Cycle power: a predictive model G. J. van Ingen Schenau

In competition cycling success usually goes to the rider most resourceful mentally and physically. Nevertheless, essentially subjective qualities can be usefully reinforced by an understanding of the various physical factors involved. These include rolling and air friction, the physiology of muscle, the weight of the machine, body posture, and air density.

There is a close correlation between the speed of a cyclist and the average power he can supply over a given distance. The primary role of this power, which can be enhanced by training, is to overcome friction. Since little power is lost via the pedals, chain, and rear wheel used to transfer the propulsive force to the ground, it is possible to make a direct correlation between stamina and the generated mechanical power, on the one hand, and those factors influencing fric- tion, on the other. Such a power equa- tion permits the calculation of factors influencing air resistance and rolling friction, thus determining their influence on the speed achieved. It is therefore possible to calculate the effect on the speed of racing cyclists, and hence their performance, of factors such as the weight of the bicycle, posture, altitude, and whether they are cycling alone or as part of a group.

The most obvious approach to the mechanics of cycling is to examine the structure of the bicycle. Techniques and methods used in mechanics can, howev- er, also be employed to identify natural laws to which the rider is subject. The scientific discipline concerned with such models of human (and animal) move- ment is known as biomechanics.

If we view the bicycle and cyclist as a single system (‘free body diagram’), we can describe it using a range of mecha- nical tools such as an external forces equation, a comparison of momenta, or a power equation. Adoption of the free body diagram means that the external and internal values are, in principle, fixed. Using Newton’s Second law to draw up an equation of forces, we can

G. J. Van lngen Schenau, Ph.D.

After graduating in physics at the Technical University, Delft, he joined the physical educa- tion department of the Free University of Am- sterdam. His research is particularly concerned with the physical factors and techniques in- volved in cyclic motion -as in skating, cycling, and swimming - and with intermuscular coor- dination during complex movements.

EURO ARTICLE (see p.ii)

This article is published in association with Natuur en Techniek, Netherlands.

Endeavour. New Series, Volume 12, No. 1,1988. 0160-9327188 $3.00 + 0.00. 0 1988. Pergamon Press pk. Printed in Great Britain.

identify the external forces acting on the system, as shown in figure 1. These are the air friction (F,) on the entire system; the rolling friction, on the front and rear wheels (F,, and Frz respectively); the normal forces on the wheels (N, and N2); gravity (G) acting on the whole system; and the propulsion (FJ on the rear wheel. Application of Newton’s Second Law to these forces means that the dynamic motion equations involved are completely determined. Muscular forces, power on the pedals, etc., are regarded as internal forces in the free body diagram and therefore play no part in the description of the system. They do, of course, influence it but find ex- pression in factors such as FP.

It is then possible to draw up two equations for the everyday situation of someone cycling up a slope with the gradient angle (r.

The horizontal vector can be express- ed as:

F, - F, - F1 - G sin (II = ma (1)

and the vertical vector as: N-Gcosa=O (2)

In these equations, the rolling friction and normal forces shown in figure 1 are consolidated as F, and N respectively.

At a given constant speed (ma = 0), propulsion F, must equal the sum of air friction, rolling friction, and the compo- nent of gravity acting in the direction of travel. As is also the case with motor- driven systems, little useful information can be gained merely from such an equation of forces. Before we can turn (1) into a more meaningful power equa- tion, we need to understand something of the nature of the various forces in- volved.

Friction Rolling friction is caused by what can be regarded as continual impacts between the tyre and the road, and in particular with irregularities in the road surface. These results in deformation of the tyre and, occasionally, of the road as well. As a first approximation, the established formula for contact friction can be ex- ploited to find the rolling friction:

F,=,uN (3) where p is the coefficient of rolling friction and N the normal force between

the tyre(s) and the road. The above is true for the whole system, irrespective of the number of wheels. The coefficient of rolling friction has been found to be inversely proportional to the diameter of the wheels and also to depend on the width of the tyre, the tyre pressure, and the nature of the road surface. Air friction comprises two components, namely frictional resistance and drag.

