Fig 1.
Force balance during banked and level flight in a circular trajectory.
For a beetle flying in the level orientation (left) the lift force (L) acts vertically to counteract the weight (W) of the beetle. However, the beetle experiences a centrifugal side force (Fc) due to the circular trajectory. For a beetle flying in the banked orientation (right) the lift force can be tilted towards the center of the circular trajectory to balance the resultant (R) of both the weight and centrifugal force. The wing colored red denotes the wing internal to the circular trajectory (closer to the center of the circle).
Fig 2.
The flight-mill used in the study.
The insert (A) shows a picture of the device and the definition of the elevation angle of the radial arm (γ) in the vertical plane. The schematic drawing (B) shows the various components and tethering orientation (level versus banked) of the insects. See text for a more elaborate description.
Fig 3.
Definition of flapping kinematics.
The area of the right wing is shown in grey. A) The motion of the wing tip relative to the body defines a stroke plane angle (β) in which the instantaneous flapping angle (φ) can be measured from the projection of the wing length onto that plane. The deviation angle is the deviation of the wing length from that plane. The geometric angle of incidence (α) is defined as the angle of the wing chord with the stroke plane. See appendix A for detailed explanation and definitions. B,T, and C are the wing base, wing tip, and a point on the trailing edge of the wing. The subscript “r” denotes that the points are on the right wing. B) The flapping motion of the wings in the stroke plane (blue circle). Dashed lines Illustrate the definition of flapping angles at the ventral and dorsal stroke reversal points (VSRP and DSRP, respectively). The angle between these two angular positions (Φ) is defined as the flapping amplitude.
Fig 4.
Wingtip trajectory in the body frame of reference during flight in the flight-mill.
The rows present the four flights by the same beetle (beetle #4) in the four variants of the flight-mill. The trajectories of the left and right wingtips are denoted by red and blue color respectively. The left wing is the wing internal to the circular flight trajectory. Left, center, and right columns are the trajectory of the wingtips in the dorsal (XY), transverse (YZ), and sagittal (XZ) planes respectively, where X,Y,Z are the body axes defined in the insert on the right. The horizontal axis in each graph is the X, Y and X body axis and the vertical axis is the Y, Z and Z body axis in the left, center, and right column, respectively. The origin is always the base of the wing and all units are meters.
Table 1.
Body mass and wing measurements of beetles used in the study.
Table 2.
Repeated measures ANOVA results (p-values) showing the significance of the effect of wing side (left /right), tethering orientation of the beetle (level/banked), and pivot (fixed/seesaw) of the radial beam on flapping kinematics.
All three-way interactions were not statistically significant (P>0.05).
Fig 5.
Changes in angular positions of the stroke reversal points in the different variants of the flight-mill.
Red and blue colors denote the left (internal) and right wings, respectively. Asterisks denote significance of the fixed/seesaw (A) or bank/level (B,C) effects reported by a repeated measurements ANOVA (* p<0.05, ** p<0.01, *** p<0.001). Capital letters denote significant statistical differences (P<0.05) and similarities in a Tukey post-hoc test when the interaction with wing side is significant. A) The dorsal stroke reversal point of the right wing in the seesaw condition was significantly lower (less dorsal) compared to the left or right wing when the flight-mill was in the fixed condition (Tukey, p<0.005 in both cases). B) The dorsal stroke reversal point was higher in the banked condition compared to the level condition (F1,9 = 10.1, p = 0.011). A significant interaction between banking condition and wing side (F1,9 = 13.5, p = 0.005) revealed that the right wing during leveled flight reached lower (less dorsal) dorsal stroke reversal angles than the left wing (Tukey, p = 0.033) and lower angles than either the left or right wings during flight in the banked orientation (Tukey, p<0.008). C) The ventral stroke reversal point was significantly lower (more dorsal) in the banked condition (F1,9 = 23.5, p<0.001).
Fig 6.
Effect of wing side and body orientation on flapping kinematics I.
The left and center columns show the change between the left and right wing and the level and banked flight orientation, respectively, for each beetle (black lines). The right column presents the interaction between wing side and body orientation. Significance is denoted by asterisks as in Fig 5. A) The Stroke plane angle (β) was significantly larger when the beetles were banked (RMANOVA, F1,9 = 14.6, p = 0.004) B) Geometric angle of incidence (α) during mid downstroke was not significantly different between the left (internal) and right (external) wings (RMANOVA, F1,9 = 4.2; P = 0.070). The angles of incidence were significantly smaller when the beetles were flying banked compared to level (F1,9 = 9.5; p = 0.013). C) Geometric angle of incidence during mid upstroke was significantly smaller in the left compared to the right wing (RMANOVA, F1,9 = 34.6, p <0.001 and this bilateral flapping asymmetry was larger when the beetles were level (F1,9 = 8.9, p = 0.015 See Fig 3A for definitions of these angles.
Fig 7.
Effect of wing side and body orientation on flapping kinematics II.
The figure is arranged as in Fig 6. A) Flapping amplitude was significantly higher in the left wing. B and C) mean speed of the wingtip relative to air during the downstroke and upstroke, respectively. The speed of the left wing was significantly higher than that of the right wing during the upstroke.
Fig 8.
Empirical measurements of the relationship between turning rate (angular speed in radians per second) and angular deceleration (radians s-2) of the flight-mill.
The deceleration is due to resistance of the flight-mill to rotation with and without a tethered flightless beetle. The relationship is used to estimate the horizontal force that the beetle must overcome to maintain the flight-mill rotating at a constant speed. The data shown are for the flight-mill in the fixed condition only.
Fig 9.
Aerodynamic vertical force generated by the tethered beetles.
The forces are calculated from the conical pendulum analysis. A) the mean vertical force (Fv) for the beetles in the level and banked orientation. Both forces are normalized by the beetle’s body weight B) The lift forces. Data as in (A) after correcting for the banking angle to give the force normal to the dorsal plane (lift). Red and blue colors denote the left (internal) and right wing, respectively.
Fig 10.
Aerodynamic forces estimated from the quasi-steady analysis.
A) Lift forces. B) Horizontal (thrust) forces. Red and blue colors denote the left (internal) and right wing, respectively. Both forces are normalized with the beetles’ body weight. Thrust/drag is defined as positive and negative when directed forwards and backwards, respectively. The insert above B shows the yaw torque due to asymmetric flapping that should lead to exiting the circular flight trajectory in free-flying beetles.
Fig 11.
Effect of tethering orientation on mass loss during prolonged flight in the flight-mill.
Each symbol denotes the % of net body mass lost due to flight during two hours of tethering to the flight mill. Black and red colors denote banked and level body orientation, respectively. A) Mass loss as a function of actual flight time. B) Mass loss as a function of flight distance.
Fig 12.
Free-flight maneuvers of females R. ferrugineus flying around a lamp in the laboratory.
Blue lines are the flight trajectories tracked in 3-dimensions. A) The trajectories in the horizontal plane (i.e. viewed from above). B) The trajectories in 3D.
Table 3.
Dimensions and mass of parts of the flight-mill used in the study.
The data is substituted in Eq 14 to find the vertical force generated by the flying beetle.