by Craig Simon © 1982
Plastic Aerodynamic Gyroscopics
Spin is the essence of the disc's life, and life, we recall, is the sum of many forms. Paraphrasing Laban, it is impossible not to recognize life in human movement. Freestylers also recognize the existence of life in the disc. It may not be animate or conscious, but it does move--this is undeniable.
Direction and magnitude of the disc's spin is tremendously important for any frisbie activity. In order to throw, manipulate or catch the disc, the player will either accelerate, maintain, or stop the spin. In freestyle, spin is the medium through which the disc and the player interact.
Z's (pronounced zeez; the apostrophe is simply a spelling convention) is a term freestylers often use synonymously with spin. For good reason. The sound of the word suggests the speediness of "zoom" and the multiplicity of any plural.
Right Z's spin clockwise around the axis that intersects the disc's center and the downward pull of gravity. If the disc is directly vertical to the ground, the direction of spin depends on the player's point of reference. The reference frame is the player and the ground rather than the top or bottom of the disc.
Left Z's spin counterclockwise.
Benign refers to situations in which the disc has life because it is balanced, but not spinning. The disc is supported directly underneath its center of gravity by a part of the body, usually the palm or the fingers, balanced either vertically or horizontally. Simply stated, benign is the absence of spin. (Using the thumb to keep the disc propped up is cheating.)
Butterfly refers to the spin of the disc about an axis from rim to rim through the center rather than around the center. Crock and clounter are the terms I have created to describe the disc's spin under these circumstances. Based on the coordinates of the spin axis with the player and the ground as the frame of reference, crock is equivalent to right Z's and clounter is equivalent to left Z's.
Arbitrary, which originally meant batting the disc back and forth between the forearms, now has a much wider meaning. Both benign and butterfly are common to arbitrary, a style that is as strange as it is popular.
Flip Flopping (also called scooping) refers to moving the nonspinning disc through the air, using air pressure to keep it from falling. In this case, the top or bottom of the disc presses against the open palm as the player waves his or her hand forward and around, often under the legs and behind the body.
The disc's composition is of high-density propylene and polyethylene thermoplastics. These two kinds of plastics tend to be softer and lighter than most plastics, which makes them much easier to handle. Thermoplastics are noted for their ability to be remolded once first molded. High densities are preferred because they are more rigid and resistant to abrasion, but there is a trade-off here because they are more likely to crack under high pressure and in cold weather.
Manufacturers often add pigment to the plastic for aesthetic reasons, but this too has its drawbacks. Pigment creates inconsistencies in the bonding of the plastic as it cools after molding and can produce flaws severe enough that the structure will eventually break under pressure. Very dark pigments are poor choices for freestyle discs because they cause the disc to soften considerably in the sunlight and rip apart.
Discs are manufactured in a process called injection molding in which the molten plastic is shot into a cavity between two pieces of metal. The sprue is the most visible remnant of the disc's birthing, located in the center of the disc, either on top or bottom, marking the spot through which the hot plastic flowed.
The shape of the disc is circular and generally flat with a short projection extended perpendicularly around the circumference. These two major sections are called the flight plate and the rim. The side of the plate from which the rim extends is called the bottom and the opposite side is called the top. The bottommost point along the rim is the edge, its outer surface is the lip, its inner surface is the cheek and the junction of the rim and the plate is the ditch (also called the recess). Most players refer to all four locations without discrimination as the rim, but these distinctions are significant. Depending upon how the disc is built, the inner portion of the rim may toe in or dish out, thus creating an acute or obtuse angle between the cheek and the flight plate. For freestyle, a rim that is toed in has advantages over one that is dished out, but the closer it is to a right angle, the better.
Slope refers to the profile of the disc from the edge to the flat part of the flight plate. The slope can he tapered or blunt, giving the disc a more or less streamlined appearance. Fred Morrison's early designs had a tapered slope and flew quite poorly. (Morrison is considered to be the inventor of the plastic flying disc. More on him later.) Through trial and error he found that somewhat blunter slopes resulted in stabler flights. Understanding why this is so leads us to a discussion of the various forces that affect a flying disc. Before moving on to this, though, it is useful to name the most important parts of the disc in flight.
