First published online March 2, 2007
Journal of Experimental Biology 210, 971-982 (2007)
Published by The Company of Biologists 2007
doi: 10.1242/jeb.02728
Limitations to maximum running speed on flat curves
Young-Hui Chang1,* and
Rodger Kram2
1 Comparative Neuromechanics Laboratory, School of Applied Physiology,
Georgia Institute of Technology, Atlanta, GA 30332-0356, USA
2 Locomotion Laboratory, Department of Integrative Physiology, University of
Colorado, Boulder, CO 80309-0354, USA
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Fig. 1. Ground reaction forces (GRF) in the frontal plane of a sprinter along a
straight path (A) and on a curved path (B). Along a straight path, lateral
forces (Flateral) are negligible and the peak vertical
component of the GRF (Fvertical) equals the peak resultant
GRF (Fresultant). When running along a curved path,
Flateral comprises a significant portion of the total
resultant force. If the upper limit to Fresultant is
achieved on the curve as Greene's theory suggested
(Greene, 1985 ), then for the
same Fresultant, Fvertical on the
curve must be smaller relative to that generated on a straight path. Note that
the axis of the fore–aft component of the GRF is coming out of the page
in both cases and the fore–aft component is negligible when
Fresultant is at its peak.
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Fig. 2. Overhead view of experimental set-up. Circular lines of 1, 2, 3, 4 and 6 m
radius were painted on the ground such that they were all cotangential with a
force platform mounted flush with the ground (counter-clockwise sprinting
direction, as indicated). We used a 30 m straight runway leading up to the
force platform for control trials. A high-speed video camera recorded lateral
views of the subjects as they stepped onto the force platform. Pipe sleeves
(gray circles) were inserted into the ground to mount a removable steel pole
at the center of each track for tethered trials.
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Fig. 3. Maximum sprint velocity as a function of radius for normal curve running
(open circles) and tethered running (filled circles). Velocity decreased with
decreasing radius. The tether reduced the need to generate centripetal force
and increased velocity on the curve, but to magnitudes less than those
predicted by Greene (Greene,
1985). Data represent means for five subjects at each condition.
Error bars are the s.e.m. for absolute velocities. The broken line indicates
mean maximum velocity on straight path (Vo) and the gray
band indicates ± s.e.m.
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Fig. 4. Representative ground reaction force (GRF) components from typical trials
for one subject sprinting at each radius of curvature normally (A) and with a
tether supplying external centripetal forces (B). Straight path sprinting is
shown for comparison (thick gray lines, symmetry assumed for straight path)
and correspondingly smaller radii are indicated by progressively thinner black
lines (6, 4, 3, 2, 1 m). The inside leg (left) and outside leg (right) are
indicated for each GRF component. (Ai,Bi) Vertical GRF. Fore–aft GRFs
(Aii,Bii) indicate negative, braking forces followed by positive, propulsive
forces in each case. Positive lateral GRFs (Aiii,Biii) indicate centripetal
forces acting at the feet. Corresponding tether forces are shown with
indicated curve radius (Biii, inset).
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Fig. 5. Mean peak ground reaction force (GRF) components (i, vertical; ii,
fore–aft; iii, lateral) for normal curve sprinting (A, open symbols) and
tethered curve sprinting (B, filled symbols) as a function of curve radius.
Data for each condition from the outside leg (triangles) and the inside leg
(inverted triangles) are given. Fore–aft GRFs indicate both peak braking
GRFs (negative) and peak propulsive GRFs (positive). Mean peak GRFs (broken
lines) and s.e.m. (gray bands) during straight path sprinting are given for
each case. Values are means ± s.e.m. for the same number of subjects
indicated for each condition as in Table
1.
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Fig. 6. Normalized peak resultant ground reaction forces (body weights, BW) for the
outside leg (triangles) and inside leg (inverted triangles) as a function of
radius (R) during normal curve sprinting (A) and curve sprinting with
a tether (B). Contrary to current curve sprinting theory
(Greene, 1985), axial leg
force (represented here by resultant GRF) decreased with decreasing radius.
During normal curve sprinting, the outside leg generates significantly greater
axial leg force than the inside leg force (A). With the addition of an
external centripetal force provided by a tether rope, however, each leg
produces the same axial leg force (B). Values are means ± s.e.m. for
all subjects at each radius. The broken line indicates average peak force on
straight path; the gray band indicates ± s.e.m. Lines represent power
fits of the outside leg (solid line) and inside leg (broken line) data. For
normal curve sprinting: resultant GRF of outside
leg=2.27R0.091 (r2=0.983); resultant
GRF of inside leg=1.87R0.156
(r2=0.985). For tethered curve sprinting: resultant GRF of
outside leg=2.16R0.155 (r2=0.976);
resultant GRF of inside leg=2.09R0.176
(r2=0.977).
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Fig. 7. Normal curve sprinting velocity data from all subjects plotted with
velocities predicted by theory (Greene,
1985; Greene,
1987). Normalized velocity (V/Vo)
plotted against a dimensionless radius
(Rg/V2o) for normal curve
sprinting (A) and the same data plotted after being transformed to negative
log–log coordinates (B). This negative log transformation allows for
ease of comparing slopes of our data against theory. Our data fit to a power
curve with a significantly higher exponent than both of Greene's 1985
predictions (P<0.05) and smaller than Greene's 1987 predictions
(P<0.05). Our data provide the following fit:
V/Vo=0.746
(Rg/V2o)0.363±0.012.
Greene's 1985 theory for small radii [for
Rg/V2o<0.25, thin broken
line; equation 42 in Greene (Greene,
1985)] predicted a relationship of:
V/Vo=(Rg/V2o)0.333.
Greene's theory for large radii [for
Rg/V2o<1, thin dotted lines;
equation 12 in Greene (Greene,
1985)] predicted a relationship of
V/Vo=0.879
(Rg/V2o)0.258.
Greene's 1987 theory [for ß=0.27, thick dotted lines; equation 20 in
Greene (Greene, 1987)]
predicted a relationship of V/Vo=0.234
(Rg/V2o)0.812 or
[for ß=1.75, equation 20 in Greene
(Greene, 1987)] predicted a
relationship of V/Vo=0.505
(Rg/V2o)0.903.
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Fig. 8. Hypothesized relationship of performance constraints between a sagittal
plane extensor muscle and a frontal plane stabilizer muscle (e.g. foot
invertor). As track radius decreases, the ratio of extensor muscle force
generation to joint stabilizer muscle force (indicated by slope of solid
lines) decreases as frontal plane stabilization becomes increasingly important
at these tighter curves. Along the straight path, joint stabilizers play a
negligible role in limiting speed and maximum sprint speed is constrained only
by a maximum extensor muscle force limit (F(ext)max,
broken horizontal line). On a curved path, however, the importance of joint
stabilization in the frontal plane becomes increasingly important with smaller
radii, and sprint speed may become increasingly limited by the ability of a
group of joint stabilizer muscles to generate force (e.g.
F(inv)max, broken vertical line). Open circles denote the
hypothetical maximum attainable sprint speed for a given track radius.
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© The Company of Biologists Ltd 2007
