3 years ago

16.338 Lab Report #2: Kapitsa's Stable Inverted Pendulum - LMPT

16.338 Lab Report #2: Kapitsa's Stable Inverted Pendulum - LMPT

0.15 0.1 0.05 θ (rad) 0

0.15 0.1 0.05 θ (rad) 0 −0.05 −0.1 −0.15 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 t (s) Figure 4: Time history of θ 10 3 ω (RPM) 10 2 10 1 |θ(jω)| 10 0 10 −1 10 −2 10 −3 0 500 1000 1500 2000 2500 3000 Figure 5: Power spectrum of θ 7

150 Simulation Theory Experiment 100 ω slow (RPM) 50 0 1000 1200 1400 1600 1800 2000 2200 2400 2600 2800 3000 ω fast (RPM) Figure 6: Frequency of pendulum oscillations vs. frequency of excitation 8 Conclusion The critical stabilizing forcing frequency has been determined by two approximate analytical methods and numerically. The analytical methods consisted of an approximate solution specific to the expected behavior of the inverted pendulum, and an approximate stability boundary taken from the broader theory of the Mathieu equation. The two analytical and the numerical predictions for the stabilizing frequency agreed very well with each other, but there was a larger than desired difference between the predictions and the experimental measurements. Frictional effects and other unmodeled phenomena have been considered by qualitative argument and by simulation. While there were a number of unmodeled behaviors occuring during the experiment, it is our conclusion that the critical component is the pivot point. The interaction of the Coulomb friction, the backlash and the forcing created very complex behavior. Simple harmonic motion of the slow oscillation was not observed as our analytical models predicted and the crude measurement devices (strobe light and a stopwatch) were not adaptable take the necessary measurements to record this complexity. We suggest that the use of a bearing at the pivot point be investigated for future incarnations of the experiment. The final word on the laboratory investigation is a good one. We were able to treat the difficult nonlinear system model by three different methods and we obtained excellent agreement and we were able to deduce the critical stabilizing frequency from a rather crude experiment, obtaining a value within 21% of the predictions. 8

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