Question

Consider the system. (1) M →1.0) M +0.1 kg, B=0.2 N-s/m Mv(1) + By(t) = 1,01) Consider a system described by the following differential equation: 0.1"WX2 +0.2v(t) = .0), where y(t) and 4.0) are the output and the input of the system. dt (la) Convert the above differential equation into the form of the typical first-order dynamic system: + ) = ), and explain the physical meaning of the two parameters 7 and v.. (5%) dv(1) (1b) According to the information of (la), find and sketch the step response of the system, which is the time response (1) due to unit step input /.(1) = u() with zero initial condition (0) = 0. On the graph, specify the value of time constant and the values of (t) at = rand . (5%) (1c) According to the information of (la), find and sketch the initial state response of the system, which is the time response (i) due to the initial state (0) = v= 5m/s with zero input 1.0=0. On the graph, specify the value of time constant r and the values of vi) at =, and 1- 0.(5%) (id) What is the pole of the system? What is the relationship between the system pole and the time constant T? How do they affect the time response? (5%)

Answer 1

2. W bat egn We have a spstem described by a differen oib. +0.2014 = falty' bie , VT -oʻz facet t 1719- 5 faces So comparing this to a duer= vosfalto we can writ- /e=1/2, Vos= 57 Here physical meaning of r is that the 7 is the time const and Uss to the thong patoloog is the de-gain or steady stal gain and Voldy opady ptali gandingdes 16) To nketch the step responsa we writ- the differential egres input/output as system . Now taking I 2 the Tapiace tras get Tuch + VII - ne ve, faces transform of both side with fa(G) = Us (4) we get KTVCOTT + [vt] = VES. Cey] SV187 - 070) + .12S) - V 65 +211232 -12o - event of i V83/27. Vos .giver, voro a 12 = 3 (574) 3.187 +

0.371551.95 taking the inverne Replace from form 7749 = Vss Tēm? ] VH frosss at t=7 (e= Von: 11-7'] = VisLi-0.637 lire t = 0.37. Vas . Tu = 1.85 and as too, TV4 Vers = 5). 16. Again we write the differential es? as. AUTH = facts Taking the haplars transform groot with allt o we get 5.178) - Veo) + 13 = 0 (3 + 2 // J V = no given 120 = 5 m/s f 175 overse taking inverne lepeace trous form we get lep faking 124 = 50" the ud 19.15- 2B Vete - - 5x0.63 = 3.15 m/ at t oo T1705.07 T -7 7

zd 3- - 2 idy To find the pole we write the first order I tram fer for as - Where Ves=5 9 5+2 2 148 -22 case fol at (3: - 27 General form the mystem has pole - therefore, is 1 1 4 relation I the negetive pole value mean the nyotem time response is exponentially decaying) n in nature