Spring Mass Characteristics of the Fastest Men on Earth

Abstract

The spring mass model has widely been used to characterize the whole body during running and sprinting. However the spring mass characteristics of the world's fastest men are still unknown. Thus the aim of this study was to model these characteristics for currently the 3 fastest men on earth (Usain Bolt, Tyson Gay and Asafa Powell). This was done by using data collected during the 2009 World championships in Berlin and the modelling method of Morin et al. 21. Even though Bolt achieved the greatest velocity (12.3 m.s - 1) over the 60-80 m split compared to his competitors, his estimated vertical stiffness (355.8 kN.m - 1) and leg stiffness (21.0 kN.m - 1) were significantly lower than his competitors. This reduction in stiffness is a consequence of Bolt's longer contact time (0.091 s) [corrected] and lower step frequency (4.49 Hz).Thus Bolt is able to run at a greater velocity but with lower stiffness compared to his competitors.

Full text

667Orthopedics & Biomechanics
Taylor MJD, Beneke R. Spring Mass Characteristics of … Int J Sports Med 2012; 33: 667–670
accepted after revision
February 04 , 2012
Bibliography
DOI http://dx.doi.org/
10.1055/s-0032-1306283
Published online:
April 17, 2012
Int J Sports Med 2012; 33:
667–670 © Georg Thieme
Verlag KG Stuttgart · New York
ISSN 0172-4622
Correspondence
Dr. Matthew JD Taylor, PhD
Centre for Sports and Exercise
Science
University of Essex
Wivenhoe Park
CO4 3SQ Colchester
United Kingdom
Tel.: +44/01206/872 818
Fax: +44/01206/872 592
mtaylor@essex.ac.uk
Key words
●
▶
sprinting
●
▶
biomechanics
●
▶
stiff ness
●
▶
athletics
●
▶
Usain Bolt
Spring Mass Characteristics of the Fastest Men on
Earth
for K
vert ) therefore leg stiff ness (K
leg ) is calculated.
These variables are usually derived from double
integration of force data. However, a method pro-
posed by Morin et al. [ 21 ] mathematically mod-
els the force curve as a sine-wave using mass,
stature, fl ight (t
f ) and contact (t
c ) times, and
velocity data. This model has allowed the SMM to
be calculated during the 100 m for novice sprint-
ers (t
100 14.21 s; ̄υ 7.06 m
. s -1 ), and the 400 m of
‘well trained athletes’ (t
400 52.67 s) whilst run-
ning on the track [ 15 , 22 ] . An intriguing applica-
tion of this approach is to use it in elite
competitive events where only running speed
and step frequency data are available for 20 m
splits. It was therefore the aim of this study to
adopt the model of Morin et al. [ 21 ] and apply it
to the top 3 male sprinters in the 100 m World
Athletics Championship fi nal of 2009. It was
hypothesized that even though UB runs at a
greater velocity than his fellow competitors his
leg and vertical stiff ness would be less due to his
reduced step frequency [ 11 ] and the requirement
for increased impulse [ 4 ] .
Method
▼
This study was performed in accordance with the
ethical standards laid out by the IJSM [ 12 ] . The
IAAF commissioned a biomechanics project,
Introduction
▼
The 100 m world record, currently held by Usain
Bolt (UB), is 9.58 s. UB is clearly a phenomenal
athlete and our work has suggested that his stat-
ure and his reduced step frequency facilitate his
success resulting in an advantage in relative
power development and mechanical effi ciency
compared to his competitors [ 3 ] . Furthermore
our past work [ 4 ] supported the concept [ 3 , 25 ]
that the longer steps of UB with longer ground
contact times and longer distances travelled dur-
ing ground contact generated higher impulses
resulting in an exceptionally fast winning time.
However we do not know what the spring mass
characteristics of the world’s fastest men are. Leg
and vertical stiff ness is often cited as increasing
as speed increases, but these studies have been at
relatively low speeds (~8 m
. s − 1 ) [ 1 , 6 , 14 , 21 ] and
not the speeds of world class sprinters.
The spring-mass model (SMM) is used to model
both the vertical motion of the centre of mass
(CoM) during contact and the stiff ness of the leg
spring. It has widely been used to characterize
the whole body during running and sprinting.
