Download 26-th ECMI Modelling Week Final Report

26-th ECMI Modelling Week
Final Report
19.08.2012—25.08.2012
Dresden, Germany
Group 5
Phantom footballs and
impossible free kicks:
modelling the flight of
modern soccer balls
Alexandra Hazard Kampmann
Dep. of Informatics and Mathematical Modelling, Technical University of
Denmark,
Lyngby, Denmark.
Dunja Arsic
Department of Mathematics, University of Novi Sad, Novi Sad, Serbia.
João Jorge Dias Neves
Department of Mathematics, Technical University of Lisbon
Lisbon, Portugal
Johan Håkansson
Department of Mathematics, Chalmers University of Technology,
Goteborg, Sweden
Pedro Filipe Carvalho Santos
Department of Mathematics, VU University Amsterdam,
Amsterdam, Netherlands
Raido Paas
Department of Mathematics, University of Tartu,
Tartu, Estonia.
Instructor: Timothy Reis
Mathematical Institute, University of Oxford,
Oxford, UK
2
Abstract
In this project we will attempt to model the trajectory of a free-kick taken
in football matches using simple Newtonian mechanics. We will model a
kick in Mathematica and Matlab, and in both two and three dimensions.
We will also attempt an analytical solution of the ODE system in order to
gain mathematical insight to the dominant forces of the free-kick. We will
furthermore use the models to investigate some of the determining factors
for the free-kick such as drag force and lift force alluded to by the analytical
solution, and examine the difference of these between the 2010 Jabulani
football by Adidas and traditionally manufactured footballs. Finally, we
will model the famous ’banana-kick’ performed by Roberto Carlos in 1997.
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5.1
Free kicks in football
Introduction
In 2010 Adidas introduced a new football named the Jabulani. Many notable
players started complaining about the ball, claiming that it seemed to defy
all known laws of physics. When kicked appropriately it appeared to bounce
in midair, seemingly oblivious of gravity. Effects of this sort are known in
baseball as knuckleballs, and are a common but curious phenomenon. In this
report we will attempt to model a free kick and recreate the gravitationally
taunting bounces and curves.
5.2
The physical model
Before we start any numerical approximations we first have to build a model
using basic Newtonian mechanics.
Forces acting on the ball
When the ball is kicked and moves through the air we assume that it is only
affected by the following forces; a gravitational force due of course to the
gravitational field of the Earth, and so-called drag and lift forces relating to
the air resistance and spin of the ball.
Figure 5.1: Forces acting on a football in flight
According to Newton’s second law these forces result in an acceleration
of the ball:
F~resulting force = F~gravity + F~drag + F~lift
(5.1)
We say that the ball is moving with velocity ~v and the resulting force is
then (again according to Newton)
F~resulting force = m~a = m~v˙
(5.2)
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where m is the mass of the ball and ~a is the resulting acceleration. The
gravitational force on the ball is just the usual F~gravity = m~g , where ~g is the
acceleration of the gravitational field on the ball.
The drag and lift forces are slightly more complicated to explain and
both relate the ball to the fluid it is travelling through. They can both only
exist when the ball moves relative to the surrounding air.
When the ball flies through the air it forces the air to move in space, instead
flowing around the ball. This flow can be either laminar (“smooth”) or
turbulent (“chaotic”) depending on the parameters of the ball and fluid.
The onset of turbulence is dependent on the surface geometry of the ball
and properties of the fluid and determined by a single, non-dimensional
number called the Reynold’s number. Laminar flow occurs at relatively
low Reynold’s numbers, and turbulent at high Reynold’s number. We will
discuss this number and the drag force in more detail when looking at the
mechanisms from a fluid dynamical point of view, but for now the drag force
is simply
1
F~drag = − ρACD~v |~v |
2
(5.3)
where ρ is the density of the fluid, A is the cross-sectional area of the ball
and CD is a dimensionless number called the drag coefficient. This number
will be discussed in further detail in the models below.