Friction resistance arises from the ex- istence of air layers moving past each other at varying speeds and exerting a force on each other. Since air is carried along on the surface of the cycle and the cyclist, there will be a gradient of varying windspeeds between this air and the surrounding air. The frictional resist- ance is strongly influenced by the rough- ness of the surface. The familiar smooth, high-speed racing clothing (also known as a ‘skin suit’) is designed to reduce this frictional resistance.

The air pressure ahead of the rider is different from that behind him, this being the result of the differing velocities of the boundary layers in front of and behind the body. Rather than merging smoothly behind the body (as they do with small objects and at low speeds), however, these boundary layers display rapid turbulent eddies. According to Bernoulli’s Law, this speed differential generates a pressure differential known as dynamic pressure. Where the cyclist’s speed relative to the air is v, the dynamic pressure (p) will be:

p = ‘/2 @ v2 (4) with (kg/m

f being the density of the air ). If this dynamic pressure is mul-

tiplied by the surface exposed to the onrushing-air (the perpendicular section A, of bicycle and rider), we arrive at the force exerted on the rider and hence a measure of air resistance. The extent of turbulence and the associated slipstream formation does, however, depend to a great degree on how streamlined the object is. There is, therefore, a need to expand the air friction equation by in- cluding a dimensionless term -the drag coefficient CD - specifically covering streamlining. The following equation is usually used for total air friction (that is, including frictional resistance):

F1 = % Q v2 CD A, = k1 v* (5)

Figure 1 The forces acting on the bicycle + rider system and appearing as external forcesinthefree bodydiagram.

Note, however, that the Co value covers not only the streamlining but also, for example, the nature of the surface.

Propulsive Forces We now come to the last of the signi- ficant forces featuring in figure 1; the force required to overcome the opposing forces, This thrust or propulsion is a reaction to the force exerted on the ground by the rear wheel. By selecting the correct gear ratio, the rider ensures optimum conversion of muscular power into thrust against the earth. Thrust against some other body is essential for any self-propelled system (thus for any form of locomotion) where friction must be overcome. It is also an essential requirement that the propulsive force (the reaction to the thrust) can never move forwards while the thrust is being exerted. Were this to happen, of course, the system would be gaining power (= force x the velocity of the focal point of the force) from its surroundings - which by definition is not the case with locomo- tion. This is rather harder to visualise with cycling than with, for example, walking. After all, it seems as if the propulsive force is constantly moving forwards. It should, however, be re- membered that the tyre is stationary, relative to the ground, at the point of contact. The wheel can in principle be viewed as a system comprising millions of tiny feet successively exerting monet- ary thrust on the road. Association with this is the wheel’s role as a thrust mechanism - ensuring a constant supply of new thrust points while itself using up little energy.

Av is the velocity of the air relative to the ground (wind). As has been pointed out, F, plays no part in this equation because the point through which it acts has zero velocity. The final term in (5) differs from the proceeding two in that it con- cerns not lost power but rather power contributing to the potential energy of the system. If the power exerted by the cyclist (PO) is not equal to Pf, the differ- ence P, - Pf will result in a change mva (where a is acceleration) in the kinetic energy (and hence in the velocity) of the system:

P,-‘/z~A,Co(v+Av)~v-@NV-N sin u.v = mva (6)

This so-called power equation for cycl- ing will be used throughout the rest of this article.

In running, by contrast, a great deal of mechanical energy is required to acceler- ate the legs as they are swung forward, largely to be lost again as they are slowed down. It is for this reason, in particular, that much higher speeds can be attained by cycling than by running.

