Ice Friction in the Sport of Bobsleigh

Home , Bobsleigh, Bobsleigh World Cup, Ice skating, Kristan Bromley, Pierre Lueders

UNIVERSITY OF CALGARY

Louis Poirier

A THESIS

SUBMITTED TO THE FACULTY OF GRADUATE STUDIES

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE

DEGREE OF DOCTOR OF PHILOSOPHY

DEPARTMENT OF PHYSICS AND ASTRONOMY

CALGARY, ALBERTA

August, 2011

°c Louis Poirier 2011 UMI Number: NR81854

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FACULTY OF GRADUATE STUDIES

The undersigned certify that they have read, and recommend to the Faculty of Graduate

Studies for acceptance, a thesis entitled “Ice friction in the sport of bobsleigh” submitted by Louis Poirier in partial fulﬁllment of the requirements for the degree of DOCTOR OF

PHILOSOPHY.

Supervisor, Dr. Robert I. Thompson Dr. Sean Maw Department of Physics and Department of Math, Physics, and Astronomy Engineering Mount Royal University

Co-Supervisor, Dr. Darren J. Stefanyshyn Dr. Walter Herzog Faculty of Kinesiology Internal-External Examiner Faculty of Kinesiology

Dr. Edward P. Lozowski Department of Earth and Dr. Robert M. Frederking Atmospheric Sciences External Examiner University of Alberta National Research Council Canada

Date Abstract

The primary objective of this work is to examine the effect of the bobsleigh runner profile on ice / runner friction. The work is centered on a computational model (F.A.S.T. 3.2b) which calculates the coefficient of friction between a steel blade and ice.

The ﬁrst step was to analyze runners used in the sport of bobsleigh. This analysis was performed using a handheld rocker gauge, a device used in speed skating. The size of the device was optimized for hockey, short and long track speed skating, and bobsleigh.

A number of runners were measured using the gauge and it was found that the portion of the runner contacting the ice generally has a rocker value of (20 − 50) m.

Next, the hardness of athletic ice surfaces was analyzed. The ice hardness was determined by dropping steel balls varying in mass from (8 − 540) g onto the ice surface, from a height of (0.3 − 1.2) m, and measuring the diameter of the indentation craters. The ice hardness was found to be P¯(T ) = ((−0.6 ± 0.4)T + 14.7 ± 2.1) MPa and the elastic recovery of the ice surface was found to be negligible.

The F.A.S.T. model was adapted from a speed skate model to calculate the coefficient of friction between a bobsleigh runner and a flat ice surface. The model predicts that maximum velocities are obtained for temperatures between −10 and −20◦C, in agreement with observations on the Calgary bobsleigh track. The model for flat ice suggests that the flattest runners produce the lowest coefficient of friction and that the rocker affects friction more than the cross-sectional radius.

The coeﬃcient of friction between runners and ice and the drag performance of 2- men bobsleighs were determined from radar speed measurements taken at the Calgary

Olympic Oval and at Canada Olympic Park: at the Ice House and on the bobsleigh track during a World Cup competition. The mean coeﬃcient of friction was found to be

−3 2 µ¯ = (5.3 ± 2.0) × 10 and the mean drag performance was CdA = (0.18 ± 0.02) m .

ii Acknowledgements

First, I would like to thank NSERC for the scholarship that allowed this project to get started and for their continued ﬁnancial support. I acknowledge support from Bobsleigh

Canada Skeleton. They provided a radar gun, runner access, and support for my trip to the World Championships in K¨onigssee. Their interest in my work, and the potential that

Canadian athletes can beneﬁt from it, make it worth while. I would like to acknowledge support from the Canadian Sport Centre Calgary for the use of their radar gun and all their support during my years as an athlete. I would like to thank the Calgary Olympic

Oval for facilities access, and access to employees, their expertise, and data from the ice surfaces. I would like to thank WinSport Canada for facilities access at Canada

Olympic Park and access to the technical drawings of those facilities. I would like to thank Alberta Bobsleigh for access to their temperature probe. I would like to thank

Foothills Bobsleigh Club for access to their equipment. I would like to thank all the people from the F´ed´erationInternationale de Bobsleigh et Tobogganing (F.I.B.T.) for supporting our work and allowing us access to the track and jury during the 2010 − 11

Calgary World Cup competition. I would like to thank Corma Enterprises for putting so much into our runner development project. I would like to thank the University of

Calgary for all of its ﬁnancial and infrastructure support. I would like to thank the Human

Performance Lab at the University of Calgary for access to their radar gun and I would like to thank Matt Nykolichuk for allowing me to use his speed skates for photographs.

I would like to thank the Department of Physics and Astronomy. First, I would like to thank Peter Gimby and the late Hugo Graumann for their help, their ideas, and access to the undergraduate laboratory equipment. Next, I would like to thank, the ladies from the main oﬃce who made every day more enjoyable. Finally, for being a top notch department, assuring support for students, and allowing us to do our work; I thank

iii everyone. It shows in our productivity and we are the envy of graduate students around campus.

For helping with my ice friction experiments at the bobsleigh track during the World

Cup, I would like to thank my volunteers: Matt Bumstead, Ian Cockerline, Nick De-

Ruyter, Julie Grant, Samuel George, Amy Johnston, Mike Kwiatkowski, Stephen Lane,

Alexis Morris, Allison Rubenok, Lohrasp Seify, Joshua Slater, Terence Stuart, and Wolf- gang Tittel. This really was a team eﬀort. The experiment would not have been possible without all of your great work. I know that the hot chocolate wasn’t enough payment or even enough to keep away the cold. Thankfully, you all have a great passion for science and sport.

I would like to thank Chris Spring and his team for the use of his truck to move bobsleighs to and from the Olympic Oval and for help with runner preparation. I would like to thank Adam Anderson for the polishing equipment. And thanks to all of those whom helped me polish runners; Adam Farley, Chris Gudzowski, Cliﬀ Iwassa, Rosalyn

Nykolichuk, Heather Patterson, Patrick Riley, Efrem Violatto, and Alex Torbert and his team. I’m sure I’ve missed both people and services provided by the local bobsleigh community. They have blown me away with all of their help and support.

I would like to sincerely thank my committee for all their help and for indulging my project. I realize that none of you really had any interest in bobsleigh research before

I came along. I thank you for having faith in my abilities and supporting my work. I would ﬁrst like to thank Ed. Your enthusiasm and constant desire to get out and make measurements inspires me. You are very a passionate scientist and I hope I can be as passionate about my work when I retire. That would be a truly blessed life. Sean, I would like to thank you for being so meticulous. You never satisfy yourself with showing me something but you take the time to teach me. Your students are very lucky. Darren, I would like to thank you for keeping me on track. I sometimes let my ideas run all over the

iv place, especially in the beginning. Thank you for helping me ﬁnd what was important and stay focused on that. Rob, it’s been eight years since I ﬁrst came to Calgary to bobsleigh. I enjoyed that career but I have enjoyed this one even more. We have been through a lot since then in both our careers and our lives. I feel truly blessed to have worked with you through so much and I hope that we can continue to work together for many years to come.

I would like to thank my family back home in the Maritimes. I miss you all a great deal. Hopefully, with real jobs, Amy and I will be able to visit more often.

Finally, for the hours spent reading and helping to edit my work, and for the countless hours helping me while freezing at the Oval, the Ice House, and the bobsleigh track, I am eternally grateful to my wife, Amy Johnston. For all your help and all your support

I would like to thank you from the bottom of my heart. I really could not have done this without you. Je t’aime ma belle.

v Table of Contents

Abstract ...... ii Acknowledgements ...... iii Table of Contents ...... vi List of Tables ...... viii List of Figures ...... ix 1 INTRODUCTION ...... 1 1.1 The Sport of Bobsleigh ...... 1 1.1.1 The Sled ...... 1 1.1.2 Basic Physics in Bobsleigh ...... 2 1.1.3 Canadian History in Bobsleigh ...... 4 1.2 Motivation ...... 4 1.3 Background ...... 5 1.4 Thesis Outline ...... 8 2 EQUIPMENT ANALYSIS ...... 10 2.1 Introduction ...... 10 2.2 Motivation for Equipment Survey ...... 11 2.3 Gauge vs. Direct Profile Measurements ...... 13 2.4 Analysis of the Handheld Rocker Gauge ...... 13 2.4.1 Theory ...... 19 2.4.2 Results and Discussion ...... 23 2.4.3 Rocker Gauge Summary ...... 25 2.5 Profile Analysis ...... 26 2.6 Equipment Survey ...... 36 2.7 Bobsleigh Runner Design Project ...... 40 2.8 Summary ...... 45 3 ICE HARDNESS ANALYSIS ...... 47 3.1 Background ...... 47 3.2 Experimental Protocols for Ice Hardness Measurements ...... 52 3.3 Dynamic Ice Hardness Data and Analysis ...... 61 3.4 Conclusions ...... 69 4 F.A.S.T. MODEL ...... 70 4.1 Introduction to Ice Friction ...... 70 4.1.1 Introduction to the Model ...... 71 4.1.2 F.A.S.T. 1.5 ...... 71 4.1.3 Extensions ...... 72 4.2 Development of the Theory ...... 81 4.2.1 Contact Between the Runners and the Ice ...... 81 4.2.2 Ploughing Force ...... 85 4.2.3 Pressure Effect on Melting Point ...... 86 4.2.4 Couette Flow ...... 86 4.3 Simulated Data and Analysis ...... 96

vi 4.4 Limitations of the F.A.S.T. Model ...... 110 4.5 Conclusions ...... 111 5 ICE FRICTION EXPERIMENTS ...... 113 5.1 Introduction ...... 113 5.2 Ice Friction Experiments ...... 115 5.2.1 Low Speed ...... 115 5.2.2 High Speed ...... 119 5.3 Theory ...... 124 5.4 Results ...... 126 5.5 Discussion and Analysis ...... 129 5.6 Comparison to F.A.S.T. Model Results ...... 134 5.7 Conclusions ...... 137 6 THESIS SUMMARY ...... 139 6.1 Runner Analysis ...... 139 6.2 Ice Analysis ...... 140 6.3 F.A.S.T. Model ...... 140 6.4 Experimental Ice Friction and Air Drag Measurements ...... 141 6.5 Future Work ...... 143 6.6 Summary ...... 143 A Supplementary Information on Equipment Analysis ...... 144 A.1 Rocker Gauge Optimization ...... 144 A.2 Bobsleigh Runner Proﬁles ...... 153 B Catalog of 2-men Bobsleigh Runner Proﬁles ...... 155 C Supplementary Information on Ice Hardness ...... 161 C.1 Integration Volume ...... 161 C.2 Crater Diameter ...... 161 C.3 Consistency of Ice Hardness Data ...... 164 D Supplementary Information on the F.A.S.T. Code ...... 168 D.1 F.A.S.T. 1.5 Code ...... 168 D.2 F.A.S.T. 3.2b Code ...... 187 D.3 Supplementary F.A.S.T. Model Results ...... 208 E Supplementary Information on Ice Friction Analysis ...... 214 E.1 Calculation of Mean Square Velocity for Air Drag Calculation ...... 214 E.2 Determination of the Slope in the Ice House ...... 214 E.3 Supplemental Lossy Acceleration Data ...... 217 E.4 Surveyor’s Reports ...... 219 E.5 Analysis of the Surveyor’s Reports ...... 228 Bibliography ...... 231

vii List of Tables

2.1 Optimizing the half width of a rocker gauge (a) for 7 different long track speed skates by minimizing average α values. Minimums are in bold font. 24 2.2 Recommended gauge sizes to reduce information loss ...... 25 2.3 For the runner design process each runner is divided into five sections along its length. Subscripts f and r designate the front or rear runners and t and h designate the toe or heel of a given runner...... 43 2.4 This table includes values for several parameters found in Table 2.3. These values help to define the five sections used in the runner design process. . 43 2.5 Parameters for equations 2.10 to 2.14 for the designed 2-men runner profile. 44

3.1 Mass and radius of the balls used for ice hardness experiments ...... 60

5.1 Parameters from a quadratic regression of the data illustrated in Fig. 5.11. 133

E.1 Acceleration of the sled up and down the slope when pushed forward . . 217 E.2 Acceleration of the sled up and down the slope when pushed in reverse . 217 E.3 Parameters from a quadratic regression for three of the sled masses used. 217 E.4 Determination of the mean slope between corners 1 and 2 at the Calgary Olympic bobsleigh track...... 228 E.5 Determination of the mean slope between corners 3 and 4 at the Calgary Olympic bobsleigh track...... 229 E.6 Determination of the mean slope between corners 5 and 6 at the Calgary Olympic bobsleigh track...... 229 E.7 Determination of the mean slope between corners 8 and 9 at the Calgary Olympic bobsleigh track...... 230

viii List of Figures

1.1 Photograph of a 2-men sled with the pilot (the author of this thesis) and the brakeman taken during a training session in Whistler, BC in 2008. . . 2 1.2 A diagram of the forces exerted on a bobsleigh descending a straight section of track with a constant slope, θ...... 3

2.1 Images 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...... 14 2.2 A photograph of an individual taking gauge data on a bobsleigh runner. . 15 2.3 Gauge data from the rear runners of one of two types of 2-men runners supplied by the F.I.B.T. (the international body responsible for bobsleigh and skeleton). Squares represent data from the right runner while triangles are from the left...... 17 2.4 Representation of rocker measurements taken with the hand gauge. The gauge pins rest at the top of the figure in the corners of the large isosceles triangle and the micrometer rests in the centre measuring g(z). The gauge half width is a and the rocker or radius value is R(z)...... 20 2.5 Profile of a long track speed skate blade, measured directly...... 27 2.6 Profile of a rear 4-men bobsleigh runner, measured directly...... 28 2.7 Zoomed in profile data from a rear 4-men bobsleigh runner, measured directly...... 29 2.8 Calculated gauge data derived from the profile measurements of a long track speed skate blade...... 30 2.9 Calculated gauge data derived from the profile measurements of rear 4-men bobsleigh runner...... 31 2.10 Calculated rocker data derived from the profile measurements of a long track speed skate blade...... 33 2.11 Calculated rocker data derived from the profile measurements of a rear 4-men bobsleigh runner...... 34 2.12 Measured gauge data from one 2-men front bobsleigh runner (circles). The runner profile and the corresponding gauge data were calculated numerically using the measured gauge data taken every 5 mm (triangles), 25 mm (squares), and 50 mm (X’s)...... 35 2.13 Measured gauge data from two separate sets of DSG front 2-men bobsleigh runners. First set, right - cross, left - circle and second set, right - square, left - triangle...... 37 2.14 Measured gauge data from a set of BCS Gesuito09 front 2-men bobsleigh runners, right - square, left - triangle...... 38 2.15 Measured gauge data from a set of LG front 2-men bobsleigh runners, right - square, left - triangle...... 39

ix 2.16 Measured gauge data from two sets of DSG front 2-men bobsleigh runners (circles) and the designed front runner profile (solid line - centre section of the runner and dotted line - front and rear sections of the runner). . . 41 2.17 Diagram of the bobsleigh runner with the axes and the five sections used for our 2-men runner construction project. The figure is not to scale. . . 42

3.1 Ice hardness for various times scales and ice temperatures. The points are taken from the line of best fit from Barnes and Tabor [1], while the lines are fitted using equation 3.1. From top to bottom the data points are for time scales of 10−4 s, 1.5 s, 10 s, 103 s, and 104 s...... 50 3.2 Thin sections of the ice to show the macrocrystalline structure at the long track ice surface at the Calgary Olympic Oval. a) vertical section transverse to the skating direction, b) horizontal section. The white bars are 1 cm scales...... 53 3.3 Drop apparatus for ice hardness experiments. A ball of mass m is rolled off a platform set to a height of hi onto the ice surface leaving an indentation crater in the ice. The ball rebounds to a height of hf and out of the indentation crater...... 54 3.4 Apparent indentation crater volume vs. impact energy used to determine ice hardness. Analysis of the raw data from Table 1 of Timco and Fred- erking [2]. Indenters of mass (33 − 64) kg with hemispherical tips were dropped from heights of (0.3 − 1.4) m...... 56 3.5 An impact crater (upper left) at the bobsleigh track in Canada Olympic Park made on December 16th, 2009. Ball C was dropped from a height of 30 cm in favorable light conditions...... 58 3.6 An impact crater from the long track rink of the Calgary Olympic Oval made on December 30th, 2009. Ball C was dropped from a height of 90 cm in poor light conditions...... 59 3.7 Plot of ice crater volumes vs. impact energy of dropped balls to determine ice hardness. Data from drop testing experiments performed on Feb. 10, 2010 in the Ice House at Canada Olympic Park. The ice surface temperature was −2.7◦C...... 61 3.8 Ice hardness vs. ice surface temperature as measured on the bobsleigh track (squares) and the ice house (X) at Canada Olympic Park, as well as all three ice surfaces at the Calgary Olympic Oval (diamonds) for ice surface temperatures greater than -5◦C...... 63 3.9 Ice hardness vs. ice surface temperature from various ice surfaces at Can- ada Olympic Park and the Calgary Olympic Oval. The weighted linear fit and the confidence limits are the central and two outer lines respectively. 64

x 3.10 Examination of the indentation track left by a bobsleigh runner on a level ice surface, major axis or rocker of 34 m minor axis of 4.75 mm. The blue arrow is the calibration scale, the pink arrows are measurements of the groove width. The sled had a weight of 123 kg and 56% of it is distributed to the rear runners. A Canadian dime was used for scale. It is difficult to precisely determine the groove width. We have made two attempts to measure the width in this figure. The arrow on the left is an attempt to make the largest measurement of the groove. The arrow on the right is an attempt to make the smallest measurement of the groove. Several similar measurements were made of multiple grooves. All the measurements were used to obtain an average result and a standard deviation...... 66 3.11 Plot of the variation between measured ice hardness and the linear fit in equation 3.6 vs. absolute humidity to determine the effect of absolute humidity of air on ice hardness ...... 68

4.1 Diagram illustrating the possible recovery of the ice surface (grey) behind a passing skate blade (black). The arrows indicate the direction of the motion. a) Complete recovery b) No recovery c) Partial recovery . . . . . 75 4.2 On the left an illustration of the geometry of a skate blade, on the right a picture of a bobsleigh runner ...... 77 4.3 The bobsleigh runner causes an elongated, semi-elliptical contact in the ice surface (top) while the skate blade causes a rectangular contact (lower). Impressions are not to scale...... 79 4.4 A diagram of the contact area in the ice surface of the front (top) and rear (bottom) runners in the F.A.S.T. 3.1b code. The figure is not to scale. . . 81 4.5 Schematic of the penetration of the bobsleigh runner into the ice. This figure depicts the geometry of the penetration from either a side view (first parameters) or head-on view (second parameters). Images are not to scale...... 83 4.6 Illustration of the squeeze flow for a section of the runner. The pressure of the runner squeezes the melt layer out the sides, following the arrows. The dashed line corresponds to the minimum melt layer thickness. The figure is not to scale...... 92 4.7 The calculated shear force between the bobsleigh runner and the ice for components of the F.A.S.T. 3.2b model. The blue curve examines the melt layer caused by shear stress only. The green curve includes the conduction of heat into the ice via slow and fast heat conduction. The red curve adds the effect of heat conduction from the runner and the black curve includes the squeeze flow to complete the F.A.S.T. 3.2b model. This is a test of the model to determine the relative significance of each of the terms included in the model. The ice surface temperature was −6◦C...... 97

xi 4.8 The resisting forces on a bobsleigh runner as calculated by the F.A.S.T. 3.2b model. The solid line is the total resisting force. The curved dotted line is the shear stress and the straight dotted line is the ploughing force. The ice surface temperature was −6◦C...... 98 4.9 The resisting forces on a bobsleigh runner as calculated by the F.A.S.T. 3.2b model for sled speeds of a) 16 m/s, b) 26 m/s, and c) 36 m/s. The dotted line is the ploughing force, the dashed line is the shear force, and the solid line is the total resisting force (ploughing + shear)...... 99 4.10 The coefficient of friction between bobsleigh runners and a flat sheet of ice for different sled speeds and ice surface temperatures from -2 to -14◦C. Both the front and rear runners follow in the same track (F.A.S.T. 3.1b). 101 4.11 The coefficient of friction between bobsleigh runners and a flat sheet of ice for different sled speeds and ice surface temperatures from -2 to -14◦C. The front and rear runner leave two parallel tracks (F.A.S.T. 3.2b). . . . 103 4.12 The relative change in coefficient of friction between bobsleigh runners and a flat sheet of ice when the cross-sectional radius of the runner is decreased from 4.75 to 4.0 mm for different sled speeds and ice surface temperatures from -2 to -14◦C. The front and rear runner leave two parallel tracks (F.A.S.T. 3.2b)...... 104 4.13 The relative change in coefficient of friction between bobsleigh runners and a flat sheet of ice when the cross-sectional radius of the runner is increased from 4.75 to 5.5 mm for different sled speeds and ice surface temperatures from -2 to -14◦C. The front and rear runner leave two parallel tracks (F.A.S.T. 3.2b)...... 105 4.14 The relative change in coefficient of friction between bobsleigh runners and a flat sheet of ice when the rocker of the runner is decreased from 34 to 20 m for different sled speeds and ice surface temperatures from -2 to -14◦C. The front and rear runner leave two parallel tracks (F.A.S.T. 3.2b). . . . 106 4.15 The relative change in coefficient of friction between bobsleigh runners and a flat sheet of ice when the rocker of the runner is increased from 34 to 48 m for different sled speeds and ice surface temperatures from -2 to -14◦C. The front and rear runner leave two parallel tracks (F.A.S.T. 3.2b). . . . 107 4.16 The coefficient of friction between bobsleigh runners and a flat sheet of ice when the cross-sectional runner radius is 4.75 mm and the rocker is 34 m for different sled speeds and ice surface temperatures from -2 to -24◦C. The front and rear runner leave two parallel tracks (F.A.S.T. 3.2b). . . . 109 4.17 The maximum (y = 0) thickness of the melt layer along the contact length of the front runner. The horizontal axis has its origin in the centre of the ice / runner contact length (z = 0). The cross-section radius of curvature is 4.75 mm, the rocker is 34 m, the sled speed is 25 m/s, and the ice surface temperature is −7◦C (F.A.S.T. 3.2b)...... 110

xii 5.1 A front view of a) the training sled used in our experiments in the Ice House and Oval and b) a typical 2-men bobsleigh similar to those used in the World Cup bobsleigh competition on December 3rd, 2010...... 117 5.2 A diagram of the experimental setup in the Ice House at Canada Olympic Park. The radar gun was aligned parallel to the motion of the sled on the constant inclined section of the push track...... 118 5.3 Sample of the radar speed data as a function of time. Six runs with different initial velocities were made at the Calgary Olympic Oval. Ice surface temperature was −2.2◦C and no weight was added to the sled...... 119 5.4 An example of the speeds measured in the Ice House at Canada Olympic Park. The points are the measured speed of the sled and the line is v(t) = vi +gt sin θ. The line represents the speed that the sled would have if there were no ice friction or air drag. The ice temperature was −4.4◦C and no weight was added to the sled...... 120 5.5 Diagram of the bobsleigh track at Canada Olympic Park in Calgary, Can- ada. The start is at the top of the figure and the four straight sections of track where the data were acquired are indicated with arrows...... 121 5.6 Radar setup during preliminary testing. The photo is taken from the entrance of corner 4, looking up into the exit of corner 3. In the photo, the radar gun setup is located on the short wall and it is connected to a computer for data acquisition. During the World Cup, the race officials required us to tie the radar guns to the outside of the track so that they could not fall into the path of the competitors...... 122 5.7 Low speed lossy accelerations at the Calgary Olympic Oval (white hollow circles) and at the Ice House (grey filled circles) with linear regression and confidence interval curves. These experiments were conducted over four days. The ice surface temperatures were between −2.2 and −4.6◦C. . . . 127 5.8 Individual sled accelerations due to lossy forces vs mean square sled speed divided by sled mass. The slope of the linear fit determines the mean sled drag coefficient, while its intercept determines the mean ice friction coefficient...... 128 5.9 The difference between the accelerations and the linear fit obtained in equation 5.5. The data is both simulated (squares), and experimental from both the Oval (white hollow circles) and the Ice House (grey filled circles) 131 5.10 Sled acceleration due to lossy forces vs mean square sled speed divided by sled mass. The slope of the linear fit determines the mean drag coefficient for the sleds, while its intercept determines the mean ice friction coefficient.132 5.11 Sled acceleration due to lossy forces vs root mean square sled speed. The linear term in the quadratic regression could represent a variation in the coefficient of friction with speed...... 134

xiii 5.12 Comparison of the linearly increasing coeﬃcient of friction with sled speed ◦ from Table 5.1 and the F.A.S.T. 3.2b results for Ti = −7.6 C. The results calculated from Table 5.1 are in circles with the conﬁdence limits indicated by the dashed lines. The F.A.S.T. 3.2b model results are indicated by the solid black line. The solid red line represents the F.A.S.T. results multiplied by a factor of 3...... 137

A.1 Zoomed-in gauge data from two separate sets of DSG front 2-men bobsleigh runners. First set, right - cross, left - circle and second set, right - square, left - triangle...... 154

B.1 Measured gauge data from two separate sets of DSG 2-men bobsleigh runners. These runners were used in our runner design project. Front runners are in the top figure, rear runners are in the bottom figure. First set, right - cross, left - circle and second set, right - square, left - triangle...... 156 B.2 Measured gauge data from a set of BCS Gesuito09 2-men bobsleigh runners, right - square, left - triangle. Front runners are in the top figure, rear runners are in the bottom figure. This is a more curved set of runners. . . 157 B.3 Measured gauge data from a set of LG 2-men bobsleigh runners, right - square, left - triangle. Front runners are in the top figure, rear runners are in the bottom figure. This is a flatter set of runners...... 158 B.4 Measured gauge data from the more curved of two types of 2-men runners supplied by the F.I.B.T. (R8), right - square, left - triangle. Front runners are in the top figure, rear runners are in the bottom figure...... 159 B.5 Measured gauge data from the flatter of two types of 2-men runners supplied by the F.I.B.T. (F1), right - square, left - triangle...... 160

C.1 Cross-section of the indentation crater left in the ice by the dropped ball. The crater volume is the rotational volume of the hashed area...... 162 C.2 Determination of the relative uncertainty in the crater diameter measurements for ball A. The circles represent the frequency of occurrence for a given range of normalized crater diameters. The line is the gaussian curve that best fits the data...... 163 C.3 Determination of the relative uncertainty in the crater diameter measurements for all other balls. The circles represent the frequency of occurrence for a range value of normalized crater diameters. The line is the gaussian curve that best fits the data...... 164 C.4 Ice hardness measured in the Olympic Oval using six different balls. Ice surface temperatures were −1.1◦C...... 165 C.5 Ice hardness in various locations within the Olympic Oval using all data. Ice surface temperatures were set at −1.1◦C...... 166 C.6 Ice hardness in various locations within the Olympic Oval using only the optimized experimental methodology. Ice surface temperatures were set at −1.1◦C...... 167

xiv D.1 The melt layer thickness across the half-width at the tail end of the front runner. The cross-sectional radius is 4.75 mm, the rocker is 34 m, the ice surface temperature is −7◦C and the sled speed is 25 m/s (F.A.S.T. 3.2b). 209 D.2 The relative change in coefficient of friction between bobsleigh runners and a flat sheet of ice when the cross-sectional radius of the runner is decreased from 4.75 to 4.0 mm for various sled speeds and ice surface temperatures from −2 to −14◦C. The front and rear runner leave one track (F.A.S.T. 3.1b)...... 210 D.3 The relative change in coefficient of friction between bobsleigh runners and a flat sheet of ice when the cross-sectional radius of the runner is increased from 4.75 to 5.5 mm for various sled speeds and ice surface temperatures from −2 to −14◦C. The front and rear runner leave one track (F.A.S.T. 3.1b)...... 211 D.4 The relative change in coefficient of friction between bobsleigh runners and a flat sheet of ice when the rocker of the runner is decreased from 34 to 20 m for various sled speeds and ice surface temperatures from −2 to −14◦C. The front and rear runner leave one track (F.A.S.T. 3.1b)...... 212 D.5 The relative change in coefficient of friction between bobsleigh runners and a flat sheet of ice when the rocker of the runner is increased from 34 to 48 m for various sled speeds and ice surface temperatures from −2 to −14◦C. The front and rear runner leave one track (F.A.S.T. 3.1b)...... 213

E.1 Radar velocity data taken on September 16, 2010 of a sled oscillating on the Ice House track, ice surface temperature was −4.4◦C...... 215 E.2 Recorded lossy accelerations of the 123 kg sled with a) 0 kg, b) 30 kg, and c) 60 kg added. The white hollow circles indicate measurements from the Calgary Olympic Oval and the grey ﬁlled circles from the Ice House. The solid lines are quadratic regressions for each mass. The experiments were conducted over four days, ice surface temperatures were between −2.2 and −4.6◦C...... 218

xv 1

Chapter 1

INTRODUCTION

1.1 The Sport of Bobsleigh

The sport of bobsleigh traces its roots to the resort town of St. Moritz in Switzerland. In the late 1800s British tourists were using wooden sleds on the road to get down the hill.

Soon they began to link the sleds together creating an articulating sled that could be steered. The ﬁrst bobsleigh course was formed in 1870 to ensure the safety of pedestrians on the road, and in 1884 the ﬁrst formal bobsleigh competition was held. To this day the track in St. Moritz is built from blocks of ice each year. It is the only remaining natural track in the world. All other tracks have a refrigerated concrete base. Bobsleigh is an

Olympic sport with three competition classes; 2-women, 2-men, and 4-men. This thesis focuses on the 2-men event.

Today, bobsleigh is seen as the F1 of winter Olympic sports. The competition has two main parts: (1) the ﬁrst 50 m where the athletes push the sled to about 40 km/h and then jump inside, and (2) the drive where the pilot navigates the sled down the remainder of the ice covered track, (1200 − 1700) m. The track is a large ice chute, like a big waterslide covered with ice.

1.1.1 The Sled

Modern world-class sleds are made by race car manufacturers and have four highly polished steel runners to create a low friction contact with the ice. In Fig. 1.1 we can see a

2-men team in training. The pilot is the front athlete who drives the sled. The brakeman, rear athlete, is well hidden behind the pilot in an aerodynamic position. The cowling is 2

Figure 1.1: Photograph of a 2-men sled with the pilot (the author of this thesis) and the brakeman taken during a training session in Whistler, BC in 2008. the aerodynamic hull of the sled and the runners are the four steel blades under the sled.

The runners are the only part of the sled that makes contact with the ice surface during normal sliding conditions.

1.1.2 Basic Physics in Bobsleigh

One common situation addressed in this thesis is that of a sled descending a straight section of track with a constant slope. While analyzing the motion of a bobsleigh in corners can be a very complex problem, examining the motion on a straight section of track with a constant slope is equivalent to an object on an inclined plane. This situation is illustrated in Fig. 1.2. By limiting the analysis to these conditions we are left with a relatively simple physical system. There are four forces (solid arrows) acting on the sled: air drag, friction, gravity, and the normal force from the ice. The force of gravity can be broken into its components (dashed arrows). The down force, which is perpendicular to the ice surface, is canceled by the normal force, while the force parallel to the ice 3

Figure 1.2: A diagram of the forces exerted on a bobsleigh descending a straight section of track with a constant slope, θ. 4 surface accelerates the sled down the course. The net force on the sled is then described by equation 1.1. −→ −→ −→ −→ −→ F net = m a = F g sin θ − F frict − F drag (1.1)

In this thesis, I analyze the friction force between the bobsleigh runners and ice, as well as the air drag, in this system.

