1. Problem 10. Inverted Pendulum

2. 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. Problem 10.

3. Introduction Inverted pendulum - center of mass is above its point of suspensionInverted pendulum - center of mass is above its point of suspension Achieving stabilisation – pendulum suspension point vibrating!Achieving stabilisation – pendulum suspension point vibrating! Principal parameters:Principal parameters: lenghtlenght frequencyfrequency amplitudeamplitude

4. Introduction cont.

5. Experimental approach ApparatusApparatus ConstructionConstruction Measurements:Measurements: Pendulum angle in timePendulum angle in time Stabilisation conditions:Stabilisation conditions: amplitude vs. pendulum length amplitude vs. frequency Double pendulumDouble pendulum

6. Speaker (subwoofer)Speaker (subwoofer) Function generatorFunction generator AmplifierAmplifier StroboscopeStroboscope Pendula (wooden)Pendula (wooden) Apparatus

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

8. Construction 44.555.566.577.5 Lengths [cm]: Density [kg/m 3 ]: 626

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

10. Double pendulum

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

12. Forces on pendulum

13. Equation of motion In noninertial pendulum system:In noninertial pendulum system: Inertial acceleration:Inertial acceleration: gravity componentgravity component periodical acceleration of suspension pointperiodical acceleration of suspension point

14. Equation of motion cont. Resistance force – estimated to be linear to angular velocityResistance force – estimated to be linear to angular velocity “inverted” inverted pendulum measurements“inverted” inverted pendulum measurements

15. Equation of motion cont.  equation of motion: Analytical solution very difficultAnalytical solution very difficult Numerical solution – Euler methodNumerical solution – Euler method

16. Equation of motion cont.

17. Stability conditions  From equation of motion solutions stability determination:

18. Stability conditions cont.

19. Agreement between model and measurements relatively goodAgreement between model and measurements relatively good Discrepancies due to:Discrepancies due to: errors in small amplitude measurementserrors in small amplitude measurements speaker characteristics (higher harmonics generation)speaker characteristics (higher harmonics generation) nonlinear damping...nonlinear damping...

20. we determined and experimentaly prove stability parameterswe determined and experimentaly prove 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 inverted pendulumwe managed to stabilise multiple inverted pendulum Conclusion