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AERODYNAMIC STABILITY AND MANEUVERABILITY OF THE GLIDING FROG POLYPEDATES DENNYSI

MICHAEL G. McCAY^*

University of California — Berkeley, Department of Integrative Biology, Berkeley, CA 94720-3140, USA
^* e-mail: mccay{at}socrates.berkeley.edu

View larger version (30K): [in a new window]   Fig. 1. Rotation angles. Rotation about the frog's center of mass can be resolved into rotations about three orthogonal axes. The origin of the axes is fixed to the animal's center of mass, and the axes themselves are fixed to the animal; the axes translate and rotate as the animal translates and rotates. (A) Rotation about the cranial—caudal axis is termed roll angle, ; rolling rotates the animal's right side up or down. (B) Rotation about a dorso-ventral axis is termed yaw angle, ; yawing rotates the animal's snout to the left or right. (C) Rotation about a lateral axis is termed angle of attack, ; pitching rotates the animal's snout up or down. Grey arrows indicate the directions of the indicated torques.

View larger version (18K): [in a new window]   Fig. 2. Aerodynamic stability determined by the slope of measured aerodynamic torque about a rotational axis (here pitch) plotted as a function of the angular orientation about the same rotational axis. (A) Measured aerodynamic torques about the pitch axis for an aerodynamically stable frog. At the equilibrium angle of rotation, no torques are acting on the frog, so it remains at equilibrium. If a gust of wind pitches the frog's nose away from its initial position in either the upward or downward direction, this change in orientation relative to the airflow induces an aerodynamic torque that tends to rotate the frog back to its initial position. (B) Measured aerodynamic torques for a neutrally stable frog. If a gust of wind pitches the frog's nose away from its initial position in either the upward or downward direction, this change in orientation relative to the airflow induces no aerodynamic torque that tends to rotate the frog back to its initial orientation or to rotate the frog further away from its initial orientation, so the frog remains at the angular orientation to which it was pitched. (C) Measured aerodynamic torques for an aerodynamically unstable frog. If a gust of wind pitches the frog's nose away from its initial position in either the upward or downward direction, this change in orientation relative to the airflow induces an aerodynamic torque that tends to rotate the frog even further away from its initial position, causing the frog to diverge from its original flight path.

View larger version (15K): [in a new window]   Fig. 3. (A) A frog gliding through the air; the airflow with respect to the frog is parallel to the frog's flight path, but in the opposite direction from that in which the frog is moving. The angle of the frog's glide path with respect to the horizontal is called the glide angle, . The angle that the frog's cranial—caudal axis makes with the frog's glide path is called the angle of attack, . (B) If a wind-tunnel is tilted to the glide angle of a frog, the airflow past a frog that is stationary in the working section of the wind-tunnel simulates the airflow relative to a frog gliding at that angle through still air. Air is blown by a 560 W fan through the transparent test section of the wind-tunnel, measuring 38 cm high, 38 cm wide and 46 cm long. The tunnel can be tilted to blow air at any orientation ranging from vertical to horizontal. The tunnel airspeed was continuously adjustable up to 18 m s-1. Observations of Polypedates dennysi gliding in free air and in the wind-tunnel show that this species glides at a glide angle of 45° and an airspeed of approximately 13 m s-1.

View larger version (20K): [in a new window]   Fig. 4. Diagram showing two types of turning technique. (A) A `banked turn', during which the frog rolls into the turn. By rolling, the frog tips the direction of the lift force towards the center of the turn; the sideways component of lift pulls the frog through the turn. (B) A `crabbed turn', during which the frog yaws into the turn; airflow asymmetries on the frog's left and right sides induce a centripetal force that pulls the frog through the turn.

View larger version (14K): [in a new window]   Fig. 5. Turning behavior of frogs. Frequency of left turns (open columns), no turns (black columns) and right turns (grey columns) in the presence of a visual stimulus are shown for three Polypedates dennysi frogs in a wind-tunnel. All frogs maneuvered significantly more in the presence of the stimulus. (A) Frog 1; this frog turned significantly more often away from the plant (2-test, P=0.02). (B) Frog 2; this individual turned significantly more often towards the plant (2-test, P=0.01). (C) Frog 3; this individual turned significantly more often towards the plant (2-test, P<<0.001). Numbers of actual occurrences of each maneuver are shown above each column on the graphs.

View larger version (15K): [in a new window]   Fig. 6. Foot position during maneuvers. Frequencies of the indicated foot positions for frog 1 (A), frog 2 (B) and frog 3 (C) are shown for left turns, straight gliding without a turn and right turns. Absolute numbers of occurrences of a behavior are shown above the columns. All frogs show significant changes in foot position while turning relative to their foot positions when gliding straight without turning (2-test, P<<0.001). All frogs held the foot opposite to the direction of the turn higher; the left foot was held higher during right turns, and the right foot was held higher during left turns.

View larger version (16K): [in a new window]   Fig. 7. Comparisons of rotation angles during two left turns. The angles of rotation about the pitch (nose up and down), roll (twisting about the cranial—caudal axis) and yaw (nose left and right) directions are shown for two typical left turns. (A) A banked turn performed by the frog with its hind feet held at the same vertical height above its body. (B) A `crabbed' turn to the left performed by a frog with its right hind foot held higher than its left hind foot above the body. The patterns shown by the pitch angle and yaw angle are comparable between the two turns. However, the roll angle for the banked turn is between -40° and -60° compared with a roll angle of less than 20° for the `crabbed' turn.

View larger version (20K): [in a new window]   Fig. 8. Aerodynamic torques measured on a full-scale physical model of Polypedates dennysi posed in the gliding posture observed during non-maneuvering gliding. Aerodynamic torques about the pitch (A), roll (B) and yaw (C) axes are shown plotted against angle of attack, roll angle and yaw angle, respectively. Lines shown on each graph are linear regressions of the data. The linear regressions of rolling and yawing torque are taken over smaller ranges of roll angle and yaw angle, respectively, than the linear regression for pitching torque; the smaller range of angles reflects the lower level of perturbations in roll and yaw angle during gliding and is consistent with aeronautical engineering practice (McCormick, 1976). (D) Summary of aerodynamic stability coefficients (see text for definitions of coefficients) about the pitch, yaw and roll axes. Stability about a given axis is denoted by negative values, and instability is denoted by positive values. At the gliding speed (13 m s-1) and angle of attack (45°) used by P. dennysi, the frogs are slightly stable about the pitch axis and roll axes, and slightly unstable about the yaw axis.

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