The calculation of the eff ective vertical stiff ness
(K
vert ) is derived from the maximum vertical
force and the displacement of the CoM (Δy
c ).
During running the leg sweeps through an angle
thus it is not directly over the CoM (as modelled
Authors M. J. D. Taylor
1 , R. Beneke
2
Affi liations
1 Centre for Sports and Exercise Science, University of Essex, Colchester, United Kingdom
2 Department of Medicine, Training and Health, Philipps-Universität Marburg, Germany
Abstract
▼
The spring mass model has widely been used to
characterize the whole body during running and
sprinting. However the spring mass characteris-
tics of the world’s fastest men are still unknown.
Thus the aim of this study was to model these
characteristics for currently the 3 fastest men
on earth (Usain Bolt, Tyson Gay and Asafa Pow-
ell). This was done by using data collected dur-
ing the 2009 World championships in Berlin
and the modelling method of Morin et al. [ 21 ] .
Even though Bolt achieved the greatest velocity
(12.3 m
. s − 1 ) over the 60–80 m split compared to
his competitors, his estimated vertical stiff ness
(355.8 kN.m − 1 ) and leg stiff ness (21.0 kN.m
− 1 )
were signifi cantly lower than his competitors.
This reduction in stiff ness is a consequence of
Bolt’s longer contact time (0.091 s) and lower step
frequency (4.49 Hz). Thus Bolt is able to run at
a greater velocity but with lower stiff ness com-
pared to his competitors.
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668 Orthopedics & Biomechanics
Taylor MJD, Beneke R. Spring Mass Characteristics of … Int J Sports Med 2012; 33: 667–670
undertaken by the German Athletic Federation, which produced
individual split times and corresponding step rates for each ath-
lete during the 100 m fi nal at the 2009 Athletics World Champi-
onships in Berlin [ 12 ] . We have used this data in previous studies
[ 3 , 4 ] to calculate physiological and biomechanical parameters
for the fi rst 3 fi nishers (UB; Tyson Gay, TG; Asafa Powell, AP). We
therefore choose to estimate the SMM characteristics for these
same athletes. These athletes were chosen for analysis because
to date they are the 3 fastest sprinters of all time (100 m personal
bests: UB 9.58 s, TG 9.69 s, and AP 9.72 s) who between them
hold the top 18 all-time 100 m times [ 29 ] . Anthropometric data
for all 3 sprinters (
●
▶
Table 1 ) were gathered from available refer-
ence sources [ 27 ] . UB’s body mass was adjusted based on a per-
sonal statement about his stature and on and off season body
mass [ 9 , 28 ] .
The velocity profi le (
●
▶
Fig. 1 ) of each sprinter was modelled
based on the distance covered after individual split times using
the integral of a bi-exponential model approximating increase
and decrease amplitudes of velocity and corresponding time
constants [ 4 ] . This resulted in 0.02 s epochs over the 100 m. The
spring mass characteristics (eq. 1–6) were estimated, using the
method of Morin et al. [ 21 , 22 ] for the 60–80 m split only, this
was when the sprinters were at their maximal velocity and
accelerations/decelerations were minimal. This model [ 21 , 22 ]
has been reported to have low bias (0.12–6 %) compared to the
reference values from the force plate and high determination
coeffi cients (0.89–0.98). Vertical force (eq.2) was calculated
from the body mass of the athlete (in kg), the contact (t
c , in sec-
onds) and fl ight (t
f , in seconds) times. Step time (t
c + t f ) was cal-
culated from step frequency. Flight time (t
f ) data were derived
from Weyand et al. [ 25 ] which along with step time allowed t
c to
be calculated. Leg stiff ness was derived from the maximum ver-
tical force and the change in leg length (eq. 4).
K vert = F max ∙ ∆y
c − 1 (1)
Fmassg
t
t
f
c
max =⋅+
⎛
⎝
⎜⎞
⎠
⎟
⭈

21
(2)
where g is the gravitational acceleration.
∆y
c (in meters) is the maximal downward displacement of the
CoM during contact and was calculated using equation 3.
