Sports Engineering

, Volume 11, Issue 4, pp 201–206

Optimization of handheld gauge sizes for rocker measurements of skate blades and bobsleigh runners

Authors

    • Department of Physics and AstronomyUniversity of Calgary
  • Sean Maw
    • Department of Mathematics, Physics and EngineeringMount Royal College
  • Darren Stefanyshyn
    • Faculty of KinesiologyUniversity of Calgary
  • Robert I. Thompson
    • Department of Physics and AstronomyUniversity of Calgary
Original Article

DOI: 10.1007/s12283-009-0019-2

Cite this article as:
Poirier, L., Maw, S., Stefanyshyn, D. et al. Sports Eng (2009) 11: 201. doi:10.1007/s12283-009-0019-2
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Abstract

This work uses numerical methods to investigate the feasibility of modifying an instrument used in speed skating to analyze blades from four different ice sports. The instrument, a handheld rocker gauge, is adapted to create a device that can effectively profile other types of skate blades and bobsleigh runners. Since there are significant differences between short and long-track blades one could expect a difference in the gauges used to study these blades. Despite this expectation, the same gauge is used in both disciplines. The usefulness of these gauges has been proven in speed skating so it is expected that they should also be useful to study hockey blades and bobsleigh runners. To optimize the gauge size for different blade types we digitize the profile of a blade, which we use to simulate gauge data. Then we use that gauge data to reconstruct the profile and compare it to the original digital profile. The result is compared for various gauge sizes and the gauge size is optimized for each of the four disciplines. The only commercially available device seems optimal for bobsleigh and long track speed skating. Smaller gauges are recommended for analyzing short track speed skates and hockey skates.

Keywords

BobsleighSkatingBladeOlympicSportFrictionIceRocker

PACS

02.60.Jh02.60.Pn02.70.-c

1 Introduction

By its very nature, performance in sliding and skating sports is inherently dependent on the interaction of the metal blade with the ice. As such, success in these activities becomes dominated by the combination of athletic performance and optimized technology. Equipment modifications can significantly affect the frictional interaction between metal and ice, thus affecting performance, particularly in timed sports. For this reason, top athletes in bobsleigh and skeleton have spent a considerable amount of time and effort in optimizing their equipment [13]. Some research has been published on the effects of the runner materials [4] but for the most part research in bobsleigh and skeleton is a trade secret of top athletes and is not shared. Recent rule changes in these sports [5] have limited the construction of their blades, called runners, to one standardized steel. This has focused runner developments on the profile of the runners.

The goal of our interdisciplinary research group is to further our understanding of the interaction of ice with metal blades, specifically in the sport of bobsleigh. We are in the process of developing a thermodynamic model that will calculate the coefficient of friction between metal and ice given a blade’s physical properties and the ice conditions. The model calculates the dynamic coefficient of friction by determining the equilibrium position of the blade on the ice surface, calculating the plowing force as well as the shear force caused by the thin liquid film between the blade and ice. It also models the thermodynamic processes that affect the thickness of that film. The realization that the exact profile of the blade had significant consequences on performance both in the actual sport and in calculations by our model have led us to catalog various runner profiles. This work is an analysis of the device that we chose to use in the runner analysis as well as an extrapolation to other ice sports.

We chose a device that could be used while competing around the world. This device is a modified handheld rocker gauge that is currently used in speed skating. Rocker is the radius of curvature along the plane of the blade or how round the surface of the blade is lengthwise. The larger the rocker measurement is, the flatter the surface. The application of a device, designed for speed skating, to the sport of bobsleigh brought about the idea to optimize the device’s dimensions for each discipline. The design of the device offers a compromise between spatial resolution, the distance over which profile variations can be resolved, and rocker measurement resolution. Therefore, a given gauge should be most valid for a certain range of rocker values. Since the analysis of the blades used in each discipline will have different requirements, it is reasonable to expect that they will each require their own gauges. This paper addresses the optimization of the gauge size for four athletic disciplines using real blade profile data and numerical optimization methods.