The so-called lift force is also a result of the ball-air interaction but related
to the spin of the ball. When the ball spins it moves air around with it due
to the no-slip condition, which states that the velocity of the fluid at the
boundary is the same as the velocity of the boundary itself. If the ball is
moving with a forward speed v and rotating at speed w the the velocity at
the top and bottom of the ball will be as seen in 5.2. This results in a higher
kinetic energy at the top of the ball and thus of the air at the boundary
layer. According to Benoulli’s principle (based on a conservation of energy)
a higher kinetic energy means a corresponding decrease in potential energy
and static pressure. Likewise, a slower speed results in a higher potential
energy and lower pressure. The situation here is analogous to this process
and causes the air to move from the high pressure area to the low pressure
area, resulting in a force pushing the ball upwards (again, please see figure
5.2 for clarification).
This force is called the Magnus force and is dependent on many of the
same things as the drag force, but instead of a drag coefficient there is now
a lift coefficient, CL
1
F~lift = ρACL~v |~v | sin θ
(5.4)
2
Here θ is the angle of spin. Summing all the forces up and setting both the
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Free kicks in football
Figure 5.2: The lift (Magnus) force acting on a ball in flight.
direction away from the Earth and the direction of the kick to positive gives
1
1
m~a = −mg − ρACD~v |~v | + ρACL~v |~v | sin θ
2
2
(5.5)
The model as a system of ODEs
We wish to trace the trajectory of the football so we convert the above forces
into the acceleration in three dimensions by dividing by the mass of the ball.
This gives us the following system of ODEs:
d2 x
dx
dx
= −gx + |~v | −kD
+ kL
(5.6)
dt2
dt
dt
d2 y
dy
dy
= −gy + |~v | −kD
+ kL
(5.7)
dt2
dt
dt
d2 z
dz
dz
=
−g
+
|~
v
|
−k
+
k
(5.8)
z
D
L
dt2
dt
dt
ρACL
D
where kL = ρAC
2m , kD = 2m with m as the mass of the ball, A is the
surface area of the ball, CD , CL are the drag and lift coefficients and ρ is
the density of the air. We can now implement these ODEs and compute the
trajectory of the ball.
Fluid mechanical formulation
In the above we looked at the forces acting on the ball as the ball moved
through space. A different approach is to view the ball as stationary and
the air moving around the ball. This enables a fluid dynamical point of
view as opposed to a mechanic point of view. We will start off by assuming
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for simplicity that the air can be modelled as an incompressible, Newtonian
fluid. These two assumptions are based on the following facts:
Compressible effects are only important when the speed of the fluid is comparable to the speed of sound in the fluid. For atmospheric air the speed of
m
sound is roughly 340 m
s . We model kicks at a speed of around 35 s , so the
assumption of incompressibility is justified.
In a Newtonian fluid the stress on the fluid is proportional to the strain with
the constant of proportionality being the dynamic viscosity.
So, we assume that air is an incompressible, Newtonian fluid. The NavierStokes equations then become


∂v
∂t
|{z}

ρ

+



v| ·{z
∇v}
(5.9)
convective acceleration
unsteady acceleration
=
−∇p
| {z }
pressure gradient
+
µ∇2 v
| {z }
+
diffusion (viscosity)
∇·v =0
f
|{z}
other body forces (e.g. gravity)
(5.10)
These equations can be non-dimensionalised which results in a parameter
called the Reynold’s number, Re = ρUµL , where ρ is the density of air, U is
some characteristic speed, L is likewise a characteristic length, and µ is the
dynamic viscosity of air. The Reynold’s number in our case is in an order
of 107 , which is very high and suggests turbulent flow. The drag and lift
coeffiecient can by non-dimensionalisation be described as functions of the
Reynold’s number, something which we will come back to later.
There are several commonly used boundary conditions for this problem, but
we will only mention the most important one in this context, namely the
no-slip boundary condition. Formally, this condition is stated as
→
−
−
v =→
vb
(5.11)
−
−
where →
v is the velocity of the ball and →
v b is the velocity of the boundary
layer of fluid. In practice this condition means that when air hits the ball the
air is pulled around with the surface of the ball (please see figure 2 again).
The point of separation between the laminar and turbulent flow determine
the size and direction of the drag force. When the ball is spinning the point
is moved up or down compared to no spin, which helps push the ball in
an upward or downward direction. Thus, the surface geometry of the ball
suddenly becomes very important.
The Jabulani ball and drag crisis
Since the surface geometry seems to be of some importance we will now
briefly describe the difference between traditional footballs and the 2010
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Free kicks in football
Figure 5.3: The construction of the Jabulani ball.
Jabulani football by Adidas.
