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differential equation applications 2119.8 Implementation: Inclusion ofTime-Dependent ForceTo extend our simulation to include an external force,F ext (t)=F 0 sin ωt, (9.45)we need to include some time dependence in the force function f(t, y) that occursin our ODE solver.1. Add the sinusoidal time-dependent external force (9.45) to the spacedependentrestoring force in your program (do not include friction yet).2. Start by using a very large value for the magnitude of the driving forceF 0 . This should lead to mode locking (the 500-pound-gorilla effect), wherethe system is overwhelmed by the driving force and, after the transientsdie out, the system oscillates in phase with the driver regardless of itsfrequency.3. Now lower F 0 until it is close to the magnitude of the natural restoring forceof the system. You need to have this near equality for beating to occur.4. Verify that for the harmonic oscillator, the beat frequency, that is, the numberof variations in intensity per unit time, equals the frequency difference(ω − ω 0 )/2π in cycles per second, where ω ≃ ω 0 .5. Once you have a value for F 0 matched well with your system, make a seriesof runs in which you progressively increase the frequency of the driving forcefor the range ω 0 /10 ≤ ω ≤ 10ω 0 .6. Make of plot of the maximum amplitude of oscillation that occurs as a functionof the frequency of the driving force.7. Explore what happens when you make nonlinear systems resonate. If thenonlinear system is close to being harmonic, you should get beating in placeof the blowup that occurs for the linear system. Beating occurs because thenatural frequency changes as the amplitude increases, and thus the naturaland forced oscillations fall out of phase. Yet once out of phase, the externalforce stops feeding energy into the system, and the amplitude decreases;with the decrease in amplitude, the frequency of the oscillator returns to itsnatural frequency, the driver and oscillator get back in phase, and the entirecycle repeats.8. Investigate now how the inclusion of viscous friction modifies the curve ofamplitude versus driver frequency. You should find that friction broadens thecurve.9. Explain how the character of the resonance changes as the exponent p inthe potential V (x)=k|x| p /p is made larger and larger. At large p, the masseffectively “hits” the wall and falls out of phase with the driver, and so thedriver is less effective.−101COPYRIGHT 2008, PRINCET O N UNIVE R S I T Y P R E S SEVALUATION COPY ONLY. NOT FOR USE IN COURSES.ALLpup_06.04 — 2008/2/15 — Page 211