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## The Inverted Pendulum

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**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 Formulating the problem Forces Acting on the Pendulum • Gravity • Force of the pivot • [Friction]**m**l θ A, ω Equation of Motion • Newton’s Second Law • Linear motion: • Circular motion: • Equation of Motion**m**l θ Torques • Gravity: • Oscillating pivot: • Force: • Torque:**m**θ θ m Equation of Motion Gravity term Pivotterm For example:**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**For small values of θ:**Reparameterization Reparameterize: Let: Mathieu's Equation**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**(fixed)**(depends on ω) Physical Region Our values of ε and δ Stability condition: (approximating A << l) ε = 0.067**How We Built It**Some of the things we originally used to stabilize the pendulum**Data**Speed 1 (8 Hz) Speed 2 (13 Hz) Speed 3 (23 Hz) Speed ~2.8 (22 Hz) 25**DATA AQUISITION**1415 RPM = 23.58 Hz**Problems we had**Finding right amplitude and frequency Finding right equipment Stabilizing apparatus Getting apparatus to right frequency Soldering 27**Thank You**• Professor Page • Wei Chen • PHY 210 Colleagues • Sam Cohen • Mike Peloso