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Brandon Brown

Professor Hodge

PHY 492

19 May 2010

A Study of Discus Flight

Background

While the discus is in flight, the only forces it experiences are the aerodynamic forces of

drag and lift and the force due to gravity (Figure 1) where these two forces are given by:

If any wind is present, it affects the relative velocity such that (Figure 2). Using

the force diagram in Figure 1 one can find the equations of motion of the discus while in flight

where is the density of air, M is the mass of the discus, is the drag coefficient, and is the

lift coefficient (Frohlich 1125).

The lift coefficient depends on the angle of attack, which is the angle between the relative

velocity of the discus and the inclination of the discus (Frohlich 1126). One can see in Figure 2

that the angle appearing in the equations of motion is given by . For the purposes of

finding the trajectory of the discus, the most important term is the relative velocity. Notice that it

is affected directly by the acceleration over time. With a changing velocity comes a changing

attack angle, and hence a changing lift and drag coefficient.

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Non-Rotating Coordinate System

Method

For a system in which the orientation of the discus is constant, the jerk along the x and y

directions can help in solving the equations of motion (Frohlich 1126).

Note that from Figure 2, . Finding the derivative of is then trivial. Also,

. The attack angle brings the orientation of the discus into the equation for jerk:

. If the orientation is constant with time, the second term is zero. It turns out that

even in a rotating frame of reference, the second term does not matter. This will be explained in

more detail later. The importance of the jerk along the x and y directions is that it allows one to

take the acceleration of the discus to be constant over a small amount of time (.01 seconds at

most). The jerk then corrects for any error in keeping the acceleration constant such that after

each time interval, the change in position is given by:

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Assuming constant acceleration and making use of equations 4, 5, 6, and 7, the equations of

motion can be solved numerically using the Euler method. A C++ program was written using

this algorithm, and the algorithm itself can be found in Appendix A.

Results

It is quite obvious that there are many variables one can now play around with. One

surprising result of the aerodynamic forces is the relationship between optimum distance and

wind velocity. One can obtain higher distances with a wind velocity opposing the direction of

flow. This is confirmed from Figures 4. Figure 4 constitutes values obtained by Frohlich using

the method described above. It is very clear that up wind going against the direction of motion

helps in achieving greater distances. A program was written to simulate these results (Figure 5).

For no wind and a wind velocity of 10 m/s going with the discus, there is good agreement.

However, for wind going against the thrower, there is a rather large difference between the

results of Frohlich and the author.

Lift

Before continuing further, it is necessary to establish why an airfoil generates lift. First,

consider an infinitesimally small cube of fluid shown in Figure 6. In this Figure, the cube is

traveling along a streamline. A streamline is a line that is tangent to the velocity of a fluid at all

points in space (Granger 422). If the pressure acting on this particle decreases along the x

, then the rear of the cube experiences a greater pressure than the front end of the cube.

The result is an acceleration in the positive x direction. Thus, if the pressure along a streamline

decreases, the velocity of the fluid particle increases along the streamline. The opposite happens

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when the pressure along the streamline increases. The mathematical representation of this is

Bernoulli’s equation.

To derive Bernoulli’s equation, consider again Figure 6. The force this cube undergoes is

given by , where v is the instantaneous velocity of the discus (Granger 503).

Because in this example, the pressure is greater on the right side of the cube than the left side,

there is a force pointing in the negative x direction. This force is related to the change in

pressure given by where A is the surface area of the cube. Realizing that the mass

of the fluid particle is given by , and that the change in pressure is equivalent given by

, where is the pressure gradient across the streamline, one can combine these

relationships to obtain . Noting that v is , the previous

equation relates the pressure at a point to the velocity of the fluid particle along the streamline at

that point:

503)

Integrating this between two points on the streamline and rearranging terms then gives

Bernoulli’s Equation:

To understand lift, one needs to perform much the same procedure as above for a fluid

moving along a curved streamline in Figure 7. Because the fluid particle is moving in a circle,

the force acting on it is the centripetal force give by . Since the force is acting down, the

pressure along the outside of the fluid particle must be less than that on the inside, or:

