Exam 4 Solutions.pdf - ESE 351-02 Spring 2017 Exam 4 Apr 13 Name 1(20 points Consider the function f(t = 2 3sin t sin3t(a Find all Fourier coefficients

# Exam 4 Solutions.pdf - ESE 351-02 Spring 2017 Exam 4 Apr 13...

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ESE 351-02, Spring 2017 Exam 4, Apr. 13 Name: ___________________________ 1 . (20 points) Consider the function f t ( ) = 2 + 3sin t + sin3 t . (a) Find all Fourier coefficients, F ( n ), of the exponential Fourier Series. (b) Sketch its Fourier spectrum, F n ( ) and F n ( ) , over 5 ω 1 ω 5 ω 1 , where ω 1 is the fundamental frequency. f t ( ) = 2 + 3 1 2 i e it e it ( ) + 1 2 i e i 3 t e i 3 t ( ) f t ( ) = 2 F 0 ( ) ! 3 2 i F 1 ( ) ! e it + 3 2 i F 1 ( ) ! e it 1 2 i F 3 ( ) ! e i 3 t + 1 2 i F 3 ( ) ! e i 3 t F n ( ) = 0 for all n 0, ± 1, or ± 3 F 0 ( ) = 2 F 0 ( ) = 2, F 0 ( ) = 0 F 1 ( ) = 3 2 i ( ) F 1 ( ) = 3 2 , F 1 ( ) = π 2 F 1 ( ) = 3 2 i F 1 ( ) = 3 2 , F 1 ( ) = π 2 F 3 ( ) = 1 2 i ( ) F 3 ( ) = 1 2 , F 3 ( ) = π 2 F 3 ( ) = 1 2 i F 3 ( ) = 1 2 , F 3 ( ) = π 2 Grading: 6 papers got full credit (even if they didn’t make clear that all other F ( n ) = 0). Most errors were unique, so I graded fairly subjectively: A = 18-19, B = 16-17, C = 14- 15, D = 12-13. The few papers that had little/no clue of the “intuitive” method of Mukai (comparing terms) and homework problem 9.4 got low F grades (mostly mercy). (When I said there was a Fourier Series problem, but you wouldn’t have to take an integral, that should have made clear that the intuitive method would be on the exam.)

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ESE 351-02, Spring 2017 Exam 4, Apr. 13 2 . (20 points) Consider the mechanical system below, where the input u ( t ) is the force applied to the mass and the output y ( t ) is the displacement of the mass.

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