Question

a) Develop the transfer function X:(s)/F(s) of the mechanical system shown in Figure 3(a). Give and explain one example the real application where you can relate with this system. (5 marks) b) Routh's stability criterion is of limited usefulness in linear control systems analysis mainly because it does not suggest how to stabilize an unstable system. Thus, we should evaluate the stability range of a parameter value. Consider the servo system with tachometer feedback as shown in Figure 3(b). Evaluate the ranges of stability for K and Kh. (Note that Kh must be positive). (5 marks) [10 MARKS] X1 X2 ki W k2 Wr m W m2 D. D2 Figure 3(a) R(s) Cs) K 20 (5+1) (+4) KA Figure 3(b)

Answer 1

() Proces on mans m, mä. f (brock) kitke) 2, D. x, + K₃ N2 m, $c-X/02 f(o) - (Kitty) X, (o) - D, X, 10) + k tla) forces . زنا an maro me & R₂n 2 Y_@) [M, At mä, Mex²X (6) Kzxon - (Kett) X (A) - Bo X (0) + (K₂ + 3) +30 Kzx, (a) kg 4,(0) Car xelor M +0 0 K tkz putting X₂ (6) is ea i M, s X, (A) f(o) - (K, 4K) X (1) - DNO, C) + Kg²x (6) x 10) M1 82 + (kitke) &D - 42 M₂0² + D₂0 t ketka flo) z 1² + O₂ At Katke

+On that Kg)) (N20² + Bot tetky) – K² PA M₂0² + Dot H + K₃ x, (6 / MM, 10* + (Mas, + M, 22) 8² (Milketk) + Ml kitten) + B, 02 JA? + (bitky sa s & k, the D2 + (kit) (kakg) - 43 he+ Dot kzt K₂ M₂0² + D28 & Katka M, M2 s² & D, &M, B) ² + (1 kgt kg) + M₂ ( kitk) TRR + (katke) 0,0 + (Kitty) + (kytk) Ketter) This mechanical system u analogow to circuit ceorent capacites a colle induches Vce LdTc acolo > Voltage CI Cooth cle Le dhe voltage acoon resistor ar CR clic alt

from 20 equation cliffesential Mx und -hex ka, – Dix. - kg (x,-x₂) or & s Mä & Kino + D + kg (xrna) differential eq Mit Kade & Kg (ran & Dare ad and from and x = Ve X C2 f, voltage soura 2 Kic ker seg M, LE, DR, kice, M22 4. O R2 D, M un mit 4 Hmmm 02 hu 11K, R Yks

(1) RW-" dok (+ (0 +42 (600 RM -((0) - Kancco) dok smo (049) (044) Dok R62 - 206 e CO2 - 20k kunCOos o lotuCott) (6) (oCott) €6+4) + 20kknd +30k)cco - Rok R (r) 2oktorola ACS+6+4) 4 Rokkantook Rcon 992 13 characterstic * +504 + 4 (5Kku+1) ++ 20k = 0 4 (Skkat!) for stable system all should have 20 K same sign in | 20 (skati) -20k coloumn i 1 s' po 2ok

00 20K >0. & ko 4 (5kkati) - 4k a no skkat! ok solat skku > Skin knis Kelsta (sk and k solo fate eh