Dynamics of the long jump
Introduction
Running and jumping are two types of fast saltatoric movements, characterised by a series of alternating aerial and contact phases. The impact occurring during each contact phase serves to negate the vertical momentum. The flight phase is determined by the initial velocity vector of the centre of gravity at take-off and the gravitational acceleration.
The function of the leg in repetitive ground contacts at a constant energy level like in hopping or running is comparable to a spring as shown e.g. by Blickhan (1989), Alexander et al. (1986), McMahon and Cheng (1990) and Farley et al. (1993). Modelling the leg as a spring is suited to describe the landing if the body mass, the leg stiffness, and the initial conditions are known.
The spring–mass model is suitable to describe conservative systems. During the human long jump energy is in fact largely conserved. Nevertheless, due to the high running speed, the first so-called passive impact immediately after touch-down strongly influences the system dynamics. In the long jump this contribution accounts to about 25% of the total momentum and cannot be neglected.
Alexander (1990) proposed a two-segment model with an Hill-type extensor to predict optimum take-off techniques of the jumpers stance leg in high and long jumping. However, to cope with observed jumping distances unrealistic muscle properties had to be chosen. Even a detailed musculo-skeletal system with 17 segments including all important muscles (Hatze, 1981) does not describe the complete ground reaction force pattern in sufficient detail.
The understanding of body dynamics during landing or falling was significantly improved by the concept of wobbling masses introduced by Gruber (Gruber, 1987; Gruber et al., 1998). She showed that the different responses of soft tissues and hard skeleton to impacts are essential for predicting dynamical loads. In long jumping high impacts occur with forces up to 10 times body weight.
Our approach to long jumping is to describe the mechanics of the centre of mass and the mechanical function of the supporting leg using a 2D lumped parameter model with a minimum number of mechanical components. The action of the leg is described by a spring, the effect of soft tissues by the introduction of a visco-elastically coupled mass. Thereby, the influence of either initial conditions such as running speed and angle of attack (measured by video analysis) or model properties (like leg stiffness) on the jumping performance are investigated. The quality of the mechanical approach is judged by comparing the experimental force records with the results of the simulation.
Section snippets
Experiments
In training competitions in 1995 and 1996, 30 long jumps (distance: [5.49±0.86 SD] m) of 18 male and female sport students (m=[75.1±5.13 SD] kg, body height: [1.81±0.06 SD] m) were filmed for later analysis with a VHS camera (50 half-frames/s). The vertical and horizontal ground reaction forces were recorded with a 3D force plate (IAT, Leipzig). Kinematic input parameters for the dynamic models were obtained by digitising the video sequences (APAS, Ariel) (Fig. 1).
The concept of leg stiffness
The leg length r is defined as the
Model description and verification
A simple spring–mass system already predicts optimum strategies for the maximum jumping distance. For quantitative descriptions leg lengthening and mass distributions must be taken into account.
Discussion
The presented mechanical model describes with a minimal set of parameters the dynamics of the long jump. As it is well known (e.g. Hay, 1993), the most influential factor for jumping distance is the running speed. The model predicts also that a certain angle of attack of the leg optimises jumping performance (Alexander, 1990). This optimum requires a relatively low minimal stiffness of the leg.
The controlled musculo-skeleton unit with its connective tissues behaves similarly to a spring with a
Acknowledgements
We thank J. L. van Leeuwen, C. Farley and D. Ferris for useful comments on a draft of this paper. G. Kluge helped to verify the equations of motion. Supported by Deutsche Forschungsgemeinschaft II3B - Bl236/7-1.
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