A mathematical model for the force and energetics in competitive running

Abstract

A simple mathematical model for competitive running is developed. This model contains the force and energy reserves as key variables and it described their relationship and dynamics. It is made up of three submodels for the biomechanics of running the energetics and the optimization. The model for the energetics is an extension of the hydraulic model of Margaria and Morton. The key geometric parameters of this piecewise linear, three compartment model are determined on the basis of well known physiological facts and data.

The critical power and related whole-body bioenergetic models

“…I have traced the development of whole-body bioenergetic models over the last 50 years, from the early simple 2-component ideas of Monod and Scherrer (1965) and Lloyd (1966) and others; through Margaria (1976) and Morton's (1986a, b) 3-component extensions; to the more complex recent models of Behncke (1993Behncke ( , 1997, Ward-Smith (1985a, b) and Harman (2002). It is useful now to view the current situation in summary with reference back to the ten assumptions of the original CP model (Origins, assumptions and development and Questioning the assumptions).…”

“…Changes in VO 2 dynamics forecast by the model as a result of such structural change may provide clues. Behncke (1993) has incorporated the M-M model as one submodel in a larger mathematical model for the force and energetics of competitive running. The other submodels include the biomechanics of running and an optimisation determination.…”

Solution and asymptotic analysis of a boundary value problem in the spring–mass model of running

“…Problem 1. Let (θ(t; K), L(t, K)) be the solution of the system (4) with (6). Find K * and the smallest time t * > 0 satisfying θ(t * , K * ) = α, L(t * , K * ) = 1.…”

How to identify the physiological parameters and run the optimal race

“…It is based on energy conservation and the fundamental principal of dynamics. With his simple model, and relying on constant maximal oxygen uptake (constantV O2), he found that he could nevertheless reasonably predict the time of world records for distances bigger than 400 m. Several improvements of Keller's model have been introduced: the effect of fatigue [18,8], the variation in maximal oxygen uptake [2], air resistance and altitude [13]. Other related works include [17,9,10,12].…”