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OPTIMAL GEOMETRY PETER R. GREENE*

FOR OVAL SPRINT TRACKS and MARK A. MONHEIT

Dept. of Biomedical Engineering, The Johns Hopkins University. Baltimore. Maryland. U.S.A. Abstract-Truck aspect rani~ is defined as the percentage of lap length devoted to turns on an oval running track. Equations based on experiments are developed to model a compositerunner with a specified top speed, during an accelercltionphase in the straightaways and a centripelol phase in the turns. We calculate velocity deficits for several common track sizes over the range of aspect ratios and predict that, under our assumptions, a perfect circle is the optimal track shape.

INTRODUCTION

NOMENCI.ATURE AR LO R L Tturn T .I, TIn,” T 1-n

TIU

% increase in lap time I: L’ &, Ar

II ‘1

T’ AT L’(l) r .W)

aspect ratio=(ZnR/L,)x 100 lap length total (in m) =2L+ZnR turn radius (in m) length of one straightaway (in m) time elapsed on both turns (in s) = ‘nR,‘r( R) time clapsed on both straightaways (in s) =?L,‘r,=Z?T minimum possible lap time (in s) = L,,‘c~,,=(ZL + ZnR)/v, total transit time (in s) for one lap. ncglcxting acceleration clTccts =(Zl./r”)+L’nR/~(R)] tot;d hp time (in s) for one transit, iucluding accclcration time delay effects = T,., + ZAT (7,,- T,,,,,) x IoO/T,,, (Tahlcs I and 2) gravitational acceleration (in ms-*) top speed (in ms-l) ~‘=reduced speed on turns (in ms-‘) (solved from: gK =o’/(ui -v*)“* for v) = L’,,- r(K) = velocity deficit on turns (in ms“) =0.1276 (szm-‘)Au (ms-‘) (derived in Appendix) average velocity (in m s- ‘) average dimensionless velocity (Figs 2.3 and 4) = T,,,/T,,,=(L,lu~)/[2Ll~~o+~~Rl~‘(R)l position at beginning of the straightaway position at the end of the straightaway (.X2-.X1 = L) time at the beginning of the straightaway time at the end of the straightaway =(l* --1,)= time elapsed along the straightaway. including acceleration clTccts acceleration time delay along straightaway = T’-T=(r,-I,)L/v,, velocity as a function of time along the straightaway accclcration time constant (in s) r - Is position as a function of time along the straightaway

RrcricTJ in jinu//orm Oc+rrhcr 1989. *Correspondence to: Peter R. Greene, Ph.D.. P.E., B.G.K.T. Consulting. I53 Main Street, Huntington. NY 11743. U.S.A. Tel. (516) 421-5541

AND MOTIVATION

The great majority ofoutdoor and indoor competition running tracks are ovals. They consist of two semicircular turn segments connected by two straightaway segments,as shown in Fig. I. Some exceptions include tracks where the turns are sections of an ellipse and tracks where straightaways have been introduced in the middle of circular turns becauseof space considerations. This report covers only the oval tracks, although the techniques prescntcd may be gcneralizcd to the exceptional casts. EOUI-DISTANCE

OVALS

Fig. I. (a) Family of equal distance oval tracks. Aspect ratio is defined as 1.0 for the perfect circle, and 0.0 for the straight line. Wide tracks with aspect ratios -75% are often used outdoors. whereas long skinny tracks with AR-25% arc sometimes used indoors. The goal of this research is to determine optimum aspect ratio as a function of track size and runner’s speed. Ac;e&;ation

Fig. I. (b) For the calculations presented in Tables I and 2 and Appendix I. acceleration time delay elTectsin the acceleration zone arc included. In the acceleration zone. the runner increases his speed from v(R) IO v,,. For the larger tracks, since c(R) - uo, acceleration effects are neglected, and these results are presented in Figs 2. 3 and 4.

447

44s

P. R. GREENE and M. A. MONHEIT

The design of oval tracks raises the question as to what percentage of lap length should be devoted respectively to turns and straightaways. We introduce the term aspect ratio (the percentage of lap length occurring in’turns) to describe this design variable (Greene and McMahon, 1979). A track with an aspect ratio of 60% is commonly referred to as a 60-40 track. In practice, aspect ratios range from 75% for very wide tracks to 25% for narrow tracks. Outdoor tracks are usually 50%-a ratio which produces proper interior space for a football field. There is greater variability in the aspect ratios of indoor tracks. Figure la shows a family of equi-distance ovals. The goal of this research is to determine which, ii‘ any, of these ovals is optimal.

