Question

3. The input and output corresponding to the steady (vibratory) response of a first order system have been recorded below. (a) Use the phase difference to find the time constant of the system. (b) Use the magnitude ratio between both curves to find the time constant of the system and compare your answer to that of part (a) if the equation is known to be ai + bx = Csin(wt) where the input is Csin(wt), the output is x, and the value of bis 5. (c) Write down a differential equation of a first order system that will yield the same response shown below. (d) For which range of input frequencies will the output of the first order system be negligible? (e) For which range of frequencies may the derivative term in the first order ODE be ignored?

Answer 1

Solution 206) Input output ya Let first order To F = H(S) 1109 | 10w) ett) From grabt imput velo con be expoess ecl xD = A sinet) as and period Ta 12-5 sec where A = 2 i: frey 211 rad/pec T 21 rad/sec 6 12-5 p0.50 rad/ope so imbut 20 - 251010.5t) CS Scanned with CamScanner

from the output grapho- phase difference in sec, t =28ec ca) Irad in rad wt=0.5*2= in degree prad X 180 ㅠ =-57.324° IC det where t= time constant HO) - ts+ 1 K HjW) JWITI 57.3240 Ha) -tan' (67) o - 57.324° a LHw/w 2015 10 T = tan (57.3249): 1559 1:553 155.9 3.180c > to ī= time Constant = 3.1 ser CS Scanned with CamScanner

(6) HCjW) Yl) =250 (15) YU = Y Sin 10:57 +57 32°) ) 2 TWT)+1 K 0.25 | from sulput graph] to :X 2 = 10.50 ²T K lo 5 + 1 - 0.125 k sk c 10-51)+1 0.125 het K=1, (0.5 t)?+1264 = 05 ) 6 3 = 0 5 0 - 1931 T-19.8 Dec 0 It means that value of K#1. by comparing from part (a). CS Scanned with CamScanner

het H(S) IC Y = IS +1 in YTS + y = xk M [idy + y = xk dt T dy dt +Liy=X K ý t Tk 07 2 slando:51) ty by comparing it with the given differential en axt bx = c sinut b=5=4k = K = 45 K= 1, 2012 CS Scanned with CamScanner

and a- Y Touz Now from ein %.2=15.8 sec T= 3.17 sec we This value of T is almost equal to what got in part @ O. 2 y (het) . HS) = 3st 1 So, 0. 2X = (38 + 1) y 38 y 3) + Y3) = 0.2 X 0.28 V dy at + y is dy It tsy = This is the differential equation. CS Scanned with CamScanner

le) HS 0,2 3st 1 0.2 So, HW) 3 jwth 0.2 THOM) Jow)2+1 . By doing Asymptotic Approximation How ldeo z zalogo.2 – 20 2 24 قال وما ما If we only consider M = -20 log Jauhti radiper or wall and 3W << 1 then MdB = - 2010g1 = 0dB for UFO < magnitude is neglible. CS Scanned with CamScanner