Physical fitness and power In man, muscles can be regarded as the engines which supply the necessary pow- er. Once activated by the central ner- vous system, muscles generate force and can contract within specific limits. Each cyclical movement of the pedals requires the exertion of a certain amount of work (A) by the muscles, particularly those of the buttocks and upper legs. If the cyclical movement is repeated at a fre- quency f, the power supplied can be expressed as P, = A X f (the minimal (approx. 3 per cent) losses through the chain and gears being disregarded). As the muscle contracts, the mechanical work required is fuelled by the energy content of the energy-rich phosphates stored in the muscle fibres. The initial source of these phosphates - via gly- cogen, fats, and in extreme cases also proteins - is the food consumed. Immi- nent exhaustion of the phosphates pre- sent in the muscles (which occurs within approximately 10 s) requires their re- plenishment. This can occur in one of two ways:

1. In the absence of oxygen via the so-called anaerobic metabolism.

2. In the presence of oxygen via the so-called aerobic metabolism.

The power balance As is the case with cars, the speed of a cyclist is governed not so much by the availability muscular force as by the average power. For this reason, we will modify (1) to yield a power equation.

Figure 1 illustrates the external forces acting on the rider + bicycle system. The external power Pf is now defined as the power exerted by the system on its surroundings. To calculate this, it is necessary to multiply all external forces by the velocity of the point through which each acts and to total these values. Where the cyclist’s speed is constant, we then find: Pf=i&ApCD(v+Av)2v+yNv+G

sin cx.v (5)

In stamina sports such as cycling, the major limit on performance is the capac- ity of these ‘suppliers’; this is contrast to power sports, for example, where the key factor is the absolute force gener- ated by the muscles.

k ’ A do I 0.5 500

T (min)

Figure 2 The power(P) generated by a top athlete, expressed as a function ofthe period(z) overwhich the effort is required (tis set out on a logarithmic scale).

The extent to which muscles can carry out the aerobic or anaerobic production of energy-rich phosphates can be ex- pressed by the terms ‘fitness’ or, some- what better, ‘stamina’.

In man, we cannot speak of a single fixed level of stamina or power. In contrast to a petrol engine, for example, the maximum average power that a human being can supply is dependent on the time span over which it is required. The aerobic metabolism, in which ox- ygen is consumed in the oxidation of food can reasonably be compared to such a petrol engine. This is, however, a much slower system than the anaerobic one which, even in the absence of ox- ygen, can also oxidise food for a limited period. This anaerobic metabolism is accompanied by the production of lactic acid. Too high a concentration of this acid in muscles and in the blood reduces the performance of the muscles. If per- formance is to be optimal, therefore, the acidification of the muscles has to remain within certain limits - which can be influenced by training. The combined effect of these factors is that a person can reach very high power levels over a short period, but that the average power level decline rapidly with increasing time. This can be seen in figure 2. In 10 seconds, trained cyclists can achieve an average of 1500 Watts or more (about 2 horsepower), but over longer distances even professionals cannot exceed 400 Watts. For the sake of comparison, a recreational cyclist has to strain to reach an average of 100 Watts over one hour.

As well as the muscles’ ability to produce energy aerobically and anaero- bically, a number of other factors also influence power output. As with en- gines, the major one is the transmission ratio. While maintaining a constant propulsive force F,, the rider can alter the force required and the repetitive rhythm of his muscles by his choice of gear. Muscles have been shown to be most efficient at a strength level well below maximum strength.

Racing cyclists, therefore, choose a

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gear at which the force exerted on the pedals will be around 50 per cent of their body weight. This is then associated with a power-dependent pedal speed varying from 60 rpm at low power outputs to approximately 110 rpm at higher power levels. Recreational cyclists usually adopt a much slower rhythm and gener- ally choose an inappropriately large gear ratio.

A further point is the range over which the muscles can contract, this being determined by body posture, saddle height, and pedal crank length. With the majority of bicycles, the saddle height and crank length recommended by the manufacturer and trainer do indeed yield greatest efficiency. Saddle height: adjust with the leg straight and the foot placed horizontally on the pedal. Crank length: 20 per cent of leg length. Pro- vided these settings are correct and the right gear is chosen, cycling can be expected to be approximately 20 per cent efficient. In other words, roughly 20 per cent of the calorific value of the food consumed can be exploited as mechanic- al power.

