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A Simple Method for Measuring Stiffness during Running

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The spring-mass model, representing a runner as a point mass supported by a single linear leg spring, has been a widely used concept in studies on running and bouncing mechanics. However, the measurement of leg and vertical stiffness has previously required force platforms and high-speed kinematic measurement systems that are costly and difficult to handle in field conditions. We propose a new "sine-wave" method for measuring stiffness during running. Based on the modeling of the force-time curve by a sine function,this method allows leg and vertical stiffness to be estimated from just a few simple mechanical parameters: body mass, forward velocity, leg length, flight time, and contact time. We compared this method to force-platform-derived stiffness measurements for treadmill dynamometer and overground running conditions, at velocities ranging from 3.33 m.s-1 to maximal running velocity in both recreational and highly trained runners. Stiffness values calculated with the proposed method ranged from 0.67 % to 6.93 % less than the force platform method, and thus were judged to be acceptable. Furthermore, significant linear regressions (p < 0.01) close to the identity line were obtained between force platform and sine-wave model values of stiffness. Given the limits inherent in the use of the spring-mass model, it was concluded that this sine-wave method allows leg and stiffness estimates in running on the basis of a few mechanical parameters, and could be useful in further field measurements.
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167
A Simple Method for Measuring
Stiffness During Running
Jean-Benoît Morin
1
, Georges Dalleau
2
, Heikki Kyröläinen
3
,
Thibault Jeannin
1
, and Alain Belli
1
1
University of Saint-Etienne;
2
University of La Réunion
3
University of Jyväskylä
The spring-mass model, representing a runner as a point mass supported by
a single linear leg spring, has been a widely used concept in studies on run-
ning and bouncing mechanics. However, the measurement of leg and vertical
stiffness has previously required force platforms and high-speed kinematic
measurement systems that are costly and difcult to handle in eld condi-
tions. We propose a new “sine-wave” method for measuring stiffness during
running. Based on the modeling of the force-time curve by a sine function,
this method allows leg and vertical stiffness to be estimated from just a few
simple mechanical parameters: body mass, forward velocity, leg length, ight
time, and contact time. We compared this method to force-platform-derived
stiffness measurements for treadmill dynamometer and overground running
conditions, at velocities ranging from 3.33 m·s
–1
to maximal running velocity
in both recreational and highly trained runners. Stiffness values calculated with
the proposed method ranged from 0.67% to 6.93% less than the force platform
method, and thus were judged to be acceptable. Furthermore, signicant linear
regressions (p < 0.01) close to the identity line were obtained between force
platform and sine-wave model values of stiffness. Given the limits inherent in
the use of the spring-mass model, it was concluded that this sine-wave method
allows leg and stiffness estimates in running on the basis of a few mechanical
parameters, and could be useful in further eld measurements.
Key Words: spring-mass, modeling, practical calculation
During running, the musculoskeletal structures of the legs alternately store
and return elastic energy, so the legs can be described as springs loaded by the
runner’s body mass, constituting the “spring-mass model” (Alexander, 1992; Blick-
han, 1989; Cavagna, Heglund, & Taylor, 1977; Cavagna, Saibene, & Margaria,
1
Physiology Laboratory, PPEH Res. Unit, Univ. of Saint-Etienne, CHU Bellevue,
42055 St-Etienne Cedex 02, France;
2
Sport Sciences Res. Center, Faculty of Science and
Technology, Univ. of La Réunion, 117 General Ailleret St., 97430 Le Tampon, France;
3
Neuromuscular Res. Center, Univ. of Jyväskylä, PO Box 35, 40014 Jyväskylä, Finland.
JOURNAL OF APPLIED BIOMECHANICS, 2005, 21, 167-180
©2005 Human Kinetics Publishers, Inc.
168 Morin, Dalleau, et al.
Measuring Stiffness During Running 169
1964; McMahon & Cheng, 1990). This model has been a widely used concept for
describing and studying the storage and return of elastic energy in the lower limbs
in humans and other animals during bouncing and running (Dalleau, Belli, Bourdin,
& Lacour, 1998; Farley, Blickhan, Saito, & Taylor, 1991; Farley & Gonzalez, 1996;
Ferris, Louie, & Farley, 1998; He, Kram, & McMahon, 1991; McMahon, Valiant,
& Fredrick, 1987; McMahon & Cheng, 1990). The model consists of a point mass
supported by a single massless linear “leg spring.”
The main mechanical parameter studied when using the spring-mass model
is the stiffness of the leg spring, dened as the ratio of the maximal force in the
spring to the maximum leg compression at the middle of the stance phase (Farley &
Gonzalez, 1996). Moreover, although it does not correspond to any physical spring,
the term vertical stiffness is used to describe the vertical motion of the center of
mass (CM) during contact (Farley & Gonzalez, 1996; McMahon & Cheng, 1990),
and is dened as the ratio of the maximal force to the vertical displacement of the
CM as it reaches its lowest point, i.e., the middle of the stance phase. Maximal
ground reaction force and CM displacement measurements during running are
needed to measure these parameters requiring dynamometers such as overground
or treadmill-mounted force platforms, or even video motion-analysis (e.g., Aram-
patzis, Brüggemann, & Metzler, 1999). Such equipment is costly and not practical
for eld measurements.
