Fractal characterization and iso-mapping of parameter plane of harmonically excited pendulum

Abstract

"The motivation for the present study is derived from the fact that time rnanagernent is an integral part of good engineering practice. The present study investigated the quantification of the required computation time using two nonlinear and harmonically excited oscillators (Pendulum and Duffing) as case studies. Simulations with personal computer were effected for Runge-Kutta schemes (RK2, RK3, RK4, RK5, RK5M) and one blend (RKB) over thirty five thousand and ten excitation periods consisting the unsteady and steady solutions. The need for validation of the develope,d FORTRAN90. codes by comparing Poincare results with their conterpart from the literature informed the choice of simulation parameters However, the simulation time was monitored at three lengths of excitation period (15000, 25000 and 35000) using the current time subroutine call command. The validation Poincare results obtained for all the schemes including RKB compare well with the counterpart available in the literature for both Pendulum and Duffing. The actual computation time increases with increasing order of scheme. But suffered a decrease for the blended scheme. The diffencerence in computation time required between RK5 and RK5M is negligible for all studied cases. The actual computational time for Duffing (5-33seconds) remain consistently higher for corresponding Pendulum (3-23seconds) with difference (2-10seconds). Interestingly, the quantitative difference betweenthe corresponding normalised computation time for systems and schemes is insignificant. It is insensitive to systems and schemes and fonmed a simple average ratio{ (1.0) :(1.5) : (2.0) : (3.1) :(3.1) : (2.4)} for RK2, RK3, RK4, RK5, RK5M and RKB respectively. It is concluded that the end justified the means provided that computation accuracy is assured using the higher order scheme (with higher computational time ratio). "