THE PHYSICS OF THE PINEWOOD DERBY

Transcription

1 THE PHYSICS OF THE PINEWOOD DERBY TABLE OF CONTENTS CHAPTER 1 - INTRODUCTION... 1 CHAPTER 2 - CONSTANTS, PARAMETERS, AND VARIABLES 2.1 ENVIRONMENTAL CONSTANTS TRACK PARAMETERS The Standard Track The Perfect Ramp CAR PARAMETERS The Standard Car The Perfect Car The Perfect Race VARIABLES CHAPTER 3 - SIMPLE MATHEMATICAL MODELS FOR THE PERFECT CAR 3.1 THE CIRCULAR ARC RAMP THE INCLINED PLANE RAMP THE CIRCULAR ARC RAMP - EXACT SOLUTION THE PERFECT RAMP Various Circular Arc ramps The Brachistichrone - A Variational Application in Space The Cycloid Ramp - Horizontal Tangent Position Unspecified The Cycloid Ramp - Horizontal Tangent Position Specified Analysis of Cycloid Ramp Motion Cycloid Properties Time Calculation The Lagrangian Formulation - A Variational Application in Time THE CENTER OF MASS EFFECT The Circular Arc Ramp The Inclined Plane Ramp Comparison of The Circular Arc and the Inclined Plane Ramp CHAPTER 4 - MORE DETAILED AERODYNAMIC AND OTHER MATH. MODELS 4.1 INTRODUCTION WHEEL MOMENT OF INERTIA The Circular Arc Ramp The Inclined Plane Ramp AIR RESISTANCE Introduction The Newtonian View The Momentum Balance Approach Energy Considerations Immersed Flow Theoretical Drag Coefficients... 62

2 4.3.7 Viscosity Effects The Boundary Layer Experimental Drag Coefficients The Circular Arc Ramp Horizontal Motion The Inclined Plane Ramp FRICTION Basics of Friction The Circular Arc Ramp The Inclined Plane Ramp SUMMARY OF RELATIVE EFFECTS REACTION FORCE EFFECT ON FRICTION A MORE EXACT VELOCITY SOLUTION CHAPTER 5 - A COMPLETE MATHEMATICAL MODEL 5.1 THE CIRCULAR ARC RAMP THE VELOCITY SOLUTION HORIZONTAL MOTION THE INCLINED PLANE RAMP Ramp Motion Ignoring the Transition Region The Inclined Plane Ramp Transition Newton s Second Law Approach to the Transition Time The CM Path and Energy Relation Approach The Exact Transition Time SUMMARY The Circular Arc Ramp The Inclined Plane Ramp Horizontal Motion The Center of Mass Effect CHAPTER 6 - THE EXACT SOLUTION FOR THE CIRCULAR ARC RAMP 6.1 INTRODUCTION THE MILNE METHOD - PERFECT CAR CASE The Maclaurin Series The Milne Solution Interpolation for Time and Velocity The Solution for Small h Values THE MILNE METHOD - GENERAL CASE Summary The Maclaurin Series The Milne Solution CORRELATION DEVELOPMENT EXTENSIONS TO THE CIRCULAR ARC RAMP MODEL Smaller Values for Starting Angle Circular Arc Ramps Not Tangent to the Horizontal

3 CHAPTER 7 - WHEELS 7.1 INTRODUCTION WHEEL MATERIAL PROPERTIES SLAB WHEELS STANDARD WHEELS Standard Wheel Standard Wheel Wheel and Axle Dimensions LATHE TOOLS AND FITTINGS Lathe Tools Standard and Segmented Wheel Fittings MODIFYING THE 1990 STANDARD WHEEL The White 12/3/90 Wheel The White 12/29/90 Wheel The White 1/20/91 Wheel The Black Widow Pace Car Wheel MODIFYING THE 1991 STANDARD WHEEL The JN No. 99 Wheel The Garcia 2/93 Wheel The JS Red 2/92 Wheel The JS Min Spec Wheel The JN Test Wheel The JN Silver Rear Wheels The JN Silver Front Wheels SMALL AXLE SEGMENTED AND UNSEGMENTED WHEELS Making Small Axles " Making Small Axles " Quench Hardening the Axle Shaft Polishing the Axle Assembly The JS Red 039 Wheel The JN Gold 042 Wheel The JN Silver 042 Wheel The JN Red 042 Wheel X - WHEELS X - Wheel Outer Shell X - Wheel Inner Shell X - Wheel Assembly REMOLDED WHEELS Mold Construction Using the Mold Tapering the Wheel The T2 Hole Wheel The T2 Calibrate Wheel The Optimum Wheel

