V5.0 July 22, 2002 Dynamics of Hand-Held Impact Weapons Sive De Motu George L Turner Association of Renaissance Martial Arts http://www.thehaca.com 1 FOREWORD 5 MOMENT OF INERTIA 7 Derivation of Mass Moment of Inertia 8 The Parallel Axis Theorem 10 The Effects of the Moment of Inertia 11 Extra, For People with Too Much Time on their Hands 15 Building a Feel Simulator 16 Minimum and Maximum Obtainable Moment of Inertia 18 Building Practical Simulators 19 SIMPLE MOTIONS 20 Acceleration of the Center of Mass 20 Applied Couples 22 Simple Maneuver Equation 24 SIMPLE IMPACTS 26 Reaction Forces at a Fixed Pivot Point 26 The Definition of Percussion 28 More on Pivot Forces 30 Hand Reaction Forces 32 COMPLEX MOTIONS 34 Centrifugal Forces During a Swing 34 Conservation of Angular Momentum 38 Energy Transfer 39 The Instantaneous Center of Rotation 39 My Earlier Thoughts on Hand Shock 41 APPLICATIONS OF PERCUSSION POINT THEORY 43 Visualizing the Feel 47 Another Insight Into Feel 48 2 Mass Impact Weapons 50 The Axe’s Percussion Point 50 Percussion Points of Weapons with Tapered Mass 53 THE POMMEL 60 The Pommel’s Effects on Moment of Inertia 61 Selecting a COM Position 64 Comparing a Pommel to a Tang Extension 66 A Pendulum’s Percussion Point 68 Adding a Pommel to a Pendulum 71 CALCULATING REQUIRED POMMEL MASS 75 The Effect of Blade Mass Taper on Pommel Mass 78 Why Balance Points Vary On Authentic Swords 82 OBJECTS IN ROTATION AND TRANSLATION 84 Physical Properties of an Object in Rotation and Translation 84 Inertia 84 Derivation of Inertia Along a Blade 85 The Pommel and Apparent Inertia 90 Effect of Blade Taper on Apparent Inertia 94 Simple Calculation of a Sword’s Tip Inertia 96 Calculation of the Inertia at Various Points on a Staff 99 Sword Edge Velocity and Apparent Momentum 101 Sword Edge Kinetic Energy 103 HANDS-FREE IMPACTS 106 Post-Impact Linear and Angular Velocities of the Sword 107 Linear Impulse 108 Graphical Analysis of Impacts 110 Applying the Collision Results 112 Variations in Closing Velocity 113 Mass of the Target 117 Elasticity of the Target 119 3 IMPACTS WITH HANDS 122 Building an Impact Model 124 Impacts Past the Hand’s Percussion Point 125 THE MODES AND NODES OF VIBRATION 130 Energy in the standing waves 131 Energy in higher harmonics 132 Velocity of the side-to-side waves 133 The Edge-to-Edge Oscillations 135 POLE-ARM DESIGN THEORY 138 Impact Pole-arms 138 Quick and dirty head placement 141 Flipping the pole-arm around 142 Which hand determines the percussion point? 143 Iron Inlay Shaft Protection 143 THE CROSS-GUARD? 145 MEASURING THE PARAMETERS 148 The gravity pendulum method 148 The Torsion Pendulum Method 149 Impact Methods 149 FURTHER RESEARCH 151 4 Foreword Hand weapons are no longer well understood from a theoretical standpoint. Actually, it seems they are no longer understood at all. For all the talk about Rockwell hardness and charts of percussion point locations, one might think we know more about swords now than ever. This is blatantly not true. Despite hundreds of years of fencing, and scores of people studing the martial arts of the East, we almost certainly know less than the average knight did, and probably the average Greek hoplite. After the Atlanta 2001 conference, some talks with John Clements nagged at me, and I realized that there are some basic assumptions about the weapon, which have been passed down to us from prevoius writers, that just aren’t matching up with what’s now coming to light in his research. I searched the web for even basic information on the physics of swords, but from what few sources there are, I couldn’t even confirm that swords are swung in rotation. I thought that there would be a great deal of information from either the fencing, Eastern martial arts, or sword making communities, but came away empty handed. Well, actually I did find information, but most of it was either very simple or just plain wrong. Even the simple certainties were wrong, such as the sword’s center of percussion always being about a third back from the tip, which has turned out to be completely false. So, in frustration, I’ve had to figure out some weapon theory myself, even though I know but very little about swords or physics. So I’m no expert at this stuff, but from the looks of things no one else has seriously bothered with it either, at least since the early 1700’s. Interestingly, the only post 1600’s tool that I needed was just the concept of conservation of energy. Any competent 17th century researcher could’ve done everything else in this work, and