THEORY AND RESULTS

The derivation ol’ lap time for a runner in terms of aspect ratio requires several simplifying assumptions: (I) Following the composite runner approach of Keller (1973) and Senator (1982). we choose a nominal

rO= 10 m s- ’ as the top speed of our runner. The analysis which follows may be performed with arbitrary values of o,,. Given cO= IO ms-‘, the minimum lap time for the indoor, 200-m track (Table I) is 20 s and for the outdoor, 400-m track (Table 2) it is 40 s. (2) There is an instantaneous deceleration from ~‘e to a reduced speed u(R) upon entering a turn. The reduced speed is a function of the radius of the turn and the top speed of the runner. (3) Upon exiting a turn, a runner reacceleratesfrom r(R) to ue in a transition region, or acceleration zone. some 5-20 m in length, beginning at the turn exit (see Fig. lb). The dynamics of this acceleration phase are accurately described by the theory of Furusawa et ul. (1927). Our calculations employ a simplified version of their equations (Greene, 1986). (4) We neglect the initial transient of the start-up phase of X0- and 400-m sprints. During this phase. the runner’s velocity goes from 0 to ue in 4-5 s, over approximately 30-40 m (Volkov and Lapin, 1979; Furusawa et al., 1927). (5) We neglect the special caseswhich occur at large aspect ratios on small tracks where the length ofthe

Table I. Distonccs and times for sprinting on a 200-m track (for ;I composite runner with IJ,,= IO.000 m s- ‘) Aspect ratio 25% Length of turns Length of straights Turn radius Dimcnsiunlcss radius Turn velocity Time elapsed on turns Time rlapscd on struights Lap time (ncghzxtingaccel.) Acceleration time delay Total lap time Heel-over angle %-incrcasc in Inp time*

2nR 2L R VW:)

(m) (m) (m)

r(R) T l”,” T .W.L,h, T,,, AT T (I’*

(ms-‘) If

%-cffwl

I:; IZg., %

50%

75%

100%

50.00

100.00

150.00

1m00

150.00 50.00 23.X7 2.34 9.36 16.02 2E

200.09 0.00 31.83 3.12 9.59 20.85 0.00 20.85 0.00 20.85 16.44 4.26

7.96 0.78 7.H5 6.37 15.00 21.37 0.55 21.92 3H.30 9.60

15.92 1.56 a.91 Il.22 10.00 21.22 0.28 21.50 26.98 7.50

O:l6 21.18 20.55 5.90

‘Relative IO idcut time bused on u0 of 20.000s. Table 2. Distances and times for sprinting on a 400-m truck ((or a composite runner with uo= lO.ooOm s- ‘1 Aspect ratio

Length of turns Length of straights Turn radius Dimcnsionlcss radius Turn velocity Time clupscd on turns Time clapscd on straights Lap time (ncglccting acccl.) Acceleration time delay Total lap time Heel-over angle %-increase in lap time*

lRclativc

ZRR 2L R (Rdr:) r(R) T,“I” T.W.l*hl T,.p AT TWI 0 %-&Cl

1:; (ml (ms-‘) 13

I:; (4 (deg) %

IO idcal time based on r0 of 40.000 s.

25%

50%

15%

100%

ltM.00 300.00 IS.92 1.56 a.91 Il.22 30.00 41.22 0.28. 41.50 26.98 3.75

200.00 200.00 31.83 3.12 9.59 20.85 20.00 40.85

300.00 100.00 47.75 4.68 9.80 30.62 10.00 40.62 0.05 40.6R Il.59 I .69

4OO.rm 0.00 63.66 6.24 9.88 40.49 0.00 40.49 0.00 40.49 a.89 1.22

0.10 40.96 16.43 2.39

Optimal

accrlrration

zone is greater

than

449

geometry for oval sprint tracks

the length

banked

of the

turns (Greene,

1987). Both of these variables

straightaways.

are included in the tables. Using the above technique,

Calculation

a function of aspect ratio for a number of runners on

we have plotted the dimensionless velocity measure as

of lap times

The calculation acceleration

200- and 400-m tracks. taking into account the centri-

of lap times given our assumptions

is as follows. Appendix

I outlines the derivation

petal effects described by (2) and neglecting accelera-

of the

tion effects on the straightaways

time delay. The delay AT is proportional

Ar(s)=O.l276Ac For the circular (Greene.

(Figs 2. 3 and 4). In

practice we may find that experimental

to the velocity deficit AI:=F,,-C(R): (ms-‘).

gR/c’

(1)

turns, radius is related

fall 5-10%

a particular

to speed as

1985):

time

track with a particular

delays

comparable The

(2)

on such a track

formulae

would

results of our calculations

however we find (2) to be more accurate. Equation is a cubic equation be inverted

aspect ratios.