Old-fashioned footballs are made up of several leather panels sewn together.
The Jabulani ball by contrast is made up of eight spherically moulded panels,
which are thermally fused together, and the surface has small bumps in it.
These bumps add a certain roughness to the surface of the ball, although
it is still comparably smoother than old-fashioned balls, and are suppose to
induce turbulence transition behind the ball and improve its flight. This is
similar to the effects of the dimples on a golf ball.
In the flight of balls one phenomenon is something called the drag crisis.
The drag crisis is a sudden change in the drag coefficient when the ball
reaches a critical Reynold’s number, which is most likely associated with the
surface texture of the ball. Before and after this drag crisis the coefficient
is nearly constant. The drag crisis occurs due to the shift from laminar to
turbulent flow in a thin boundary layer on the surface of the ball.
For the Jabulani ball experimental results indicate that the drag crisis occurs
at a different speed compared to the other balls. If this is the case then
it could hypothetically lead to the erratic and unpredictable behavior of
the Jabulani as reported by many professional football players. We will
investigate this in our three-dimensional model.
5.3
Two-dimensional model results
We will start by examining our model in two dimensions. We have built the
model up by first considering gravity only, then adding drag, and finally a
lift force to the ball trajectory. We did this in both Mathematica and Matlab
to be able to compare results. In Matlab we used an ode45 solver, which is
the built-in 4th order Runge-Kutta solver with a 5th order correction. This
is assuming that the problem is non-stiff, which seems reasonable given the
simplifications we have made regarding air as a fluid and the comparable
time-scales of the drag, Magnus and gravitational forces acting on the ball.
Since the graphics take up a lot of space we have chosen to put them
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Figure 5.4: The results of our model in two dimensions showing the sidespin
of the ball.
in an appendix, but will include the trajectory of a ball in two dimensions
with the lift and drag forces (please see figure 5.3).
Analytical solution
We will not attempt to solve this system of ODEs but it is beneficial to
approximate a mathematical solution in order to give us some hint on which
terms could be dominant in the trajectory of the ball and thus where to
begin our experiments.
By assuming that the velocity in equation 5.6 in the direction perpendicular
to the direction of the free-kick is much smaller than this direction, i.e.
ẋ << ẏ when y indicates the direction of the free-kick, we can ignore the
smaller terms in the ODEs and use asymptotics of the equations to get the
following result 1 :
kL sin θ
kD [vt − y]
ln (kD tv + 1)
→
−
y (t) =
kD
→
−
x (t) =
(5.12)
(5.13)
where kD , kL are the non-dimensionalised drag and lift coefficients and θ is
the spin angle. We see that the there are several important factors influencing the trajectory of the ball: the amount of spin (the angle θ, the initial
velocity v with which the ball is kicked, and the distance vt − y from the
goal. This result also suggests a strong dependence on the drag coefficient
for the main kicking direction, as we would expect since the drag force in
essens “pulls” the ball back. The amount of sidekick depends also on the
1
Due to space constraints we were unfortunately not able to show
pthe intermediate
steps in this derivation, but important approximations include |v| = ẋ2 + ẏ 2 ' ẏ and
ẍ ' |v|KL ẏ sin θ
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Free kicks in football
drag coefficient, but more interestingly it depends linearly on the lift coefficient. This means that we should potentially investigate the lift coefficient
more closely, although we will not have the time in this project. We note,
however, that this is only an an asymptotic solution and we have not had
the time in this project to test the robustness of this solution.
5.4
Three-dimensional model results
For modelling the trajectory of the football in three dimensions, we used
the model we deduced above. We notice that when taking θ = π/2 or
θ = −π/2 in equations (5.6), our ball will have the so-called “side-spin”
(only the spinning directions are different). When taking θ = 0, our ball
will have so-called “top-spin”, which lifts the ball more up while in the air
(the kick, which goalkeepers usually take).
For the lift coefficient we use the following approximation (see [1], page
510):
ωR
Cl =
,
v
where R is the radius of the ball, v is the velocity of the ball and ω is
the angular velocity of the ball. The angular velocity ω has the following
equation (see [1], page 510, equation (4)):
ω = ω0 e−t/7 ,
where ω0 is the initial angular velocity of the ball. We notice that when the
ball is not spinning, then ω0 = 0 and we have no acceleration due to Magnus
force in equations (5.4).