apparently did so, if the Philosphical Transactions of the Royal Society, circa 1660-1670, are any indication. So I’ve been bouncing ideas off John Clements, Matt Hauser, Richard Boswell, and G. Wade Johnson, all with ARMA, and here’s the result. Few of the results were anticipated, and the research has been a constant eye opener. I’m very irritated that someone hadn’t done this kind of work some time during the past two hundred years. As it requires nothing but the simplest Newtonian mechanics, or even pre-Newtonian mechanics, I can think of no excuse for neglecting it, other than most people being content with pat answers to complex questions. It is becoming obvious that if a modern reenactor were transported back to the 1600’s, both his sword and his explainations of it would meet with derisive laughter. But if you think the following are true, then this essay will present a startling different view of the sword. • Weight and balance point are the primary determinants of a sword’s feel. • The pommel is primarily used to balance the sword, setting the location for the center of mass. • The percussion point is well determined by striking the side of the blade, and is always about 1/3 of the blade length back from the tip. • Strikes with the percussion point leave the least vibration in the blade. • Strikes with the percussion point do the maximum damage, and leave no energy in the blade. • When struck, a sword will rotate around its center of mass. • The cross guard is primarily to protect your fingers. • Hand shock drains energy from the blow. • Heavier swords stike harder than light swords. This research makes use of simple high school algebra, but many of the most startling results require no math at all. After all, back then most swordsmen weren’t mathematicians, either. If you hate math please feel free to skip the math sections and look for the red underlined passages that highlight the findings. All the physics will be in the metric system just to keep the 5 units simple. I will, however, also give a measure called “the moment of inertia” in English units, as it’s as fundamental to a sword as mass is, and can be used to communicate information about a sword’s feel over the web. In English units, the moment of inertia also matches up pretty well with the sword’s weight, which makes it easy to intuitively get an idea of how a sword will behave from just a verbal description. At several points in this article, I will bring up some commonly held but invalid beliefs that are circulating in some parts of the sword community. This is done to examine preconceived, but invalid, notions in detail. Unless they are closely scrutinized they will continue to circulate in the back of your mind, popping up to provide a misunderstanding of what you are both seeing and feeling in your practice. I will also bring up some other intuitive ideas that don’t necessarily match up with reality, but might bring some insight into what ancient smiths and swordsmen may have thought was happening in a blow. Then again, they may have truly known what happens, since they had so much empircal information to work with. Notation used in this article Symbol Meaning Units m mass kg COM center of mass I moment of inertia kg-m2 2 ICOM moment of inertia calculated kg-m around the center of mass 2 IX moment of inertia calculated kg-m around the point at x H angular momentum kg-m2/sec F force newtons (newton = 1kg-m/sec2) Γ gamma, torque newton meters (N-m) a acceleration m/sec2 v velocity m/sec s distance m α alpha, angular acceleration radians/sec2 ω omega, angular velocity radians/sec θ theta, angle radians 6 Chapter 1 Moment of Inertia A sword has many physical properties that can’t be fudged, such as mass. If your replica weighs significantly more or less than the original, it will not handle the same. A heavier sword is harder to accelerate and decelerate in thrusting. Another physical property is the location of the center of mass (COM). When multiplied by the sword’s mass this gives the moment of force required from your hand to counter accelerations perpendicular to the blade, including gravity. If you have a pair of 3-pound swords where one has twice the distance to the balance point as the other, it will take twice as much torque to keep it pointed straight ahead when you shift it side to side. It will also take twice the torque from your forearms to hold it level against the force of gravity. A third parameter is called the moment of inertia (MOI), a more complicated, yet still fundamental measure.