We define the profile h(x) as the height of the blade’s surface as a function of the position along the length of the blade as illustrated in Fig. 1. In order to study the effect of blade and runner profiles on performance, it is necessary to accurately measure these profiles both in the laboratory and at athletic venues around the world. For this purpose speed skaters use a portable handheld rocker gauge shown in Fig. 2.
https://static-content.springer.com/image/art%3A10.1007%2Fs12283-009-0019-2/MediaObjects/12283_2009_19_Fig1_HTML.jpg
Fig. 1

Profiles of the four types of blades examined in this study. From top to bottom, hockey, short track, long track and bobsleigh. The same 15 cm calibration bar is in each photo for a comparison scale

https://static-content.springer.com/image/art%3A10.1007%2Fs12283-009-0019-2/MediaObjects/12283_2009_19_Fig2_HTML.jpg
Fig. 2

A photograph of an individual taking gauge data on a bobsleigh runner

The rocker is not measured directly; it is calculated from the gauge data. The rocker gauge works using a 3 point measurement of the curvature of the blade. The gauge consists of a fixed pin at each end, with the two pins separated by a distance of 2a. These pins are placed against the surface of the blade. Halfway between the two pins, a micrometer measures a deviation g(x) relative to a flat line joining the two pins. This type of gauge is currently used in both short and long track speed skating and is supplied by at least three companies; Maple, 1 Marchese,2 and Shoei Creations. 3 In principle, a similar device could also be used in bobsleigh and hockey. However since the radius of curvature for blades and runners in each of these sports is quite different 4, we hypothesize that the gauge size for each discipline should be different as well. Despite the use of rocker gauges in both short and long track speed skating only one gauge size, a half width of a = 50 mm, currently exists on the market.

In the sport of skeleton, the rocker of a runner can be changed simply by applying tension to the runner. The four disciplines we are studying have blades in which the profile is not easily altered. In bobsleigh, the runners are cut to profile and ideally do not change. The runners are polished individually by hand with many different grades of sandpaper. If we look down the length of the runner we observe the width of the blade to be spanned by a convex surface with a radius of no less than 4 mm for 2-man and 6 mm for 4-man bobsleigh. Due to the width of the surface to be polished, the profile does not change a great deal. That said, over time and with hours of work, the profile can vary from its original state. There is no effort made by athletes to ensure that the original profile is maintained. Furthermore, since runners are often copied from a set that produces successful results, the deviations that develop over time can be reproduced in a new set of runners.

A typical example of bobsleigh runner gauge data is shown in Fig. 3. This data illustrates the blade profile variations described in the previous paragraph. In this case, the rocker is mostly constant through the middle of the blade. An exception is seen with a flatter spot near 100 mm which is accompanied by an area of smaller rocker or larger gauge data at 200 mm. This is a classic example of the modification of a profile over time which has then been reproduced in the manufacturing of a new set of runners.
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Fig. 3

Gauge data from the rear runner of one of two types of 2man runners supplied by the F.I.B.T (the international body responsible for bobsleigh and skeleton)

Speed skates are sharpened by putting a pair of skate blades in a jig such that the gliding surfaces of the blades face upwards. Smooth diamond or sand stones are then pushed back and forth along the length of the blades to polish the running surfaces and to form sharp 90° edges. Symmetry of the blades, and the consistency of the grinding forces applied to them along their length, is very important for quality sharpening that does not alter the profile or rocker. The process for (re)rockering blades is the same, except that forces are now varied along the length of the blades. For example, heavier grinding in one area of the blades will flatten that area while rounding adjacent areas. All polishing and rockering is performed manually.

Hockey blades are sharpened and rockered one skate at a time as a sharpener holds the boot and presses the blade up against a rotating grinding wheel. The degree of rockering is generally not monitored, and the rockers between two skates are often quite different as a result. Unlike the flat profile of the speed skating blade’s running surface, a hockey blade’s is concave across the width such that the skater essentially glides on two edges per blade when standing upright.

It should be noted that the most complete method of determining information about a blade is not with a handheld rocker gauge, but via direct measurement of the blade’s profile. A direct measurement of the profile requires a relatively large and costly apparatus with very rigid construction restraints. Such a device uses a micrometer that can travel the entire length of the blade. Our group has access to such a device that was designed and used for skate blade analysis. Since the bobsleigh runners are longer than skate blades, the device we have cannot accommodate the runners. An instrument capable of analyzing bobsleigh runners was found at a precision machining shop and we have access to a limited number of digitized profiles from the device for this study. To catalogue runner data in the laboratory and at athletic venues around the world we chose to adapt the handheld gauge used in speed skating. The adaptation was to simply remove the lip used to steady the device against the side of a skate blade. The lip blocks the micrometer on the wider rounded bobsleigh runners. For athletes traveling for competitions, the size and cost of a direct measurement apparatus is not practical. Therefore, the hand gauge that is widely used in the sport of speed skating should be adapted for use in bobsleigh and hockey as well.

