A simple qualitative physical explanation is suggested for the phenomenon of dynamic stabilization of the inverted rigid planar pendulum whose pivot is constrained to oscillate with a high frequency in the vertical direction. A quantitative theory based on the suggested approach is developed. A computer program simulating the physical system supports the analytical investigation. The simulation reveals subtle details of the motion and aids the analytical study of the subject in a manner that is mutually reinforcing.

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A similar stability diagram for the parametrically driven pendulum is given by Fig. 4 of Ref. 16. The curves 2 and 3 of this diagram are given by the approximate equations a/l=0.450∓1.799(ω0/ω)2 respectively, and tend to 0.450 in the limit of high driving frequencies (or zero gravity). However, the corresponding curves 2 and 4 in Fig. 12 of the present paper are given by exact equations, Eqs. (18) and (19). Approximate expansion of these equations for high driving frequencies yields a/l=0.454∓1.681(ω0/ω)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.
24.
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.
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