Considering the descriptive and predictive power of the spring-mass model
in running mechanics and the serious technical issues mentioned, the aim of this
study was to propose and validate a simple calculation method for assessing leg and
vertical stiffness during running. The computations inherent in the proposed method
are based on a few simple mechanical parameters: contact and ight times, forward
running speed, leg length, and body mass. We validated the method by comparing
leg and vertical stiffness values measured with a force platform to those obtained
using the proposed method during treadmill and overground running.
Methods
For practical reasons our study was divided into two different protocols, one aiming
to validate the presented method during treadmill running, the other during over-
ground running. However, in both cases the force platform and sine-wave method
stiffness values were each calculated in the same way. All participants gave their
informed consent to take part in those protocols.
Treadmill running: Eight young men (age 24 ± 2 yrs, height 1.78 ± 0.07 m,
body mass 76.0 ± 7.0 kg; mean ± SD) volunteered to participate in this study. They
were physical education students and experienced in treadmill running. After a 5-min
warm-up at 3.33 m·s
–1
, they performed 30-s running bouts at 3.33, 3.89, 4.44, 5, 5.56,
6.11, and 6.67 m·s
–1
at their preferred step frequency (separated by 2 min of rest)
on a treadmill dynamometer (HEF Techmachine, Andrézieux-Bouthéon, France).
Using the same calibration procedure as Belli, Bui, Berger, Geyssant, and Lacour
(2001), we determined the treadmill’s static nonlinearity to be less than 0.5% and 1%,
respectively, in the vertical and horizontal directions. Natural vibration frequency
(treadmill hit with a hammer) were 147 Hz in the vertical direction and 135 Hz in
the anterior-posterior and mediolateral directions. Vertical ground reaction forces,
belt velocity, and ight and contact times were measured at a sampling frequency
of 500 Hz. All values were averaged for 10 consecutive steps for each velocity.
168 Morin, Dalleau, et al.
Measuring Stiffness During Running 169
Overground running: 10 young men (age 23 ± 3 yrs, height 1.80 ± 0.05 m,
body mass 66.4 ± 5.3 kg) who were elite middle-distance runners volunteered to
run over a 10-m force platform (Kistler, Winterthur, Switzerland; natural frequency
higher than 150 Hz, horizontal to vertical cross-talk lower than 2%) at 4, 5, 6, and
7 m·s
–1
and at their maximal running velocity. Velocity was measured by two pairs
of photocells placed at each end of the force platform. Vertical ground reaction
forces and ight and contact times were measured for one step at each velocity at
a sampling rate of 1.8 kHz.
Reference method. Vertical stiffness: The vertical stiffness k
vert
(in kN·m
–1
)
was calculated as:
k
vert
= F
max
· y
c
–1
(1)
with F
max
the maximal ground reaction force during contact (in kN) and y
c
the
vertical displacement of the CM when it reaches its lowest point (in m), determined
by double integration of the vertical acceleration over time, as proposed by Cavagna
(1975). Figure 1 shows a typical example of force and vertical displacement of the
CM evolutions during contact.
Leg stiffness: The stiffness of the leg spring k
leg
(in kN·m
–1
) was calculated
as follows:
k
leg
= F
max
· L
–1
(2)
with L the peak displacement of the leg spring (in m) calculated from the initial
leg length L (great trochanter to ground distance in a standing position), running
velocity v (in m·s
–1
), and the contact time t
c
(in s), as per Farley and Gonzales
(1996; Eq. 2 and 3):
L = L L
2
(––)
2
+ y
c
(3)
vt
c
2
Figure 1 Typical measured (thick line) and modeled (dotted line) vertical force over
time during contact on treadmill running, and corresponding downward vertical CM
displacement (thin line) (velocity of 5 m·s
–1
, body mass of 68 kg).
170 Morin, Dalleau, et al.
Measuring Stiffness During Running 171
Sine-wave method. Model proposed: This method is based on a model used
by Dalleau et al. (2004) for vertical jumps, which considers the force as a func-
tion of time during contact to be a simple sine function (Alexander, 1989; Kram
& Dawson, 1998):
F(t) = F
max
sin(— t) (4)
The validity of this postulate was checked by comparing areas under the mea-
sured and modeled F(t) curves for all steps analyzed (Figure 1). The mean absolute
bias was calculated as (Modeled – Force platform)/Force platform×100.
Calculations: Complete calculations and assumptions are shown in the
Appendix section of this paper.
The modeled vertical stiffness k
vert
was calculated as the ratio of the modeled
maximal force F
max
over the modeled vertical CM displacement y
c
:
k
vert
= F
max
· y
c
–1
(5)
with F
max
= mg — (— +1) (6)
m being the participant’s body mass (in kg), and t
f
and t
c
, respectively, being the
ight and contact times (in s)
and y
c
= —— + g — (7)
The modeled leg stiffness k
leg
was calculated as follows:
k
leg
= F
max
· L
–1
(8)
with L = LL
2
– (—)
2
+ y
c
(9)
the modeled leg length variation (in m).
The participant’s leg length was also modeled as: L
mod
= 0.53h where h is the
participant’s height (in m), according to the anthropometric equations of Winter
(1979), in order to check the validity of the presented stiffness calculation method
using leg length values obtained from participant’s height data.
In order to quantify the force platform-model values difference, we calculated
an absolute mean error bias as follows: Bias = (Modeled – Force platform)/Force
platform×100. The force platform-model relationships were further described by
means of linear regressions and calculation of the determination coefcient R².
