PHY 306
Project Report
Double Pendulum
by
Arnab Dhabal and Raziman T V
(Y7081)
(Y7355)
Mentor:
Dr. M.K. Verma
1
Contents
1 Introduction
3
2 Theory
3
3 Construction
5
4 Numerical Analysis
5
5 Future work
7
6 Acknowledgement
8
References
8
4.1 Non-Chaotic Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
4.2 Chaotic Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2
1
Introduction
A double pendulum consists of one pendulum attached to another[1]. This is a simple mechanical
system that shows chaos for some initial conditions. A double pendulum has two degrees of freedom
and a four dimensional state space.
A pair of double pendulums can serve as a good demonstrative apparatus to show chaos, as
visibly identical initial conditions result in very dierent evolution of the system. In this project we
build a double pendulum and demonstrate chaotic and non-chaotic evolution of the system based
on dierent initial conditions. A numerical treatment of an ideal undamped double pendulum is
also done to verify experimental results.
2
Theory
Here is a schematic diagram of the double pendulum. Both limbs of the double pendulum are
restricted to move in the vertical plane.
The upper limb of the double pendulum has
length L. Its centre of mass is at a distance
l from the point of suspension and the moment
of inertia about the centre of mass is I . The
centre of mass of the lower limb is situated at
a distance l from its point of suspension and
its moment of inertia around the centre of mass
is I . The masses of the limbs are m and m
respectively.
Here we do the analysis for an ideal - ie,
undamped - double pendulum. This is a conservative system. Equations of motion are derived here
using the Lagrangian formalism.
Translational kinetic energies of the centres of mass of the two limbs are given by:
1
1
2
2
1
T1,trans
=
T2,trans
=
=
=
2
1
2
2
2 m1 x˙1 + y˙1
1
2 ˙ 2
2 m1 l1 θ1
1
2
2
2 m1 x˙2 + y˙2
1
1
2 ˙ 2
2 ˙ 2
2 m2 L θ1 + 2 m2 l2 θ2
+ m2 Ll2 cos(θ1 − θ2 )θ˙1 θ˙2
Rotational kinetic energies of the limbs around their respective centres of mass are given by
T1,rot
=
T2,rot
=
3
1
˙2
2 I 1 θ1
1
˙2
2 I 2 θ2
Hence the total kinetic energy of the system is
2
2
2
2
T = 12 m1 l12 θ˙1 + 12 m2 L2 θ˙1 + 12 m2 l22 θ˙2 + m2 Ll2 cos(θ1 − θ2 )θ˙1 θ˙2 + 12 I1 θ˙1 + 12 I2 θ˙2
2
The gravitational potential energies of the two limbs are
V1
V2
−gm1 l1 cos(θ1 )
−gm2 L cos(θ1 ) − gm2 l2 cos(θ2 )
=
=
Hence the total potential energy of the system is
V = −gm1 l1 cos(θ1 ) − gm2 L cos(θ1 ) − gm2 l2 cos(θ2 )
and the Lagrangian is
L
=
where
=
c1
=
c2
c3
c4
c5
=
=
=
=
T −V
2
2
c1 θ˙1 + c2 θ˙2 + c3 θ˙1 θ˙2 cos(θ1 − θ2 ) + c4 cos(θ1 ) + c5 cos(θ2 )
m1 l12
2
m2 l22
2
+
I1
2
I2
2
+
m2 L2
2
+
m2 Ll2
g (m1 l1 + m2 L)
gm2 l2
The evolution of the system is determined by the Euler-Lagrange equations
d ∂L
dt ∂ θ˙i
−
∂L
∂θi
=0
For the current system, this gives us two coupled second order ordinary dierential equations
2
c4 sin(θ1 ) + 2c1 θ¨1 + c3 θ¨2 cos(θ1 − θ2 ) + c3 θ˙2 sin(θ1 − θ2 )
2
c5 sin(θ2 ) + 2c2 θ¨2 + c3 θ¨1 cos(θ1 − θ2 ) − c3 θ˙1 sin(θ1 − θ2 )
=
0
=
0
Decoupling the second order parts,
2
2
θ¨1 =
2c2 c4 sin(θ1 )+c23 θ˙1 sin(θ1 −θ2 ) cos(θ1 −θ2 )+2c2 c3 θ˙2 sin(θ1 −θ2 )−c3 c5 cos(θ1 −θ2 ) sin(θ2 )
c23 cos2 (θ1 −θ2 )−4c1 c2
θ¨2 =
2
2
2c1 c5 sin(θ2 )−c23 θ˙2 sin(θ1 −θ2 ) cos(θ1 −θ2 )−2c1 c3 θ˙1 sin(θ1 −θ2 )−c3 c4 cos(θ1 −θ2 ) sin(θ1 )
c23 cos2 (θ1 −θ2 )−4c1 c2
4
3
Construction
The experimental apparatus was constructed in the Physics Workshop at IIT Kanpur. The apparatus consists of two identical double pendulums supported by the same structure. Two double
pendulums are used to see how the evolution is dierent for nearly identical initial conditions.
The upper limb of each double pendulum
is made out of two thin rectangular aluminium
rods. Two rods are used to maintain symmetry. The rods are 30 cm long, 2.5 cm wide and
5 mm thick. the two rods of the upper limb
were separated by 8 cm. The lower limb of each
double pendulum is an aluminium rod 25 cm
long but identical otherwise to the rods making
up the upper limbs. The upper limbs are suspended from a mild steel rod of diameter 6 mm.