Nose and tail refer, in turn, to the disc's leading and trailing edges. The disc also has two shoulders (sometimes called ears): the skip shoulder and the roll shoulder. The roll shoulder is the side that is touching the ground when the disc is rolling forward and the skip shoulder is the side directly opposite. The plastic on the roll shoulder is always moving back toward the tail while the plastic located at the skip shoulder is always rotating forward toward the nose.
Frisbie Physics
For many years disc players have used the terms "overstable" and "understable" to describe a disc that wouldn't fly in a straight line. An understable disc is one that drops its roll shoulder during flight, banks to the side (toward the right for clockwise and toward the left for counterclockwise spin), and eventually hits the ground and rolls. An overstable disc tends to drop its skip shoulder, bank to the side (left for clockwise, etc.) and dives toward the ground. Experience, our best teacher, shows that discs with tapered slopes are understable and discs with large rims and blunt slopes are overstable.
Unless the slope has the correct shape, as the disc moves forward through the air, it will tend to nose up or down. Since the disc is spinning, this tilt shifts to the side and the disc will start to bank. To understand this, we must journey through the domain of pilots and astronauts, physicists and mathematicians. It is there that we will find the words to explain what happens when a disc is alive.
Flight. What a beautiful thing flight is to see. The way a glider hangs as if held up by wires suspended from a heavenly ceiling. The way a bird looks so relaxed, so in control, like gravity turns off when its feet leave the ground. It must be sacrilegious, at least irreverent, to take flight from its blue setting and put it in a musty laboratory or a thick textbook, breaking it into little pieces for study, like plucking the legs off an insect. Better, perhaps, in order to know what it is, but not as pretty when disassembled. Don't despair. Whole again, it is the same, but you can see more of it, and the beauty becomes even greater than it was at first glance.
As we sort through the particulars, keep in mind that we are looking for an explanation of how the disc's shape affects its flight; in other words, how form is connected to movement.
Relative Airstream & Laminar Flow. When the disc is flying, it is moving through what is called the relative airstream, so named because, in flight, the disc's movement becomes relative to both the ground and the air. If there is a breeze, the groundspeed will probably not equal the airspeed. When the disc, or any wing, moves through the relative airstream, the air becomes disturbed and breaks into sheets that allow the disc to pass through them. When the sheets are smooth, the condition is called laminar flow.
Drag. When the laminar flow "separates," or comes apart, as often happens, it means air that had been moving in smooth layers has become turbulent. This turbulence has a force, called drag, which works directly opposite the disc's velocity through the air, slowing it down and reducing its capacity to generate lift. There are three types of drag to consider:
Form Drag, or pressure drag, is the most significant kind of drag acting upon a flying disc and is produced at the tail. Birds, airplanes and fish all have tapered tails to cut form drag as much as possible. Tapering a disc's slope would certainly reduce form drag, but as we will soon see, it would severely impair stability.
Induced Drag is the price paid for lift. This is very
high on a flying disc. The reason why is that the disc is round, giving
it a very low aspect ratio, which is the relation of a wing's span
to its area. The bottom line here is that wings don't generate lift at
their tips. High altitude aircraft and gliders have aspect ratios as high
as 15 to 1 because their wings are so long and narrow. Faster, highly maneuverable
planes have aspect ratios as low as 4 to 1. A disc's aspect ratio is 1.27
to 1. The aspect ratio is only one of the variables of induced drag, which
must increase as the amount of lift increases. For example, a wing will
stall when it has too much lift. Induced drag becomes so high that velocity
soon drops to nothing.
Profile Drag is produced by the friction of the air against the disc's leading surfaces including the rim, the slope, and any discontinuities on top of the flight plate. For a flying disc, this is not as significant as form drag and induced drag.
Lift is a force generated at right angles to the airstream, as opposed to drag, which is in line with the airstream. Both forces are a function of the disc's airspeed, diameter, profile, angle of attack, and the density of the air, but lifting forces are much more significant in the context of disc stability than drag forces. We'll see why in a moment.
