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George Mason UniversityZakaria Tarik ZerhouniSource: Home > Project 3 > Step 3Math 447: Numerical Analysis
The oscillating pivot. The aim of Computer Problem 6.3.6 was to alter the differential equation modeling our pendulum to create a damped pendulum with an oscillating pivot. The differential equation used was, \(y''+dy'+(\frac gl +A\cos 2\pi t)\sin y = 0\). Setting \( y_1 = y \) and \( y_2 = y'\) our system of differential equations is \( y_1' = y_2 \), \(y_2' = - (\frac gl +A\cos 2\pi t)\sin y - dy' \). This oscillating pivot allows us to choose the forcing parameter \(A\) in order to control the pendulum motion. Imagine balancing a baton in one's palm. The oscillating pivot will work to accomplish the two dimensional likeness of this notion. When this is achieved for an inverted (upright) pendulum, it will never have angular positions outside of the interval \( (\frac{\pi}{2}, \frac{3\pi}{2} )\). And when our pendulum is hanging or facing downwards, stability will denote that no angular position will be attained outside of \( (\frac{-\pi}{2}, \frac{\pi}{2} )\). By sampling for parameters which caused stability relative to our initial angular displacement, we were able to use interval halving to discover where the forcing parameter \(A\) changed from causing stability to unstable behaviour. This was on the interval of time from \(0\) to \(60\) and using a step size of \(h = 0.005\). However, the programming used allows any interval to be judged by this criteria (though the computational time required for a longer time interval at this step size may lead to some irritation). What was discovered was that when the pendulum was initiated at the initial conditions of \(y = 3.1\) with initial velocity \(0\), then the interval on which the inverted pendulum was stabilized was in radians \( (18.361495814968876, 25.737092048344476 ) \). The code used to compute this interval can be found here. Videos at each end of the interval are available below and provide the associated MatLab code used to create them. Stable at A = 18.361495814968876Click here to see the code used to generate these animations. Click here to download the video. Stable at A = 25.737092048344476Click here to see the code used to generate these animations. Click here to download the video.
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