Staff Site Universitas Negeri Yogyakarta

SEE 2113 KAWALAN: PEMODELAN DAN SIMULASI

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The General Second-Order System

• Two important quantities that describes the
response of second order systems:

o Natural frequency, ωn : The frequency of
oscillation of the system without damping.
o Damping ratio, ζ : Parameter that describes
the damped oscillations of the 2nd order
response.
Bigger, means more ‘damped’
response, i.e. less oscillations.

ζ =
=

Exponential decay frequency

Natural frequency (rad/second )

1 Natural period (seconds)
2π Exponential time sonstant

• Define general 2nd order response in terms of ωn
and ζ as:
• Hence the pole is given as:

ZHI

SEE 2113 KAWALAN: PEMODELAN DAN SIMULASI

4-22

• Four time responses based on ζ :

ZHI

SEE 2113 KAWALAN: PEMODELAN DAN SIMULASI

Example:

Given

the

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transfer

ωn 2
G(s) = 2
, find ζ and ωn .
2
s + 2ζω n s + ωn

function

Example: Find the value of ζ , and sketch the

kind of response expected.

ZHI

SEE 2113 KAWALAN: PEMODELAN DAN SIMULASI

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Underdamped Second-Order Systems
• A common model for physical problems.

• A detailed description of the underdamped
response is necessary for both analysis and design.

ωn 2
C ( s) =
s ( s 2 + 2ζω n s + ωn 2 )
=

K 2 s + K3

K1
, ζ