Airspeed, or the relative velocity of the disc to the air, is potentially the most important factor, since the lift (as well as the drag) generated by the disc is proportional to the airspeed squared (doubling the airspeed quadruples the lift, and so on). The longer the span of a wing--or the diameter of the disc--the more air will flow around it. Observation of different sized objects in the wind easily reveals that total force will be greater on larger surfaces. By the same reasoning, the denser the air, the more force will be generated.
Angle of Attack serves to deflect the air at right angles to the relative airstream. Increasing the angle of attack increases lift, but it also increases induced drag, leading to stalls. Reducing the angle of attack reduces lift, but the angle of attack could be zero, or even slightly negative, and the disc would still have some lift because of its profile. Before we go on to profile, which is also the key to the problem of disc stability, we need to look a little more closely at angle of attack and how it's defined.
Mung is a term coined by Stancil Johnson to describe the disc's attitude along its lateral axis (the line between the shoulders) in relation to the ground. Positive Mung means that the nose is higher than the tail. Neutral Mung means that the nose and tail are horizontal, and so on. Johnson loosely described Mung as angle of attack, but a pilot would be more precise and call this the "pitch angle." For a pilot, the angle of attack is defined in relation to the relative airstream and it is measured from the zero lift line. This is a line from the bottom edge of the tail through some point on the slope of the nose. If the disc were flying along this line, the upward lift generated by the profile would equal the downward pressure created by deflecting air off the top of the flight plate. Therefore, a disc can have positive Mung and a negative angle of attack at the same time. The point to all this is to show that lift can only be generated by the flow of air. It's possible to throw disc into the sky so that it ascends, not because of lift, but because of sheer force.
Profile. Like a wing, a flying disc is cambered, or curved, to produce lift. Early aircraft, especially biplanes and triplanes, were designed on the assumption that lift was generated only by deflecting air downward off the bottom of the wing. The Wright brothers, for example, experimented with various cambered shapes to maximize this effect. Now it is understood that as much as two thirds of the lifting force on an airfoil comes from an upward pull (you can think of it as a suction) on top of the wing.
Bernoulli's Theorem explains this force. It states that fluid pressure (air in a stream is essentially a fluid) is inversely proportional to its velocity squared. To restate it more simply: fluid pressure decreases as its velocity increases. While the disc moves forward, the air slips around it in predictable ways. As the air hits the slope and moves over the top of the flight plate it has to speed up. Given a neutral angle of attack, the air pressure underneath the disc stays about the same, but the air above now exerts less pressure. This difference in pressures causes lift.
Now we have a clue as to why flying discs with blunt or tapered slopes behave differently. It is the curvature of the slope that causes air to speed up and its pressure to drop, resulting in lift. Either type of slope will create lift, but a tapered slope produces a more gradual change in pressure over the flight plate than a blunt slope will.
The aerodynamic center is the point on an airfoil at which the lifting force is balanced between the leading and trailing edges. On an airplane or a glider this is typically located about a quarter of the way past the front. The leading edge is much blunter than the trailing edge and the air's velocity is more dramatically affected there. Since a disc is symmetrical, the aerodynamic center will be very close to midpoint, but this will vary depending on whether the slope is relatively blunt or tapered. The blunter the slope, the farther forward the aerodynamic center will be.
Pitching Moment is the tendency of a wing to tilt forward or back on its lateral axis. This is directly related to the location of the aerodynamic center. A positive pitching moment (the aerodynamic center is somewhere along the leading half of the wing) means that the nose will come up and a negative pitching moment means that the tail will come up. Birds and planes have tails to counteract this tendency: flying discs, of course, do not. The problem is even more complicated because the disc is spinning.
Overstability and Understability in a disc's flight corresponds
to the excessive bluntness or taperedness of a disc's slope and, therefore,
to its manifestation of a positive or negative pitching moment. At first
glance, knowing that the aerodynamic center on a disc is imbalanced, one
might expect the disc to either pitch straight up until it stalled or nose
right into the ground. Instead, the poorly designed disc will exhibit a
strong tendency to fade to one side or another. (We think of this as getting
a visit from Danny Divebomber or Teddy Turnover.) The thrower can counteract
this by making the disc spin more quickly or simply by tilting the disc
during the throw to compensate for the fade. The forces at play here were
explained generations ago; understanding them makes disc flight much more
predictable for the player and a lot more fun too.
