From: bradleyt@spot.Colorado.EDU (Todd Bradley)
Subject: On the Dynamics of Rotating Sport Discs
Organization: University of Colorado, Boulder
Date: Sun, 15 Nov 1992 22:45:28 GMT


I promised several people that when I finished this paper,
I would make it publicly available.  I want to thank the
participants of this newsgroup who helped steer me toward
the sources I used in writing the paper.  

As a bit of background, I am receiving an MS in Aerospace 
Engineering next month at the University of Colorado and 
have been specializing in fluid mechanics.  This department 
does not require a thesis for a Masters degree, although you
can write one if you want.  However, it is required that
degree recipients must write a "formal" paper which will
be kept on file and sent to prospective employers who
request information on the student.

I intentionally wrote this paper in simple terms so that
a larger audience (mostly my Ultimate friends) than just
aerodynamicists could understand it.  If you understand
basic college physics, you should have no problem.


Enjoy,
Todd.
.........................................................
 
            On the Dynamics of  
            Rotating Sport Discs
 
 
 
                                        
                Todd Bradley
             November 13, 1992
          Master of Science Paper
 Aerospace Engineering Sciences Department
           College of Engineering
     University of Colorado at Boulder



 [ This plain text version of the paper doesn't contain the
   figures or table found in the original version.  It uses
   the so-called IBM extended ASCII character set, which
   includes some nonstandard characters such as fractions
   and exponents. ]


 Introduction

 Perhaps one of the most interesting fluid flows which occurs in 
 sports and recreation is that of air over rotating discs and disc-
 like objects.  The variety of recreational games that include 
 thrown disc-shaped objects such as the Frisbee, discus, and the 
 increasingly popular Aerobie is perhaps the greatest testament 
 to human fascination with this type of fluid flow.  In fact, more 
 Frisbees are bought each year than footballs, basketballs, and 
 baseballs combined [1].
 
 The flow past a rotating disc is deceptively complex.  Addi-
 tionally, the analysis of the dynamic forces on and stability of 
 rotating discs is somewhat complicated because the problem 
 requires a rotational coordinate system.  This paper summarizes 
 research performed to date into the aerodynamics of flying 
 sport discs and explains, in simple terms, the fundamental 
 forces and moments which cause these discs to fly in a stable 
 manner.


 Background and History

 Since the time of the ancient Greeks, athletes have thrown
 heavy discs, called discuses, through the air in an effort to 
 achieve maximum distance.  This sport has spread through 
 Western civilization and is still a popular Olympic event.  
 Although a discus might appear to be an uninteresting aerody-
 namic model because of its high weight, the flight path of a 
 discus is highly influenced by the fluid flow around it.  It may 
 be surprising at first to note that a discus can fly farther if 
 thrown into the wind than if thrown with the wind.
 
 Recreational discs throwing was limited to discuses until the 
 nineteenth century.  In 1871, William Russel Frisbie started a
 pie baking business near Yale University.  The most frequent 
 patrons of the Frisbie Pie Company were students from the 
 university, who discovered that an upside-down pie tin could be 
 thrown with reasonable distance and accuracy.  Throwing pie 
 tins turned into a popular pastime at the pie company, but it 
 wasn't until after World War II that Frederick Morrison, an avid 
 pie tin thrower, used new plastics technology to develop what 
 became known as the Frisbee [2].  Officially, "Frisbee" is a
 trademark name owned by American toy manufacturer 
 Wham-O, although flying plastic discs of all kinds are now 
 commonly called Frisbees.  During the 1960s and 1970s, other 
 toy manufacturers joined in the flying disc business as the 
 popularity of disc sports greatly increased in the U.S. and 
 Europe. 
 
 Among the organized sports that evolved from casual pie tin 
 throwing are Ultimate Frisbee, a team sport similar to soccer 
 which uses a plastic disc instead of a ball; disc golf, in which 
 plastic discs replace balls and clubs; freestyle, where contes-
 tants try to keep the disc aloft for long durations by hitting and 
 spinning the disc; distance throwing; and several other sports.  
 The variety of games which use flying plastic discs has created 
 a market for a wide variety of types of discs.
 
 Common Frisbee-type discs are between 80 and 200 grams in 
 mass and 20 to 30 centimeters in diameter.  The common 
 design has a very slightly curved top surface which blends into 
 a stiff rim, where the disc is held.  The thickness of the disc 
 varies from about 0.2 to 0.6 cm.
 
