# Ritz and finite element methods application to aeroelastic vibrations computation of a cruise missile

### Аuthors

Blagodyreva O. V.

Tactical Missile Corporation, 7, Lenin str., Korolev, Moscow region, 105005, Russia

e-mail: OksanaBlag@yandex.ru

### Abstract

The article studies the aeroelastic stability of a cruise missile performing longitudinal short-period motion in incompressible airflow. The missile is modeled using a beam structure, including the fuselage, two straight outer wings and flight control surfaces — elevators. The wing of the missile is considered as an elastic beam operating in bending with transverse shear and torsion, while the fuselage and the control surfaces of the missile are considered as rigid. It is assumed, that the elastic vibrations of the wing panels occur according to the hypothesis of flat sections. Control drives with wiring are considered disconnected, and their effect on the structure is replaced with the unknown control forces. Aerodynamic loads are determined based on the quasi-stationary theory of plane-parallel flow cross sections of the wing. The longitudinal compression of the missile body under the effect of thrust force of an engine is also accounted for.
The equations of small vibrations of the system are represented in the form of Lagrange equations of the second kind in generalized coordinates. The generalized coordinates are considered as normal coordinates, representing the movement over the eigenvibration mode of a free structure with fixed operating controls.
Based on the Ritz method the unknown functions of transverse displacements of the axis of the fuselage, the transverse displacements of the axis of the wing and the angle of twist of the wing are presented in the form of expansions along the generalized coordinates.
For a more accurate study of the behavior of a wing’s flexural-torsional oscillations the finite element method (FEM) was used. The wing is separated into a number of compartments — finite elements (FE), within each the geometrical, stiffness and mass properties are assumed to be average and constant.
The article presents a comparative calculation of the flutter of an elastic missile using the Ritz method and the finite element method (FEM). The graphs of the variation of natural frequencies of the missile depending on the changes of the flight speed and engine thrust power are plotted. The critical speed and the stability region of the missile were defined respectively for each method.
All the calculations were performed with the software “Wolfram Mathematica 8”.

### Keywords:

aeroelastic vibrations, flutter, Ritz method, finite elements method, flying vehicle vibrations modeling

### References

1. Obraztsov I.F., Savel’ev L.M., Khazanov Kh.S. Metod konechnykh elementov v zadachakh stroitel’noi mekhaniki letatel’nykh apparatov (Finite element method in problems of structural mechanics and aircraft), Moscow, Vysshaia shkola, 1985, 392p.

2. Grossman E.P. Kurs vibratsii chastei samoleta (Course of aircraft parts vibration), Moscow, Gosudarstvennoe izdatel’stvo oboronnoi promyshlennosti, 1940, 312 p.

3. Zamlins’kii O.S. Informatsionnye sistemy, mekhanika i upravlenie, 2011, no.6, pp. 89-95.

4. Kicheev V.E. Trudy MAI, 2007, no. 27, available at: http://trudymai.ru/eng/published.php?ID=34006

5. Blagodyreva O.V. Trudy MAI, 2013, no. 68, available at http://trudymai.ru/eng/published.php?ID=41717

6. Blagodyreva O.V. Trudy MAI, 2014, no. 74, available at http://trudymai.ru/eng/published.php?ID=49345

7. Martynova A.D. Molodezhnyi nauchno-tekhnicheskii vestnik, 2014, no. 10, available at: http://sntbul.bmstu.ru/archive.html

8. VoityshenV.S., Semenov V.N. Izvestiia Komi nauchnogo tsentra UrO RAN, 2013, vol. 16, no.4, pp. 68-72.

9. Zheltkov V.I., Chyong Van Khuan. Izvestiia Tul’skogo gosudarstvennogo universiteta. Estestvennye nauki, 2014, no. 3, pp. 71-80.

10. Grishanina T.V. Izvestiia vuzov. Aviatsionnaia tekhnika, 2004, no. 2, pp. 10-13.

11. Razbegaeva I.A. Trudy MAI, 2011, no. 45, available at: http://trudymai.ru/eng/published.php?ID=25552

12. Nedelin V.G. Trudy MAI, 2012, no. 52, available at: http://trudymai.ru/eng/published.php?ID=29424

13. Parkhaev E.S., Semenchikov N.V. Trudy MAI, 2015, no. 80, available at http://trudymai.ru/eng/published.php?ID=56884

14. Komissarenko A.I. Vestnik Moskovskogo Aviatsionnogo Instituta, 2015, vol. 22, no. 2. pp. 36-41.

15. Grishanina T.V., Shkliarchuk F.N. Izvestiia vuzov. Aviatsionnaja tekhnika, 2009, no. 2, pp. 3-7.

16. Shkliarchuk F.N. Aerouprugost’ samoleta (Aeroelasticity of the aircraft). Moscow, Izd-wo MAI, 1985, 77 p.

17. Grishanina T.V., Shkliarchuk F.N. Dinamika uprugikh upravliaemykh konstruktsii (Dynamics of elastic manageable designs), Moscow, Izd-wo MAI, 2007, 328 p.

18. Grishanina T.V., Shkliarchuk F.N. Izbrannye zadachi aerouprugosti (Selected problems of aeroelasticity), Moscow, Izd-wo MAI, 2007, 48 p.

19. Rabinovich B.I. Prikladnye zadachi ustoichivosti stabilizirovannykh ob’ektov (Applied problems of stability of the stabilized objects), Moscow, Mashinostroenie, 1978, 232 p.

20. Chetaev N.G. Ustoichivost’ dvizheniia (Stability of motion), Moscow, «NAUKA» Farmalit, 1990, 176 p.

21. Mikishev G.N., Rabinovich B.I. Dinamika tonkostennykh konstruktsii s otsekami, soderzhashchimi zhidkost’ (Dynamics of thin-walled structures with compartments containing fluid), Moscow, Mashinostroenie, 1971, 564 p.

22. Lebedev A.A., Chernobrovkin L.S. Dinamika poleta bespilotnykh letatel’nykh apparatov (Flight dynamics of unmanned aerial vehicles), Moscow, Mashinostroenie, 1973, 616 p.

23. Grishanina T.V., Shkliarchuk F.N. Kolebaniia uprugikh sistem (Vibrations of elastic systems), Moscow, Izd-wo MAI, 2013, 100 p.