We then performed an ANOVA with the Scheffé post hoc test in order to
check the eventual effect of velocity on the above-mentioned force platform-model
bias. The statistically signicant level was set at 0.05.
Results
The bias between force platform and modeled F(t) areas was 5.33% (ranging from
11.7% at 3.33 m·s
–1
to 1.7% at 6.67 m·s
–1
) for treadmill running, 542 steps analyzed,
and 2.93% (ranging from 3.17% at 4 m·s
–1
to 2.33% at 7 m·s
–1
) for overground
running, 50 steps analyzed.
π
t
c
^
^
^
^
^
^
^
^
^
^
^
t
f
t
c
π
2
F
max
t
c
2
m π
2
t
c
2
8
^
^
^
^
vt
c
2
170 Morin, Dalleau, et al.
Measuring Stiffness During Running 171
The values of the different mechanical parameters measured and modeled and
the reference-model bias are shown in Tables 1 and 2 for treadmill and overground
running, respectively. For the vertical and leg stiffness, we obtained reference-model
error biases of 0.12% (ranging from 1.53% at 6.67 m·s
–1
to 0.07% at 6.11 m·s
–1
) and
of 6.05% (from 9.82% at 3.33 m·s
–1
to 3.88% at 6.67 m·s
–1
), respectively, during
treadmill running. During overground running the bias was 2.30% (from 3.64% at
5 m·s
–1
to 0.25% at 6 m·s
–1
) for the vertical stiffness, and 2.54% (from 3.71% at 5
m·s
–1
to 1.11% at maximal velocity) for leg stiffness.
Further, the reference-model linear regressions were signicant (p < 0.01; R²
= 0.89 to 0.98) for both vertical and leg stiffness either on the treadmill or during
overground running (Figures 2 and 3, respectively).
The changes in reference and modeled values of stiffness with running veloci-
ties are reported in Figure 4 for treadmill and overground running. The ANOVA
and the Scheffé post hoc test demonstrated for treadmill running a signicant dif-
Table 1 Main Mechanical Parameters Measured With Reference Method During
Treadmill Running (M ± SD)
Reference Modeled Bias (%)
y
c
(m) 0.05 ± 0.01 0.05 ± 0.02 3.28 ± 1.10
L (m) 0.20 ± 0.03 0.20 ± 0.03 0.93 ± 0.43
F
max
(kN) 2.05 ± 0.34 1.91 ± 0.32 6.93 ± 2.52
k
vert
(kN·m
–1
) 37.70 ± 8.84 37.74 ± 8.87 0.12 ± 0.53
k
leg
(kN·m
–1
) 10.37 ± 2.34 9.75 ± 2.19 6.05 ± 3.02
Note: Measured with reference method, corresponding values calculated with proposed
method and error bias values in between. Values presented are averaged for all partici-
pants and all velocities.
Table 2 Main Mechanical Parameters Measured With Reference Method During
Overground Running (M ± SD)
Reference Modeled Bias (%)
y
c
(m) 0.05 ± 0.01 0.05 ± 0.01 2.34 ± 2.42
L (m) 0.16 ± 0.02 0.16 ± 0.02 0.67 ± 1.09
F
max
(kN) 2.13 ± 0.21 2.06 ± 0.24 3.24 ± 2.08
k
vert
(kN·m
–1
) 51.39 ± 21.46 50.21 ± 20.40 2.30 ± 1.63
k
leg
(kN·m
–1
) 13.28 ± 1.85 12.95 ± 2.13 2.54 ± 1.16
Note: Measured with reference method, corresponding values calculated with proposed
method and error bias values in between. Values presented are averaged for all partici-
pants and all velocities.
172 Morin, Dalleau, et al.
Measuring Stiffness During Running 173
Figure 2 Signicant reference-model linear regressions (p < 0.01) obtained during
treadmill running for vertical stiffness (upper panel) and leg stiffness (lower panel)
compared to the identity line. Each dot represents a mean value of stiffness for a par-
ticipant at the corresponding velocity.
Reference vertical stiffness (kN·m
–1
)
Modeled leg stiffness (kN·m
–1
)
Modeled vertical stiffness (kN·m
–1
)
Reference leg stiffness (kN·m
–1
)
172 Morin, Dalleau, et al.
Measuring Stiffness During Running 173
Figure 3 Signicant reference-model linear regressions (p < 0.01) obtained during
overground running for vertical stiffness (upper panel) and leg stiffness (lower panel)
compared to the identity line. Each dot represents the value of stiffness of the step
analyzed for each participant at the corresponding velocity.
Modeled leg stiffness (kN·m
–1
)
Modeled vertical stiffness (kN·m
–1
)
Reference vertical stiffness (kN·m
–1
)
Reference leg stiffness (kN·m
–1
)
174 Morin, Dalleau, et al.
Measuring Stiffness During Running 175
Figure 4 — Changes in the modeled and reference vertical and leg stiffness with run-
ning velocity during treadmill running (upper panel) and overground running (lower
panel). *Signicant (p < 0.05) difference in reference-model bias obtained by ANOVA
and Scheffé post hoc test.
Velocity (m·s
–1
)
Stiffness (kN·m
–1
)
Velocity (m·s
–1
)
Stiffness (kN·m
–1
)
174 Morin, Dalleau, et al.
Measuring Stiffness During Running 175
ference (p < 0.05) between the bias in leg stiffness assessment at 3.33 m·s
–1
and
that obtained at the four highest velocities. For overground running, a signicant
difference was observed on the vertical stiffness bias (p < 0.05) obtained at 6 m·s
–1
and at the maximal velocity (Figure 4).