1.1.3 Canadian History in Bobsleigh

The first Olympic team to represent Canada in the sport of bobsleigh won the 4-men gold medal in 1964. After those athletes retired from the sport, Canada had difficulty obtaining significant results until 1988. After hosting the 1988 Olympics in Calgary, Canada had a home track and an opportunity to train more athletes. The 1990s brought a great deal of success to the Canadian bobsleigh team, culminating with a gold medal in the 2-men bobsleigh event at the 1998 Olympics. At that time, the team was coached by the great

Swiss bobsledder and sled builder, Hans Hiltibrand. Shortly after this victory, Hiltibrand left the Canadian program. Since his departure, the Canadian team has continued to achieve excellent results in both bobsleigh and skeleton. However, they have consistently fallen behind top nations in their ability to produce top quality equipment for the sport of bobsleigh.

1.2 Motivation

Research in the sport of bobsleigh, like in other sports, is generally undertaken by indi- viduals or organizations seeking to gain an advantage over other competitors. For this reason, the majority of the work is secretive. The work is not held to the standards of open debate and peer-review, and it goes unpublished. This makes knowledge retention very diﬃcult. In Canada, we have only one sled builder, Dave Hugill and Pierre Lueders, 5 in collaboration with Stamco, an Edmonton based machining company, have been the only recognized bobsleigh runner makers in the country over the past decade. Our top athletes gain considerable knowledge on tracks and equipment over the course of their careers. However, that knowledge can often be lost upon retirement.

Bobsleigh athletes have long realized that the quality of their runners has a great deal of impact on overall performance in the sport. Canadian athletes regularly spend thou- sands of dollars importing runners and / or sleds from Europe in the hope of improving their results. Much of the development in this sport could be classiﬁed as more of an art than a science. Athletes copy equipment that works without understanding why and information is carefully guarded as a trade secret. The new rules [3] implemented after the

2006 Olympics limit the construction material for runners to one standardized steel, and forbid coating or treating those runners. This has considerably limited the factors that influence runner performance. This led our group to be interested in studying ice friction in bobsleigh at that time. The main variable influencing performance is now the profile or shape of the runner. We therefore decided to focus on developing a computational model to calculate the friction between the runners and ice. The main goal of this work is to examine the relationship between bobsleigh runner profile and ice friction performance for a variety of sled velocities and ice surface temperatures. This goal led to a wide range of initiatives in computational, theoretical, and experimental physics, as the remainder of this thesis will demonstrate.

1.3 Background

Although the majority of research done in high performance sport is secret, there is still a considerable amount of published research from various sports that is relevant to our work. Balakin et al. [4, 5] have studied sled dynamics in luge; a sliding sport where the 6 athletes travel down the course feet first in a supine position. In the work by Balakin et al., the coefficient of friction was determined using the internal timing system on the track. It was found to be 0.0133 to 0.015. More recently Fauve and Rhyner [6] have optimized the properties of luge blades. The factors that impact performance in the sport of skeleton have been studied by Bromley [7]. This work was focused on sled design and control. In this sliding sport, athletes descend the track head first in a prone position. The effect of athlete physique on performance has been studied [8, 9] to determine an ideal body type for skeleton athlete recruitment. Studies examining the interaction of skate blades with ice [10, 11, 12] can also inform our work; however, the contact dynamics for all of these sports are different than in the sport of bobsleigh. Therefore, they cannot tell us the effect of bobsleigh runner profile on ice friction performance.

Factors aﬀecting performance in the sport of bobsleigh were studied by Baumann

[13]. Ice friction was not studied in detail, only the eﬀect of reducing ice friction on

ﬁnish times. Bobsleigh aerodynamics have undergone considerable study using numerical simulations and wind tunnel models [14, 15, 16, 17, 18, 19]. This work has advanced the aerodynamic design of sleds over recent years. The motion of a sled descending a track was studied [20, 21, 22] and a training simulator was constructed [23]. A computational model similar to that behind the simulator is used when designing new tracks to determine maximum speeds obtained by competitors. Again, ice friction was not studied in detail only the impact of friction on the velocity and thus the trajectory of the sled. Multi-body dynamics have been used to improve the design and performance of bobsleighs [24, 25].

Speciﬁcally, in this work the optimization of certain sled components can be examined.

This optimization is difficult to perform on the track due to the limited number of available runs to acquire data, the inconsistencies of a human pilot, and varying weather and ice conditions. A telemetry system for bobsleigh was developed [26] and ice friction in the sport has been studied at low speeds [27]. In the ice friction study, the coefficient 7 of friction of bobsleigh-like runners was measured by observing the deceleration of a sled on flat ice. The coefficient of friction varied from 0.006 to 0.012. Three types of steel were examined for the runner and ice surface temperatures ranged from -1 to -10◦C. The details of the runner preparation and the runner profiles were not included in the paper.

Ice friction has been studied extensively by tribologists. The most common setups involve rotating rings [28, 29, 30] or a pin on a rotating ice surface [31, 32, 33]. Evans et al. [34] placed curved skate blades onto a rotating drum to measure ice friction. All of these results are interesting and can tell us a great deal about ice friction. The contact dynamics however, are considerably different than in the sport of bobsleigh. In order to study the effect of the bobsleigh runner profile on ice friction, real bobsleigh runners interacting with ice must be examined.

My analysis of ice friction in the sport of bobsleigh revolves around a computational model that calculates the coeﬃcient of friction between a blade and ice (F.A.S.T. or

Frictional Algorithm using Skate Thermohydrodynamics [12]). Preliminary results from the model suggested that the ice hardness has a large impact on ice friction. For this reason, the hardness of ice was studied as a subproject in my research. Hardness of a material is deﬁned as its resistance to permanent deformation [35]. There are three main methods for measuring hardness; scratch hardness, static indentation hardness, and dynamic hardness. There are two main methods for determining scratch hardness. The

Mohs scale is a ranking system developed in 1812, based on the ability of a harder material to scratch a softer one. The Turner sclerometer, invented in 1896, draws a diamond point over a smooth surface and uses the weight required to produce a standard scratch to determine the hardness. To determine the static indentation hardness a deﬁned force is applied to a material via an indentor and the impression is used to determine hardness.

These indentation measurements are the most common hardness tests. Examples include

Brinell, Vickers, Knoop, and Rockwell. The form of the indentor and / or the application 8 of pressure varies between these four tests. Certain combinations are preferred for speciﬁc applications. In dynamic testing, the indentor is dropped and the kinetic energy of the indentor deforms the surface of the material. The material hardness can be determined by measuring the rebound height of the indentor and / or the volume of the indentation left in the material.

Ice hardness has often been studied as a slow process, examining plastic ﬂow or creep

[35, 36, 37, 38, 1, 39, 40, 41]. It has also been studied in terms of ice fracture [35, 42, 43, 41].

Jordaan and Timco [44] have examined the ice crushing process. Dynamic ice hardness measurements have been made on small scale laboratory ice [1], larger scale laboratory ice

[2], and icebergs [45]. These measurements vary from 10 to 65 MPa at -12◦C suggesting that the ice hardness depends on the properties of the ice. Ice from athletic facilities varies considerably from the ice surfaces mentioned here. The key characteristics of an athletic ice surface are that it is approximately 25 mm thick, it is artiﬁcially cooled from a concrete surface at its base, and it is regularly scraped and resurfaced with water to ensure a constantly smooth surface.

1.4 Thesis Outline

The main goal of this thesis is to examine the effect of runner profile on performance in the sport of bobsleigh. The project is centered around a computational model (F.A.S.T.) that calculates the coefficient of friction between the runners and ice. This model is examined in detail in chapter 4. Before discussing the model, I begin by analyzing the profiles of runners used in the sport of bobsleigh in chapter 2. The hardness of athletic ice surfaces is examined by dropping balls onto the ice surface and measuring the size of the indentation crater in chapter 3. Finally, in chapter 5, I use radar speed data to determine the coefficient of friction and aerodynamic drag of 2-men bobsleighs in an 9 effort to validate the F.A.S.T. model.

The runner profile is a central key in this thesis and I begin there. In the following chapter, I examine profiles used in the sport of bobsleigh today. We must first understand what exists before we can suggest improvements. 10

Chapter 2

EQUIPMENT ANALYSIS

2.1 Introduction

The primary goal of this chapter is to determine the necessary information on bobsleigh runner profiles to be incorporated into the F.A.S.T. model. A second goal was to acquire as much information as possible on runner profiles in an effort to construct runners for Canadian bobsleigh athletes. A third goal was to develop tools and techniques to monitor bobsleigh runner profiles on an ongoing basis in order to optimize equipment performance. As a bobsleigh athlete, I had access to a variety of bobsleigh runners from both Foothills Bobsleigh Club and Bobsleigh Canada Skeleton. One set of 2-men runners was constructed in partnership with Foothills Bobsleigh Club.

The first step is to determine an effective way to acquire information on the bobsleigh runner profiles. The conventional method of acquiring this information is to take the runners into a machining shop and have them digitize the profile. The equipment used to perform this measurement is very large and costly. Speed skating athletes use a handheld rocker gauge in order to determine their profiles. This gauge has two carbide pins separated by a fixed distance and a micrometer in the center. The micrometer provides an evaluation of the average curvature over the width of the gauge.

In this chapter, I discuss the differences between direct profile measurements and the gauge data as well as the different information that can be obtained from the profile.

Due to differences between speed skate profiles and bobsleigh runners it is not evident that a speed skate gauge could adequately determine bobsleigh runner profiles. Using digitized profile data, I have performed a numerical optimization of the width of the 11 rocker gauge for hockey, short and long track speed skating, and bobsleigh. This portion of the chapter is taken directly from our paper published in Sports Engineering in 2009

[46]. The digitized blade proﬁles analyzed in this paper were acquired from Professor

Maw and Bobsleigh Canada Skeleton. The project was proposed, executed, and the paper manuscript was mostly written by myself. The coauthors acted primarily in a supervisory and editorial role.

After optimizing the dimensions of the handheld rocker gauge it was used to catalogue the proﬁles of a number of runners. This allowed us to determine a range of typical rocker values, or radii of curvature along the length of the blade. During the project, I was approached by the Foothills Bobsleigh Club to aid them in the design and construction of a new set of 2-men bobsleigh runners. That process has also been described at the end of this chapter.

2.2 Motivation for Equipment Survey

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 [47, 7, 48]. Some research has been published on the effects of the runner materials [27], 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 [3] have limited the construction materials for the runners to one standardized steel. This has focused runner developments on the profile of the runners. This chapter will focus 12 on the determination and analysis of bobsleigh runner profiles.

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 have developed a thermohydrodynamic model that calculates the coefficient of friction between metal and ice given a blade’s physical properties and the ice conditions [49]. The model has been used to calculate the dynamic coefficient of friction between bobsleigh runners and ice. The realization that the exact profile of the blade had significant consequences on performance, both in the actual sport and in the calculations with our model, has led to our desire to catalog various runner profiles. This data will provide us with the current state of runners being used in the sport. It will also be our starting point in an effort to improve runner profiles.

A survey of athletes and Bobsleigh Canada Skeleton (BCS) staff provided little insight into quantitative knowledge of bobsleigh runner profiles. Analyzing profiles of runners was not something performed by athletes or staff. On occasion, the task was contracted out in order to reproduce a given runner profile on a newly machined set. This newly cut set of runners was not generally verified to ensure consistency with the original set and it was not monitored to verify whether or not any variations in the profile occur as the runners are used and maintained over time. Athletes in the sport of speed skating pay much more attention to the profile of their blades. Most high performance speed skaters have a handheld rocker gauge in order to determine the rocker, or radius of curvature along the length of the blade. In this thesis a larger rocker signifies a flatter runner. This notation is consistent with the usage of the term in speed skating but contrary to the common usage of the term in the sport of skeleton. In speed skating, the handheld rocker gauge is used to ensure that the rocker is not modified as they polish, or maintain, their blades. Given that speed skate athletes pay more attention to the profile of their blades,

I have decided to adopt the rocker gauge from speed skating in order to perform the 13 bobsleigh equipment analysis.

2.3 Gauge vs. Direct Proﬁle Measurements

In section 2.2, the examination of bobsleigh runners by an outside contractor was mentioned. This is a direct measurement of the runner profile that, for 4-men bobsleigh runners, requires a device that can travel up to 1.3 m along the length of the runner, with millimeter precision, and determine the height of the running surface to micrometer precision. This is a highly specialized, expensive, and large piece of equipment that can not easily be moved. There is a similar, but smaller, device designed for skate blade analysis at the Olympic Oval. While a direct profile measurement provides more information on the blade profile, it is not often used by athletes. For portability, and ease of use in a competition setting, speed skating athletes generally use the much more affordable and compact handheld rocker gauge. While the rocker gauge is useful, it does not provide the complete profile. The rocker gauge provides us with a measurement of the curvature of the blade at a given point. This information can be used to reconstruct the runner profile, although some information is lost.

2.4 Analysis of the Handheld Rocker Gauge

In an effort to minimize the amount of information lost using the handheld rocker gauge we analyzed the device and its suitability for use in analyzing bobsleigh runners, long and short track speed skates, as well as hockey skates. Rocker is the radius of curvature in the plane of the blade or how curved the surface of the blade is lengthwise. The use of a device designed for speed skating for 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, 14 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 section addresses the optimization of the gauge size for four athletic disciplines using real blade profile data and numerical optimization methods.

We define the profile b(z) as the height of the blade’s surface as a function of the position along the length of the blade as illustrated in Fig. 2.1. 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, as shown in Fig. 2.2. The

Figure 2.1: Images 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. 15

Figure 2.2: A photograph of an individual taking gauge data on a bobsleigh runner. rocker is not measured directly. It is calculated from the gauge data. The rocker gauge works using a three-point measurement of the curvature of the blade. It consists of a fixed pin at each end of the gauge. The two pins are 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(z) relative to a flat line joining the two pins. Precision in the rocker measurement is limited by the resolution of the micrometer and the half width of the gauge a. The rocker precision increases with a. An increase in a also results in a decrease in the spatial resolution. This type of gauge is currently used in both short and long track speed skating and is supplied by at least three companies; Maple1, Marchese2, and Shoei Creations3. In principle, a similar device could also be used in bobsleigh and hockey. However, since the radius of curvature for the blades and runners in each of these sports is quite different4, 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 can be changed simply by applying tension to

1http://www.mapleskate.com/ 2http://www.marcheseracing.com/ 3http://shoeicreations.com/ 4hockey 0.5 − 6 m, short track 6 − 12 m, long track 18 − 30 m, bobsleigh 10 − 150 m 16 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-men and 6 mm for 4-men 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 bobsleigh 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 in the profile over time can be reproduced in a new set of runners.

We can see from the example of bobsleigh runner gauge data in Fig. 2.3 that the rocker is mostly constant through the middle of the blade. In this runner, an exception is seen with a flat, even concave, 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. Since the location of the front and / or rear of a runner can vary largely between different sets of bobsleigh runners the zero in the horizontal or z-axis corresponds to the middle of the blade. Specifically, it is the mid way point between the bolt holes used to attach the runner to the sled. The location of the holes is standard and set by the

F.I.B.T.5. In 2-men bobsleigh the front runner holes are separated by 585 mm and the rear are separated by 743 mm. These holes are the attachment points, covered by bolts, for the bobsleigh runner in Fig. 2.1. Positive z values signify the front of the runner.

6Speed skates are sharpened by putting a pair of skate blades in a jig such that

5http://www.ﬁbt.com/ﬁleadmin/Rules/Bob Drawings all 2007.pdf 6This paragraph was written by coauthor Professor Maw, for our paper [46] 17

Figure 2.3: Gauge data from the rear runners of one of two types of 2-men runners supplied by the F.I.B.T. (the international body responsible for bobsleigh and skeleton). Squares represent data from the right runner while triangles are from the left. 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 degree 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 18 is generally not monitored, and the rockers between two skates are often quite diﬀerent as a result. Unlike the ﬂat surface 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. Hockey blades can be rerockered by removing the edges and cutting the blade to a constant radius. This is an uncommon practice and must be followed with cutting a groove in the blade to recreate the edges [50]. Creating the edges can modify the rocker.

It should be noted that the most complete method for 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 bobsleigh runners are longer and wider than skate blades, the device we have cannot accommodate the runners. An instrument capable of performing a direct profile measurement of bobsleigh runners was found at a precision machining shop and we have access to a limited number of digitized profiles from that instrument 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 could be adapted for use in bobsleigh and hockey as well. 19

2.4.1 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 that we wish to recreate. Using this digital profile we will simulate gauge data using the digital profile data at the positions of the pins and micrometer of a theoretical hand gauge, and then apply a gaussian distribution of uncertainty comparable to the measurement uncertainty observed in practice while using the real hand gauge. We will then use the simulated gauge data to recreate a profile iteratively from the center of the runner or blade out to each end. Finally, we will 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 1000 times and then averaged for each gauge width.

Proﬁle Analysis

The hand gauge provides a measurement that represents the curvature along the length of the skate blade or bobsleigh runner. Fig. 2.4 demonstrates how we calculate rocker data from the gauge measurements. The two outside points are ﬁxed and a micrometer takes a reading g(z) in the middle. The gauge has a half width of a. We should point out that the curvature in Fig. 2.4 is an exaggeration; in reality a/R ∼ 10−2 and this approach works well with very small angles.

Assuming that we are working with small angles, equation 2.1 gives us the calculated gauge data g(z) from the known proﬁle b(z).

b(z + a) + b(z − a) g(z) = b(z) − + δ (2.1) 2

The measurement uncertainty is represented by δ. In the simulations, δ has a value between ± 5σ and it follows a normal distribution of width σ. The magnitude of σ is equivalent to the standard deviation of gauge data that has been observed in practice 20

Figure 2.4: Representation of rocker measurements taken with the hand gauge. The gauge pins rest at the top of the ﬁgure in the corners of the large isosceles triangle and the micrometer rests in the centre measuring g(z). The gauge half width is a and the rocker or radius value is R(z). 21 using the handheld rocker gauge to analyze bobsleigh runners. The uncertainty observed in practice appears to be constant for most values of g(z). The uncertainty does increase for very large values of g(z). These large values can only be observed at the ends of the blades. For simplicity, this study has used a constant value of σ for each simulation.

The resolution of the micrometer is one micron and in our simulations we use σ values from one to two microns. The horizontal position along the blade or runner where the micrometer measurement is taken is z.

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(z). This allows us to use the Pythagorean theorem to create equation 2.2.

a2 + g2(z) R(z) = (2.2) 2g(z)

A larger gauge width results in a larger measurement and therefore a smaller relative uncertainty in g(z). However, the larger width also leads to a greater loss of information through low pass ﬁltering [51] 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.

Proﬁle Reconstruction

The reconstruction of the new proﬁle br(z) is an iterative process that requires comparing successive data points. To reﬂect the spatial resolution of the data collected with the handheld rocker gauge, ∆z, we express equation 2.1 without the measurement uncertainty and with ∆z replacing the gauge half width a, resulting in equation 2.3.

b (z + ∆z) + b (z − ∆z) g (z) = b (z) − r r (2.3) ∆z r 2 g∆z(z) is not an actual gauge measurement but a mathematical tool used to reconstruct the runner proﬁle. 22

In speed skating, gauge measurements are taken by continuously moving the gauge along the length of the blade. For this study we will assume the same type of analysis for hockey skates as well. The spatial resolution, ∆z, 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, ∆z is the distance between measurements. To simplify the reconstruction algorithm we chose ∆z to be an integer fraction of the gauge half width a. By observing Fig. 2.4 and replacing a with

∆z we can use the Pythagorean theorem and the quadratic equation to produce an alternative expression for g∆z, q 2 2 g∆z(z) = R(z) − R (z) − (∆z) (2.4) where R(z) is calculated from equation 2.2 using the measured, or in this case simulated, gauge data. Now by equating equations 2.3 and 2.4 we arrive at equation 2.5.

b (z + ∆z) + b (z − ∆z) q b (z) − r r = R(z) − R2(z) − (∆z)2 (2.5) r 2

At this point, equation 2.5 can be re-arranged into the forward or backwards recursive relationships, both described in equation 2.6. q 2 2 br(z ± ∆z) = 2br(z) − br(z ∓ ∆z) − 2(R(z) − R (z) − (∆z) ) (2.6)

Once the gauge data g(z) is collected along the length of the runner we note that reconstructing the proﬁle br(z) from curvature data is eﬀectively a discrete 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 br(−∆z) = br(∆z) = 0, which sets the position and slope of the blade about the center to zero. We can then solve equation 2.6 for z = 0, as shown in equation 2.7. q 2 2 br(0) = R(0) − R (0) − (∆z) (2.7) 23

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 equation 2.6.

Proﬁle Comparison

Once we have reconstructed the profile, both the original profile b(z) and the new reconstructed profile br(z) must be reoriented so that they overlay for comparison. While it is possible to simply adjust the original profile so that b(∆z) = b(−∆z) = 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.

N 2 1 X (b(zj) − br(zj)) α = 2 (2.8) N j=1 σ

Due to the random nature of the uncertainty the process is repeated 1000 times and averaged α values are obtained. This process was executed using a C++ code which can be found in Appendix A.1. Minimization of α occurs when the two curves are best matched.

2.4.2 Results and Discussion

The gauge width is optimized using digital profiles and minimizing the averaged α value for multiple profiles of each blade type and averaging the result. In Table 2.1, we have shown some of the results from the averaged α calculations from five pairs of long track 24

Table 2.1: Optimizing the half width of a rocker gauge (a) for 7 diﬀerent long track speed skates by minimizing average α values. Minimums are in bold font. 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 speed skating blades. It should be noted that although Table 2.1 only shows some typical data as an example of the results of these calculations, the additional data that was not shown has the same averages as the data that was displayed. The data from Table 2.1 are calculated with a spatial resolution of ∆z = 5 mm and with a standard deviation for the gauge data of σ = 1.5 microns.

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 − 10 mm for the three types of skate blades, 10 − 25 mm for the bobsleigh runners and standard deviations from 1 − 2 microns for all. We are conﬁdent that this parameter space fully encompass the uncertainty in these parameters and thus properly reﬂects our uncertainty in the optimum gauge sizes. 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 2.2.

It should be noted that the optimum gauge size is the same for bobsleigh and long track speed skating, despite the rocker being greater in bobsleigh. This is likely due to the fact that athletes in speed skating monitor their blades more closely. Because of this we observed less spatial variations along their blades reducing the need for spatial resolution 25

Table 2.2: Recommended gauge sizes to reduce information loss blade type gauge 1/2 width hockey 10 ± 3 mm short track 25 ± 5 mm long track 50 ± 10 mm bobsleigh 50 ± 20 mm and allowing for a larger gauge. If bobsleigh athletes were to monitor their runner proﬁles more carefully perhaps a larger gauge could be recommended in the future. The blades we examined required the extra spatial resolution in order to reproduce the original proﬁle.

2.4.3 Rocker Gauge Summary

The handheld rocker gauge is a very useful instrument for blade analysis in speed skating.

This work suggests that 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 acquired a modified 5 cm gauge to analyze bobsleigh runners. Following this analysis, we feel confident using it for a survey of runners used in the sport. I have also had a gauge constructed with a 1 cm half width and it can be used to effectively study hockey skates.

Athletes suggest that certain bobsleigh runners work diﬀerently in certain conditions

(i.e. cold vs. warm weather). We hope that by analyzing this equipment with our 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. 26

2.5 Proﬁle Analysis

The purpose of optimizing the rocker gauge is to use that gauge to analyze the profiles of multiple runners used in the sport of bobsleigh. There are several different ways of looking at the profile of a runner. In this section, we examine one long track speed skate and one 4-men bobsleigh runner using three different methods. This will illustrate the different ways to examine blade profiles as well as the differences between the profiles of speed skates and bobsleigh runners.

In our ﬁrst example, we examine the data obtained from a direct proﬁle measurement.

This data tells us the height of the blade for each position along the length of the blade.

We have examples of this for long track speed skating (Fig. 2.5) and bobsleigh (Fig. 2.6).

Despite the different scales in the vertical and horizontal axes, it is still very difficult to extract meaningful information from this type of graphical representation. We can certainly see some differences between the bobsleigh and speed skate blades but it is difficult to extract quantitative information from these figures. A trained eye can observe that the rocker is relatively constant for the speed skate, and we could estimate an average value over the entire blade. On the bobsleigh runner, an experienced eye can note that the rocker is quite high in the center of the blade |z| < 200 mm but decreases significantly towards the front and rear of the runner. The bobsleigh runner is more than twice as long as the skate blade and there is much more variation in the height of the running surface. However, if we examine only the middle portion of the runner, |z| < 200 mm, the differences are not as pronounced. This portion of the runner is much longer than the contact length of the runner on a flat ice surface [52]. Therefore, it is the only portion important to consider for that type of contact. This section of the runner is illustrated in greater detail in Fig. 2.7. By examining this zoomed in figure we can estimate that the bobsleigh runner’s rocker is about twice that of the skate blade and it is not as 27

Figure 2.5: Proﬁle of a long track speed skate blade, measured directly. 28

Figure 2.6: Proﬁle of a rear 4-men bobsleigh runner, measured directly. 29

Figure 2.7: Zoomed in profile data from a rear 4-men bobsleigh runner, measured directly. symmetrical. It is difficult to quantify the asymmetries by examining the profile data.

Further information can be obtained from the same blades by examining their gauge data. Specifically, this approach allows us to quantify variations in the rocker along the length of the blade. The gauge data will provide us with an evaluation of the curvature of the blade. Examples of this are again provided for long track speed skating (Fig. 2.8) and bobsleigh (Fig. 2.9). Examining the gauge data allows us to see a great deal of information that we were not able to observe by examining the profile data. For example, while the gauge data for the speed skate blade are very consistent we can see that the blade has a slightly flatter section, lower gauge data, around z = 300 mm. For the bobsleigh runner, we can see a number of variations in the rocker all over the blade, including the significant increases in rocker, or decreases in gauge data for |z| < 200 mm, 30

Figure 2.8: Calculated gauge data derived from the proﬁle measurements of a long track speed skate blade. 31

Figure 2.9: Calculated gauge data derived from the profile measurements of rear 4-men bobsleigh runner. 32 which we noted in Fig. 2.6. We can also see that there is an extremely flat section on the blade between -100 and 20 mm and we can note where significant changes in the rocker occur at the front and rear of each blade.

A third way to examine the blades is by plotting the rocker as a function of position along the blade. Like the gauge data, the rocker is an evaluation of the curvature of the blade. The relationship between the gauge data and the rocker is expressed in equation 2.2. Noting that7 a2 À g2, the relationship between gauge data and rocker can be simplified to equation 2.9. a2 R(z) = (2.9) 2g(z) Generally, the rocker data are more sensitive to variations in the profile of the runner than the gauge data. While this extra sensitivity can be an added benefit, in some cases it is simply too much information. It can simply look like noise. The rocker data are illustrated in Fig. 2.10 (long track speed skating) and Fig. 2.11 (bobsleigh). Although the gauge data and the rocker data both represent the curvature, it is clear that the different graphical representations provide a different perspective. Examining Fig. 2.10, we note that the rocker is much more sensitive to minor variations along the length of the blade. Due to this noise however, it is more difficult to notice the higher rocker around z = 300 mm that we noticed in Fig. 2.8. On the other hand, by examining the long track blade rocker data we can see how the rocker changes at the front of the blade. In

Fig. 2.8, the gauge data quickly jumps oﬀ the scale around 370 mm. This illustrates how the rocker data are more sensitive to variations than gauge data at larger rocker values but less sensitive for small rocker values. Because athletes in speed skating monitor their blades, speed skate proﬁles tend to be more consistent along the length of the blade.

Bobsleigh runner proﬁles are generally less consistent. For this reason, it is easier to examine gauge data when studying bobsleigh runners. The data are less erratic and it is

7a ∼ 10−2 m; g ∼ 10−5 m 33

Figure 2.10: Calculated rocker data derived from the proﬁle measurements of a long track speed skate blade. 34

Figure 2.11: Calculated rocker data derived from the proﬁle measurements of a rear 4-men bobsleigh runner. 35

Figure 2.12: Measured gauge data from one 2-men front bobsleigh runner (circles). The runner proﬁle and the corresponding gauge data were calculated numerically using the measured gauge data taken every 5 mm (triangles), 25 mm (squares), and 50 mm (X’s). easier to compare two runners which have similar proﬁles.

If we use the gauge data to reproduce a proﬁle, it acts like a low pass ﬁlter. The more data we have the better; however, at a certain point we have diminishing returns.

In Fig. 2.12 we have measured the gauge data every 5 mm along the length of a runner to determine the optimum spatial frequency of gauge measurements. Assuming that the calculated gauge data from the measurements taken every 5 mm is the ideal, we can see that the reproduction from the data taken every 50 mm differs by more than the measured gauge data variation from point to point. The reproduced gauge data from the data taken every 25 mm however is effectively equivalent to that from every 5 mm. This suggests there is no benefit in acquiring data more frequently than every 25 mm. The 36 gauge half width is 50 mm, in order to simplify the reconstruction algorithm we used an integer fractions of that distance.

2.6 Equipment Survey

In this section, we show a number of different bobsleigh runner profiles and the differences between them. First, we illustrate what differences can be observed in runners even when they are expected to be the same. Because athletes in the sport of bobsleigh do not monitor the profiles of their runners, the profiles can change over time. This can happen when the athletes run over debris in the track or while polishing their runners during regular maintenance. In Fig. 2.13 we show two older sets of runners that have been used for many years. In this section, we will only include the front runners. The rear runner profiles are very similar to the front except that they are longer and therefore the sharp increase in gauge data at the front and rear of the runner occurs further from its centre.

The rear runner proﬁles are included in Appendix B. While we believe that the runners in

Fig. 2.13 were originally identical, over the years the profiles have changed and they are no longer the same. In Fig. A.1 we examine the profile more closely. Despite considerable differences between the four runners, the majority of the gauge data in the center of the blade is concentrated between 25 − 50 microns. Because this is an older set of runners, it is an extreme case of differences between runners that should be similar.

One example of a small rocker set of runners is a set of Gesuito runners reproduced by Bobsleigh Canada Skeleton in 2009. The proﬁle can be found in Fig. 2.14. The gauge data in this example reaches a minimum of ∼ 45 microns. An example of a large rocker set of runners is the set of LG runners in Fig. 2.15. In this case, the gauge data reaches a minimum of ∼ 20 microns, if we ignore the one ﬂat point. Because the runners in Figs.

2.14 and 2.15 have been used for less time than the runners in Fig. 2.13, the diﬀerences 37

Figure 2.13: Measured gauge data from two separate sets of DSG front 2-men bobsleigh runners. First set, right - cross, left - circle and second set, right - square, left - triangle. 38

Figure 2.14: Measured gauge data from a set of BCS Gesuito09 front 2-men bobsleigh runners, right - square, left - triangle. 39

Figure 2.15: Measured gauge data from a set of LG front 2-men bobsleigh runners, right - square, left - triangle. 40 between the squares and triangles in those two examples are more typical of differences in profiles between supposedly identical runners. Further examples of bobsleigh runner profiles are included in Appendix B.

2.7 Bobsleigh Runner Design Project

Early in my process of examining bobsleigh runner proﬁles, I was asked by Foothills

Bobsleigh Club to provide them with a 2-men runner profile that they could use to machine a new set of runners. At the time, I decided to base the runners on the two sets of DSG 2-men runners that I had analyzed. These runners are not of World Cup calibre but they are known as very good quality runners and both sets had a reputation for producing excellent results on the Europe and America’s Cup tours. Because we had two sets of runners, there were four front and four rear runners (see Fig. B.1) on which to base the runner profiles. Unfortunately, there were considerable differences between each of those four runners. In fact, there were more differences between the four runners than between many aspects of the front and rear runners.