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. As was done when deriving Bernoulli’s Equation,

, where h is the height of the cube, and , where n is the direction normal to the

curve. Combining these gives . This simplifies to a very important

concept when considering why airfoils generate lift:

This is relates the gradient of the pressure change across a curved streamline, to the local radius

of curvature, and the local velocity along the streamline. It can now be shown as to why airfoils

generate lift. To see this, consider Figure 8. Here is a simple airfoil moving through the

atmosphere at a specific point in time. At point A, far away from the airfoil, the pressure of the

atmosphere is just the atmospheric. As one moves along the line connecting point A with the

airfoil, the pressure gradually decreases (as it has already been established that for any given

point along a curved streamline, the pressure outside is greater than the pressure inside). This

means that the pressure at B is less than the pressure at A. The opposite holds if one goes from

point C to point D. Again, the pressure at point C is the pressure of the atmosphere. But as one

goes up the line, the pressure gradually increases such that the pressure at point D is greater than

the pressure of the atmosphere. Therefore, if . We have

the required pressure difference to generate lift.

So we have shown that even the simplest objects can generate lift. But how can we

improve lift performance? The first thing to consider is the shape of the airfoil itself (Figure 9).

Figure 9 shows two airfoils. Both airfoils show similar streamline patterns above and below

their lower surface. However, the thicker airfoil has streamlines with little curve below the

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bottom surface. Keep in mind that for a perfectly straight streamline, , meaning that the

pressure change is 0. So while the pressure directly under the airfoil may still be greater than the

pressure above the airfoil, it’s not going to be by much. For the thin airfoil however, there is a

well-defined radius of curvature and thus a greater lift force directly under the airfoil.

The second way to improve lift performance has already been discussed but now shall be

discussed further. Consider Figure 9, where three airfoils are moving through the atmosphere at

varying attack angles. For the first airfoil, there is a low angle of attack. Notice the curvature of

the streamlines. The low angle of attack is associated with very little curvature, meaning little

lift is acting on the airfoil. However, in the case of the second airfoil, the high angle of attack

has allowed for well defined curvature meaning more lift is generated. The third case illustrates

the complexity of the relationship between angle of attack and lift. What is present here is

turbulence on the tail end of the airfoil. This is absolutely not conducive for flying.

Why is all this important? It is important because it has illustrated that even the slightest

difference in the change in pressure can have interesting effects on objects considered to be

airfoils. Such an object is the discus considered in this paper. The nature of the force acting on

the discus has now been explained. It is straightforward to see how minute pressure changes

across the bottom of a discus could change the orientation of the discus during flight and in fact,

this is exactly what happens (Figure 11). The aerodynamic lift force is larger on the front half of

the discus causing the front half to pitch up about 1.1 degrees per second over the course of the

flight. This change in pitch is also the cause of a roll (due to Force B in Figure 11). In discus

competition, there are strict guidelines on the mass, but none on the mass distribution of a discus

(Frohlich 1128). Because any torque generated is given by where I is the inertia tensor

and the angular velocity in some arbitrary direction, it is very clear that the change in

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orientation of the discus over time very much so depends on the mass distribution or moment of

inertia. Ultimately, I’d like to test this given how the change in orientation affects the equations

of motion, and what the forces causing the torque are. There have been no laboratory

measurements of torque and until tests are conducted, no reliable scientific answer can be given

on how different mass distributions affect the trajectory of the discus in flight (Frohlich 1132).

Rotating Coordinate System

Method

Because we have just established that the goal of the project is to determine the effect of

the moment of inertia on discus trajectories, one must now realize a cold hard reality. The

moments of inertia given the current coordinate system are much too hard to find. The solution

then, is to work with a rotated coordinate system, one such that the axis is aligned with the

primary axis of the discus. After each run of the loop described in Appendix A, a rotation vector

is then used to determine the velocities relative to the new coordinate system. If a rotated

coordinate system is used, the equations of motion are essentially the same except for a change

in the angle:

Notice from Figure 2 that in order to align the initial axis with the primary axis of the discus, the

initial axis needs to be rotated by . This, however, changes the components (although

obviously not the magnitude) of g. It also changes the components of any wind velocity present.

