written 7.7 years ago by | • modified 4.9 years ago |
(i) Gain
(ii) Damping ratio
(iii) Natural Frequency
(iv) Transfer function
written 7.7 years ago by | • modified 4.9 years ago |
(i) Gain
(ii) Damping ratio
(iii) Natural Frequency
(iv) Transfer function
written 7.7 years ago by |
From the given figure,
$M_p$=3.45-2
=1.45
$t_p$=0.55 sec
$t_s$=6 sec
$M_p$=e$^\frac{-πξ_2}{{\sqrt{(1-ξ_2 2))}}}$
1.45=e$^\frac{-πξ_2}{\sqrt{(1-ξ_2 2)}}$
Log 1.45=$\frac{-πξ}{\sqrt{(1-ξ2)}}$ log e
0.161=$\frac{-πξ}{\sqrt(1-ξ2)}$ x 0.4342
0.3707=$\frac{-πξ}{\sqrt{(1-ξ2)}}$
0.118=$\frac{ξ}{\sqrt{(1-ξ2}}$
Squaring both the sides
0.0139=$\frac{ξ^2}{(1-ξ^2 )}$
0.0139-0.0139$ξ^2$=$ξ^2$
1.0139$ξ^2$=0.0139 ξ= 0.117
Peak time,$t_p$=0.55 sec=$\frac{π}{w_d}$
$w_d$ =5.70 rad/sec
$w_d$=$w_n \sqrt{(1-ξ2)}$
$w_n$=$\frac{5.70}{\sqrt{(1-5.70^2 )}}$
$w_n$=5.789 rad/sec
$w_n$->Natural frequency
T.F=$\frac{(w_n^2)}{(S^2+2ξw_n S+w_n^2 )}$
T.F= $\frac{32.94}{(S^2+1.342S+32.94)}$
Gain of system=$\frac{1}{(1+G(S)H(S))}$
=$\frac{\frac{1}{1+32.94}}{(S^2+1.342S+32.94) x 1)}$
Gain of system=$\frac{(S^2+1.342S+32.94)}{(S^2+1.342S+65.88)}$