Wing kinematics: Rotational axes, angles, and length of relative moment arms for pitching within the stroke cycle.
(A) Instantaneous stroke angle 
, stroke amplitude 
; (B) body angle 
, heaving angle 
during wing translation and inclination of total force vector 
. (C) Lift normal to the wing surface (black) is the vector sum between vertical force Fv and a radial force component Fr. See text for more details. (D–E) Location of the animal’s centre of mass in the z-plane and length of moment arm for pitching moments in the horizontal dlx and vertical direction dly. (F) Horizontal force of the flapping wing is the vector sum of a horizontal component Fh(x) and a force component Fh(z) in z-direction. COF, centre of force producion. (G) Simplified hypothetical alteration in length of the moment arm for each angular position of the longitudinal wing axis within a horizontal stroke plane and without heaving motion. Total pitching moments (black) are the sum of moments produced by vertical force Fv (red) and horizontal force Fh(x) (blue). Moments were calculated using Eqn 13 and Eqn 14 and plotted for a complete stroke cycle with 180° stroke amplitude. In the example shown body angle is 30°, normalized distance 
between wing hinge and COG is 0.2 wing length, and wing length R, horizontal Fh(x) and vertical force Fv are 1.0, respectively. (H) Examples of changes in length of moment arm for pitching plotted at various body angles (=0–60°, R=1, Fh(x)=1, Fv=1, =0.2, =0°). (I) Examples of alterations in length of moment arm for pitching at various distances between wing hinge and COG (=0–0.5, R=1, Fh(x)=1, Fv=1, =30°). Positive and negative arm for pitching moments produce pitching down and up moments, respectively. *Point of attack for mean force vector acting on the wing at 0.65 wing length; circled cross, centre of body mass (COG) and filled circle, wing hinge (WH) of the virtual insect.
Credits: Fritz-Olaf Lehmann and Simon Pick
Source: The aerodynamic benefit of wing–wing interaction depends on stroke trajectory in flapping insect wings, The Journal of Experimental Biology