The leg length calculation using Winter’s equation led to a mean error bias of
1.94 ± 1.51%, and the linear regression between the measured leg length and that
obtained using Winter’s model was signicant (R² = 0.89; p < 0.01).
Discussion
The values of the different mechanical parameters obtained in this study—verti-
cal ground reaction forces, displacements, and stiffness—are in line with those of
studies about the spring-mass model in running (Arampatzis et al., 1999; Blickhan,
1989; Farley & Gonzales, 1996; Ferris et al., 1998; He et al., 1991; McMahon et al.,
1987; McMahon & Cheng, 1990). It should be noted that we obtained slight differ-
ences in displacements and vertical and leg stiffness between our two experimental
protocols (treadmill and overground running), due to the different running abilities
of the two populations in these two parts of the study.
Indeed, highly trained middle-distance runners who performed the force plat-
form protocol showed lower CM displacement and leg length change and higher
vertical and leg stiffness than the nonspecialists who volunteered to perform the
treadmill protocol (e.g., leg stiffness: 10.37 ± 2.34 kN·m
–1
for the treadmill protocol
group vs. 13.28 ± 1.85 kN·m
–1
for overground runners). This is in line with the study
of Mero and Komi (1986), who observed higher “apparent spring constants” of the
support leg and lower CM displacement during running in highly trained sprinters
compared to less-skilled ones (Luhtanen & Komi, 1980) or marathon runners (Ito,
Komi, Sjödin, Bosco, & Karlsson, 1983).
The stiffness vs. velocity patterns (Figure 4), i.e., the increase in vertical
stiffness and the constancy of leg stiffness with increasing velocity either on the
treadmill or in overground conditions, has been previously noted in the literature
(He et al., 1991; McMahon et al., 1987).
Validity of the proposed sine-wave method: The aim of this study was to
provide a calculation method based on a few simple parameters allowing vertical
and leg stiffness to be assessed during treadmill and overground running without
a force platform. The low bias obtained between force platform and model values
(from 0.12% to 6%), and the high determination coefcients (from 0.89 to 0.98, p
< 0.01), demonstrate the validity of this calculation method during both treadmill
and overground running. Furthermore, the validity of this method was tested for a
wide range of velocities (i.e., from 3.33 to 6.67 m·s
–1
on the treadmill and 4 m·s
–1
to maximal velocity on the force platform) with runners of different ability levels,
from nonspecialists to highly trained middle-distance runners, all giving acceptable
results (Figures 2 and 3). This may allow us to use the proposed method during
submaximal to maximal velocity running, with either nonspecialists or elite athletes,
in eld conditions. It should be also noted that the mechanical parameter input
to the model (maximal force, CM displacement, leg length change) also showed
acceptable reference-model bias (0.67 to 6.93%, Tables 1 and 2).
Basis postulate and assumptions: The basis postulate of this study was that
the F(t) curve can be tted by means of a simple sine function. The validity of this
sine modeling, recently used in a study aimed at validating a stiffness measuring
176 Morin, Dalleau, et al.
Measuring Stiffness During Running 177
method during hopping (Dalleau, Belli, Viale, Lacour, & Bourdin, 2004), was
checked by comparing areas under F(t) curves for all steps analyzed on the treadmill
and on the force plate, and we obtained low error bias values (5.33% and 2.93%,
respectively). On the treadmill the accuracy of this sine modeling was improved
at faster velocities, the bias ranging from 11.7% at 3.33 m·s
–1
to only 1.7% at 6.67
m·s
–1
. This was probably due to the alteration of the passive impact peak in the
vertical ground reaction force at low velocity and to an F(t) curve more closely
approximating a sine curve at faster velocities. This could explain the signicant
effect of velocity on reference-model bias in leg stiffness observed between that
obtained at 3.33 m·s
–1
and that at the four highest velocities (Figure 4, upper panel),
the accuracy of the basic sine-wave postulate increasing at faster velocities on the
treadmill. To the contrary, this velocity effect on the bias was not observed during
overground running.
The bias in F(t) curve tting by the sine function was rather constant, and the
previously mentioned velocity effect on reference-model bias was only signicant (p
= 0.043) in vertical stiffness between that obtained at 6 m·s
–1
and that at the maximal
velocity (Figure 4, lower panel). This smaller effect of velocity may be explained
by the fact that F(t) curves of elite level “forefoot striking” runners showed smaller
or no passive impact peaks, whatever the velocity, with shorter contact times and
F(t) plot shapes closer to a sine-wave (Nilsson & Thorstensson, 1989).
Sensitivity analysis: In order to further determine the inuence of the dif-
ferent mechanical parameters constituting the presented model on the vertical and
leg stiffness calculated, we performed a sensivity analysis (Figure 5). It was then
possible to observe the relative inuence of each mechanical parameter on the
calculated stiffness values. It should be noted that the most sensitive parameter, for
both vertical and leg stiffness estimates, is the contact time. Its variation inuences
the stiffness in a proportion of about 1 to 2, i.e., for instance a 10% reduction in
contact time leads, according to this model, to a 20% increase in vertical stiffness
or even 25% in leg stiffness. All the other parameters have a 1 : 1 weight or even
less, especially anthropometrical parameters of body mass and leg length.
