Post on 20-Jan-2016
Problem 10.Problem 10.
Inverted Inverted PendulumPendulum
It is possible to stabilise an inverted It is possible to stabilise an inverted pendulum. It is even possible to stabilise pendulum. It is even possible to stabilise inverted multiple pendulum (one pendulum inverted multiple pendulum (one pendulum on the top of the other). Demonstrate the on the top of the other). Demonstrate the stabilisation an determine on which stabilisation an determine on which parameters this depends.parameters this depends.
Problem 10.Problem 10.
IntroductionIntroduction
• Inverted pendulum - center of mass is Inverted pendulum - center of mass is
above its point of suspensionabove its point of suspension• Achieving stabilisation – pendulum Achieving stabilisation – pendulum
suspension point vibrating!suspension point vibrating!• Principal parameters:Principal parameters:
• lenghtlenght• frequencyfrequency• amplitudeamplitude
Introduction Introduction cont.cont.
Experimental Experimental approachapproach• ApparatusApparatus• ConstructionConstruction• Measurements:Measurements:
• Pendulum angle in timePendulum angle in time• Stabilisation conditions:Stabilisation conditions:
amplitude vs. pendulum lengthamplitude vs. pendulum lengthamplitude vs. frequencyamplitude vs. frequency
• Double pendulumDouble pendulum
• Speaker Speaker (subwoofer)(subwoofer)
• Function Function generatorgenerator
• AmplifierAmplifier
• StroboscopeStroboscope
• Pendula Pendula (wooden)(wooden)
ApparatusApparatus
• Speaker – low harmonics generationSpeaker – low harmonics generation
• Audio range amplifierAudio range amplifier
• Stroboscope – accurate frequency Stroboscope – accurate frequency measurementmeasurement
• Point of support amplitude measured Point of support amplitude measured with (šubler) with (šubler)
• Multiple measurements for error Multiple measurements for error determinationdetermination
Apparatus Apparatus cont.cont.
ConstructionConstruction
4 4.5 5 5.5 6 6.5 7 7.5
Lengths Lengths [cm]:[cm]:
Density [kg/mDensity [kg/m33]: ]:
626
MeasurementsMeasurements
• Stability – pendulum returns to Stability – pendulum returns to upward orientationupward orientation
• measurements of boundary conditions:measurements of boundary conditions:frequency vs. amplitudefrequency vs. amplitude
length vs. amplitudelength vs. amplitudeangle in time (two cases); angle in time (two cases);
• inverted penduluminverted pendulum• ““inverted” inverted inverted” inverted
pendulum – for drag pendulum – for drag determinationdetermination
Double pendulumDouble pendulum
Theoretical approachTheoretical approach
• Pendulum – tends to state of minimal Pendulum – tends to state of minimal
energyenergy• Upward stabilisation possible if enough Upward stabilisation possible if enough
energy is given at the right timeenergy is given at the right time• Formalism – two possibilities:Formalism – two possibilities:
• equation of motionequation of motion• energy equation – Lagrangian energy equation – Lagrangian
formalism formalism • Forces approach – more intuitive: Forces approach – more intuitive:
mass ofcenter on the
acting force inertial
resistance
point suspension
ofon accelerati
axisy and pendulum
between angle
lenght pendulum
y
r
F
F
h
l
Forces on pendulumForces on pendulum
Equation of motionEquation of motion
• In noninertial pendulum system:In noninertial pendulum system:
• Inertial acceleration:Inertial acceleration:• gravity componentgravity component• periodical acceleration of periodical acceleration of
suspension pointsuspension point
lFlFI rys 2
1sin
2
1
mass ofcenter on the
acting force inertial
resistance
inertia
ofmoment pendulum
axisy and pendulum
between angle
lenght pendulum
y
r
s
F
F
I
l
time [s]
-0,5 0,0 0,5 1,0 1,5 2,0 2,5 3,0
an
gle
[°]
-30
-20
-10
0
10
20
30
Equation of motion Equation of motion cont.cont.
• Resistance force – estimated to be linear Resistance force – estimated to be linear
to angular velocityto angular velocity• ““inverted” inverted pendulum inverted” inverted pendulum
measurementsmeasurements
teff
e 2max ~
amplitudeangular
s 0.3
tcoefficien damping
max
1-
eff
eff
Equation of motion Equation of motion cont.cont.
equation of motion:equation of motion:
0sinsin12
20
tg
Aeff
frequencyangular point suspension
amplitudepoint suspension
2
3 :parameter 2
020
A
g
l
• Analytical solution very difficultAnalytical solution very difficult• Numerical solution – Euler methodNumerical solution – Euler method
time [s]
0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,6
angl
e [r
ad]
-0,6
-0,4
-0,2
0,0
0,2
0,4
0,6
Equation of motion Equation of motion cont.cont.
mm 5.42
rad/s 685
cm 0.5
A
l
2A [m]0,001 0,002 0,003 0,004 0,005 0,006
freq
uenc
y [H
z]
40
60
80
100
120
140
160
180
200
Stability conditionsStability conditions From equation of motion solutions From equation of motion solutions
stability determination:stability determination:
cm 0.5l
Stability conditions Stability conditions cont.cont.
2A [mm]
1,8 2,0 2,2 2,4 2,6 2,8 3,0 3,2 3,4
leng
th [c
m]
3
4
5
6
7
8
9
Hz 100freq
Stability conditions Stability conditions cont.cont.
• Agreement between model and Agreement between model and
measurements relatively goodmeasurements relatively good• Discrepancies due to:Discrepancies due to:
• errors in small amplitude errors in small amplitude
measurementsmeasurements• speaker characteristics (higher speaker characteristics (higher
harmonics generation)harmonics generation)• nonlinear damping...nonlinear damping...
• we determined and experimentaly we determined and experimentaly
prove stability parametersprove stability parameters• mass is not a parametermass is not a parameter• theoretical analisis match with resultstheoretical analisis match with results• we managed to stabilise multiple we managed to stabilise multiple
inverted penduluminverted pendulum
ConclusionConclusion