Young Won Lim10/27/14

Time Domain Analysis (1A)

Young Won Lim10/27/14

Copyright (c) 2014 Young W. Lim.

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Time Domain Analysis (1A) 3 Young Won Lim

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2nd Order Systems

9

s2+9 s+9

9

s2+2 s+9

9

s2+9

9

s2+6 s+9

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Step Responses

9

s2+9 s+9

9

s2+2 s+9

9

s2+9

9

s2+6 s+9

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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

Time Domain Analysis (1A) 6 Young Won Lim

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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

Time Domain Analysis (1A) 7 Young Won Lim

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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

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Standard Form: varying a (1)

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Standard Form: varying a (2)

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Standard Form: varying b (1)

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Standard Form: varying b (2)

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Standard Form: varying zeta (1)

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Standard Form: varying zeta (2)

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Standard Form: varying omega (1)

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Standard Form: varying omega (2)

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”