite element method in the problems of structural mechanics of aircraft), Moscow, Vysshaya shkola, 1985, 390 p
11. Banichuk N.V., Karpov I.I., Klimov D.M. et al. Mekhanika bol’shikh kosmicheskikh konstruktsii (Mechanics of large space structures), Izd-vo «Faktorial», 1997, 302 p.
12. Tsurkov V.I. Dinamicheskie zadachi bol’shoi razmernosti (Dynamic problems of large dimension), Moscow, Izd-vo «Nauka», 1998, 288 p.
13. Biderman V.L. Teoriya mekhanicheskikh kolebanii (Theory of mechanical vibrations), Moscow, Vysshaya shkola, 1980, 408 p.
14. Shklyarchuk F.N., Bobylev D.S. Voprosy prochnosti tonkostennykh konstruktsii (Issues of strength of thin-walled structures, collection of articles), Moscow, Izd-vo MAI, 1989, pp. 52-56.
15. Vlasov V.Z. Izbrannye Trudy (Selected works). Vol. I. Moscow, Izd-vo AN SSSR, 1962, 528 p.
16. Vlasov V.Z. Izbrannye trudy (Selected works). Vol. III. Moscow, Izd-vo AN SSSR, 1964, 472 p.
17. Grishanina T.V., Shklyarchuk F.N. Kolebaniya uprugikh system system (Vibrations of elastic systems), Moscow, Izd-vo MAI, 2013, 100 p.
18. Grishanina T.V. Zadachi po teorii kolebanii uprugikh system (Problems in the theory of oscillations of elastic systems), Moscow, Izd-vo MAI, 1998, 48 p.
19. Gnezdilov V.A., Grishanina T.V., Nagornov A.Yu. Trudy MAI, 2017, no. 95. URL: http://trudymai.ru/eng/published.php?ID=84435
20. Babakov I.M. Teoriya kolebanii (Theory of vibrations), Moscow, Nauka, 1968, 560 p.

Download