2 Theory

The analysis of the hand gauge to optimize its size begins with a direct profile measurement and the assumption that this digital profile is the actual blade or runner we wish to recreate. Using this digital profile we simulate gauge data by applying a gaussian distribution of uncertainty comparable to the measurement uncertainty observed in practice while using the real hand gauge. We then use the simulated gauge data to recreate the profile iteratively from the center of the runner or blade out to each end. Finally, we compare the reconstructed profile with the original profile measurement. Due to the random nature of the uncertainty, the process was automated so that it could be repeated 1,000 times and then averaged for each gauge width.

2.1 Profile analysis

The hand gauge provides a measurement that represents the curvature along the length of the skate blade or bobsleigh runner. Figure 4 demonstrates how we calculate rocker data from the gauge measurements. The two outside points are fixed and a micrometer takes a reading g(x) in the middle. The gauge has a half width of a. We should point out that the curvature in Fig. 4 is an exaggeration; in reality a/R ∼ 10−2 and this approach works well with very small angles.
https://static-content.springer.com/image/art%3A10.1007%2Fs12283-009-0019-2/MediaObjects/12283_2009_19_Fig4_HTML.gif
Fig. 4

Representation of rocker measurements taken with the hand gauge

Assuming that we are working with small angles (Eq. 1) gives us the gauge data g(x) from the known profile h(x). The measurement uncertainty is represented by δ; it is a value between ±5σ and it follows a normal distribution. The σ value is the standard deviation of gauge data as observed in practice. The uncertainty observed in practice appears to be constant for most values of g(x). It does, however, increase for very large values of g(x) that can be observed at the ends of the blades. For simplicity, this study has used a constant value of σ for each simulation. The horizontal position along the blade or runner where the micrometer measurement is taken is x.
$$ g(x)=h(x)-{\frac{h(x+a)+h(x-a)} {2}}+\delta . $$
(1)
In order to make use of the gauge data, we make the assumption that the radius of curvature within the width of the gauge is a constant value R(x). This allows us to define Eq. 2.
$$ R(x)={\frac{a^2+g^2(x)} {2g(x)}} . $$
(2)
A larger gauge width results in a larger measurement and therefore a smaller relative uncertainty in g(x). However, the larger width also leads to a greater loss of information through low pass filtering [6] due to the constant radius assumption. This is an ever present trade off and by optimizing the gauge size we should be able to reduce the total information loss from both sources.

2.2 Profile reconstruction

The reconstruction of the new profile hr(x) is an iterative process that requires comparing successive data points. To reflect the spatial resolution of the data collected with the handheld rocker gauge, Δx, we express Eq. 1 without the measurement uncertainty and with Δx replacing the gauge half width a, resulting in Eq. 3.
$$ g_{\Updelta x}(x)=h_r(x)-{\frac{h_r(x+\Updelta x)+h_r(x-\Updelta x)} {2}} $$
(3)
In speed skating, gauge measurements are taken by continuously moving the gauge along the length of the blade and for this study we will assume the same type of analysis for hockey skates as well. The spatial resolution Δx is the distance along the blade over which we can differentiate between gauge readings. Since we are cataloging data and the bobsleigh runners are significantly longer than skate blades we take discrete measurements at fixed points for the bobsleigh runners. In that case Δx is the distance between measurements. To simplify the reconstruction algorithm we chose Δx to be an integer fraction of the gauge half width a. By observing Fig. 4 we can note that Eq. 3 can also be written as Eq. 4.
$$ g_{\Updelta x}(x)=R(x)-\sqrt{R^2(x)-(\Updelta x)^2} $$
(4)
Now by equating Eqs. 3 and 4 we arrive at Eq. 5.
$$ h_r(x)-{\frac{h_r(x+\Updelta x)+h_r(x-\Updelta x)} {2}}= R(x)-\sqrt{R^2(x)-(\Updelta x)^2} $$
(5)
At this point, Eq. 5 can be re-arranged into the forward or backwards recursive relationship, both described in Eq. 6.
$$ h_r(x \pm \Updelta x)=2h_r(x)-h_r(x \mp \Updelta x)-2\left(R(x)-\sqrt{R^2(x)-(\Updelta x)^2}\right) $$
(6)
Once the gauge data g(x) is collected along the length of the runner we note that reconstructing the profile hr(x) from curvature data is effectively a second order integration. Therefore, it requires two data inputs, similar to integration constants, that set the position and orientation of the blade in space. We set hr(−Δx) = hrx) = 0, which sets the position and slope of the blade about the center to zero. Then we can solve Eq. 6 for x = 0, as shown in Eq. 7.
$$ h_r(0)=R(0)-\sqrt{R^2(0)-(\Updelta x)^2} $$
(7)
We now have all the information required to calculate the entire runner profile iteratively from the center out to each end of the blade or runner using Eq. 6.

