# Mathematical model for estimation of the effect of aerodynamic aircraft control surfaces compensation on the level of their vibrations in case of transonic flutter

 1Safronov, AV, 2Semon, BY, 1Nedilko, AN1The National Defence University of Ukraine named after Ivan Cherniakhovsyi, Kyiv, Ukraine2The National Defence University of Ukraine named after Ivan Cherniakhovsyi, Kyiv, Ukraine Space Sci.&Technol. 2018, 24 ;(4):14-23 https://doi.org/10.15407/knit2018.04.015 Publication Language: Ukrainian Abstract:  There are cases of intense fluctuations of supersonic aircraft and aviation-space 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 self-oscillations 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 right-hand 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
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