Page - 262 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python 3.6, Volume Second Edition

262 8 SolvingOrdinaryDifferentialEquations Fig. 8.30 Effect of lineardamping 8.4.10 IllustrationofLinearDampingwithSinusoidalExcitation Wenowextendthepreviousexampletoalso involvesomeexternaloscillatingforce on the system:F(t) = Asin(wt). Driving a car on a road with sinusoidal bumps might give such an external excitation on the spring system in the car (w is related to thevelocityof thecar). WithA=0.5andw=3, import math w = 3 A = 0.5 F = lambda t: A*math.sin(w*t) we get the graph in Fig. 8.31. The striking difference from Fig. 8.30 is that the oscillationsstartoutasadampedcost signalwithoutmuchinfluenceof theexternal force, but then the free oscillations of the undamped system (cost) u′′ +u = 0 die out and the external force 0.5sin(3t) induces oscillations with a shorter period 2π/3.Youareencouragedto playaroundwith a largerAandswitch froma sine to acosine inF andobserve theeffects. Ifyou lookthisup inaphysicsbook,youcan findexactanalytical solutions to thedifferentialequationproblemin thesecases. A particularly interesting case arises when the excitation force has the same frequency as the free oscillations of the undamped system, i.e., F(t) = Asint. With the same amplitudeA= 0.5, but a smaller dampingb= 0.1, the oscillations inFig.8.31becomesqualitativelyverydifferentastheamplitudegrowssignificantly larger over some periods. This phenomenonis called resonance and is exemplified in Fig. 8.32. Removing the damping results in an amplitude that grows linearly in time.