Figure 1.
A) a beetle just after leaving the ground showing the flexed body. B) a modeled beetle seen from the side. ‘a’, and ‘p’ denote the tip of the head and abdomen respectively. ‘o’ denotes the hinge. θ is the flexion angle, α and β are the angles between the long axes of the subunits and the horizontal. C) a model beetle in the pre-jump position (grey) and after flexing the body (black). Circles denote the center-of-mass of the subunits and red rectangles with an x denote the center-of-mass (CM) of the entire body in the pre-jump posture and after flexing.
Figure 2.
3D kinematics of the aerial maneuver during the jump of the click-beetle.
Top and lower panels are two examples of data extracted from movies. Figures on the left show the ballistic trajectory through the 3D positions and orientation of the tip of the head (a, red), tip of the abdomen (p, black) and the lines connecting them through the hinge (blue) in each video frame (See Fig. 1B). Figures on the right show the instantaneous flexion of the body (green), yaw (red), pitch (blue) and roll (black) angles as a function of time for the jumps on the left. A is an example for a jump in which the beetle mostly rolled and B shows a jump where the beetle somersaulted (pitch) in the air.
Figure 3.
The mean rotation rate observed in 14 jumps made by three different beetles. Left figures (A) show the number of revolutions per jump and the same data is represented on the left (B) as revolutions per second. The upper and lower figures refer to somersaults and rolls respectively. Each column represents a single jump.
Figure 4.
Observed jump angles (A) and takeoff speeds (B).
Each column represents the mean of one out of 9 beetles. The error bars is 1 s.d. The number of jumps observed per beetle is denoted at the top of the figure.
Figure 5.
Model simulation of the angular velocity.
A) The modeled beetle in figure 1C showing that the point of contact with the ground should be within the red rectangle. The x axis of the rectangle is magnified in B. B) The model calculation for angular speed of somersaulting () as a function of the point of contact with the ground along the x axis. C) Model calculation of angular speed for rolling in the air (
) as a function of roll angle ψ of the body in the pre-jump posture. The longitudinal position of the point of contact with the ground is assumed to be x = 2 mm posterior to the hinge. The insert in C shows the beetle in cross-section, defining ψ and the moment arm. Red circle denote the center-of-mass.
Figure 6.
Proportional change in moment-of-inertia for pitch (IΦ) and roll (IΨ) as a function of θ.
The moment of inertia of the body for a given flexion angle (0<θ<55°) is divided by the moment of inertia of the beetle at the pre-jump posture (θ = 0°). Two alternative calculations are shown. Data points and the thick lines represent calculated estimates using the instantaneous center of mass. The thin lines are the same estimates calculated assuming the center of rotation stays at the initial position of the center of mass as the beetle left the ground (i.e. θ = 55°).
Figure 7.
Simulated takeoff speed (V0) and jump angles (γ) for a wide range of θ values.