In order to create a runner profile from the existing data, I divided the runner into three sections. The centre section had a constant rocker, the front, and the rear both had quadratically increasing gauge data. This model appeared to best fit the experimental measurements. The fit was performed using a least squares method. In Fig. 2.16 we show how the designed runner profile compares to the experimental gauge data.

A similar analysis was done simultaneously for the rear runners and this gauge data was used to create a runner proﬁle. The process of creating a runner proﬁle from the gauge data is similar to the blade reconstruction used in the gauge optimization process in section 2.4.1. As our two data inputs, the blade heights aligned with the runner holes were set to a height of zero and this zero was set at a distance of 37 mm from the surface 41

Figure 2.16: Measured gauge data from two sets of DSG front 2-men bobsleigh runners (circles) and the designed front runner proﬁle (solid line - centre section of the runner and dotted line - front and rear sections of the runner). 42

Figure 2.17: Diagram of the bobsleigh runner with the axes and the five sections used for our 2-men runner construction project. The figure is not to scale. of the runner opposite to that contacting the ice (see Fig. 2.17). The numerical code used to convert the gauge data to a runner profile was adapted from the code used to optimize the size of the hand gauge (see Appendix A.1).

The measured gauge data did not extend to the front or rear tips of the runner because the handheld rocker gauge was not able to make measurements much larger than 1000 microns. The calculated runner profiles had to be extrapolated out to the runner tip and heel in order to conform to F.I.B.T. regulations8. To execute this, the runner profile was partitioned into two extra sections. The five sections of the runner are defined in Table

2.3 and 2.4. The subscripts f and r designate the front or rear runners and t and h designate the toe or heel of a given runner. The surface proﬁle of the newly designed runners is described in equations 2.10 to 2.14 where z is in mm and x(z) is in microns.

Heel,

x(z) = e0z − 1/a0 · ln |a0z + b0| + c0 (2.10)

8http://www.ﬁbt.com/ﬁleadmin/Rules/Bob Drawings all 2007.pdf 43

Table 2.3: For the runner design process each runner is divided into ﬁve sections along its length. Subscripts f and r designate the front or rear runners and t and h designate the toe or heel of a given runner.

Heel zmin < z < zmin + fh(f/r) Back zmin + fh(f/r) < z < d − l(f/r) Centre d − l(f/r) < z < l(f/r) + d Front l(f/r) + d < z < zmax − ft Toe zmax − ft < z < zmax

Table 2.4: This table includes values for several parameters found in Table 2.3. These values help to deﬁne the ﬁve sections used in the runner design process.

lf = 178 mm lr = 264 mm d = 16.7 mm ft = 150 mm fhf = 70 mm fhr = 50 mm front runners: zmax = 500 mm zmin = −420 mm rear runners: zmax = 604 mm zmin = −501 mm

Back,

4 2 x(z) = a1(z + l(f/r) − d) + b1(z + l(f/r) − d) + c1(z + l(f/r) − d) + e1 (2.11)

Centre,

2 x(z) = a2(z − d) + b2(z − d) + c2 (2.12)

Front,

4 2 x(z) = a3(z − l(f/r) − d) + b3(z − l(f/r) − d) + c3(z − l(f/r) − d) + e3 (2.13)

Toe,

4 4 2 x(z) = g3(z −zmax +ft(f/r)) +a3(z −l(f/r) −d) +b3(z −l(f/r) −d) +c3(z −l(f/r) −d)+e3 (2.14)

The parameters for equations 2.10 to 2.14 are located in Table 2.5. A number of these parameters are equal. In certain cases this was to ensure continuity of the proﬁle, the slope, and the curvature. In other cases, the least squares solutions of two parameters were found to be equivalent within uncertainty. In this case, they were ﬁxed as equivalent 44

Table 2.5: Parameters for equations 2.10 to 2.14 for the designed 2-men runner profile. Front runners 0 1 2 3 a 0.59 −2.79 × 10−9 −1.94 × 10−5 −2.79 × 10−9 b 203 −1.94 × 10−5 −1.58 × 10−3 −1.94 × 10−5 c 135 5.33 × 10−3 2.11 −8.48 × 10−3 e 0.40 1.78 - 1.22 g - - - −2.96 × 10−8 Rear runners 0 1 2 3 a 6.00 −2.79 × 10−9 −1.94 × 10−5 −2.79 × 10−9 b 2700 −1.94 × 10−5 −1.26 × 10−3 −1.94 × 10−5 c 197 −9.01 × 10−3 3.04 −1.15 × 10−2 e 0.45 2.01 - 1.35 g - - - −1.40 × 10−8 and the least squares fit was repeated. This was the case for the constant rocker on the front and rear runners a2 as well as the quadratic increase on the front and back of the front and rear runners a1, a3. The runner profile describes the height of the runner along the length of the blade.

The cross-sectional radius of the blade was set to a constant 4.1 mm. Using this radius, close to the minimum allowed by the F.I.B.T., was common practice by BCS at the time. This is no longer the case. I learned this through discussions with BCS coaches and staﬀ and by measuring runners constructed both at that time and more recently.

Athletes have found that a larger cross-sectional radius resulted in improved velocities for most conditions. In 2-men bobsleigh a cross-sectional radius of 4.5 − 4.75 mm would be considered normal today.

This profile was designed early in my research project. In retrospect, too much focus was placed on fitting the gauge data on the front and rear of the runner, sections that rarely touch the ice. In section 4.2, I show the derivation of the contact length between the blade and the ice ls. This length is dependent on the ice surface temperature and 45 approximately 5 cm. This suggests that the only time the runner for |z| > 150 mm touches the ice is when it is intersecting uneven ice, or while entering or exiting a curve. It is not relevant to normal gliding. By fitting the designed runner profile to all of the gauge data, the data from |z| > 150 mm pulled the constant rocker section to a higher gauge data value. If the constant rocker were fit to only the gauge data between |z| < 100 mm, the runner would be flatter over that section. I believe that would better reflect the original runners and thus, be a better product. Despite these suggestions for the future, I feel that this was a good first attempt at runner construction. I am confident that I would have just as many regrets on a second generation set of runners as it is a learning process.

An option for the future would be to vary the rocker linearly from a minimum point out to the point where there is a large increase in the gauge data. This type of proﬁle can be observed in the LG runners (Fig. 2.15), which are considered to be of top quality.

For the LG runners, the gauge data, in microns, is described by equation 2.15, with z given in mm.

g(z) = 0.342 · |z + 25.5| + 14.2 (2.15)

This relationship is valid from -225 mm to 275 mm for the front runners and -325 mm to 350 mm in the rear runners. However, I would also consider forcing g(z) ≥ 20 microns because the experimental data does not dip signiﬁcantly below this value. The gauge data is almost constant for |z + 25.5| < 50 mm. Further analysis would be required to describe the front and rear ends of the runners. Since the ends are so far from the contact area with a ﬂat ice surface, those ends are not a great concern.

2.8 Summary

After examining 12 sets of 2-men bobsleigh runners, I feel comfortable that the majority of the runners have gauge readings in the range of 65 − 25 microns on the section of the 46 runner contacting the ice when sliding on a ﬂat section of track. This range of gauge readings corresponds to rockers of 20 − 50 m. Cross-sectional radius was not examined in detail in this study. However, it was often noted in our measurements. The average radius appeared to be between 4.5 − 4.75 mm. The runners we used for ice friction experiments in the ice house had a rocker of 34 m [53] and a radius of 4.75 mm. For this reason our numerical model (F.A.S.T.) uses those values as our normal runner rocker and cross-sectional radius. I also consider a range from R = 20 − 48 m and r = 4.0 − 5.5 mm.

In designing runner proﬁles in the future, I suggest placing a larger focus on the center of the runner where the runner generally contacts the ice. With the runner survey complete, we have completed our analysis of the runner proﬁle. In our next chapter, we tackle the other side of the interface, ice. 47

Chapter 3

ICE HARDNESS ANALYSIS

3.1 Background

In order to further our examination of the interaction between a steel blade and ice in the context of winter sports, we are now moving to the other side of this contact. This chapter will focus on ice surfaces used in sport, speciﬁcally their hardness. In order to determine the dynamic ice hardness, we dropped steel balls ranging from (8 − 540) g onto several diﬀerent ice surfaces from heights between (0.3 − 1.2) m. This chapter is largely taken from our paper published in Cold Regions Science and Technology [52].

The measurements, analysis, and the writing of the paper were performed by me. The coauthors acted primarily in an advisory role. Professor Lozowski and Amy Johnston also assisted with some of the experimental measurements.

Athletes in the sliding sports (bobsleigh, skeleton, and luge) observe that ice quality, and therefore their maximum speed, is dependent on ice surface temperature and air humidity. The objective of this work is to improve our understanding of how these parameters affect the physical properties of ice, specifically its hardness. Athlete experience and the F.A.S.T. model [12, 49] both suggest that ice hardness affects the coefficient of friction between a blade and ice because it affects the contact area.

Previous experiments have shown that ice hardness depends on the time scale of the measurement process [54]. While previous work focused on slower processes, with applications to subjects such as glacier ﬂow, our work focuses on athletic performance and thus requires short timescale measurements. Despite a detailed investigation of ice surfaces in speed skating, the ice hardness analysis performed by Kobayashi [10] was static 48 and performed in a controlled isothermic environment. It was performed in a laboratory and not on an athletic surface or at time scales relevant to winter sports. A review of the relevant literature indicates that one set of data is consistently accepted to be representative of ice hardness as a function of ice temperature and contact time [1, 39,

54, 55]. While that work considered contact times from (10−4 −104) s, there is only one set of dynamic hardness data. It is measured at a time scale of 10−4 s. In dynamic hardness measurements the ice indentation cannot be described by plastic deformation. There is failure of the ice and fracturing or crushing can be observed. The dynamic hardness data is obtained by dropping a ball onto the ice surface and measuring the diameter of the indentation crater. Their measurements were made in an isothermal environment.

Hence, they may not be representative for ice in an athletic facility that has an internal temperature gradient.

In order to understand whether these previous results might be relevant to our work, we decided to estimate the contact times associated with the sports of speed skating and bobsleigh. First, we assume that the apparent contact area, the indentation left in the ice surface after the contact, is the ratio of the applied load to the ice hardness. We use the dynamic ice hardness data from Fig. 3 in Hobbs [54], which is approximately 50 MPa at −5◦C. Taking typical masses and blade geometries for speed skating (75 kg, 1.1 mm width) and bobsleigh (390 kg or 92.5 kg per runner, 4 mm cross-sectional radius, and 35 m rocker), the contact length between the blade and ice is on the order of (1 − 10) cm for both sports. Considering a velocity range of (10 − 40) m/s, a point on the ice surface is in contact with the blade for ∼ 1 ms. This is a substantially longer contact time than the 10−4 s of Barnes et al. [39].

Barnes and Tabor [38] propose a model for ice hardness as a function of ice contact time and temperature. The indentation hardness of ice can be computed from equa- 49 tion 3.1. µ ¶ 1 µ ¶ 1 n Q P (t, Tk) = B exp 0 (3.1) t nR Tk In this equation, the contact time of an indenter with the ice is t seconds, the temperature

0 of the ice surface is Tk Kelvins. The ideal gas constant is R (cal/K/mol), Q is the activation energy for self diﬀusion (kcal/mol), n is the unitless stress exponent, and B is a model constant. Equation 3.1 is based on the work of Mott [56], which describes the deformation of metals by steady-state creep.

We have analyzed the data from Barnes and Tabor [38, 1], and have observed that equation 3.1 matches all four of their experimental data sets with contact times of 1.5 s or greater. However, it fails for the dynamic hardness data set. Equation 3.1 predicts an ice hardness six times greater than measured for a contact time of 10−4 s, as shown in

Fig. 3.1. The parameters B, n, and Q used in Fig. 3.1 were optimized to ﬁt the data with contact times greater than 1.5 s, presented in the two papers by Barnes and Tabor, while considering the conﬁdence intervals published in their analysis. The values we obtained are B = 15.5 Pa, n = 4.15, and Q = 32.1 kcal/mol.

It is apparent that dynamic ice hardness is a diﬀerent problem that cannot be represented by equation 3.1. Ice is a brittle material and we observe crushed particles at the surface once the rate of energy transmission has exceeded a critical value. Jordaan and Timco [44] conducted experiments pushing a ﬂat indenter into a sheet of ice. The energy required in 0.05 s to create this crushed ice was 0.006 J. A skate blade or bobsleigh runner applies about half that amount of energy in about 10−3 s and the balls in our experiments transfer between (0.02 - 3.16) J to the ice surface in about 10−4 s. In all of our cases, crushed ice was observed. Another method for dynamic hardness testing would be high strain scratch testing [57]. With a maximum speed of 1 mm/s, the strain rate would not likely be high enough to observe the crushed ice mentioned in the previous experiments. Because the contact area is so small, this method could have contact times 50

Figure 3.1: Ice hardness for various times scales and ice temperatures. The points are taken from the line of best ﬁt from Barnes and Tabor [1], while the lines are ﬁtted using equation 3.1. From top to bottom the data points are for time scales of 10−4 s, 1.5 s, 10 s, 103 s, and 104 s. 51 on the order of 10−3 s and it would be interesting to see a hardness measurement based on sliding instead of the top down impacts from other experiments.

Past dynamic ice hardness measurements vary. There are two previous experiments where crater diameter measurements were used and the ice hardness was determined at a temperature of −12◦C. Analyzing the raw data from Table 1 of Timco and Frederking

[2], we determined that the ice hardness was 10 MPa for S1 and S2 freshwater ice. The dynamic hardness was 65 MPa when Barnes et al. [39] examined randomly oriented polycrystalline ice formed from double distilled water. Gagnon and Gammon [45] used a pressure transducer to obtain peak center pressure when dropping a spherical indenter onto ice samples from an iceberg. They observed an ice hardness of 40 MPa. We suspect that the differences may be due to different preparations of the ice, such as temperature control, air content, and / or grain size and orientation. Other factors that could affect the results are the amount of crushed ice, fracture and whether the ice sample had backing such as concrete or was suspended like ice freezing on a body of water. Due to the variability of previously published results, we decided to make our own measurements of dynamic ice hardness specifically for athletic surfaces.

While there are many variations in the preparation of athletic ice surfaces, they are generally built to a thickness of approximately 25 mm, layer by layer, by spraying about

1 mm of water at a time onto a smooth surface. Maintenance includes regular scraping and resurfacing with water. In this study, we have considered only artificially cooled ice surfaces; however, we have examined both flat and non-flat ice, as well as indoor and outdoor venues. The three flat surfaces were the long and short track speed skating facilities and the hockey rink at the Calgary Olympic Oval1. Fig. 3.2 shows thin sections of an ice sample taken from the long track oval, illustrating the macrocrystalline structure of the ice. The figures show that the ice is made up of individual crystals with dimensions

1http://www.ucalgary.ca/oval/facility 52 on the order of 1 cm. The smaller crystals at the bottom of Fig. 3.2a are located at the base of the ice in contact with the artiﬁcially cooled concrete surface. This eﬀect was also observed in Fig. 10 of Kobayashi [10].

Another two surfaces, a mixed use bobsleigh, skeleton and luge track2 and an indoor training facility for bobsleigh and skeleton called the Ice House3, were located at Canada

Olympic Park. The bobsleigh track, our only outdoor facility, resembles a roller coaster course formed of ice. The Ice House is a slide with an almost ﬂat section of ∼ 15 m where the slope is < 1◦, transitioning to a steeper slope of 6.8◦ for about 50 m before climbing to an end point. We limited our testing to relatively level (< 1◦) portions on the non-ﬂat ice surfaces, and measured the hardness over a range of ice temperature and humidity that encompasses most normal competition conditions.

3.2 Experimental Protocols for Ice Hardness Measurements

Steel balls of varying size and mass were dropped from heights of (0.3 − 1.2) m onto the ice surface. As illustrated in Fig. 3.3, each ball was rolled oﬀ a platform ﬁxed at a height hi. The energy from the inelastic collision between the ball and ice formed a crater in the ice with maximum depth δ and diameter dc.

In our experiments, we measured the diameter of the impact crater dc in order to calculate the crater volume and thus the ice hardness. Ice extrusions above the ice surface were not included in measurements. This is a potential source of uncertainty since the extruded ice fragments could potentially support some of the mass of the ball. This eﬀect appears to be negligible due to the small size of the ice extrusions above the ice surface.

Typically, six repetitions of the same drop conditions were used to obtain a mean value of the crater diameter. The ice hardness analysis was based on two key assumptions. The

2http://www.ﬁbt.com/index.php?id=150 3http://www.winsportcanada.ca/facilities/icehouse.cfm 53

Figure 3.2: Thin sections of the ice to show the macrocrystalline structure at the long track ice surface at the Calgary Olympic Oval. a) vertical section transverse to the skating direction, b) horizontal section. The white bars are 1 cm scales. 54

Figure 3.3: Drop apparatus for ice hardness experiments. A ball of mass m is rolled oﬀ a platform set to a height of hi onto the ice surface leaving an indentation crater in the ice. The ball rebounds to a height of hf and out of the indentation crater. 55

ﬁrst is that the steel ball is not signiﬁcantly deformed during the collision with the ice.

This condition is valid in our situation since the dynamic hardness of ice [2, 39] is ∼ 1% of that of an alloyed steel [58, 59] and ∼ .01% of the Young’s modulus of steel [60]. This assumption allows the use of the work-energy theorem in equation 3.2, which equates the loss of energy during the collision, Ei − Ef , and the work required to deform the ice surface. Z δ Ei − Ef = mg(hi − hf ) = − P (x, y, z) · a(z) dz (3.2) 0 where m is the mass of the indenter, g is the gravitational acceleration, hi is the initial drop height, hf is the bounce height, and a(z) is the maximum horizontal cross-section of the impact crater when the penetration depth is z. P (x, y, z) is the resisting pressure of the ice over the entire crater surface.

Martel [61] demonstrated that, in metals, the volume of the indentation is proportional to the energy of the indenter. Using the mean value theorem, an average P (x, y, z) can be removed from the integral in equation 3.2 leading to equation 3.3. We will denote P¯ as the ice hardness. ¯ Ei − Ef = mg(hi − hf ) = PV (dc) (3.3)

The apparent volume V (dc) is the integration volume of the impact crater, assuming that the crater has the radius of curvature of the dropped ball Rb. This assumption can be made if the elastic recovery of the ice surface is negligible. We verify this assumption in our experiments by observing that the bounce height is negligible compared to the drop height4. Hence, the integration volume is: Ã ! 3 2 q 2R 1 ³ ´3/2 R V (d ) = π b + 4R2 − d2 − b 4R2 − d2 (3.4) c 3 24 b c 2 b c

The derivation of this integral is in Appendix C.1. The apparent volume is used in equation 3.3 to calculate the ice hardness P¯.

4 hi = (0.3 − 1.2) m; hf ∼ 1 mm 56

Figure 3.4: Apparent indentation crater volume vs. impact energy used to determine ice hardness. Analysis of the raw data from Table 1 of Timco and Frederking [2]. Indenters of mass (33 − 64) kg with hemispherical tips were dropped from heights of (0.3 − 1.4) m.

In an eﬀort to verify the assumption that V ∝ Ei for ice, we analyzed the raw data (mass, drop height, and crater diameter) from Timco and Frederking [2] and show the results in Fig. 3.4. While the energies from this experiment are about two orders of magnitude larger than those used in our experiments, the experimental data appears to suggest that the apparent volume of their indentation craters increases linearly with the impact energy. While we will need to verify this linearity in our experiments, these results suggest that our assumption is reasonable.

In Fig. 3.4, we observe that a linear fit to the data does not appear to intersect the origin. The negative y-intercept presumably reflects the amount of kinetic energy that does not go into the formation of the impact crater. That energy is represented by Ef 57 in equation 3.3. Given the large masses used in Timco and Frederking’s experiment, we hypothesize that this offset was possibly caused by deformation and oscillation of the entire ice sheet. Fig. 5 from Timco and Frederking [2] appears to support this hypothesis.

In their experiment, accelerometer data from the ice surface shows oscillations after the impact with a period of (3 − 4) milliseconds.

We suspect that our collisions involve plastic deformation with crack formation, crushing, and extrusion of crushed particles. Hence, ignoring Ef and hf (there is a null y- intercept in our experimental data), equation 3.3 becomes:

E V (d) = i (3.5) P¯

Using equation 3.5, the ice hardness P¯ was determined by plotting the apparent crater volume V (dc) versus the kinetic energy Ei of the indenter. The diameter of the impact crater was measured either by digital photographic analysis using the software package

Tracker [62], or by direct measurement using a caliper. The crater photographs for this analysis were taken with a Nikon Coolpix P6000, in close up mode, from a height of 20 cm above the ice. A coin (Canadian dime5) with a diameter of 18.03 mm was used for scale. The protocols were optimized for best photo resolution and minimal distortion, by taking photographs of a sheet of graph paper. Photographic analysis worked well under favorable light conditions, Fig. 3.5; but it proved more diﬃcult in low light conditions,

Fig. 3.6. In the end, measuring with the caliper proved more eﬃcient. Both methods were compared during one series of measurements and they produced equivalent data.

In Figs. 3.5 and 3.6, the crater is not perfectly circular because of irregular ice extrusion around the perimeter and cracking at the ice surface. Ice extrusions above the surface were not included in measurements. Due to the irregular shapes of the impact craters, each one was measured along two perpendicular axes, as shown in Fig. 3.6. For each ice

5http://www.mint.ca/store/mint/learn/10-cents-5300008 58

Figure 3.5: An impact crater (upper left) at the bobsleigh track in Canada Olympic Park made on December 16th, 2009. Ball C was dropped from a height of 30 cm in favorable light conditions. 59

Figure 3.6: An impact crater from the long track rink of the Calgary Olympic Oval made on December 30th, 2009. Ball C was dropped from a height of 90 cm in poor light conditions. 60

Table 3.1: Mass and radius of the balls used for ice hardness experiments ball mass (g) radius (mm) A 8.366 ± 0.002 6.346 ± 0.001 B 28.17 ± 0.01 10.028 ± 0.001 C 151.3 ± 0.1 16.67 ± 0.02 D 286.8 ± 0.1 20.65 ± 0.02 E 358.5 ± 0.1 22.24 ± 0.02 F 535.5 ± 0.1 25.41 ± 0.02 temperature, ball and height, drops were repeated six times in order to obtain a mean value for each case. The ball diameters were measured with a caliper or micrometer and their masses were determined using a digital scale (ACCULAB VIC-303 or VIC-5101).

Six balls were used in these experiments; their radii and masses are listed in Table 3.1.

We attempted to use the greatest possible range in size and mass that would produce a measurable indentation, without significantly damaging the ice for future users of the facilities. In Appendix C, we verified that the ice hardness does not vary with different balls used in the analysis. Ice surface temperature measurements were made throughout the data collection period with an AMR (Ahlborn) THERM 2280-2 controller and a

T122-1 150 11K temperature probe.

We measured more than 850 impact craters. While all six balls were used initially, the procedure was optimized to improve eﬃciency. The number of balls was eventually reduced to four and the number of drop heights was also reduced in order to avoid overlapping impact energies. This had the unanticipated beneﬁt of reducing experimental variability. By performing all of the drops over a shorter period of time, we reduced the possible changes in air and ice temperatures during the experiment. Balls A, B, C, and E were dropped from 30 cm, all of those except A from 90 cm, and only ball E from 60 cm. These conditions correspond to energies of (.02, .08, .25, .44, 1.05, 1.33, 2.10, and

3.16) Joules. Ball D was eliminated to avoid overlapping energies. Ball F was not used 61

Figure 3.7: Plot of ice crater volumes vs. impact energy of dropped balls to determine ice hardness. Data from drop testing experiments performed on Feb. 10, 2010 in the Ice House at Canada Olympic Park. The ice surface temperature was −2.7◦C. in the later experiments because it produced more damage to the ice surface, including radial cracks. While the number of measurements changed during our experiments, all measurements are valid and are included in our analysis.

3.3 Dynamic Ice Hardness Data and Analysis

In Fig. 3.7 we illustrate one typical example of the data generated from drop testing hardness measurements performed by our group. This particular example used the optimized experimental protocols discussed in the previous section to avoid the overlapping initial energies. This data set was generated through the analysis of 48 drop craters. The 62 uncertainties are extrapolated from the uncertainties in the mass, radius, drop height, and the crater diameter. The uncertainty in both axes is indicated. However, the uncertainty in the impact energy is very small. The crater diameter is the dominant source of uncertainty. The relative uncertainty in the diameter of the crater is a constant for all but the smallest ball, A. Because the sample size for each energy is small, six measurements, uncertainties for the crater diameters were analyzed using data from every drop. ¯ ¯ The result was ∆dc/dc = 0.10 for ball A and ∆dc/dc = 0.066 for all the other balls. The detailed analysis can be found in Appendix C.2. Due to varying uncertainties, a weighted linear regression [63] was used to ﬁt the point data. The linearity of the data conﬁrms that P¯ is constant over this range of impact energies. The zero intercept suggests that ¯ Ef is indeed negligible. Hence P behaves as if it were a property of the ice, namely its hardness, which is simply the inverse slope of the line. The impact crater volume was calculated with equation 3.4.

We have performed experiments on five different ice surfaces. In an initial effort to observe differences between similar ice surfaces, we set three ice surfaces within the

Calgary Olympic Oval at the same temperature, -1.1◦C. These were the hockey rink, the long track oval and three different locations on the short track rink that experience different amounts of skating traffic; the curve, the straightaway and the center of the ice.

This analysis is illustrated in Appendix C.3. There was no significant difference between the five measurements. Therefore we have compiled this data together as one ice surface temperature.

The indoor ice facilities are limited to ice surface temperatures of about −5◦C or higher. We have shown that range of temperatures in Fig. 3.8. The displayed temperatures are averages of multiple measurements taken over the course of the drop tests. The error bars are the standard deviation of those measurements. While most points are the result of analyzing between 40 and 100 ice craters, the point that examined ﬁve diﬀerent areas 63

Figure 3.8: Ice hardness vs. ice surface temperature as measured on the bobsleigh track (squares) and the ice house (X) at Canada Olympic Park, as well as all three ice surfaces at the Calgary Olympic Oval (diamonds) for ice surface temperatures greater than -5◦C. within the Olympic Oval is based on over 400 craters. For each point, the data were plotted as in Fig. 3.7 and the slope measured in order to determine the ice hardness. The scatter in the data from the bobsleigh track appears greater than any difference between the ice surfaces. Because it is difficult to identify any systematic difference between the various venues, we decided to compile the results from all of our ice surfaces together in a single graph. That data was used to determine the variation of ice hardness with ice surface temperature. We have made additional measurements over a range of ice surface temperatures typically encountered during bobsleigh and speed skating competitions.

These ice hardness measurements were added to the data from Fig. 3.8 and are shown in Fig. 3.9. 64

Figure 3.9: Ice hardness vs. ice surface temperature from various ice surfaces at Canada Olympic Park and the Calgary Olympic Oval. The weighted linear ﬁt and the conﬁdence limits are the central and two outer lines respectively. 65

Most of the data points in Fig. 3.9 are from the bobsleigh track at COP. We observed a greater temperature range there because it is outdoors. Due to varying uncertainties in both axes, the data points were ﬁtted using a weighted linear regression [63] yielding:

P¯(T ) = ((−0.6 ± 0.4)T + (14.7 ± 2.1)) MPa (3.6) where T is the ice surface temperature in degrees Celsius. It is unlikely that the ﬁt will be valid for temperatures warmer than −1◦C because ice hardness decreases rapidly as the melting point is approached. This point is illustrated in Fig. 3 of Barnes et al. [39].

The applicability of these experiments to the sport of bobsleigh and our computational model (F.A.S.T.) was veriﬁed by measuring the width of the track left by a bobsleigh training sled sliding on the ice. The bobsleigh runner surface contacting the ice can be approximated by a torus with a major axis (rocker) R = 34 m much greater than the minor axis (cross-sectional radius) rc = 4.75 mm. The measurements of the tracks left in the ice were performed by photographic analysis using the Tracker software package. Due to the small size of the tracks, we felt that it was too diﬃcult to analyze with a caliper.

We examined images from the Ice House at Canada Olympic Park and the hockey rink at the Calgary Olympic Oval. One example of the tracks left in the ice is illustrated in Fig.

3.10. The ice surface temperature was −(2.5 ± 0.5)◦C and the sled was moving ∼ 1 m/s for these measurements. We determined that the contact width was (0.8 ± 0.2) mm.

The intersection of a torus (runner) and a plane sheet of ice can be approximated as an ellipse. Given that the elastic recovery of the ice has been shown to be negligible in this experiment (Ef = 0), we have assumed that the rear portion of the blade does not contact the ice when it is in motion. This leads to a semi-elliptical contact area.

µ1 w ¶ F = m g = PA¯ = P¯ πl (3.7) r 2 s 2

The fraction of the mass distributed to each rear runner was mr = 35 kg. The mass of the entire sled as well as the distribution to the front and rear runners was determined with 66

Figure 3.10: Examination of the indentation track left by a bobsleigh runner on a level ice surface, major axis or rocker of 34 m minor axis of 4.75 mm. The blue arrow is the calibration scale, the pink arrows are measurements of the groove width. The sled had a weight of 123 kg and 56% of it is distributed to the rear runners. A Canadian dime was used for scale. It is difficult to precisely determine the groove width. We have made two attempts to measure the width in this figure. The arrow on the left is an attempt to make the largest measurement of the groove. The arrow on the right is an attempt to make the smallest measurement of the groove. Several similar measurements were made of multiple grooves. All the measurements were used to obtain an average result and a standard deviation. 67 a Mettler Toledo 2155 floor scale, the distribution left to right was assumed to be equal.

Since the front runners carried less of the mass, it was assumed that their indentation would be smaller. Both the contact length ls and the contact width w can be related to the maximum depth the runner penetrates into the ice δmax.

q 2 ls = 2Rdmax − dmax (3.8)

q 2 w = 2 2rcdmax − dmax (3.9)

Since we can measure w, equation 3.9 can be re-arranged into, s w2 d = r − r2 − (3.10) max c c 4

Knowing the maximum depth of penetration, we can rearrange equation 3.7 using equations 3.8 and 3.9 in order to determine the ice hardness.

m g 2m g P¯ = r = q r (3.11) A 2 2 π (2Rdmax − dmax)(2rcdmax − dmax)

The track width analysis results in an ice hardness of 16 ± 9 MPa. Our drop testing experiments provide an ice hardness of 16 ± 3 MPa over the same temperature range.

This suggests that our drop experiments are appropriate to determine the ice hardness for this sport.

Feedback from athletes in sliding sports suggests that the ice quality, and therefore peak velocities, can be reduced by high air humidity. In an eﬀort to explain the scatter in

Fig. 3.9, we have attempted to determine whether air humidity affects ice hardness. We attempted to remove the temperature effect so that any effect of humidity could be seen.

In Fig. 3.11 we plotted the diﬀerence between the linear ﬁt and the point data in Fig. 3.9 versus the absolute humidity. Despite the appearance of a decrease in ice hardness with increasing humidity, the uncertainty in the slope from the regression of the data in Fig.

