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详细说明：The two main physical concepts behind the Frisbee are aerodynamic lift
(or the Bernoulli Principle) and gyroscopic inertia . A spinning frisbee
can be viewed as a wing in free flight with the Bernoulli Principle being
the cause of the lift and the angular momentum of the disc providing
its stability.The Physics of Frisbees
The coefficient Cn) is a drag coefficient that varies with the object
and is given in Hummel(2003)as being a quadratic function solely
dependent on the angle of attack a. a is the angle formed between the
plane of the frisbee and the relative velocity vector
Thc cocfficicnts CD0, ao and Cpa arc constants and and dcpcnd on thc
physica l aspects of the Frisbee
The lift force felt by a frisbee is very similar to the lift force on
airplane wings and is calculated using the Bernoulli principle. The
Bernoulli Principle is a well known principle that states that there
s a relationship between the velocity, pressure and height of a fuid
at any point on the same stream line. Fluids fowing at a fast velocity
have a lower pressure Chan fluids llowing al a slower velocily. This can
be written mathematically as
+2+gh
p2
+2+gh2,
2
2
where v is the velocity of the fluid, p is the pressure of the fluid, p is the
density of the fluid, g is the acceleration of gravity and h is the height
of the fluid. The subscripts 1 and 2 refer to different points in the fuid
along the same streamline. This equation is commonly referred to as
Bernoulli's equation. For our purposes, the height difference between
the air flowing above and the air flowing below the frisbee is negligible
therefore the two height dependent terms cancel out. We will also as
sume that the velocity of the air fowing above is direct ly proportional
to the vclocity of the air bclow bccausc the difference in path length is
constant(i.e. 71=C22). We now have the equation
tting FL/A= pi
here Fr is the lift force and A is the area of
Lhe Frisbee)and solving for FL gives
AC
Throughout the steps needed to determine(6), the coefficient C was
incorporated into the coefficient CL Cl is given in Hummel (2003)as
being a linear function of the angle of attack, a
CL-Clo + c
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4
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where Clo and Clo are constants that depend on the physical proper
ties of the frish
2.2. GYROSCOPIC STABILITY
The rotation of a frisbee is a necessary component in the mechanics
of how a Frisbee flies. Without rotation, a Frisbee would just futter
to the ground like a falling leaf and fail to produce the long distance
stable flights that people find so entertaining. This is caused by the
fact the aerodynamic forces described in the previous section are not
directly centered on the frisbee. In general, the lift on the front half of
the disc is slightly larger than the lift on the back half which causes a
torque on the Frisbee(See Figure 1)
CoM
a
mg
Figure 1. Diagram of the off-center center of pressure(COP)and the center of mass
(COM) that results in a torque exerted on the Frisbee
When a Frisbee isnt spinning, this small torque fips the front of
the disc up, and any chance for a stable fight is lost. When a Frisbee is
thrown with a large spin, it has a large amount of angular momentum
that has a vector in either the positive or negative vertical direction
When the small torque is exerted, the torque vector points to the right
side of the frisbee(when viewed from behind. This can be determined
using the righthad rule with
r×F
(8)
MorrisonFin. tex; 20/08/2005;8: 48; p 4
The Physics of Frisbees
5
L
dt
the angular momentum vector will begin to precess to the right. This
phenomenon can easily be viewed when throwing a Frisbee, this is the
reason that many thrown frisbees bank to either the left of the right
Due to this, the greater the initial angular momentum given to the
Frisbee, the more stable it's fight will be
3. Numerical Modelling of a Frisbee in Flight
To model the fight of a Frisbee, a Java program was written that used
the numerical technique Euler's method applied to the forces described
in the previous section(see code in Appendix). To accomplish this, the
different forces were separated into horizontal and vertical components
and Euler's method was applied each component. It should be noted
that in the model it is assumed that the frisbee is given enough initial
spin so as to maintain a stable flight. In applying Fuler's method, the
trajectory of thc Frisbee is divided into discrctc timc steps, At, and
at each step a new horizontal velocity, v, and horizontal position, T, is
clinc
0+1=v+△U,
x;+△
(11
where Av and Ar are the changes in velocity and position respectively
A similar equation to equation (11)can be used with the vertical
position, y, used instead of The Av's are obtained by solving the
following relationships
Fr= FD
(12)
n At=2DU2ACD
(13)
ACn△t
(14
where fp is the drag force on the Frisbee. also
Fu= Fa+ Fr
(15)
1y g+ SPU2ACL
(16)
9+aPACE
(17)
MorrisonFin. tex; 20/08/2005;8: 48; p5
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where the subscripts x and y denote the horizontal and vertical velocit
respectively and Fa is the force of gravity. A c and Ay are simply stated
as
(18
=℃
9
The program written contains a. method simulate which takes five
input paramctcrs, initial y position and velocity, initial x vclocity(thc
initial x position always set to zero), the angle of attack(in degrees)
and thc At. All units other than that of angle of attack arc in si units
In all of the trials a at=0.001s was used. trials with At=0.001s and
At=0.002 s were both tested and the difference between the results
was unnoticeable. (Note: In the simulation the values of the coefficients
used were: CDo=0.08, CDa =2.72, Clo =.15, Cla= 1.4
4. Results
When conducting the simulations, all trials had
tial height of 1
m, an initial x velocity of 14 m/s which is considered that standard
