Chem Differential Eq HW Solutions Fall 2011 22

Chem Differential Eq HW Solutions Fall 2011 22 - 22 Chapter...

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22 Chapter 2 Fourier Series 9. (a) Natural frequency of the spring is ω 0 = ± k μ = 10 . 1 3 . 164 . (b) The normal modes have the same frequency as the corresponding components of driving force, in the following sense. Write the driving force as a Fourier series F ( t )= a 0 + n =1 f n ( t ) (see (5). The normal mode, y n ( t ), is the steady-state response of the system to f n ( t ). The normal mode y n has the same frequency as f n . In our case, F is 2 π -periodic, and the frequencies of the normal modes are computed in Example 2. We have ω 2 m +1 =2 m + 1 (the n even, the normal mode is 0). Hence the frequencies of the ±rst six nonzero normal modes are 1, 3, 5, 7, 9, and 11. The closest one to the natural frequency of the spring is ω 3 = 3. Hence, it is expected that y 3 will dominate the steady-state motion of the spring. 13. According to the result of Exercise 11, we have to compute y 3 ( t ) and for this purpose, we apply Theorem 1. Recall that y 3 is the response to
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This note was uploaded on 12/22/2011 for the course MAP 3305 taught by Professor Stuartchalk during the Fall '11 term at UNF.

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