ΔyF
mass
tgt
c
cc
=+᎐⭈ ⭈
max
2
2
2
8
(3)
K leg = F max ∙ ∆L
− 1 (4)
where L is leg length (in metres) and is modelled from each ath-
lete’s stature according to Winter [ 26 ] . Leg length obtained this
way has no signifi cant eff ect on the stiff ness values obtained
using the method of Morin et al. [ 21 ] ,
L = 0.53h (5)
The change in leg length (ΔL, in meters) was calculated from
''LL L ty
cc
c
 
§
©
¨·
¹
¸
2
2
2
%
(6)
Statistical analysis
▼
Descriptive statistics (means and standard deviations) and a non-
parametric Kruskal-Wallis-Test with a post-hoc Mann-Whitney-
Test were used to determine changes in spring mass variables
between the 3 sprinters. Statistical signifi cant level was set at
p < 0.05. Statistical analysis was performed on SPSS (v 16.0).
Results
▼
The average velocity achieved by UB, TG, and AP over the
60–80 m splits was 12.19 (0.26) m
. s − 1 . Step frequency was mark-
edly less for UB compared to TG and AP (
●
▶
Table 2 ). Estimated
F
max was signifi cantly greater for UB compared to AP and TG, and
AP was signifi cantly greater than TG (
●
▶
Table 2 ). Estimated K vert
Table 1 Anthropometric and performance data from the 2009 World
Championship.
UB TG AP
age (yrs) 24 28 28
stature (m) 1.96 1.83 1.90
body mass (kg) 95 73 88
t 100 (s) 9.58 9.71 9.84
v 100 (m.s − 1 ) 10.44 10.30 10.16
Time (s)
012345678910
Velocity (m s–1)
0.0
2.5
5.0
7.5
10.0
12.5
Fig. 1 Velocity profi le derived from the bi-exponential model for UB
(black line), TG (light grey line) and AP (dark grey line).
Table 2 Estimated SMM characteristics for 60–80 m.
Parameter UB TG AP
t c ( s ) 0.091 ± 0.001 *# 0.070 ± 0.001 * 0.080 ± 0.001
t f ( s ) 0.132 ± 0.001 *# 0.132 ± 0.001* 0.131 ± 0.001
v c ( m . s − 1 ) 12.3 ± 0.02 *# 12.1 ± 0.02* 11.9 ± 0.01
F max ( kN ) 3.60 ± 0.001 *# 3.25 ± 0.002* 3.59 ± 0.001
K vert ( kN . m − 1 ) 355.8 ± 0.46 *# 541.8 ± 0.89* 457.0 ± 0.47
K leg ( kN . m − 1 ) 21.0 ± 0.05 *# 31.0 ± 0.05* 28.4 ± 0.05
∆y c ( m ) 0.01 ± 0.0001 *# 0.006 ± 0.0001 * 0.008 ± 0.0001
∆L ( m ) 0.17 ± 0.0001*# 0.10 ± 0.0001* 0.13 ± 0.0001
SF ( Hz ) 4.49 4.96 4.74
All data signifi cant to < p 0.001; * Signifi cant diff erence to AP; # signifi cant diff er-
ence to TG. ± standard deviation
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669Orthopedics & Biomechanics
Taylor MJD, Beneke R. Spring Mass Characteristics of … Int J Sports Med 2012; 33: 667–670
and K
leg for UB were signifi cantly less compared to TG and AP.
Estimated K
vert for UB was 52 % and 28 % less than TG and AP
respectively and estimated K
leg was 47 % and 35 % less compared
to TG and AP, respectively. TG had the greatest stiff ness charac-
teristics of all the sprinters and this was in part due to the sig-
nifi cantly shorter t
c and reduced Δy
c and ΔL compared to UB and
AP (
●
▶
Table 2 ). t f was statistically signifi cantly diff erent between
all athletes, however this was due to the calculation method of
fl ight time and it is in practical terms not a signifi cant fi nding.
Discussion
▼
The spring mass characteristics of world class sprinters, in com-
petition, who run at velocities in the region of 12.19 (0.26) m
. s − 1
have not previously been reported. Morin et al. [ 21 ] reported the
spring mass characteristics of novice sprinters running at a
velocity of 8.24 (0.24) m
. s − 1 (during 60–80 m split), the sprinters
of Arampatzis et al. [ 1 ] ran at 6.59 (0.24) m
. s − 1 , while the sprint-
ers of He et al. [ 14 ] and Cavagna et al. [ 8 ] ran at 6 m
. s − 1 and
5 m
. s − 1 , respectively.