Gyroscopics
Precession is probably the best-known mechanical rule among people who call themselves freestylers, even among those who disdain the technical side of the sport. Although many definitions have been offered, Dan Roddick's was probably the briefest: "the 90 degree advancement of a net effect of a force on the gyroscopic plane." Steve Longley wrote, "The effect which the applied force would have produced on a nonspinning disc will be carried by the spin to a point a quarter of a revolution (90 degrees) farther along."
Both these definitions are sufficient to explain why a disc is overstable or understable. A disc with a positive pitching moment (caused by a blunt slope) experiences an upward force on its nose. Precession shifts this tilt to the roll shoulder and the disc will very likely fly into the ground on its skip shoulder. Conversely, a negative pitching moment places an upward force on the discs tail. The tilt manifests itself at the skip shoulder and the disc will tend to fly into the ground and begin to roll.
Precession also explains why the skip shoulder is appropriately named. If the disc is flying forward with the skip shoulder tilted down (but not vertical), when it finally hits a surface beneath it, it will bounce back up, creating an upward force at that point. Due to precession, the nose will come up, changing the disc's angle of attack. Given sufficient airspeed forward, the disc will lift up, giving the appearance of a rock skimming across water.
Perpendicularity. It's a simpler matter to know what precession does than to understand why it does it. It seems logical that any downward or upward force on the rim of a spinning object will be carried forward, but why does the maximum effect always show itself a quarter turn ahead? Does this tell us anything else that's important to know about freestyle? For the answers, we must refer to one of the most important mathematicians of our civilization.
Isaac Newton has left us with so much useful information about the mechanics of spinning bodies that you'd think he was hit by a disc instead of an apple. Newton was concerned with explaining "Precession of the Equinoxes," or why the Earth's axis wobbles as the planet travels through space. If you're familiar with astronomy, you're probably aware that the Earth's axis is tilted in relation to its orbit around the Sun. This tilt results in an uneven heating of our planet's surface, creating seasonal weather changes. Right now, the axis points North toward Polaris, but because of precession, it is shifting away. 13,000 years from now the axis will point to a star called Vega, but 26,000 years from now it will point back toward Polaris again. Newton lived in an age when explaining the circumstances of such facts constituted the most difficult puzzles of science.
Without the benefit of computers or refracting telescopes, the astronomers of the 17th Century painstakingly calculated the movements of the heavenly bodies. When they discovered the steady shift in the stellar longitude caused by precession, they were mystified. But even as they plotted the journeys of the planets around the Sun, and the moons around the planets, they had no good way to explain what kept everything in place. They didn't even know what kept the Earth spinning. The problem was, before Newton came along, most people thought that, given enough time, everything in the universe would someday just slow down and stop moving.
Inertia. By developing the concepts of force and gravity in his three Laws of Motion, Newton completely changed the way people thought about the world (as well as explaining how frisbies work). The first law concerns the inertia, or laziness, of bodies: an object at rest will remain at rest and an object in motion will sustain that motion unless other forces come into play. Therefore, if a disc is moving, it will keep on moving until something, say, someone's hand or particles of air, gets in its way. This is easy enough for us to accept in the Space Age, but in those days it was as revolutionary as Rock and Roll.
Force. The second law of motion provided a way to measure force, with the formula "Force equals Mass times Acceleration (F=Ma)." Since "Acceleration times Time equals Velocity" (V=AT), this means that, given two objects of different masses, it will take more force or time to accelerate the larger mass to the same speed as the smaller one (V=FT/M). For a disc, or any spinning object, the second law is a little more complex, because its spinning inertia is a function, not only of its mass, but also the distribution of that mass. For example, if two discs have the same rate of rotation and the same mass, but one has more of its mass concentrated toward the center, the other one has more energy (E=VM, or what is properly called angular momentum. This is logical because the second disc has more mass moving at greater velocities. Over the past few years, some manufacturers have begun to produce "two-piece" discs with exceptionally lightweight flight plates and heavier rims, in order to maximize the potential angular momentum of the spinning disc.