Therefore, although the legs are not stiff either at landing or takeoff, resulting
in a slight overestimation of leg length (Arampatzis et al., 1999), the inuence of
such a phenomenon on the accuracy of the stiffness calculations was not important.
In addition, the results of the present study showed that the leg length value used
in this sine-wave model can be obtained using anthropometric equations, on the
basis of the individual’s height, without signicant changes in the stiffness values
obtained. Furthermore, a recent study did not nd any signicant variation in the
stiffness values obtained with the proposed method, using an estimated value of leg
length from the participant’s height according to Winter’s model (Winter, 1979),
showing that this parameter is not crucial for improving the accuracy of the sine-
wave model (Jeannin, 2003).
This model also assumes a constant point-of-force application on the ground
during the entire contact phase. However, the location of the point-of-force applica-
tion was shown to move forward by about 0.16 m (Lee & Farley, 1998), and this
constitutes another limitation of the simplest spring-mass model.
Finally, it is worth noting that the limits of the proposed sine-wave method are
also those inherent in the use of any theoretical spring-mass model, i.e., the human
lower limb is not a true linear spring in a physical sense. This point has been widely
discussed elsewhere (Blickhan, 1989; Farley & Gonzales, 1996; Ferris, Louie, &
176 Morin, Dalleau, et al.
Measuring Stiffness During Running 177
Figure 5 Sensitivity analysis: Variations of the different mechanical parameters
constituting the equation’s model plotted against the corresponding leg stiffness (upper
panel) and vertical stiffness (lower panel) variations.
Farley, 1998; He et al., 1991; McMahon & Cheng, 1990). Even if one prefers the
term “quasi-stiffness” (Latash & Zatsiorsky, 1993), the simplicity and predictive
power of the spring-mass model in studies aimed at understanding and analyzing
human running performance outweigh the limitations.
The equations based on a force-time curve sine modeling allow these calcu-
lations from simple mechanical parameters of ight and contact times, leg length,
body mass, and running velocity. The reference-model biases on calculated stiff-
ness were acceptable for a wide range of velocities, different levels of runners, and
Vertical stiffness variation (%)
Leg stiffness variation (%)
Parameters variation (%)
Parameters variation (%)
178 Morin, Dalleau, et al.
Measuring Stiffness During Running 179
either on a treadmill dynamometer or overground. To conclude, this new method
may allow researchers to measure and to understand the inuence of leg stiffness
on performance in eld running conditions and also to improve track & eld coach-
ing and training.
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Kram, R., & Dawson, T.J. (1998). Energetics and biomechanics of locomotion by red kan-
garoos (Macropus rufus). Comparative Biochemistry and Physiology, 120, 41-49.
Latash, M.L., & Zatsiorsky, V.M. (1993). Joint stiffness: Myth or reality? Human Movement
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Measuring Stiffness During Running 179
Lee, C., & Farley, C.T. (1998). Determinants of the center of mass trajectory in human walk-
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APPENDIX
Modeled vertical stiffness computations
The modeled vertical stiffness k
vert
was calculated as:
k
vert
= F
max
· y
c
–1
(5)
with F
max
the modeled maximal force and y
c
the modeled vertical peak displace-
ment of the center of mass (CM) during contact.
F
max
computations
The pattern of vertical ground reaction force over time was modeled using the
following equation:
F(t) = F
max
· sin(— · t) (4)
with F
max
the peak force value and t
c
the contact time.
From this equation, the momentum change during contact is:
tc
[F(t) – mg] · dt = mu = mgt
f
(A10)
with m the participant’s body mass, u the vertical velocity, g the gravity accelera-
tion, and t
f
the mean ight time (mean of ight times before and after contact).
Substituting Eq. A4 in Eq. A10 gives:
tc
[F
max
· sin(— · t) – mg] · dt = mu = mgt
f
(A11)
[F
max
— cos (— · t)]mgt
c
= mgt
f
(A12)
2 F
max
mg (t
f
+ t
c
) (A13)
The modeled peak force during contact is then obtained as:
F
max
= mg — (— + 1) (6)
t0
t0
π
t
c
π
t
c
t
c
π
π
t
c
t
c
t0
t
c
π
π
2
t
f
t
c
^
^ ^
^
^
^
^
^
180 Morin, Dalleau, et al.
y
c
computations
Based on Eq. A4 and according to the fundamental law of dynamics, vertical velocity
is obtained by integrating the vertical acceleration of the CM during contact:
u(t) =
tc
[—– – g] · dt + u(t0) (A14)
u(t0) being the downward vertical velocity at the beginning of contact.
u(t) =
tc
[—–– sin (— · t) – g] · dt + u(t0) (A15)
u(t) = [– —– — cos (— · t)] – gt + u(t0) (A16)
Knowing that the vertical velocity is nil at the time of half-contact:
u() = —– — – g— + u(t0) = 0 (A17)
—– — + u(t0) = g— (A18)
The nal expression of vertical velocity during contact being:
u(t) = – —– — cos (— · t) – gt + g— (A19)
Integrating this vertical velocity over time, vertical displacement can be obtained:
y(t) =
tc
u(t) · dt + y(t0) (A20)
with y(t
0
) the vertical position of the CM at the beginning of contact.
