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Computer simulations in physics

The parametrically driven pendulum

An ordinary rigid planar pendulum whose axis is driven periodically in the vertical direction is a paradigm of contemporary nonlinear dynamics. This rather simple mechanical system is also interesting because the differential equation of the pendulum is frequently encountered in various problems of modern physics. Mechanical analogues of physical systems allow a direct visualization of motion and thus can be very useful in gaining an intuitive understanding of complex phenomena.

Depending on the frequency and amplitude of forced oscillations of the suspension point, this seemingly simple mechanical system exhibits a rich variety of nonlinear phenomena characterized by amazingly different types of motion. Some modes of such parametrically excited pendulum are quite simple indeed and agree well with our intuition, while others are very complicated and counterintuitive. A highly interactive stand-alone computer program that simulates the parametrically driven pendulum is developed for a PC running under Windows 95/98/ME/2000/XP operating system. This program is included in the software package Nonlinear Oscillations which includes a set of programs that visualize the motion of several simple nonlinear mechanical oscillatory systems.

We note that the program Pendulum with the Vertically Driven Pivot is supplied with several sets of predefined examples illustrating many interesting modes of the pendulum's behavior. These include, in particular, the ordinary parametric resonance, the phenomenon of dynamic stabilization of the inverted pendulum described in the paper On the dynamic stabilization of an inverted pendulum (PDF), many other simple and complicated, regular and chaotic, rotational, oscillatory and combined modes. There are examples that show destabilization of the (dynamically stabilized) equilibrium position (the "flutter" mode), and subharmonic resonances of different orders ("multiple-nodding" oscillations) described in the paper Subharmonic resonances of the parametrically driven pendulum (PDF).

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