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![Page 1: Time Domain Analysis (1A) - Wikimedia Commons · 2014-10-27 · Time Domain Analysis (1A) 7 Young Won Lim 10/27/14 2nd Order Transfer Function: Standard Form s =−ζωn ± j√1−ζ](https://reader033.vdocument.in/reader033/viewer/2022050315/5f77a689204fd92bed52d8a4/html5/thumbnails/1.jpg)
Young Won Lim10/27/14
Time Domain Analysis (1A)
![Page 2: Time Domain Analysis (1A) - Wikimedia Commons · 2014-10-27 · Time Domain Analysis (1A) 7 Young Won Lim 10/27/14 2nd Order Transfer Function: Standard Form s =−ζωn ± j√1−ζ](https://reader033.vdocument.in/reader033/viewer/2022050315/5f77a689204fd92bed52d8a4/html5/thumbnails/2.jpg)
Young Won Lim10/27/14
Copyright (c) 2014 Young W. Lim.
Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled "GNU Free Documentation License".
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![Page 3: Time Domain Analysis (1A) - Wikimedia Commons · 2014-10-27 · Time Domain Analysis (1A) 7 Young Won Lim 10/27/14 2nd Order Transfer Function: Standard Form s =−ζωn ± j√1−ζ](https://reader033.vdocument.in/reader033/viewer/2022050315/5f77a689204fd92bed52d8a4/html5/thumbnails/3.jpg)
Time Domain Analysis (1A) 3 Young Won Lim
10/27/14
2nd Order Systems
9
s2+9 s+9
9
s2+2 s+9
9
s2+9
9
s2+6 s+9
![Page 4: Time Domain Analysis (1A) - Wikimedia Commons · 2014-10-27 · Time Domain Analysis (1A) 7 Young Won Lim 10/27/14 2nd Order Transfer Function: Standard Form s =−ζωn ± j√1−ζ](https://reader033.vdocument.in/reader033/viewer/2022050315/5f77a689204fd92bed52d8a4/html5/thumbnails/4.jpg)
Time Domain Analysis (1A) 4 Young Won Lim
10/27/14
Step Responses
9
s2+9 s+9
9
s2+2 s+9
9
s2+9
9
s2+6 s+9
![Page 5: Time Domain Analysis (1A) - Wikimedia Commons · 2014-10-27 · Time Domain Analysis (1A) 7 Young Won Lim 10/27/14 2nd Order Transfer Function: Standard Form s =−ζωn ± j√1−ζ](https://reader033.vdocument.in/reader033/viewer/2022050315/5f77a689204fd92bed52d8a4/html5/thumbnails/5.jpg)
Time Domain Analysis (1A) 5 Young Won Lim
10/27/14
2nd Order Transfer Function: Standard Form
G(s) =ωn
2
s2+2ζωn s+ωn
2
s2+2ζωn s+ωn
2= 0
s =−ζωn ± √ζ2ωn
2−ωn
2
=−ζωn ± √ζ2−1ωn
=−ζωn ± j √1−ζ2ωn
ζ = 0s = ± jωn
s =−ζωn ± j√1−ζ2ωn 0 < ζ < 1
s =−ωn ζ = 1
s =−ζωn ± √ζ2−1ωn ζ > 1
![Page 6: Time Domain Analysis (1A) - Wikimedia Commons · 2014-10-27 · Time Domain Analysis (1A) 7 Young Won Lim 10/27/14 2nd Order Transfer Function: Standard Form s =−ζωn ± j√1−ζ](https://reader033.vdocument.in/reader033/viewer/2022050315/5f77a689204fd92bed52d8a4/html5/thumbnails/6.jpg)
Time Domain Analysis (1A) 6 Young Won Lim
10/27/14
2nd Order Transfer Function: Standard Form
G(s) =ωn
2
s2+2ζωn s+ωn
2
s2+2ζωn s+ωn
2= 0
ζ = 0s = ± jωn
s =−ζωn ± j√1−ζ2ωn 0 < ζ < 1
s =−ωn ζ = 1
s =−ζωn ± √ζ2−1ωn ζ > 1
−ζωn
√1−ζ2ωn
(−ζωn)2 + (√1−ζ2ωn)
2
= ζ2ωn2 + (1−ζ2)ωn
2
= ωn2
ωn2
+ j√1−ζ2ωn
![Page 7: Time Domain Analysis (1A) - Wikimedia Commons · 2014-10-27 · Time Domain Analysis (1A) 7 Young Won Lim 10/27/14 2nd Order Transfer Function: Standard Form s =−ζωn ± j√1−ζ](https://reader033.vdocument.in/reader033/viewer/2022050315/5f77a689204fd92bed52d8a4/html5/thumbnails/7.jpg)
Time Domain Analysis (1A) 7 Young Won Lim
10/27/14
2nd Order Transfer Function: Standard Form
s =−ζωn ± j√1−ζ2ωn 0 < ζ < 1
ζ = 0.2, ωn = 100
ζ = 0.4, ωn = 50
ζ = 0.1, ωn = 200 s2+4 s+20√0.99
s2+4 s+10√0.96
s2+4 s+5√0.84