Assuming y(t
0
) = 0 and substituting Eq. A19 in Eq. A20:
y(t) =
tc
[– —– — cos (— · t) – gt + g—] · dt (A21)
y(t) = [– —– — sin (— · t) – — gt
2
]
tc
+ g—t (A22)
y(t) = – —- — sin (— · t) – — gt
2
+ g—t (A23)
The total CM displacement at the time of half contact, i.e., for t = t
c
/ 2 is then:
y
c
= – —- — + g— (7)
Modeled leg stiffness computations
The modeled leg stiffness k
leg
was calculated as:
k
leg
= F
max
· L
–1
(8)
with L the modeled leg peak displacement during contact.
L computations
L was obtained on the basis of the spring-mass model’s typical equations and
assumptions (Farley & Gonzales, 1996; McMahon & Cheng, 1990):
L = L L
2
(—–)
2
+ y
c
(9)
L being the participant’s leg length and v the constant horizontal velocity.
F(t)
m
F
max
m
F
max
m
π
t
c
t
c
π
π
t
c
t0
t
c
2
F
max
m
t
c
π
t
c
2
F
max
m
t
c
π
t
c
2
F
max
m
t
c
π
π
t
c
t
c
2
t0
t0
F
max
m
t
c
m
π
t
c
t
c
2
F
max
m
t
c
2
π
2
π
t
c
1
2
t
c
2
F
max
m
t
c
2
π
2
π
t
c
1
2
t
c
2
F
max
m
t
c
2
π
2
t
c
2
8
vt
c
2
t0
^
^
^
^
^
^
^
^
^
^
^
^
^ ^ ^
^
^
^
^
^
t
c
t0
t0
... 7 k leg is associated with leg spring deformation when it comes to horizontal motion, while k vert is a characteristic of the leg spring pattern associated with the vertical motion of the center of mass. 6,7,9 Greater lower-limb stiffness has been shown to be associated with a better RE in adult runners. 5,10,11 To the best of our knowledge, no studies have been found investigating the relationship between stiffness and RE in children. ...
... Relative SL was calculated as the ratio of SL to leg length. k vert and k leg were calculated from the variables of peak ground reaction force, vertical displacement of the center of mass and the change in leg length previously identified by Morin et al. 9 (Equations 1-5). ...
Article
Full-text available
Introduction: Lower limb stiffness has been shown to be associated with running economy (RE) in adults, but this relationship in children remains unclear. Objectives: The purpose of this study was to investigate the relationship between lower limb stiffness, RE, and repeated-sprint ability in child soccer players. Methods: Twenty-eight male child soccer players (mean age 11.8 ± 0.9 years) participated in the study. RE was determined by measuring the steady-state oxygen uptake (ml/min/kg) at submaximal running speeds of 8 and 9 km/h. Vertical and leg stiffness were calculated from the flight and contact time data obtained during two submaximal running tests. Additionally, vertical stiffness was measured during the maximal and submaximal hopping tests. All participants performed the repeated sprint test consisting of 10 × 20-m all-out sprints interspersed with 20-s active recovery. Results: During both submaximal running tests, vertical (r=-0.505 to-0.472) and leg stiffness (r=-0.484 to-0.459) were significantly correlated with RE (p< 0.05). Maximal (r=-0.450) and submaximal hopping stiffness (r=-0.404) were significantly correlated with RE at 8 km/h (p< 0.05). Maximal hopping stiffness was significantly correlated with the best sprint time (r=-0.439) and mean sprint time (r=-0.496) (p< 0.05). Vertical (r=-0.592 to-0.433) and leg stiffness (r=-0.612 to-0.429) at 8 and 9 km/h and submaximal hopping stiffness (r=-0.394) were significantly correlated with the fatigue index (p< 0.05). Conclusions: Current findings indicate that the lower limb stiffness may be an important determinant of both RE and repeated-sprint ability in child soccer players. Level of Evidence II; Diagnostic Studies-Investigating a Diagnostic Test.
... K vert and K leg measurements using the sinewave method on a treadmill showed high ICC (0.99 and 0.86) (Pappas et al., 2014). They were calculated according to the recommendations of Morin et al. (2005). K vert is defined as the ratio of the maximum vertical GRF to the COM vertical displacement (Farley and González, 1996), as in Eq. 1. ...
... K vert and K leg as global stiffnesses describe the ability of the entire lower limb to reduce energy consumption and to utilise elastic energy during vertical and horizontal movements (Dalleau et al., 1998;Heise and Martin, 2001;Struzik et al., 2021). In our study, the K leg values (13.04 kN·m −1 ) and K vert values (25.07 kN·m −1 ) reported are similar to those of previous studies at similar speeds among recreational runners (Morin et al., 2005;García-Pinillos et al., 2019b). We found that K vert , but not K leg , was significantly negatively associated with RE at 10 km·h −1 . ...
... K vert and K leg measurements using the sinewave method on a treadmill showed high ICC (0.99 and 0.86) (Pappas et al., 2014). They were calculated according to the recommendations of Morin et al. (2005). K vert is defined as the ratio of the maximum vertical GRF to the COM vertical displacement (Farley and González, 1996), as in Eq. 1. ...
... K vert and K leg as global stiffnesses describe the ability of the entire lower limb to reduce energy consumption and to utilise elastic energy during vertical and horizontal movements (Dalleau et al., 1998;Heise and Martin, 2001;Struzik et al., 2021). In our study, the K leg values (13.04 kN·m −1 ) and K vert values (25.07 kN·m −1 ) reported are similar to those of previous studies at similar speeds among recreational runners (Morin et al., 2005;García-Pinillos et al., 2019b). We found that K vert , but not K leg , was significantly negatively associated with RE at 10 km·h −1 . ...