A predictive model A number of calculations can be made on the basis of the power balance equa- tion (6). First, it is possible to determine the power a rider must supply in order to maintain a given average speed. In the case of a 7.5 kg rider, 1.80 m tall, adopting a racing posture on his 9 kg bicycle and travelling in windless weath- er over a flat road, (6) can be simplified to:

P, = 0.19 v3 + 4 v (7) where P is in Watts and v in m/s. This equation is the source of the values in figure 3, where power is expressed as a function of speed. For cycling uphill, of course, this equation can be sup- plemented by a value 750 sin (x x v. Further data probably of interest to the reader and already used to derive (7) are: e = 1.2 kg/m3; A, = 0.4 m2; Co = 0.8; p

= 0.0055. Wind tunnel experiments have shown a close correlation between CD X A, for a given rider and that rider’s height 1 and weight m, the relationship being Co X A, - 1%. As an approximation, moreov- er, 0.15 v3 of the air friction value 0.19 v3 can be regarded as caused by the rider and the rest by the cycle. Finally, it can be noted that the air density Q is prop- ortional to the air pressure and falls with increasing height h above sea level in accordance with power of e (e-“~“oo125 h). The above data enable predictions to be made. The influence of individual fac- tors on speed is calculated, all other factors being assumed to remain con- stant.

Should a reader checking through the calculations for the examples below en-

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counter small discrepancies, these will usually be because the original calcula- tions were done on the basis of’s rather more complex model which also in- cluded the (experimentally determined) dependency of Co on the velocity.

Weight of bicycle and rider Until the Italian Moser (figure 4) broke the world one-hour record a few years ago using a drastically modified bicycle (small front wheel, large rear wheel, enclosed wheels, raised ends to the handlebars), the world of cycling con- centrated on reducing the weight and streamlining the frame. Although this may have helped psychologically, it is not an’ obvious consideration from a biomechanical point of view. Other fac- tors - such as body posture and clothing - influence speed to a much greater extent. For example, with a superlight bicycle weighing 6 kg, a rider will be able to travel roughly 0.15 per cent faster than with one weighting 9 kg. The rider’s own weight is rather more significant in that the same weight difference (3 kg) will yield a speed differential of 0.6 per cent air friction being taken into account. Both figures are for cycling on flat roads. Weight is of much greater significance when cycling uphill.

Skin suits After the successful introduction of the smooth, close-fitting (skin) suit for ska- ters, this clothing is increasingly being worn by racing cyclists (see figure 4). As has been pointed out, performance will be enhanced at speeds above 7 m/s. A typical racing cyclist producing 350 Watts can expect to achieve a decrease in air friction at 42 km/hour equivalent to l-l.5 per cent increase in speed. While not earth-shattering, it is nevertheless considerably greater (and much cheap- er) than that gained with a superlight bicycle.

Body posture Body posture with respect to the vertical greatly influences the value A, Co. Wind-tunnel experiments on skaters can be used as the basis for order-of- magnitude estimates of how cycling speed will be influenced by variations in

Speed (m/s)

Figure 3 The external power(P) that a typical racing cyclist in a racing posture (see text) has to supply in order to maintain a given speed.

body posture. The wind-tunnel tests showed that a 10 per cent deviation from a skater’s optimal horizontal trunk posi- tion increased the air friction by over 18 per cent. Given the relatively small contribution made by cylists’ legs to the frontal surface, the trunk posture being the same, it is fair to assume that a cyclist (without cycle) will be affected at least as much by alterations in the position of the trunk. In speed terms, this means a loss of 1 km/hour (that is, almost 2.5 per cent) for every 10” deviation from the optimal posture. Since riders of normal racing cycles frequently assume postures up to 30” from the horizontal, it is obvious that there is often considerable potential for increasing speed. Many, however, find it difficult to keep the trunk so low for long periods on a normal racing bicycle (the thighs hit the trunk and the arms must be kept bent). Never- theless, a great deal can be achieved by slightly the whole frame forward, as is illustrated in figure 5 (and see figure 4). One disadvantage of such a design, with its large rear wheel and small front wheel, is, of course the cyclist’s need to have two different tyre sizes available. It was on a bicycle of this kind that Moser achieved a world one-hour record (over 51 km).