4 7.11 WHEEL FORCES AND PERFORMANCE Wheel Friction Wheel Types Approximate Analysis - One Wheel Sliding on Axle Detailed Analysis Summary of Wheel Forces WHEEL PROPERTY SUMMARY CHAPTER 8 - CAR BODIES 8.1 INTRODUCTION CAR BODY DESIGN Pinewood Derby Body Blocks The Standard Car Body The White Car Body The JSRG Car Body The JNRG Car Body The JNTS Car Body The JSTR Car Body The JNSN Car Body The T2 Car Body The JNGE Car Body THE ESTIMATION OF AERODYNAMIC DRAG COEFFICIENTS The Reynold s Number Skin Friction Drag Pressure Drag Drag Coefficient Estimation for Key Symmetrical Shapes Various Drag Contributions Shapes 1B and 2B, R D =7.3 x 10 4, L/D = Skin Friction Calculations Streamlined Noses and Negative Nose Drag Jet Pump Effect Shapes 1A and 2A, R D =7.3 x 10 4, L/D = Shapes 1B, 2B, 1A, 2A for R D = 2.9 x Drag for Shapes 3A, 3B Drag for Shapes 4A, 4B Drag for Shapes 5A, 5B Drag for Shapes 6A, 6B Estimation of Drag Coefficients for Unsymmetrical Shapes Application of Derived Drag Coefficients to Car Body Design EXPERIMENTAL DETERMINATION OF DRAG COEFFICIENTS Introduction Wind Tunnel Types Wind Tunnel Construction and Operation Wind Tunnel Velocity Calibration Standard Test Objects Car Body Drag Coefficient Measurement SUMMARY OF BODY AND WHEEL DRAG COEFFICIENTS

5 CHAPTER 9 - FRICTION AND LUBRICATION 9.1 INTRODUCTION THE NATURE OF FRICTION CHARACTERISTICS OF LUBRICANTS Liquid Lubrication - Hydrodynamic Liquid Lubrication - Boundary Dry Solid Lubricants LUBRICANT SCREENING TESTS General Experimental Setup The Total Spin Time (TST) Method The Spin Time Decay (STD) Method Cylindrical Tins The Small Tin The Large Tin Thin Discs Thin Steel Disc - Top Spun The Navier-Stokes Solution The T2 Trimmed Wheel Liquid Lubricant Friction Tests Wheel Bore and Axle Preparation Fitting FDS Liquid Lubricant Preparation Experimental Procedure Liquid Lubricant Friction Results Solid Lubricant Friction Tests - Low Velocity Average Coefficient of Friction Wheel Bore and Axle Preparation Lubrication Procedures Solid Lubricant Screening Results X-Lube Graphite Lubricant Results PTFE Powder Results Molybdenum Disulfide Powder Results Effect of Axle Roughness Solid Lubricant Friction Tests - Coefficient of Friction as Function of Velocity Discussion of Results Average Coefficient of Friction by TST Variable Friction Effect on Race Time Brass Bore on Steel X-lubed Axle COEFFICIENT OF FRICTION FOR TRANSVERSE WHEEL SLIDING CHAPTER 10 - THE PERFECT RACE 10.1 INTRODUCTION CENTER OF MASS VERSUS AIR RESISTANCE OPTIMIZATION WHEEL RADIUS VERSUS MOMENT OF INERTIA OPTIMIZATION PERFORMANCE OF RACE600Z.BAS SENSITIVITY OF TIME TO NET MASS OF RACE CAR BALL BEARING RUNS ACTUAL PERFECT RACE ANALYSES FOR VARIOUS CARS

