Fig. 3. 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 F_v and a radial force component F_r. 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 dl_x and vertical direction dl_y. (F) Horizontal force of the flapping wing is the vector sum of a horizontal component F_h(x) and a force component F_h(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 F_v (red) and horizontal force F_h(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 F_h(x) and vertical force F_v are 1.0, respectively. (H) Examples of changes in length of moment arm for pitching plotted at various body angles (=0–60°, R=1, F_h(x)=1, F_v=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, F_h(x)=1, F_v=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.