3.11 is greater than the slope itself. This provides inconclusive results. Our humidity data 68

Figure 3.11: Plot of the variation between measured ice hardness and the linear ﬁt in equation 3.6 vs. absolute humidity to determine the eﬀect of absolute humidity of air on ice hardness 69 were taken from the Environment Canada weather station located at Canada Olympic

Park. Only ice hardness data from the bobsleigh track was used in Fig. 3.11. It should be noted that there is a correlation between low temperatures and low humidity and that accurate humidity measurements at temperatures below zero can be quite diﬃcult [64].

Because of these facts and the absence of an unequivocal trend in Fig. 3.11, these results are inconclusive. Further work will be required to determine whether a relationship exists between humidity and ice hardness, or whether the eﬀect observed by athletes might be the result of deterioration of the ice surface quality due to frost build up.

3.4 Conclusions

This study has provided new data on ice hardness for athletic ice surfaces. We have observed that the ice hardness is ((−0.6 ± 0.4)T + (14.7 ± 2.1)) MPa and we have incorporated it into the F.A.S.T. model (see chapter 4). Ice hardness at athletic ice surfaces was 22 ± 7 MPa at −12◦C, a value that falls between the results of previous studies

[39, 2, 45] where diﬀerent types of ice were used as well as diﬀerent controls on ice fracture. The temperature dependence of ice hardness observed here is ∼ 20% of that observed in the dynamic impact experiments of Barnes et al. [39] and ∼ 62% of that observed by Gagnon and Gammon [45]. We have observed, with a null y-intercept in Fig.

3.7, that elastic recovery of the ice surface was insignificant in our impact tests. We verified that our experiments match observations with real sports equipment. An attempt to extract a systematic effect on ice hardness due to air humidity was inconclusive.

Now that we have examined the ﬁner points of both the blades used in the sport of bobsleigh and athletic ice surfaces, we are able to examine the interaction between the two. 70

Chapter 4

F.A.S.T. MODEL

4.1 Introduction to Ice Friction

The very low friction coeﬃcient of ice can be experienced on any winter day. While it is very well known, it is not nearly as well understood. In surveying the relevant subject matter, Rosenberg [65] covers the major bases of the problem. He does not however, put the puzzle together to calculate the coeﬃcient of friction of an object on ice.

In this Chapter, I attempt to put the puzzle pieces together. Rosenberg states that since wet surfaces are more mobile and thus more slippery than dry, the explanation for the low friction of ice is as simple as determining what leads to melting on the surface of ice. The dry ice friction measurements of Bluhm et al. [66] suggest the same conclusion.

One piece of the puzzle is pressure melting, a component in the low friction of ice that was studied in detail by Colbeck [67]. While on its own the eﬀect cannot explain melting beneath a skate blade at ice temperatures below −2◦C, it does contribute to the process.

Frictional heating alone cannot explain the low coeﬃcient of static friction on ice but it too contributes to the low friction on ice [68, 69]. The key starting point to the low friction on ice is the quasi-liquid layer that exists on the surface of ice below its melting point [70].

I am following up on the work of Penny et al. [12] by adapting their numerical model to simulate bobsleigh runners sliding on ice. I use this model to calculate the coefficient of friction between the ice surface and bobsleigh runners of a given profile. The ultimate goal is to optimize the profile in order to reduce friction and improve performance. 71

4.1.1 Introduction to the Model

The model I am currently using is called the Frictional Algorithm using Skate Thermo- hydrodynamics (F.A.S.T.) version 3.2b, where the b is for bobsleigh. The model from which version 3.2b was developed was originally written to describe ice friction in speed skating [12]. The 3.2b model calculates the dynamic coefficient of friction of a blade already in motion. The model accounts for three main points. First, it calculates the force required to plough a groove through the ice. Second, the melting point of ice is adjusted to account for the pressure applied by the blade. Finally, the Couette flow between the blade and ice is used to determine the shearing force between the blade and the liquid melt layer. The shearing force is inversely proportional to the thickness of the melt layer between the blade and ice. This thickness depends on six factors, including a pre-existing quasi-liquid layer, present at the front of the blade where it contacts the ice surface. As the blade glides over the surface of the ice, the thickness of the melt layer varies along the length of the blade according to the other five factors. These factors include the shear from the Couette flow, three heat conduction terms, and the squeeze flow which escapes from under the blade’s surface at the sides. The model calculates the thickness of the melt layer in sections along both the length and width of the blade-ice contact area. It is a finite-difference calculation from the front of the runner to the rear. Each section across the width is considered independently from the others.

In the remainder of section 4.1, I discuss the adaptation of F.A.S.T. 1.5 obtained from Penny et al. I will brieﬂy cover the major changes that I have made to the model.

A detailed derivation of the full model will follow in section 4.2 and beyond.

4.1.2 F.A.S.T. 1.5

The F.A.S.T model was originally published by Penny et al. [12]. Their code, as I received it, was written in Fortran and is included as Appendix D.1. This model was used to 72 calculate the coeﬃcient of friction between a vertical speed skate blade and ice. Penny et al. refer to the model as 1.0. When I received the code, certain changes had been made and it was labeled F.A.S.T. 1.5. These changes include a term to calculate the eﬀect of heat conduction from the blade into the melt layer and an attempt to account for the stride duration of the skater. The model was compared to the experimental friction measurements of DeKoning et al. [11] and was found to be in reasonable agreement [12].

4.1.3 Extensions

Speed Skate Models

Upon receiving the F.A.S.T. 1.5 model, I translated the code into C++. This exercise was an eﬀort to familiarize myself with the code. Once translated, the two codes were compared by running a number of simulations calculating the coeﬃcients of friction for various ice surface temperatures and skater speeds. All results from the two models were equivalent. At this point, the Fortran model was put aside and work continued with the new C++ code.

The second step in adapting the F.A.S.T. model was a verification of the parameters used in the model. A number of the parameters in the model were adjusted, such as the altitude where the experiment was performed, the latent heat of fusion of ice, and the thickness of the quasi-liquid layer on the ice surface. I believe that the adjustment of these and other parameters allow them to better reflect reality. However, the adjustments had little to no affect on the results from the model. It was not sensitive to these factors.

I also removed the code describing the skating stride. My ultimate goal was to model the motion of bobsleigh runners and strides are not relevant to that problem.

With the help of Oval staﬀ, ice surface and basal ice temperatures from the Olympic

Oval were compared. After discussions with Tracy Seitz [71], the former track manager at Canada Olympic Park and current manager at the Whistler Sliding Centre, I changed 73 the basal ice temperature in the model from a constant -18◦C to 2◦C less than the ice surface temperature. Because this change did impact the results, this parameter may need further analysis in the future. Further study could be performed by placing temperature probes at diﬀerent depths and measuring the temperature gradient in the ice.

At the beginning of this project, the composition of the standardized F.I.B.T. steel was not known. For proprietary reasons, there is no published literature on the runner steel composition. On October 28, 2008, I had the opportunity to speak with the runner inspector during the America’s Cup competition in Park City, Utah [72]. The inspector informed me that the F.I.B.T. standardized steel was based on a 17-4 steel. He stated that the steel is a 17-4 with dopants, which the officials can pick up in their analysis, and the physical properties are equivalent to a 17-4 steel. With this knowledge I was able to find the steel’s properties (thermal conductivity, density, and specific heat) from a number of suppliers [59, 73] and adapt the code accordingly.

The next step in the evolution of the model was the determination of the variation in the melt layer thickness. The thickness of the melt layer in the model is h. This thickness is equal to the quasi-liquid layer thickness at the front of the blade and it increases along the length of the runner. The F.A.S.T. 1.5 model calculated the variation in melt layer thickness in terms of ∆(h2) along the length of the runner. It was believed that this avoided a singularity in the calculation [12]. This is not necessary. The singularity can be avoided by beginning the calculation not where the runner first contacts the ice at the front of the blade (x = 0) but at the first step in the model. A second required assumption is that the melt layer thickness must always be greater than zero. I converted the model to calculate the variation in the melt layer thickness in terms of ∆h. This simplified the calculations. The F.A.S.T. 1.5 code included a condition which prohibited the melt layer thickness from decreasing towards the rear of the blade. In the original model this was required to prohibit the melt layer thickness from decreasing to zero 74 thickness at the front edge of the blade resulting in unrealistically high friction values.

This was not justified by Penny et al. [12]. I modified the condition to state that the melt layer thickness could never be less than the quasi-liquid layer thickness because we assume this melt layer thickness on the ice surface. This clause was not called upon in the calculations presented in this thesis. It may no longer be required since the basal ice temperature is no longer -18◦C. The modified clause permits the melt layer thickness to decrease but it cannot decrease to zero. With these changes, I felt the model had taken on a significantly new form and I began referring to it as F.A.S.T. 2.0.

The hardness of the ice surface came into question while reviewing the model. The

F.A.S.T. 1.5 code used indentation lengths resulting from a skater stepping onto the ice to determine the ice hardness. This type of measurement is very sensitive to motion of the skater and diﬃcult to execute. Further, this process takes substantially longer (seconds) than the time required to pass over a given piece of ice while skating (milliseconds).

I was concerned that the ice hardness was being underestimated, because ice hardness decreases with contact time [54]. For this reason, a new analysis of ice hardness was undertaken and is described in chapter 3. In the F.A.S.T. 1.5 code, the static contact length between the blade and ice ls is calculated using the force balance equation 4.1.

F l = g (4.1) s P¯ w

¯ Fg is the force the blade applies to the ice surface, P is the dynamic ice hardness, and the blade width is w. Chapter 3 also addresses the elastic recovery of the ice surface after impact. Assuming that impacts and blade impressions deform and damage the ice similarly, this elastic recovery is the possible recovery of the ice surface behind the blade.

The eﬀect is illustrated in Fig. 4.1. A complete recovery of the ice surface behind a blade moving towards the right is illustrated in a). It could also illustrate a motionless blade. An ice surface with negligible recovery is illustrated in b) and a partial recovery is illustrated 75

Figure 4.1: Diagram illustrating the possible recovery of the ice surface (grey) behind a passing skate blade (black). The arrows indicate the direction of the motion. a) Complete recovery b) No recovery c) Partial recovery 76 in c). In the F.A.S.T. 1.5 code, it was assumed that, if the recovery of the ice was less than complete, the contact length would be less than that of a static blade. The contact length of a blade in motion would be a value between 0.5 and 1 times ls, as illustrated in Fig. 4.1 c). In chapter 3, I have shown that the elastic recovery of ice for low impact energies is negligible in Fig. 3.7. Despite this observation, I maintain that the dynamic contact length must always be ls as calculated in equation 4.1. If the contact length ¯ is less than ls the ice hardness would have to be greater than P . There is no physical justiﬁcation for the ice hardness to be greater than P¯. Therefore, the contact length is ls and the contact has the form of Fig. 4.1 b). This results in a deeper penetration into the ice surface and a longer contact length than the F.A.S.T. 1.5 code. This leads to an increase in both the ploughing and shear forces.

F.A.S.T. 2.0b

F.A.S.T. 2.0b is the first version of the model to address the differences between skate blades and bobsleigh runners interacting with ice. The first major difference between the two disciplines is the shape of the blade. In speed skating, the surface of the blade is flat across the width, in bobsleigh it is convex with a cross-sectional radius rc, as shown in Fig. 4.2. The cross-sectional radius of a 2-men bobsleigh runner is greater than 4 mm; for a 4-men bobsleigh it is greater than 6 mm. The blades in speed skating are also much smaller in both width (∼ 1 mm) and length (∼ 0.4 m) than in bobsleigh (∼ 14 mm and ∼ 1 m). This difference in the size and shape of the blades means that the shape of the contact with the ice in bobsleigh is different than in speed skating. The shape of a bobsleigh runner contact with the ice is a segment of a toroidal surface. It should be noted that I assume the rocker, or the major axis of the toroid, is a constant. Runner profiles from chapter 2 show that this is not the case over the entire runner. However, since the contact length on flat ice is only about 5 cm, the rocker can be considered 77

Figure 4.2: On the left an illustration of the geometry of a skate blade, on the right a picture of a bobsleigh runner 78 constant over that length. The recovery of the ice behind the passing blade is negligible.

Therefore, I have approximated the projection of the contact area onto the ice surface as a semi-ellipse. The force balance equation was used to calculate its major and minor axes. The F.A.S.T. 1.5 code was written in Cartesian coordinates. For bobsleigh, it was useful to reorient the coordinate system so that cylindrical coordinates could be used to calculate the squeeze ﬂow with the same z axis as the Cartesian system I use for the rest of the calculations. The squeeze ﬂow escapes from under the sides of the blade.

Other parameters needed to be changed to reflect the change to bobsleigh. These include the weight distributed to each runner, the rocker, and cross-sectional radius of the runner. These modifications forced a re-evaluation of the ploughing force, the contact width of the runner with the ice, and the squeeze flow in this new model. Like the speed skate model, this version of the model only examines the contact of one blade with the ice. This model assumes that the runner is gliding on flat ice of constant slope. The model does not currently address the problem of a bobsleigh in a turn.

F.A.S.T. 3.0b

Fig. 4.3 illustrates the projection of the contact area onto the surface of the ice, caused by a bobsleigh runner and a speed skate. Assuming that the motion is parallel to the orientation of the blade, along the z axis (x in the F.A.S.T. 1.5 code), for a bobsleigh runner, a point on the ice under the center of the blade is in contact with the blade longer than a point near the edges (max and min y positions). This causes the melt layer to be thicker in the center of the blade than at the edges, for a given value of z. For the skate blade, this phenomenon does not occur since the contact time is independent of y. The thickness of the melt layer is independent of y for a skate blade. For a bobsleigh runner, the contact time between a point on the ice and the runner is a function of y.

Consequently, the thickness of the melt layer is a function of both y and z for a bobsleigh 79

Figure 4.3: The bobsleigh runner causes an elongated, semi-elliptical contact in the ice surface (top) while the skate blade causes a rectangular contact (lower). Impressions are not to scale. runner. This added an extra dimension to the numerical calculation.

F.A.S.T. 3.1b and 3.2b

These two versions of the model address the fact that a bobsleigh has four runners contacting the ice at all times. First, I assume that the problem is symmetrical. The right and left sides of the sled are assumed to react in the same manner and have the same weight distribution. Discussions with athletes [74] and measurements performed on the training sled used in the Calgary Ice House suggest that the weight distribution to the rear runners is generally 55 to 60% of the total sled weight. This weight distribution can vary from sled to sled and with athlete preference. I have used the ratio of the runner carrier lengths for the front and rear runners to determine the weight distribution to the front and rear runners in the model. The ratio of the runner carriers ﬁts the approximate weight distribution used by athletes. In a 2-men bobsleigh, the holes used to attach the front runners to the runner carriers, and thus the sled, are separated by a distance of

585 mm. For the rear runners, the separation is 743 mm [75]. In Fig. 2.1 these holes 80 are used to bolt the runner to a stand. Using these distances, 44% of the weight is distributed to the front runners and 56% to the rear. The total weight of a 2-men sled with competitors is 390 kg.

F.A.S.T. 3.1b considers front and rear runners which are perfectly aligned so that the rear runner passes in the track of the front runner leaving only one groove in the ice. If this occurs, as the rear runner ﬁrst reaches the groove left by the front runner, it immediately contacts the entire width of the groove left by the front runner. However, the contact length will be inﬁnitesimal. In order to increase the contact length, the runner will descend deeper into the ice until the runner is occupying a contact area large enough to support the weight distributed to that runner. The contact area of the rear runner can no longer be approximated by a semi-ellipse. If we examine the top image in Fig.

4.3, the actual contact area is the left half of the semi ellipse. I approximate it as a trapezoid (see Fig. 4.4). The minimum width corresponds to the maximum width left by the front runner ymax and the length lsr and maximum width ymaxr are determined by the force balance equation. In this version of the model, it is possible for some of the melt layer from the front runner to be present when the second runner arrives. To determine whether this occurs, I use the distance between the front and rear runner contact points with the ice and the speed of the sled to calculate the time between the two contacts.

I then use the heat ﬂux into the ice surface to determine whether the melt layer has the time to refreeze before the passing of the second blade. In my simulations refreezing occurs for all cases where the ice surface temperature is less than -2◦C.

F.A.S.T. 3.2b considers a front and rear blade which are slightly out of alignment so that the two runners leave two independent tracks in the ice. In this case, the code is simply two versions of F.A.S.T. 3.0b, with slightly diﬀerent weight distributions, working independently to calculate the total ice friction on the sled. 81

Figure 4.4: A diagram of the contact area in the ice surface of the front (top) and rear (bottom) runners in the F.A.S.T. 3.1b code. The ﬁgure is not to scale.

4.2 Development of the Theory

Now that I have addressed the new contributions to the model I will provide a detailed description of the current state of the model (F.A.S.T. 3.1b and 3.2b).

4.2.1 Contact Between the Runners and the Ice

The ﬁrst step in the model is to establish the contact proﬁle between the runners and ice.

The contact area in the plane of the ice surface, Af (front runner) and Ar (rear runner) is determined by the ice hardness, P¯ and the fraction of the gravitational force of the sled, Ffg (front) or Frg (rear) exerted by the runner onto the ice.

F A = fg (4.2) f P¯

The weight distribution to the front and rear runners was addressed in section 4.1.3.

The maximum depth of penetration into the ice, dmax can then be calculated with the 82 geometry of the blade as well as the angle of tilt of the runner with respect to the surface of the ice. The tilt of the runner is determined by the recovery of the ice surface after the blade has passed. In chapter 3, it was observed that recovery of the ice surface is negligible on athletic surfaces such as those used in the sport of bobsleigh. Fig. 4.1 b) illustrates the shape of the contact from a side view. For the front runner in F.A.S.T.

3.1b and both runners in F.A.S.T. 3.2b, the top view of the contact area is a semi-ellipse as illustrated in Fig. 4.4. These three runners will be addressed before examining the rear runner in F.A.S.T. 3.1b. Examining Fig. 4.4, equation 4.2 can be expressed as equation

4.3, where ymax is the maximum half width of the front runner contact and ls is the maximum contact length. F π A = fg = · y · l (4.3) f P¯ 2 max s

A schematic of the maximum penetration depth dmax as seen from both the side (z, R, ls) or front (y, rc, ymax) is found in Fig. 4.5. As mentioned in chapter 2, for the F.A.S.T. model, rocker values of R = (20−48) m and cross-sectional radii of rc = (4.0−5.5) mm are examined. Using Fig. 4.5, the Pythagorean theorem, and simple algebra we can determine the maximum contact length and half-width in equations 4.4 and 4.5 respectively.

q 2 ls = 2Rdmax − dmax (4.4)

q 2 ymax = 2rcdmax − dmax (4.5)

This relationship can be simpliﬁed by examining the orders of magnitude for the contact

−3 −5 parameters (R ∼ 10 m, rc ∼ 10 m, dmax ∼ 10 m). Thus, equations 4.4 and 4.5 can be simpliﬁed into equations 4.6 and 4.7 respectively.

q ls = 2Rdmax (4.6)

q ymax = 2rcdmax (4.7) 83

Figure 4.5: Schematic of the penetration of the bobsleigh runner into the ice. This ﬁgure depicts the geometry of the penetration from either a side view (ﬁrst parameters) or head-on view (second parameters). Images are not to scale. 84

These two equations can be inserted into equation 4.3, resulting in equation 4.8.

F π q q fg = 2r d 2Rd (4.8) P¯ 2 c max max

Equation 4.8 can be re-arranged to solve for dmax in equation 4.9. F √fg dmax = ¯ (4.9) πP Rrc This describes the penetration depth of the front runners in both F.A.S.T. 3.1b and 3.2b.

The solution for dmax can also be inserted into equations 4.6 and 4.7 to determine ls and ymax. For the rear runners in F.A.S.T. 3.2b, we simply replace the weight distributed to the front runner Ffg with that distributed to the rear Frg to obtain the rear runner penetration depth dmaxr. The rear runners in F.A.S.T. 3.1b follow in the track left by the front runners. As the rear runner ﬁrst contacts the groove left by the front, it covers the entire width of the groove but has no contact length. To increase the contact length the runner begins to penetrate deeper into the ice until the contact area is large enough to support the weight distributed to the rear runner. This contact area, as viewed from above the ice, will be approximately trapezoidal (see Fig. 4.4). It will have a minimum half-width of ymax from the front runner, a maximum half-width of ymaxr and a contact length of lsr. This results in a contact area described by equation 4.10.

F A = rg = (y + y ) · l (4.10) r P¯ max maxr sr The contact width can be found using the same method as for the front runners resulting in equation 4.11. q 2 ymaxr = 2rcdmaxr − dmaxr (4.11)

The contact length can also be found using the same method. However, the penetration depth begins at dmax at the front of the runner. q 2 lsr = 2R (dmaxr − dmax) − (dmaxr − dmax) (4.12) 85

In this case, there is no analytical solution for dmaxr. Therefore the F.A.S.T. model solves for it numerically. We begin the calculation by posing dmaxr = 2dmax. We then calculate a test parameter in equation 4.13 and evaluate equation 4.14. µ q ¶ q 2 2 test = ymax + 2rcdmaxr − dmaxr 2R (dmaxr − dmax) − (dmaxr − dmax) (4.13) ¯ ¯ ¯ Frg ¯ ¯test − ¯ ¯ P > 0 (4.14) test

If the inequality in equation 4.14 is true, we decrease the value of dmaxr, if it is false we

Frg increase its value. This is repeated until the desired precision (|test − P¯ |/test < 0.001), is obtained.

4.2.2 Ploughing Force

The cross-sectional area, perpendicular to the motion, being ploughed by the runners is af (front) and ar (rear). This area is used with the ice hardness to calculate the ploughing force resisting the motion of the sled. Since the ice hardness measurements were made using vertical drops and this is a transverse shear, we are assuming an isotropy in the ice hardness. Any anisotropy is assumed to be negligible at the current precision of ice hardness measurements. For the F.A.S.T. 3.1b code, the rear runner ploughs through the entire front groove so only the cross-section of the rear runner is required to calculate the ploughing force in equation 4.15.

−→ ¯ Fp = −P arvˆ (4.15) ar is the area of a circle of radius rc that is truncated at a height of dmaxr and is described by the integral in equation 4.16. √ Z Z 2 2 rc rc −x ar = √ dy · dx (4.16) 2 2 rc−dmaxr − rc −x This solves as equation 4.17. Ã Ã !! q 2 π dmaxr 2 ar = rc − arcsin 1 − − (rc − dmaxr) 2rcdmaxr − dmaxr (4.17) 2 rc 86

For the F.A.S.T. 3.2b model there are two distinct runner grooves. In that case, the rear runner ploughing area is also described by equation 4.17. However, dmaxr has a diﬀerent value than in the F.A.S.T. 3.1b code. The front runner ploughs through an area described in equation 4.18. Ã Ã !! q 2 π dmax 2 af = rc − arcsin 1 − − (rc − dmax) 2rcdmax − dmax (4.18) 2 rc

The ploughing force for the F.A.S.T. 3.2b model is described by equation 4.19.

−→ ¯ Fp = −P (af + ar)ˆv (4.19)

4.2.3 Pressure Eﬀect on Melting Point

While pressure melting is not sufficient to explain low friction on ice at temperatures below −2◦C, the pressure does affect the melting point of ice. That effect is described by the Clausius-Claperyon relation in equation 4.20 [76].

−8 ◦ Tm = −7.37 × 10 C/Pa (4.20)

The model takes into account this depression of the melting point due to pressure.

4.2.4 Couette Flow

Couette flow is the laminar flow of liquid between two plates which move relative to one another. In the case of a bobsleigh runner gliding on ice with a thin melt layer, this flow causes a drag between the runner and the ice. For a front runner this drag is described in equation 4.21 [77]. −→ −→ X µw v Fsf = − ∆y · ∆z (4.21) j,k hj,k The velocity of the runner over the surface of the ice is −→v . The dynamic viscosity of

−3 water is µw = 1.79 × 10 kg/m/s [78, 77]. j and k are integer steps in the y and z directions. The thickness of the melt layer between the runner and the ice at a given 87

point of the contact area (j, k) is hj,k. The step size along the width of the runner is ∆y and the step size along its length is ∆z. The Couette ﬂow is summed for every of j and k in the contact area and an identical calculation is performed for the rear runner. The values of ∆y and ∆z were reduced until the model results converged. The values used in the model were ∆y = 10−7 m and ∆z = 10−6 m. The calculation of the melt layer thickness along the length of the runner is described in the following subsections.

The Couette Flow for both the front and rear contact surfaces are summed and added to ploughing force to calculate the total resistance caused by the interaction of the runners with the ice surface. This force is divided by the total weight applied to the ice (Ffg +Frg) to calculate the coeﬃcient of friction between the runners and the ice.

Pre-Existing Melt Layer

Previous studies suggest that ice below its melting point is covered by a quasi-liquid layer

[79, 80]. This model assumes that the thickness of the lubricating melt layer at the front edge of the runner, where it ﬁrst contacts the ice, has the thickness of this quasi-liquid layer.

◦ −1/2.4 hj,k = 3.5 (−Ti/ C) nm (4.22)

The temperature of the ice surface is Ti. In this work, we examine ice surface temperatures ranging from −(2−24)◦C. This film should be considered as a supercooled fluid. However, it is a small amount of fluid that is warmed quickly by the passing blade. Thus, when calculated in the model, the effect of warming this fluid was too small to cause any variation in the calculated coefficient of friction. For this reason, we consider the layer as water at the freezing point during the entire calculation. The quasi-liquid layer provides the thickness of the melt layer where the front edge of the runner contacts the ice surface z0(y). Examining Fig. 4.3, we observe that this front edge contact occurs at different positions along the length of the runner for each y. The thickness of the melt layer varies 88 along the length of the runner due to shear stress, heat conduction, and squeeze flow. In the model, the calculations for each value of y are performed independently.

Melt Due to Shear Stress

The shear stress caused by the Couette ﬂow generates power that increases the thickness of the melt layer. Using equation 4.21 we can determine the power generated by the shear stress in equation 4.23. 2 −→ −→ µwv Pshear = Fsf · v = ∆y · ∆z (4.23) hj,k This power transfer term will be added to the conduction terms to determine the total power transfer into the melt layer. All the following terms are calculated for both the front and rear runners.

Three Conduction Terms

There are three heat conduction terms that determine the eﬀect of heat transfer into or out of the melt layer: slow conduction in the ice, fast conduction from the melt layer into the ice surface, and conduction from the runner into the melt layer. The slow conduction term arises because artiﬁcial ice has a cooler temperature at its base than at its surface.

The power transferred from the surface of the ice [81, 82] is described by equation 4.24.

ki (Ti − Tb) Pslow = − ∆y · ∆z (4.24) hice

◦ The thermal conductivity of ice is ki = 2.25 W/m/K. This is for Ti = −5 C [83]. The ice surface temperature and the ice base temperature are represented by Ti and Tb and the thickness of the ice is hice = 25 mm [71, 84]. After discussion with Tracy Seitz, former track manager at Canada Olympic Park in Calgary and current manager at the Whistler

Sliding Centre, I have decided to set the ice base temperature at two degrees less than the ice surface temperature in the model. This parameter could beneﬁt from further study. 89

The second conduction term is the fast conduction. It explains the power transfer caused by the temperature diﬀerence between the liquid melt layer at Tm and the ice surface at Ti [81, 82]. This relationship is described in equation 4.25. s ki (Tm − Ti) vρiciki Pfast = − √ ∆y · ∆z = − (Tm − Ti) ∆y · ∆z (4.25) πκit π(z0(y) − z)

The local contact time between the runner and the ice is t and κi is the thermal diffusivity of ice. The position along the length of the blade is z and the z position corresponding to the first contact with the ice at the front of the blade for a given y is z0(y). The specific

3 ◦ heat of ice is ci = 2.04 × 10 J/kg/K, set for −9 C [85, 83], and the density of ice is

3 ◦ ρi = 917.5 kg/m , set for −4 C [86, 83]. A temperature dependence could be added to ρi, ci, and ki. However, in sensitivity testing of the F.A.S.T. model, results were not found to be very sensitive to these parameters, so this was not seen as necessary.

The third conduction term examines the heat transfer at the surface of the runner.

This term is very similar to equation 4.25 and is described in equation 4.26. s ks (Ts − Tm) vρscsks Prunner = √ ∆y · ∆z = (Ts − Tm) ∆y · ∆z (4.26) πκst π(z0(y) − z)

The temperature of the runner is Ts. After discussions with an experienced international bobsleigh oﬃcial [87] and examination of the runner temperatures from the December

rd ◦ 3 , 2010 World Cup 2-men bobsleigh race, Ts was set to 0.2 C greater than Tm. This parameter could be the hardest to determine. It is highly dependent on the weather, and more specifically the sun. The athletes attempt to keep their runners as warm as possible without exceeding the reference runner temperature by more than the 4◦C permitted in the rules [75]. In the event that the reference runner is too warm, the race officials are permitted to cool all the teams’ runners on the ice before they start the competition. On a cool evening, the runners can be as cold as the ice. On a warm sunny day, the runners will be warm as well. In sensitivity testing the model results have been sensitive to variations of the runner temperature. This parameter could benefit from further analysis. The steel 90

3 ◦ ◦ parameters (ρs = 7800 kg/m , cs = 460 J/kg/ C, and ks = 14.0 W/m/ C) were set for a 17-4 steel [59, 73].

By combining the four power transfer terms we arrive at equation 4.27

P = ∆y · ∆z · ... Ã s s ! 2 µwv ki (Ti − Tb) vρiciki vρscsks ... − − (Tm − Ti) + (Ts − Tm) hj,k hice π(z0(y) − z) π(z0(y) − z) (4.27)

3 5 We use ρw = 1000 kg/m as the density of water and lf = 3.34 × 10 J/kg for the latent heat of fusion for ice. We can then describe the power required to melt a slab of ice in equation 4.28. ∆z · ∆y · ∆h P = ρ l j,k = ρ l v · ∆y · ∆h (4.28) w f ∆t w f j,k By equating equations 4.27 and 4.28, we can determine the eﬀect of the power transfer terms on the thickness of the melt layer along the length of the runner in equation 4.29. ∆h 1 j,k = · ... ∆Ãz ρwlf v s s ! 2 µwv ki (Ti − Tb) vρiciki vρscsks ... − − (Tm − Ti) + (Ts − Tm) hj,k hice π(z0(y) − z) π(z0(y) − z) (4.29)

Squeeze Flow

The final term affecting the thickness of the melt layer is the squeeze flow. The runner is compressing a liquid layer. Thus, we expect that some of this liquid will escape from under the blade’s surface to the sides. This is called the squeeze flow and it reduces the thickness of the melt layer.

In describing the contact area between the bobsleigh runner and the ice surface I assumed that the ice hardness was a constant. Therefore, the ice applied constant pressure to the entire contact surface of the blade. This assumption was supported by the results of Martel [61] and Fig. 3.7, which demonstrate that the average ice hardness remains constant despite a variation of orders of magnitude in the volume of the indentation. If 91 we use this assumption for the squeeze flow, we obtain a constant pressure in the melt layer under the blade and a null pressure where the blade is no longer present. Under this description, near the side of the runner the pressure in the melt layer varies from the ice hardness to zero in one unit step (∆y) of the model. This steep gradient creates a very large squeeze flow. In fact, using these assumptions, the squeeze flow eliminates the entire melt layer resulting in unrealistically high friction values. To avoid this dilemma, we returned to the assumptions made in F.A.S.T. 1.5. This was option was chosen because of the success of the F.A.S.T. 1.5 model in predicting the coefficient of friction in speed skating. Under those assumptions, the pressure is maximum at the centre of the blade and null at the edges. This leads to a realistic solution. However, we may be underestimating the squeeze flow and thus, the coefficient of friction.

