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
Density [kg/m3]:
626
7
7.5
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
&− acceleration of
h&
suspension point
Fr − resistance
Fy − inertial force acting
on the center of mass
Equation of motion
• In noninertial pendulum system:
1
1
&= − Fy l sin ϕ − Fr l
I sϕ&
2
2
l − pendulum lenght
ϕ − angle between
pendulum and y axis
I s − pendulum moment of
inertia
Fr − resistance
Fy − inertial force acting
on the center of mass
• Inertial acceleration:
• gravity component
• periodical acceleration of suspension
point
Equation of motion cont.
• Resistance force – estimated to be linear to
angular velocity
• “inverted” inverted pendulum measurements
30
20
ϕ max ~ e
angle [°]
10
−
β eff
2
t
0
β eff − damping coefficien t
β eff = 3 . 0 s -1
-10
ϕ max − angular amplitude
-20
-30
-0,5
0,0
0,5
1,0
1,5
time [s]
2,0
2,5
3,0
Equation of motion cont.
⇒ equation of motion:
2


ω
A
2
&+ β eff ϕ&+ ω0 1 −
ϕ&
sin ωt  sin ϕ = 0
g


3 l
2 g
point amplitude
point angular frequency
ω 02 − parameter : ω 02 =
A − suspension
ω − suspension
• Analytical solution very difficult
• Numerical solution – Euler method
Equation of motion cont.
0,6
0,4
angle [rad]
0,2
l = 5.0 cm
ω = 685 rad/s
0,0
2 A = 4.5 mm
-0,2
-0,4
-0,6
0,0
0,2
0,4
0,6
0,8
time [s]
1,0
1,2
1,4
1,6
Stability conditions
⇒From equation of motion solutions stability
determination:
200
180
l = 5 . 0 cm
frequency [Hz]
160
140
120
100
80
60
40
0,001
0,002
0,003
0,004
2A [m]
0,005
0,006
Stability conditions cont.
9
8
length [cm]
7
6
5
4
freq = 100 Hz
3
1,8
2,0
2,2
2,4
2,6
2A [mm]
2,8
3,0
3,2
3,4
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