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First published online June 6, 2005
Journal of Experimental Biology 208, 2333-2346 (2005)
Published by The Company of Biologists 2005
doi: 10.1242/jeb.01661

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Effect of an increase in gravity on the power output and the rebound of the body in human running

G. A. Cavagna^1,*, N. C. Heglund² and P. A. Willems²

¹ Istituto di Fisiologia Umana, Università degli Studi di Milano, 20133 Milan, Italy
² Unité de Physiologie et Biomécanique de la Locomotion, Université catholique de Louvain, 1348 Louvain-la-Neuve, Belgium

View larger version (19K): [in a new window]   Fig. 1. Diagrammatic representation of the effective aerial and contact times, and vertical displacements. The vertical displacement of the centre of mass during the time of contact with the ground tc (continuous line) and during the aerial phase ta (broken line) is divided into a lower part Sce (red) taking place when the vertical force is greater than body weight, and into an upper part Sae (blue) taking place when the vertical force is less than body weight. Running speed increases from top to bottom. Note that in all cases Sce (red) represents the amplitude of the oscillation of the spring–mass system from its equilibrium point and its duration tce represents a half period of the oscillation (neither the peak-to-peak vertical displacement nor the vertical displacement during contact represent the amplitude of the oscillation). Sae (blue) represents the amplitude of the oscillation in the opposite direction, and its duration tae the half period of the oscillation, only at the lowest running speed (A) when the whole vertical displacement takes place during contact Sc. Only A, when no aerial phase takes place, is consistent with the spring–mass model. With increasing speed a progressively greater fraction of the vertical displacement takes place during the aerial phase Sa. The resonant frequency of the spring–mass system fs=1/(2tce) equals the step frequency f only when tce=tae, i.e. when the rebound is symmetric (A,B). At high running speeds (C) the rebound is asymmetric (tce<tae) and the step frequency is lower than the resonant frequency of the system.

View larger version (25K): [in a new window]   Fig. 2. Experimental records of running on Earth (A) and during a flight profile simulating 1.3 g (B). (A) Fore–aft (Ff) and vertical (Fv) components of the force exerted by the foot on the force platform during running between the two photocells at the indicated speeds on Earth. (B) From top to bottom at each speed are shown the vertical (av), lateral (al), and fore–aft (af) components of the acceleration recorded on the aircraft during the run; the force signals Ff and Fv from the force platform (noise is due to vibrations of the aircraft) and the output of the photocell circuit. The vertical dotted lines delimit the time interval corresponding to the steps illustrated in Fig. 3, with expansion of the energy records to include the previous valley of potential energy as described in the Materials and methods. Subject A, mass 72 kg.

View larger version (18K): [in a new window]   Fig. 3. Mechanical energy changes of the centre of mass of the body during the running step. (A) 1 g, (B) 1.3 g. At each speed the curves show from bottom to top: the changes in the gravitational potential energy of the centre of mass of the body (Ep, dotted line), the sum of the kinetic energy of vertical motion (Ekv) plus Ep (continuous line), the kinetic energy of forward motion (Ekf) and the total translational energy of the centre of mass in the sagittal plane (Ecm=Ep+Ekv+Ekf). The records were obtained as described in the Materials and methods from the Fv and Ff signals for the steps indicated by the vertical interrupted lines in Fig. 2, expanded to the left to include the previous valley of Ep. At each speed, the zero line corresponds to the minimum attained by the Ep curve. The continuous line below each panel indicates the ground contact time.

View larger version (25K): [in a new window]   Fig. 4. Step duration and displacements of the centre of mass in vertical and forward direction as a function of running speed. (A–C) 1 g, (D–F) 1.3 g. Diamonds (black) indicate, from top to bottom, the step period (), 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.

View larger version (22K): [in a new window]   Fig. 5. Stiffness (A,C), resonant frequency of the bouncing system and freely chosen step frequency (B,D) as a function of running speed. (A,B) 1 g, (C,D) 1.3 g. In A and C, filled circles give the vertical stiffness of the bouncing system [kvert=Mb(/tce)2], whereas open circles give the leg stiffness (kleg), calculated as described in the Materials and methods. The kvert,1g line in C is drawn for comparison to show the similarity of the two stiffness at intermediate speeds. In B and D, filled squares indicate the freely chosen step frequency (f) for comparison with open squares, the resonant frequency of the bouncing system [fs=1/(2tce)], calculated assuming that the effective contact time corresponds to one half-period of the oscillation of the elastic system. Note that the increase in gravity increases the maximum speed where f=fs. The two lower lines indicate the frequency fk,leg=(kleg/Mb)0.5/(2), calculated from the leg stiffness and the frequency fc=1/(2tc), calculated assuming that the time of contact corresponds to one half-period of the oscillation of the elastic system: both give false indication in their relation with the actual step frequency. Lines are least-squares linear regressions or weighted mean of all the data (Kaleidagraph 3.6.4).

View larger version (25K): [in a new window]   Fig. 6. External power and work per unit distance. (A,B) 1 g, (C,D) 1.3 g. The mass-specific power (A,C) and the mass-specific work done per unit distance (B,D) to maintain the movement of the centre of mass in the sagittal plane (filled diamonds, ext) to accelerate it forwards (open squares, f) and to lift it against gravity (open triangles, v) are given as a function of the running speed. In C, x and + give the mean values plotted in B x1.3: the agreement with the experimental data shows that a 1.3x increase in gravity causes an 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).

View larger version (40K): [in a new window]   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.

View larger version (13K): [in a new window]   Fig. 8. The mass-specific external work done at each step (A) and the step frequency (B) are plotted as a function of running speed for the experiments made at 1 g (open circles) and at 1.3 g (filled circles). It can be seen that, on average, step frequency is increased by gravity more than the work per step: in other words, an increase in gravity increases the mechanical power output mainly through an increase in step frequency. Lines are least-squares linear regressions (1 g) or weighted mean (1.3 g) of all the data (Kaleidagraph 3.6.4).

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