The Wing Divergence Problem in a Supersonic Gas Flow
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Journal of Mathematical Sciences, Vol. 250, No. 1, October, 2020
THE WING DIVERGENCE PROBLEM IN A SUPERSONIC GAS FLOW A. N. Kulikov P. G. Demidov Yaroslavl’ State University 14, Sovetskaya St., Yaroslavl’ 150003, Russia anat [email protected]
UDC 517.9
We study the boundary value problem for a nonlinear hyperbolic type evolution equation governing torsional oscillations of a wing in a hypersonic gas flow in the Bolotin statement. We study the existence, stability, and asymptotic representation of nontrivial equilibrium states. Using elliptic functions, we obtain in some cases explicit formulas for the equilibrium states and clarify whether these equilibrium states are stable in the sense of Lyapunov. Bibliography: 9 titles.
1
Introduction
The physical statement of the problem on the behavior of a wing in a hypersonic gas flow can be found in [1], where this problem is mentioned as one of the problems of the elastic stability theory in a nonlinear setting. The equation governing the torsional oscillations of the wing has the form 2κ/(κ−1) p∞ (b0 + κ0 ) κ−1 JΘt1 t1 + g0 Θt1 = GJd Θx1 x1 + M1 Θ 1+ 2 2 2κ/(κ−1) κ−1 − 1− M2 Θ , (1.1) 2 where Θ = Θ(t1 , x1 ) is the torsion angle, x1 ∈ [0, l], t1 0, l is the wing length, GJd is the torsional rigidity, J is the running moment of inertia with respect to the rod axis, b0 is the wing width, x0 is the leading edge flap width, κ is the polytrope coefficient, p∞ is the gas pressure in an unperturbed medium, M = U/c∞ is the Mach constant, U is the gas flow velocity (the aircraft velocity), c∞ is the sound speed in a given medium, and g0 is the damping coefficient. To take into account the aerodynamic forces, the piston theory (the Il”yushin law of plane sections) [2]–[4] is used. The right-hand side of (1.1) contains two positive constants M1 and M2 such that M1 = U1 /c and M2 = U2 /c, i.e., the Mach constants M1 and M2 correspond to the flow velocities U1 and U2 . The corresponding flow velocities U1 and U2 above and below the wing can be different. The simplified variant U1 = U2 = U is considered in the Bolotin model problem. Translated from Problemy Matematicheskogo Analiza 104, 2020, pp. 29-37. c 2020 Springer Science+Business Media, LLC 1072-3374/20/2501-0031
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We set x1 = γ1 x, t1 = γ0 t, Θ = γ2 u, where the constants γ0 , γ1 , γ2 are chosen as follows: l 2 J l . γ1 = , γ02 = π π GJd Then we obtain the equation utt + gut = uxx + au + ac2 u2 + ac3 u3 ,
(1.2)
where γ0 p∞ (b0 + x0 )x0 , a = γ0 κ(M1 + M2 ), J 2J κ+1 κ+1 (M1 − M2 ), c3 = γ22 (M12 + M22 − M1 M2 ). c 2 = γ2 2 12 g = g0
We can assume that γ2 (κ + 1)/2 = 1. Consequently, c2 = M1 − M2 ,
c3 =
M12 + M22 − M1 M2 >0 3(1 + κ)
for all values of the parameters of the problem. To obtain Equation (1.2), we renormalize Equation (1.1) and apply the Taylor formula, where only terms of the first, second, and third order of smallness are left. The motivation of such a simplification is explained in [1]. Equation (1.2) is considered together with the boundary conditions u(t, π) = ux (t, 0) = 0,
(1.3)
i.e., we ass
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