This analysis assumes a constant pressure along the length of the blade so that the

flow is only out the sides. Because ls À ymax, the sides will be the dominant source of squeeze flow. For a particular section of the runner, the squeeze flow is illustrated in

Fig. 4.6. Because the centre of the runner is the initial point of contact with the ice, the melt layer will always be thickest at its centre. The melt layer thickness across the runner width is calculated in Appendix D.3. An exact solution of the squeeze flow would be very computationally expensive. As an approximation of the squeeze flow, we assume that the melt layer is of constant thickness across its width, equivalent to the minimal melt layer thickness hmin,k. Because it is the aperture for the squeeze flow this thickness is the limiting factor in the flow. Only one calculation is performed for each ∆z step in the model and it will be applied for all the y values for that value of z. The minimum thickness is illustrated in Fig. 4.6 by the dashed line and this assumption is equivalent to supposing the fluid is stagnant below the dashed line. 92

Figure 4.6: Illustration of the squeeze ﬂow for a section of the runner. The pressure of the runner squeezes the melt layer out the sides, following the arrows. The dashed line corresponds to the minimum melt layer thickness. The ﬁgure is not to scale. 93

Analyzing the squeeze flow begins with the Navier-Stokes equation 4.30. Ã ! ∂−→v ρ f + −→v · ∇−→v = −∇p + µ ∇2−→v + f (4.30) w ∂t f f w f −→ Here vf indicates fluid velocity in the melt layer, p is pressure in the melt layer, f indicates external forces, and t is time. In an effort to calculate the squeeze flow, we assume negligible accelerations, external forces and that the velocity gradient is continuous throughout the melt layer. We also assume that the pressure is constant both radially from the runner surface and along the length of the runner. The Navier-Stokes equation can then be reduced to equation 4.31. We are using cylindrical coordinates that share the same center as the cross-sectional radius of the runner surface.

2−→ ∇p = µw∇ vθ (4.31)

Where vθ is the velocity component in the liquid which is parallel to the runner surface in Fig. 4.6. By expanding equation 4.31 and using the boundary conditions vθ(r = rc) = vθ(r = rc + hmin,k) = 0, we can solve the second order diﬀerential equation to obtain the velocity of the liquid in equation 4.32. Ã ! 1 ∂p hmin,k · ln(r/rc) vθ(r) = r − rc − (4.32) µw ∂θ ln(1 + hmin,k/rc)

While hmin,k corresponds to the dashed line in Fig. 4.6, we are assuming that line corresponds to the ice surface. The continuity equation for this problem can be reduced to equation 4.33. Ã ! 1 ∂(rv ) ∂v r + θ = 0 (4.33) r ∂r ∂θ Using equation 4.33 and the boundary condition that the velocity at the bottom of the melt layer is null, vr(r = rc + hmin,k) = 0; we can solve the diﬀerential equation and determine the radial velocity in the ﬂuid at the surface of the runner in equation 4.34. Ã Ã !! 2 3 2 hmin,k ∂ p 1 1 hmin,k ∂ p vr(r = rc) = − 2 rc + hmin,k − ≈ − 2 2 µwrc ∂θ 2 ln(1 + hmin,k/rc) 12µwrc ∂θ (4.34) 94

−6 −3 The approximation is the result of stating hmin,k ∼ 10 m is much less than rc ∼ 10 m. Using the approximation, equation 4.34 can be rearranged into the constant second order diﬀerential equation 4.35. 2 3 ∂ p vr(rc)hmin,k 2 = − 2 (4.35) ∂θ 12µwrc

Using the boundary conditions, p(±θmax) = 0, where the farthest width of the runner begins to touch the ice, and Ã ! ∂p = 0 (4.36) ∂θ θ=0 due to symmetry in the middle of the runner, we can solve the diﬀerential equation for the pressure within the melt layer below the surface of the runner in equation 4.37.

2 ³ ´ 6vrµwrc 2 2 p(θ) = − 3 (arccos(1 − d(z)/rc)) − θ (4.37) hmin,k

The penetration depth of the runner at θ = 0 is d(z). We can now use the force balance equation 4.38. Z θ0 Afr(z) ¯ p(θ) · cos θ · dA = mg = P · Afr(z) (4.38) −θ0 At The positions on each side of the runner where the runner begins to contact the surface

0 of the ice are ±θ = arccos (1 − d(z)/rc). At is the total contact area between the runner ¯ and ice and Afr(z) is the fraction of the surface between z and z + ∆z. P is the ice hardness. By completing the integration and solving for the velocity of penetration into the melt layer we arrive at equation 4.39. This is equivalent for all values of j across the width of the runner. ∆h ∆h ∆z v (r ) = j,k = j,k · = ... r c ∆t ∆z ∆t ¯ 3 (4.39) P hmin,k sin (arccos (1 − d(z)/rc)) ... = − 2 12µwrc (sin (arccos (1 − d(z)/rc)) − (1 − d(z)/rc) arccos (1 − d(z)/rc)) Now, using the angle θ0, we obtain equation 4.40.

¯ 3 0 ∆hj,k ∆hj,k ∆z P hmin,k sin (θ ) vr(rc) = = · = − 2 0 0 (4.40) ∆t ∆z ∆t 12µwrc (sin (θ ) − (1 − d/rc) θ ) 95

This equation can then be re-arranged to give the squeeze ﬂow eﬀect on the thickness of the melt layer in equation 4.41.

¯ 3 0 ∆hj,k P hmin,k sin (θ ) = − 2 0 0 (4.41) ∆z 12vµwrc (sin (θ ) − (1 − d(z)/rc) θ )

It is clear that my initial supposition that the pressure in the melt layer is a constant is not possible. It is however possible that the pressure in the melt layer is constant over some domain and drops off over some distance greater than the step size used in the model. This would lead to a larger squeeze flow than we predict in the model and thus, higher friction. The presented solution for the squeeze flow was a first attempt to quantify this phenomenon for the sport of bobsleigh. This aspect of the model could use further study.

To recap the operation of the F.A.S.T. 3.2b model, we begin by using the ice temperature and runner proﬁle to determine the geometry of the contact between the runner and the ice. Next, we use the contact geometry to calculate the ploughing force. Then we adjust the point of fusion of ice according to the Clausius-Claperyon equation. The quasi-liquid layer thickness is then calculated and it is used as the thickness of the melt layer at the front of the blade. The next step is to begin the loop to calculate the melt layer thickness hj,k for every value of y and z in the contact area between the ice and the front runner. This is done by computing the shear stress, the three conduction terms, and the squeeze ﬂow. The hj,k values are used to calculate the shear stress between the runner and the melt layer, and that is summed for every value of y and z. The process is then repeated for the rear runner. Next, the ploughing forces for the front and rear runners and the total shear force over both the front and rear runner contact areas are added to determine the total resistive force from the interaction with the ice surface.

Finally, this total resistive force is divided by the total down force applied to the ice by the runners (Ffg + Frg) to obtain the coeﬃcient of friction between the runners and the 96 ice surface.

4.3 Simulated Data and Analysis

After examining the theory behind the F.A.S.T. model we will now examine some of the results obtained from those calculations. We begin by examining the constituent components of the F.A.S.T. 3.2b model in Fig 4.7. It should be noted that, it is not possible to have a shear force that is not affected by heat conduction into the ice, or any of the other factors that affect the melt layer thickness. This is a computational exercise to verify the relative effect of the various components included in the F.A.S.T. 3.2b model. The slow conduction is less than 0.1% of the heat conduction into the ice surface; therefore, this factor could be ignored. The squeeze flow calculated in this model also has little impact on the calculated shear stress. The maximum observed effect of the squeeze

ﬂow in Fig. 4.7 is 0.7% of the shear stress at 41 m/s. The two largest contributors to the shearing force are the melting caused by the shear stress and the heat conduction into the ice. The ﬁrst is monotonically increasing with sled speed, while the second decreases monotonically with speed. At low speeds, the heat conduction into the ice appears to limit the melt layer thickness, leading to high friction.

In Fig. 4.8, we illustrate the eﬀect of the ploughing force in the F.A.S.T. 3.2b model.

The ploughing force is only dependent on the ice hardness and the blade profile. It is independent of sled speed. The resisting forces on a bobsleigh runner are plotted as a function of ice surface temperature in Fig. 4.9. The ploughing force increases with increasing temperature. This suggests that the effect of ploughing through a larger area is greater than the effect of ploughing through softer ice. As observed in Fig. 4.8, the ploughing force is independent of the sled speed. It is the same in all three figures. The temperature dependence of the shear force varies with sled speed. At low speed, the shear 97

Figure 4.7: The calculated shear force between the bobsleigh runner and the ice for components of the F.A.S.T. 3.2b model. The blue curve examines the melt layer caused by shear stress only. The green curve includes the conduction of heat into the ice via slow and fast heat conduction. The red curve adds the effect of heat conduction from the runner and the black curve includes the squeeze flow to complete the F.A.S.T. 3.2b model. This is a test of the model to determine the relative significance of each of the terms included in the model. The ice surface temperature was −6◦C. 98

Figure 4.8: The resisting forces on a bobsleigh runner as calculated by the F.A.S.T. 3.2b model. The solid line is the total resisting force. The curved dotted line is the shear stress and the straight dotted line is the ploughing force. The ice surface temperature was −6◦C. 99

Figure 4.9: The resisting forces on a bobsleigh runner as calculated by the F.A.S.T. 3.2b model for sled speeds of a) 16 m/s, b) 26 m/s, and c) 36 m/s. The dotted line is the ploughing force, the dashed line is the shear force, and the solid line is the total resisting force (ploughing + shear). 100 force decreases with temperature at a greater rate than the ploughing force increases leading to a total resisting force that decreases with increasing temperature. At 26 m/s, the slopes of the ploughing and shearing forces nearly cancel each other leading to an almost constant resisting force over the temperature range. At 36 m/s, there is a reversal in the total resisting force. The shearing force begins to level over the temperature range.

Thus, the total resisting force increases with increasing temperature.

Next, we calculate the coefficient of friction between runners and ice of a constant slope, for typical ice temperatures, sled velocities, cross-sectional runner radii, and runner rocker. First, we examine the F.A.S.T. 3.1b model. In this model the front and rear runners are perfectly aligned so that the rear runner passes in the groove left by the front runner. In my experience as an athlete in the sport of bobsleigh, we attempted to align the front and rear runners perfectly. The two front runners were aligned with steel shims as small as 0.13 mm. The runner alignment was verified with an alignment bar to assure they were running parallel. The same was done for the rear runners. According to Fig 3.10, the track width left by a runner is about 0.8 mm wide. This was a lighter training sled. The F.A.S.T. model suggests that the front runners of a competition sled with athletes leaves a track of approximately 1.2 mm. While it was more difficult to align the rear runners to the front, using a straight edge or a thin wire, I would expect significant overlap. In my experience observing grooves left in the ice by a sled, I observed only one track. My experience observing these grooves is very limited and took place at the Ice House (see Fig. 3.10) and the Calgary Olympic Oval. On a bobsleigh track there are generally too many grooves to identify those left by an individual sled.

The coefficient of friction between bobsleigh runners and a flat ice surface, calculated using the F.A.S.T. 3.1b code, is shown in Fig. 4.10. This figure uses standard values for rocker (R = 34 m) and cross-sectional radius (rc = 4.75 mm). We observe that for Ti = −2◦C, the coefficient of friction is significantly reduced compared to all other cases. This 101

Figure 4.10: The coefficient of friction between bobsleigh runners and a flat sheet of ice for different sled speeds and ice surface temperatures from -2 to -14◦C. Both the front and rear runners follow in the same track (F.A.S.T. 3.1b). 102 ice temperature is the only one that was tested where the melt layer created by the front runner had not completely refrozen before the arrival of the rear runner. In my experience in the sport of bobsleigh, I have never witnessed such low friction performance at warm temperatures. Consultation with Olympic bobsleigh coaches suggest that this does not occur in practice [88, 89]. It should be noted that the athletes push the sled to speeds exceeding 10 m/s. Velocities below that are not relevant to the sport, but they could be interesting to help us learn more about ice friction and understand the correlation between low speed ice friction measurements and performance in sport [10, 27, 90, 53].

The front to rear alignment is more diﬃcult than aligning left and right runners to parallel. Further, the track width of approximately 1.2 mm is very small compared to the distance between the front and rear runners (> 1 m). For these reasons, it is reasonable to examine the possibility that the rear runner follows in a parallel track not intersecting the groove left by the front runner. This would eliminate the possibility of the melt layer remaining for the rear runner at warmer ice surface temperatures. This calculation is performed by the F.A.S.T. 3.2b code and the results are found in Fig. 4.11.

The runner parameters are the same as for Fig. 4.10. The results from F.A.S.T. 3.1b and

◦ 3.2b are very similar, with the exception of Ti = −2 C. The coeﬃcient of friction values from F.A.S.T. 3.1b are about 3.5% less, on average, than those from F.A.S.T. 3.2b for velocities exceeding 10 m/s. Because the results from F.A.S.T. 3.2b appear to correlate better with observations by athletes and coaches, speciﬁcally for warm temperatures, I have focused on this model for further analysis.

Next, the eﬀect of cross-sectional runner radius on ice friction in the sport of bobsleigh is examined using the F.A.S.T. 3.2b model. Simulations were performed similar to Fig.

4.11, varying only the cross-sectional radius and reducing the amount of data obtained and thus, the velocity resolution. The results were then compared to those from Fig. 4.11.

Decreasing the cross-sectional radius from rc = 4.75 to 4.0 mm produces Fig. 4.12. By 103

Figure 4.11: The coefficient of friction between bobsleigh runners and a flat sheet of ice for different sled speeds and ice surface temperatures from -2 to -14◦C. The front and rear runner leave two parallel tracks (F.A.S.T. 3.2b). 104

Figure 4.12: The relative change in coefficient of friction between bobsleigh runners and a flat sheet of ice when the cross-sectional radius of the runner is decreased from 4.75 to 4.0 mm for different sled speeds and ice surface temperatures from -2 to -14◦C. The front and rear runner leave two parallel tracks (F.A.S.T. 3.2b). 105

Figure 4.13: The relative change in coefficient of friction between bobsleigh runners and a flat sheet of ice when the cross-sectional radius of the runner is increased from 4.75 to 5.5 mm for different sled speeds and ice surface temperatures from -2 to -14◦C. The front and rear runner leave two parallel tracks (F.A.S.T. 3.2b). decreasing the cross-sectional radius of the runner to 4.0 mm we have increased runner friction for all velocities greater than 10 m/s. On average the effect is about 0.75%.

The effect of increasing the cross-sectional radius is shown in Fig. 4.13. The results show that by increasing the cross-sectional radius of the runner from 4.75 to 5.5 mm, we have decreased runner friction for all velocities greater than 10 m/s. On average the effect is about 0.5%. It can be noted that the effect of varying the cross-sectional radius is minimized for lower temperatures. By increasing the cross-sectional radius we are also decreasing the contact length which can impact sled control. This is a greater concern on ice which is cold and hard since controlling the sled is already more difficult. This could 106

Figure 4.14: The relative change in coefficient of friction between bobsleigh runners and a flat sheet of ice when the rocker of the runner is decreased from 34 to 20 m for different sled speeds and ice surface temperatures from -2 to -14◦C. The front and rear runner leave two parallel tracks (F.A.S.T. 3.2b). explain why athletes prefer a smaller cross-sectional radius in colder conditions.

Next, we examine the eﬀect of varying the rocker of the runner, the radius of curvature along its length. Again we perform the simulations using the F.A.S.T. 3.2b model. All of the parameters are the same as for Fig. 4.11 except the rocker. The rocker was decreased from R = 34 to 20 m, the results were compared to those found in Fig. 4.11 and their relative diﬀerences are found in Fig. 4.14. When the runner rocker is reduced to 20 m, ice

/ runner friction increases for all velocities. On average the eﬀect of reducing the rocker from 34 to 20 m is about 25%.

We have also analyzed the eﬀect of increasing the runner rocker in Fig. 4.15. Increasing 107

Figure 4.15: The relative change in coefficient of friction between bobsleigh runners and a flat sheet of ice when the rocker of the runner is increased from 34 to 48 m for different sled speeds and ice surface temperatures from -2 to -14◦C. The front and rear runner leave two parallel tracks (F.A.S.T. 3.2b). 108 the rocker from 34 to 48 m has resulted in a decrease in runner friction for all velocities.

On average, for sled velocities exceeding 10 m/s, the effect is about 12%. Increasing the rocker also reduced the coefficient of friction in the speed skate model [91]. In view of all these results from the F.A.S.T. 3.2b model, we conclude that the ice / runner friction is generally reduced by having a flatter runner. Increasing the rocker has a much greater impact on ice friction than increasing the cross-sectional radius over a reasonable range for both. It must be noted that all of these results have been computed for a flat ice surface. The kreisel corner in Calgary has a radius of curvature of approximately 30 m. If runners with a rocker of 48 m slide through kreisel, the contact is not that described by the F.A.S.T. 3.2b code. There will be two contacts, one at the front and one at the rear of each runner. This would be similar to having a negative rocker while gliding on a flat ice surface. Sean Maw, an expert on the sport of speed skating, suggests that a negative rocker is undesirable in skating [92]. While the F.A.S.T model results suggest that flatter blades result in a reduction of friction the minimum radius of curvature of the track surface may need to be taken into consideration when deciding on the optimum rocker to be used in a competition. Examining ice friction of bobsleigh runners on a curved track could be an interesting future topic of study for the F.A.S.T. model. A varying radius such as that observed in Fig. 2.15 may be an attempt to have a flatter profile on straight sections of the track but a more curved profile in the curves where the contact length is greater due to the centrifugal force. This could be an interesting topic for further study.

The results in this section have generally focused on the F.A.S.T. 3.2b model. Results from the F.A.S.T. 3.1b model were very similar and can be found in Appendix D.3.

There is no universal consensus on what ice conditions create the fastest sled velocities.

Olympic coaches do agree [88, 89] that if the ice is too warm, and therefore soft, the maximum velocities will be reduced. The same is true if the ice is too cold and hard. The fastest times are believed to occur at ice surface temperatures of −5 to −15◦C. In order 109

Figure 4.16: The coefficient of friction between bobsleigh runners and a flat sheet of ice when the cross-sectional runner radius is 4.75 mm and the rocker is 34 m for different sled speeds and ice surface temperatures from -2 to -24◦C. The front and rear runner leave two parallel tracks (F.A.S.T. 3.2b). to determine whether F.A.S.T. 3.2b predicts a minimum in the coefficient of friction at a particular temperature, a greater range of low temperatures is examined in Fig. 4.16.

At lower speeds, the F.A.S.T. model always predicts a minimum coeﬃcient of friction for warmer ice. However, at 31 m/s the minimum occurs at −10◦C, and at 41 m/s it occurs at −19◦C. Since maximum velocities in many bobsleigh tracks range from 30 − 40 m/s these model results would suggest that minimum friction at top speed, and therefore maximum velocities, could be achieved at temperatures ranging from −10 to −20◦C. In predicting the potential for maximum velocities, the F.A.S.T. model does appear to agree with observations by athletes and coaches [89, 88]. 110

Figure 4.17: The maximum (y = 0) thickness of the melt layer along the contact length of the front runner. The horizontal axis has its origin in the centre of the ice / runner contact length (z = 0). The cross-section radius of curvature is 4.75 mm, the rocker is 34 m, the sled speed is 25 m/s, and the ice surface temperature is −7◦C (F.A.S.T. 3.2b).

4.4 Limitations of the F.A.S.T. Model

Every computational model has limitations due to its assumptions. One of the limitations of the F.A.S.T. 3.2b model is in predicting performance at warm temperatures. Previous studies show that ice hardness decreases rapidly at ice temperatures above −1◦C [39, 76].

At these temperatures, the ice hardness used in the model is no longer valid.

Fig. 4.17 illustrates the maximum (y = 0) melt layer thickness along the contact length of the runner. Because the melt layer was only microns thick, the eﬀect of capillary bonding was ignored in the F.A.S.T. model. On a warm and wet ice surface the model could be improved by considering the eﬀect of capillary bonding [93, 94]. These warm 111 conditions are not common and this was not a priority for this study.

The F.A.S.T. model assumes that both the runner and the ice surface are perfectly smooth. Snow and / or frost could increase friction between a bobsleigh runner and ice but they are not accounted for in the current model. It is assumed that runner preparation by the athletes leaves the surface of the runner significantly smoother than the thickness of the melt layer. This is not likely the case at the front and side edges of the runner where the melt layer is thinner. This has not yet been considered because it is a very small fraction of the total runner surface. It is also assumed that minor asperities in the ice surface are ploughed through by the runner. Larger irregularities in the ice surface can cause chatter, or vertical movement, in the runners and the entire sled. This could affect performance by using the sled’s kinetic energy to perform work on the sled, displacing it vertically. Since the original design of the F.A.S.T. model rule changes now permit officials in the sport of bobsleigh to pass over runners with a fine abrasive. The head coach of Canada’s bobsleigh team has informed me that at recent F.I.B.T. meetings the use of more coarse abrasives has been approved to reduce speed on certain tracks [88]. If the use of coarse abrasives becomes common, the surface roughness of the runners may need to be taken into account in the F.A.S.T. model.

The F.A.S.T. 3.2b model has not yet been adapted to study a curved ice surface. This would be a logical extension of this work. Another interesting area of study would be non-parallel front and rear runners. This is analogous to examining the eﬀect of steering on friction in bobsleigh.

4.5 Conclusions

The F.A.S.T. 3.2b model has been able to reproduce a number of qualitative observations made by athletes and coaches in the sport of bobsleigh. Specifically, the maximum veloc- 112 ities occurring at ice surface temperatures of −10 to −20◦C and the athletes preferring a smaller cross-sectional radius on colder harder ice. The model was used to calculate the effect of cross-sectional runner radius and rocker on the coefficient of friction. The model suggests that flatter runners have reduced friction and that rocker has much larger impact on coefficient of friction than cross-sectional radius for a reasonable range of both.

The model performs ice friction calculations on ﬂat ice. At this time, the model cannot predict the performance of a runner in a curve. For this reason, the ﬂattest runners may not result in the best performance on an actual track. Ice / runner friction in curves should be studied further. In the next chapter, we leave the computer model behind and study ice friction where it counts, on the ice. 113

Chapter 5

ICE FRICTION EXPERIMENTS

5.1 Introduction

With the computer simulations completed, we are interested in the agreement between the F.A.S.T. model results and data from real bobsleigh runners on ice. Much of the material from this chapter has been taken from our paper accepted with minor revisions by Sports Engineering [53] and our commissioned report submitted to the F.I.B.T. [95].

Since our submission of the report to the F.I.B.T., further data on the proﬁle of the track has been obtained (Appendix E.4) which has changed some of the results. I designed and oversaw the experiments, analyzed the results, and wrote the papers. I was aided in the low speed data acquisition by Amy Johnston, Professor Lozowski, and Professor

Thompson and by 14 volunteers, whom I trained and supervised for experiments during the 2-men bobsleigh World Cup. My coauthors acted primarily in an advisory role.

In the sport of bobsleigh, maximum speeds are limited by the coeﬃcient of friction between bobsleigh runners and ice, and the aerodynamic drag. Understanding these two physical parameters can help us to better predict the movement of a sled down a track and the maximum speeds that can be obtained on newly designed tracks. The last three bobsleigh tracks were designed for the 2002, 2006, and 2010 Olympics. On these tracks, athletes have exceeded computer simulated maximum speeds [96]. This result, along with a review of the literature, led us to believe that there is a lack of scientiﬁc data on drag and friction for real equipment in competitive situations. Our group, with the support of the international body for bobsleigh and skeleton (F.I.B.T.), sought to contribute more data on this subject, in order to improve future track design. 114

As is commonly found across the breadth of research in sport, much of the work in this field is based on improving performance and is confidential. Our knowledge of previous experiments is limited to published work. A number of groups [15, 16, 17, 18, 19] have used numerical simulations and wind tunnel testing of scaled models to study the aerodynamics of bobsleighs. Gibertini et al. [18] have also performed wind tunnel testing on full-sized sleds. The drag coefficients of various sleds and models range from (0.2 -

0.5).

Over the years, other groups have been motivated to learn more about ice friction in sport. A world champion in the sport of skeleton, Kristan Bromley, has discussed the issues relevant to ice friction in that sport without reporting any measurements [7].

Experiments have been performed to measure ice friction in a laboratory setting [34, 30].

The dynamics of those experiments are diﬀerent than using real sports equipment since the slider is not sliding on virgin ice. Penny et al. [12] formulated a numerical model,

F.A.S.T. 1.0, with a fully lubricated or hydrodynamic contact, in order to compute the coeﬃcient of friction between a speed skate blade and ice. For ice temperatures of −1 to −10◦C and average speeds of (1 − 10) m/s, the friction coeﬃcients computed by the

F.A.S.T. 1.0 model are comparable to measurements of actual skate blades performed by

Kobayashi [10] and Koning et al. [11]. In those speed skating experiments, the coeﬃcient of friction varied from (3 − 10) × 10−3.

The coeﬃcient of friction between bobsleigh-like runners and ice has been previously measured at low speeds (∼ 1.5 m/s) [27] and the results ranged from (6 − 12) × 10−3.

However, it is not clear that these low speed measurements are valid at higher speeds.

In fact, Figs. 4.10 and 4.11 suggest the coefficient of friction at those lows speeds can be quite different than at 15 to 35 m/s, speeds more typically observed in bobsleigh. At much higher speeds, a track’s built-in timing system was used to determine the coefficient of friction in luge (14 × 10−3) [4]. Radar guns providing a single velocity measurement 115 of a 4-men bobsleigh at multiple locations along a track were also used to determine the drag performance (0.56 m2) and the coefficient of friction (30 × 10−3) [20] in 1979.

Both these methods provide very little data over a very complex track. There were up to

12 measurements taken over an entire bobsleigh course. A bobsleigh course can be from

(1200 − 1750) m in length and have between 13 and 20 curves. The analyses from these two examples require using data that included sled movement in curves and assumptions made to describe the motion of the sled in the curves are diﬃcult to validate. The motion is diﬃcult to analyze because the amount of steering in the curves is not known and steering is widely believed to have a large impact on the ice friction.

In this work, we performed both low and high speed ice friction and air drag experiments. The low speed experiments were performed on the Hockey rink at the Calgary

Olympic Oval1 and the Ice House2, a bobsleigh training facility at Canada Olympic Park.

The high speed measurements were performed during a World Cup 2-men bobsleigh competition in Calgary, Canada3. To simplify the problem, curves were excluded from the present experiment. Radar guns were set up to determine the acceleration of the sleds in the Oval, the Ice House, and several straight sections of the bobsleigh track, each with constant inclination.

5.2 Ice Friction Experiments

5.2.1 Low Speed

The low speed experiments were conducted on four separate days in a one month period, two days at the Oval on ﬂat ice and two in the Ice House on an inclined ice surface.

The inclination of the ice surface in the Ice House was (6.80 ± 0.02)◦. Details of the slope

1http://www.ucalgary.ca/oval/facility 2http://www.winsportcanada.ca/facilities/icehouse.cfm 3http://www.ﬁbt.com/index.php?id=150 116 measurement can be found in Appendix E.2. For each run, an athlete pushed the training sled to an initial speed, then released it to glide freely for about 30 m on either a ﬂat or a constant-incline sheet of ice. The training sled, which is shown in Fig. 5.1 a), is the frame of an actual bobsleigh. It is used by athletes for push training in the Ice House. A real bobsleigh is shown in Fig. 5.1 b) for comparison. The cylindrical box in the centre of the training sled is made of rigid plastic. It covers any additional weight added to the sled so that the cross-sectional area remains constant as the weight of the sled is varied in these experiments. In both the Oval and the Ice House, the radar gun was pointed at the back of the sled and aligned parallel to its motion in order to acquire the speed measurements.

The Ice House setup is illustrated in Fig. 5.2. The ice surface temperature was measured at multiple times during each session with an AMR (Ahlborn) THERM 2280-2 controller and a T122-1 150 11K temperature probe. The ice surface temperatures were between

−2.2 and −4.6◦C for all the measurements. Our work with the F.A.S.T. model suggests that the main effect of temperature on ice friction is its effect on ice hardness. According to Fig. 3.8, this limited variation in temperature would have no significant effect on ice hardness at our level of measurement precision. Therefore, the ice surface temperature was assumed to be the same for all experiments.

The masses of the sled and supplementary weight plates were measured with a Mettler

Toledo 2155 ﬂoor scale. The mass of the training sled was 123 kg, and up to 70 kg of weight plates were added during the experiments. The portion of the runner that contacts the ice surface can be approximated as a segment of a torus. The four runners had cross-sectional radii of rc = 4.75 mm and rocker of R = 34 m. These values are typical for bobsleigh runners. The runner surfaces were polished twice during the month by bobsleigh athletes using twelve grades of silicon carbide ﬁnishing abrasives ranging from (80 − 1) micron in roughness. This is a typical race preparation for the runners.

A Stalker Pro ATS radar gun [97] was used to measure the sled speed as a function of 117

Figure 5.1: A front view of a) the training sled used in our experiments in the Ice House and Oval and b) a typical 2-men bobsleigh similar to those used in the World Cup bobsleigh competition on December 3rd, 2010. 118

Figure 5.2: A diagram of the experimental setup in the Ice House at Canada Olympic Park. The radar gun was aligned parallel to the motion of the sled on the constant inclined section of the push track. time. Example data from the experiments on ﬂat ice are shown in Fig. 5.3. On ﬂat ice, the slopes of the data represent the acceleration due to non-conservative forces. In both the Ice

House and the Oval, the athletes pushed the sled to initial speeds ranging from vi = (1.5− 6) m/s. The sled then continued, unmanned, down the course during the measurement period. The speed data were taken with a precision of 0.03 m/s at a sampling rate of

31.25 Hz, with the digital filter option of the system disabled. Before every use, the gun’s calibration was verified according to the manufacture’s specifications. Generally, the data acquisition time was longer for the slower speeds, the notable exception being the slowest because we did not record speeds much less than 1 m/s.

Fig. 5.4 is an example of measured sled speeds in the Ice House. The diﬀerence between the line, which describes friction free motion on a θ = 6.80◦ slope, and the point data in Fig. 5.4, which are the sled speed measurements v, is the deceleration of the sled due to non-conservative forces. The speeds in the Ice House are generally higher because the sled is traveling down a slope. 119

Figure 5.3: Sample of the radar speed data as a function of time. Six runs with diﬀerent initial velocities were made at the Calgary Olympic Oval. Ice surface temperature was −2.2◦C and no weight was added to the sled.

5.2.2 High Speed

In a similar experiment, the radar guns were set up in four locations along the Olym- pic bobsleigh track in Calgary, during the 2-men World Cup competition on December

3, 2010. We used four radar guns obtained from Stalker4 (2 ATS, 1 Pro, and 1 newer

Pro II). The three older guns collect data at 31.25 Hz while the newer gun has a measurement frequency of 46.875 Hz. The four locations observed were the straight sections of track between corners 1-2, 3-4, 5-6, and 8-9 (see Fig. 5.5). The track inclinations were 6.64◦, 4.59◦, 8.61◦, and 6.79◦ respectively. The track inclination measurements were performed using two Leica TCR total stations and two Leica Sprinter 250M levels by

4http://www.stalkerradar.com/ 120

Figure 5.4: An example of the speeds measured in the Ice House at Canada Olympic Park. The points are the measured speed of the sled and the line is v(t) = vi + gt sin θ. The line represents the speed that the sled would have if there were no ice friction or air drag. The ice temperature was −4.4◦C and no weight was added to the sled. personnel from the Geomatics Engineering Department at the University of Calgary.

