Adaptive Time Steps Runge-Kutta Methods: Comparative Analysis of Simulation Time in Nonlinear and Harmonically Excited Pendulum and Duffing Oscillators
Time management without integrity compromise is an integral part of good engineering practice. The present study investigated for the required computation time in nonlinear and harmonically excited oscillators (Pendulum and Duffing). Adaptive time steps simulation of their governing equations with personal computer were performed by Runge-Kutta methods (RK2, RK3, RK4, RK5, RK5M) and one blend (RKB) comprising unsteady and steady solutions. The respective Pendulum and Duffing Poincare result at damp quality (4, 0.168), excitation amplitude (1.5, 0.21) and excitation frequency (2/3, 1.0) were used to validate the codes developed in FORTRAN environment. Actual simulation time was monitored at three different lengths of excitation periods (40000, 80000 and 120000) using the current time subroutine call command. Except for RK2, the validation Poincaré results compare well with the counterpart available in the literature for the oscillators. The actual computation time decrease rapidly with increasing order of Runge-Kutta method, but suffered relative increase for the blended method. The difference in computation time required between RK5 and RK5M is negligible for all studied cases. The Pendulum required longer actual computation time (4-115 seconds) than Duffing (3-52 seconds). The respective normalized range of time step for Pendulum and Duffing formed a simple average ratios {(1.0): (7.5): (13.3): (26.0): (24.0): (29.7)} and {(1.0): (3.7): (5.1): (9.1): (8.0): (11.2)} for RK2, RK3, RK4, RK5, RK5M and RKB. It is concluded that substantial time management can be achieved by the Runge-Kutta methods except RK2 that permitted strictly shorter simulation time steps leading to longer actual simulation time and consequently simulated largely an unacceptable Poincare result.