Prior to discussing the results it should be made clear that these
data are modelled and are therefore estimates. We acknowledge
that there may be limitations in our approach, but the opportu-
nity to take direct measurements in a world championship fi nal
to calculate SMM characteristics is, at the moment, unlikely to
happen. The sine-wave model of calculating stiff ness [ 21 ] has
been shown to be a valid method of estimating stiff ness. How-
ever it does have limitations. Namely the model assumes a con-
stant point of force application on the ground during stance
phase, when the location actually moves 0.16 m [ 17 ] . The limita-
tions associated with spring-mass models per se are also appli-
cable to the sine-wave model [ 21 ] .
The sine-wave model requires fl ight (t
f ) and contact time (t
c )
which were not directly measured during the 100 m fi nal. We
estimated t
f and t
c via step frequency, which allows the calcula-
tion of step time. t
f data were derived from Weyand et al. [ 25 ]
which along with step time allowed t
c be to calculated. Both t
f
and t
c were comparable to those measured via kinematics, accel-
erometry or on an instrumented treadmill [ 10 , 15 , 22 ] . Further-
more our data was also comparable to world class sprinters (Ben
Johnson (11.76 m
. s − 1 )/Carl Lewis (11.63 m
. s − 1 )) recorded at the
1987 World Athletic championships with t
c of 0.082/0.085 s and
a t
f of 0.122/0.138 s at the 60 m split for both sprinters [ 20 ] .
Therefore the calculated t
f and t
c data can be used in this context
to estimate SMM. Lastly, we were unable to take direct measure-
ments of mass, stature and leg length. Therefore we used anthro-
pometric data freely available in the public domain. These data
have been used in previous sprinting, and sprinting and anthro-
pometric studies [ 3 , 9 ] . Anthropometrical measures have a 1:1
weighting (or less) on stiff ness measures [ 21 ] . A percentage
change in leg length has virtually no impact on vertical or leg
stiff ness whereas a 10 % reduction in body mass has a 10 % reduc-
tion in stiff ness and vice versa [ 21 ] .
Running velocity is the product of step frequency (SF) and step
length (SL). An inverse relationship exists between SF and SL at
maximum eff ort, thus an increase in SF for example will lead to
a decrease in SL and vice versa. UB had a longer step length to
accompany his slower step frequency whereas TG and AP had
shorter step lengths to accompany their faster step frequency
[ 12 ] . To accommodate the higher step frequencies during run-
ning the leg spring becomes stiff er [ 11 ] . The size of the sprinter
will impact upon these spatial parameters which in turn will
impact upon the stiff ness characteristics. We only simulated the
spring mass characteristics for 3 sprinters with UB perhaps
exhibiting an extreme in morphology compared to his competi-
tors and that there may be greater variance in body size among
elite sprinters. However, Charles and Bejan stated that world
record holders in the 100 m sprint are becoming taller and heav-
ier [ 9 ] . This agreed with Watts et al. [ 24 ] who studied not just
the world record holders but the top 10 athletes spanning 10
decades of recorded competition. They found that the reciprocal
ponderal index (indicating that athletes have become taller and
more linear) is a more signifi cant factor of success. Bejan et al.
further explored the evolution of height and sprinting speed and
suggested that it is the height which the CoM falls from which is
indicative of sprinting performance – if the CoM falls from a
greater height the more advantageous [ 5 ] . The location of the
CoM is dependent upon the morphology of the body, thus an
athlete with longer limbs and narrower circumferences of body
segments (i. e., the shanks) will result in a higher position of the
CoM. The results support that UB’s tall stature enabled for longer
steps with longer ground contact times and longer distances
travelled during ground contact [ 4 ] . This required lower force
and power to generate higher impulses during ground contact
under favourable conditions of force generation [ 3 ] and biome-
chanical effi ciency [ 3 ] .
Estimated F max was greater than that reported for slower sprint-
ers [i. e., 22] and comparable to sprinters running at 10.37 m
. s −1
[ 6 ] . The greater F ma x is indicative of sprinting performance –
faster top speeds are achieved by applying increased vertical
forces [ 25 ] . Weyand et al. suggested that faster runners had
briefer contact times which made greater vertical forces possible
[ 25 ] . The simulation in this present work however suggests that
greater F
max was achieved for UB who actually had a longer con-
tact time and ran at a faster velocity than his competitors. This
present study and others [i. e., 14, 22, and 25] have all focused on
the vertical force component. However it should be noted that
anterior-posterior forces are also manipulated to improve sprint
performance [ 18 , 19 ] .