Further details are included in Appendix E.4. The radar guns were pointed up the track as shown in Fig. 5.6. In these high speed experiments, the sleds have athletes in them.

At low speeds, it was beneﬁcial to acquire data with no athletes in the sled. This allowed us to access a greater range of relative velocities. Further, at low speeds, movement of the athletes in the sled could aﬀect the movement of the sled. In the bobsleigh track, the athletes have had time to settle in the sled. Therefore, they should not be moving.

Also, because their velocities are much greater, any effect of athlete movements would have a smaller effect on the relative sled velocity. It is also important to acquire data from real sleds in real situations. Because the radar guns measure the fastest object in the field, high speed measurements are not affected by motion outside of the track. This 121

Figure 5.5: Diagram of the bobsleigh track at Canada Olympic Park in Calgary, Canada. The start is at the top of the ﬁgure and the four straight sections of track where the data were acquired are indicated with arrows. 122

Figure 5.6: Radar setup during preliminary testing. The photo is taken from the entrance of corner 4, looking up into the exit of corner 3. In the photo, the radar gun setup is located on the short wall and it is connected to a computer for data acquisition. During the World Cup, the race officials required us to tie the radar guns to the outside of the track so that they could not fall into the path of the competitors. 123 fact is possibly the reason we have not been able to measure velocities below 1 m/s. Since bobsleighs never move at near that speed in competition, we made no efforts to extend low speed measurements below 1 m/s. There should be no difference in targeting the front or the rear of the sled. Our radar gun positioning was chosen first, to best align the radar gun to the sled motion and second, to stay out of the view of TV cameras and officials.

The raw data obtained from the radar guns were reduced to remove data obtained while the sled was not on the constant slope section between two corners. The data were truncated in order to analyze only the constant acceleration portion. The truncated data are very similar to the data in Fig. 5.4, the major diﬀerence being that the speeds are higher on the bobsleigh track.

Sled masses with competitors and ice surface temperatures were obtained after the competition from F.I.B.T. oﬃcials. The measurement methods used were similar to those implemented for the low speed experiments.

There were two heats of 16 competitors in the Calgary World Cup 2-men competition.

A few data points were lost because of errors in data acquisition. Some were discarded because the sleds did not achieve a constant acceleration. Elimination of these data was justiﬁed for each case by visual observations of skids or wall strikes at the track or by subsequent analysis of the race video. A total of 118 out of a possible 128 data sets from the four stations were included in the analysis. Attempts were made to further reduce the size of the analyzed data set by using more stringent video analysis exclusion criteria or by examining only the top ﬁnishers in the competition. However, neither method provided a reduction in the data scatter. 124

5.3 Theory

According to the literature [13, 21, 22, 5], the net forces acting on a sled descending a straight section of track with an inclination θ are described by equation 5.1.

1 F = m a = m g sin θ − µm g cos θ − ρv2C A (5.1) net s s s 2 d

The first term is the acceleration due to gravity on an inclined track, the second term is the ice / runner friction, and the third term is the aerodynamic drag. The mass and acceleration of the sled are ms and a respectively, gravitational acceleration is g = 9.81 m/s2, and the coefficient of friction between the runners and ice is µ, typically assumed to be constant. The density of air is ρ, the drag coefficient of the sled is Cd, the cross-sectional area of the sled is A, and the speed of the sled is v. Cd and A are not the same for our low and high speed experiments, because different types of sleds were used for the experiments (see Fig. 5.1).

Equation 5.1 assumes that the normal force of the sled on the ice results purely from the gravitational force on the sled. Centrifugal eﬀects and lift forces are thus neglected in this equation of motion. Hence, this equation focuses on motion along straight sections of track and it assumes that aerodynamic lift has a negligible eﬀect on the normal force.

Lift forces were typically ignored in the previously mentioned studies, because they reach only 2% of the sled weight at the maximum speeds in our study [16]. Due to their small magnitude and the lack of published literature on the subject, we have chosen to ignore the eﬀect of the lift force in this work.

In our experiments, we measured sled speed as a function of time. The measurements were taken at a frequency of (30 - 50) Hz, depending on the radar gun, and the data were taken over an interval of (1 - 3) s. We used the mean squared speed v¯2 to infer the average air drag during the acceleration of the sled on a particular section of track. By deﬁning initial and ﬁnal speeds, v(0) = vi and v(tf ) = vf , and assuming that the acceleration of 125 the sled is constant, we can determine: R tf v2(t)dt (v − v )2 ¯2 0 2 f i v = = vi + vi (vf − vi) + (5.2) tf 3

The integral is described in detail in Appendix E.1. A constant acceleration of the sled can be assured by limiting the data acquisition time.

In order to analyze the data, we make some assumptions about air drag. These were more significant sources of uncertainty outdoors in the high speed experiments than in the low speed experiments. The low speed experiments were performed with one single sled, removing any variability between sleds. They are also performed indoors, sheltered from any wind. On the bobsleigh track, since we are examining a World Cup field, we assume that all the sleds have a similar air drag performance (CdA). This would be a requirement to be competitive at this level. We also assumed that the atmospheric conditions were similar along the track. Specifically, we assumed that the air density was constant along the track and that the effect of wind was insignificant. The variation in elevation of the locations considered in the high speed experiment was about 80 m, and hence, air density variations were ignored. The orientation of each section of track was different and we did not measure wind velocity in the track. Therefore, variations in wind conditions could be a small source of uncertainty. Wind speeds from (2 - 7) km/h were recorded at the top of the hill during our experiment, but the wind data were used only for the uncertainty analysis (climate data, COP Upper - December 3, 2010 from 18:00 -

20:005). The wind was blowing from the Southwest and its speed may have been reduced in the track by the track walls and because the track is built on the North face of a hill.

With these assumptions, a new drag constant, α, is deﬁned in equation 5.3.

1 α ≡ ρC A (5.3) 2 d 5http://www.climate.weatheroﬃce.gc.ca/ 126

This new drag constant is diﬀerent for the low and high speed experiments because two diﬀerent types of sleds were used.

We can calculate the mean acceleration due to lossy forcesa ¯l by using a small angle approximation (cos θ = 0.99 ' 1) and averaging equation 5.1 to produce equation 5.4.

This approach is key to comparing results from ice surfaces with diﬀerent inclinations.

v¯2 a¯l =a ¯ − g sin θ = −µg¯ − α (5.4) ms

The mean acceleration of the sleda ¯ was determined from a linear ﬁt to the experimental speed data. This allowed us to determine average values for the coeﬃcient of friction between the four bobsleigh runners and the ice as well as the new drag constant for the

2-men bobsleighs in the competition.

5.4 Results

Fig. 5.7 displays the low speed accelerations from lossy forces from both the Olympic

Oval and the Ice House. Using various combinations of vi and ms, the lossy accelerations

¯2 a¯l were plotted as a function of v /ms. A least squares linear regression with standard errors [98, 99] was used to ﬁt the data points. The linear regression curve and the 95% conﬁdence limits are included in Fig. 5.7 and equation 5.5.

kg v¯2 m a¯ = (−0.20 ± 0.02) · − (0.041 ± 0.005) 2 (5.5) m ms s

Using equation 5.4, and 5.5, the slope of the best-ﬁt line was used to determine α =

(0.20 ± 0.02) kg/m for the training sled and the y-intercept was used to determineµ ¯ =

−3 (4.2 ± 0.5) × 10 . Since ms and g are both known to better than 1% accuracy and the speed data are very linear (see Fig. 5.3), the uncertainty in the Oval data was dominated by the variability in the acceleration data. On the inclined ice surface, there was an additional uncertainty due to the uncertainty in the slope measurement. However, it is 127

Figure 5.7: Low speed lossy accelerations at the Calgary Olympic Oval (white hollow circles) and at the Ice House (grey filled circles) with linear regression and confidence interval curves. These experiments were conducted over four days. The ice surface temperatures were between −2.2 and −4.6◦C. less than the scatter in the data. Only the observed data scatter was used to determine the confidence intervals in the linear fit.

On the bobsleigh track, using the speed data obtained from the radar guns, the measured track inclinations, and the sled masses with competitors, the lossy accelerations

¯2 were plotted as a function of v /ms in Fig. 5.8. The results are concentrated into four blocks. These corresponded to the four stations along the track where the measurements were taken. Each point corresponds to one sled in one section, on one particular run. The second block of data has less scatter than the other three. This block of data corresponds to measurements on the section of track between corners 3 and 4, which is the easiest for the athletes to drive. It is also the section of track with the most consistent slope (see

Appendix E.4). Hence, it is not surprising that the results are most consistent there. A 128

Figure 5.8: Individual sled accelerations due to lossy forces vs mean square sled speed divided by sled mass. The slope of the linear fit determines the mean sled drag coefficient, while its intercept determines the mean ice friction coefficient. least squares linear regression with standard errors [98, 99] was used to fit the data points.

This analysis yielded a y-intercept of −µg¯ = (0.035 ± 0.011) m/s2 and a slope of −α =

−0.110 ± 0.005 kg/m. This leads to a coeﬃcient of friction between the blades and ice of

µ¯ = (3.6 ± 1.1) × 10−3. We used data obtained from Environment Canada, speciﬁcally, air pressure p = 88.64 kPa from Springbank6 weather station and air temperature T =

−10.6◦C from the COP Upper weather station, along with a speciﬁc gas constant for dry air of R0 = 287 J/kg/K, to determine the air density from the ideal gas law. p ρ = = 1.18 kg/m3 (5.6) R0(273.15 + T/◦C)K We used this air density and an estimated cross-sectional area of the sleds, A = 0.342 m2 6Air pressure information was not available at the COP weather station. 129

[16], along with equation 5.3, to determine a mean drag coeﬃcient for the sleds used in the competition: 2α C¯ = = 0.55 ± 0.03 (5.7) d ρA

5.5 Discussion and Analysis

In order to combine all the measurements in Fig. 5.7 we assumed that the ice friction was the same in the Olympic Oval and the Ice House and on the various days of data collection, but that is not necessarily the case. In the Olympic Oval, the sled glides on a

flat sheet of ice. In the Ice house the sled runs in two grooves. This causes a difference in the contact area with the ice between the two locations. Since the grooves are cut to a radius of 6.5 mm, the contact area in the Ice House will be wider and shorter than for flat ice. The magnitude of this effect is negligible with 2-men runners but it would be important for 4-men runners, since their radius is approximately 6.5 mm. Another potential difference is the fabric of the ice. The size and orientation of the crystals forming the ice sheet may not be the same at the two locations. However, since the ice surfaces are prepared in a similar manner they are assumed to be the same. Finally, because the runners are not constrained to grooves in the Oval, it is possible that the sled could skid slightly sideways. Qualitatively, this effect appeared to be minor; however, it may explain why there is more scatter in the Oval data than in the data measured in the Ice House.

Given the linearity of the data in Fig. 5.7 (R2 ' 0.95), these eﬀects can be assumed to be negligible at our current level of experimental precision.

The fully lubricated friction model, proposed by Penny et al. [12] suggests that the coeﬃcient of friction can vary with slider speed. The measurements made on speed skates by Koning et al. [11] suggest that the coeﬃcient of friction increases linearly with speed from (4−6)×10−3 over the range v = (4.5−10) m/s, at an ice temperature of -4.6◦C. the 130

F.A.S.T. 3.2b model results suggest that the increase in coeﬃcient of friction with velocity is much less important in bobsleigh (see Fig. 4.11). The maximum predicted increase is

20% between (16 − 36) m/s at an ice surface temperature of −2◦C. Kietzig [100] states that an increase of the coeﬃcient of friction with increasing speed is a characteristic of a fully lubricated contact between an object and ice. Due to the random scatter in our data measurements and the limited range of speeds in our study, linear increases in the coeﬃcient of friction are not easily observed in this work.

In an effort to determine our ability to detect linear variations in the coefficient of friction with speed, we plotted the difference between simulated data and equation 5.5.

Assuming that the coeﬃcient of friction has a mean value of 4.2×10−3 and varies linearly with v at the same rate as observed by Koning et al. 0.33×10−3/◦C [11], we can simulate data such as observed in Fig. 5.7 by using equation 5.4. Subtracting both experimental

(Fig. 5.7) and simulated accelerations from the linear regression, equation 5.5, we obtain

Fig. 5.9. These simulations used the most common parameters from our experiments, mean speeds fromv ¯ = (1 − 9) m/s and weights added to the training sled of 0, 30, and

60 kg. Any further increase in the variation of the coefficient of friction would push the simulated data beyond our experimental scatter. This analysis shows that our confidence in the coefficient of friction is decreased toµ ¯ = (4.2 ± 0.9) × 10−3.

If we assume that the coefficient of friction with ice is constant with velocity, the low speed ice friction results can be used to complement the measurements we have obtained at higher speeds. Possible differences between these two experiments are: variations in the ice quality, ice temperature, and runner surface quality. It has been our observation that the ice quality in the Ice House is more similar to that of the bobsleigh track than that of the Oval. Since the ice friction measurements from the Oval and the Ice House were indistinguishable, we do not believe that the difference in ice quality is a significant factor. However, ice temperature can significantly affect the coefficient of friction. For 131

Figure 5.9: The difference between the accelerations and the linear fit obtained in equation 5.5. The data is both simulated (squares), and experimental from both the Oval (white hollow circles) and the Ice House (grey filled circles) the experiments in the Olympic Oval and the Ice House, ice surface temperatures ranged from −2.2 to −4.6◦C. At the top of the bobsleigh track the ice surface temperatures ranged from −6.1 to −8.1◦C during the competition. Over this temperature range, the coefficient of friction has been shown to increase with temperature in speed skating by

10% [11]. In both experiments, the runners were prepared in a similar manner. However, before an international competition, oﬃcials pass over the runners with sandpaper [3].

In the experiments at the Olympic Oval and the Ice House this was not done. During the Ice House and Oval experiments we ran up to four hours of experiments, or about

60 runs, between runner preparations. We do not believe that the runner surface quality was better than during the World Cup competition. However, without a detailed analysis of the runner surfaces this is diﬃcult to state with conﬁdence.

The drag coeﬃcient was not calculated for the experiments in the Ice House and 132

Olympic Oval because a training sled with no cowling was used. Since the experiments in the Olympic Oval used similar runners, and the measurements were made at speeds as low as 1 m/s, it is possible to extrapolate those accelerations to zero speed. As the speed approaches zero, the aerodynamic drag on the sled has no eﬀect on the result. Hence, it is possible to add these low speed data to the data from the bobsleigh track. This is shown in Fig. 5.10. The error bars were increased by adding the uncertainties observed at the

Figure 5.10: Sled acceleration due to lossy forces vs mean square sled speed divided by sled mass. The slope of the linear fit determines the mean drag coefficient for the sleds, while its intercept determines the mean ice friction coefficient.

Ice House and Olympic Oval in quadrature with 10% of the measurement, which is the expected variation due to the temperature diﬀerence. The four non-zero velocity points correspond to the average lossy acceleration from each station where measurements were taken during the Calgary World Cup competition on December 3, 2010. Error bars are a 133 result of the uncertainty of the track inclination, the standard deviation of the mean of the lossy acceleration data for each station, and the uncertainty caused by the maximum observed wind, all added in quadrature.

We used a weighted linear regression [63] to determine the slope −α = −(0.108 ±

0.014) kg/m and y-intercept −µg¯ = 0.052 ± 0.019 m/s2. This leads to a mean coefficient of friction between the runners and ice ofµ ¯ = (5.3 ± 2.0) × 10−3 and a drag coefficient of ¯ Cd = 0.54 ± 0.07. As previously suggested, other studies show that ice friction in sport increases with speed. If the coefficient of friction did increase linearly with sled speed, it would have the √ ¯2 form of equation 5.8, where vrms = v .

µ(v) = µ1vrms + µ0 (5.8)

Introducing equation 5.8 into equation 5.4 produces equation 5.9.

α ¯2 a¯l =a ¯ − g sin θ = −µ0g − µ1gvrms − v (5.9) ms

The data from Fig. 5.10 was also plotted as a function of vrms in Fig. 5.11 in order to determine whether the coeﬃcient of friction increases with speed. The results from a quadratic regression to the data from Fig. 5.11 is found in Table 5.1. A linear dependance

Table 5.1: Parameters from a quadratic regression of the data illustrated in Fig. 5.11. −4 −4 −2 2 α/ms (10 /m) µ1g (10 /s) µ0g (10 m/s ) −2.70 ± 0.15 −6.5 ± 5.2 −2.9 ± 0.4

of the coeﬃcient of friction with speed is indicated by the linear term (µ1g) in the quadratic regression. These results will be compared to the F.A.S.T. model results in the following section. 134

Figure 5.11: Sled acceleration due to lossy forces vs root mean square sled speed. The linear term in the quadratic regression could represent a variation in the coeﬃcient of friction with speed.

5.6 Comparison to F.A.S.T. Model Results

The F.A.S.T. model predicts a coeﬃcient of friction of approximately 1.5×10−3 for the ice surface temperatures and velocities observed in our experiments. This is approximately one third of the coeﬃcient of friction observed in our experiments. There are a number of factors not included in the F.A.S.T. model that can increase friction in the sport of bobsleigh. The F.A.S.T. model assumes that the runners and ice are perfectly smooth.

However, just before the bobsleigh competition the runners are scratched by oﬃcials in order to inspect for illegal coatings. The runner polish can also be damaged as the sled descends the track. These scratches could be greater than the melt layer thickness 135 calculated by the F.A.S.T. model and increase the coeﬃcient of friction. Runner surface roughness could be an interesting topic for future study.

In section 4.2.4, I discussed the possibility that the squeeze ﬂow may be underestimated in the F.A.S.T. model. If this is the case, it would lead to an underestimate of the coeﬃcient of friction.

The ice surface of a bobsleigh track is scraped by hand and it is not as smooth as ice prepared by a Zamboni. As a sled slides down the track it is constantly chattering up and down. This could be caused by the weight of the sled not being evenly distributed on all four runners due to uneven ice. This chatter is a source of lost energy that translates into an increase in the experimental coeﬃcient of friction observations. If we assume the chatter is caused by a displacement perpendicular to the sled motion, we can determine the increase in the observed coeﬃcient of friction ∆µ. This is done by equating the work performed by the increased friction to the energy required to displace the sled in equation

5.10.

∆µ · msgds = msghs (5.10)

The displacement of the sled along the track is ds and the perpendicular displacement due to chatter is hs. The difference between the F.A.S.T. 3.2b coefficient of friction and the experimental coefficient of friction is about 3 × 10−3. If we assume that the chatter has a frequency of 100 Hz and that the sled is traveling at 20 m/s we can calculate an average perpendicular displacement of 0.6 mm, if this is the only source of the increase in the coefficient of friction. This chatter could also be an interesting topic for future study.

When the runner encounters larger irregularities in the ice surface the groove left in the ice by the runner is occasionally discontinued for a short distance. This has been noted by visual observations of grooves left in the ice by bobsleigh runners at the Ice

House. Restarting the liquid melt layer would also cause an increase in the experimental coeﬃcient of friction. 136

Another potential source of uncertainty in our experiments is the track walls. Since previous studies have shown that track walls can affect the drag coefficient [17], it is likely that proximity to the track walls will also affect the air drag. Therefore, results could be affected by typical drive lines. For example if the preferred drive line is close to one wall the air drag may be higher than if the typical drive line were in the middle of the track, or crossing from one side to another.

Two physical mechanisms: the squeeze flow and the sled chatter, have been suggested as potential sources for the difference between the F.A.S.T. 3.2b model results and the experimental ice friction analysis. It is likely that a combination of both, combined with other factors account for the difference between the model results and experimental observations. We compare the linear variation in the coefficient of friction proposed in Table

5.1 to the F.A.S.T. 3.2b model results for the same average temperature (−7.6◦C) in Fig.

5.12. Because I have identified factors that can account for the difference in the results between the F.A.S.T. model and experimental observations I have included scaled results from the F.A.S.T. 3.2b model. The multiplicative factor of 3.0 was chosen to best overlay both data sets for sled speeds exceeding 10 m/s. The data is suggestive of a possible increase in coefficient of friction with sled speed. The slope of the scaled F.A.S.T. data appears closer to the minimum slope then the mean slope. The F.A.S.T. model results are also very non-linear, especially for sled speeds below 10 m/s. Because of the high degree of uncertainty in the analysis, the fact that any possible quadratic dependence of the experimental coefficient of friction on sled speed is lost in the aerodynamic drag, and that previously published models for sled dynamics [13, 21, 22, 5] assume that the coef-

ficient of friction is a constant; I have decided against including any dependence of the coefficient of friction on sled speed in my conclusions. Further data would be required to determine if a dependence truly exists. Specifically, because of varying centripetal forces, analyzing bobsleighs in curves may allow us to separate a quadratic dependence of the 137

Figure 5.12: Comparison of the linearly increasing coefficient of friction with sled speed ◦ from Table 5.1 and the F.A.S.T. 3.2b results for Ti = −7.6 C. The results calculated from Table 5.1 are in circles with the confidence limits indicated by the dashed lines. The F.A.S.T. 3.2b model results are indicated by the solid black line. The solid red line represents the F.A.S.T. results multiplied by a factor of 3. coefficient of friction on velocity from the aerodynamic drag.

5.7 Conclusions

I have used the acceleration and deceleration of a bobsleigh training sled to determine the coeﬃcient of friction between the steel blades (runners) and ice over a speed range of

(1 − 10) m/s. The coefficient of friction was determined to be (4.2 ± 0.9) × 10−3. These results were added to data taken during a World Cup bobsleigh competition where sleds were observed at speeds from (15−36) m/s. I determined an average coefficient of friction 138 between steel blades and ice, and an average drag coefficient. The measurements were taken on straight sections of track with a constant incline. The coefficient of friction was

−3 2 µ¯ = (5.3 ± 2.0) × 10 . The drag performance was CdA = 0.18 ± 0.02 m , leading to a ¯ drag coefficient of Cd = 0.54 ± 0.07. Attempts to determine a correlation between the coefficient of friction and sled speed indicated a possible increase of the coefficient of friction with increasing sled speed. However, they were not conclusive.

These coefficients of friction are lower than values currently accepted for the sport of bobsleigh and the drag coefficient is higher than any previously published results for bobsleigh. This suggests to me that scientists cannot prepare bobsleigh runners as well as the athletes and that athletes in competition do not behave as well aerodynamically as in a wind tunnel. I recommend that additional data be taken at other tracks in order to avoid systematic errors caused by typical drive lines on a given course. Data of sled movement in curves may help determine if a dependence between coefficient of friction and sled speed exists. Since these parameters are used to predict maximum velocities at newly designed tracks I hope that these new data will allow track designers to better predict the velocities that competitors will achieve on new tracks.

The coeﬃcient of friction results are approximately three times greater than the results produced by the F.A.S.T. 3.2b model. I have discussed a number of factors which are not included in the F.A.S.T. model which could explain this diﬀerence. These results suggest that the F.A.S.T. model still requires further development before it can completely explain the interaction between a bobsleigh runner and ice. 139

Chapter 6

THESIS SUMMARY

This thesis has focused on ice friction in the sport of bobsleigh, a topic that has not received much attention from the academic community in the past. The work has re- volved around four main topics: (1) analyzing the runners, (2) analyzing the ice, (3) a computational model to predict ice friction, and (4) experimental measurements of ice friction and air drag in bobsleigh.

6.1 Runner Analysis

Despite consulting many sources in the bobsleigh community, it was difficult to find any information on what made runners fast. Copying profiles seemed to be the recommended approach. I adapted a handheld rocker gauge from speed skating and developed a protocol for measuring bobsleigh runner profiles in order to move towards a more quantitative understanding of this component of the sport. The half width of the rocker gauge was optimized for long (50 mm) and short (25 mm) track speed skating, hockey (10 mm), and bobsleigh (50 mm). I used the bobsleigh gauge to catalog a number of 2-men bobsleigh runner profiles, allowing me to analyze a variety of runners used in the sport today.

Cross-sectional runner radius was found to range from 4.0 to 5.5 mm and rocker values in the centre of the runners ranged from 20 to 50 m.

I used the catalog of runners to design a set of 2-men bobsleigh runners for Foothills

Bobsleigh Club. If asked to design runners again in the future, I would put more emphasis on the centre of the runner and less on the tips which rarely contact the ice. I recommend that bobsleigh athletes adopt the use of a rocker gauge so that they can maintain their 140 runner proﬁles as speed skaters do with their blades.

6.2 Ice Analysis

The dynamic hardness of ice was measured on five different athletic ice surfaces using dropped spherical indenters. I was unable to measure a difference in ice hardness between the various ice surfaces. The data from all five surfaces was compiled to determine the ice hardness as a function of temperature, P¯(T ) = ((−0.6 ± 0.4)T + 14.7 ± 2.1) MPa, where

T is in degrees Celsius. The width of a track left in the ice by a bobsleigh runner was also used to estimate the ice hardness in the Ice House and the Olympic Oval, but for only one ice temperature. These measurements were in agreement with our analysis, but they were not as precise. I was able to determine that elastic recovery was negligible for our dynamic contact experiments. This suggests that there will also be no recovery of the ice surface behind the blade. The original F.A.S.T. 1.0 model assumed a partial recovery of the ice surface behind the blade. This result forced me to reconsider the contact dynamics in the model.

The uncertainties in my ice hardness analysis are still quite high. I would like to obtain further data, speciﬁcally at temperatures below −10◦C, where we have few results.

I would also like to measure the temperature gradient within the ice surface. Coaches suggest that optimum performance is obtained when the ice is cold and the ice surface is warm. This suggests that the temperature gradient in the ice may not be linear as we assume in the F.A.S.T. 3.2b model.

6.3 F.A.S.T. Model

I have adapted the Frictional Algorithm using Skate Thermohydrodynamics (F.A.S.T.) model to calculate the coefficient of friction between bobsleigh runners and a flat ice 141 surface. The version of the model that assumes the front and rear runners are perfectly aligned (F.A.S.T 3.1b) predicts that the coefficient of friction is lowest for the warmest ice surface temperature calculated (−2◦C). This disagrees with the experience of athletes and coaches. The model that assumes the runners are slightly offset (F.A.S.T. 3.2b) suggests the maximum velocity is obtained for ice surface temperatures from −10 to −20◦C. This result agrees with observations by athletes and coaches. The F.A.S.T. 3.2b model suggests that the dependence of the coefficient of friction on speed is not as important in the sport of bobsleigh as previous results suggest it is for speed skating [11, 12].

On a ﬂat ice surface, F.A.S.T. 3.2b suggests that ﬂatter runners result in a lower coef-

ficient of friction. The rocker appears to have a greater effect on the coefficient of friction than the cross-sectional radius, for a realistic range of both. A flatter runner may also affect sled control. This may explain why athletes prefer a smaller cross-sectional radius on colder, and thus harder, ice. In these conditions, improved sled control may be more important than a reduction in coefficient of friction. Because the flat ice model (F.A.S.T.

3.2b) suggests that flatter runner provides a lower coefficient of friction; the profile of the track, specifically its minimum radius of curvature, may ultimately determine the best rocker to optimize performance. The next step for the F.A.S.T. model is to adapt the code to simulate sliding in curves.

6.4 Experimental Ice Friction and Air Drag Measurements

I have established new experimental protocols for measuring ice friction and air drag in the sport of bobsleigh. Radar speed guns were used in straight sections of track with a constant inclination to measure sled acceleration. The protocols were ﬁrst tested in low speed experiments at the Calgary Olympic Oval and in the Ice House at Canada Olym- pic Park. Further experiments were performed during the Calgary World Cup 2-men 142 bobsleigh race on December 3rd, 2011. Attempts to determine whether or not a dependence exists between the coeﬃcient of friction and the sled velocity were inconclusive.

Assuming velocity independence, the mean coeﬃcient of friction between the bobsleigh runners and the ice surface wasµ ¯ = (5.3 ± 2.0) × 10−3 and the air drag performance was

¯ 2 ¯ CdA = (0.18 ± 0.02) m resulting in a drag coefficient of Cd = 0.54 ± 0.07. The coefficient of friction was much lower than that determined in previous experiments. There were two types of experiments in the past. In the first, researchers examined the deceleration of sleds on flat ice at low speeds. The details of the runner preparation were not clear in that study and it is not clear that low speed results (∼ 1 m/s) can be extrapolated to higher speeds. In the second type of experiment, data was taken over the entire bobsleigh track in order to determine both air drag and ice friction. In these cases, it is very difficult to determine local ice friction and air drag since we do not know how much steering is occurring in the corners. Steering is believed to significantly increase the coefficient of friction. In the future, our results from only the straight sections of a track could possibly be used with previous studies of an entire track to better understand sled movements in corners.

The air drag performance was considerably higher than previous experiments. Pre- vious estimates of air drag for 2-men bobsleighs were based on numerical models and wind tunnel tests. It is not surprising that athletes descending a track do not perform as well aerodynamically as in a wind tunnel or a computer model. On the track, athletes move due to vibrations of the sled and drivers will likely sit higher because they are focused on navigating the course. The air conditions, such as natural wind, are also less controlled at the bobsleigh track. The primary objective of wind tunnel testing and numerical models has not been to obtain absolute values for sled aerodynamics but to determine methods to reduce aerodynamic drag in the sport. In the same way, the F.A.S.T. model may never be able to determine absolute values of ice / runner friction but it can help us ﬁnd ways 143 to reduce friction in the sport.

6.5 Future Work

In the previous four sections there were a number of suggestions to continue working on each of these topics. In another direction, an interesting project could be to attempt to acquire GPS data from a bobsleigh descending the track. Lachapelle et al. [101] have developed a GPS system for use in alpine skiing. The system is light-weight and has the precision required (centimeter) to make accurate three dimensional position, velocity, and acceleration measurements of a sled descending a track. Because of the added equipment, these measurements could not be made on sleds in competition. The best opportunity to test the capacity of the system might be during tour bob rides. They could provide a large body of data (∼ 10 runs) in a single session. The antenna must have a clear line of sight to satellites. Sun shades, roofs in the corners, and even the sides of the sled could limit our ability to connect with satellites. It is possible that the GPS could connect with multiple satellites despite being in the corners. Even a system that only worked in the straight sections of the track could provide an alternative to our ice friction measurements and it could provide us with data on a larger portion of the track.

6.6 Summary

Ice friction in the sport of bobsleigh has been analyzed from runner to ice, with computational models and experimental measurements. This research has provided a new tool to help athletes analyze their runners. Experimental data was acquired on sled dynamics for the international body for bobsleigh. These results can be used in the design of future bobsleigh tracks. We have improved the intellectual capital surrounding bobsleigh runner proﬁles in Canada and we have raised many questions to be addressed in the future. 144

Appendix A

Supplementary Information on Equipment Analysis

A.1 Rocker Gauge Optimization

Provided below is the C++ code that was used to optimize the size of the handheld rocker gauge in section 2.4. The code was compiled using Borland C++ Builder 3.

Gauge optimization program

X Profile determination 2.1 X

*************************** ABSTRACT ********************************

This program uses gauge data to calculate the local curvature around

a specific point. That curvature is used to calculate the adjacent

points and recreate the blade profile. This code uses a fixed width

for the gauge data. The code has been adapted to optimize the gauge

size for a given blade profile.

************************* AUTHOR **********************************

Louis Poirier

------

*** DECLARATION OF VARIABLES ***

a half width of the runner gauge (mm)

b h[0] for the linear regression (mm) 145 bn h_n[0] for the linear regression (mm) chi2sum the sum of all the chi2 del standard deviation of the gauge data (mm) g[n] gauge measurements (mm) gstep resolution of gauge measurments (mm) h[n] actual runner profile (mm) hbar mean value of the runner profile hr[n] renormalized runner profile (mm) h_nbar average value of the h_n function h_n[n] calculated runner profile (mm), position at...

