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First published online July 23, 2003

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Flexural stiffness in insect wings II. Spatial distribution and dynamic wing bending

S. A. Combes^* and T. L. Daniel

Department of Biology, University of Washington, Seattle, WA 98195, USA

View larger version (114K): [in a new window]   Fig. 1. Unloaded (A) and loaded (B) Manduca sexta wing illuminated with laser lines. A Matlab program was used to find the center of a laser line running from the base of the wing to the tip before and after applying a point force at the tip (yellow and red lines, respectively). The change in position of the laser line y(x) was then used to find displacement of the wing (x).

View larger version (46K): [in a new window]   Fig. 2. Finite element models based on Manduca sexta wings. (A) Model wing with membrane elements (blue) and vein elements (pink), each of homogeneous material stiffness E. (B) Model wing in which declining material stiffness of membranes and veins results in an exponential decline in flexural stiffness. Each color represents a different value of material stiffness.

View larger version (36K): [in a new window]   Fig. 3. Flexural stiffness distribution in wings of Manduca sexta and Aeshna multicolor, and in finite element models of M. sexta wings. In each graph, spanwise flexural stiffness is shown above (longer lines) and chordwise flexural stiffness below; dorsal measurements are in black and ventral measurements in gray. Each line within these groups represents the flexural stiffness distribution estimated from wing displacement measurements performed on a different individual. (A) Flexural stiffness distribution of male Manduca sexta forewings (N=9 spanwise, N=10 chordwise), and of finite element models with homogeneous (blue) and exponentially declining (red) vein and membrane material stiffness. (B) Flexural stiffness distribution of female Manduca sexta forewings (N=4 spanwise, N=9 chordwise). (C) Flexural stiffness distribution of male Aeshna multicolor forewings (N=8). (D) Flexural stiffness distribution of male Aeshna multicolor hindwings (N=8).

View larger version (25K): [in a new window]   Fig. 4. Results of static bending tests on finite element model (FEM) wings. Wings are fixed at the base, and displacement from original position (black outline) in the z-direction is indicated by the color bar. The FEM wing with homogeneous membrane and vein material stiffness is shown on the left and the wing with exponentially declining material stiffness is shown on the right. (A) Displacement due to a point force of 0.003 N (green arrow) at the wing tip. (B) Displacement due to a normal face load of -14.43 Pa (green arrows), the approximate pressure on a wing during steady flight.

View larger version (23K): [in a new window]   Fig. 5. Sequence of images from flapping finite element model wings with homogeneous (blue) or exponentially declining (red) vein and membrane material stiffness. Wings are moving to the right, viewed from their leading edge. Models are shown near the end of the simulation at the same time steps, beginning at 0.4325 s andproceeding in 0.0029 s intervals. For a movie of the model wings in motion, see http://faculty.washington.edu/danielt/movies.