PS1Fa2023

.pdf
School
University of Wisconsin, Madison **We aren't endorsed by this school
Course
M E 440
Subject
Electrical Engineering
Date
Sep 19, 2023
Pages
1
Uploaded by AdmiralFang986 on coursehero.com
Problem Set 1 1) In the lectures, we use three different representations of free vibrations of a simple harmonic oscillator with natural frequency ࠵? ! : i) Trigonometric: ࠵?(࠵?) = ࠵? " cos ࠵? ! ࠵? + ࠵? # sin ࠵? ! ࠵? ii) Exponential: ࠵?(࠵?) = ࠵? " ࠵? $% ! & + ࠵? # ࠵? '$% ! & iii) Combined trigonometric: ࠵?(࠵?) = ࠵? sin (࠵? ! ࠵? + ࠵?) Express ࠵? ",# , ࠵? and ࠵? in terms of ࠵? ",# . 2) A harmonic motion has a frequency of 15 Hz and maximum velocity of 3 m/s. Calculate this motion's amplitude, period, circular frequency, and maximum acceleration. 3) Consider two harmonic motions ࠵? " (࠵?) = cos(2࠵?࠵?) and ࠵? # (࠵?) = cos(2࠵?(1 + ࠵?)࠵?) where ࠵? = 0.1 . Using a software of your choice (Matlab, Mathematica, Origin, Python, Excel, etc.), plot the motion ࠵?(࠵?) = ࠵? " (࠵?) + ࠵? # (࠵?) for ࠵? ∈ [0, 30] . Calculate the beat frequency (see pg. 92 of the textbook for the beat phenomenon). 4) Conduct Fourier series expansion of the triangular wave shown below and sketch by hand or plot the amplitude spectrum. 5) Using an engineering software of your choice (Matlab, Mathematica, Origin, Python, Excel, etc.), plot the Fourier series expansions you found in Problem 4 truncated at fundamental (M=1), fifth (M=5) and ninth (M=9) harmonics and calculate the mean-square-error between the actual triangular wave and different expansions over a period. Do you observe Gibbs' phenomena in the Fourier series expansion? Briefly explain why or why not in reference to the reading assignment titled "Gibbs' Phenomena".
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