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Fig. 7. Relationship between vertical acceleration and vertical displacement of the
centre of mass. (A) 1 g, (B) 1.3 g. In each
group the top set of speeds refers to subject A and the lower set to subject
D. In each column graphs are arranged in couples, with speed increasing from
top to bottom, as indicated. In each couple the graph on the right is the
experimental record of vertical acceleration (av)
vs vertical displacement (Sv) of the centre of
mass during the step. These curves are disturbed by a large oscillation during
the fall (McMahon et al.,
1987 ), and are more consistent with the spring–mass model
during the lift (arrows directed rightward and downward). Additional
oscillations at 1.3 g are due to the vibrations of the
aircraft: these are clearly visible in Fig.
2 in the force platform records, but not in the acceleration
records due to the high damping of the accelerometers (see Materials and
methods). The left graph of each couple is constructed using three points on
the ordinate: +av,mx, av=0
–av,mx, corresponding to bottom, half and top of the
vertical oscillation, and the lift–fall average of the measured values
of Sce, Sae and Sa
on the abscissa. The zero on the abscissa corresponds to the bottom of
Sv when the upward acceleration, on the ordinate, is at a
maximum, av,mx (measured as the av
peak following the early peak due to rapid deceleration of the foot after
contact; McMahon et al.,
1987). The end of Sce (the beginning of
Sae) corresponds to av=0 by
definition, i.e. to
Fv=Mbg. The end of
Sae corresponds to Fv=0 and to
–av,mx=1 g (Ai–iii) or 1.3
g (Bi–iii). The mass-specific vertical stiffness
measured during the lower half of the oscillation,
kvert,ce/Mb=+av,mx/Sce
(the slope of the line from av,mx to
av=0), is similar to and
kvert/Mb=( /tce)2
calculated on the assumption that tce represents one half
oscillation of the bouncing system (Fig.
5). The mass-specific vertical stiffness measured during the upper
half of the oscillation when the foot is in contact with the ground,
kvert,ae-a/Mb=|–av,mx|/(Sae–Sa)
(the absolute value of the slope of the line from av=0 to
–av,mx), differs, in some conditions, from the
mass-specific vertical stiffness during the lower half of the oscillation,
kvert,ce/Mb. In particular,
kvert,ae–a>kvert,ce in some
speeds at 1 g (subject D) and in a wider speed range at 1.3
g: the significance of this finding is described in the
text.
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), the vertical
displacement of the centre of gravity of the body during the step
(Sv) and the step length (L) as a function of the
running speed (
f).
Triangles (blue) indicate the duration (tae) of the
effective aerial phase, and the displacement of the centre of gravity during
this phase in the vertical direction (Sae) and in the
forward direction (Lae). Similarly, squares (red) indicate
the duration (tce) of the effective contact phase and the
corresponding displacements in the vertical direction
(Sce) and in the forward direction
(Lce). The red broken line in each panel indicates the
actual contact time (tc), and the vertical
(Sc) and forward (Lc) displacement of
the centre of mass during it. The blue broken line in each panel indicates the
actual aerial time (ta), and the vertical
(Sa) and forward (La) displacement of
the centre of mass during it. The vertical bars indicate the standard
deviation of the mean calculated in each velocity class; the figures near the
symbols in the upper panels indicate the number of items in the mean. Note
that the step divisions based on the effective contact time and aerial time
correspond to about half of total duration and displacements of the step,
whereas the fraction of the step occupied by the actual contact and aerial
phases varies widely with speed. Note also that the speed beyond which
tae=tce,
Sae=Sce and
Lae=Lce is greater at 1.3
g than at 1 g.
1.3x increase in work.
Lines for external power,
ext, are least-squares
linear regressions of all the data, the other lines are weighted means of all
the data (Kaleidagraph 3.6.4).