Question

4 C(s) حرام | A system has a block diagram as shown. The input is "R(s) and the output is C(s). a) Using only the block diagram reduction method, R(S) find the transfer function of the system. b) Determine the characteristic function and the order of the system c) Find the characteristic roots of the system. d) Find the natural frequency of the system. e) Find the damped natural frequency of the system. 8 * NOTE: All stages of block diagram reduction must be clearly shown until you arrive at the last simple closed loop diagram and then the final open-loop equivalent with the transfer function in question. In addition, the final transfer function must be simplified, i.e., no fraction(s) over fraction(s).

Answer 1

Solution! Block Diagram reduction rule! Rulet! check for the block connected in series & simply Rule2 check for the block connected for parallel & simply. Rules! check for the blocks connected in feedback loop a simply Rule 4! If there is difficulty with stake-off point while simplfying, shift it towards right. Rulesi If there is difficulty with summing point while simplifying 3 shift it towards left. Rules! Repeat the above steps fill you get the eimplified farm; le sigle block (a) (4k R(s) co S sk use rule 3 for block '4' and to the modified block R(S) 1 1 5 8

RIS) C(s) 1 st4 S 8 use rule 1, modified block is! 1 8 use rule 3, the modified block is 1 s(+4) 09 1+ 5(8+4) RES) s(s+4) +8 Transfer function is, 1 1 S(St) +8 74518 R(S) Characteristic equation of a system can be obtained by equating the denominator of the polynomial of a trons for function is The characteristic equation of the equation is: s²745+8=0 Cero.

©) order of the sypten = second order system Characteristic roots of the system! 3+45+8=0 put s=x - x² + 4x+8=0 1= -442—4x8x1 2x1 d= -474 i 2 x = -2±2i characteristic roots are di= (2+1) & dz=-2-2i (a ) (544578 544248 standard form of and order transfer function C(s) Ko one RCS) s²4 25 wnst wir compare the both equations , we get natural frequency con=218 rad see K= = 0.125

and 28 4 $= 2001 4 2x2,8 = 0.7142 Damped natural freqreency (w) We wo dinge w = 2.81-8171423 >W= 1.9898 sadleee w