See answer: Resonance and Beats. In Section 3.8 we observed that an undamped harmonic

Chapter 6, Problem 23

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Resonance and Beats. In Section 3.8 we observed that an undamped harmonic oscillator (such as a springmass system) with a sinusoidal forcing term experiences resonance if the frequency of the forcing term is the same as the natural frequency. If the forcing frequency is slightly different from the natural frequency, then the system exhibits a beat. In 19 through 23 we explore the effect of some nonsinusoidal periodic forcing functions.Consider the initial value problemy + y = h(t), y(0) = 0, y(0) = 0,wheref(t) = u0(t) + 2nk=1(1)ku11k/4(t).Observe that this problem is identical to except that the frequency of theforcing term has been increased somewhat.(a) Find the solution of this initial value problem.(b) Let n 33 and plot the solution for 0 t 90 or longer. Your plot should show aclearly recognizable beat.(c) From the graph in part (b) estimate the slow period and the fast period for thisoscillator.(d) For a sinusoidally forced oscillator, it was shown in Section 3.8 that the slow frequencyis given by | 0|/2, where 0 is the natural frequency of the system and is the forcing frequency. Similarly, the fast frequency is ( + 0)/2. Use these expressionsto calculate the fast period and the slow period for the oscillator in this problem. Howwell do the results compare with your estimates from part (c)?