Dynamics of the long jump

Abstract

A mechanical model is proposed which quantitatively describes the dynamics of the centre of gravity (c.g.) during the take-off phase of the long jump. The model entails a minimal but necessary number of components: a linear leg spring with the ability of lengthening to describe the active peak of the force time curve and a distal mass coupled with nonlinear visco-elastic elements to describe the passive peak. The influence of the positions and velocities of the supported body and the jumper's leg as well as of systemic parameters such as leg stiffness and mass distribution on the jumping distance were investigated. Techniques for optimum operation are identified: (1) There is a minimum stiffness for optimum performance. Further increase of the stiffness does not lead to longer jumps. (2) For any given stiffness there is always an optimum angle of attack. (3) The same distance can be achieved by different techniques. (4) The losses due to deceleration of the supporting leg do not result in reduced jumping distance as this deceleration results in a higher vertical momentum. (5) Thus, increasing the touch-down velocity of the jumper's supporting leg increases jumping distance.

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.