For estimating the drag coefficient Cd in equations (5.3), we found the
following equations from the source [4] (pages 776-777, equations (3) and
(4)), which were achieved by using the “Teamgeist” football:
(
cSpd
if v > vc and Sp > 0.05,
Cd =
(5.14)
b
a + 1+e(v−vc )/vs otherwise.
In the equations (5.14), a = 0.155, b = 0.346, vc = 12.19 m/s, vs =
1.309 m/s, c = 0.4127, d = 0.3056 and Sp , the “spinning parameter”, has
the same equation as Cl in our model, that is, Sp = ωR/v. In the case if the
ball is not spinning (Sp = 0), the relationship between the velocity of the
ball and the drag coefficient for the ball used (“Teamgeist”) can be seen in
figure 5.5 on page 9.
The sudden change of the drag coefficient is, as mentioned, called the
“drag crisis” (see figure 5.5 on page 9). Our aim was to see, how big an
effect the “drag crisis” has in playing with different footballs. To do this we
looked for the relationship between velocity and drag coefficients for various
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Figure 5.5: Velocity of the ball versus drag coefficient for various footballs.
balls. We have that relationship already for the “Teamgeist” football. From
the sources [6] and [2] (page 1022, figure 6) we obtained approximate data
for the “Jabulani” and 32-panel football, which data we interpolated with
cubic spline (the difference between the graphs can be seen in figure 5.5 on
page 9).
We ran two simulations with three different balls (“Jabulani”, “Teamgeist”
and 32-panel ball). The diameter for these balls is the same (0.22m), while
“Jabulani” weighs 0.44 kg and 32-panel ball weighs 0.425 kg (the weight of
“Teamgeist” was 0.442 kg). With air density ρ = 1.225 kg/m3 and with
inital velocity v0 = 15 m/s in the first simulation and with initial velocity
v0 = 25 m/s, with no spin in both simulatons (w0 = 0 rad/s), α = 35◦
(please see figure 5.9) in the first simulation and α = 16◦ in the second
simulation, β = 0◦ in both simulations, we obtained the results shown in
figures 5.6 and 5.7.
Figure 5.6: Comparison of balls, initial velocity 15 m/s.
By looking at the graphs we can deduce that when the velocities of the
balls are high (around 25 m/s), then it does not matter much with what ball
you are taking the free-kick. As the average free-kick is taken with initial
10
Free kicks in football
velocity from about 25 m/s up to 35 m/s, then it does not actually matter
much which ball you use, because the trajectories are pretty much the same.
The difference of the balls has an effect when the velocities are not so high
as we can see from the figure 5.6, which result is probably because of the
“drag crisis”.
Figure 5.7: Comparison of balls, initial velocity 25 m/s.
Unfortunately we were not able to compare the balls when they were
kicked and were spinning in the air. This was because we did not find
enough information about how the drag coefficient changes during the flight
for various balls in that case (in the case when the balls are spinning in
the air). Figure 1.5 describes the relationship between the velocity of the
ball and the drag coefficient when the balls are not spinning. If we kick the
“Teamgeist” with initial velocity v0 = 25 m/s, with initial angular velocity
w0 = 14*pi rad/s, alpha = 20 deg, beta = 0 deg, then the drag coefficient,
which is calculated according to the equations, varies between 0.245 and
0.265, which is different from that shown on the figure 1.5. We do not
have the information how “Jabulani”, for example, would behave when it
is also spinning in the air. Nevertheless, from the tests we performed, we
made a rough assumption, that it does not matter much with which ball the
professional football players take the free kick and they therefore have no
reason to complain about the balls.
Figure 5.8: Roberto Carlos’ 1997 free kick in Mathematica.
26th ECMI modelling week
5.5
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Modelling a real free kick
In 1997 Roberto Carlos performed a legendary free kick against France2 ,
where not even the goalkeeper had seen the fantastic swerve of the ball
coming. Using our simple model we have been able to reproduce this free
kick (please see figure 5.8). We have also modelled the free-kick with our
three-dimensional Matlab model. We chose our coordinate system so that
the zero point (0, 0, 0) is the point where the free-kick is taken.
Figure 5.9: Initial velocity vector and angles.
