1. For a second order system RIS) win² (5²+ 2gunstun²³² verify when RIS)= $ (1) Wh: Undamped natural frequency >C(5) 1:

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1 For A Second Order System Ris Win 5 2gunstun Verify When Ris 1 Wh Undamped Natural Frequency C 5 1 1 (184.03 KiB) Viewed 40 times

1. For a second order system RIS) win² (5²+ 2gunstun²³² verify when RIS)= $ (1) Wh: Undamped natural frequency >C(5) 1: damping ratib, >0. ocfel, underdamped system Cits = 1- e "swit (cos wat + ·Sin wat) à funt √1-9² e =/- - cos (wat-p) S funt =1- √1-32 Sin (wat + p) 7-9² Wd = | T-pawn damped natural frequency. Wn #= tant ² of = tant 17² . Critically damped system گراد where (2) f=1, c(t) = 1-et (1+wnt) (3) f>1, overdamped system c(t) = 1 + 21/07/²2/1 (eft e' eBt Pi P₂ where P₁=(1+1=1) w/n, B2=lf-19²1) Wn. (4) Find when t→∞o, c(t) = ? For 11) (2) (3). 2. From 1. please check the following answers (a)-(e).
(a) (b) (4) (e) R(x)= R(1) R(1) R()- System G(x) General GO) 9 Overdamped G(₂) 9 ²+28+9 Underdamped G(x) 9 +9 Undamped G(₂) Critically damped C(s) C(₂) CU) C(₂) Pole-zero Plot ja Response c00) c) 1 0.171754- 0.5 "" c() c(n)-1-(cos/Br+sin√81) 141 -1-1.06 cos(√81-19.47") 1.2) 0.4 0.2 c() c()=1-cos 3 M. -13 plane 08 +6 0.4) 0.2) 0 plane X -7.854 -1.146 -plane, -0 jas X √8 X -plane -JNK 3/3
3. For each of the transfer functions shown below, find the locations of the poles and zeros, plot them on the s-piane, and then write an expression for the general form of the step response without solving for the inverse Laplace transform. State the nature of each response (overdamped, underdamped, and so on). 2 a. T(s) s+2 5 b. T(s): = (5+3)(5+6) 10(s+7) c. T(s) = (s + 10) (s+20) 20 d. T(s)=32 +65 +144 s+2 e. T(s): = f. T(s): (5+10)² 4 Calculate the exact response of each system of Prob- lem 3 using Laplace transform techniques, and com- pare the results to those obtained in that problem. S Find the damping ratio and natural frequency for each second-order system of Problem 3and show that the value of the damping ratio conforms to the type of response (underdamped, overdamped, and so on) predicted in that problem. = +9 (5+5)