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The Physics and Technology of Tennis
Table of Contents
By Howard Brody, Rod Cross & Crawford Lindsey
Preface
PART 1: RACQUETS
1 Choosing A Racquet
Rod Cross
Choosing a Racquet
How Does It Feel?
Grip Size
Racquet Weight and Power
Sweetspots
Balance Point and Swingweight
Headsize
Frame Stiffness
String Type and Diameter
String Tension
2 Racquet Materials
Crawford Lindsey
Graphite
Modulus and Strength
Making High Modulus Carbon
Making a Racquet
Racquet Lay-up
Titanium
Piezoelectric Materials
3 Weight, Balance & Swingweight
Crawford Lindsey & Howard Brody
Force, Mass and Inertia
Weight
Balance Point
Pickup Weight
What Balance Point Means to Tennis Players
Customizing the Balance Point
Swingweight
What Swingweight Means to the Player
Customizing Swingweight
The Relationship of Weight, Balance and Swingweight
Is It the Weight of the Swing, or the Swing of the Weight?
As Weight Goes Down, Balance Goes Up
4 Racquet Stability
Crawford Lindsey & Howard Brody
Weight In Past Racquet Developments
Stability By Weight Polarization
What, How, When, Where and Why of Polarization
Twistweight: The Stability of the Racquet Against Twisting
Interpreting the Stability Table
5 Measuring Swingweights
Rod Cross & Crawford Lindsey
Measuring Swingweight
Measuring Twistweight
Measuring Recoilweight
6 Sweetspots
Howard Brody, Rod Cross, & Crawford Lindsey
The Center of Percussion
The Node
The Power of the Tennis Racquet: COR and ACOR
The Deadspot
7 Introduction To Shock & Vibration
Howard Brody & Crawford Lindsey
Introduction
What Is Vibration?
Bending Shapes (Modes)
Amplitude of Vibration and the Node
Frequency
Stiffness
Dampening
Injury
What Is Shock and What Is Its Relation to Vibration?
Lessons From Vibrations
How Does Shock Absorption Work?
Conclusion
8 Racquet Vibration Theory
Rod Cross
Frame Bending and Vibrations3
Vibration Shapes
Racquet Motion During Impact
Vibration Frequency
Vibration Modes of a Rod or Beam
Impact with a Beam or Racquet
Bounce Speed Off a Vibration Node
9 Racquet Feel
Rod Cross
Effects of String Tension on Jarring
Simple Stiffness and Vibration Experiments
Measurements of Shock and Vibration
Vibration Frequency and Amplitude
Vibration Frequency of a Racquet
Sweetspot Softness and Feel
Sweetspot Size
Shock and Feel
Recoil Speed of the Handle
Impact Duration, Force, and Feel
Effect of Frame and String Stiffness on Frame Vibrations
String Vibrations
Measurements of the Force on the Hand
String Vibrations and Feel
10 Force and Momentum
Crawford Lindsey
Introduction
Newton's First Law of Tennis
Newton's Second Law of Tennis
Newton's Third Law of Tennis
Momentum
Impulse = Change in Momentum Theorem
Law of Conservation of Racquet and Ball Momentum
Conclusion
11 Energy and Coefficient of Restitution
Crawford Lindsey
Introduction
Kinetic Energy
Law of Conservation of Racquet and Ball Energy
Difference between Racquet Energy and Momentum
Energy Return and Coefficient of Restitution (COR)
Final Ball and Racquet Velocities -- The Hard Way
How To Use the Speed Equations
Apparent Coefficient of Restitution (ACOR)
Final Ball Speed Formulas -- The Easy Way
Conclusion
12 Collisions
Rod Cross
Effect of Racquet Mass on Serve Speed
Elastic Energy Stored During a Collision
Calculation of the COR
Rotation
Relation Between Angular Speed and Actual Speed
Moment of Inertia
Angular Momentum
Collisions Involving Rotation of the Racquet
Effective Racquet Mass
Estimating the ACOR from the Effective Racquet Mass
Axis of Rotation of a Racquet
Notes
Appendix
Formula for Effective Racquet Mass for Impact Along Center Line
Formula for Effective Racquet Mass for Impact Anywhere
13 Tennis Springs
Rod Cross
Stiff Springs and Soft Springs
Springs in Series or Parallel
Potential Energy
Potential Energy Stored in a Mass
Potential Energy Stored in a Spring
The Tennis Ball As a Spring
Strings
How the Elastic Energy Is Shared
Effect of String Tension on Ball Speed
Sideways Stiffness of a Single Stretched String
Impact of a Ball on a Stretched String
14 Racquet Power
Rod Cross
Summary
Introduction
Factors Affecting Power
Physics of Power
Maximum Possible Racquet Power
Estimating Racquet Power
Moving Racquet vs Free Racquet at Rest
Effects of Frame Stiffness on Racquet Power
15 Determining On Court Ball Speed
Howard Brody
Measuring Serve Speeds
Measuring Groundstroke Speeds
16 Customizing Racquets
Rod Cross
Introduction
Balance Point
Swingweight
Racquet Power
Rotation About the Long Axis
Shifting the Sweetspots
Summary
PART 2: STROKES
17 Racquet Technology and Strokes
Howard Brody
Traditional Racquets
Modern Racquets
Classic Strokes
New Strokes
Sweetspots
Stability
Power
Court Surfaces and Strokes
'Bring Back the Wood Racquet'
18 Impact Location
Howard Brody
Where on the Racquet Head Should You Hit the Ball?
