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Journal of Biomechanics 40 (2007) 3341–3348

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Effects of altered stride frequency and contact time on leg-springbehavior in human running

J.B. Morin�, P. Samozino, K. Zameziati, A. Belli

UR Physiologie et Physiopathologie de l’Exercice et Handicap, Universite de Saint-Etienne, Medecine du Sport – Myologie, CHU Bellevue,

42055 Saint-Etienne Cedex 2, France

Accepted 2 May 2007

Abstract

Many studies have demonstrated that contact time is a key factor affecting both the energetics and mechanics of running. The purpose

of the present study was to further explore the relationships between contact time (tc), step frequency (f) and leg stiffness (kleg) in human

running. Since f is a compound parameter, depending on both contact and aerial time, the specific goal of this study was to independently

vary f and tc and to investigate their respective effects on spring–mass characteristics during running, seeking to determine if the changes

in kleg observed when running at different f are mainly due to inherent changes in tc. We compared three types of constant 3.33m s�1

running conditions in 10 male subjects: normal running at the subject’s freely chosen f, running with decreased and increased f, and

decreased and increased tc at the imposed freely chosen f. The data from the varied f trials showed that the variation of tc was strongly

correlated to that of kleg (r2¼ 0.90), and the variation of f was also significantly correlated to that of kleg (r

2¼ 0.47). Further, changes in

tc obtained in various tc conditions were significantly correlated to changes in kleg (r2¼ 0.96). These results confirm that leg stiffness was

significantly influenced by step frequency variations during constant speed running, as earlier demonstrated, but our more novel finding

is that compared to step frequency, the effect of contact time variations appears to be a stronger and more direct determinant of kleg.

Indeed, 90–96% of the variance in kleg can be explained by contact time, whether this latter parameter is directly controlled, or indirectly

controlled through its close relationship with step frequency. In conclusion, from the comparison of two experimental procedures, i.e.

imposing various step frequency conditions vs. asking subjects to intentionally vary contact time at their freely chosen step frequency, it

appears that changes in leg stiffness are mainly related to changes in contact time, rather than to those in step frequency. Step frequency

appears to be an indirect factor influencing leg stiffness, through its effect on contact time, which could be considered a major

determinant of this spring–mass characteristic of human running.

r 2007 Elsevier Ltd. All rights reserved.

Keywords: Spring–mass model; Stiffness; Time of force application; Stride rate

1. Introduction

When bipeds or quadrupeds hop, skip, trot, gallop orrun, the forces produced by their muscular machinery areapplied to the ground and oriented in order to increase,maintain or decrease their momentum, during the time thata foot, hoof or paw is in contact with the ground, i.e.during the ‘‘contact time’’ (tc). In addition to mechanics, ithas been shown that the inverse of contact time (tc

�1), alsoreferred to as the rate of force application is an important

e front matter r 2007 Elsevier Ltd. All rights reserved.

iomech.2007.05.001

ing author. Tel.: +33477 127985; fax: +33 477 127229.

ess: jean.benoit.morin@univ-st-etienne.fr (J.B. Morin).

determinant of the energetic cost of running across speed invarious species of terrestrial mammals and birds (e.g. Kramand Taylor, 1990; Roberts et al., 1998). Indeed, Kram andTaylor (1990) showed that the rate of force application wasstrongly related to and could primarily explain themetabolic cost of locomotion in small to large mammals,i.e. in 30 g kangaroo rats (Dipodomys merriami) to 140 kgponies (Equus caballus). These findings were furtherconfirmed by Roberts et al. (1998), who included humansand running birds in their study. Other studies have shownthat tc

�1 could estimate mass-specific rates of oxygenuptake in freely running humans (Hoyt et al., 1994;Weyand et al., 2001).