...h[\pm1] set to 0 h_nsum sum of all the new h_n values (mm) hsum sum of all the h values (mm) i an integer j an integer k an integer m slope for the regression (unitless) mn slope for the regression of the new data (unitless) mt top sum in the slope calculation mtn top sum in the slope calculation for the new data mb bottom sum in the slope calculation n number of data points within the half width of the gauge

N the step that corresponds to the position x=0 nmax maximum value of n, this limits the size of the gauge o the centre point of the blade phi probability function, gaussian dist. 146

q random number btw -5*del and 5*del

q2 random number btw 0 and 1

R[n] radius of curvature of the runner (m)

R1 initial calculation or previous point for radius of...

...curvature of the runner (m)

R2 radius of curvature of the runner (m)

rel relative uncertainty of the radius

x position along the runner (mm)

xbar mean value of the position (mm)

xmax initial position where a gauge measurement was taken (mm)

xmin final position where a gauge measurement was taken (mm)

xsteps distance between gauge measurements (mm)

xsum sum of all x values (mm)

#include

double a,amax,chi2sum,del,g[150],gstep,h[150],hr[150],h_n[150],h_nr[150]; double mb,phi,pi,q,q2,R[150],R1,R2,rel,xbar,xstep,xsum,xmax,xmin; int i,j,k,l,lmax,n,N,nmax,o;

//This function normalizes both profiles to zero slope and zero offset 147

//then calculates the difference squared double chi2(void)

{ double hsum = .0; double h_nsum = .0;

for (i=nmax-1;i<=2*o-(nmax-1);i++)

{

hsum += h[i];

h_nsum += h_n[i];

}

double hbar = hsum/(2*(o-nmax)+3); double h_nbar = h_nsum/(2*(o-nmax)+3);

double mt = .0; double mtn = .0; for (i=nmax-1;i<=2*o-(nmax-1);i++)

{

mt += (i*gstep-xbar)*(h[i]-hbar);

mtn += (i*gstep-xbar)*(h_n[i]-h_nbar);

}

double m = mt/mb; double b = hbar - xbar*m; double mn = mtn/mb; 148 double bn = h_nbar - xbar*mn;

double sum = .0; for (i=nmax-1;i<=2*o-(nmax-1);i++)

{

hr[i] = h[i] - m*i*gstep - b;

h_n[i] = h_n[i] - mn*i*gstep - bn;

sum += (hr[i]-h_n[i])*(hr[i]-h_n[i]);

} return sum/del/del/(2*(o-nmax)+3);

}

//This is a random number generator double aleat()

{ return ((double)rand()/RAND_MAX);

}

//This function returns the sign of the argument double sign(double z)

{ if (z >= .0)

return 1.0; else

return -1.0;

} 149

//*************************** runner profile ************************* ifstream f1; ofstream f2; main()

{ f1.open("c:\\data.txt",ios::in); f2.open("c:\\output.txt",ios::out); randomize();

/*cout<<"Input xmax in mm"<<’\n’; cin>>xmax; cout<<"Input xmin in mm"<<’\n’; cin>>xmin; */ xmax = 448; xmin = 27;

//uncertainty on blade runner data del = .0015;

//blade runner resolution xstep = 1.0;

//gauge data resolution gstep = 5.0;

//number of data entries

N = floor((xmax - xmin) / xstep);

//Define pi 150 pi = 2.0*acos(.0);

j = 0; //gstep/xstep; k = 0;

//Input gauge data from file for (i=0;i<=N;i++)

{

f1>>h[k];

if (j == gstep/xstep)

{

k++;

j = 0;

}

j++;

}

//Determine the centre point of the calculation o = floor((k-1)/2);

//This parameter limits the maximum size of the gauge amax = 70.0; nmax = amax/gstep;

//This calculates the bottom half of the linear regression xsum = .0; for (i=nmax-1;i<=2*o-(nmax-1);i++)

xsum += i*gstep; xbar = xsum/(2*(o-nmax)+3); 151 mb = .0; for (i=nmax-1;i<=2*o-(nmax-1);i++)

mb += (i*gstep-xbar)*(i*gstep-xbar);

a = 6.0*gstep; while (a <= amax) //amax

{

n = a/gstep;

chi2sum = .0;

lmax = 1000;

for (l=0;l

{

//Calculate the gauge data

for (i=nmax;i<=2*o-nmax;i++)

{

q2 = 1.0;

phi = .0;

while (phi < q2)

{

q = (2.0*aleat() - 1.0)*5.0*del;

phi = exp(-q*q/2.0/del/del)/100/sqrt(2.0*pi);

q2 = aleat();

}

g[i] = h[i] - (h[i+n] + h[i-n])/2.0 + q;

// f2<

} 152

//Calculate radius of curvature

for (i=nmax;i<=2*o-nmax;i++)

{

if (g[i] == .0)

R[i] = 1.0e5;

else

R[i] = (a*a + g[i]*g[i])*1.0e-3/2.0/g[i];

// f2<

}

h_n[o+1] = h_n[o-1] = .0;

h_n[o] = 1e3*(R[o]-sqrt(R[o]*R[o]-gstep*gstep*1e-6));

for (i=1;i<=o-nmax;i++)

{

h_n[o+i+1] = 2.0*h_n[o+i] - h_n[o+i-1] - 2.0*1e3*(R[o+i]-...

...sqrt(R[o+i]*R[o+i]-gstep*gstep*1e-6));

h_n[o-i-1] = 2.0*h_n[o-i] - h_n[o-i+1] - 2.0*1e3*(R[o-i]-...

...sqrt(R[o-i]*R[o-i]-gstep*gstep*1e-6));

}

chi2sum += chi2();

// for (i=nmax-1;i<=2*o-(nmax-1);i++)

// f2<

}

f2<

cout<

a += gstep;

} f1.close(); f2.close(); return 0;

}

A.2 Bobsleigh Runner Proﬁles

This section includes supplemental details on bobsleigh runner proﬁles to complement the discussion in section 2.6. The supplemental data is a zoomed-in view of Fig. 2.13.

The zoomed-in image can be found in Fig. A.1. Despite considerable diﬀerences between the four runners the majority of the gauge data in the center of the blade is concentrated between 25 − 50 microns. 154

Figure A.1: Zoomed-in gauge data from two separate sets of DSG front 2-men bobsleigh runners. First set, right - cross, left - circle and second set, right - square, left - triangle. 155

Appendix B

Catalog of 2-men Bobsleigh Runner Proﬁles

In an effort to provide an overview of the bobsleigh runner profiles used in 2-men bobsleigh I have included all the profiles from my thesis along with their rear runners in this appendix. We have also included some extra profiles in order to provide a broader perspective. The profiles include: the runners used in our runner design project, one set that is more curved (lower rocker, larger gauge data), one set that is flatter (higher rocker, smaller gauge data), and the two standard 2-men runner cuts provided by the F.I.B.T.

The ﬁrst set of runners (R8) is more curved than the second (F1). All the gauge data was acquired with the handheld rocker gauge and the proﬁles are all in pairs with the front runners on top and the rear runners on the bottom. 156

Figure B.1: Measured gauge data from two separate sets of DSG 2-men bobsleigh runners. These runners were used in our runner design project. Front runners are in the top ﬁgure, rear runners are in the bottom ﬁgure. First set, right - cross, left - circle and second set, right - square, left - triangle. 157

Figure B.2: Measured gauge data from a set of BCS Gesuito09 2-men bobsleigh runners, right - square, left - triangle. Front runners are in the top ﬁgure, rear runners are in the bottom ﬁgure. This is a more curved set of runners. 158

Figure B.3: Measured gauge data from a set of LG 2-men bobsleigh runners, right - square, left - triangle. Front runners are in the top figure, rear runners are in the bottom figure. This is a flatter set of runners. 159

Figure B.4: Measured gauge data from the more curved of two types of 2-men runners supplied by the F.I.B.T. (R8), right - square, left - triangle. Front runners are in the top ﬁgure, rear runners are in the bottom ﬁgure. 160

Figure B.5: Measured gauge data from the ﬂatter of two types of 2-men runners supplied by the F.I.B.T. (F1), right - square, left - triangle. 161

Appendix C

Supplementary Information on Ice Hardness

C.1 Integration Volume

We calculate the integration volume as the apparent volume of the indentation crater.

The crater has a width of dc and we assumed it to be spherical with a radius of the q 2 2 indenter Rb, the maximum depth of penetration is therefore δ = Rb − Rb − dc /4. The volume we are calculating is the rotational volume of the hashed area in Fig. C.1. The rotational volume can be described by the integral in equation C.1. √ Z Z 2 2 Z Rb Rb −y 2π V (dc) = √ r · dθdrdy (C.1) 2 2 Rb −dc /4 0 0

Integrating with respect to θ we obtain equation C.2, √ Z Z 2 2 Rb Rb −y V (dc) = 2π √ r · drdy (C.2) 2 2 Rb −dc /4 0

Which leads to equation C.3 when we integrate with respect to r, Ã ! Z R 2 b 2 y V (dc) = π √ R − dy (C.3) 2 2 b Rb −dc /4 4

Integrating this quadratic equation we obtain equation C.4. Ã ! 3 2 q 2R 1 ³ ´3/2 R V (d ) = π b + 4R2 − d2 − b 4R2 − d2 (C.4) c 3 24 b c 2 b c

C.2 Crater Diameter

The uncertainty in the diameter of the crater was not a reﬂection of the precision of the measurement but of the consistency of the result. To obtain an average result the drop was repeated six times for every; location, ice surface temperature, ball, and drop 162

Figure C.1: Cross-section of the indentation crater left in the ice by the dropped ball. The crater volume is the rotational volume of the hashed area. 163 height. Due to the irregular shape of the crater the diameter was also measured across two perpendicular axes resulting in 12 measurements of dc. For each set of 12 measurements ¯ I took the average dc and then compared each individual measurement to its twelve ¯ point average resulting in a normalized crater diameter dc/dc. This was done for every dc measurement, over 1700 in all. In this way we were able to determine the statistical uncertainty on the diameter measurement. The results were compared for diﬀerent ice surfaces, and for each ball the result was constant except for the smallest ball, A. The craters formed by ball A had a higher uncertainty and those results are shown in Fig. ¯ C.2. The relative uncertainty in the crater diameter for ball A is σ = ∆dc/dc = 0.10. A

Figure C.2: Determination of the relative uncertainty in the crater diameter measurements for ball A. The circles represent the frequency of occurrence for a given range of normalized crater diameters. The line is the gaussian curve that best ﬁts the data. similar analysis was performed for the craters from all the other balls. Those results are 164 shown in Fig. C.3. The relative uncertainty in the crater diameter for the other balls is

Figure C.3: Determination of the relative uncertainty in the crater diameter measurements for all other balls. The circles represent the frequency of occurrence for a range value of normalized crater diameters. The line is the gaussian curve that best ﬁts the data.

¯ σ = ∆dc/dc = 0.066.

C.3 Consistency of Ice Hardness Data

In November 2009, we performed measurements to determine whether the ice hardness measurements would be the same for each of the six balls we were using for the measurements. Each of the balls was dropped from four diﬀerent heights between 0.3 and 1.2 m.

The data was analyzed by determining the slope of a four data point plot equivalent to

Fig. 3.7 for each ball. In Fig. C.4 we observe that the ice hardness analyses are equivalent 165

Figure C.4: Ice hardness measured in the Olympic Oval using six diﬀerent balls. Ice surface temperatures were −1.1◦C. for each of the six balls. For this reason, data from the diﬀerent balls were used together in order to increase the range of impact energies thus reducing the uncertainty of the ice hardness analysis.

In a separate experiment carried out on December 30, 2009, we set all three ice surfaces at the Olympic Oval to −1.1◦C in order to compare the ice hardness on the diﬀerent surfaces. Balls A, B, C, and E were dropped from four diﬀerent heights between

0.3 and 1.2 m. Fig. C.5 is a result of analyzing all the data. The data appears to suggest the straight section on the North rink has a harder surface. If we limit the data to the optimized experimental procedure described in section 3.2 the results are diﬀerent (see

Fig. C.6). This analysis suggests that the ice hardness is equivalent at all the locations 166

Figure C.5: Ice hardness in various locations within the Olympic Oval using all data. Ice surface temperatures were set at −1.1◦C. within the Oval. 167

Figure C.6: Ice hardness in various locations within the Olympic Oval using only the optimized experimental methodology. Ice surface temperatures were set at −1.1◦C. 168

Appendix D

Supplementary Information on the F.A.S.T. Code

D.1 F.A.S.T. 1.5 Code

This code was obtained from Edward Lozowski. This model simulates the contact between a vertical speed skate blade and ice. It was the starting point for our bobsleigh ice friction model. c A numerical model to provide a quantitative estimate of skate c blade friction coefficient - FAST 1.5 c **Frictional Algorithm using Skate Thermohydrodynamics** c c *************************** ABSTRACT ******************************** c A novel numerical model has been formulated to predict the kinetic c ice friction coefficient for speedskates. The model is based on c lubrication thermohydrodynamics and accounts for: Couette flow c viscous dissipation (frictional melting), heat conduction into c ice, freezing point depression and the premelted layer. c c The model predicts an increasing depth of the lubrication layer c along the length of the contact zone, with maximum thickness of c less than one micron. c c Friction is separated into two components, the ploughing force and c shear stress force. The ploughing force is computed using the 169 c geometry of the skate blade, mass of the skater and empirically c derived ice resistance pressure. The shear stress force is c obtained by computing the thickness of the lubrication layer along c the length of the dynamic contact zone. The thickness of the c lubrication layer depends on frictional melting due to viscous c dissipation (term 1 from Equation 5), heat conduction into the ice c (terms 3 and 4 from Equation 5) and lateral squeeze flow of the c liquid (term 2 from Equation 5). In this latest version, FAST 1.5, c heat conduction at the interface between the skate blade and c lubrication layer has been included in calculation of the shear c stress force. c c According to the model, ice friction variation with ice c temperature depends strongly on skating velocity - warm ice is c fast at low speeds, cold ice at high speeds. c c For speedskating, ice friction is dominated by frictional melting c - "pressure melting" and the premelted layer play a minor role. c c Comparison with experiments (de Koning et al., 1992) shows c encouraging agreement, with some discrepancies yet to be c explained. c c ************************** AUTHORS ********************************** c Colin Fong c Edward Lozowski 170 c Andrew Penny c NSERC c August 25, 2006 c c ------c *** DECLARATION OF VARIABLES *** c air_switch - switch to turn on air conduction c c_i - specific heat capacity of ice [J/kg/K] c c_s - specific heat capacity of skate blade [J/kg/K] c c_t - specific heat capacity of skate blade tube material c [J/kg/K] c check - records whether d(h**2)/dx is negative and disregarded c ARRAY c d - maximum penetration depth of the blade [m] c delta_T_i - difference between melting temperature and initial c temperature of ice surface [degrees C] c delta_T_s - difference between melting temperature and initial c temperature of skate blade base [degrees C] c down_force - m*g [N] c dt - time step used to calculate heat conduction for skating c stroke [s] c dx - grid interval along blade [m] c D_star - effective diameter of skate blade tube and blade [m] c elev - elevation with respect to sea level [m] c force_total - total resistive force [N] c frict_coeff - coefficient of ice friction 171 c g - gravity (allowed to vary according to elevation) [m/s^2] c gravconst - universal gravitational constant [m^3*kg^(-1)*s^(-2)] c h - total lubrication layer thickness [m] ARRAY c hbar - average value of h [m] c h_cond - lubrication layer thickness with ONLY frictional melting c and heat conduction (terms 1, 3 and 4 on rhs of Equation 5) c - [m] ARRAY c h_cond_sq - h_cond squared - [m^2] ARRAY c h_max - maximum value of h [m] c h_melt - lubrication layer thickness with ONLY frictional melting c (term 1 on rhs of Equation 5)- [m] ARRAY c h_melt_sq - h_melt squared - [m^2] ARRAY c h_neg_sq - h_cond_sq + h_squeeze_sq [m^2] ARRAY c h_sq - total lubrication layer thickness squared [m^2] ARRAY c h_squeeze - lubrication layer thickness with ONLY frictional c melting and squeeze flow (terms 1 and 2 on rhs of Equation c 5) - [m] ARRAY c h_squeeze_sq - h_squeeze squared - [m^2] ARRAY c h_sum - used to compute the avg lubrication layer thickness [m] c h_xfer - heat transfer coefficient [W*m^(-2)*K^(-1)] c i_cond - counter variable for skate conduction c ice_switch - switch to turn on ice conduction c kappa_i - thermal diffusivity of ice [m^(2)/s] c kappa_s - thermal diffusivity of skate blade [m^(2)/s] c kin_visc_air - kinematic viscosity of air [m^(2)/s] c k_a - thermal conductivity of air [W/m/K] 172 c k_i - thermal conductivity of ice [W/m/K] c k_s - thermal conductivity of skate blade [W/m/K] c Lfract - ratio of dynamic and static contact lengths c l_b - total length of skate blade [m] c l_d - length of dynamic contact zone of skate on ice [m] c l_f - latent heat of freezing at O degrees Celsius [J/kg] c l_s - length of static contact zone of skate on ice [m] c m - mass of the skater [kg] c massearth - mass of earth [kg] c n - number of strokes c Nu - Nusselt number c num_iter - number of iterations based on dynamic contact length c and dx c pbar - mean resistive stress over contact area of blade [Pa] c pi - pi c plough_force - ploughing force [N] c qll - quasi-liquid layer thickness based on ice temperature [m] c radearth - radius of earth [m] c rcurve - rocker radius of curvature [m] c Rey_num - Reynolds number c rho_i - density of ice [kg/m^(3)] c rho_s - denisty of skate blade [kg/m^(3)] c rho_t - denisty of skate blade tube material [kg/m^(3)] c rho_w - density of water [kg/m^(3)] c s - skate blade height [m] c shear_force - shear stress force for length of dx along skate 173 c blade [N] ARRAY c shear_force_sum - sum of shear stress force along skate blade [N] c switch - variable to switch between ice and air conduction c tau_i - contact time of a location on the ice with the blade [s] c tau_period - the period time of one stroke [s] c tau_s - contact time of the blade with the ice [s] c time - total time for skate blade conduction with strokes [s] c time_stroke_half - cumulative time per half of stroke period [s] c time_stroke_full - cumulative time per full stroke period [s] c T_a - air temperature near skate blade [degree C] c T_b - ice basal temperature [degree C] c T_i - ice surface temperature [degree C] c T_m - melting temperature due to pressure melting [degrees C] c T_s - skate blade temperature [degree C] c T_s_new - new skate blade temp after heat conduction with ice and c air, stored temporarily then passed to T_s [degree C] c v - velocity of skater [m/s] c visc - dynamic viscosity of water [kg/m/s] c volume_tube - volume of the skate blade tube [m^3] c w - width of skate blade [m] c x - position along the skate blade [m] ARRAY c x_max - position along skate blade where maximum value of h c occurs [m] c z - total ice thickness [m]

real air_switch, c_i, c_s, c_t, check(2000000), d, delta_T_i, 174

& delta_T_s, down_force, dt, dx, D_star, elev, force_total,

& frict_coeff, g, gravconst, h(2000000), hbar, h_cond(2000000),

& h_cond_sq(2000000), h_max, h_melt(2000000), h_melt_sq(2000000),

& h_neg_sq(2000000), h_sq(2000000), h_squeeze(2000000),

& h_squeeze_sq(2000000), h_sum, h_xfer, ice_switch, kappa_i, kappa_s,

& kin_visc_air, k_a, k_i, k_s, m, l, Lfract, l_b, l_d, l_f, l_s,

& massearth, Nu, pbar, pi, plough_force, qll, radearth, rcurve,

& Rey_num, rho_i, rho_s, rho_t, rho_w, s, shear_force(2000000),

& shear_force_sum, tau_i, tau_period, tau_s, time, time_stroke_half,

& time_stroke_full, T_a, T_b, T_i, T_m, T_s, T_s_new, v, visc,

& volume_tube, w, x(2000000), x_max, z

integer i, i_cond, j, k, n, num_iter

c ****** Define Constants ********

gravconst = 6.673E-11

massearth = 5.9742E24

pi = 2.*acos(0.)

radearth = 6378.1E3

c ******** Define Numerical Constants **********

dt = 0.01

dx = 0.000005

Lfract = 0.5

c ********** Standard Values ************ 175 c_i = 2.04E3 c_s = 460.0 c_t = 1012.0

D_star = 0.01168 elev = 250.0 kin_visc_air = 0.0000133 k_a = 0.0243 k_i = 2.3 k_s = 19.0 l_b = 0.42 l_f = 3.5E5 m = 75.0 n = 100 rcurve = 25.0 rho_i = 920.0 rho_s = 8.10E3 rho_t = 1.2 rho_w = 1000.0 s = 0.026 tau_period = 1.5 tau_s = 0.75

T_i = -9.0

T_b = -18.0

T_s = -3.0 v = 10.0 visc = 1.8E-3 176

volume_tube = 0.00002086

w = 1.1/1000.0

z = 25.0/1000.0

c *************************** FAST 1.5 ********************************* c **********************************************************************

c *** create files for output ***

open(unit=12,file=’output.txt’)

open(unit=13,file=’hvsx.txt’)

open(unit=14,file=’stroke_conduction.txt’)

c *** set initial values of parameters to be varied and start looping **

T_i = -15.0

do i=1, 8

v = 1.0

do j=1, 40

c *** determine g based on elevation above sea level ***

g=gravconst*massearth/((radearth + elev)**2.0)

c *** determine force directed downwards with increase factor to c account for skater thrust ***

down_force = m*g

c *** compute the mean ice resistive stress - pbar *** 177

pbar = 3755134.06 + (-181182.74 * T_i)

c *** compute static contact length of skate blade on ice ***

l_s = down_force/(w*pbar)

c *** compute dynamic contact length ***

l_d = Lfract * l_s

c *** compute dynamic value of pbar based on dynamic contact length ***

pbar = down_force/(l_d*w)

c *** compute max penetration depth of the skate blade into the ice ***

d=l_d**2.0/(2.0*rcurve)

c *** compute ploughing force (Equation 3) ***

plough_force = pbar*w*d

c *** compute melting temperature due to pressure melting - T_m ***

T_m = pbar*(-1.0/134.0)/101300.0

c *** compute the thermal diffusivity of ice - kappa_i ***

kappa_i = k_i/(rho_i*c_i)

c *** compute the thermal diffusivity of skate blade - kappa_s ***

kappa_s = k_s/(rho_s*c_s) 178 c *** compute difference between blade-ice interface temperature and c initial ice surface temperature - delta_T_i ***

delta_T_i = T_m - T_i

c *** compute difference between blade-ice interface temperature and c initial skate blade temperature - delta_T_s ***

delta_T_s = T_m - T_s

c *** contact time for a point on the ice with skate blade - tau_i ***

tau_i=l_d/v

c *** compute the number of iterations based on grid interval size ***

num_iter = int(l_d/dx)

c *** initialize arrays and values for computation of lubrication layer c thickness ***

check(1) = 0.0

h(1) = 0.0

h_cond(1) = 0.0

h_cond_sq(1) = 0.0

h_max = 0.0

h_melt_sq(1) = 0.0

h_melt(1) = 0.0

h_neg_sq(1) = 0.0

h_sq(1) = 0.0

h_squeeze_sq(1) = 0.0 179

h_squeeze(1) = 0.0

h_sum = 0.0

shear_force(1) = 0.0

shear_force_sum = 0.0

x_max = 0.0

x(1) = 0.000005

c *** compute quasi-liquid layer thickness and set h(1) equal to qll - c source: Dash, J. G., Surface melting, Contemporary c Physics, 1989, 30, 99. ***

qll = 1.0E-9 + (T_i - (-20.0))*(0.205263157E-9)

h(1) = qll

c *** transfer initial value of h based on qll to components of h ***

h_sq(1) = h(1)**2.0

h_melt(1) = h(1)

h_cond(1) = h(1)

h_squeeze(1) = h(1)

c *** compute initial value of shear stress force and start sum ***

shear_force(1) = visc*v*(w+d)*dx/h(1)

shear_force_sum = shear_force_sum + shear_force(1)

c *** main loop to compute lubrication layer thickness and shear stress c force ***

do k=2, num_iter 180

c * compute terms used to find fluid thickness - h *

h_melt_sq(k) = dx*2.0*visc*v/(rho_w*l_f)

h_cond_sq(k) = dx*2.0*sqrt(k_i*rho_i*c_i)*delta_T_i*h(k-1)/

& (sqrt(v*pi*x(k-1))*rho_w*l_f) + dx*2.0*k_i*(T_i-T_b)*h(k-1)/

& (z*v*l_f*rho_w) + dx*4.0*h(k-1)*sqrt(rho_s*c_s*k_s)*delta_T_s/

& (rho_w*l_f*v*sqrt(pi*tau_s))

h_squeeze_sq(k) = dx*2.0*down_force*h(k-1)**4.0/

& (v*visc*l_d*(w+d)**3.0)

h_neg_sq(k) = h_cond_sq(k) + h_squeeze_sq(k)

c * check if d(h**2)/dx is negative, if so, do not update value of h *

if(h_melt_sq(k).le.h_neg_sq(k))then

h_sq(k) = h_sq(k-1)

h_melt(k) = h_melt(k-1)**2.0

h_cond(k) = h_cond(k-1)**2.0

h_squeeze(k) = h_squeeze(k-1)**2.0

check(k) = 1.0

endif

c * check if d(h**2)/dx is positive, if so, update value of h *

if(h_melt_sq(k).gt.h_neg_sq(k))then

h_sq(k) = h_sq(k-1) + h_melt_sq(k)-h_cond_sq(k)-h_squeeze_sq(k)

h_melt(k) = h_melt(k-1)**2.0 + h_melt_sq(k)

h_cond(k) = h_cond(k-1)**2.0 + h_melt_sq(k) - h_cond_sq(k)

h_squeeze(k) = h_squeeze(k-1)**2.0+h_melt_sq(k)-h_squeeze_sq(k) 181

check(k) = 0.0

endif

c * update values for lubrication layer thickness and shear stress c force loop *

x(k) = x(k-1) + dx

h(k) = sqrt(h_sq(k))

h_melt(k) = sqrt(h_melt(k))

h_cond(k) = sqrt(h_cond(k))

h_squeeze(k) = sqrt(h_squeeze(k))

shear_force(k) = visc*v*(w+d)*dx/h(k)

shear_force_sum = shear_force_sum + shear_force(k)

h_sum = h_sum + h(k)

c * output h vs x data according to certain velocity and ice temp *

if(T_i.eq.-9.0.and.v.eq.10.0)then

write(13,200)T_i, int(v), k-1, (x(k-1)-dx)*100.0,

&h_melt(k-1)*1000000, h_cond(k-1)*1000000, h_squeeze(k-1)*1000000,

&h(k-1)*1000000, int(check(k-1)), int(check(k-1))*h(k-1)*1000000

200 format(F5.1,’,’,I2,’,’,I5,’,’,F12.8,’,’,F12.8,’,’,F12.8,

&’,’,F12.8,’,’,F12.8,’,’,I1,’,’,F12.8)

endif

c * check to see if max values (x and h) need to be updated *

if(h(k).gt.h_max)then 182

h_max = h(k)

x_max = x(k)

endif

enddo

c *** compute avg value for h ***

hbar = h_sum/num_iter

c *** compute the total resistive force - force_total ***

force_total = plough_force + shear_force_sum

c *** compute coefficient of friction ***

frict_coeff = force_total/down_force

c *** output coefficient of friction and varied parameters ***

write(12,210)T_i, v, h_max*1000000.0,

&hbar*1000000.0, plough_force, shear_force_sum,

&plough_force/down_force, shear_force_sum/down_force, frict_coeff

210 format(F6.2,’,’,F6.2,’,’,F12.8,’,’,F12.8,’,’,F12.8,’,’,

&F16.10,’,’,F16.10,’,’,F16.10,’,’,F16.10)

c *** update varied parameters ***

v = v + 1.0

enddo 183

T_i = T_i + 2.0

enddo

c ******* compute skate blade temperature vs number of strokes ********* c **********************************************************************

c *** initialize values for blade conduction stroke loop ***

v = 10.0

T_i = -9.0

T_s = 3.0

time = 0.0

time_stroke_half = 0.0

time_stroke_full = 0.0

air_switch = 0

ice_switch = 1

i_cond=1

c *** compute the Reynolds number for skate blade ***

Rey_num = v * D_star / kin_visc_air

c *** compute Nusselt number ***

Nu = 0.683 * Rey_num**(0.466)

c *** compute heat transfer coefficient - h_xfer ***

h_xfer = k_a * Nu / D_star 184 c *** compute air temperature near skate based on ice surface temp ***

T_a = T_i + 10.0

c *** enter into loop to calculate change in skate blade temperature as c a function of time ***

c *** return to this point for new stroke period where skate initially c makes contact with the ice ***

300 continue

c *** return to this point when half way through stroke period, where c skate is lifted off the ice ***

400 continue

c *** calculate new skate blade temperature as a result of heat c conduction to/from the ice and air ***

T_s_new = T_s + 2.0*ice_switch*dt*sqrt(rho_s*c_s*k_s/(pi*tau_s))*

& delta_T_s*w*l_d/((c_s*rho_s*s*w*l_b) + (volume_tube*c_t*rho_t)) -

& dt*h_xfer*air_switch*(T_s - T_a)*((2.0*s) + w)/ ((c_s*rho_s*s*w*

& l_b) + (volume_tube*c_t*rho_t))

c *** output of updated skate blade temperature ***

write(14,230)time, time_stroke_half, time_stroke_full, i_cond,

&h_xfer, air_switch, ice_switch, T_s 185

230 format(F7.2,’,’,F4.2,’,’,F4.2,’,’,I4,’,’,F8.2,’,’,F3.1,’,’,F3.1,

&’,’,F7.4)

c *** check to see if new skating stroke has started - if so, reset c values ***

if(time_stroke_full.ge.2.0*tau_s-dt)then

air_switch = 0

ice_switch = 1

i_cond = i_cond + 1

c *** update values for time step ***

T_s = T_s_new

delta_T_s = T_m - T_s

time = time + dt

time_stroke_half = dt

time_stroke_full = dt

c *** check whether n strokes have been completed ***

if(i_cond.gt.n)then

goto 900

endif

goto 300

endif

c *** check to see if half of stroke period has expired - if so, reset 186 c values***

if(time_stroke_half.ge.tau_s-dt)then

air_switch = 1

ice_switch = 0

time_stroke_half = dt

c *** update values for time step ***

T_s = dT_s

delta_T_s = T_m - T_s

time = time + dt

time_stroke_full = time_stroke_full + dt

goto 400

endif

c *** update skate blade temperature and values if not start of half or c full stroke period ***

T_s = T_s_new

delta_T_s = T_m - T_s

time = time + dt

time_stroke_half = time_stroke_half + dt

time_stroke_full = time_stroke_full + dt

goto 400

900 continue

close(12) 187

close(13)

close(14)

stop

end

D.2 F.A.S.T. 3.2b Code

This is the source code used to simulate the contact between one front and one rear bobsleigh runner and ice. The runners are slightly oﬀset, leaving two parallel grooves in the ice. This C++ code was compiled using Borland C++ Builder 3.

