Scaling of maximum net force output by motors used for locomotion

Abstract

Biological and engineered motors are surprisingly similar in their adherence to two or possibly three fundamental regimes for the mass scaling of maximum force output (F_max). One scaling regime (Group 1: myosin, kinesin, dynein and RNA polymerase molecules; muscle cells; whole muscles; winches; linear actuators) comprises motors that create slow translational motion with force outputs limited by the axial stress capacity of the motor, which results in F_max scaling as motor mass^0.67 (M^0.67). Another scaling regime (Group 2: flying birds, bats and insects; swimming fish; running animals; piston engines; electric motors; jets) comprises motors that cycle rapidly, with significant internal and external accelerations, and for whom inertia and fatigue life appear to be important constraints. The scaling of inertial loads and fatigue life both appear to enforce F_max scaling as M^1.0 in these motors. Despite great differences in materials and mechanisms, the mass specific F_max of Group 2 motors clusters tightly around a mean of 57 N kg^-1, a region of specific force loading where there appears to be a common transition from high- to low-cycle fatigue. For motors subject to multi-axial stresses, the steepness of the load-life curve in the neighborhood of 50-100 N kg^-1 may overwhelm other material and mechanistic factors, thereby homogenizing the mass specific F_max of grossly dissimilar animals and machines. Rockets scale with Group 1 motors but for different mechanistic reasons; they are free from fatigue constraints and their thrust is determined by mass flow rates that depend on cross sectional area of the exit nozzle. There is possibly a third scaling regime of F_max for small motors (bacterial and spermatazoan flagella; a protozoan spring) where viscosity dominates over inertia. Data for force output of viscous regime motors are scarce, but the few data available suggest a gradually increasing scaling slope that converges with the Group 2 scaling relationship at a Reynolds number of about 10². The Group 1 and Group 2 scaling relationships intersect at a motor mass of 4400 kg, which restricts the force output and design of Group 2 motors greater than this mass. Above 4400 kg, all motors are limited by stress and have F_max that scales as M^0.67; this results in a gradual decline in mass specific F_max at motor mass greater than 4400 kg. Because of declining mass specific F_max, there is little or no potential for biological or engineered motors or rockets larger than those already in use.