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First published online March 30, 2006
Journal of Experimental Biology 209, 1502-1515 (2006)
Published by The Company of Biologists 2006
doi: 10.1242/jeb.02146

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Passive mechanical properties of legs from running insects

Daniel M. Dudek^* and Robert J. Full

Department of Integrative Biology, University of California at Berkeley, Berkeley, CA 94720-3140, USA

View larger version (29K): [in a new window]   Fig. 1. The distal end of the tibia was attached to the arm of a servo-motor via a stainless steel pin. In both preparations, the angle between the steel pin and tibia was 110° because the body–coxa joint is held at a constant 20° during locomotion (Kram et al., 1997). The hindlimb was chosen due to its vertically oriented joint axes, where vertical deflections of the leg are absorbed by either the body–coxa joint or passive deflection of the exoskeleton. The servo-motor input sinusoidal oscillations from 0.01 to 100 Hz and 0.1 to 1.0 mm and recorded the induced forces. (A) Ventral view of the joint axes of rotation in the meta-thoracic leg. (B) Sagittal view of the joint axes of rotation in the meta-thoracic leg. (C) In the fixed-coxa preparation, the leg was removed and affixed with epoxy resin to 0.95 cm-thick Plexiglas. The tarsus (gray broken line) was removed. (D) In the free-coxa preparation, the cockroach was tethered to a bronze rod via the metanotum. These two preparations were chosen to bound the effect muscle activation at the body–coxa joint could have on leg properties. The tarsus (gray broken line) was removed.

View larger version (25K): [in a new window]   Fig. 2. Leg force as a function of displacement amplitude and oscillation frequency (blue line, 0.25 Hz; green line, 12 Hz; red line, 40 Hz). (A) Fixed coxa with small-amplitude oscillations (0.1 mm shown) resulted in nearly linear hysteresis loops. (B) Free coxa with small-amplitude oscillations (0.1 mm shown) showing near-linear hysteresis loops. (C) Fixed-coxa oscillations with large amplitudes exceeding 0.3 mm (1.0 mm shown) resulted in hysteresis loops with pronounced non-linearities. (D) Free-coxa oscillations with large amplitudes exceeding 0.3 mm showing non-linearities. The relatively larger areas of the loops in the free-coxa preparation (B,D) compared with the fixed-coxa legs (A,C) show the decreased resilience of a leg with a freely rotating coxa.

View larger version (14K): [in a new window]   Fig. 3. Leg impedance as a function of frequency at amplitudes of 0.1 (circles), 0.5 (squares) and 1.0 mm (triangles). Leg impedance increased significantly as frequency increased from 0.05 to 25 Hz. Impedance of the fixed-coxa leg (A) was significantly greater than in the free-coxa leg (B).

View larger version (37K): [in a new window]   Fig. 4. Phase shift as a function of frequency at amplitudes of 0.1 (circles), 0.5 (squares) and 1.0 mm (triangles). (A) As the leg underwent deflection (solid line), the induced force (broken line) peak lagged behind the maximum displacement. (B) This phase shift, , can be seen as the angle between the maximum force and displacement of the hysteresis loop. Energy lost (Elost) or hysteresis is shown in stipples. Energy of unloading (Eunloading) is shown as hatched. (C) In the fixed-coxa preparation, tan() remained constant as frequency increased, and decreased as amplitude increased. (D) In the free-coxa preparation, tan() remained constant as frequency increased, and increased as amplitude increased.

View larger version (24K): [in a new window]   Fig. 5. Energy storage, returned and lost as a function of frequency at amplitudes of 0.1 (circles), 0.5 (squares) and 1.0 mm (triangles). (A) Loading energy in fixed coxa increased with increasing amplitude and as oscillation frequency increased from 0.05 to 25 Hz. (B) Loading energy in free coxa required 45–60% less energy than the fixed leg. (C) Unloading energy in the fixed coxa increased with increasing amplitude and as oscillation frequency increased to 25 Hz. (D) Unloading energy in the free coxa returned 50–70% less energy than the fixed leg. (E) Hysteresis (or lost energy) in the fixed-coxa preparation decreased as frequency increased, but was only significant at 1.0 mm oscillations. (F) Hysteresis in the free-coxa preparation increased as frequency increased significantly for 0.5 and 1.0 mm oscillations and was only about 30% less than the fixed-coxa legs.

View larger version (11K): [in a new window]   Fig. 6. Resilience (R) as a function of frequency at amplitudes of 0.1 (circles), 0.5 (squares) and 1.0 mm (triangles). (A) Fixed-coxa resilience was significantly greater than (B) free-coxa resilience. (C) Using tan() to calculate resilience (black line and open circles) overestimates the actual values (gray line and filled circles) and produces an inverse dependence on frequency because the calculation is based on a linear model. While leg data are well fit by a linear model, the leg data are not linear, and care should be used when applying linear calculations to any biomaterial.

View larger version (23K): [in a new window]   Fig. 7. Viscous damping model stiffness and damping coefficients as a function of frequency at amplitudes of 0.1 (circles), 0.5 (squares) and 1.0 mm (triangles). (A) Stiffness (k) of the fixed coxa. (B) Stiffness of the free-coxa leg. (C) Viscous damping coefficient (c) of the fixed-coxa leg decreased with oscillation frequency and amplitude. (D) Viscous damping coefficient of the free-coxa leg also decreased with frequency and amplitude. Viscous damping in the free-coxa leg was 40–50% less than in the fixed-coxa leg.

View larger version (24K): [in a new window]   Fig. 8. Hysteretic damping model stiffness and damping coefficients as a function of frequency at amplitudes of 0.1 (circles), 0.5 (squares) and 1.0 mm (triangles). (A) Stiffness of the fixed-coxa leg increased with frequency from 0.05 to 25 Hz. The slopes and intercepts of the 0.3–1.0 mm amplitude oscillations were not significantly different from each other but were significantly lower than the 0.1 mm oscillations. (B) Stiffness of the free-coxa leg increased linearly with increasing frequency. As in the fixed-coxa preparation, the slopes and intercepts of the 0.3–1.0 mm amplitude oscillations were not significantly different from each other but were significantly lower than the 0.1 mm oscillations. The slopes and intercepts were 50–60% lower for the free-coxa leg than for the fixed-coxa leg. (C) Structural damping factor of the fixed-coxa leg was independent of oscillation frequency and amplitude. (D) Structural damping factor of the free-coxa leg also was independent of frequency and amplitude. Hysteretic damping in the fixed-coxa leg was 30% less than in the free-coxa leg.

View larger version (20K): [in a new window]   Fig. 9. Hysteretic damping model fitting. (A) Hysteresis loops recreated using the stiffness (kh) and damping () parameters from the hysteretic damping model fit (red line) closely matched the actual data (blue line) at low amplitudes. (B) Nonlinearities of the leg were not captured by the linear lumped parameter model, but the hysteretic model's peak-to-peak displacement, force and area inside the loop were within 10% of the actual data.