Angular Momentum is a function of a spinning body's mass times the distribution of that mass (which together equal the moment of inertia) times its acceleration over time (which in this case is called angular velocity). Since a disc is a rigid body, the distribution of its mass does not change; but this does happen to some spinning bodies, like merry-go-rounds and ice skaters. Once the ice skater starts turning, pulling his or her arms in speeds up the rate of turning. If passengers on a free spinning merry-go-round lean to the outside, the spin will slow. These phenomena are explained by a rule called conservation of angular momentum (energy). Given that angular momentum remains constant, any increase or decrease in the moment of inertia (as happens when the distribution of mass changes) must create a corresponding decrease or increase in angular velocity. Keep this in mind if you're practicing moves that involve turning and pivoting.
Gravity. Newton's third law, which says that every action has an equal and opposite reaction, is subtler than the other two. It implies that, as the earth pulls an apple toward the ground, the apple pulls the earth up. Since the two masses are so different in size, when the apple falls, the rise of the earth is imperceptible. The third law helped explain gravity and set the stage for Newton's explanation of precession. He correctly guessed that the Earth tends to flatten out as it spins around its axis, developing an equatorial bulge. Newton showed that both the Sun and the Moon exert force on the equator, pulling on it as the Earth moves about its orbit. If the Earth weren't spinning, the equator would line up with the plane of the Sun and the Moon, and the poles would be perpendicular to it, but this is not the case. Instead, the Earth has to compensate for this gravitational assault by precessing. The angle of its tilt doesn't change, but the direction does. The result is that the Earth's axis effectively marks out a cone every 26,000 years. Likewise, given the proper conditions, a disc's spin axis will describe a cone.
Remember that we're trying to find out why precession always works at 90 degrees. We'll need a few more definitions to continue, but first, consider this:
To those who study the progress of exact science, the common spinning-top is a symbol of the labours and the perplexities of men who had successfully threaded the mazes of planetary motions. The mathematicians of the last age, searching through nature for problems worthy of their analysis, found in this toy of their youth, ample occupation for their highest mathematical powers. (From James Clerk Maxwell, On a Dynamical Top)
Vector is a term used in physics to represent a force that has a known direction and magnitude. (It is also the trademark for AMF Voit's flying disc.) Vectors are expressed geometrically as arrows on a graph. The pictures at the head of this chapter used vectors to portray the disc's spin and velocity. A flying disc's velocity vector points exactly toward its line of flight. In a moment we'll discuss how to depict spin vectors.
Vectors in the same plane can be added together in order to show mathematically what happens when distinct forces either collide together or pull on the same point. If two coinciding vectors are not aligned, that is, if they are not headed straight into each other or straight apart, their sum can he calculated geometrically by laying them head to tail and measuring the diagonal from the tail of the first to the head of the second. This is the same thing as graphing a parallelogram with the known vectors as two of the sides. The diagonal connecting the two is the vector sum.
Torque is a force that tends to rotate something. To give a disc Z's, you must apply torque. This concept is closely related to leverage. The longer your lever, the more torque you can apply. (By the way, another phrase that means the same thing as torque is moment of force. When something is turning, it is said to have a moment; recall that pitching moment is the rotation of the disc along the axis between its shoulders.)
Torque Vector is the mathematical representation of the axis
along which the torqued body rotates. The magnitude of this vector depends
on the energy of the initial force as well as its distance from the axis
(this is where leverage comes in). As the pictures on the opposite page
show, this vector points through the center the same direction a right-handed
screw (like a corkscrew) would advance if it matched the rotation produced
by the torque. Consider the example of a disc that is supported at its
center so that it can tilt and rotate freely. If the rim is pressed down
at one point, say, 3 o'clock, the nonspinning disc will tilt along the
axis running through the center from 6 to 12. Using the right hand rule,
the torque vector points from the center through 12. If the rim is pressed
down at 9 (equivalent to up at 3) the torque vector passes through 6. Note
that the torque vector is independent of the spin vector. The determination
of the magnitude of the torque vector is quite complex, as it is dependent
upon the radial distance position of the support (or applied force).
