A novel graphic presentation and fractal characterisation of poincare solutions of harmonaically excited pendulum

Abstract

"The extensive completed research and continuous study of pendulum is due to its scientific and engineering importance. The present study simulate the Poincare solutions of damped, nonlinear and harmonically driven pendulum using FORTRAN 90 coded form of the popular fourth and fifth order Runge-Kutta schemes with constant time step. Validation case studies were those reported by Gregory and Jerry (I 990) for two damping qualities q1,q2= 2,4), fixed drive amplitude and frequency (g = 1.5,ώD= 2/3). A novel graphic presentation of the displacement and velocity components of the Poincare solutions for 101-eases each drawn from the parameters spaces 2≤q≤4 and 0.9≤ g≤ 1.5 at 100-equal steps were characterised using the fractal disk dimension analysis. Corresponding validation results compare well with reported results of Gregory and Jerry (1990). There is observed quantitative variations in the corresponding consecutive Poincare solutions prescribed by Runge-Kutta schemes with increasing number of excitation period however the quality of the overall Poincare section is hard to discern. Non uniform variation of scatter plots per area of solutions space characterised chaotic and periodic responses as against average uniform variation for a random data set The plots of periodic response distribute restrictedly on the solutions space diagonal while probabilities of chaotic responses on the studied parameters space is between 21.5% and 70.6%. Estimated fractal disk dimension variation is in the range 0.00≤ Df≤ 1.8 I for studied cases. The study therefore has demonstrated the utility of the novel graphic plots as a dynamic systems characterising tool.

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