A Performance Analysis System for the Sport of Bowling

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$Course Syllabus - Penn State Harrisburg Math/Computer Sciences ...$

$A Simulator for the MARIE Architecture - Penn State Harrisburg Math ...$

3.4.3 Finding the Coefficient of Friction The coefficient of friction can be recovered from the previous calculations. Under the assumptions, the only active force is the force due to friction acting between the ball and the lane, and that force acts solely to transfer linear momentum to angular momentum. Since the changes in angular and linear momentum for each sample are known, the frictional force required to generate those changes can be found. For any revolution i, the frictional force F µi acting during i is given by Fµi = m / 2 · k(ω 2 i - ω 2 i-1 ) / (D i - D i-1 ) = m / 2 · (v 2 i - v 2 i-1 ) / (D i - D i-1 ). The coefficient of friction µ i between the ball and the lane for revolution i is then µ i = Fµi / mg, where g is the acceleration due to gravity. Similarly, the frictional force for any sample j of revolution i can be found from Fµi,j = m / 2 · k(ω 2 i,j - ω 2 i,j-1 ) / (d i,j - d i,j-1 ) = m / 2 · (v 2 i,j - v 2 i,j-1 ) / (d i,j - d i,j-1 ). The coefficient of friction µ i,j between the ball and the lane for any sample j of revolution i is then given by µ i,j = Fµi,j / mg. Using this result, there is now enough information to graph the coefficient of friction between the ball and the lane with respect to the distance from the foul line. 54
3.5 ASSUMPTIONS AND ERROR ANALYSIS The derivations behind the MASTER calculations presented in the previous discussion rely heavily on several basic assumptions, some of which have already appeared in other sections throughout the course of the paper. This section presents the analysis assumptions in detail, along with an assessment of their possible error contributions. 3.5.1 Distance Throughout the paper, it has been assumed that the ball travels 60 feet - the distance from the foul line to the head pin. Under normal circumstances, release of the ball occurs close to the foul line, and the first pin the ball encounters is the head pin (at least on the first ball of any frame). It is time to take a closer look at what this assumption entails. • The ball is released at the foul line: An error factor is introduced in the average linear velocity calculation if the ball is released behind the foul line (the actual velocity will be higher than calculated), or beyond the foul line (the actual velocity will be lower than calculated). Generally, the bowler releases the ball at, or just beyond, the foul line. For a legal delivery, no part of the bowler's body may touch the lane beyond the foul line. This presents a physical limitation on release beyond the foul line of approximately 12 inches. Additional error is possible from behind the foul line, usually resulting from a noticeable lapse in execution on the bowler's part (dropping the ball, releasing the ball early, or stopping short of the foul line). Although the module can not detect the absolute point of release, it does indicate when the ball hit the lane, in relation to release. In addition, the bowler easily recognizes when these major lapses in execution occur, and can take note of them. The nominal margin of error at the point of release is estimated to be ±6", with the vast majority of releases occurring within a ±3" range. • The ball hits the head pin at a fixed location: At the other end of the lane, the ball can hit the head pin in different spots - head on, or on the right or left side. A pin is 4½" in diameter at the height at which the ball contacts it, and the center of the pin is located 60' ±½" from the foul line. A ball is nominally 8½" in diameter. Assuming the ball is released with its center over the foul line and it hits the head pin dead center, it must travel 59' 5" to 59' 6" from the point of release. If the ball barely grazes the right or left side of the head pin, the ball traveled at least 59' 11½" to 60' ½". A solid pocket hit (the goal of any potential strike delivery) falls halfway between these two ranges, so the expected distance from the foul line that the initial pin impact should occur is 59' 8¾". Therefore, the margin of error introduced in hitting the head pin is ±3¾" If the ball misses the head pin, it must travel at least an additional 10.4" to get to the next pin, and the same error factors as for hitting the head pin come into play. The bowler could take note of the first pin that was hit - the head pin, the 2/3 pins, the 4/6 pins or the 7/10 pins, and the analysis software could again adjust for the difference. Bowlers readily identify the location of the ball based on the board it is on when it hits the pins. This information could be used to better locate the ball at pin impact, along with the first pin hit, and this error could be reduced to approximately ±1". 55
Page 1 and 2:
The Pennsylvania State University T
Page 3 and 4:
TABLE OF CONTENTS Section I: Introd
Page 5 and 6:
LIST OF FIGURES Figure 2-1 SMARTDOT
Page 7 and 8:
SECTION I: INTRODUCTION, BACKGROUND
Page 9 and 10: eceiver that stores, processes, and
Page 11 and 12: 1.3.2 Product Development Phase The
Page 13 and 14: SECTION II: COLLECTING THE DATA - T
Page 15 and 16: 2.2 Design Constraints The feasibil
Page 17 and 18: 2.3 SENSING REQUIREMENTS The module
Page 19 and 20: Pin Deck Light Foul Line Figure 2-2
Page 21 and 22: 2.4.1 Ambient Light Sensor The ambi
Page 23 and 24: 9 28 8 RST V CC V CC 5 6 hyst 7 3 +
Page 25 and 26: 2.6 HARDWARE/SOFTWARE INTERACTION D
Page 27 and 28: 2.6.3 Baud Rate The module communic
Page 29 and 30: Configuration settings and the samp
Page 31 and 32: 2.7.5 The Ball is Rolling If the wa
Page 33 and 34: covered by the wand). When the modu
Page 35 and 36: 2.8.1 Detecting Release Detecting t
Page 37 and 38: Light Level 100 90 80 70 60 50 40 3
Page 39 and 40: By filtering the data, and then cou
Page 41 and 42: The 'F300x can write to any portion
Page 43 and 44: One possibility is to use a transfo
Page 45 and 46: 2.10 SUMMARY This portion of the re
Page 47 and 48: The ball's angular velocity increas
Page 49 and 50: Light Level 100 90 80 70 60 50 40 3
Page 51 and 52: the fundamental frequency has been
Page 53 and 54: Since the fundamental frequency and
Page 55 and 56: Figure 3-7: MASTER "Analysis" Scree
Page 57 and 58: known, the linear momentum (and thu
Page 59: In order to find the value for v 1
Page 63 and 64: 3.5.3 External Forces and Friction
Page 65 and 66: 3.6.1 Implementation and Performanc
Page 67 and 68: Figure 3-11a: Effects of Phase Shif
Page 69 and 70: 3.6.3 The "Perfect" Game to Analyze
Page 71 and 72: Since the change in angular velocit
Page 73 and 74: BIBLIOGRAPHY [1] United State Paten
Page 75 and 76: APPENDIX A - SMARTDOT MODULE EMBEDD
Page 77 and 78: Appendix A: SMARTDOT Module Embedde
Page 85 and 86: APPENDIX B - SMARTDOT MODULE SOURCE
Page 87 and 88: Appendix C: SMARTDOT Module Command
Page 89 and 90: Figure C-3 Appendix C: SMARTDOT Mod
Page 91 and 92: APPENDIX E - 300 GAME MASTER SCREEN
Page 93 and 94: Appendix E: 300 Game Analysis Frame
Page 95 and 96: Appendix E: 300 Game Analysis Frame
Page 97: Appendix E: 300 Game Analysis Frame
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A Performance Analysis System for the Sport of Bowling

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