 More recently, in the late 1970s, an engineering professor from 
 Stanford University named Alan Adler began a serious inves-
 tigation into the possibilities of flying plastic sport discs.  He 
 was concerned with producing a flying disc which would cor-
 rect some of the problems of the Frisbee-style disc.  The result 
 of his research was a flying ring which was named the Aerobie.  
 This ring, which is slightly larger in diameter than a Frisbee-
 type disc, consists of metal core surrounded by soft rubber 
 which forms an interesting airfoil shape.  The outer edge of the 
 Aerobie is an unusual, somewhat blunt rim shape while the 
 inner edge is tapered.  Because the Aerobie is a ring and not a 
 disc, each edge is a trailing edge on one side of the Aerobie and 
 a leading edge on the other side.                                                            


 Motivation for Study

 The majority of studies into the aerodynamics of rotating discs
 has been motivated by the desire to understand turbine 
 machinery.  Unfortunately, because this research is often con-
 cerned with compressible flows which have significant heat 
 transfer, it is of only minor relevance to the study of rotating 
 sport discs.  A very small body of research into the aerody-
 namics of sport discs does exist, however.  A few studies have 
 been done on the flight path of discuses, in order to understand 
 how to increase range.  Mac Shuurmans conducted some water 
 tunnel tests of the Frisbee shape while studying in Switzerland 
 and has written three papers on the subject of Frisbee flight.  
 And the U.S. Army funded a research project in the early 1970s 
 to study the feasibility of attaching flares to Frisbees and other 
 flying discs as a way of increasing the flares' time aloft.


 Lift and Drag

 Under normal conditions, the flow field around a sport disc can
 be considered incompressible, but unsteady and rotational.  
 Fortunately, because the lift and drag forces of a flying disc are 
 quite similar to the lift and drag on a simple airfoil section 
 passing through incompressible, inviscid air, the unsteady and 
 rotational effects can be ignored and the analysis can parallel 
 that of airfoil sections.
 
 Like a lifting airfoil, the top surface of a Frisbee-style disc is a 
 curved surface, concave downward.  This camber accelerates 
 the flow on the top of the disc to a speed higher than the flow 
 below the disc.  According to Bernoulli's principle, this speed 
 difference causes a pressure difference, which causes lift.  The 
 bottom surface, although also concave down, is slightly more 
 complicated.  Most recreational flying discs have a rim which is 
 used to grip the disc.  This rim causes much of the drag on the 
 disc, because it is generally perpendicular to the direction of 
 flight.  In addition to causing drag due to its frontal area, the 
 rim disrupts the flow under the disc.  
 
 Ignoring this disruption and the effects of rotation, the Frisbee 
 disc can be treated as an airfoil with the lift and drag calculated 
 by the standard expressions
 
 Fdrag = 1/2 Cd r A Vrel^2
 Flift = 1/2 CL r A Vrel^2
 
 where Cd  and CL  are the drag and lift coefficients, r is the air 
 density, A is the disc surface area, and Vrel is the relative wind 
 speed (airspeed) passing over the disc.  The relative airspeed 
 takes wind into account, so that a disc thrown into the wind has 
 a higher relative speed than one thrown with the wind.  Also, 
 the drag and lift coefficients are functions of the angle of attack 
 of the disc.  In contrast to the Frisbee shape, a discus has a 
 symmetrical, wedge-shaped cross section.  Because of this, a 
 discus will not generate lift if thrown at an angle of attack of 
 zero.  Therefore, a discus requires a high angle of attack to 
 produce enough lift to keep its weight aloft.  
 
 Because a higher airspeed results in a higher drag force, one 
 might be tempted to believe that it is harder to throw a disc into 
 the wind.  However, an increase in airspeed also increases the 
 lift on the disc.  In fact, there is a range of wind speeds under 
 which a discus thrown into the wind at the proper angle will 
 actually travel farther than one thrown with the wind or in no 
 wind.  This fact was discovered through practice by expert dis
 cus throwers long ago and now is a factor to be considered in 
 analyzing discus world records.  Presumably, Frisbee and 
 Aerobie distances can be increased by throwing into a low-
 speed wind as well.
 
 Frohlich [5] analyzed two-dimensional discus flight under the
 assumption that the two basic equations mentioned previously 
 represent the only forces on the discus.  Using lift and drag 
 coefficients determined experimentally, he constructed a com
 puter simulation of discus flight.  The parameters of the 
 computer experiment were initial speed and initial angle of 
 attack.  His results indicate numerically that the shortest dis
 tances will occur when the discus is thrown with a 7.5 m/s wind 
 (a tailwind).  With higher tailwinds, the wind blows the discus 
 further.  With no tailwind or a headwind of up to about 20 m/s, 
 the additional lift will increase the total distance.  However, to 
 take advantage of this phenomenon, one must throw the discus 
 at the optimal angle of attack, which depends on the wind 
 speed.


