Homework Problems

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Problem 1. Consider the system

y''(t) + cy'(t) + y(t) = 0, y(0)= , y'(0) = 1 .

For each of the following pictures try to guess the corresponding value of c. Verify your guesses by solving and plotting the solutions corresponding to the values that you have chosen. Make sure that the frequency and the amplitude of your choices matches closely the pictures given below.

(a)

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(b)

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Problem 2. Undamped Vibrations for Different Input Frequencies. Consider the following undamped system with the periodic external driving force sin( w t):

y''(t) + 4y(t) = sin( w t) , y(0) = 0, y'(0) = 0.

(a) Find corresponding to the natural frequency of the system.

(b) Investigate the behavior of the system for a couple of input frequencies w close to , for example: w = 1.8,1.9. Find and plot the corresponding solutions. Apply the command combine(expression,trig) to the output of the dsolve command to see clearly the nature of solutions. In your plots, choose long enough interval for t to see global behavior of solutions, for example, t=0..40*Pi . The solutions look neater if you enter fractions instead of decimals, i.e. 18/10 instead of 1.8.

(c) Note the characteristic "beat" behavior of solutions. Comment on the amplitude and the duration of beats as gets closer to .

(d) Find and plot the solution corresponding to w equal to . What do you observe about the amplitude of oscillations in this case? What is this phenomenon called?

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Problem 3. Consider the mass-spring system

my''(t)+cy'(t)+ky(t)=Fcos(wt)

in which c, k, F, and w can't be changed but we are free to vary the mass m. Specifically, assume that c=1,k=2,w=3,F=1. Find the exact value of m for which the steady-state amplitude is as large as possible and find the maximum value of the amplitude.

Hints: 1) Remember what the steady-state solution is. Use the formula for the solution derived in Sec.2 of this worksheet. Substitute the correct values of c, k, w, F. The coefficients A, B become functions of m only.

2) Consider the square of the amplitude rather than the amplitude itself. The square of the amplitude is maximal when the amplitude is maximal. Use the command simplify to simplify the square of the amplitude.

3) Find the exact maximum of the square of the amplitude by calculating its derivative.

Or do all of it differently, as long as your way is correct!

Note: Do not denote any of your expressions as " y " or " ", etc. . This causes Maple to misunderstand your intentions when you enter ODE's.

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Problem 4. Consider the function

f(t)=3.23sin(0.4t)-5.56cos(0.4t).

Find the representation f(t)=Ccos(0.4t-d) for suitable C and d. Plot graphs to check your answer.

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