2.3 Profile comparison

Once we have reconstructed the profile, both profiles h(x) and hr(x) must be reoriented so they overlay for direct comparison. While it would be possible to simply adjust the original profile to hx) = h(−Δx) = 0, that would put a greater emphasis on matching of the center of the profile. In this work, the overlay has been balanced over the entire range by taking a linear regression of both profiles and readjusting it to set the slope and intercept of each profile to zero.

Now that the orientations of the two profiles are matched we can compare them using α, a measure of the accuracy of the reconstruction that is based on a statistical chi-squared type of measurement. The major difference between this and a real chi-square test is in the uncertainty. We do not have the real uncertainty of our profile data points so we use σ. It is the standard deviation as recorded from experimental gauge data and the same value that is used to calculate δ.
$$ \alpha={\frac{1} {N}}\sum_{j=1}^{N}{\frac{(h(x_j)-h_r(x_j))^2} {\sigma^2}} $$
(8)
Due to the random nature of the uncertainty the process is repeated 1,000 times and averaged α values are used. Minimization of α occurs when the two curves are best matched.

3 Results and discussion

The gauge width is optimized by taking a digital profile and minimizing the averaged α value for various profiles of each blade type and averaging the result. In Table 1, we have shown some of the results from the averaged α calculations for five pairs of long track speed skating blades. It should be noted that although Table 1 only shows some typical data as an example of the results of these calculations, the additional data that was not shown has effectively the same averages as the data that was displayed. The data from Table 1 are calculated with a spatial resolution of Δx = 5 mm and with a standard deviation for the gauge data of σ = 1.5μm.
Table 1

Optimizing the half width of a rocker gauge for seven different long track speed skates by minimizing α values

a (mm)

1

2

3

4

5

6

7

30

1.70

1.58

2.18

2.07

1.44

3.00

2.40

35

1.03

0.90

1.70

1.74

0.84

2.47

1.75

40

0.71

0.61

1.50

1.64

0.57

2.36

1.44

45

0.62

0.48

1.45

1.63

0.39

2.47

1.31

50

0.60

0.37

1.51

1.82

0.35

2.77

1.21

55

0.62

0.34

1.64

2.13

0.33

3.12

1.13

65

0.64

0.30

2.46

3.26

0.44

4.28

1.39

70

1.02

0.69

4.93

7.27

1.00

5.57

6.37

Minimums are in bold font

For the complete analysis of each gauge many such tables were generated covering a broader spectrum of parameters. For this broader analysis we have included spatial resolutions from 3 to 10 mm for the three types of skate blades, 10–25 mm for the bobsleigh runners and standard deviations from 1 to 2 μm for all. We are confident that this parameter space fully encompasses the uncertainty in these parameters and thus properly reflects our uncertainty in the optimum gauge sizes themselves. However, that uncertainty could possibly be narrowed by observing a larger number of blades or runners. This analysis yields the optimum gauge sizes to analyze hockey skates, short and long track speed skates as well as bobsleigh runners. These results are summarized in Table 2.
Table 2

Recommended gauge sizes to reduce information loss with a handheld rocker gauge

Blade type

Gauge 1/2 width (mm)

Hockey

10 ± 3

Short track

25 ± 5

Long track

50 ± 10

Bobsleigh

50 ± 20

4 Conclusions

The handheld rocker gauge is a very useful instrument for blade analysis in speed skating. This work suggests a similar gauge could be used for hockey and bobsleigh as well. Numerical analysis was used to minimize the loss of information from micrometer resolution and low pass filtering, thus optimizing the size of the rocker gauges for all four disciplines. The gauge currently on the market, 5 cm half width, was found to be a good gauge for long track speed skating but we suggest that a gauge of only half that size should be used to analyze short track blades. Our group has used a modified 5 cm gauge to analyze bobsleigh runners with success and this study shows that that gauge size is appropriate to study bobsleigh runners. We have also found that a gauge with a 1 cm half width could be used to effectively study hockey skates. We are already aware that certain equipment works differently in certain conditions (ie: cold vs. warm weather). We hope that by analyzing this equipment with the gauge and comparing results with our thermodynamic model, we will be able to improve our understanding of ice friction as it relates to these sports.

Footnotes
4

hockey 5–6 m, short track 6–12 m, long track 18–30 m, bobsleigh 10–150 m.

 

Acknowledgments

We would like to thank NSERC and the Calgary Olympic Oval for their support.

Copyright information

© International Sports Engineering Association 2009