Page - 146 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python

146 4 SolvingOrdinaryDifferentialEquations 4.3.10 IllustrationofLinearDampingwithSinusoidalExcitation Wenowextendthepreviousexampletoalso involvesomeexternaloscillatingforce on the system: F.t/ D Asin.wt/. Driving a car on a roadwith sinusoidal bumps mightgive suchanexternal excitationon the spring system in thecar (w is related to thevelocityof thecar). WithAD0:5andwD3, from math import pi, sin w = 3 A = 0.5 F = lambda t: A*sin(w*t) we get the graph in Fig. 4.28. The striking difference from Fig. 4.27 is that the oscillationsstartoutasadampedcost signalwithoutmuchinfluenceoftheexternal force, but then the free oscillations of the undamped system (cost) u00 Cu D 0 die out and the external force0:5sin.3t/ inducesoscillationswith a shorter period 2 =3.Youareencouraged toplayaroundwitha largerAandswitch fromasine to acosine inF andobservetheeffects. Ifyoulookthisupinaphysicsbook,youcan findexact analytical solutions to thedifferential equationproblemin thesecases. Aparticularly interestingcaseariseswhen theexcitation forcehas thesamefre- quencyas the freeoscillations of theundamped system, i.e.,F.t/DAsint. With the same amplitudeA D 0:5, but a smaller dampingb D 0:1, the oscillations in Fig. 4.28becomesqualitatively verydifferent as the amplitudegrowssignificantly largerover someperiods. Thisphenomenonis called resonanceand is exemplified inFig. 4.29. Removing the damping results in an amplitude that grows linearly in time. Fig.4.28 Effect of lineardamping incombinationwitha sinusoidal external force