Question

A faulty ball bearing assembly is fitted inside a heavy machinery gear box. It generates vibrations due to this rotating motion in the form of a force with a magnitude of 2000 [N] this is every to = 0.03 [s] period as shown on the figure below. Determine the Fourier series representation for the periodic excitation. F(N) 2000 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12 0.13 0.14 0.15 0.16 (s)

Answer 1

Fouricy slics presentation for puriodic cicitation FIN) 2000 t(s 0.09 0.01 0.02 0.04 0.05 From the graph. Time period, T = 0.04 SEC frequency, to = 25 HZ т angular frequency, w = 2T f = 50T 351 27 f(3) = 2, + (an cos(nwi) + bn sin(wa)) - W - angular frequency = 2 AI = 507 T 20 = + +(1) de an = f(1) cos[nwa) di N=1,2.... 0 T 2 T f(1) sin (nwi) di 1,2... D Haic independent variable is 7' ic, ast - FLL) = a + < ( an cos (nwt) + by sin (nwt)) n=1 For over one time period, T[=0.04 50 f(t) = 2000 oct<0.01 0.01<t<0.04

T 0.04 0.04 0.01 1 2 + +Lt) at 0.DE 0.04 f(t) at 2000 dt + 0.04 0,01 0.01 x 2 DOO 000[t] 500 a = D 0.04 2000X0.01 0.04 -0,= 500 w = 50 TT an - FLL) COS(nwt) at =1,2... In 0.01 to 0.04 interval f(t) = 0 0.04 2 n=1,2,3 f[t) Cos ( 5ontit) at 0.04 D 0.04 10 0.01 13) ar 0.cos(son Tt) at 2 0.04 2000 COS(50nlit) dt + 0.01 20.01 n=1,2.. sin(son Tit) 2 * 2000 x 0.04 50n 0 2000 sin(n n TI n=1,2,3... 4000 Sin (0.500) חת 27 0.01 0.04 2000 - 5in (50n Tĩ ) dt For bn 2 2 0.04 F[) sin(50nīt)dt D 0.04 0.01 n=1,2,3.... 200D X 2000 x 2 0.04 COS( 50 nūt) 50 ni הח 0 חח ... 2000 n TT 2000 [-C05( a = 500 sin (I), bn n=1,2,3 Coefficients po 500 + E n=1 ... f(t) 2000 n TT sin (4) cos(soữnt) + (1-cos( 2 ) sin(50nTt) Fouries series representation of periodic crcitation.
I hope you understand. Comment if you have any doubt.