Fractal Characterization and [so-mapping of Parameter Plane of Harmonically Excited Pendulum

Abstract-

This study utilized fractal disk dimension characterization to investigate the system response of a nonlinear, harmonically driven pendulum of mass, m (or weight. W) and length I over a well-defined parameter plane of excitation amplitude versus damping factor. The system response was simulated at three drive ffequencies (symbol,*) of 1/3, 2/3 and 313. A Runge-Kutta fourth order algorithm, coded in FORTRAN programming language was used to simulate the System response over a 101 by 101 parameter plane with the forcing amplitude (symbol, g) varying between 0.9 and 1.5 and the damping factor (symbol, q) varying between 2.0 and 4 0 Initial conditions for both angular displacement and angular velocity for the simulation at each of the 10,201 points of the plane was set at 0 0 The Po~ncare section obtained at each node on the parameter plane at the end of 2000 steady-state drive periods was characterked to estimate the fractal disk dimension of the system response at that node. The fractal dimensions of three randomly selected nodes on the parameter plane are returned by computer code. Further, the nodes whose fractal dimensions re within an upper and lower tolerance of 1 % of the frectal dimensions of these randomly selected nodes are also returned by code and they, along with the random fractal dimensions of the three randomly selected nodes are plotted on the parameter plane In the form of an equal potential (system response) plot This plotting on the parameter plane to show the points that correspond to each of the three bands is done in order to achieve the iso-mapping on the plane. The steady-state response of a nonlinear, harmonically driven system depends on the drive frequency, the forcing amplitude and the damping factor. With a lower drive frequency of IM, higher fractal dimensions (above 1.0) were obtained with a combination of h~gh damping factor and low to medium forcing amplitude; with the median drive frequency of 213, fractal dimensions above 1.0 were obtained with a combination of low damping factor and medium to high forcing amplitude; while with the higher drive frequency of 313, a fractal dimension that is up to 1.0 was not obtainable at any of the 10.201 nodes on the parameter plane. We note that as the forcing amplitude 1s increased from 0.9. In order to assure early onset (as from 04.044) of higher fractal dimensions that are above 1.0, lower drive frequency (113) must be used The resub of this study could be used for engineering design, education and fashion design