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