Shielding Riding immediately behind another cyc- list cuts air friction significantly (by up to 40 per cent). For this reason, a group can travel faster for the same average power output than a single cyclist (figure 6). Provided all members of the group take an equal share in the front-runner’s role, and make optimal use of the slipstream of the rider in front of them, it is evident from the equation of power that groups of two and five cyclists will travel approx- imately 5 per cent and 10 per cent faster respectively than a cyclist unable to use the slipstream of a preceding rider. One need look no further for an explanation of the failure of most solo breaks from the bunch, even by very powerful riders.

Air density At sea level, high-pressure and low pressure zones can create considerable variations in atmospheric pressure. Time trials on covered tracks, in particu- lar, will be faster during low-pressure periods. Given the variations in air pressure that can occur in the Nether- lands, up to 2 per cent of the speed differences recorded on such a track can be attributed to variations in air pressure (use equation (6) to compare speeds at 970 and 1040 millibars).

Air pressure falls rapidly with increas- ing altitude. From the relevant data, it can be calculated that speeds will in- crease by almost 0.2 per cent for every 1OOm increase in altitude. Up to a height of around 1500-2000 m, no matching drawback is encountered (provided an

Figure4 Francesco Moser riding a low profile bicycle with rear disc wheel and wearing ‘skin’clothing. (Photograph courtesy of J. Bishop, MidOxon Cycle Racing Team.)

Figure 5 By tilting the frame, it is possible to achieve an aerodynamically more favourable trunk position without hindering muscle movement.

Figure 6 The’bunch’during the 1986Tour de France. (Photograph courtesyofJ. Bishop, Mid Oxon Cycle Racing Team.)

acclimatisation period is observed) but further increases in altitude will result in a fall in the aerobic contribution to power output as the oxygen concentra- tion declines. Nevertheless, the optimal altitude for speed cycling, even at the rather longer distances, probably lies between 2500 and 3000 m.

Human-powered vehicles Enormous increases in speed can be achieved if the entire rider and bicycle are enclosed in a cigar- or tear-drop shaped housing. Although it increases both the weight and the vertical cross- section, this kind of ideally streamlined housing (CD in the order of 0.15) never- theless yields a major decrease in air friction. In particular, designs which reduced the vertical cross-section by placing the rider in a horizontal position make it possible to reach speeds very significantly higher than those achiev- able with an ordinary racing bicycle. The 200 m record, with a flying start, for racing bicycles stands at approximately 72 km/hour, yet the same record for a fully enclosed cycle with a supine rider is already over 91 km/hour. With this kind of cycle even the relatively low power output of 150 watts is sufficient to main- tain a speed of over 50 km/hour. It is difficult to predict whether such vehicles will be found on the roads of the future. Some solution will have to be found to the problem of high susceptibility to side winds.

Conclusions There are, of course limits to the practic- al application of the above calculations. Cycle racing is predominantly a group sport and the winner is usually the cleverest rider with the best race sense rather than the one maximizing power output and minimizing frictional losses. Nevertheless, it is clear from the above that biomechanical principles can be exploited to identify mechanical laws governing human movement.

Bibliography For comparable examples of power equations and their application the read- er is referred to: [I] Clarijis, J. P. and Ingen Schenau, G. J.

van (ed.), Wielrennen. Een konfrontatie tussen de wetenschap en de praktijik van het wielrennen. De Tijdstroom Lochem/ Gent, 1985.

[2] Faria, I. E. and Cavanagh, P. R. ‘The Physiology and Biomechanics of Cycling’. Wiley, New York. 1978.

[3] Ingen Schenau, G. J. van. ‘The influence of air friction in speed skating’. Jnl. Biomechan. 15,449-58, 1952.

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