 Gyroscopic and Magnus Forces

 Because Frisbee-type discs are much lighter (with a corre
 spondingly lower rotational moment of inertia) than discuses, 
 two additional dynamic phenomena are important--gyroscopic 
 precession and the Magnus force.  The first of these, gyroscopic 
 precession, is the cause of most of the frustration for inexperi-
 enced Frisbee players.  The leading and trailing edge rims cause 
 the air passing under the disc to slow down more than the air 
 passing over the disc.  Because of this, the difference in the 
 fluid speed above and below the disc is greater at the back half 
 of the disc.  Therefore, the lift is greater over the trailing sec-
 tion.  This greater lift generates a "lift torque," a moment whose 
 direction vector points to the left (from the perspective of the 
 thrower).  However, because the disc is spinning and has rota
 tional momentum, it reacts to the moment in a direction 
 perpendicular to both the rotational momentum vector and the 
 lift torque vector.  The result is that the right side of the disc 
 dips, causing the disc to veer off course to the right or to com
 pletely flip over onto its side and crash.  This frustrating effect 
 can be countered by releasing the disc tilted slightly to the left.
 
 In contrast to a Frisbee-type disc, the Aerobie suffers almost no 
 gyroscopic precession:  the Aerobie's inner edge is shaped dif-
 ferently than its outer edge in such a way as to equalize the 
 uneven lift pattern the Frisbee has.  The combination of this 
 even lift distribution and the Aerobie's negligible rim means 
 that the Aerobie suffers almost no gyroscopic precession.  
 Thus, it is much easier to throw an Aerobie long distances 
 without the disc turning over (see Table 1).
 
 The other force relevant to Frisbees is the Magnus force, which 
 is caused directly by the rim.  As air passes around the curved 
 rim, the air must accelerate.  This acceleration will again 
 decrease the air pressure, resulting in a force perpendicular to 
 the direction of flight and in the plane of the disc.  If a flying 
 disc, travelling in a straight and level path, did not spin, the 
 forces caused on either side of the disc would exactly balance 
 out.  However, because a flying disc spins while in the air, one 
 side of the disc has a higher effective speed through the air.  
 The Magnus force caused by the difference in speeds should 
 cause the disc to drift horizontally.
 
 However, studies of the Magnus force have come to conflicting 
 conclusions.  Some researchers believe it to have an important 
 effect on Frisbee flight, but flow visualization studies (see ref-
 erence [6]) reveal no visible evidence of Magnus effects.  
 Because the Aerobie has a negligible rim width, it has no sig-
 nificant Magnus force.


 Other Airfoil Type Phenomena

 Nakamura and Fukamachi, in their recent Frisbee flow visual
 ization report [6], cite clear evidence of the downwash effect
 caused by the lifting ability of the Frisbee shape.  This down
 wash increases with rotation speed, suggesting that the rotation 
 of the disc aids in lift production.  Flow visualization also 
 clearly reveals the analogue of wingtip vortices.  A pair of 
 longitudinal vortices, trailing from the outer disc edges, appears 
 to enhance the downwash effect.  Nakamura and Fukamachi 
 conclude that this pair of vortices "is responsible for the gen-
 eration and enhancement of lift force in the forward flight of a 
 Frisbee."


 Stability

 One of the reasons that Frisbee-style discs are interesting from
 an aerodynamics standpoint is that they are stable under a very 
 wide range of conditions.  It is easy to discover that very thin, 
 flat discs are inherently unstable (try throwing the lid of a cof-
 fee can more than a few yards).  They have a natural tendency 
 to turn on their sides and dive erratically into the ground.  
 However, the Frisbee, when given a range of initial conditions 
 (rotational and translational speed and angle of attack) the 
 average player can attain, remains generally stable and upright 
 throughout its flight path.
 
 The differential equations governing the dynamic motion of 
 flying rotating discs are nonlinear  and the stability criteria 
 which result from them are also given in complicated, nonlinear 
 equations (see [3] or [4]).  Linearization of these equations 
 simplifies matters greatly and allows the general nature of the 
 stability to be understood.

