TMHP51 Servomechanisms (HT2012)
Lecture 10
Signal Analysis
Hydraulic System Evaluation
Magnus Sethson
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1
1
“Real World”
“Math & Physics”
Analysis
f
8
pL Ap Bp ẋp FL = Mt ẍp
>
>
>
<
q = Ap ẋp + 41 Vet ṗL
>
q
>
>
: q = Cq wxv 1 (Ps pL )
⇢
Transformation
t
Synthesis
“A Better World”
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2
2
f
Frequency Domain
Transformation
f
t
f
End-Value Theorem
t
f
Loop Gain
t
f
Fourier Transform
t
f
Digital Filtering
t
f
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Laplace Transform
t
Time Domain
Inverse Fourier Transform
t
3
3
Signal Analysis
A Short Repetition
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4
4
Useful Transforms
The Fourier Transform
F (!) =
Z
1
f (t) e
i!t
dt
1
The Laplace Transform
F (s) = lim
✏!0+
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Z
1
f (t) e
st
dt
✏
5
5
Sampling Theorem
Source: Signaler och System, Anders Svärdström
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6
6
Nyquist Frequency & Filtering
Source: Signaler och System, Anders Svärdström
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7
7
Ideal Signal vs. Sampled Signal
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8
8
Sampling Part of a Signal
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9
9
A Window Frame of a Signal
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10
10
Windows
My Favorites
Hanning
Blackman
Square
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11
11
Example
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12
12
Simulation model
t
f
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13
13
Model Disturbance Response
−3
−8.2
x 10
−8.4
−8.6
−8.8
−9
−9.2
−9.4
−9.6
6
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8
10
12
14
16
18
20
14
14
The “Real” Signal Response
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1
15
15
Windowed Signal and Removed Mean-Value
−4
x 10
Output
5
0
−5
6
8
10
12
14
16
18
20
14
16
18
20
Time
−4
Windowed Output
x 10
2
1
0
−1
6
8
10
12
Time
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16
16
Result, Resonance vs. Stick Length
5
Resonance frequency [Hz]
4.5
4
3.5
3
2.5
2
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0
2
4
6
r2
8
10
12
17
17
Result Damping
0.08
Relative damping [−]
0.07
0.06
0.05
0.04
0.03
0.02
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0
2
4
6
r2
8
10
12
18
18
Result Comparison
5
10
0
10
−5
10
0
10
1
2
10
10
2
10
0
10
−2
10
0
10
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1
2
10
10
19
19
Some Synthesis Issues
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20
20
“Worst case”
!h =
h
s
Kce
=
Ap
2
e Ap
Mt
s
✓
2
2
+
(Vt + Ap x) (Vt Ap x)
e Mt
✓
◆
2
2
+
(Vt + Ap x) (Vt Ap x)
Lc
Lc
x
2
2
◆
10
9
Frequency and damping versus displacement xp
of a symmetric cylinder
8
7
*+
6
;+
*+
,(
20 +0
+1 21
./
345'"67%84
5
9:
-
4
h/
!"#$#%&'"()
2
p
eA
3
h
Mt
1
V1
h0
1
V2
2
h
1
0
−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
Symetric Cylinder Position
0.2
0.3
0.4
K ce
Ap
e
Mt
1
V1
1
V2
h
0.5
Karl-Erik Rydberg, Linköping University, Sweden
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15
21
21
Minimize Jerk
F =ma
Ap p L
Bp ẋ
ks (x
x0 ) + Fexternal = mẍ
A sudden change in forces is similar
to a “hit”. Minimizing sudden
changes is the same thing as limiting
the acceleration or minimizing jerk.
It is a good strategy for increasing
the life time and endurance of
machinery
... dF
x /
dt
...
min x ! ramping ẍ
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22
22
Magnus Sethson
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23
23