/* c A numerical model to provide a quantitative estimate of skate c blade friction coefficient - FAST 3.2b c **Frictional Algorithm using Skate Thermohydrodynamics** c c *************************** ABSTRACT ******************************** c A numerical model has been formulated to predict the kinetic c ice friction coefficient for speedskates, and modifed for the c calculation of bobsleigh runners. The model is based on c lubrication thermohydrodynamics and accounts for: Couette flow c viscous dissipation (frictional melting), heat conduction into and c out of the melt layer, freezing point depression and the premelted c layer. c c The model predicts an increasing depth of the lubrication layer 188 c along the length of the contact zone. This version calculates along c the width as well as the length of the runner. c c Friction is separated into two components, the ploughing force and c shear stress force. The ploughing force is computed using the c geometry of the skate blade, mass applied to the ice and empirically c derived ice resistance pressure. The shear stress force is c obtained by computing the thickness of the lubrication layer along c the length of the dynamic contact zone. The thickness of the c lubrication layer depends on frictional melting due to viscous c dissipation (term 1 from Equation 5), heat conduction into the ice c (terms 3 and 4 from Equation 5) and lateral squeeze flow of the c liquid (term 2 from Equation 5). In this latest version, FAST 3.2b, c heat conduction at the interface between the skate blade and c lubrication layer has been included in calculation of the shear c stress force. c c This version of the model is fully 3D and incorporates both

C the front and rear runner tracks running parallel to one another. c c ************************** AUTHORS ********************************** c Louis Poirier c Colin Fong c Edward Lozowski c Andrew Penny c NSERC 189 c May. 2011 c c ------c *** DECLARATION OF VARIABLES *** c A - Ploughing area of the runner perpendiculer to the sled motion c Aw - Total volume of of water ahead of the rear runner divided by c the forward step size [m^2] c Aw1 - calculated volume of of water ahead of the rear runner c divided by the forward step size [m^2] c c_i - specific heat capacity of ice [J/kg/K] c c_s - specific heat capacity of skate blade [J/kg/K] c d - penetration depth of the blade [m] c da - distance between the front and rear axels [m] c dh - size of the water column displaced in front of the rear c runner [m] c delta_h - variation in the height of the melt layer thickness [m] c delta_T_i - difference between melting temperature and initial c temperature of ice surface [degrees C] c delta_T_s - difference between melting temperature and initial c temperature of skate blade base [degrees C] c down_force - m*g*g_force [N] front runner c down_forcer - m_r*g*g_force [N] rear runner c down_force_tot - down_force + down_forcer [N] both runners c dmax - maximum penetration depth of the front runner [m] c dmaxr - maximum penetration depth of the rear runner [m] c dstep - variable step size for the numerical calculation of dmax 190 c dx - displacement of the water column at the front of the rear c runner [m] c g_force - factor multiplied by m*g c dz - grid interval along the length of the blade [m] c dy - grid interval along the width of the blade [m] c elev - elevation with respect to sea level [m] c f - the function to solve numerically for penetration depth c force_total - total resistive force [N] c frict_coeff - coefficient of ice friction c g - gravity (allowed to vary according to elevation) [m/s^2] c gravconst - universal gravitational constant [m^3*kg^(-1)*s^(-2)] c h - total lubrication layer thickness [m] c hbar - average value of h [m] c HI - indentation hardness [Pa] c hp - maximum thickness of the pooled melt layer behind the front c runner - [m] c h_cond - lubrication layer thickness with ONLY heat conduction c (Equation 5) - [m] c h_conds - lubrication layer thickness with ONLY slow heat c conduction (Equation 5) - [m] c h_condf - lubrication layer thickness with ONLY fast heat c conduction (Equation 5) - [m] c h_condb - lubrication layer thickness with ONLY heat conduction c from the blade (Equation 5) - [m] c h_max - maximum value of h [m] c h_melt - lubrication layer thickness with ONLY frictional melting 191 c (term 1 on rhs of Equation 5)- [m] c h_squeeze - lubrication layer thickness with ONLY frictional c melting and squeeze flow (terms 1 and 2 on rhs of Equation 5) - [m] c h_sum - used to compute the avg lubrication layer thickness [m] c kappa_i - thermal diffusivity of ice [m^(2)/s] c kappa_s - thermal diffusivity of skate blade [m^(2)/s] c k_a - thermal conductivity of air [W/m/K] c k_i - thermal conductivity of ice [W/m/K] c k_s - thermal conductivity of skate blade [W/m/K] c l_b - total length of skate blade [m] c l_d - contact length of the front runner with the surface of the c ice [m] c l_dr - contact length of the rear runner with the surface of the c ice [m] c l_f - latent heat of freezing at O degrees Celsius [J/kg] c m - mass fraction carried on the runner [kg] c massearth - mass of earth [kg] c num_iter - number of iterations based on dynamic contact length c and dz c pi - pi = 3.141593 c plough_force - ploughing force [N] c r - cross sectional radius of the runner [m] c radearth - radius of earth [m] c rcurve - rocker radius of curvature [m] c rho_i - density of ice [kg/m^(3)] c rho_s - density of skate blade [kg/m^(3)] 192 c rho_w - density of water [kg/m^(3)] c shear_force - shear stress force for length of dz along skate c blade [N] c shear_force_sum - sum of shear stress force along skate blade [N] c tau_i - contact time of a locaton on the ice with the blade [s] c tau_s - contact time of the blade with the ice [s] c T_a - air temperature near skate blade [degree C] c T_b - ice basal temperature [degree C] c T_i - ice surface temperature [degree C] c T_m - melting temperature due to pressure melting [degrees C] c T_s - skate blade temperature [degree C] c th - total ice thickness [m] c test - test parameter to determine convergence of water vol. calc. c v - velocity of skater [m/s] c visc - dynamic viscosity of water [kg/m/s] c wforce - Force associated with moving the water settled in the c runner groove back into the melt layer in front of the rear runner [N] c x - height of the bottom of the runner grove [m] c y - position along the width of the runner [m] c y_max - half contact width of the front runner with the surface c of the ice [m] c y_maxr - half contact width of the rear runner with the surface c of the ice [m] c z - position along the length of the skate blade [m] ARRAY c z0 - first point of contact of the runner at the width y [m] ARRAY c z_max - position along skate blade where max value of h occurs [m] 193

#include

double* h; double* z0;

double h_cond,h_melt,d,da,delta_h,dh,dx,h_squeeze,shear_force,qll; double A,c_i,c_s,delta_T_i,delta_T_s,dmax,dmaxr,down_force,down_forcer; double down_force_tot,dT_i,dy,dv,dz,elev,force_total,frict_coeff,g_force; double g,gravconst,hlast,h_conds,h_condf,h_condb,HI,hp,k_a,k_i,k_s,m,m_r; double l_d,l_dr,l_f,massearth,pi,plough_force,r,radearth,rcurve,rho_i; double rho_s,rho_w,s,sc,shear_force_sum,tau_i,t,T_a,T_b,T_i,T_imin,T_m; double T_s,test,th,v,visc,vmax,x,y,y_max,y_maxr,z;

//double hbar,dstep,f,h2,h_max,h_sum,kappa_i,kappa_s,l_b,tau_s,test1;

//double wforce,z_max;

int i,j,jlast,k,l,o,zsteps,ysteps;

//int n,p,print,yprint;

//*************************** FAST 3.1b *********************************

//ofstream fp1; ofstream fp2; main()

{ 194 cout<<"press any key to begin"<<’\n’; getch(); cout<<"running..."<<’\n’;

//fp1.open("c:\\hvsx.txt",ios::out); fp2.open("c:\\fast32.txt",ios::out);

//****** Experiment parameters ********

T_i = -2.0; //ice surface, temp. (Mean sample)

T_b = T_i - 2.; //ice base, temp.

//should be adapted into a function of ice temp. I discussed this with

//Tracy at one point before he went to Whistler, This parameter used to be

//-18 @ -15

delta_T_s = .2; //T_s - T_m;

//*** compute difference between blade-ice interface temperature and

//initial skate blade temperature - delta_T_s ***

r = 4.75/1000.0; //2man (varied from 4-5.5 mm)

// r = 6.0/1000.0; //4man

th = 25.0/1000.0; //ice thickness

//from discussions with Tracy and Mark Messer

//****** Define Constants ********

gravconst = 6.672e-11; //Gravitational constant

massearth = 5.9742e24; //Mass of the Earth

pi = 2.0*acos(0.0); 195

radearth = 6372.8e3; //mean radius from wikipedia (Earth)

//******** Define Numerical Constants **********

g_force = 1.0; //cos(6.5/180*pi);

da = 1.69;

dT_i = 4.;

dz = 1.e-6;

dy = 1.e-7;

T_imin = -15.;

// T_imin = -2.;

vmax = 43.;

// vmax = 21.;

//********** Standard Values ************

c_i = 2.04e3; //appropriate for ice, a curve would be better

c_s = 460.0; //corresponds to a cast iron.

//A lab at Dal can measure this. I contacted them they have not replied.

//No time to do this in PhD

elev = .0; //1050.0; //from wikipedia (Calgary)

//reasonable according to www.engineeringtoolbox.com

// k_a = 0.0243;

k_i = 2.25; //set for -5 C www.engineeringtoolbox.com

// should be a function of T 196

k_s = 14.0; //sandmeyer:2003a

//latent heat of fusion for ice

l_f = 3.34e5; //I have corrected from 3.5e5 J/kg

//www.engineeringtoolbox.com and world of physics

//Runner holes are 585/743, leads to 86/109; 40/60 would be 78 & 117

m = 86.0; //approx weight dist. to each front runner in 2man

//(I discussed weight balance w/ Pierre we discussed both options)

m_r = 109.0; //approx weight dist. to each rear runner in 2man

rcurve = 34.0;

//(does not vary over 10 cm, contact length according to the code, a

//function of ice hardness)

//(I believe varying this from 20-48 m is a good sample)

rho_i = 917.5; //engineering toolbox: changed from 920 which is

//for -30, now set for -3 degrees, will try and make an equation

// rho_s = 7.65e3; //measured experimentally, May 2, 2007

rho_s = 7.8e3; //sandmeyer:2003a (7800)

rho_w = 1000.0;

visc = 1.79e-3; //This is set for 0 degrees Hallet and

//engineering toolbox (supercooled fluid?)

//************** 197

//determine g based on elevation above sea level

g = gravconst*massearth / (radearth + elev) / (radearth + elev);

i=0;

//start looping while (T_i >= T_imin) //Temperature loop

{

dv = 1.25;

v = 1.;

// v = 42.;

d = .0;

//determine downward force with increased factor for g-forces

down_force = m*g*g_force;

down_forcer = m_r*g*g_force;

down_force_tot = down_force + down_forcer;

//compute the indentation hardness of ice Poirier et al., 2011 ***

// dm = .4 and db = 2.1

HI = (-.6*T_i + 14.7) * 1.0e6;

//calculation for depth of front runner penetration, Feb. 4, 2011

dmax = down_force/pi/HI/sqrt(rcurve*r);

//For a rear runner running parallel to the front runner’s track

dmaxr = down_forcer/pi/HI/sqrt(rcurve*r);

//for a parallel track allignment, front and rear tracks do not over lap 198

//The 1/2 contact width of the front runner with the ice surface

//Sept. 22, 2009

y_max = sqrt(2.0*r*dmax-dmax*dmax);

//Apr.11, 2011 : for a rear runner following in the front runner’s

//track

/* dmaxr = 2.*dmax;

test = sqrt(2.*rcurve*(dmaxr-dmax)-(dmaxr-dmax)*(dmaxr-dmax))*y_max...

...+ sqrt(2.*r*dmaxr-dmaxr*dmaxr));

dh = dmax/2.;

while(fabs(test-down_forcer/HI)/test > 1e-3)

{

if (test-down_forcer/HI > 0)

dmaxr -= dh;

else

dmaxr += dh;

test = sqrt(2.*rcurve*(dmaxr-dmax)-(dmaxr-dmax)*(dmaxr-dmax))*...

...(y_max + sqrt(2.*r*dmaxr-dmaxr*dmaxr));

dh = dh/2.;

} */

//The 1/2 contact width of the rear runner with the ice surface

//Apr. 6, 2011

y_maxr = sqrt(2.0*r*dmaxr-dmaxr*dmaxr);

//The contact length of the front runner with the surface of the ice 199

//Sept. 22, 2009

l_d = sqrt(2.0*rcurve*dmax-dmax*dmax);

//The contact length of the rear runner with the surface of the ice

//Apr. 6, 2011

//For a rear runner following the front

//l_dr = sqrt(2.0*rcurve*(dmaxr-dmax)-(dmaxr-dmax)*(dmaxr-dmax));

//For a rear runner parallel to the front runner track

l_dr = sqrt(2.0*rcurve*dmaxr-dmaxr*dmaxr);

if (i==0)

{

ysteps = y_maxr/dy;

h = new double[ysteps];

z0 = new double[ysteps];

i++;

}

//compute ploughing force (Equation 3) ***

//Derived Apr.1, 2008 redone Sept.22, 2009

//A = r*r*(pi/2.0-asin(1.0-dmax/r)) + (dmax-r)*sqrt(2.0*r*dmax-dmax*dmax);

//This ploughing area is for a parallel track allignment, front and

//rear, tracks do not over lap

A = r*r*(pi/2.0-asin(1.0-dmax/r)) +...

...(dmax-r)*sqrt(2.0*r*dmax-dmax*dmax) +...

...r*r*(pi/2.0-asin(1.0-dmaxr/r)) +...

...(dmaxr-r)*sqrt(2.0*r*dmaxr-dmaxr*dmaxr); 200

//Derived Apr.1, 2008 redone Sept.22, 2009

//This ploughing area is for a rear runner following the front

//A = r*r*(pi/2.0-asin(1.0-dmaxr/r)) +...

//...(dmaxr-r)*sqrt(2.0*r*dmaxr-dmaxr*dmaxr);

plough_force = HI*A;

//compute melting temperature due to pressure melting - T_m

//(Fully agrees with, The Physics of Hockey, p.11)

T_m = -HI/101300.0/134.0;

//compute difference between blade-ice interface temperature and

//initial ice surface temperature - delta_T_i ***

delta_T_i = T_m - T_i;

// *compute the number of steps depending on the grid interval size*

zsteps = floor(l_d/dz);

//*** compute quasi-liquid layer thickness source: R.R. Gilpin, 1980

//(wire@lowT) and Dash, J. G., Surface melting, Contemporary Physics,

//1989, 30, 99. ***

qll = 3.5e-9 * pow(-T_i , -1.0/2.4);

while (v <= vmax) //Velocity loop

{

if (v>10) 201

dv = 2.5;

//transfer initial value of h based on qll or h(1) to components

//of h transformed to an array Sept. 22, 2009

//if we look down the length the left and right halves are

//assumed to be the same, only one half is calculated

for (k=0;k<=y_maxr/dy;k++)

h[k] = qll;

//initialize arrays and values for lubrication layer thick. calc.

sc = dy;

shear_force_sum = .0;

z = l_d/2.0;

//**compute initial value of shear stress force and start sum**

shear_force = visc*v*sc*dz / h[0];

shear_force_sum += shear_force;

//for output analysis, interim process (debugging)

// fp1<

jlast = 0;

// l = 100;

//********************************************************************

//main loop to compute lubrication layer thickness of the

//front runner contact 202 for (k=1;k

{

z -= dz;

//July 24, 2010

d = sqrt(rcurve*rcurve-(l_d/2.+z)*(l_d/2.+z)) -...

...sqrt(rcurve*rcurve - l_d*l_d);

//d = (z - l_d/2.0)*(z - l_d/2.0) / 2.0 / rcurve;

//The half contact width between the runner and the ice at z

y = sqrt(2.0*r*d-d*d);

ysteps = floor(y/dy);

for (j=0;j

{

//compute contact length of section from front view ***

//Mar.29, 2011

sc = dy/sqrt(1.-(j*j*dy*dy/r/r));

if (j>=jlast)

z0[j] = z + dz/2.;

//* compute terms used to find fluid thickness - h *

h_melt = dz * visc*v / (h[j]*rho_w*l_f);

h_condb = -dz * k_i*(T_i-T_b) / (th*v*l_f*rho_w);

//ref., Carslaw (p.97); Ingersoll (p.91) 203

h_condf = -dz * delta_T_i / (rho_w*l_f) *...

...sqrt(k_i*rho_i*c_i / (v*pi*(z0[j]-z)));

h_conds = dz * delta_T_s / (rho_w*l_f) *...

...sqrt(rho_s*c_s*k_s / (v*pi*(z0[j]-z)));

h_cond = h_conds + h_condf + h_condb;

delta_h = h_melt + h_cond;

//* update value of h *

h[j] += delta_h;

}

//Jul. 24, 2010

jlast = j-1; //+1;

hlast = h[jlast];

//Changed this see (May 24, 2009) (Mar. 30, 2011)

h_squeeze = -dz * HI*hlast*hlast*hlast / 4.0/visc/v/r/r *...

...sin(acos(1.0-d/r))/acos(1.0-d/r)/acos(1.0-d/r)/acos(1.0-d/r);

//July 28, 2010

for (j=0;j<=jlast;j++)

{

h[j] += h_squeeze;

//I may want to output this to identify those conditions

if(h[j] < qll)

h[j] = qll; 204

//update values for lubrication layer thickness and

//shear stress force loop

shear_force = visc*v*sc*dz / h[j];

shear_force_sum += shear_force;

}

//End of front runner contact

//Calculate the amount of refreeze between front and rear runner

//contacts. This may need correction due to settling of the melt

//layer, the force to moved the melt layer back in front of the

//rear runner is negligible 4-12-11.

//If front and rear runners are on parallel tracks limit loop to

//h[j] = qll;

for (j=0;j<=jlast;j++)

h[j] = qll;

//initialization of variables before the rear runner loop

zsteps = l_dr/dz;

z = l_dr/2.0;

jlast = 0;

//main loop to calc. lubrication layer thickness for the

//rear runner 205

for (k=1;k

{

z -= dz;

//For rear runner following in front runner track

//Apr.11, 2011 : Based or front runner result

//d = dmax + sqrt(rcurve*rcurve-(l_dr/2.+z)*(l_dr/2.+z)) -...

...sqrt(rcurve*rcurve - l_dr*l_dr);

//For rear runner running parallel to front runner track

d = sqrt(rcurve*rcurve - (l_dr/2.+z)*(l_dr/2.+z)) -...

...sqrt(rcurve*rcurve - l_dr*l_dr);

//The half contact width between the runner and the ice at z

y = sqrt(2.0*r*d-d*d);

ysteps = floor(y/dy);

for (j=0;j

{

//compute contact length of section from front view ***

//Mar.29, 2011

sc = dy/sqrt(1.-(j*j*dy*dy/r/r));

if (j>=jlast)

z0[j] = z + dz/2.;

//* compute terms used to find fluid thickness - h * 206

h_melt = dz * visc*v / (h[j]*rho_w*l_f);

h_condb = -dz * k_i*(T_i-T_b) / (th*v*l_f*rho_w);

//ref., Carslaw (p.97); Ingersoll (p.91)

h_condf = -dz * delta_T_i / (rho_w*l_f) *...

...sqrt(k_i*rho_i*c_i / (v*pi*(z0[j]-z)));

h_conds = dz * delta_T_s / (rho_w*l_f) *...

...sqrt(rho_s*c_s*k_s / (v*pi*(z0[j]-z)));

h_cond = h_conds + h_condf + h_condb;

delta_h = h_melt + h_cond;

//* update value of h *

h[j] += delta_h;

}

//Jul. 24, 2010

jlast = j-1; //+1;

hlast = h[jlast];

////Changed this see (May 24, 2009) (Mar. 30, 2011)

h_squeeze = -dz * HI*hlast*hlast*hlast / 4.0/visc/v/r/r *...

...sin(acos(1.0-d/r))/acos(1.0-d/r)/acos(1.0-d/r)/acos(1.0-d/r);

//July 28, 2010

for (j=0;j<=jlast;j++)

{

h[j] += h_squeeze; 207

//I may want to output this to identify those conditions

if(h[j] < qll)

h[j] = qll;

//output of layer thickness

//July 24, 2010

//update values for lubrication layer thickness and

//shear stress force loop

shear_force = visc*v*sc*dz / h[j];

shear_force_sum += shear_force;

}

//*** compute avg value for h ***

// hbar = h_sum / zsteps;

//*** compute the total resistive force - force_total ***

force_total = plough_force+ shear_force_sum;

//*** compute coefficient of friction ***

frict_coeff = force_total / down_force_tot;

//*** output coefficient of friction and varied parameters ***

fp2<

..."<

cout<

v += dv;

}

T_i -= dT_i;

// if (T_i < -5.0)

T_b -= dT_i;

}

//delete arrays

//delete[] h;

//delete[] z0;

//close program

//fp1.close(); fp2.close();

cout<<"press any key to end program"<<’\n’; getch(); return 0;

}

D.3 Supplementary F.A.S.T. Model Results

In this section supplemental results from the F.A.S.T. model are examined. In Fig. D.1 the melt layer thickness across the half-width at the tail end of a front runner is shown. 209

The opposite half-width would be a mirror image. The melt layer at the very edge is

Figure D.1: The melt layer thickness across the half-width at the tail end of the front runner. The cross-sectional radius is 4.75 mm, the rocker is 34 m, the ice surface temperature is −7◦C and the sled speed is 25 m/s (F.A.S.T. 3.2b). about 28 nm which is about 20 times the quasi-liquid layer thickness at −7◦C.

Next, we examine the relative change in the coefficient of friction when we decrease the cross-sectional radius of curvature from 4.75 to 4.0 mm in the F.A.S.T. 3.1b model in Fig. D.2. In a similar analysis, we examine the effect of increasing the cross-sectional radius from 4.75 to 5.5 mm in Fig. D.3. In Figure D.4 we examine the effect of decreasing the rocker from 34 to 20 m. The effect of increasing the rocker from 34 to 48 m is analyzed in Fig. D.5. All of these F.A.S.T. 3.1b model results are very similar to the results from the F.A.S.T. 3.2b model. 210

Figure D.2: The relative change in coeﬃcient of friction between bobsleigh runners and a ﬂat sheet of ice when the cross-sectional radius of the runner is decreased from 4.75 to 4.0 mm for various sled speeds and ice surface temperatures from −2 to −14◦C. The front and rear runner leave one track (F.A.S.T. 3.1b). 211

Figure D.3: The relative change in coeﬃcient of friction between bobsleigh runners and a ﬂat sheet of ice when the cross-sectional radius of the runner is increased from 4.75 to 5.5 mm for various sled speeds and ice surface temperatures from −2 to −14◦C. The front and rear runner leave one track (F.A.S.T. 3.1b). 212

Figure D.4: The relative change in coeﬃcient of friction between bobsleigh runners and a ﬂat sheet of ice when the rocker of the runner is decreased from 34 to 20 m for various sled speeds and ice surface temperatures from −2 to −14◦C. The front and rear runner leave one track (F.A.S.T. 3.1b). 213

Figure D.5: The relative change in coeﬃcient of friction between bobsleigh runners and a ﬂat sheet of ice when the rocker of the runner is increased from 34 to 48 m for various sled speeds and ice surface temperatures from −2 to −14◦C. The front and rear runner leave one track (F.A.S.T. 3.1b). 214

Appendix E

Supplementary Information on Ice Friction Analysis

E.1 Calculation of Mean Square Velocity for Air Drag Calculation

The calculation of the mean squared velocity of the sled v¯2. We suppose that the acceleration of the sled is constant in the range, v(0) = vi to v(tf ) = vf . R tf v2(t)dt v¯2 = 0 (E.1) ∆t

Since v(t) is linear, it can be expanded resulting in,

³ ´2 R tf vf −vi 0 vi + · t dt v¯2 = ∆t (E.2) ∆t which can be expanded to, Ã ! R 2 tf 2 2vi(vf −vi) (vf −vi) 2 0 vi + ∆t t + (∆t)2 t dt v¯2 = (E.3) ∆t

By performing the integral we obtain,

2 2 vi(vf −vi) 2 (vf −vi) 3 vi tf + t tf + 3t2 tf v¯2 = f f (E.4) tf

Which can be simpliﬁed to the solution,

(v − v )2 v¯2 = v2 + v (v − v ) + f i (E.5) i i f i 3

E.2 Determination of the Slope in the Ice House

The precision required on the measurement of the slope in the Ice House was demanding.

A precision around 0.1◦ was needed in order to compare the data with results from 215 experiments at the Olympic Oval on flat ice. A carpenter’s square with a height of 57 cm was used with a string and needle in an attempt to construct a large protractor. While this should have provided the required precision the scatter in the data meant that the desired precision was not obtained. The Ice House is like a V shaped slide. The radar gun was set to measure the acceleration down one side of this V. An experiment was setup where the oscillation of the sled was examined in the V. Given the symmetry of the blades over the contact area it was assumed the coefficient of friction µ was the same in both directions. The aerodynamic drag is a function of the shape of the object, the cross-sectional area, and the surface texture. This was significantly different in each direction therefore another drag constant was named for the back of the sled αb. An example of the oscillation data is illustrated in Fig. E.1. The Stalker ATS radar gun

Figure E.1: Radar velocity data taken on September 16, 2010 of a sled oscillating on the Ice House track, ice surface temperature was −4.4◦C. measures only positive data whether the object is moving closer or further. In Fig. E.1, 216 the sled is moving forward down the main slope from (0 − 10) s, it moves up and down the opposite slope from (10 − 27) s. To determine the angle of the main slope in the Ice

House the data from the following two sections were used. First, from (27 − 36) s the sled is moving backwards up the slope. Extending from equation 5.1, Newton’s second law for this portion of the movement was:

¯2 ma = mg sin θ + µmg cos θ + αbv (E.6)

Second, from (36 − 44) s the sled returns down the hill in the forward direction, its movement can be described by:

ma = mg sin θ − µmg cos θ − αv¯2 (E.7)

In this manner the deceleration and acceleration of the sled are observed on the same incline of the hill. In the same manner, the sled can be turned around and the same experiment can be performed. The movement of the reversed sled is described by two new equations, for the movement up the slope:

ma = mg sin θ + µmg cos θ + αv¯2 (E.8)

And for the return down the hill:

¯2 ma = mg sin θ − µmg cos θ − αbv (E.9)

Now by measuring the acceleration data in each of these four cases and taking the average result we can determine an average slope of the hill with a very high repeatability.

a = g sin θ (E.10)

In all the cases the acceleration was determined over the same range of velocities, from

(2.2 − 7.1) m/s. Finally, averaging the acceleration values from the forward and reverse directions we obtain, a = g sin θ = 1.161 m/s2 which we can use to determine θ =

(6.80 ± .02)◦. 217

Table E.1: Acceleration of the sled up and down the slope when pushed forward run # up (m/s2) down (m/s2) mean (m/s2) 1 1.249 1.076 1.163 2 1.254 1.076 1.165 3 1.289 1.041 1.165 4 1.246 1.087 1.167 mean 1.165 ± 0.002

Table E.2: Acceleration of the sled up and down the slope when pushed in reverse run # up (m/s2) down (m/s2) mean (m/s2) 1 1.226 1.083 1.161 2 1.230 1.080 1.155 3 1.238 1.085 1.154 mean 1.157 ± 0.004

E.3 Supplemental Lossy Acceleration Data

Lossy accelerations for our 123 kg sled with three diﬀerent amounts of added mass are shown in Fig. E.2. The quadratic regression curves have been included with the data.

2 The regression parameters are included in table E.3 where, al = Avrms + Bvrms + C. In

Table E.3: Parameters from a quadratic regression for three of the sled masses used. Additional mass (kg) A (10−3) B (10−3) C (10−3) 0 -2.0 4.4 -54 30 -1.4 1.6 -49 60 -1.1 -.7 -38

Fig. E.2 and Table E.3 we see that the lossy acceleration is proportional to the square of vrms and it decreases monotonically with mass. 218

Figure E.2: Recorded lossy accelerations of the 123 kg sled with a) 0 kg, b) 30 kg, and c) 60 kg added. The white hollow circles indicate measurements from the Calgary Olympic Oval and the grey ﬁlled circles from the Ice House. The solid lines are quadratic regressions for each mass. The experiments were conducted over four days, ice surface temperatures were between −2.2 and −4.6◦C. 219

E.4 Surveyor’s Reports

This section includes the surveyor’s reports of the bobsleigh track at Canada Olympic

Park in Calgary, Alberta. The report was prepared by Andrew Hunter, Assistant Profes- sor in the Geomatics Engineering Department of the University of Calgary. 220 221 222 223 224 225 226 227 228

E.5 Analysis of the Surveyor’s Reports

In the following section I will cover the analysis of the data from the two surveyor’s reports. The bobsleigh track is about 25 years old so we expect some movement has occurred since construction. A slope, designed to be constant, may now have some minor variations. I used the pre-construction drawings to help guide the analysis.

In Table E.4, I examine the section of track between corners 1 and 2. According to the track drawings [102] the slope was constant for 18.5 m and then leveled slightly. After examining the surveyor’s report I decided that the sections 0 − 2 were in that constant slope. This corresponds to 14.9 m.

Table E.4: Determination of the mean slope between corners 1 and 2 at the Calgary Olympic bobsleigh track. Track Start End Slope (◦) Slope (◦) Slope (◦) Segment Position Position 1st report 2nd report Average 0 -5.0 0.0 6.57 6.57 1 0.0 4.9 6.57 6.58 6.57 2 4.9 9.9 6.79 6.79 6.79 3 9.9 14.9 6.33 6.33 6.33 4 14.9 21.5 6.05 6.05 6.05 mean slope sections 0 − 4 6.46 ± 0.28 sections 0 − 2 6.64 ± 0.13

In Table E.5, I examine the section of track between corners 3 and 4. According to the track drawings the slope was constant for 52.5 m and then leveled slightly. The surveyors were unable to access the start of this section of track due to the track coverings in corner

3 [103]. After examining the surveyor’s report I decided that the sections 1 − 4 were in that constant slope. This corresponds to 39.9 m.

In Table E.6, I examine the section of track between corners 5 and 6. According to the track drawings this entire slope was constant and measured 52.5 m. After examining the surveyor’s report I decided that the entire 35.4 m section examined was in a relatively 229

Table E.5: Determination of the mean slope between corners 3 and 4 at the Calgary Olympic bobsleigh track. Track Start End Segment Position Position Slope (◦) 1 267.0 277.0 4.59 2 277.0 286.9 4.59 3 286.9 296.9 4.59 4 296.9 306.9 4.59 5 306.9 312.8 5.30 mean slope sections 1 − 5 4.73 ± 0.32 sections 1 − 4 4.59 ± 0.00

Table E.6: Determination of the mean slope between corners 5 and 6 at the Calgary Olympic bobsleigh track. Track Start End Segment Position Position Slope (◦) 1 447.6 456.5 8.62 2 456.5 466.4 8.54 3 466.4 476.3 8.54 4 476.3 483.0 8.73 mean slope sections 1 − 4 8.61 ± 0.09 constant slope.

In Table E.7, I examine the section of track between corners 8 and 9. According to the track drawings this slope was constant over 103.0 m. After examining the surveyor’s report I decided that the entire 103.1 m section examined was in a relatively constant slope. 230

Table E.7: Determination of the mean slope between corners 8 and 9 at the Calgary Olympic bobsleigh track. Track Start End Segment Position Position Slope (◦) 1 744.5 759.4 6.65 2 759.4 774.3 6.81 3 774.3 789.2 6.81 4 789.2 806.1 6.81 5 806.1 821.0 6.81 6 821.0 835.9 6.81 7 835.9 847.6 6.81 mean slope sections 1 − 7 6.79 ± 0.06 Bibliography

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