Spin Vector is a convenient mathematical device to represent angular momentum geometrically. Every physical point on the disc has its own vector, theoretically quantifying the direction and magnitude of its inertial force, but naming all the points in the same gyroscopic plane would be very cumbersome. The spin vector, then, is an imaginary line along the axis of the rotating body. Again, this vector points the same direction that a right turning screw would advance if it matched the body's rotation. For a clockwise spinning disc this is down; for counterclockwise it is up. For computational purposes, the length of the spin vector is related to the angular momentum.
Precession (Encore). The original question remains: Why does a force at the disc's rim, perpendicular to the spin plane, travel 90 degrees to show its maximum downward or upward effect? One answer is that the spin vector is perpendicular to the torque vector, and the vector sum that results both shares the same plane as those vectors and defines the new location of the spin vector.
For example, if the disc is spinning clockwise in a horizontal plane, the spin vector points straight down through the center. If a downward force is applied at 3, the torque vector points from 6 to 12. The only point the two vectors have in common is the center. Those two vectors also lie in the same plane. Remember that the diagonal of a parallelogram containing those two lines as its sides defines the vector sum. Since the disc can rotate freely about its center, the spin vector must move to coincide with the vector sum. In effect, the spin vector tilts up toward the torque vector in the 6 to 12 plane. This is called "chasing" the torque vector. Therefore, 6 comes down.
If you were to pursue the subject, you'd find out that the previous example is an oversimplification of the mechanics involved, although the result is correct. To be exactly precise would call for a discussion of "Dot Products" and "Cross Products" (different ways of multiplying vectors) and "Orthogonal" mathematics (rules for right angles and perpendiculars). But this is a book about frisbie, after all, not calculus. There won't be any homework assignments; however, I do want to suggest a project.
You can do an interesting experiment by taking a disc and putting a small hole through the center so that you can support it on some sort of a device, like a needle or the tip of a pen, so that the disc can rotate freely. A Pyradisc, which is a sportdisc with a hollow pyramid at the center, works well for this. If you try to support the nonspinning disc on the device while it's upside-down (while the center of gravity is above the support point) it will tend to tip over and fall. The other way (with the center of gravity beneath the support point) the disc will tend to balance itself, even if you try to tip it. If you spin the disc, you'll find that the upside-down disc precesses in the same direction as the spin while the upside-up disc precesses opposite the spin. Confusing? Remember that the spin vector always chases the torque vector. When the center of gravity is above the support point, gravity creates a torque that pushes the lowest side down, but when the disc's center of gravity is below the support point, the effect is reversed. It turns out that freestylers rarely control the disc exactly at the center, so during play, it behaves like a top--like the upside-down disc of the example--whether it is upside-down or not, always precessing the same direction as the spin.
There are two more things about precession worth mentioning. Each illustrates a point that most freestylers understand intuitively.
More Z's! The first is that the rate of precession of a free-spinning body is inversely proportional to angular momentum. In other words, the greater the angular momentum, the smaller the rate of precession. Since the Earth has such a large moment of inertia, its angular momentum is large and its rate of precession is small. By the same token, the faster a disc spins, or the more mass it has distributed away from the center, the more slowly it will tend to wobble.
Go with the flow! The second is that the rate of precession changes if the support point moves in space. As just shown, a disc usually behaves like a top, precessing the same direction as the spin, because gravity's torque constantly pushes the lowest side down. The principle of a "nail delay," which l will soon describe in detail, is to "head off" gravity, so to speak. This is similar to the principle involved in keeping a long pole balanced vertically. If the pole tilts, you must move your end of the pole right under the top of it, or it will fall over. (As we will see, precession makes this a little trickier on a spinning disc; you don't want to move directly under the falling side to restore its balance.) The point here is to show that moving in the opposite direction will only make the problem worse, permitting gravity more leverage, therefore increasing the rate of precession and making it more difficult to restore balance. This is especially important for understanding "rim delays" and "against the spin" moves.