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hw10soln

# hw10soln - ME 365 HOMEWORK 10 Out Due Fall 2011 Problem 1 A...

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ME 365 HOMEWORK 10 Fall 2011 Out: November 10, 2011 Due: November 17, 2011 Problem 1 A wheel is rolling with linear velocity V over an infinite set of speed bumps as shown in the figure below. The bumps are H x W in size and spaced D apart. As the wheel rolls over a bump, the vertical position of the wheel hub rises then fall s. The hub’s position, x(t), follows a sinusoid during this motion, as shown in the bottom portion of the figure. The wheel’s vertical displacement x(t) is given by, x ( t ) 0 for T 2 t   2 T D V H cos t for 2 t   2 where 0 for 2 t   T 2 W V (a) Show through simulation that the Fourier coefficients and fundamental frequency for the periodic signal x(t) are 1 2 V D A o 2 2 HW D A k HW D sinc kW D 1 2 sinc kW D 1 2 B k 0 where sinc ( a ) sin( a ) a and sinc (0) 1 (note there is a sinc function built into Matlab.)

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ME 365 HOMEWORK 10 Fall 2011 Out: November 10, 2011 Due: November 17, 2011 (b) To approximate x(t) use a finite number of terms (N), i. e. x ( t ) A o 2 M k cos( k 1 t k ) k 1 N If the error is the approximation is defined as error x ( t ) ) x ( t ) . At what point (in time) during the signal will the error be the greatest? Why? (c) Determine the formula for the maximum error when N terms are used in the approximation. The formula should only be a function of H, W, D and N. (d) If H = 0.1 and W = 0.2 and D = 1, plot the maximum approximation error versus N. How many terms are required to achieve an error less then 1mm?
ME 365 HOMEWORK 10 Fall 2011 Out: November 10, 2011 Due: November 17, 2011 Solution: (a) Implementing the Fourier series in Matlab with the first 20 terms using the values H = 0.1, W = 0.2, D = 1 and V = 1 generates the following plot. This matches expected result giving us confidence the Fourier coefficients provided are correct.

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ME 365 HOMEWORK 10 Fall 2011 Out: November 10, 2011 Due: November 17, 2011 (b) The Fourier approximation with a limited number of terms will have the largest errors at point of discontinuity in position (step change) or first derivative (sharp corner).
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