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Nomenclature

F vertical forceFmax peak vertical forcetc contact timeta aerial timef step frequencyFCf freely chosen step frequencyVARI-f reduced or increased step frequency running

conditionsVARI-tc reduced or increased contact time running

conditionsCOM center of massGRF ground reaction forcePFA point of force application to the ground

PFAT point of force translation during contactd point of force translation distanceDy vertical downward peak displacement of the

center of mass during contactDL leg length variation (compression) during con-

tact in the classical spring–mass modelDL0 leg length variation (compression) during con-

tact taking into account the displacement of thepoint of force application

Dy vertical downward peak displacement of thecenter of mass during contact

L initial leg lengthv forward running velocityDF duty factorkleg leg stiffness

Table 1

Physical characteristics of subjects, and freely chosen step frequency (FCf)

values at 3.33m s�1

Age

(years)

Body

mass (kg)

Stature

(m)

FCf (Hz)

Mean 28.6 75.6 1.75 2.75

SD 6.4 10.4 0.10 0.13

SD: standard deviation.

J.B. Morin et al. / Journal of Biomechanics 40 (2007) 3341–33483342

Biomechanically, tc is a critical parameter in the variousspring–mass models for running (Alexander, 1992; Blic-khan, 1989; Dalleau et al., 1998; Dickinson et al., 2000;Farley et al., 1991, 1993; Farley and Gonzalez, 1996; Ferriset al., 1998; He et al., 1991; McMahon and Cheng, 1990;McMahon et al., 1987). In these models, widely used todescribe and study the mechanics of human and otheranimals bouncing and running gaits, the musculo-skeletalstructures of the legs alternately store and return elasticenergy (though one should keep in mind that the springterm used in this model does not imply a pure linearelasticity of the leg). The main mechanical parameterderived from spring–mass models is the stiffness of the legspring, defined as the ratio of the maximal force applied tothe leg-spring to the maximum leg compression. Recently,Morin et al. (2005) developed and validated a simplecomputational method (based on a sine-wave modeling offorce–time curves) that estimates leg stiffness from fewsimple mechanical parameters including contact time. Asensitivity analysis of that theoretical model showed thatcontact time is the parameter with the greatest relativeinfluence on leg stiffness. Indeed, tc has a 1:2.5 effect in thissensitivity analysis (a 10% decrease in contact time leadingto a 25% increase in leg stiffness and vice versa) whereas allother mechanical and anthropometrical parameters testedshowed a 1:1 or lower relative influence (Morin et al.,2005).

The purpose of the present study was to further explorethe relationships between contact time, step frequency andleg stiffness in human running. Indeed, the basic kinematicsof running state that tc is related to step frequency (f)

f ¼ ðtc þ taÞ�1, (1)

where ta is the aerial or flight time. The relationshipsbetween step frequency and leg stiffness were studied byFarley and Gonzalez (1996) who clearly demonstrated thatalthough leg stiffness did not significantly change withrunning speed, it did when step frequency varied at a givenconstant speed. They reported that leg stiffness increased

by about two-fold when step frequency was increased by65% (Farley and Gonzalez, 1996). However, as seen inEq. (1), f is not only depending on tc, but also on ta.Consequently, we sought to examine these frequency–stiff-ness relationships in more detail since frequency is acompound parameter, and to determine if the changes inleg stiffness observed when running at different stepfrequencies are mainly due to inherent changes in contacttime. Therefore, the specific goal of this study was toindependently vary step frequency and contact time and toinvestigate their respective effects on spring–mass char-acteristics during running.

2. Materials and methods

2.1. Subjects and protocol overview

Ten male subjects gave their written informed consent to participate in

this study. They were all involved in regular physical activities, but not

specialized in running. Their main anthropometric data and their freely

chosen step frequency (FCf) are shown in Table 1. The reference period

considered in the present study regarding running kinematics was the step

(i.e. the time period from the onset of one foot contact, to the onset of the

contralateral foot contact). We compared three types of constant sub-

maximal velocity running conditions summarized in Table 2: normal

running (Normal), running with decreased and increased step frequencies

(VARI-f), and decreased and increased contact times at the imposed FCf

(VARI-tc).

