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First published online April 26, 2005
Journal of Experimental Biology 208, 1653-1664 (2005)
Published by The Company of Biologists 2005
doi: 10.1242/jeb.01483

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Scaling of maximum net force output by motors used for locomotion

James H. Marden

208 Mueller Laboratory, Department of Biology, University Park, PA 16802, USA

View larger version (62K): [in a new window]   Fig. 1. Cross section of a Drosophila melanogaster (fruit fly) thorax (ca. 0.0003 g) and a diesel engine (118 681 kg). Along with superficial shape similarities, vastly different motors have very similar performance in terms of force output per motor mass. (Drosophila image courtesy of R. Ordway.)

View larger version (20K): [in a new window]   Fig. 2. Maximum force output by translational motion motors (Group 1) as a function of their mass. The least squares fit (solid line) is: Fmax=891 M0.67. The dashed line is the scaling fit for Group 2 motors in which Fmax scales as M1.0 (Fig. 3). Maximum force output (N) and motor mass (kg).

View larger version (23K): [in a new window]   Fig. 3. Maximum force output by Group 2 motors as a function of their mass. The least squares fit (solid line) is: Fmax=55 M0.999. The dashed line is the scaling fit for Group 1 motors for which Fmax scales as M0.67 (Fig. 2). Inset: the frequency distribution of mass specific force output for all of the Group 2 motors on the main plot. Maximum force output (N) and motor mass (kg).

View larger version (16K): [in a new window]   Fig. 4. Maximum force output in relation to motor mass by a selection of Group 2 motors (flying insects; Marden, 1987), along with data from two studies that predicted either vastly different (microjet; Epstein et al., 2000) or subtly different (euglossine bees; Dillon and Dudley, 2004) results. These comparisons demonstrate the robustness of the M1.0 scaling and a mean specific force of 57 N kg-1 for these types of motors. Maximum force output (N) and motor mass (kg).

View larger version (21K): [in a new window]   Fig. 5. Force outputs of small Group 3 motors (red curve) that operate in the viscous regime. For comparison, data are included for Group 1 motors, along with a dashed line representing Group 2 motors. The scaling curve for Group 3 motors connects their performance with that of the smallest Group 2 motor (fruit fly). Maximum force output (N) and motor mass (kg).

View larger version (15K): [in a new window]   Fig. 6. Load-life curves for steel bearings (solid circles) and entire turbine engines (open circles).

View larger version (16K): [in a new window]   Fig. 7. Predicted lifespan (N cycles) over a range of mass specific force outputs for a generalized 1 kg motor that can withstand a maximal force load of 890 N. This curve assumes that a=1 and b=3 in the load-life equation (Eqn 2). Also shown is a curve relating specific load to the observed number of wingbeat cycles for hummingbirds over one day of normal flight (a highly conservative estimate of cycles to failure) and during experimentally imposed maximal loading.

View larger version (34K): [in a new window]   Fig. 8. Specific force output (N kg-1 motor mass) for three electric motors used with different combinations of propellers, gear ratios, voltage and current. Dark bars show combinations that are indicated by the manufacturer to cause rapid failure. Superimposed images show a schematic and a photograph of one of these motors, consisting of the motor, gears and propeller nose cone.

View larger version (24K): [in a new window]   Fig. 9. A general model for the load-life curve of motors, with transitions from high-cycle to low-cycle fatigue occurring in the region between about 55-120 N kg-1. In the background is the frequency distribution of specific force output of all Group 2 motors from Fig. 3.

View larger version (11K): [in a new window]   Fig. 10. Scaling of maximum force output (A) and ratio of total piston cross sectional area to stroke length (B) in relation to motor mass over the entire size range of piston engines. The vertical dashed line indicates the crossing point of the scaling fits for Group 1 (dashed line) and Group 2 (solid line) motors. Maximum force output (N), motor mass (kg), total piston area/stroke length (m2 -1).

View larger version (40K): [in a new window]   Fig. 11. Cross sectional schematic of one of the largest piston engine (Burmeister and Wain K98MC-C) and three of its human operators. The geometry of piston engines larger than 4400 kg departs radically from what would be predicted from the scaling of smaller piston engines; this departure begins at the crossing point of the scaling relationships for Group 1 and Group 2 motors. Reproduced with permission from B&W Engine Selection Guide, February 2000.