1 TÜRKİYE PROFESYONEL LİGLERİNDE GÖREV ... - Spor Bilim

MATHEMATICAL MODELLING OF CRAWL STROKE SWIMMING
Tolga Akış a , Yusuf Orçan b and Ahmet N. Eraslan b
a Department of Civil Engineering, Atılım University, İncek, 06836 Ankara, Turkey
b Department of Engineering Sciences, Middle East Technical University,
06531 Ankara, Turkey
Objectives
In the study of biomechanics of swimming, a fundamental goal is to determine the
propulsive force developed by the swimmer and the opposing drag force and their relationship
to the technique and performance of the swimmer. To obtain the magnitude of these forces
during free swimming is difficult as the mechanisms and conditions of locomotion are highly
complex in the case of movement through water. In order to overcome this complexity,
different experimental and analytical techniques have been developed. The aim of the present
study is to investigate, by mathematical means, the forces acting on the body during crawl
stroke swimming and their relations to stroke rate, velocity and arm position of the swimmer.
A three dimensional analytical model is developed in which the swimmer is modeled as a
body and two arms connected to the body at the shoulder joints. Each arm is assumed to
consist of three segments. In order to check the performance of the mathematical model
developed, the numerical results are compared with the results obtained in experimental
studies.
Methods
There are a number of mathematical models of crawl stroke swimming available in the
literature [1-5]. The one developed by Martin et al. [4] consists of three parts: the body and
the two rotating arms. Each arm is modeled as a single straight segment representing the
upper arm, the forearm and the hand. The equation of motion of the swimmer is derived and
solved analytically using the mean values of some variables, which change with time during
the stroke. In this study, a three dimensional mathematical model for crawl stroke swimming
is developed which is an extension of the two dimensional model introduced by Martin et al.
[4]. In the mathematical model of the present study, the propulsive force developed by the
three dimensional motion of the arm is considered. The model consists of three parts: the
body and the two arms connected to the body by the shoulder joints. Each arm consists of
three parts: the segment from the shoulder to the elbow (upper arm), the segment from the
elbow to the wrist (forearm), and the hand. The two segments are modelled as circular
cylinders and the hand is modeled as an elliptical plate. The shoulder joint is simulated by a
ball and socket connection and the elbow is considered as a rotational joint about an axis
perpendicular to the plane of the two arm segments. The model is shown in Figure 1 where m
is the mass of the body, a 1 is the length of the upper arm a 2 is the length of the forearm, q is
the angle between x direction, which is the swimming direction, and the projected axis of the
upper arm on x-y plane, g is the angle between z axis and upper arm, and b is the angle
between z axis and the forearm. It should be pointed out that, due to the three dimensional
motion of the arm, the observed lengths a 1 and a 2 in the front view and the projected length
in the right side view are not the true lengths of the arm segments.