After a standardized 5min warm-up run at 2.78m s�1, subjects ran

2min on a treadmill dynamometer at 3.33m s�1 allowing the computation

of their FCf from force signals. Then, subjects were asked to run at

3.33m s�1 at various step frequencies (set by a metronome tone) for six

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Table 2

Summary of the running conditions imposed to the subjects

Normal VARI-f VARI-tc

Velocity 3.33m s�1 3.33m s�1 3.33m s�1

Frequency FCf

(measured)

0.7–1.3 FCf

matched to

audio signal

FCf matched

to audio signal

Contact time Freely chosen Freely chosen Intentionally

reduced or

increased

Duration 2min (1 trial) 2min per trial

(6 trials)

3min per trial

(2 trials)

J.B. Morin et al. / Journal of Biomechanics 40 (2007) 3341–3348 3343

randomized 2min trials under imposed 30% decreased to 30% increased

FCf. Lastly, subjects performed two randomized 3min running trials at

3.33m s�1 while matching a metronome set to their FCf with either shorter

(Stc) or longer (Ltc) contact time. During these two conditions, no contact

time value was accurately imposed to the subjects (by audio tone, or real

time feedback for instance), they were verbally asked before the trials to

reduce and increase contact time as much as possible.

2.2. Stride kinematics and dynamics

Mechanical parameters were measured for each step using a treadmill

dynamometer (HEF Techmachine, Andrezieux-Boutheon, France), for

details, see Belli et al. (2001). Vertical ground reaction force (F) data and

belt velocity were sampled at a rate of 1000Hz. Contact and aerial times

were measured from F(t) signals, as shown in Fig. 1. Duty factor was

computed as

DF ¼ tcðtc þ taÞ�1. (2)

The different decreased (f�30%, f�20% and f�10%) and increased (f+30%,

f+20% and f+10%) step frequency conditions were calculated from FCf

(measured after 2min during the first run) and imposed by an audio signal.

In VARI-f conditions, force data were collected after 1min, and in VARI-

tc conditions, measurements were performed at 30 s, 1min 30 s and 3min,

in order to study the ability of the subjects to maintain either the increased

or reduced contact times consistently. All mechanical parameters

measured were averaged for 10 consecutive steps.

2.3. Spring–mass characteristics

A spring–mass model paradigm (for details see Alexander, 1992;

Blickhan, 1989; Farley et al., 1991, 1993; Farley and Gonzalez, 1996;

He et al., 1991; McMahon and Cheng, 1990; Morin et al., 2005) was used

to investigate the main mechanical parameters characterizing the lower

limbs behavior during running. In these studies, the stiffness of the leg

spring (kleg in kNm�1) was calculated from F(t) measurements as follows:

kleg ¼ FmaxDL�1 (3)

with DL the maximum leg spring compression (in m) calculated from

values of initial leg length L (great trochanter to ground distance in a

standing position), running velocity v (in m s�1), contact time tc (in s) and

vertical maximal downward displacement of the center of mass (COM)

during contact Dy, as per (Farley and Gonzalez, 1996; He et al., 1991;

McMahon and Cheng, 1990; Morin et al., 2005)

DL ¼ L�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiL2 �

vtc

2

� �2rþ Dy. (4)

However, during level running the point of force application (PFA) is

not a fixed point over a typical stance phase (Lee and Farley, 1998), and it

has recently been shown that the distance of the PFA translation (PFAT)

should be taken into account when computing DL, in order to increase the

accuracy of the spring–mass model for describing the lower limbs

mechanics during human running (Bullimore and Burn, 2006). Leg length

variation during contact was therefore calculated on the basis of Eq. (4),

incorporating the distance of PFAT into the ‘‘traditional’’ planar

spring–mass model:

DL0 ¼ L�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiL2 �

vtc � d

2

� �2s

þ Dy. (5)

The distance of PFAT (d) was shown to be 0.15770.006m in subjects

running from 1.5 to 5.0m s�1 (Lee and Farley, 1998), and this value was

estimated to be equal to about 18% of the mean leg length of these

subjects (Bullimore and Burn, 2006). As we could not measure d in the

present study, this parameter was assumed to be

d ¼ 0:18L. (6)

Therefore, leg stiffness was calculated in the present study, from Eqs. (3)

and (5) as

kleg ¼ FmaxDL0�1. (7)

Dy being determined by double integration of the vertical acceleration

of the COM over time, as proposed by Cavagna (1975).

