Mathematical model for estimation of the effect of aerodynamic aircraft control surfaces compensation on the level of their vibrations in case of transonic flutter
Heading:
^{1}Safronov, AV, ^{1}Semon, BY, ^{1}Nedilko, AN ^{1}The National Defence University of Ukraine named after Ivan Cherniakhovsyi, Kyiv, Ukraine 
Space Sci.&Technol. 2018, 24 ;(4):1423 
https://doi.org/10.15407/knit2018.04.015 
Publication Language: Ukrainian 
Abstract: There are cases of intense fluctuations of supersonic aircraft and aviationspace aircrafts structures elements around the sound numbers M in the current practice of their exploitation. Some of these cases ended in serious flight events. That is why the possibility of reducing oscillations level of aerodynamic control surfaces for supersonic and hypersonic airplanes in the event of transonic flutter is actual.
Our purpose is to evaluate the possibility of reducing the level of oscillation due to the aerodynamic compensation of the surface of the control and to develop a mathematical model for assessing the impact of aerodynamic compensation of the surfaces of control on the level of their oscillations in the event of a transonic flutter. In order to solve this problem, an analysis of publications devoted to this topic has been conducted, in which the materials of laboratory and field studies of this phenomenon are presented. Since the fluctuations of aerodynamic control surfaces in the event of a transonic flutter are referred to as selfoscillations with the amplitude of the boundary cycle, so they are referred to as nonlinear oscillations. The mathematical model for evaluating the effect of aerodynamic compensation of the control surface on the level of their oscillations is presented in the form of an ordinary differential equation of the second order with a nonlinear righthand side. To solve the differential equation of this type, the energy balance method was used to determine the amplitude of the oscillations of the aerodynamic surfaces of control. According to the results of calculations, which were conducted with the help of the proposed model, the dependence of the oscillations level of aerodynamic surfaces of control on the magnitude of their aerodynamic compensation is obtained. In our opinion, it is useful to take it into account for the preliminary approximation of the aerospace characteristics of supersonic and hypersonic aircraft.

Keywords: aerodynamic aircraft control surface, aerodynamic compensation, mathematical model, supersonic aircraft, the hinge moment, transonic flutter 
References:
1. Aerodynamic study of an oscillating control surface at transonic velocities, Moscow: TsAGI, 1975, No. 456, 105 р. [in Russian].
2. Goshek I. (1954). Aerodynamics of high speeds, Moscow: IL, 547 p. [in Russian].
3. Kalmukov L. A., Levkin V. F., Nushtaev P. D. (1968). Investigation of aerodynamic derivatives of the aerodynamic moment of control elements, Papers of TsAGI, 32 p. [in Russian].
4. Keldish M. V. (1985). Izbrannie trudi. Mechanika. [Selected works. Mechanics], Moscow: Science, 568 p. [in Russian].
5. Lebedev A. A., Сhernobrovkin L. S. (1962). Dynamics of flight of unmanned aerial vehicles, Moscow: Oborongiz, 548 p. [in Russian].
6. Levkin V. F. (1982). Experimental studies of nonstationary aerodynamic characteristics of control surfaces at transonic speeds, Papers of TsAGI, issue 2132, 16 p. [in Russian].
7. Panovko J. G. (1980). Introduction to the theory of mechanical oscillations, Moscow: Science, 272 p. [in Russian].
8. Safronov A. V., Nedilko A. N., Safronov V. A. (2014). Аdaptive mathematical model of assessment of excited hinge moments of supersonic aircraft aerodynamic control surfaces on transonic airspeed, Research Papers Collection of the Center of military and strategic studies of the National Defence University of Ukraine named after Ivan Chernyahovsky, No. 3(52), pp. 28—33 [in Ukrainian].
9. Safronov A. V., Nedilko A. N., Safronov V. A (2015). Сomparative analysis of theoretical and calculation and experimental methods to assess transonic flutter characteristics, Research Papers Collection of the Center of military and strategic studies of the National Defence University of Ukraine named after Ivan Chernyahovsky, No. 1(53), pp. 41—48 [in Ukrainian].
10. Safronov A. V., Nedilko A. N. (2016). Mathematical model of estimation of maximum possible values of excited hinge moments of aerodynamic surfaces of control in the occurrence of transonic flutter, Science and Technology of the Air Force of Ukraine, No. 4(25), pp. 19—23 [in Ukrainian].
11. Semon B. I., Safronov A. V., Nedilko A. N. (2016). Transonic Flutter: from MiG25 to Space Ship Two, Science and Defense, No. 3, pp. 32—35 [in Ukrainian].
12. Strelkov C. P. (1964). Introduction to the theory of oscillations, Moscow: Nauka, 440 p. [in Russian].
13. Chapligin C. A. (1976). On the pressure of a planeparallel flow on barrier bodies, Selected Works, Mechanics of Fluids and Gas. Mathematics. General mechanics, pp. 97—130 [in Russian].
14. Jakovlev K. P. (1960). Brief physicotechnical handbook, volume II, Moscow: Fizmatgiz, 412 p. [in Russian].