Limit cycles in aircraft lateral dynamics are called wing rock. Wing rock prevention and/or control is an important objective for aircraft that need to fly and maneuver at moderate to high angles of attack. This requires a characterization of wing rock periodic motions, which is our aim. Normal and large-amplitude types of wing rock are distinguished. Onset of normal wing rock occurs at a Hopf bifurcation followed by limit cycle oscillations of gradually growing amplitude. Onset of large-amplitude wing rock is similar, but the limit cycles following the Hopf bifurcation show a jump to a large-amplitude oscillation at a periodic saddle-node bifurcation. Large-amplitude wing rock is shown to be the result of lateral-longitudinal coupling in conjunction with a resonance condition. A continuation algorithm is used to track periodic solutions with varying parameters and to locate bifurcation points. A strategy to avoid large-amplitude wing rock is indicated on the basis of the study. Nomenclature b = wing span F = Jacobian matrix of / / = function of state variables g = vector of nonlinear terms ix»iz > ixz = moments of inertia l,m,n = roll, pitch, and yaw moment coefficients, respectively m = mass /?, r = roll and yaw rates, respectively S = reference area T = time period (of periodic motion) t = time V = velocity jc = vector of states y = sideforce coefficient a, ft, 0 = angle of attack, sideslip, and roll, respectively 8 -oscillator damping parameter e = coupling parameter [i -perturbed angle of attack p = air density r -oscillator variable co = oscillator frequency Subscripts p, r,... = stability derivatives with respect to p, r,... 1,3 = linear and cubic stability derivatives, respectively, with respect to ft (when used with y,l,n)
This paper describes a model that has been developed to study the stability characteristics of aerostats. This model incorporates the concepts of apparent mass, dynamic tether and allows 6 degrees of freedom for the motion of the aerostat. Estimation of aerodynamic coefficients is based on empirical relations and curves available in literature. Weight and buoyancy are calculated based on geometry of the aerostat. Appropriate values for operational altitude and desired angle of attack of the aerostat are assumed. Moment balance about confluence point gives the optimal location of the confluence point. Equations of motion for the aerostat and dynamic tether are simulated and appropriate boundary conditions are applied. Force balance gives the tether tension force and its orientation at the confluence point. Based on the tether tension and its orientation at the confluence point, the tether profile is estimated by breaking up the tether into several elastic segments, each in equilibrium. Once equilibrium is established, the wind is perturbed and the response of the aerostat is simulated. The paper reports the results of simulation carried out for the TCOM 365Y aerostat and the aerostat response to various ambient velocity profiles.
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