3. Statistical analysis

After checking the normal distribution of the values foreach parameter using a Shapiro–Wilk test, an ANOVAwith Tukey’s honestly significant difference post hoc testwas performed to investigate differences in the previouslydescribed parameters between Normal, VARI-f and VARI-tcconditions.We investigated the influence of both contact time and

step frequency variations on changes in leg stiffness usingPearson’s correlation test, and linear regressions were usedto further describe significant correlations. Statisticalsignificance was accepted at Po0.05.

4. Results

For all conditions, subjects matched the imposed stepfrequencies quite well, as shown by the non-significantANOVA comparing imposed and actual frequenciesthrough the entire protocol. Typical curves of verticalground reaction force and vertical displacement of theCOM during contact in VARI-f and VARI-tc conditions areshown in Figs. 1 and 2, respectively.Mechanical parameters of stride kinematics and dy-

namics, as well as spring–mass parameters are shown inTable 3. In the VARI-f conditions, tc, Dy and DL0

significantly decreased (Po0.05) with increasing f, Fmax

and ta remaining unchanged. This corresponded to asignificant increase in kleg (Po0.05). To the contrary, withdecreasing f, neither tc nor stiffness changed significantly,whereas Fmax, Dy and DL0 significantly increased (Po0.05).These changes observed when f ranged from f�30% to

f+30% are shown in Fig. 3 concerning step kinematics andin Fig. 4 for step dynamics and spring–mass modelparameters.As the changes in tc in VARI-tc conditions show (Table 3),

the significant changes in tc ranged from �14.374.19% inStc condition to+27.078.2% in Ltc condition. During

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Fig. 1. Typical curves of vertical ground reaction force (bottom) and vertical displacement of the center of mass (top) as a function of time during normal

(line), decreased step frequency (dashes) and increased step frequency (dots) running.

Fig. 2. Typical curves of vertical ground reaction force (bottom) and vertical displacement of the center of mass (top) as a function of time during normal

(line), decreased contact time (dashes) and increased contact time (dots) trials.

J.B. Morin et al. / Journal of Biomechanics 40 (2007) 3341–33483344

these VARI-tc running trials, kleg increased significantly(Po0.05) by 44.1713.8% in Stc, and decreased signifi-cantly (Po0.05) by 45.979.3% in Ltc.

The data from the varied f and tc trials show that thevariation of tc was strongly correlated to that of kleg

(r2 ¼ 0.90; Po0.05, Fig. 5A), and the variation of f wasalso significantly correlated to that of kleg (r2 ¼ 0.47;Po0.05, Fig. 5B).

Lastly, Fig. 6 shows changes in tc obtained both inVARI-f or VARI-tc conditions (i.e. either depending on f

variations, or directly and intentionally controlled bythe subjects). Changes in tc observed in both VARI-tcand VARI-f conditions were significantly correlated tochanges in kleg (r2 ¼ 0.96; Po0.05), that ranged fromabout �50% to about +150% of that measured in Normalcondition.

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Table 3

Grand averages of contact (tc) and aerial (ta) times, duty factor (DF), maximal vertical ground reaction force (Fmax), downward displacement of the center

of mass (Dy), leg compression (DL0), vertical (kvert) and leg (kleg) stiffness, and the absolute percentage difference between the step frequency of the

condition and the freely chosen step frequency (fdiff)

Condition tc (s) ta (s) DF Fmax (bw) Dy (m) DL0 (m) kleg (kNm–1) fdiff (%)

Normal 0.24 0.12 0.67 2.52 0.06 0.12 16.6 –

70.02 70.02 70.04 70.21 70.01 70.01 72.9

f+30% 0.20* 0.09 0.7 2.42 0.04* 0.08* 25.7* +26.8*

70.03 70.03 70.09 70.42 70.01 70.01 78.5 72.3

f�30% 0.25 0.25* 0.50* 3.08* 0.10* 0.16* 15.4 �26.0*

70.01 70.03 70.03 70.26 70.01 70.02 72.5 76.1

Stc 0.21* 0.17* 0.56* 3.15* 0.06 0.1 23.7* +2.48

70.01 70.02 70.04 70.22 70.01 70.01 72.8 72.88

Ltc 0.31* 0.07* 0.82* 1.92* 0.06 0.17* 8.86* +4.36

70.02 70.02 70.05 70.14 70.01 70.02 71.3 72.75

The five running conditions presented are normal running (Normal), running with 30% increased (f+30%) and 30% decreased (f�30%) step frequency, and

running with short (Stc) and long (Ltc) contact times. During the two latter conditions, the frequency imposed was the normal freely chosen step frequency.