(2)

trends

in the variable w= r*, and can thus

either approximately

Detailed

reported

continuously

or exactly to solve

for

increase

by a

are presented

in

I and 2. Table I shows the results for a 200-m

Tables

presents the results for a 400-m

are suggested in Jain (1980).

prediction

bank angle. The

amount.

track at aspect ratios of 25.50.75 Alternative

data for V& vs

below the theoretical

in the

with

track

and 100%.

calculations tables

Table 2

track with the same vary

reveal

that

smoothly

aspect ratio.

These

the and

results

for u = tj( R). The exact solution for specific values has

vary in magnitude,

been calculated

runners with different top speeds. The most surprising

by a computer

results are presented in Tables Equation

program

and

the

finding is that the circular (100%

I and 2.

(2) can be re-cast in dimensionless

but are qualitatively

similar

for

aspect ratio) track is

predicted to be fastest in all cases considered (with our

coord-

(gR/a& ~‘/t+,) space or (gR/o*. c&) space. This technique is useful for summarizing the

set of assumptions). with or without ctccc+rcrtion efiv?s.

data from many individuals

of the ccntripetal

inates in either

Figures 2 and 3, though considering only the eficcts

with dilTercnt top speeds

ation

radius based on peak velocity (gR/a,$) most useful for

function of aspect ratio. The speed function attains a

describing

flat turns,

based

current

velocity

the more complex

problem

on

and

the dimensionless

(gR/r*)

concerning

phase. revcal a good deal of inform-

v,, on the same set of axes. WC find the dimcnsionlcss

relative

radius

minimum

the behavior

of track

speed as a

at an aspect ratio which depends

convenient

for

both on the speed of the runner

of speed attenuation

on

track. Faster runners and smaller tracks have minima

and the size of the

“0

0.901 0

I 20

I

I

I 80

I too

Fig. 2. Dimensionless velocity C/r, vs aspect ratio for typical indoor 200-m tracks, with top speed LJ,,q a parameter. Results show that the worst performance occurs with aspext ratio in the 1540% range, and best performance occurs with AR = 100%. i.e., the perfect circle.

P. R. GREENE and M. A. MONHEK

450

1.00

r

-

8m/s

0.99-

I

“.a4

0

20

I 40 Aspect

I

60 ratio

I 80

1 100

(%I

Fig. 3. Dimensionless velocity C/r0 vs aspect ratio for the typical outdoor 400-m lap length, with top speed v,, as a paramclcr. Results show that the worst performance occurs with aspect ratio in the IQZO% range. and best performance again occurs with AR = lOO%. i.e.. the perfect circle.

0.99

-3 ;s 0.9E A

Y g

0.97

5 2

0.9E

E

0

0.94

0.93

O

I 20

I 40 Aspect

I 60 ratio 1%)

I

80

1 too

Fig. 4. Dimensionless velocity C/u,,vs aspect ratio for the typical composite runner with u,, = IO ms-’ with lap length L, as a parameter, comparing the indoor 200 m with the outdoor 400 m case Rcsul~sshow that track size can have a -6% elTectindoors, and a - 3% effect outdoors; aspect ratio has - 2% elTectindoors or outdoors.

Optimal geometry for oval sprint tracks

451

at higher aspect ratios. All runners. on both tracks,

an important

role in determining

have local maxima

well be that

speed penalties

longitudinal

and centripetal

maxima

near 0%

100%.

however

and we expect

at 100%

(a perfect circle). The

aspect ratio tracks

that

may exceed those at

this narrow

are impractical

our assumptions

would

down for values in this range. Figure 4 demonstrates another important Again considering centripetal deficit is much greater

break finding.

track. We believe that the turn radius and acceleration

200 m) fall short

reason why

tracks (often much smaller of the outdoor

records.

Al-

though the absolute speed penalties are greater on the smaller

tracks.

will

be so

great as to argue for lower aspect ratios (e.g.. in the 50%

range).

Nevertheless,

the current

assumptions

sprint runners, regardless of lap length.

effects only, the velocity

effects described here are the fundamental than

by combined

acceleration

predict that a perfect circle is the optimal geometry for

on a 200- than on a 400-m

sprint records on indoor

track speed. It may

imposed

Fig. 4 suggests that

optimizing

Acknowfedyements-Funding for this work was partially provided by the Dept. of Biomedical Engineering and the Whiting School of Engineering at Johns Hopkins University; by En-tout-cas, Ltd., L&ester, England; and by B.G.K.T. Consulting Huntington, New York. Special thanks to Stan Corrsin. Bob Giegengack. Floyd Hightill. Tom McMahon. George Pratt, and Dick Taylor for excellent suggestions.

the

for the 200-m track (i.e., the differences

shape of the track is no more critical

track than the 400-m between the local minima and the maxima at 100% are similar for both tracks).