The equations for jerk are then:

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The procedure is then much the same except instead of outputting x and y, one needs to output

the following:

In the steps prior, it was and that was obtained. After the initial program was updated,

there was excellent agreement found for no wind and a wind opposing the thrower using both

programs.

The next step is to allow the orientation of the disc to change. In this paper, the discus

will only change orientation by way of pitch (changing alpha; Figure 2). The equations for

acceleration and jerk are essentially the same when allowing the orientation angle to change with

the exception of the second term in the equations for jerk not being zero. The biggest change

between non-changing alpha and changing alpha is an extra step after outputting the rotated

values of x and y. This constitutes the end of the first time increment, and thus the orientation

angle needs to be changed by . As mentioned before, a value was found for this to be

1.1 degrees per second tilted upward. Because the orientation angle is tilting upward, the attack

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angle is now decreasing and (of course) each time the attack angle changes, so too must the

coefficient of drag and lift.

Results

Moment of Inertia

As mentioned before, the torque causing the change in orientation of the discus depends

on the mass distribution or moment of inertia. While it may not be possible at the moment to

find what this force is, it is good to know how to find the moments for the three primary axes.

To begin, consider a sphere of constant density centered at an origin. The moment of inertia of

an infinitesimal disc about an axis perpendicular to its center (not the discus we are considering)

is given by , where A is the radius of the disc. In the case of this sphere, it is

useful to recognize that . Plugging this result into

and recognizing that , one gets . Integrating this from z to r

and multiplying by two then gives the moment of inertia about all three axes for this sphere.

Now, instead consider an oblate spheroid. This shape is very similar to the shape of a discus

with the exception of a minor correction or cut off at the very top (Figures 1 or 2). This shape

makes it possible to model the shape of a discus. Notice from Figure 13 that . First,

however, it’s helpful to consider the case closest to that of finding the moment of inertia of a

sphere. To find , again, consider that , except this time, r is not

constant. For z = 0, r = y so at first, the moment of inertia of the infinitesimal disc is given by

. But as z increases in value, r gradually becomes less and less. What one

can do then, is find the contribution from each infinitesimal element, and sum them up. Taking

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. To correct for the slight cut off in the discus, one can measure the value of x at the

very top of the discus, setting this as a limit by which any contribution exceeding that value is 0.

Results

After changing alpha by 1.1 degrees per second, the results were quite surprising.

Changing the orientation angle actually helped the trajectory and seemed to help the trajectory of

the discus going in the direction of the wind more so than the discus not experiencing any wind.

There may be some error in the program that may need to be fixed for future research. As the

attack angle increases, the lift coefficient should decrease while the drag coefficient increases

(Figure 3). Nonetheless, it is clear that changing alpha in any way has a profound effect on the

trajectory. Further study should no doubt show that the effect is negative and not to be desired.

Conclusions

While the main objective of this study was not met, it is still clear that changing the

orientation of the discus has an impact on the trajectory. While this effect may not be accurate

due to program error, the point is still clear. There is obviously much left to do with this

research. The one thing I wish I had gotten to was changing the angle describing roll and

implementing it into the equations of motion. I devoted hours to this but I spent too much time

on trying to figure out the force associated with the torque. And this is the main item left to

accomplish in this research. Again, there have not been any measured torques on a discus while

in flight and it is most necessary to do so. One final thing is that the differential equations need

to be solved with using a better approximation than the Euler method. Overall though, I know I

learned a lot from this project. I have a much greater understanding concerning why things fly.

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I’ve also taken lessons from this research that I will use for many research endeavors to come.

And it may be possible to continue this very research in graduate school.

Appendix A

Algorithm for finding the trajectory using Euler’s Method

1. Declare constants (gravity, discus mass, atmospheric pressure, discus cross-sectional

area).

