Fourier Series Pt 4

Fourier Series Pt 4 - University of Waterloo CivE 331 Periodic Loading of a Railway Foundation Consider a stationary train in which every other car

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Unformatted text preview: University of Waterloo CivE 331 Periodic Loading of a Railway Foundation Consider a stationary train in which every other car is loaded with cargo: The train and its cargo exert a force (per unit length) on the track bed and this causes a vertical deflection of the track. Excessive loading can cause unacceptable deflection of the track resulting in a bumpy and unstable ride that could possibly derail the train. Therefore, we wish to model the deflection of the track in response to the periodic loading depicted in the figure. Assume the following boxcar characteristics: w L = 40000 N/m (the loading of a loaded boxcar) w E = 12000 N/m (the loading of an empty boxcar) w G = 0 N/m (the loading of a gap between boxcars) 20 m = length of each boxcar 1 m = the gap between each boxcar Assume the following characteristics of the track bed: EI = 4.30×10 6 N-m 2 (the flexural rigidity) k = 6.58×10 7 N/m 2 (the stiffness coefficient) If we model the track bed as an elastic foundation, then the deflection of the track, u ( x ), is governed by the following ODE: ( 29 x w ku EIu xxxx = + Where w ( x ) is the periodic loading of the boxcars. If the x-axis is aligned with the centre of a loaded boxcar, then one period of the loading can be sketched as follows: empty empty empty . . . . . . Track Bed Track 5 10 15 20 25 30 35 40-21-18-15-12-9-6-3 3 6 9 12 15 18 21 Loading (kN/m) w(x) University of Waterloo CivE 331 To solve the ODE, we first represent the loading as a Fourier Series expansion. By inspection of the loading sketch, w ( x ) is an even function with a period ( T ) of 42 and 21 2 / = = T l ....
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Fourier Series Pt 4 - University of Waterloo CivE 331 Periodic Loading of a Railway Foundation Consider a stationary train in which every other car

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