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First published online January 25, 2005
Journal of Experimental Biology 208, 447-459 (2005)
Published by The Company of Biologists 2005
doi: 10.1242/jeb.01407

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Dynamic flight stability of a hovering bumblebee

Mao Sun^* and Yan Xiong

Institute of Fluid Mechanics, Beijing University of Aeronautics and Astronautics, Beijing 100083, People's Republic of China

View larger version (20K): [in a new window]   Fig. 1. (A) A sketch of the rigid body approximation. (B) Definition of the state variables u, w, q and . The bumblebee is shown during a perturbation (u, w, q and are zero at equilibrium).

View larger version (96K): [in a new window]   Fig. 2. (A) Portions of the grid for the bumblebee wing. (B) The bumblebee body planform and a portion of the grid.

View larger version (23K): [in a new window]   Fig. 3. Sketches of the reference frames and wing motion. o1x1y1z1 is an inertial frame, with the x1,y1 plane in the horizontal plane. o'x'y'z' is another inertial frame, with the x'y' plane in the stroke plane. , positional angle of the wing; min and max, minimum and maximum positional angle, respectively; , angle of attack of the wing; ß, stroke plane angle; R, wing length. CL,w and CT,w, coefficients of vertical and thrust of wing, respectively.

View larger version (14K): [in a new window]   Fig. 4. The u-series (A), w-series (B) and q-series (C) force and moment data. X+ and Z+, non-dimensional x- and z-components of the total aerodynamic force, respectively; M+, non-dimensional pitching moment; u+ and w+, non-dimensional x- and z-components of velocity of center of mass, respectively; q+, non-dimensional pitching rate (the equilibrium value is subtracted from each quantity).

View larger version (21K): [in a new window]   Fig. 5. Time courses of CL,w (A), CT,w (B) and CM,w (C) in one flapping cycle. CL,w, CT,w and CM,w are the difference between CL,w, CT,w and CM,w at u+=0.05 (w+=q+=0) and their counterparts at reference flight, respectively. u+ and w+, non-dimensional x- and z-components of velocity of center of mass, respectively; q+, non- dimensional pitching rate; , non-dimensional time.

View larger version (19K): [in a new window]   Fig. 6. Time courses of CL,w (A), CT,w (B) and CM,w (C) in one flapping cycle. CL,w, CT,w and CM,w are the difference between CL,w, CT,w and CM,w at w+=0.05 (u+=q+=0) and their counterparts at reference flight, respectively. u+ and w+, non-dimensional x- and z-components of velocity of center of mass, respectively; q+, non-dimensional pitching rate; , non-dimensional time.

View larger version (17K): [in a new window]   Fig. 7. Time courses of CL,w (A), CT,w (B) and CM,w (C) in one flapping cycle. CL,w, CT,w and CM,w are the difference between CL,w, CT,w and CM,w at q+=0.07 (u+=w+=0)) and their counterparts at reference flight, respectively. u+ and w+, non-dimensional x- and z-components of velocity of center of mass, respectively; q+, non- dimensional pitching rate; , non-dimensional time.

View larger version (25K): [in a new window]   Fig. 8. (A) Characteristic transients of disturbance quantities in the unstable oscillatory mode; (B) sketches showing the combinations of the pitching and horizontal motions in this mode. u+, q+ and , disturbance quantities in non-dimensional x-component of velocity, pitching rate and pitch angle, respectively.

View larger version (22K): [in a new window]   Fig. 9. (A) Characteristic transients of disturbance quantities in the fast subsidence mode; (B) sketches showing the combinations of the pitching and horizontal motions in this mode. u+, q+ and , disturbance quantities in non-dimensional x-component of velocity, pitching rate and pitch angle, respectively.

View larger version (6K): [in a new window]   Fig. 10. Characteristic transients of disturbance quantities in the slow subsidence mode; w+, disturbance quantities in non-dimensional z-component of velocity.

View larger version (14K): [in a new window]   Fig. 11. Characteristic transients of disturbance quantities (first half-cycle) in the unstable oscillatory mode. u+, q+ and , disturbance quantities in non-dimensional x-component of velocity, pitching rate and pitch angle, respectively.

View larger version (25K): [in a new window]   Fig. 12. Diagram of the bumble attitude, horizontal speed and pitching rate, and the forces and moments during the first cycle of the disturbed motion (unstable oscillatory mode). u+, q+ and , disturbance quantities in non-dimensional x-component of velocity, pitching rate and pitch angle, respectively; M+, non-dimensional pitching moment; mg, insect weight; F0, total aerodynamic force at equilibrium.