2. Declare variables to change (wind velocity, attack angle, directional angle ( ),

orientation angle, horizontal position, vertical position, drag/lift coefficients, drag/lift

gradients, acceleration along horizontal and vertical, velocity along horizontal and

vertical, jerk along horizontal and vertical, and relative velocity)

3. Write four programs. The first two taking attack angle as input and returning the change

in lift and drag coefficients with respect to change in attack angle. The remaining two

programs take the newly found change in lift and drag coefficients with the attack angle

as input and return the updated lift and drag coefficients. One can model the relationship

between attack angle and lift/drag coefficient linearly (Figure 3).

4. Find drag and lift coefficients using initial attack angle.

5. Loop over several runs (this number will depend on the desired time increment, this

should be at most .01 seconds).

6. Using initial conditions of release angle (initial directional angle), height, inclination,

wind velocity, discus velocity, and lift/drag coefficients, find acceleration along the

horizontal and vertical.

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7. Multiply accelerations by desired time increment to obtain the discus velocity along the

horizontal and vertical. Find new relative velocity using desired wind velocity.

8. Using new relative velocity, find new directional velocity. Use this value to find new

attack angle.

9. Repeated step 4 using new attack angle.

10. Use drag/lift coefficients, acceleration along the horizontal and vertical, relative velocity

along the horizontal and vertical, and remaining constants from step 2 to obtain the jerk

along the horizontal and vertical.

11. Use equations 6 and 7 to find the change in position along the horizontal and vertical.

Data and Figures

Fig. 1- Force diagram on discus. This is a cross-sectional view of the discus. All forces are acting on the center of mass.

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Fig. 2 Wind velocity, discus velocity, and relative velocity with attack angle, orientation angle, and

directional angle. Horizontal plane represents the x-direction with the directional angle.

Fig. 3- Coefficients of lift and drag versus attack angle. Relationship can be interpolated linearly between angles

(Frohlich 1128).

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Fig 4. – Trajectories for no air and wind velocities of 10 m/s (B), 0 m/s (C), and -10 m/s (D). Release velocity is 25

m/s, discus area is 0.038 , mass is 2.0 kg, density is 1.29 , g is 9.8 m/ , and release height is 1.8m.

Fig. 5- Showing the trajectory from the program the author built using a method very similar to that of Frohlich

(Figure 4). There is good agreement (off by about 4 meters) between what Frohlich obtained and what the author

obtained. It is very clear that having a wind in the direction of motion does not help the trajectory of the discus but

rather decreases the trajectory relative to having no wind.

0

2

4

6

8

10

12

14

16

18

20

0 10 20 30 40 50 60 70

y (m

ete

rs)

x (meters)

y versus x

No wind

10 m/s

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Fig 6.- Motion of a fluid particle of length l and surface area A along a streamline undergoing a change in pressure

and thus an acceleration (Babinski 503).

Fig 7.- Motion of a fluid particle of length l and surface area A along a curved streamline undergoing a centripetal

acceleration. Because of the centripetal acceleration, pressure on the outside of the particle is greater than that on the

inside (Babinsky 503).

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Fig. 8- A basic airfoil. As shown above, it too can generate lift (Babinsky 500).

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Fig. 9 – Diagram showing an airfoil with curvature producing much lift due to increased curvature of streamlines

(top) and showing an airfoil producing less lift due to less curvature of streamlines (bottom) (Babinsky 501).

A. Low angle of attack B. High angle of attack C. Stalled flow

Fig. 10- Streamlines represented by smoke flowing past the airfoil at varying angles of attack (Babinsky 502).

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Fig. 11- (Frohlich 1127)

Fig. 12-

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Fig. 13-

3

2

1

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Works Cited Page

Frohlich, Cliff. “Aerodynamic effects on discus flight” Am. J. Phys. 49.12 (1981): 1125-1132.

Babinsky, Holger. “How do wings work?” Physics Education. 38.6 (2003): 497-503.

Granger, Robert. “Fluid Mechanics”. New York: Dover Publications, 1995.