On the dynamic stabilization of an inverted pendulum

American Journal of Physics

Volumn 69, Issue 7, 2001, Pages 755-768

  Butikov, Eugene I ^a  

^a ST PETERSBURG STATE UNIVERSITY   (Russian Federation)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0035595759     PISSN: 00029505     EISSN: None     Source Type: Journal    
DOI: 10.1119/1.1365403     Document Type: Article

References (27)

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26. 2. Figure 12 also shows the lower-frequency boundary of parametric resonance (curve 1, or the left branch of curve 3 in the presence of friction), which is absent in Fig. 4 of Ref. 16.
27. The time variations of the force of inertia give a clear physical explanation of the growth of initially small oscillations at conditions of parametric resonance. When the oscillating pivot is below its middle position, this additional force is directed downward, and vice versa. We can treat the effect of this varying force as a periodic modulation of the gravitational force. Let the pendulum move from the utmost deflection toward the lower equilibrium position while the pivot in its constrained oscillation is below the midpoint. Due to the additional apparent gravity the pendulum gains a greater speed than it would have gained in the absence of the pivot's motion. During the further motion of the pendulum away from the equilibrium position, the pivot is above its midpoint, so that the force of inertia reduces the apparent gravity. Thus the pendulum reaches a greater angular displacement than it would have reached otherwise. During the second half-period of the pendulum's motion the swing increases again, and so on, until the stationary motion is established due to violation of the resonance conditions at large swing.