Question

Problem #2 (%25) (a) For the following mechanical system, obtain the equation of motion and the transfer function X,(s)/F (s). (b) Obtain the loop, the node equations and the transfer function Vow(s)/V...(). +

Please solve the questions legibly and descriptive

Course Name: Control System Design

Answer 1

Problem? :- DATE Hi uz We LMm PM XXX by Now Free there Body are 2 displacements Diagrams of both & we drew For body I force lix, alting opposite to Fe bivi - acting opposite V - KX bivi to, pposite AAA acting - 4 4 using Mechanical laws I on both sides mia, ka (x-x2) m, a, and balancing forces Fe = kait bivit lez(x - x2 + mia, 9,= d²xi ; Vi = dxi e = ki xit ka at bi dex, & mid²xi - k2u dt m, at bi xit (kithe) a - kq ns 2 at 2 Taking els) = Laplace transform both sides. m. s² & X, (s) + bis x, (s) + kith) x, w X2 Ls) f

The ABD for and displacement Similiarly will be:- Mylez x2 kg (21-x2) baye Now, balancing forces both sides ke (xi-kr) = K₂X2 + b2 V₂ - bo vzt ikat k3) 2 - kexico - (2) - at bz sert kat kez) X2 - kzx, o Taling hap lace Trausform both sides- bzs X2(e) t llat lez) x.) - K2X, (s) =0. - kz X768) + (625+ katky) Xz(s) -01-27 Now, using Cross-multiplication or subst we can solve this. Substituting X, (s) in all from (2) eg. we can get our solution.

X₂ (s) (best katkez) = X, (3) ke New putting this value of x, (o) Dece) = (mis²+ bist (with2) & ( bz st (karkez)) x2(6) 42 you - kz X2(W) Fels) = (m, 5²+ bist kitks) (bast (leatles)) - Kr] Xals) Fels)= mist bist (kither) (brst karlz) - 622X24) - uz . Xz(s) = Feli) K2 (mist bist kitka) (best katkez)) -kes If we want, we can open the bracelet although it will become more complicated on opening the brackets, but we can do just by cumple mathematics and ter I will get cancelled fu the end. Cs Scanned with CamScanner

Vinets SR Vout DATE TOOTD (A) Loop Apply 2 11) in equations. mesh currents the loop or it in (1) and mesh 2. am I + V Viult ( 28R1 2 { R vout ( H using simple I voltages KUL in a Rule that Sum of Loop is zero. .: What Vin- V - ₂ co. Vi = 1 fiyat ; V2 = Rich - 1) CIJ I Vin= 1 lidt & R, (1-12) This is loop a une cqui :- for +₂ - V₂ - Vout to V = Reli, 12) z = L diz Vout - Riz iz de Tvout = V₂ V3 ) Canned with

. Rain - R, (1, -12) & L die til R2 12 tl dir + R, 12 - Rigao. T-R, & CH + (Ritke) 12 (H + L de (H20 | 12 This the and KV equation. Medal equatica 2 me camblue It we want form to write them in ordered Vin= tflidt & Ri (it) Vout a fer R. (1, -12) - L din or vout - RillH+ R ir R2) I ₂ 14 t & die - ou-2 last equation is obtained when we replace Vb (t) with Ral and te arrange the temus.

Nodal: - Since there are 2 elementary wode assume 2 voltages V, LH and solu an Nodel and Node 2 respectively, Therefore, now use Nodal equations or Kirchoff's current law that curnent entering is equal to current leaving Vout At nodel : 11, H₂ # iz=of | 1 = ra c d (v - Vin) 12 = Vi ; 13 = 1 / - V 2) at Node oqu (1) - V + 1 (sv - U₂) dt zo C & (i - Vin) + dt canned with PAGETTI

for node 2 Hd VLH-V21H dt =o. - 142 J These are the 2 undlal equations given by (3) and (4) Now, for obtaining Vix Vout) we will linis I use the wash equations 10 and 12 derived earlier Mesh equation or loop equation (1) Vin= 1 n dtt Rielt) - R,L 1H Taking Lito both sides! Vinls) = 1 as IiLs) + R, S, (o) - Ro Pals) Rath.s 2 I2 W ) -2 20 Suiliarly equ (2) Vout (s) = d B R. I, ls) - BS For we can use substitution solve the equation method in to - Scanned with an Scanner

Voutts) = I2 (5) R2 Vout - DATE DIT Rai - Ilu Pals Putting this value in equ (211 Voutis) & Rat Ls²) | . Voutls) = R, L, (s) R2 I Ls) = Voutls) + b Vout + L s² Voutly RIRL BL Ii (s) = 12 + - TRI L s²). Vout RIR values of I and I in eq" (1 10-11 + R las /2 RI + 2 s2) Vout RIR2 / - & Vout R2 Vinks) = 2 + RIGS 2+ L s t G R, R2 Ls2 - 1 ve R Voulo-2R2 Leet Ls3 GRASGRR SG RI R SCR R2 Voutis) = et s²L + SGR R & Q R. Vials Sented with CamScanner