Article
Full-text available
Neuromuscular characteristics, such as lower-limb joint strength, the ability to reuse elastic energy, and to generate force are essential factors influencing running performance. However, their relationship with running economy (RE) remains unclear. The aim of this study was to evaluate the correlations between isokinetic lower-limb joint peak torque (PT), lower-limb stiffness, isometric force-time characteristics and RE among recreational-trained male runners. Thirty male collegiate runners (aged 20-22 years, VO 2max : 54.02 ± 4.67 ml·kg −1 ·min −1) participated in test sessions on four separate days. In the first session, the body composition and RE at 10 km·h −1 were determined. In the second session, leg and vertical stiffness (K leg and K vert), knee and ankle stiffness (K knee and K ankle) were evaluated. In the third session, isokinetic knee and ankle joint PT at velocity of 60°s −1 were tested. The force-time characteristics of isometric mid-thigh pull (IMTP) were evaluated in the final session. The Pearson's product-moment correlations analysis shows that there were no significant relationships between knee and ankle joint concentric and eccentric PT, K knee and K ankle , K leg , and RE at 10 km·h −1. However, K vert (r = −0.449, p < 0.05) and time-specific rate of force development (RFD) for IMTP from 0 to 50 to 0-300 ms (r = −0.434 to −0.534, p < 0.05) were significantly associated with RE. Therefore, superior RE in recreational runners may not be related to knee and ankle joint strength and stiffness. It seems to be associated with vertical stiffness and the capacity to rapidly produce force within 50-300 ms throughout the lower limb.
... The average of the five jab-cross punch trials on the force plate was used in the subsequent analysis. For the lower extremity sports performance assessment, participants were directed to perform the countermovement jump (CMJ), emphasizing a quick jump, as a sharp transition between eccentric to concentric movement is essential to the stretchshortening cycle (Morin et al., 2005;Nicol et al., 2006;Anthony, 2010). Participants performed 10 min dynamic warm-up, including light jogging and dynamic stretching. ...
... It was calculated by dividing the peak vertical GRF to the center of mass (COM) displacement between the highest and the lowest point during the CMJ trial. All markers calculated COM from the body (Helen Hayes model) during jumping with Orthotrak program (ORTHOTRAK 6.2.4; Single Trial Processing Module, Motion Analysis System; Clinical Gait Analysis Software) (Morin et al., 2005;Miyaguchi et al., 2014b;Miyaguchi et al., 2015). ...
Article
Full-text available
Improving lower extremity sports performance may contribute to punching performance in boxers. We compared the effects of two typical boxing routines for developing lower extremity sports performance and subsequent punching performance. Twenty-four high school amateur boxers between the ages of 12 and 18 performed training at least 3 days per week. All Athletes had 3–5 years of experience in boxing training. The participants separated into two groups to receive an 8-week plyometric or jump rope training program. They performed each training program for 30 min on 3 days/week. Lower extremity sports performance in countermovement jump (leg stiffness, jump power, and rate of force development) and jab-cross punching performance (punch velocity, punch force, reaction time, movement time, and ground reaction force) were assessed at pre-and post-training. The data were analyzed using a two-way mixed-design analysis of variance (ANOVA) (group × time). Both training programs improved the rate of force development in countermovement jump, the reaction time of punch, the peak ground reaction force of the rear leg during the jab punch, and the velocity of the jab punch. There were no group differences and interaction effects in all variables analyzed. It is concluded that 8 weeks of plyometric and rope jumping programs had a similar impact on improving lower extremity strength and punching performance. Both training programs may improve muscle strength and power, rate of force development, and reaction time. These improvements may contribute to lower extremity strength for driving a punch at the target with excellent performance.
... Contact time and flight time were used to calculate step frequency and duty factor. Leg stiffness and vertical stiffness were calculated using running speed, contact time, flight time and participant mass and height (Morin et al., 2005). ...
Thesis
The objectives of this thesis were to investigate the performance determinants of trail running, and to evaluate the changes in running economy following prolonged endurance running exercise. First, we tested elite road and trail runners for differences in performance factors. Our results showed that elite trail runners are stronger than road runners, but they have greater cost of running when running on flat ground. In the second study, we evaluated the performance factors that predicted performance in trail running races of different distances, ranging from 40 to 170 km. We found that maximal aerobic capacity was a determinant factor of performance for races up to 100 km. Performance in shorter races, up to approximately 55 km, was also predicted by lipid utilization at slow speed, while performance in the 100 km race was also predicted by maximal strength and body fat percentage. The most important factors of performance for races longer than 100 km are still debated. We also tested the effects of trail running race distance on cost of locomotion, finding that cost of running increased after races up to 55 km, but not after races of 100-170 km. Finally, we tested the. effects of two different exercise modalities, cycling and running, on cost of locomotion, after 3 hours of intensity-matched exercise. Cost of locomotion increased more following cycling than running, and the change in cost of locomotion was related to changes in cadence and loss of force production capacity.
... Indeed, DF and t f are related to the average vertical ground reaction force during t f (Beck et al., 2020) and effective vertical impulse during t f (Dorn et al., 2012), respectively. Both the average vertical ground reaction force during t f and effective vertical impulse during t f are proportional to the peak vertical ground reaction force, as supports the sine wave model of the vertical ground reaction force (Morin et al., 2005) and experimental data (Bonnaerens et al., 2021). The present study reported lower association between relative t c and DF values (correlations were moderate to high; Table 3 and Figure 4) than relative t c to DF values. ...