6 CHAPTER 11 - STATISTICS AND RACE VARIABILITY 11.1 INTRODUCTION TO STATISTICS CHAPTER OUTLINE, INCLUDING APPENDICES STATISTICAL DEFINITIONS TIME MEASUREMENTS IN DATA SET Data acquisition for Data Set Graphical Depiction of Data Set The Distribution of the Mean Values The Distribution of the Deviations from Time Heat Mean Values The z Variable Goodness-of-Fit for Normal Distributions of Data Set Data Set 1- Goodness-of-Fit Time Heat and Randomly Selected 3-Time Averages - Goodness-of-Fit Time Heat Deviates - Goodness of Fit Other Applications of the $ 2 Distribution The $ 2 Distribution for df = Testing $ 2, df = The $ 2 Distribution for df = 2 and df = The Standard Deviation Distribution APPLICATIONS TO DATA SET APPLICATIONS TO DATA SETS 3 AND ESTIMATION APPLICATIONS USING NORMAL AND T -DISTRIBUTIONS Given 1, µ and a Normal Distribution Given 1, x and a Normal Distribution The Difference Between Time Heat Averages Given s, x n, and a Normal Distribution CORRELATION USING LINEAR REGRESSION Simple Linear Regression - Derivation Simple Linear Regression - Correlation Coefficient A Statistical Model for Linear Regression Multiple Linear Regression - Introduction Multiple Linear Regression - Derivation LINEAR REGRESSION APPLICABILITY CHAPTER 12 - OTHER CAR DESIGNS 12.1 FUNNY CARS The Falling Weight Car The Rocket Powered Car Car Parts Car Construction Wiring the Circuit Operation Performance THE NEARLY PERFECT CAR Introduction Body Design Wheel Design Air Drag Body Assembly and CM Calculations Performance of the NP Car

7 APPENDIX 1 - PHOTOGRAPHS TABLE OF CONTENTS (CONTINUED) Photo A1.P1 Cars JNGE, JNSN, and T2 at the gate Photo A1.P2 The Standard Car as described in Chapter 1 and in Fig Photo A1.P3 Views of the BW Pace Car and the JSRG Car described in Chapter Photo A1.P4 Views of the JNRG Car and the JNTS Car as described in Chapter Photo A1.P5 Views of the JNSN Car and the JSTR Car as described in Chapter Photo A1.P6 Views of the T2 Car and the JNGE Car as described in Chapter Photo A1.P7 Photos of the first 12 wheels from Table Photo A1.P8 Photos of the last 12 wheels from Table Photo A1.P9 Body hole drill guide Photo A1.P10 Body lead mold as shown in Figure Photo A1.P11 Compression mold for forming T2 type wheel Photo A1.P12 Various aluminum and brass fittings used in wheel modification Photo A1.P13 Fitting FDS as shown in Figure Photo A1.P14 An old Craftsman 4-in lathe used in this work Photo A1.P15 Brass tubing " OD by 0.036" ID Photo A1.P16 Brass tubing shown in A1.P15 crimped towards the end Photo A1.P17 Lathe tools as shown in Figure Photo A1.P18 The T2 Calibrate wheel Photo A1.P19 Parts for the JS 039 axle nail according to Figures 7.36 through Photo A1.P20 The complete JS 039 wheel and axle assembly according to Figure Photo A1.P21 The graphite samples studied in Appendix Photo A1.P22 A 30 power magnified view of X-lube next to the fine sample G Photo A1.P23 A sample of X-lube spread out in a plastic dish Photo A1.P24 A neodymium iron boron (NIB) magnet placed beneath the sample Photo A1.P25 A 30X magnified view of a sparse sample of X- lube spread out Photo A1.P26 The magnetic field from a NIB magnet causes the individual APPENDIX 2 - TRACK CONSTRUCTION A2.1 INTRODUCTION A2.2 MATERIALS A2.3 PLYWOOD AND GUIDE STRIP PREPARATION A2.3.1 Cutting the Pieces A2.3.2 Sanding and Varnishing A2.3.3 Assembly of the Plywood Track Sections A2.4 SECTION 1 CONSTRUCTION AND ASSEMBLY A2.4.1 Leg Assembly and Attachment A2.4.2 The Starting Bar A2.4.3 The Trigger Assembly A2.4.4 Guide Strip Attachment and Track Setup APPENDIX 3 - THE BRUSH STOPPER