DISCCSSION

Actual competition

tracks require a straightaway

reasonable length for passing. The theoretical we present. and fundamental

experiments,

of

results

support the

belief common among athlctcs and track coaches that wider tracks arc more dcsirablc. The optimal for sprints (the only type of race analyxcd) wide turns and short straightaways ratio). For a particular

installation,

tracks

will have

(i.e., high aspect the estimated

lap

times will vary with lap length. projected top speed of runners, aspect ratio, turn radius, bank angle, etc. The equations presented here are most appropriately to estimate

REFERENCFS

Furusawa, K., Hill. A. V. and Parkinson. J. L. (1927) The dynamics of ‘sprint’ running. Proc. R. Sot. E 102, 29-41. Greene. P. R. (1985) Running on flat turns: experiments. theory. and applications. A.S.hf.E. J. hiomech. Enyr. 107, 96103. Greene. P. R. (19R6) Predicting sprint dynamics from maximum-velocity measurements. hfurh. Biosci. 80. I-IX. Greene. P. R. (1987) Sprinting with banked turns. J. Bitzmechattics 2Q. 667-680. Greene. P. R. and McMahon. T. A. (1979) Running in circles. The Physiokyi.sr 22. S35-S36. Jain. P. C. (19x0) On a discrepancy in track races. RCS. Quarr. Exercise Sporr 51. 432-436. Keller. J. B. (1973) A theory of competitive running. Physics rocluy 26, 43 -47. Senator. M. (19X2) Extending the theory of dash running. A.S.M.E. J. hiomech. &r/r. 104, 20Y -2 I 3. Volkov, N. I. and Lapin, V. I. (I97Y) Analysis of the velocity curve in sprint running. Medicine Sci. Sport I I, 332-337.

used

the time delays of one trnck relative

to

another.

APPENDIX Accelrration

Limifuf ions u/ fhr ryuufions

We have not succeeded in providing

a fully dimen-

sionless solution to the well-posed problem izing the aspect ratio for arbitrary speed

L),,. With

max-min

a dimensionless

problem,

of optim-

lap length Land top solution

all combinations

to

the

of track length,

top speed. aspect ratio, etc., could be projected with a single nomogram-like

chart. It appears impossible

to

find such a solution since the physical scaling laws for the acceleration

phase are ditferent from those of the

circular turn phase, yet net lap time is a combination of these phases-requiring

each case to be calculated

A second limitation of our equations not account for the initial run-up

the relevant

occurs partly

on

Combining

the

on straightaways.

mechanics of longitudinal

those of centripetal

acceleration

experiments

symmetric acceleration

is that they do

phase (the accclcr-

ation from 0 to u,). This generally turns and partly vectorial

rime delay eJficrs

We would like to arrive at a simple expression for the time delay imposed by acceleration efTectson the straightaways. Assuming that the runner enters the straight at the reduced turn speed v(R). then some distance later, on the order of S-20 m, he has re-accelerated up to top speed u,,. If. instead, the runner had traversed this acceleration zone&c Fig. I b) at top speed, then a time T= L/u, would be required IO cover this distance. However, acceleration effects demand that a longer time T’= T+ AT is required. From basic principles, we would like to derive an expression for the time delay AT. Furusawa er al. (1927) present the following results, confirmed by experiments. for velocity and position as a function of elapsed time for an accelerating runner: u(r)=u,(l

separately.

acceleration

with

is highly complex and

have not been done. The

zone approach

lects the initial transient acceleration,

used here ncgwhich may play

I

-es”‘)

(A))

~(t)~~~f-u~~(l-e~“‘).

(A2)

These results are derived from Newton’s Laws, and require only that we know the top speed u,, of the runner, and the time constant r in order to predict position and velocity as a function of time. We can rewrite (AZ) twice, once for the beginning of the acceleration mne x,, and once for the end position x2 (x,-x, = L): x, =u,t,

-u,r(l

-e-‘1”)

~~~u~r~-u~r(l-e-‘~“).

(A3) (A4)

45’

P. R.

GREEF~E

and M. A. MCINHEIT

Subtracting (A3) from (Ad), and taking the limit t2+zu yields: L=uo(t2 -t()--L’oTc-““. (A%

Using the notation AC = rO- u(R), we find:

AT=(mg,'fJ(Av/gL

(A9J

We can solve equation (Al) for t, as: r,=-rIn(l-u(t,)/c,).

(A@

Substituting (A61 into (AS) and solving for the time elapsed yields: l*-t,=(L/l’o)+r(l--P(f,)/VO). (A7) Greene (1986) derives the time constant r in terms of the top speed v,, and the horizontal force F&g as: r=(L’,/g)(mg/FJ

(A@

For Olympic-class athletes, we set (F,/mg)z0.8 at:

AT(s)=O.l276Av(ms-').

to arrive

(AlO)

Equation (AlO) is our central result. which is used to calculate the acceleration time delay effects presented in Tables I and 2.