Locomotion: dealing with friction

Abstract

To move on land, in water, or in the air, even at constant speed and at the same level, always requires an expenditure of energy. The resistance to motion that has to be overcome is of many different kinds depending on size, speed, and the characteristics of the medium, and is a fascinating subject in itself. Even more interesting are nature's stratagems and solutions toward minimizing the effort involved in the locomotion of different types of living creatures, and humans' imitations and inventions in an attempt to do at least as well.

Figure 1

Using rollers to move heavy stones as was presumably done when building the pyramids. As each roller comes out at the rear it has to be reintroduced in front, a tricky operation that must have added not a little to the agony of the slaves.

Figure 2

The wheel and axle, among the greatest inventions ever. (Inset) Sliding friction has to be overcome at one place, but which can now be easily lubricated. The ultimate form of the wheel. Rollers (or balls) between the axle and the wheel hub make it all rolling friction. The ratio of lift to drag could easily be as high as 100:1.

Figure 3

Wave-making drag increases catastrophically with speed as discovered by William Froude (14). It effectively limits the speed (measured in knots) of displacement hulls to the square root of twice their waterline lengths (measured in feet).

Figure 4

A comparison of Cayley’s sketch of the cross-section of a trout with a modern low-drag airfoil section. Dots indicate trout (adapted from figure 4 in ref. 1).

Figure 5

Engine power plotted against maximum speed for four makes of automobiles. The diagonal is steep because power increases as the third power of the speed. To go twice as fast requires an 8-fold increase in engine power. [Reproduced with permission from ref. . (Copyright 1992 and 1996, Henk Tennekes. Reprinted by permission of MIT Press.)]

Figure 6

Speed record over the years for a cyclist over a 200-m distance after a flying start: steady slow improvement, followed by the revolution of streamlining. [Reproduced with permission from ref. . (Copyright 1987, Philip and Phylis Morrison. Reprinted by permission of Random House, Inc.)]

Figure 7

The energy cost of transport in different types of single vehicles. [Reproduced from ref. with permission. (Copyright 1950, American Society of Mechanical Engineers International.)]

Figure 8

The minimum energy costs of transport for a variety of swimmers, fliers, and runners as well as some man-made devices and different forms of human locomotion. [Reproduced from ref. with permission. (Copyright 1975. Reprinted with permission from American Scientist.)]

Figure 9

The energy cost of transport over unit distance versus speed for swimming fishes. For transport of mass M over unit distance at uniform speed V, the basal cost is A/V and that for motion is BV², where A is assumed proportional to M^3/4 (metabolic rate), and B proportional to M^2/3 (surface area for self-similar shapes) and includes conversion efficiencies. Then, for unit body mass, the minimum energy cost is proportional to M^−5/18 (the slope in Tucker’s plot) and is achieved at an optimum speed, which, for the above assumptions, is effectively independent of the mass, actually proportional to M^1/36. (It would be truly independent of the mass if the dependences of A and B on mass had a common exponent, say γ. The minimum energy cost then would be proportional to M^(γ−1).)

Figure 10

(Upper) Vance Tucker’s budgerigar in the wind tunnel. [Reproduced with permission from ref. . (Copyright 1992 and 1996, Henk Tennekes. Reprinted by permission of MIT Press.)] (Lower) The curves show the dependence on speed of power input (upper solid line), and energy cost of transport (bottom solid line), for the budgerigar in level flight. The energy cost of transport has its minimum at a higher speed than for the power input, and corresponds to the value where a dashed line drawn through the origin of the axes for power input and flight speed is tangent to the corresponding curve. [Reproduced from ref. with permission. (Copyright 1975. Reprinted with permission from American Scientist.)]