velocity of a thrown frisbee, and an initial y velocity of 0 m/. Trials
were conducted using angles of attack ranging from 0 to 45. This was
the only parameter that was changed because the coefficients of lift and
drag depend solely on angle of attack. It can be seen from figures 2, 3
and 4 that the angle of attack has a large effect on the trajectory of the
Frisbee. With low angles of attack(generally less than 5 degrees) the
lift force was very small and the frisbee dropped quickly to the ground
after a short distance, usually less than 20 m. With larger angles of
attack, a larger lift force was apparent and the frisbee reached greater
heights and travelled much further, up to 40 m. The maximum distance
travelled was obtained with an angle of attack of approximately 12 and
it travelled 40 m with a maximum height of 7. 7 m. At larger angles of
attack the Frisbee went significantly higher, but due to the much larger
drag force travelled a smaller distance. Trials that were conducted with
different initial velocities followed a trend similar to those with an initial
velocity of 14 m/s. At lower velocities the lift force was greatly reduced
and the Frisbees just dropped to the ground faster. At higher velocities
the lift force was greater and their trajectories were higher and longer
MorrisonFin. tex; 20/08/2005;8: 48; p6
The Physics of Frisbees
10
15
20
3
Figure 2. Plot of height(m) versus distance(m) for a Frisbee with initial velocity 14
m/s and angle of attack 5
25
40
Figure 3. Plot of height(m) versus distance(m)for a Frisbee with initial velocity 14
m/s and angle of a ttack 7.5
4
10
15
40
Figure f. Plot of height(m) versus distance(m)for a Frisbcc with initial velocity 14
m/ s and angle of attack 10
MorrisonFin. tex; 20/08/2005;8: 48; p7
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5. Discussion
Although simplistic in naturc, the results obtaincd from the program
written provide a realistic simulation of the trajectory of an actual
Frisbcc. It was shown (using information from Hummcl(2003)and
Motoyama(2002) what the various forces that act on a Frisbee are
and what they depend on, as well as how different angles of attack can
ary the distance and height a Frisbee reaches greatly. In the future
urther research may include developing a three dimensional model that
includes the precession and rolling of the frisbee, as well as looking into
the various physical properties of the Frisbee. These may include the
different thicknesses of the Frisbee edges which varies the moment of
inertia, Hying rings which travel great distances and ridges placed on
the Frisbee lo reduce drag. By incorporating these properties it Inlay
be possible to design better Frisbees
A
ppendIx
A. Java code for plotting the trajectory of a Frisbee
The following code has been slightly modified so as to make it fit on
the page.
import java. lang Math;
import Java. l0.*,
The class frisbee contains the method simulate which uses euler>s
method to calculate the position and the velocity of a frisbee in
two dimensions
Author vance morrison
version march 4 2005
public class Frisbee t
private static double x
//The x position of the frisbee
pr1
tatic double y
//The y position of the frisbee
private static double vx;
/ /The x velocity of the frisbee
private static double vy
//The y velocity of the frisbee
private static final double g =-9.81;
MorrisonFin. tex; 20/08/2005;8: 48; p 8
The Physics of Frisbees
/The acceleration of gravity (m/s"2)
private static final double m=0.175;
//The mass of a standard frisbee in kilograms
private static final double rho = 1. 23;
//The density of air in kg/m"3
private static final double area =0. 0568
//The area of a standard frisbee
private static final double clo=0.1;
/The lift coefficient at alpha =0
private static final double Cla =1. 4
//The lift coefficient dependent on alpha
private static final double CDo =0.08
//The drag coefficent at alpha =0
private static final double CDa =2.72;
/The drag coefficient dependent on alpha
private static final double aLPha =-4
a method that uses Euler>s method to simulate the flight of a frisbee in
two dimensions, distance and height (x and y, respectively)
public static void simulate(double yo, double vxo, double vyo,
double alpha, double deltaT
//Calculation of the lift coefficient using the relationship given
//by S. A. Hummel
double cl =CLO CLA*alpha*Math. PI/180;
//Calculation of the drag coefficient (for Prantl's relationship)
//using the relationship given by S. A. Hummel
double cd =CDO CDA*Math. pow((alpha-ALPHAO)*Math. PI/180, 2);
//Initial position x =0
0
//Initial position y= yO
//Initial x velocity vx
//Initial
elocity v
0
try
20/08/2005;8:48;p.9
Ⅵ orrison
//A PrintWriter object to write the output to a spreadsheet
PrintWriter pw =new PrintWriter (new BufferedWriter
(new FileWriter("frisbee. csv"))
//A loop index to monitor the simulation steps
int k =0:
//A While loop that performs iterations until the y position
//reaches zero (i. e. the frisbee hits the ground)
while(y>0)t
//The change in velocity in the y direction obtained setting the
//net force equal to the sum of the gravitational force and the
//lift force and solving for delta v
double deltav =(RHO*Math. pow(vx, 2)*AREA*c1/2/m+g)*deltaT;
//The change in velocity in the x direction, obtained by
//solving the force equation for delta v. (The only force
//present is the drag force)
double deltav =-RHO*Math. pow(vx, 2)*AREA*cdxdeltaT
//The new positions and velocities are calculated using
//simple introductory mechanics
Vx vx del tax
vy deltav;
x = x vx*deltar
y =y+ vy*deltat;
/ /Only the output from every tenth iteration will be sent
//to the spreadsheet so as to decrease the number of data points
if(k%10==0){
pw. print(x+","+y+", "t vx)
pw. println
pw. flush)
k++
se();
catch(Exception e)t
System. out. println("Error, file frisbee. csv is in use. ") H
Morrisonfin.tex;20/08/2005;8:48;p.10
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