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The Inverted Pendulum

The Inverted Pendulum. 1. Arthur Ewenczyk Leon Furchtgott Will Steinhardt Philip Stern Avi Ziskind PHY 210 Princeton University. m. l. θ. A, ω. Formulating the problem. Pendulum Mass ( m ) Length ( l ) Oscillating Pivot Amplitude ( A ) Frequency ( ω). m. l. θ. A, ω.

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The Inverted Pendulum

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  1. The Inverted Pendulum 1 Arthur Ewenczyk Leon Furchtgott Will Steinhardt Philip Stern Avi Ziskind PHY 210 Princeton University

  2. m l θ A, ω Formulating the problem • Pendulum • Mass (m) • Length (l) • Oscillating Pivot • Amplitude (A) • Frequency (ω)

  3. m l θ A, ω Formulating the problem Forces Acting on the Pendulum • Gravity • Force of the pivot • [Friction]

  4. m l θ A, ω Equation of Motion • Newton’s Second Law • Linear motion: • Circular motion: • Equation of Motion

  5. m l θ Torques • Gravity: • Oscillating pivot: • Force: • Torque:

  6. m θ θ m Equation of Motion Gravity term Pivotterm For example:

  7. Physical Intuition Pivot term (dominates for large values of ω) Inertial (lab)frame Non-inertialframe (of pendulum) Butikov, Eugene. On the dynamic stabilization of an inverted pendulum. Am. J. Phys., Vol. 69, No. 7, July 2001

  8. For small values of θ: Reparameterization Reparameterize: Let: Mathieu's Equation

  9. Stability stable stable Regular Pendulum: Inverted Pendulum: unstable unstable Smith, Blackburn, et. al.Stability and Hopf bifurcations in an inverted pendulum. Am J. Phys. 60 (10), Oct 1992 Matlab Plot

  10. (fixed) (depends on ω) Physical Region Our values of ε and δ Stability condition: (approximating A << l) ε = 0.067

  11. Sample Plots

  12. HOW WE BUILT IT

  13. HOW WE BUILT IT 13

  14. HOW WE BUILT IT

  15. Why The Driver Didn’t work

  16. HOW WE BUILT IT

  17. HOW WE BUILT IT

  18. How We Built It Some of the things we originally used to stabilize the pendulum

  19. How We Built It

  20. How we built it 20

  21. HOW WE BUILT IT

  22. DATA AQUISITION

  23. Data Acquisition

  24. DATA AQUISITION

  25. Data Speed 1 (8 Hz) Speed 2 (13 Hz) Speed 3 (23 Hz) Speed ~2.8 (22 Hz) 25

  26. DATA AQUISITION 1415 RPM = 23.58 Hz

  27. Problems we had Finding right amplitude and frequency Finding right equipment Stabilizing apparatus Getting apparatus to right frequency Soldering 27

  28. Thank You • Professor Page • Wei Chen • PHY 210 Colleagues • Sam Cohen • Mike Peloso

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