Inverted Pendulum

Problem 10.
Inverted Pendulum

Problem 10.
It is possible to stabilise an inverted
pendulum. It is even possible to stabilise
inverted multiple pendulum (one pendulum
on the top of the other). Demonstrate the
stabilisation an determine on which
parameters this depends.

Introduction
• Inverted pendulum - center of mass is
above its point of suspension
• Achieving stabilisation – pendulum
suspension point vibrating!
• Principal parameters:
• lenght
• frequency
• amplitude

Introduction cont.

Experimental approach
• Apparatus
• Construction
• Measurements:
• Pendulum angle in time
• Stabilisation conditions:
amplitude vs. pendulum length
amplitude vs. frequency
• Double pendulum

Apparatus
• Speaker
(subwoofer)
• Function generator
• Amplifier
• Stroboscope
• Pendula (wooden)

Apparatus cont.
• Speaker – low harmonics generation
• Audio range amplifier
• Stroboscope – accurate frequency
measurement
• Point of support amplitude measured with
(šubler)
• Multiple measurements for error
determination

Construction
Lengths [cm]:
4
4.5
5
5.5
6
6.5
7
7.5
Density [kg/m 3 ]:
626

Measurements
• Stability – pendulum returns to upward
orientation
• measurements of boundary conditions:
frequency vs. amplitude
length vs. amplitude
angle in time (two cases);
• inverted pendulum
• “inverted” inverted pendulum –
for drag determination

Double pendulum

Theoretical approach
• Pendulum – tends to state of minimal energy
• Upward stabilisation possible if enough energy
is given at the right time
• Formalism – two possibilities:
• equation of motion
• energy equation – Lagrangian formalism
• Forces approach – more intuitive:

Forces on pendulum
l − pendulum lenght
ϕ − angle between
pendulum and y axis
h&
− acceleration of
F
F
r
y
suspension point
− resistance
− inertial force acting
on the center of mass

Equation of motion
• In noninertial pendulum system:
I
s
1
ϕ&= − Fyl
sinϕ
−
2
• Inertial acceleration:
• gravity component
• periodical acceleration of suspension
point
1
2
F l
r
l − pendulum
ϕ − angle
I
s
F
F
r
y
pendulum
− pendulum
inertia
−
resistance
− inertial force acting
on the
lenght
between
and
moment
center of
y axis
of
mass

Equation of motion cont.
• Resistance force – estimated to be linear to
angular velocity
• “inverted” inverted pendulum measurements
30
20
10
eff
−
e 2
ϕ max ~ β
t
angle [°]
0
β
eff
−
damping
coefficien
t
-10
-20
β
ϕ
eff
max
= 3.0 s
-1
− angular
amplitude
-30
-0,5 0,0 0,5 1,0 1,5 2,0 2,5 3,0
time [s]

Equation of motion cont.
⇒ equation of motion:
2
2⎛ Aω
⎞
& ϕ + β & ϕ + ω0 ⎜1
− sin ω
⎟
eff
t sin ϕ =
⎝ g ⎠
0
ω
2
0
− parameter
A − suspension
ω − suspension
:
ω
2
0
point
point
=
3
2
l
g
amplitude
angular
frequency
• Analytical solution very difficult
• Numerical solution – Euler method

Equation of motion cont.
0,6
0,4
angle [rad]
0,2
0,0
l = 5.0 cm
ω = 685 rad/s
2A
= 4.5 mm
-0,2
-0,4
-0,6
0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,6
time [s]

Stability conditions
⇒From equation of motion solutions stability
determination:
200
180
l
=
5.0
cm
160
frequency [Hz]
140
120
100
80
60
40
0,001 0,002 0,003 0,004 0,005 0,006
2A [m]

Stability conditions cont.
9
8
7
length [cm]
6
5
4
freq
= 100
Hz
3
1,8 2,0 2,2 2,4 2,6 2,8 3,0 3,2 3,4
2A [mm]

Stability conditions cont.
• Agreement between model and measurements
relatively good
• Discrepancies due to:
• errors in small amplitude measurements
• speaker characteristics (higher
harmonics generation)
• nonlinear damping...

Conclusion
• we determined and experimentaly prove
stability parameters
• mass is not a parameter
• theoretical analisis match with results
• we managed to stabilise multiple inverted
pendulum