Translational and rotational traction and stiffness

INTRODUCTION
TRANSLATIONAL AND ROTATIONAL TRACTION AND STIFFNESS
Bruce Barry 1 and Peter Milburn 2
1 Dept Human Movement Studies, Royal Melbourne Institute of Technology, Melbourne, Australia
2 School of Physiotherapy, University of Otago, Dunedin, New Zealand
brucebarry@bigpond.com
In field sports, a player who is able to stop, accelerate, or change direction more rapidly than the opponent because of superior
traction on the surface would have an advantage over their opponent. Traction is the required level of braking or propulsive force
parallel to the field surface that is needed by the athlete to successfully achieve a particular manoeuvre on the sports field. Whether
the athlete gains sufficient traction depends on the physical properties of the two sliding surfaces and the limiting traction that can
be developed between them. However, there is no established theory to calculate traction forces given the material properties of
the two sliding surfaces (Czichos, 1986; Suh & Sin, 1981). Most of these experimental investigations have modelled the
mechanism of traction using the principles of dry friction established by Coulomb. However, Valiant (1987) and van Gheluwe et
al. (1983), both found the laws of dry friction were not followed and Valiant (1994) stated there was a need for a more valid
evaluation of outsole traction. Furthermore, most of the research has involved court shoes and hard or artificial surfaces rather than
field shoes and natural surfaces. In this study, exponential curve fitting procedures were developed and applied to traction data to
produce two measurable parameters to compare translational and rotational traction and stiffness.
METHODOLOGY
The method used to analyse the data was novel to footwear traction research due to the displacement control provided by the
traction device (Barry, et al., 2000) and based on methods used in terramechanics. Research into the performance of off-road
terrain vehicles (Wong, 1989) provided different exponential functions that were used for the non-linear regression analysis of the
traction data depending on the characteristics of the response curve.
Figure 1: RMIT Traction test device (Barry, et al, 2000)
Traction response curves collected from each test were fitted with exponential functions by non-linear regression using the
Levenberg-Marquardt method to obtain traction parameters that fully defined the response curve. The parameters included the
asymptotic, or the peak and residual traction forces or torques, and corresponding slide or rotation displacements that enabled the
stiffness of the shoe/surface combinations to be calculated, along with the relevant statistics on goodness of fit. The translation
traction-displacement relationship for footwear/surface combinations could be characterised by one of two exponential functions
used in terramechanics (Wong, 1989). They are both graphed showing all quantities in Figure 2.
The asymptotic translation traction model was fitted by
Fx = Fa [ 1-e (- Dx/Da) ] (1)
The residual translation traction model was fitted by
Fx = Fr [ 1 + { 1/ (Kr ( 1-1/e ))- 1 } e (1 – Dx/Dp) ] [1 – e (- Dx/Dp ] (2)
Where the quantity Fx was the traction force (dependent variable) at any slide distance Dx (independent variable). The parameters
for the asymptotic model were the asymptotic traction force Fa, and the asymptotic slide distance Da to where the asymptotic
force line Fa intersected the tangent to the origin of the curve. The parameters for the residual model were: the residual traction
force Fr, the residual ratio Kr equal to the residual traction force divided by the peak traction force (Fr/Fp), and the slide distance
to the peak traction force Dp. Each of the parameters depended on the materials used by the footwear/surface combination and the
nature of sliding mechanism at their interface.

0
Force (N)
-50
-100
Fp Fa Fr
-150
Da Elasto-plastic model
Dp
-200
0 5 10 15 20
Slide distance Dx (mm)
Residual model
Asymptotic model
Elasto-plastic model
Figure 2: The translation traction-displacement residual and asymptotic exponential models for a footwear/surface combination
showing all defined quantities in Equations (1) and (2).
The rotation traction torque-angular displacement relationship were characterised by the same two expressions with the quantities,
Fx replaced by Tz, and Dx by θz. The parameters, Fa replaced by Ta, Da by θa, Fr by Tr, Dp by θp, and Kr would equal Tr/Tp.
Therefore, the rotational exponential functions would be as follows,
Asymptotic rotation traction model was given by
Tz = Ta [ 1-e (- θz/θa) ] (3)
Residual rotation traction model was given by
Tz = Tr [ 1 + { 1/ (Kr ( 1-1/e ))- 1 } e (1 – θz/θp) ] [1 – e (- θz/θp ] (4)
The elasto-plastic model shown in Figure 2 can approximate these models. For the asymptotic model in translation:
For rotation they would be:
Fx = (Fa / Da) Dx for (0 < Dx < Da), and (5)
Fx = Fa for (Dx ≥ Da) (6)
Tx = (Ta / θa) θx for (0 < θx < θa), and (7)
Tx = Ta for (θx ≥ θa) (8)
The ratio Fa/Da is the stiffness of the footwear/surface combination having units of N/mm. The rotation model stiffness was Ta/θa
having units equal to Nm/deg.
CONCLUSIONS
The exponential curve fitting procedures developed were applied to traction data to produce two measurable parameters to
compare translational and rotational traction and stiffness on a variety of surfaces. These two parameters provide a more valid and
reliable explanation of the mechanism by which traction at the shoe-surface interaction is generated.
REFERENCES
Barry, EB, Kummer, R, Milburn, PD (2000) in The Engineering of Sport. Oxford, England: Blackwell Scientific, 103-112.
Czichos, H (1986) in Friction and Wear of Polymer Composites, Amsterdam; Elsevier, 1-11.
Suh, NP, Sin, HC (1981) The genesis of friction. Wear, 69, 91-114
Valiant, GA (1987) in Proc Int Symp Biomech Sports III & IV., 29-37.
Valiant, GA (1994) in Proc Eighth Canadian Soc Biomech. Conf. Calgary, 326-327.
Van Gheluwe, B, Depone, E, Hebbelink, M (1983) in Biomechanical Aspects of Sports Shoes and Playing Surfaces. University of
California, 161-168.
Wong, JY (1989) Terramechanics and Off-Road Vehicles. Amsterdam: Elsevier