The estimated peak displacement of the leg spring (∆L) was
comparable to that reported previously using the same model
[ 22 ] . The maximal downward displacement of the CoM was
markedly reduced in these elite sprinters compared to novice
sprinters [ 22 ] indicating at maximum speed elite sprinters
exhibit reduced vertical displacement. This along with the
markedly increased F
max (which is comparable (~ 3.0kN) to
sprinters running at 10.37 m
. s − 1 [ 6 ] ) resulted in an increase in
K
vert compared to slower sprinters. As speed increases K
vert has
also been shown to increase [ 10 ] . The K vert reported here is 3.8–
5.7 times greater than for slower sprinters [ 22 ] . These results
also show that even though UB achieved the greatest velocity his
K
vert was signifi cantly lower than his competitors. This paradox
is partly due to the signifi cantly increased t
c for UB compared to
TG and AP. Similarly t
c along with signifi cantly greater ∆L for UB
resulted in a signifi cantly lower K
leg compared to TG and AP.
The data suggest increased t
c for UB results in a decreased K
vert
and K
leg compared to his competitors. This agrees with Morin et
al. who manipulated t
c at running speed of 3.33 m
. s − 1 and
showed that K
vert and K
leg decreased when t
c was increased and
when t
c was decreased K
vert and K
leg increased [ 23 ] . Arampatzis
et al. [ 2 ] showed that a reduction in t
c during drop jumps resulted
in an increase in stiff ness. Furthermore Arampatzis et al. [ 2 ]
reported an increase in maximum vertical ground reaction force
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670
Taylor MJD, Beneke R. Spring Mass Characteristics of … Int J Sports Med 2012; 33: 667–670
Orthopedics & Biomechanics
and a greater vertical displacement of the center of mass when
t
c was increased agreeing with the results for the sprinters in
this current study.
As velocity increases from slow (2.0 m
. s − 1 ) to moderate
(8.24 m . s − 1 ) K
leg also increases by 60 % (11.2 kN/m–17.0 kN/m)
[ 7 ] . For the 3 sprinters in this present study, UB K
leg was compa-
rable to sprinters running at slower velocities [ 15 , 22 ] however
K
leg for TG and AP were 1.6 and 1.5 times greater, respectively,
compared to slower sprinters reported in the literature [ 15 , 22 ] .
The greater vertical and leg stiff ness seen for TG and AP may
help resist the collapse of the body during contact and enhance
force production during push-off , ultimately resulting in
increased step frequency [ 7 ] . For UB step frequency is less than
his competitors and the lower K
leg and K
vert and increased ∆L
and ∆y
c suggest greater compliance for UB thereby facilitating
the storage and utilization of elastic energy during the stretch
shortening cycle [ 7 ] . The greater t c and lower step frequency for
UB appears to be advantageous as it allows an increase in
impulse and distance travelled during contact [ 4 ] . The increased
t
c suggests that UB is able to run at a greater velocity but with
lower stiff ness compared to his competitors.
Conclusion
▼
In this present study the SMM characteristics were estimated for
world class athletes whilst in competition. Even though UB
achieved the greatest velocity (12.3 m
. s − 1 ) over the 60–80 m
split, compared to his competitors, his K
vert and K
leg were signifi -
cantly lower. This reduction in stiff ness is a consequence of the
increased contact time and lower step frequency. The opportu-
nity to take direct measurements in a world championship fi nal
to calculate SMM characteristics is unlikely to happen; therefore
these data provide a unique estimation of the SMM at the elite
level of competition.
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Notice:
This article was changed according to the following erratum on July 26th, 2012
Erratum
The article contains an error in the Abstract.
Instead of:
This reduction in stiff ness is a consequence of Bolt’s longer contact time (0.91 s) and lower step frequency (4.49 Hz).
It should read….
This reduction in stiff ness is a consequence of Bolt’s longer contact time (0.091 s) and lower step frequency (4.49 Hz).
This document was downloaded for personal use only. Unauthorized distribution is strictly prohibited.