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Duty factor (DF) and step frequency (SF) are key running pattern determinants. However, running patterns may change with speed if DF and SF changes are inconsistent across speeds. We examined whether the relative positioning of runners was consistent: 1) across five running speeds (10-18 km/h) for four temporal variables [DF, SF, and their subcomponents: contact (t c) and flight (t f) time]; and 2) across these four temporal variables at these five speeds. Three-dimensional whole-body kinematics were acquired from 52 runners , and deviations from the median for each variable (normalised to minimum-maximum values) were extracted. Across speeds for all variables, correlations on the relative positioning of individuals were high to very high for 2-4 km/h speed differences, and moderate to high for 6-8 km/h differences. Across variables for all speeds, correlations were low between DF-SF, very high between DF-t f , and low to high between DF-t c , SF-t c , and SF-t f. Hence, the consistency in running patterns decreased as speed differences increased, suggesting that running patterns be assessed using a range of speeds. Consistency in running patterns at a given speed was low between DF and SF, corroborating suggestions that using both variables can encapsulate the full running pattern spectrum. ARTICLE HISTORY
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The goals of this study were to examine the following hypotheses: (a) there is a difference between the theoretically calculated (McMahon and Cheng, 1990. Journal of Biomechanics 23, 65-78) and the kinematically measured length changes of the spring-mass model and (b) the leg spring stiffness, the ankle spring stiffness and the knee spring stiffness are influenced by running speed. Thirteen athletes took part in this study. Force was measured using a "Kistler" force plate (1000 Hz). Kinematic data were recorded using two high-speed (120 Hz) video cameras. Each athlete completed trials running at five different velocities (approx. 2.5, 3.5, 4.5, 5.5 and 6.5 m/s). Running velocity influences the leg spring stiffness, the effective vertical spring stiffness and the spring stiffness at the knee joint. The spring stiffness at the ankle joint showed no statistical difference (p < 0.05) for the five velocities. The theoretically calculated length change of the spring-mass model significantly (p < 0.05) overestimated the actual length change. For running velocities up to 6.5 m/s the leg spring stiffness is influenced mostly by changes in stiffness at the knee joint.
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A mathematical model for terrestrial running is presented, based on a leg with the properties of a simple spring. Experimental force-platform evidence is reviewed justifying the formulation of the model. The governing differential equations are given in dimensionless form to make the results representative of animals of all body sizes. The dimensionless input parameters are: U, a horizontal Froude number based on forward speed and leg length; V, a vertical Froude number based on vertical landing velocity and leg length, and KLEG, a dimensionless stiffness for the leg-spring. Results show that at high forward speed, KLEG is a nearly linear function of both U and V, while the effective vertical stiffness is a quadratic function of U. For each U, V pair, the simulation shows that the vertical force at mid-step may be minimized by the choice of a particular step length. A particularly useful specification of the theory occurs when both KLEG and V are assumed fixed. When KLEG = 15 and V = 0.18, the model makes predictions of relative stride length S and initial leg angle θ0 that are in good agreement with experimental data obtained from the literature.
The relationships between ground reaction forces, electromyographic activity (EMG), elasticity and running velocity were investigated at five speeds from submaximal to supramaximal levels in 11 male and 8 female sprinters. Supramaximal running was performed by a towing system. Reaction forces were measured on a force platform. EMGs were recorded telemetrically with surface electrodes from the vastus lateralis and gastrocnemius muscles, and elasticity of the contact leg was evaluated with spring constant values measured by film analysis. Data showed increases in most of the parameters studied with increasing running speed. At supramaximal velocity (10.36±0.31 m×s−1; 108.4±3.8%) the relative increase in running velocity correlated significantly (P<0.01) with the relative increase in stride rate of all subjects. In male subjects the relative change in stride rate correlated with the relative change of IEMG in the eccentric phase (P<0.05) between maximal and supramaximal runs. Running with the towing system caused a decrease in elasticity during the impact phase but this was significant (P<0.05) only in the female sprinters. The average net resultant force in the eccentric and concentric phases correlated significantly (P<0.05−0.001) with running velocity and stride length in the maximal run. It is concluded that (1) increased neural activation in supramaximal effort positively affects stride rate and that (2) average net resultant force as a specific force indicator is primarily related to stride length and that (3) the values in this indicator may explain the difference in running velocity between men and women.
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Walking and running on the level involves external mechanical work, even when speed averaged over a complete stride remains constant. This work must be performed by the muscles to accelerate and/or raise the center of mass of the body during parts of the stride, replacing energy which is lost as the body slows and/or falls during other parts of the stride. External work can be measured with fair approximation by means of a force plate, which records the horizontal and vertical components of the resultant force applied by the body to the ground over a complete stride. The horizontal force and the vertical force minus the body weight are integrated electronically to determine the instantaneous velocity in each plane. These velocities are squared and multiplied by one-half the mass to yield the instantaneous kinetic energy. The change in potential energy is calculated by integrating vertical velocity as a function of time to yield vertical displacement and multiplying this by body weight. The total mechanical energy as a function of time is obtained by adding the instantaneous kinetic and potential energies. The positive external mechanical work is obtained by adding the increments in total mechanical energy over an integral number of strides.