Dynamics of hinged wings in strong upward gusts

Abstract

A bird's wings are articulated to its body via highly mobile shoulder joints. The joints confer an impressive range of motion, enabling the wings to make broad, sweeping movements that can modulate quite dramatically the production of aerodynamic load. This is enormously useful in challenging flight environments, especially the gusty, turbulent layers of the lower atmosphere. In this study, we develop a dynamics model to examine how a bird-scale gliding aircraft can use wing-root hinges (analogous to avian shoulder joints) to reject the initial impact of a strong upward gust. The idea requires that the spanwise centre of pressure and the centre of percussion of the hinged wing start, and stay, in good initial alignment (the centre of percussion here is related to the idea of a 'sweet spot' on a bat, as in cricket or baseball). We propose a method for achieving this rejection passively, for which the essential ingredients are (i) appropriate lift and mass distributions; (ii) hinges under constant initial torque; and (iii) a wing whose sections stall softly. When configured correctly, the gusted wings will first pivot on their hinges without disturbing the fuselage of the aircraft, affording time for other corrective actions to engage. We expect this system to enhance the control of aircraft that fly in gusty conditions.

Keywords: bird flight; centre of percussion; gust rejection; hinged wings; soft stall; suspension system.

Figure 1.

The modelled system. The depicted motion is illustrative; the arrow lengths are only intended to give a qualitative sense of the relative magnitude of the loads. (a) Free-body diagrams for the fuselage (left) and wing (right) masses, horizontal forces omitted (they do not affect the vertical dynamics). The fuselage can translate vertically, while each wing can hinge about its root end on a pin joint (open circle). The fuselage is regarded as dimensionless, and the aerodynamic force upon it is neglected. (b) Motion occurs when extra vertical force on the wings pushes the system away from equilibrium. The CoM of the overall system (filled orange circle) lies, by definition, between those of the wings and that of the fuselage (filled black circles).

Figure 2.

Pseudo free-body diagrams of load transmission from wing to fuselage. Hinge torque is constant and equal to the static value. (a) Linearized dynamics of the hinged wing. The direction of the fuselage reaction increment ΔR varies with the spanwise location of the applied force ΔF. Force through P produces no immediate reaction. (b) Dynamics of the fixed wing. ΔR acts in the same direction as ΔF with magnitude ΔR = μ_fΔF. Note that ΔR < ΔF because μ_f < 1, i.e. some of the applied force is always required to accelerate the wing mass. (c) A hinged wing with a linear mass distribution transmits less reaction to the fuselage than a fixed wing, provided ΔF acts within the wide, symmetrical interval (blue, to scale) centred on P.

Figure 3.

Gust response of the hinged, fixed and immobile systems with an LLC in the 30% gust. Plot lines become dotted at the instant the wing angle crosses 20 degrees (approximate onset of nonlinearity). (a) External force ΔF = F − F₀ (thinner lines) on each wing versus dynamic fuselage reaction ΔR (thicker lines). For each case in turn, the reaction increment is $\mathrm{\Delta }R=m_f\ddot{z}/2$ (hinged), ΔR = μ_fΔF (fixed) and ΔR = ΔF (immobile). The gust velocity profile (shaded grey) is shown for reference. (b) The normalized centre of pressure (l_F/l) on each wing. On the hinged wing, it moves inboard at first, departing from P. (c) Vertical system velocity $\dot{G}$ of the hinged (plus component masses, broken lines) and fixed wing systems. The ascending wings control fuselage motion. (d) Rejection terms for the hinged system. The potential line indicates the maximum achievable rejection, i.e. the amount necessary to keep the fuselage perfectly level.

Figure 4.

Gust response of the hinged and fixed systems with an LLC/NLC in the 30% gust. Plot lines become dotted at the instant the wing angle crosses 20 degrees (the approximate onset of nonlinearity). (a) The NLC lift coefficient is capped at unity at the stall AoA. (b) Spanwise AoA distributions at six evenly spaced instants in time during the first half of the linear period, from gust onset (t = 0 ms) to approximate maximum force saturation (t ≃ 0.05 ms), for the NLC hinged system. The arrow identifies forward chronology as the AoA enters the stall region (shaded grey). Note that the spanwise position has been normalized. (c) Spanwise force distributions at the same six instants in time for the NLC hinged system. The arrow identifies forward chronology as the initial force distribution flattens out. Note that the spanwise position has been normalized. (d) Normalized centre of pressure l_F/l. (e) Dynamic fuselage reaction ΔR. The departure from the LLC line coincides with the onset of force saturation. (f) Resulting fuselage velocity $\dot{z}$. (g) Rejection terms for the NLC hinged system. The potential line indicates the maximum achievable rejection (that is necessary to keep the fuselage on a perfectly level trajectory).

Figure 5.

Gust response of the hinged system for three different wing mass fractions, versus the fixed-wing case, with an NLC in the 30% gust. Line thickness denotes the mass of the hinged wing, from 0.2 M to 0.5 M. Plot lines become dotted at the instant the wing angle crosses 20 degrees (approximate onset of nonlinearity). (a) Normalized centre of pressure l_F/l. Notice that, with increased mass, the wing is slower to exceed the linear threshold; the model therefore captures the initial dynamics even better. (b) Resulting fuselage velocity $\dot{z}$. (c) Rejection terms for the hinged system. The higher the relative wing mass, the greater the inertial rejection at first.

Figure 6.

Gust response of the hinged system for three different hinge stiffnesses, versus the fixed-wing case, with an NLC in the 30% gust. Plot lines become dotted at the instant the wing angle crosses 20 degrees (approximate onset of nonlinearity). Dynamic torque comes from a hinge that behaves as a linear torsional spring, for which ΔT_h = k_tθ. The legend gives the stiffness constant k_t (N m rad⁻¹) for each case. (a) Fuselage velocity $\dot{z}$. (b) Wing angle θ. For the stiffest hinged system, oscillations begin as the lift coefficient hits the stall plateau. This case is illustrative; no well-tuned inertial rejection system would actually be so stiff or permitted to oscillate in this way.