8 APPENDIX 4 - SIMPLE HIGH PERFORMANCE CARS A4.1 THE STANDARD CAR A4.2 PREPARING THE AXLE A4.3 LUBRICATION A4.4 PERFORMANCE IMPROVEMENTS A4.4.1 Modification B - Use 3 Wheels Rather than A4.4.2 A Simple but Fast Cub Scout Car A General Construction A Modification C - Streamline Body A Modification D - Moving Weight to Rear A Modification E - Trim Wheels to Minimum Allowed A Modification F - Change to X-Lube Graphite A Modification G - Change to a Low Profile Car A Modification H - Move More Weight to the Rear A4.4.3 Modification I - Shave Wheel as a Semi-Pro Modification A4.5 A BODY DRILL GUIDE ASSEMBLY A4.6 A SIMPLE INEXPENSIVE BALANCE APPENDIX 5 - MILNES.BAS APPENDIX 6 - RACE600Z.BAS APPENDIX 7 - RACE600Z.BAS APPLIED TO PERFECT RACE DATA- FIG APPENDIX 8 - AXLE/WHEEL FRICTION FORCES A8.1 NEWTON S SECOND LAW APPROACH - SMALL CLEARANCE A8.2 WORK-ENERGY APPROACH - LARGE CLEARANCE APPENDIX 9 - AIR DRAG FOR PROJECTED AREA ASPECT CHANGES APPENDIX 10 - AIR RESISTANCE OF ROTATING DISC ABOVE A PLANE APPENDIX 11 - MAGNETIC PROPERTIES OF GRAPHITES A11.1 INTRODUCTION A11.2 EXPERIMENTAL PROCEDURE A11.3 DATA ACQUISITION A11.4 MOLAR CONCENTRATION OF ADSORBED OXYGEN A11.5 DISCUSSION OF RESULTS A11.6 RELATED AND FUTURE WORK APPENDIX & 1999 STANDARD WHEELS APPENDIX 13 - AIR RESISTANCE OF A ROLLING SPHERE A13.1 INTRODUCTION A13.2 EXPERIMENTAL SETUP A13.3 EXPERIMENTAL RESULTS A13.4 DISCUSSION OF RESULTS APPENDIX 14 - THE CHI-SQUARED DISTRIBUTION A14.1 INTRODUCTION A14.2 THE ISOTHERMAL ATMOSPHERE A14.3 TWO DEGREES OF FREEDOM

9 A14.4 THREE DEGREES OF FREEDOM A14.5 THE GAMMA FUNCTION APPENDIX 15 - NORMAL PARAMETERS AND THE STUDENT-T DISTRIBUTION A15.1 INTRODUCTION A15.2 AVERAGE OF A NORMAL RANDOM VARIABLE A15.3 VARIANCE OF A NORMAL RANDOM VARIABLE A15.4 DISTRIBUTION OF THE SAMPLE VARIANCE A15.5 THE SAMPLE VARIANCE AVERAGE VALUE A15.6 THE STUDENT - T DISTRIBUTION APPENDIX 16 - DISCRETE RANDOM VARIABLES & THE GOODNESS-OF-FIT TEST A16.1 INTRODUCTION A16.2 DISCRETE PROBABILITY DISTRIBUTIONS A16.3 THE NATURE OF RANDOM VARIABLES A16.4 COMBINATIONS OF TWO RANDOM VARIABLES A16.5 BERNOULLI TRIAL A16.6 BINOMIAL RANDOM VARIABLES A16.7 MULTINOMIAL RANDOM VARIABLES A16.8 THE GOODNESS-OF-FIT TEST - NO PARAMETERS A16.9 GOODNESS-OF-FIT - NORMAL DISTRIBUTION APPLICATION APPENDIX 17 - SAMPLING AND THE CENTRAL LIMIT THEOREM A17.1 INTRODUCTION A17.2 RANDOM SAMPLES AND POPULATIONS A17.3 THE LAW OF LARGE NUMBERS A17.4 THE CENTRAL LIMIT THEOREM A17.5 AN EMPIRICAL COIN TOSS EXPERIMENT A17.6 AN EXPERIMENT WITH A SIMULATED NORMAL DISTRIBUTION A17.7 RELATION BETWEEN VARIANCES OF POPULATIONS, AVERAGES, DEVIATES. 494 APPENDIX TIMES IN 135 TIME HEATS APPENDIX 19 - APPLICATION NOTES A19.1 VARIABLE AIR DRAG COEFFICIENTS A The Euler numerical method A Application to Falling Objects A Application to a Race Car A19.2 THE BALLISTIC PENDULUM REFERENCES INDEX