With the assumption that the free-kick was shot with an 16◦ angle with
xy-plane and with an 5◦ angle with y-axis (see figure 5.9, in our model
α = 16◦ and β = 5◦ ), with initial velocity v0 = 37 m/s and initial angular
velocity ω0 = 14π rad/s, spinning angle of the ball θ = −90◦ and with the
air density ρ = 1.225 kg/m3 , mass of the ball m = 0.442 kg, diameter of the
ball d = 0.22 m, we obtain the result shown in figure 5.10.
5.6
Discussion and conclusion
The purpose of this project was to model a free-kick as performed most
notably by Roberto Carlos in 1997 and Christiano Ronaldo in pretty much
every game, and to investigate the difference between the 2010 Jabulani ball
and traditional footballs. We found that it was possible to create a realistic
model with simple Newtonian mechanics, where the three forces acting on
the ball were gravity, drag and lift.
We tried to see whether the so-called drag crisis had any effect on the ball
trajectory since the drag crisis hits at a different velocity for the Jabulani
ball as compared to the traditional balls. We found that it did make a
2
http://www.youtube.com/watch?v=Pl0LHM-33Io
12
Free kicks in football
Figure 5.10: Our simulation of the famous free-kick from Roberto Carlos
from 1997 in Matlab in three dimensions.
notable difference at lower speeds, but that this difference diminshed when
the speed of the ball neared an average free-kick speed. Therefore, we concluded that the drag crisis did have some effect but was probably not the
main cause of the weird Jabulani behavior. We also concluded that the
players’ moaning was unjustified in our model, but perhaps not in a more
detailed model.
There are plenty of possibilities for a further investigation of this topic.
Seeing as the drag coefficient and drag crisis do not appear to have much
varying effect on the trajectory a natural place to continue research is of the
lift coefficient. There have been articles about a changing lift coefficient,
and these changes could be applied readily in the model in the same way as
the drag coefficient was changed.[4]
Another possibilty is looking more closely at the surface geometry of the
ball. Since, as discussed earlier, the surface of the Jabulani is radically different then previous balls, but does not otherwise seem to differ in weight
or water retention, than this factor is definitely worth looking into [3], [5],
[7].
Also, the density is present in our model. It is worth noting that the 2010
World Cup took place sometimes quite a bit over sea level meaning a lower
air density. We did not consider these effects in our model.
We were trying to model the so-called knuckling ball effect where the ball
appears to “bounce” in midair, but we did not succeed in producing this
effect. Other authors have tried to varying degrees of success [8],[9] and
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implementing some of their observations in the model could be fun as well.
The above further possibilites are all related to experimenting with the
model, but it could also be beneficial to investigate an analytical solution to
the system of ODEs that includes more terms to better understand which
mechanism dominate the process.
All in all this project has been a lot of fun, with the only negative side effect
being the inability to ever watch a football match as innocently unaware of
the physics behind it ever again!
Bibliography
[1] Vassilios M. Spathopoulos, A Spreadsheet Model for Soccer
Ball
Flight
Mechanics
Simulation,
2009,
Greece,
http://onlinelibrary.wiley.com/doi/10.1002/cae.20331/pdf
[2] John Eric Goff and Matt J Carré, Trajectory Analysis of a Soccer Ball,
2009
http://goff-j.web.lynchburg.edu/Goff Carre AJP 2009.pdf
[3] S. Barber and Matt J. Carré, The effect of surface geometry on soccer
ball trajectories, 2010,
Sports Eng 13:47-55
[4] John Eric Goff and Matt J Carré, Soccer Ball Lift Coefficients Via
Trajectory Analysis, 2010,
http://goff-j.web.lynchburg.edu/Goff Carre EJP 2010.pdf
[5] Firoz Alam et.al., Aerodynamics of Modern Footballs, December 2010,
Proceedings of the 13th Asian Congress of Fluid Mechanics
[6] Jabulani, a Ball in Crisis? ,
http://engineeringsport.co.uk/2010/06/25/jabulani-a-ball-in-crisis/
[7] Rabindra D. Mehta, Aerodynamics of Sports Balls, 1985, Ann. Rev.
Fluid Mech.
[8] Sungchan Hong et.al., Ball impact dynamics of knuckling shot in soccer,
March 2012, Proceedings of the 9th Conference of the ISEA
[9] Takeshi Asai et.al., A Study of Knuckling Effect of Soccer Ball, The
Engineering of Sport 7 - Vol. 1
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