Values are means7SD. *Significantly different from the normal running condition (Po0.05).

Fig. 3. Changes in step kinematics (contact times, aerial times and duty

factor) as a function of changes in step frequency, from 30% decreased to

30% increased step frequency compared with the normal running

condition performed at the freely chosen step frequency (FCf). *:

significantly different from the normal running condition (Po0.05).

Fig. 4. Changes in the main spring–mass parameters of maximal vertical

ground reaction force (Fmax), downward displacement of the center of

mass (Dy), leg compression (DL0), leg stiffness (kleg) induced by changes in

step frequency, from 30% decreased to 30% increased step frequency

compared with the normal running condition performed at the freely

chosen step frequency (FCf). *: significantly different from the normal

running condition (Po0.05).

J.B. Morin et al. / Journal of Biomechanics 40 (2007) 3341–3348 3345

5. Discussion

The results of the present study confirm that kleg wassignificantly influenced by f variations during constantspeed running, as earlier demonstrated by Farley andGonzalez (1996). Our more novel and important finding isthat compared to f, the effect of tc variations appears to bea stronger and more direct determinant of kleg. Indeed, inVARI-f conditions, 47% of the variance in kleg can beexplained by variation in f (Fig. 5B), whereas 90% of thisvariance in kleg is explained by tc variations, when takingonly the latter parameter into account (Fig. 5A). While it

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Fig. 5. (A) Changes in leg stiffness (kleg) as a function of contact time

variations in VARI-f conditions. The coefficient of determination is

r2 ¼ 0.90 (Po0.05). Black dots: mean values for each subject during each

VARI-f condition. White dots: grand means for all subjects during each

VARI-f condition. (B) Changes in leg stiffness (kleg) as a function of step

frequency variations in VARI-f conditions. The coefficient of determination

is r2 ¼ 0.47 (Po0.05). Black triangles: mean values for each subject during

each VARI-f condition. White triangles: grand means for all subjects during

each VARI-f condition.

Fig. 6. Linear inverse relationship between changes in contact time and in

leg stiffness (kleg). Contact time variations were obtained in two ways:

either induced by various step frequencies conditions (VARI-f conditions,

black dots) or intentionally increased or decreased by subjects (VARI-tcconditions, white dots). Dashed line: significant linear regression

(r2 ¼ 0.96; Po0.05) between contact time and leg stiffness observed in

VARI-tc conditions.

J.B. Morin et al. / Journal of Biomechanics 40 (2007) 3341–33483346

was to be expected that kleg is closely related to tc, this is toour knowledge the first study to investigate kleg whilesubjects vary their tc. The predominance of tc effect on kleg

variations was further confirmed by the VARI-tc part of the

experiment and the significant linear regression observedbetween tc variations (induced by varying f and directly inthe trials where the subjects ran at their FCf) and kleg

variations. Overall, these results demonstrate that inhuman running, 90–96% of the variance in kleg can beexplained by that in tc, whether this latter parameter isdirectly controlled, or indirectly controlled through itsclose relationship with f (Eq. (1)). This interdependency isemphasized if one considers that the significant effect of tcon kleg is about twice as great as that of f in VARI-fconditions (r2 ¼ 0.90 vs. 0.47, Fig. 5A and B), and evenmore so when tc is considered independently, as in the VARI-tc conditions of this study. The results shown in Table 3also make this point: in Stc and Ltc conditions, tc valueswere significantly shorter and longer, respectively, whichcorresponded to significant increase and decrease in kleg