![Page 8: Time Domain Analysis (1A) - Wikimedia Commons · 2014-10-27 · Time Domain Analysis (1A) 7 Young Won Lim 10/27/14 2nd Order Transfer Function: Standard Form s =−ζωn ± j√1−ζ](https://reader033.vdocument.in/reader033/viewer/2022050315/5f77a689204fd92bed52d8a4/html5/thumbnails/8.jpg)
Time Domain Analysis (1A) 8 Young Won Lim
10/27/14
Standard Form: varying a (1)
![Page 9: Time Domain Analysis (1A) - Wikimedia Commons · 2014-10-27 · Time Domain Analysis (1A) 7 Young Won Lim 10/27/14 2nd Order Transfer Function: Standard Form s =−ζωn ± j√1−ζ](https://reader033.vdocument.in/reader033/viewer/2022050315/5f77a689204fd92bed52d8a4/html5/thumbnails/9.jpg)
Time Domain Analysis (1A) 9 Young Won Lim
10/27/14
Standard Form: varying a (2)
![Page 10: Time Domain Analysis (1A) - Wikimedia Commons · 2014-10-27 · Time Domain Analysis (1A) 7 Young Won Lim 10/27/14 2nd Order Transfer Function: Standard Form s =−ζωn ± j√1−ζ](https://reader033.vdocument.in/reader033/viewer/2022050315/5f77a689204fd92bed52d8a4/html5/thumbnails/10.jpg)
Time Domain Analysis (1A) 10 Young Won Lim
10/27/14
Standard Form: varying b (1)
![Page 11: Time Domain Analysis (1A) - Wikimedia Commons · 2014-10-27 · Time Domain Analysis (1A) 7 Young Won Lim 10/27/14 2nd Order Transfer Function: Standard Form s =−ζωn ± j√1−ζ](https://reader033.vdocument.in/reader033/viewer/2022050315/5f77a689204fd92bed52d8a4/html5/thumbnails/11.jpg)
Time Domain Analysis (1A) 11 Young Won Lim
10/27/14
Standard Form: varying b (2)
![Page 12: Time Domain Analysis (1A) - Wikimedia Commons · 2014-10-27 · Time Domain Analysis (1A) 7 Young Won Lim 10/27/14 2nd Order Transfer Function: Standard Form s =−ζωn ± j√1−ζ](https://reader033.vdocument.in/reader033/viewer/2022050315/5f77a689204fd92bed52d8a4/html5/thumbnails/12.jpg)
Time Domain Analysis (1A) 12 Young Won Lim
10/27/14
Standard Form: varying zeta (1)
![Page 13: Time Domain Analysis (1A) - Wikimedia Commons · 2014-10-27 · Time Domain Analysis (1A) 7 Young Won Lim 10/27/14 2nd Order Transfer Function: Standard Form s =−ζωn ± j√1−ζ](https://reader033.vdocument.in/reader033/viewer/2022050315/5f77a689204fd92bed52d8a4/html5/thumbnails/13.jpg)
Time Domain Analysis (1A) 13 Young Won Lim
10/27/14
Standard Form: varying zeta (2)
![Page 14: Time Domain Analysis (1A) - Wikimedia Commons · 2014-10-27 · Time Domain Analysis (1A) 7 Young Won Lim 10/27/14 2nd Order Transfer Function: Standard Form s =−ζωn ± j√1−ζ](https://reader033.vdocument.in/reader033/viewer/2022050315/5f77a689204fd92bed52d8a4/html5/thumbnails/14.jpg)
Time Domain Analysis (1A) 14 Young Won Lim
10/27/14
Standard Form: varying omega (1)
![Page 15: Time Domain Analysis (1A) - Wikimedia Commons · 2014-10-27 · Time Domain Analysis (1A) 7 Young Won Lim 10/27/14 2nd Order Transfer Function: Standard Form s =−ζωn ± j√1−ζ](https://reader033.vdocument.in/reader033/viewer/2022050315/5f77a689204fd92bed52d8a4/html5/thumbnails/15.jpg)
Time Domain Analysis (1A) 15 Young Won Lim
10/27/14
Standard Form: varying omega (2)
![Page 16: Time Domain Analysis (1A) - Wikimedia Commons · 2014-10-27 · Time Domain Analysis (1A) 7 Young Won Lim 10/27/14 2nd Order Transfer Function: Standard Form s =−ζωn ± j√1−ζ](https://reader033.vdocument.in/reader033/viewer/2022050315/5f77a689204fd92bed52d8a4/html5/thumbnails/16.jpg)
Young Won Lim10/27/14
References
[1] http://en.wikipedia.org/[2] M.L. Boas, “Mathematical Methods in the Physical Sciences”[3] E. Kreyszig, “Advanced Engineering Mathematics”[4] D. G. Zill, W. S. Wright, “Advanced Engineering Mathematics”