The inverted pendulum

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Abstract
The condition for stability of a pendulum, maintained in an inverted position by vertical oscillation of the point of support, is derived by replacing sinusoidal motion of the support, as usually assumed, by a sawtooth or a parabolic approximation. The physical principles are unchanged but the mathematics is so much simpler that other customary approximations are unnecessary, and the complete range of oscillatory amplitude and frequency can be surveyed. The lower limit of stability agrees with other treatments and the upper limit, usually neglected, is determined.

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