Question

7. Suppose the input f(t) to the following model is a ramp function: f(t)= at. Assuming that the model is stable, for what values of a, m, c, and k will the steady-state response be parallel to the input? For this case, what is the steady- state difference between the input and the response? mž +ci+hor = ft) ution:

Answer 1

Page no difference between Answers Objective is to determine the steady sale the input and the response. Given that Input FCH) to the given model is ramp funch on :- U zat. given differential equation is mz+ci+k2 = PO Assuming the model is stable and values of amc sk. will the steady state response be parallel to the input. ginven input is (t) = at. Laplace of Plt) i F(s): ģe Now take, given differential equation mie + cxt ke = f(t) Apply taplace transform for this by taking initial conditions to be gero, Then ms x (s) + GSX(S) + K XCS) = Taling X (s) ay common. XG) [ms? +cs + x) = (5) - a 54ms?+ S +) This is the response equation. - ê V partial fraction for this equation is X (s) Ys+ Z 55 ms +cStk + where Ystz 2 ms?+c9 + . transient siate response ☺ + B = steady state funcion. using these equations, finding the values for UEU. Finding v:- v- xG) 8490 - substituting een 0 in een ®

Page no Aenean coming comedor V m3 24 stk. 1 s.o. after substituling sao va acoget eros et > = Finding value of us at (x(s) sºls=0 - 0 suosliluk ceno in ogni us at Comet ost k3 org loco. US Consor] loro Como consegue loro Us - a(2m(0) +c) (702)+(60+) U - a (2mstc). (ms? +65 +k) 250 substitute u and value in equalron *(s) = [order] Is + ]/62 + 4672 ms?+cs+K. X (S) - fake + YS + Z - ac K25 steady ks² ms²+c5tk. state part of the response Y skady (5)= ve siz

Inverse laplace foranspoorm for this. page no 3 esteady(t) = qt - uCt). - © Comparing the slope Poor This steady stale response and To the given input. They should have same slopes to be parallel slope of the input = a slope of x(1) S equalizing both a = a. kaal ie, [ks] substitute kulion eqn . Tobody (1) - 우 - A), Iskady (t) = .at - acult') if so then ult)= 1) " ZSteady (t) = at -ac. ,: The steady state response difference between input le response is di Herence = f(t) - z slady substitule equations f(t)= at . Xoleado t) = at- ac diffe at - Cat -ac) . dift = at - at tac diff = acl difference between the input The and response is ac.