Underdamped Unstable - Michigan State University€¦ · · 2007-02-19Second order system...
Transcript of Underdamped Unstable - Michigan State University€¦ · · 2007-02-19Second order system...
Second order system response.
Unstable
Re(s)
Im(s)
Und
ampe
d
Overdamped or Critically damped
Underdamped
Underdamped
Polar vs. Cartesian representations.
System transfer function :
Significance of the damping ratio : … Overdamped
… Critically damped… Underdamped
… Undamped
All 4 cases Unless overdamped
… Cartesian overdamped
Polar vs. Cartesian representations.
System transfer function :
Significance of the damping ratio : … Overdamped
… Critically damped… Underdamped
… Undamped
All 4 cases Unless overdamped
… Cartesian overdamped
Polar vs. Cartesian representations.
System transfer function :
Significance of the damping ratio : … Overdamped
… Critically damped… Underdamped
… Undamped
All 4 cases Unless overdamped
Overdamped case:
… Cartesian overdamped
Second order impulse response – Underdamped and Undamped
Increasing / Fixed
Impulse Response
Time (sec)
Ampl
itude
0 0.5 1 1.5 2 2.5 3-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Impulse Response
Time (sec)
Ampl
itude
0 0.5 1 1.5 2 2.5 3-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Second order impulse response – Underdamped and Undamped
Increasing / Fixed
Impulse Response
Time (sec)
Ampl
itude
0 0.5 1 1.5 2 2.5 3-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Second order impulse response – Underdamped and Undamped
Increasing / Fixed
Impulse Response
Time (sec)
Ampl
itude
0 0.5 1 1.5 2 2.5 3-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Second order impulse response – Underdamped and Undamped
Increasing / Fixed
Second order impulse response – Underdamped and Undamped
Increasing / Fixed
Impulse Response
Time (sec)
Ampl
itude
0 0.5 1 1.5 2 2.5 3-0.1
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
0.1
Second order impulse response – Underdamped and Undamped
Increasing / Fixed
Impulse Response
Time (sec)
Ampl
itude
0 0.5 1 1.5 2 2.5 3-0.1
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
0.1
Second order impulse response – Underdamped and Undamped
Increasing / Fixed
Impulse Response
Time (sec)
Ampl
itude
0 0.5 1 1.5 2 2.5 3-0.1
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
0.1
Second order impulse response – Underdamped and Undamped
Increasing / Fixed
Impulse Response
Time (sec)
Ampl
itude
0 0.5 1 1.5 2 2.5 3-0.1
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
0.1
Second order impulse response – Underdamped and Undamped
Increasing / Fixed
Impulse Response
Time (sec)
Ampl
itude
0 0.5 1 1.5 2 2.5 3 3.5 4-1
0
1
2
3
4
5
6
-6 -4 -2 0 2-10
-5
0
5
10
Second order impulse response – Underdamped and Undamped
Increasing / Fixed
Impulse Response
Time (sec)
Ampl
itude
0 0.5 1 1.5 2 2.5 3 3.5 4-1
0
1
2
3
4
5
6
-6 -4 -2 0 2-10
-5
0
5
10
Second order impulse response – Underdamped and Undamped
Increasing / Fixed
Impulse Response
Time (sec)
Ampl
itude
0 0.5 1 1.5 2 2.5 3 3.5 4-1
0
1
2
3
4
5
6
-6 -4 -2 0 2-10
-5
0
5
10
Second order impulse response – Underdamped and Undamped
Increasing / Fixed
Impulse Response
Time (sec)
Ampl
itude
0 0.5 1 1.5 2 2.5 3 3.5 4-1
0
1
2
3
4
5
6
-6 -4 -2 0 2-10
-5
0
5
10
Second order impulse response – Underdamped and Undamped
Increasing / Fixed
Impulse Response
Time (sec)
Ampl
itude
0 2 4 6 8 10 12-4
-3
-2
-1
0
1
2
3
4
5
-5 0 5-6
-4
-2
0
2
4
6
Second order impulse response – Underdamped and Undamped
Increasing / Fixed
Impulse Response
Time (sec)
Ampl
itude
0 2 4 6 8 10 12-4
-3
-2
-1
0
1
2
3
4
5
-5 0 5-6
-4
-2
0
2
4
6
Second order impulse response – Underdamped and Undamped
Increasing / Fixed
Impulse Response
Time (sec)
Ampl
itude
0 2 4 6 8 10 12-4
-3
-2
-1
0
1
2
3
4
5
-5 0 5-6
-4
-2
0
2
4
6
Second order impulse response – Underdamped and Undamped
Increasing / Fixed
Impulse Response
Time (sec)
Ampl
itude
0 2 4 6 8 10 12-4
-3
-2
-1
0
1
2
3
4
5
-5 0 5-6
-4
-2
0
2
4
6
Second order step response – Underdamped and Undamped
0 5−2
−1
0
1
2
3
0 5−2
−1
0
1
2
3
0 5−2
−1
0
1
2
3
0 5−2
−1
0
1
2
3
0 5−2
−1
0
1
2
3
0 5−2
−1
0
1
2
3
0 10 200
0.5
1
1.5
0 10 20−10
0
10
20
30
40
−2+j0.5zeta=0.97
−10+j0.5zeta=0.998 +j0.5
zeta=0
0.2+j0.5
0.2+j5+j5zeta=0
−2+j5zeta=0.3714
−10+j5zeta=0.8944
time sec.
output
time sec. time sec. time sec.
time sec. time sec. time sec. time sec.
output
Second order impulse response – Underdamped and Undamped
Unstable
Faster response Slower response
Higher frequency oscillations
Lower frequency oscillations
Second order step response – Time specifications.
0 0.5 1 1.5 2 2.5 30
0.2
0.4
0.6
0.8
1
1.2
1.4
… Steady state value.… Time to reach first peak (undamped or underdamped only).… % of in excess of .… Time to reach and stay within 2% of .… Time to rise from 10% to 90% of .
Second order step response – Time specifications.
… Steady state value.
More generally, if the numerator is not , but some :