(Po0.05). If one considers the extreme VARI-f conditions,kleg only increased significantly for f+30% reaching25.778.5 kNm�1 vs. 16.672.9 kNm�1 in Normal run-ning, and f�30% was not marked by a decrease in kleg, or intc. When comparing results in VARI-tc and VARI-f condi-tions, it can be noticed that changes in kleg followed thosein tc, and that running with 30% reduced f did not induce asignificant increase in tc in this study (0.2570.01 s vs.0.2470.02 s in Normal condition), nor did it affect kleg.This point contrasts with the results of Farley andGonzalez (1996), who found ‘‘symmetrical’’ variations ofkleg when f varied similarly below and beyond FCf (i.e.about 2.66 steps s�1, and �26 to+36% of this FCf ,respectively). They reported that kleg doubled between thelowest and the highest f, whereas in this study, kleg onlyincreased at the highest f, and did not vary with decreasingf, as shown in Fig. 4. Both main spring–mass parametervalues and their changes with f were quite similar to thoseobtained by Farley and Gonzalez (1996), i.e. Fmax

decreased by about 20% (vs. from 3.0870.26 to2.4270.42 bw in the present study) and DL0 decreased byabout 65% (vs. from 0.1670.02 to 0.0870.01m in thisstudy), but our values for kleg differ from Farley andGonzalez (1996). In one sense, this could be explained bytc, which linearly decreased from 0.365 to 0.248 s in theirstudy (Fig. 3 in Farley and Gonzalez, 1996), subjectsrunning at 2.5m s�1) whereas tc only decreased in thepresent study from FCf to f+30% (0.24–0.20 s), with oursubjects running at a faster velocity: 3.33m s�1. Therefore,it may be hypothesized that the decrease in kleg that couldhave been expected was not observed because tc did notsignificantly increase at low f in our study, remainingconstant at about 0.25 s in f+30% condition (Fig. 3).Concerning the Normal and VARI-f conditions, our

results are in line with the spring–mass model literature(e.g. Alexander, 1992; Arampatzis et al., 1999; Blickhan,1989; Dutto and Smith, 2002; Farley et al., 1993; Farleyand Gonzalez, 1996; He et al., 1991; Lee and Farley, 1998;McMahon and Cheng, 1990; Morin et al., 2005). It ishowever worth noting that tc values reported here inNormal conditions at 3.33m s�1, i.e. 0.2470.02 s, are lower

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Fig. 7. Theoretical changes in leg stiffness as a function of normalized

30% increases and decreases in contact time (below and beyond reference

unit values). The central black point represents a typical kleg value

obtained for a dataset measured in Normal condition. These typical

reference values are kleg ¼ 10.9 kN/m, tc ¼ 0.25 s, ta ¼ 0.12 s, f ¼ 2.70Hz,

m ¼ 75 kg, L ¼ 0.94m, v ¼ 3.33m/s and Dy ¼ 0.06m.

J.B. Morin et al. / Journal of Biomechanics 40 (2007) 3341–3348 3347

than those reported by Farley and Gonzalez (1996). Thisdiscrepancy can be mainly assumed to be due to thedifferent running speeds, as reported for instance by Hoytet al. (2000). Indeed, for instance, similar tc values of about0.26 s were observed in subjects running at 3.33m s�1

(Chang and Kram, 1999).The influence of tc on kleg can also be seen in light of the

linear regression obtained between tc variations (bothdepending on f variations and intentionally performed bythe subjects) and the related kleg variations, as shown inFig. 6. This figure shows that 96% of the variance in kleg

can be explained by tc variations, but also that this linearrelationship is beyond the line of identity, meaning that therelative weight of tc variations on kleg changes duringrunning is greater than 1:1. In this study, the absoluterelative slope of this relationship was 2.06, which meansthat during running, a 10% decrease in tc corresponded toa �20% increase in kleg, and inversely, the decrease in kleg

was worth twice the increase in tc. These experimental datacan be compared to those obtained theoretically by Morinet al. (2005), who computed that (all other parametersremaining equal) a 10% decrease in tc would correspond toa 25% increase in kleg and vice versa, which correspondedto the experimental results shown in Fig. 6.

