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MAE 384. Advanced Mathematical Methods for Engineers. Homework Assignment 5. Due March 15. 1. The following data represent the vibration amplitude in mm at a location on a helicopter fuselage panel during one-half rotation of the (two-bladed) rotor. y = h 3:5 6:0 9:0 11:7 14:1 15:3 8:5 8:0

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MAE 384. Advanced Mathematical Methods for Engineers. Homework Assignment 5. Due March 15. 1. The following data represent the vibration amplitude in mm at a location on a helicopter fuselage panel during one-half rotation of the (two-bladed) rotor. y = h 3:5 6:0 9:0 11:7 14:1 15:3 8:5 8:0 9:1 10:1 10:9 8:6 1:05 1:9 3:2 4:7 4:8 3:9 1:7 0:5 i (a) Find the real discrete Fourier transform for this data set. (b) Any term in the Fourier series can be written: ak cos(k!t) + bk sin(k!t) = ck cos(k!t + k) where ck = q a 2 k + b 2 k and k = tan1 bk ak Find the ckís (including c0) and plot their amplitude on a bar graph vs. k to illustrate the relative size of each term in the series. Explain the signiÖcance of the plot. (c) Plot the function (inverse Fourier transform) and the original data on the same plot. Would you feel comfortable using this function to interpolate the data? Why or why not? (d) Estimate the maximum velocity amplitude at this location on the fuselage panel. 2. The density of the earth varies with radius, r. The following table gives the approximate density at di§erent radii: r (km) 0 800 1200 1400 2000 3000 3400 3600 4000 5000 5500 6370 (kg/m3 ) 13000 12900 12700 12050 11650 10600 9900 5500 5300 4750 4500 3300 The mass of the earth can be calculated by integrating the density: m = Z R 0 4r2 dr where R = the maximum radius of the earth = 6370 m. (a) Find the mass of the earth using the trapezoidal rule. (b) Find the mass of the earth by Örst interpolating the data using a cubic spline to obtain 50 equally spaced intervals. Then use both trapezoidal rule and Simpsonís rule to Önd the mass of the earth. (c) Discuss your results, and compare with the actual mass of the earth. In order to explain your results, you may wish to plot both the data and the interpolat

 
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