A similar mild steel rod is xed to the lower
end of either upper limb from which the lower
limbs hang. All free joints are connected using
ball bearings to minimise damping. The steel
rod from which the upper limbs are suspended
is supported by two vertical aluminium rods of
square cross section of side 2 cm, xed to the
mild steel base.
A heavy base was used to counter the large hammering torques that come about when the lower
limbs ip over.
4
Numerical Analysis
The two second order ODEs were solved using Runge-Kutta 4th order integration method by converting them to four rst order ODEs in θ , p , θ , p .
1
θ˙1
=
pθ˙ 1
θ˙2
=
pθ˙ 2
=
=
2
θ1
θ2
pθ1
2c2 c4 sin(θ1 )+c23 p2θ sin(θ1 −θ2 ) cos(θ1 −θ2 )+2c2 c3 p2θ sin(θ1 −θ2 )−c3 c5 cos(θ1 −θ2 ) sin(θ2 )
1
2
c23 cos2 (θ1 −θ2 )−4c1 c2
pθ2
2c1 c5 sin(θ2 )−c23 p2θ sin(θ1 −θ2 ) cos(θ1 −θ2 )−2c1 c3 p2θ sin(θ1 −θ2 )−c3 c4 cos(θ1 −θ2 ) sin(θ1 )
2
1
c23 cos2 (θ1 −θ2 )−4c1 c2
5
The time step used for integration was 0.001 seconds. As the expression in the denominators
of the ODEs for p˙ never approaches zero and energy conservation bounds the values of p , the
integration is not singular. Hence this time step is good enough for evolving the system based on
RK4 routine.
Approximate values of c s were found for the current apparatus and were used in the numerical
simulation. Here we tabulate the normalised values (normalised wrt c = 1)
θi
θi
i
3
c1
2.16
c2
0.27̇
c3
1
c4
172.5
c5
32.6̇
Time evolution of the system was noted for dierent initial conditions. To see if a given initial
condition results in chaos, evolution of the system for nearly identical initial conditions (diering in
one of the angles by 0.0001 radians) was also done and the results were compared.
Some initial conditions were found to be non-chaotic while some others were found to be chaotic.
As an example, we show here two sets of initial conditions with the same energy (E = 0), one
of which results in non-chaotic evolution of the system while the other shows chaos.
4.1
Non-Chaotic Evolution
A set of initial conditions that shows non-chaotic evolution is θ = θ = , p = p = 0. Total
energy of the system in this case is zero. Evolution of θ and θ with time is shown for two closely
separated (4θ = 0.0001) initial conditions.
1
1
2
π
2
θ1
10
12
θ2
2
2
2
1.5
1.5
1
1
0.5
0.5
Angle (rad)
Angle (rad)
2
0
0
-0.5
-0.5
-1
-1
-1.5
-1.5
-2
-2
0
2
4
6
8
10
12
14
16
18
20
0
Time (s)
2
4
6
8
14
Time (s)
16
18
20
Evolution of θ
Evolution of θ
We see that the two evolution curves virtually lie on top of each other. This shows that the
system is non-chaotic for this inital state
1
2
6
4.2
Chaotic Evolution
Another initial state of the system that has the same value of energy is θ = , θ = , p =
p = 0. Even though the initial energy is the same, the evolution of the system is qualitatively
very dierent. Evolution of θ and θ with time is shown for two closely separated (4θ = 0.0001)
initial conditions.
1
π
2
2
3π
2
θ1
35
40
θ2
2
2
2
100
1.5
0
1
-100
0.5
-200
Angle (rad)
Angle (rad)
1
0
-300
-0.5
-400
-1
-500
-1.5
-600
-2
-700
0
5
10
15
20
Time (s)
25
30
35
40
0
5
10
15
20
Time (s)
25
30
Evolution of θ
Evolution of θ
We see that the evolutions for the initial states are very dierent, even qualitatively. This shows
that the evolution is chaotic.
1
5
2
Future work
The current apparatus provides scope for a lot more of work.
The theoretical treatment could be expanded to include damping. This would give a more
realistic model for the system. Once this is developed, at least for the non-chaotic initial conditions,
comparison could be done between theoretical and experimental evolution of the system.
Experimental evolution of the system could also be studied quantitatively. Two possible ways for
analysis involve image processing and the use of motion sensors. Tips of the limbs could be marked
with easily identiable - for example reective - material. A video could be made of the evolution
with a camera having suciently high frame rate. Tracking software could be used to nd the
values of the state variables from the videos. Alternatively, motion sensors such as accelerometers
and gyroscopes could be attached to the limbs to record the positions as a function of time.
Also, in the current work we have not investigated the sets of initial conditions that result in
chaos. A more detailed analysis could be done to see what initial angular momenta and energies
result in chaos.
7
6
Acknowledgement
We would like to thank Dr. M.K. Verma for guiding us through the project. We also express
our heartfelt gratitude for Mr. Omprakash and others at the Physics Workshop for their eort in
constructing the experimental apparatus.
References
[1] Wikipedia. Double pendulum wikipedia, the free encyclopedia, 2010. [Online; accessed
8-November-2010].
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