These relationships could have been expected from atheoretical computation. Indeed, spring–mass theory ap-plied to running mechanics states that kleg ¼ FmaxDL�1,and from impulse considerations, Fmax can be expressed asa function of DF (hence tc and f), and a constant parameter(p) describing the assumed sine shape of the force profile(Alexander, 1989; Kram and Dawson, 1998):

Fmax ¼pmg

2DF(8)

and with

p ¼ pmg=2, (9)

Fmax ¼p

DF¼

p

tcf. (10)

Therefore, kleg can be expressed theoretically as afunction of tc and f, rewriting Eqs. (3) and (4)

kleg ¼p

tcfL�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiL2 �

vtc

2

� �2rþ Dy

" #�1. (11)

From typical mean values which reflect our data set(Normal condition), we computed �30% to +30%changes in tc, to study their effects on kleg, all otherparameters remaining equal (Fig. 7).

From the unit value of tc at a constant f, it appears that a25% increase in kleg can be induced by a �10% decrease intc, which tends to confirm the �1:2 to 1:2.5 weight ofchanges in tc on those in kleg, which approximatelycorresponds to the experimental data we obtained throughour measurement protocol. The results of this theoreticalanalysis lead us to the following remarks: (I) ourexperimental protocol allowed us to actually observe and

measure the tc�f�kleg relationships that could have beenexpected from a theoretical consideration of the spring–mass model mechanics, and (II) the spring–mass modelassumptions (e.g. lower limb geometry during contact andsine shape of F(t) curves) reflect a lower limb behaviourthat actually matches the reality of human running.Another way to show that kleg was highly related to tc,

rather than directly to f, is to observe experimental valuesof kleg obtained either in VARI-tc and VARI-f conditions,focusing in both cases, on similar tc values. For instance, asshown in Table 3, a kleg value of 16.672.9 kNm�1 wasmeasured in Normal running (for a tc of 0.2470.02 s),and increased to 23.772.8 kNm�1 in the Stc condition(f remaining unchanged), for a corresponding tc of0.2170.01 s. In order to obtain such high kleg values inthe VARI-f conditions, subjects had to run with a 30%greater f, characterized by a tc of 0.2070.03 s, i.e. almostexactly the one previously described in the Stc condition.Thus, in constant running speed conditions, similar kleg

values can be observed independently of f variations,provided that similar tc are performed, reinforcing thedominant effect of tc (rather than f) on kleg.From the partial analysis reflected in a schematic

summary (Fig. 8), it becomes clear that the main factorrelated to kleg seems to be tc, whether it is influenced by f

(Fig. 8A), or not (Fig. 8B).Furthermore, it must be pointed out that the significant

relationship between tc and kleg may not hold acrossrunning speed. Indeed, it is commonly observed that kleg

stays nearly constant across various speeds (from 2m s�1 tomaximal sprint speed) yet tc changes with running speed(e.g. Farley et al., 1993; Farley and Gonzalez, 1996;He et al., 1991; McMahon and Cheng, 1990; Morin et al.,2005). Lastly, this study shows that subjects were able to

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Fig. 8. Schematic summary of the linear regressions obtained between

changes in step frequency (Df), contact time (Dtc) and leg stiffness (Dkleg)

in VARI-fq conditions (A) and in VARI-tc conditions (B). Determination

coefficients (r2) of the linear regressions are adjoining the arrows.

J.B. Morin et al. / Journal of Biomechanics 40 (2007) 3341–33483348

voluntarily increase or decrease tc during running at aconstant sub-maximal velocity (without changing f), in aconsistent manner over several minutes (up to 3min in thisprotocol). Thus, it would be possible to conduct studies inreduced and increased tc conditions and steady-stateenergetics measurements, providing interesting experimen-tal evidences to the fact that energetic cost of humanrunning may strongly depend on tc, as previouslymentioned.

In conclusion, it appears from these experimental resultsthat changes in leg stiffness are mainly related to changes incontact time, rather than to those in step frequency. Stepfrequency appears to be an indirect factor influencing legstiffness, through its effect on contact time. This lattercould be considered a major determinant of this spring–mass characteristic of human running.

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