Transverse Direction
Conclusion
19 Improving Your Serve
Howard Brody
Introduction
Vertical Angular Acceptance (Window)
Ball Speed
Impact Height
Topspin
Higher Ball Toss
Location on Court
Racquet Parameters
Summary
20 Serve Speed
Rod Cross & Howard Brody
Serve Speed vs Racquet Speed
Serve Speed Formulas
Impact Duration
Arm Rotation
Serve Speed Strategy
Case Study 1: Second Serve Speed
Case Study 2: Serve Speed
Summary
21 Receiving Serve
Howard Brody
22 Strokes and Spin
Howard Brody
Topspin vs Backspin
Watch That Spin
Slicing the Slice
The Approach Shot -- Why Keep It Low and Deep?
23 The Drop Shot
Howard Brody
Hitting a Drop Shot
Returning a Drop Shot
24 Time and Strokes
Howard Brody
25 Error Reduction
Howard Brody
Lateral Errors
Errors of Depth
Equipment and Error Reduction
PART 3: STRINGS
26 Overview of Strings
Howard Brody
Introduction
Air Resistance
The String Plane Deformation
The Ping of the String
Dwell Time
Power
The Feel of the Strings
Elasticity
Gauge
Wear and Breakage
String Patterns
Control
Rebound Angle
Grommets
27 String Chemistry
Crawford Lindsey
Polymers
Bonding
Bonding on a Chain
Bonding Between Polymer Chains
String Materials
Copolymers
Polymers Are Silly
Conclusion
28 String Materials
Crawford Lindsey
What Materials?
What Is Strength?
What Is Stiffness?
Tension and Stretch
Strength and Stiffness by Weight
Conclusion
29 String Construction
Crawford Lindsey
Core Principles
Natural Gut Construction and Play
Wraps: To Serve and Protect
Twists, Angles and Fiber Counts
Resin: The Glue that Bonds
Coatings: Slip Sliding Away
How Are Strings Different?
Why the Cost Differences?
30 String Properties
Rod Cross & Crawford Lindsey
Dynamic Stiffness -- It Determines Almost Everything
Impact Force and Duration -- It's What You Feel
Calculations
Do Strings Ever Lose Power or Resilience?
Player Perceptions of Power
31 Power and Strings Part 1
Crawford Lindsey
String Tension and Power
String Length and Power
String Gauge and Power
32 Power and Strings Part 2
Rod Cross
Serve Speed
Typical ACOR Values
Increasing Power by Increasing Bounce
Increasing Bounce Speed by Lowering Tension
Stored Energy vs Lost Energy in Strings and Balls
What if the Ball and Strings Are Equally Stiff?
Effect of Lowering Tension on Stored Energy and Power
How Does This Affect the Serve Speed?
Energy Calculations
33 String Tension
Crawford Lindsey
What Is Tension?
String Tension Is a Moving Target
Where Does It Go and by How Much?
Why Does It Go?
Tension Loss While Stringing
Tension Loss and Pre-stretching
Time: Every Second Counts
Consistency and Attention to Detail
Pull Angle and Tension
Tension Head Location and Final Tension
Temperature and Tension
Conclusion
34 Measuring String Tension
Rod Cross
Racquet Diagnostic Machines
The Stringmeter
The ERT700
Artificial Tension Calibration
Four Reasons for Tension Loss During Stringing
String Tension: Can Players Tell the Difference?
Determining Actual Stringbed Tension: Method 1
Determining Actual Stringbed Tension: Method 2
Determining Tension in Each String
Comparing Pull Tension, ERT, Pacific RA, and Calculations
Comparing the Stringmeter with Calculations
Where To Go From Here?
The Stringmeter Revisited
Conclusion
35 String Gauges
Rod Cross
Summary
Introduction
Controlled Experiment
Conclusions
PART4: BALLS AND COURTS
36 The Tennis Ball
Howard Brody
Pressurized and Pressureless Balls
High Altitude Balls
Big Balls
Air Resistance and the Big Ball
37 Ball Testing for Bounce
Rod Cross
Compression Test
Slow vs Fast Compression
Bounce of Spinning Ball
38 The Bounce of the Ball
Howard Brody
Vertical Velocity
Horizontal Velocity
Fast Courts -- Slow Courts
39 Ball Spin and Bounce Off the Court
Rod Cross
Summary
Introduction
Ball Spin
Friction Force
Rolling
Spaghetti Strings
Ball Speed and COR
Horizontal Speed of a Court
Measurements of Court Speed
Measurements of Court Speed on Different Surfaces
Measurements of Ball Spin
Measurements of Vertical Speed (COR)
Measurements of Court Speed on Actual Courts
Measurements of the Coefficient of Friction
40 Ball Bite and Bounce
Rod Cross
Introduction
Apparatus
Friction Force Results
Explanation of Friction Force Results
Slamming on the Brakes
Perception of Court Speed
41 Ball Spin and Bounce Off A Racquet
Rod Cross
Introduction
Bounce Basics
Topspin or Backspin Into the Strings?
Ball Speed, Spin and Angle Measurements
How To Calculate the Bounce Off a Moving Racquet
Example of Bounce Calculation
Serving with Topspin
42 Ball Trajectories
Rod Cross
Introduction
Examples of Ball Trajectories
Drag Force
Magnus Force
Serve Trajectories
Appendix 1: Trajectory Equations
Appendix 2: Simple Ball Speed Measurement
APPENDICES
Appendix A: Glossary
Appendix B: Units and Conversions
Appendix C: Equipment Rules
Appendix D: The Mathematics of Ball Bounce
Brody's Bounce Model
Case Where Ball Slides Throughout the Bounce
Case Where Ball Rolls
Appendix E: Tennis Court Dimensions
Appendix F: References
Aerodynamics
Ball Bounce
Biomechanics
General Interest
Measurements
Racquets
Related Sports
Statistics
Strings
Strokes
Sweetspots
Tennis Courts