 One advantage the Frisbee shape has over flat discs is camber.  
 However, the camber creates stability in a roundabout way.  
 The camber of a Frisbee causes a pitch-down moment at low 
 positive angles of attack and a pitch-up moment at high angles 
 of attack.  As might be expected, there is an equilibrium angle 
 at which the pitching moment is zero.  This angle is around 10 
 degrees for the Frisbee [4].
 
 The other important factor in stable flight is angular momen-
 tum.  A nonspinning Frisbee is unstable even at the equilibrium 
 angle of attack.  However, a spinning disc will resist destabiliz
 ing moments in the same manner as a spinning top.  As 
 mentioned previously, the angular momentum of a spinning 
 disc will cause it to react to imposed moments by rotating in a 
 direction which is perpendicular to both the spin axis and the 
 imposed moment.
 
 The result of the pitch-down moment at low angles of attack, 
 then, is a roll to the right.  Likewise, the pitch-up moment at 
 high angles of attack is a roll to the left.  If the disc is thrown 
 with an initial roll angle, called the "hyzer angle" [2], in the direc-
 tion opposite to the known roll tendency, the Frisbee will roll to 
 the horizontal position during flight and keep a roughly flat 
 trajectory throughout its flight path.  Adjusting the hyzer angle 
 to the preferred angle of attack and forward speed appears to be 
 the key to accurate, long Frisbee throws.  The ideal hyzer angle 
 is a function of disc rotation speed, horizontal speed, and air 
 density.  For example, Frisbee players who travel to different 
 altitudes find that they must adjust their throwing technique to 
 take the change in air density into account.


 Conclusion

 The fundamentals of the complex flow of air around a flying
 sport disc can be easily understood in terms of common airfoil 
 analogies and basic physics.  It is the combination of this flow 
 and the special stability characteristics of the Frisbee that 
 makes the spinning flying disc one of the most popular recre
 ational toys in America.  However, the author has yet to find a 
 complete, detailed quantitative study of the aerodynamics of 
 these discs; other than simple stability investigations, little sci-
 entific investigation has been pursued in this field.  Very likely, 
 this lack of research is due to the subject's inapplicability to 
 improving aircraft design and to the dearth of research funds 
 available for the improvement of recreational toys.  Regardless 
 of this lack of broad scientific relevance, though, wind tunnel 
 testing of the Frisbee and Aerobie shapes and calculation of 
 their aerodynamic properties (rotational moment of inertia, 
 coefficient of moment, zero lift angle of attack, coefficients of 
 lift and drag as functions of angle, and so on) would prove 
 intellectually rewarding.


 References
 
 1.   Academic American Encyclopedia. 1991. Grolier Elec-
      tronic Publishing.
 
 2.   Mace Schuurmans. "Flight of the Frisbee". New Scien-
      tist, July 28,1990, pp. 37-40.
 
 3.   Paul Katz. "The Free Flight of a Rotating Disc". Israel 
      Journal of Technology, 1968, Vol. 6 No. 1-2,  pp. 150-155.
 
 4.   G.D. Stilley & D.L. Carstens. "Adaptation of the 
      Frisbee Flight Principle to Delivery of Special Ordi-
      nance". AIAA Paper 72-932, presented to the
      Atmospheric Flight Mechanics Conference, 2nd, 1972,
      Palo Alto, CA.
 
 5.   Cliff Frohlich. "Aerodynamic effects on discus flight". 
      American Journal of Physics, Dec. 1981, 49(12).
 
 6.   Y. Nakamura & N. Fukamachi. "Visualization of the 
      flow past a Frisbee". Fluid Dynamics Research 7, 1991,
      pp. 31-35.
 
 7.   Guiness Book of Records 1992. Donald McFarlan, ed., 
      Facts on File, New York.


 Additional References

 The following sources were not used in this paper.  However,
 they are listed here for completeness in the event the reader is 
 interested in further research.
 
 1.   Mace Schuurmans. "Frisbee--Geschichte und Aerody-
      namik". Schweizer Jugend forscht, 1985, pp 161-167,
      ISBN 3-908504-03-1.
 
 2.   Mace Schuurmans. "Frisbee--Geschichte und Aerody-
      namik". Unpublished 51-page paper, 1985.
 
 3.   Stancil E. D. Johnson. Frisbee: A Practitioner's Manual 
      and Definitive Treatise. Workman Publishing Com
      pany, New York, 1975, ISBN 0-911104-53-4.
 
 4.   Craig Simon. Frisbee: Beyond Catch and Throw. 1982, 
      65-page pamphlet available from Box 1502, Santa Cruz,
      CA 95061.
 


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