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.
Control
ch4
ch8
ch10
aem-ch4
aem-ch5
Control Systems
Interactive control systems using
Next.js
Resolution 20
a 0.02
b 0.04
Natural Frequency Omegan 1, 越大反應時間越短
Damping ratio 0.7, 越大overshoot越小
T
p
=
π
ω
n
1
−
ζ
2
=
Tp = \frac{\pi}{\omega_n \sqrt{1 - \zeta^2}} = \
T
p
=
ω
n
1
−
ζ
2
π
=
4
.
3
9
9
4.399
4
.
3
9
9
%
o
v
e
r
s
h
o
o
t
=
e
−
(
ζ
π
/
1
−
ζ
2
)
∗
1
0
0
=
\%overshoot = e^{-(\zeta\pi / \sqrt{1 - \zeta^2})} * 100 = \
%
o
v
e
r
s
h
o
o
t
=
e
−
(
ζ
π
/
1
−
ζ
2
)
∗
1
0
0
=
4
.
5
9
9
4.599
4
.
5
9
9
T
s
=
4
ζ
ω
n
=
Ts = \frac{4}{\zeta \omega_n} = \
T
s
=
ζ
ω
n
4
=
5
.
7
1
4
5.714
5
.
7
1
4
T
r
=
0
.
8
+
2
.
5
∗
ζ
ω
n
=
Tr = \frac{0.8 + 2.5 * \zeta} {\omega_n} = \
T
r
=
ω
n
0
.
8
+
2
.
5
∗
ζ
=
2
.
5
5
0
2.550
2
.
5
5
0
-10
-8
-6
-4
-2
0
2
4
6
8
10
-10
-8
-6
-4
-2
0
2
4
6
8
10
m
y
′
′
+
c
y
′
+
k
y
=
0
my'' + cy' + ky = 0
m
y
′
′
+
c
y
′
+
k
y
=
0
y
′
′
+
c
m
y
′
+
k
m
y
=
0
y'' + \frac{c}{m}y' + \frac{k}{m}y = 0
y
′
′
+
m
c
y
′
+
m
k
y
=
0
[
y
1
′
y
2
′
]
=
[
0
1
−
k
m
−
c
m
]
[
y
1
y
2
]
\begin{bmatrix} y_1' \\ y_2' \end{bmatrix} = \begin{bmatrix} 0 & 1 \\ -\frac{k}{m} & -\frac{c}{m} \end{bmatrix} \begin{bmatrix} y_1 \\ y_2 \end{bmatrix}
[
y
1
′
y
2
′
]
=
[
0
−
m
k
1
−
m
c
]
[
y
1
y
2
]
p
=
a
1
1
+
a
2
2
=
−
c
m
p = a_{11} + a_{22} = -\frac{c}{m}
p
=
a
1
1
+
a
2
2
=
−
m
c
q
=
d
e
t
(
A
)
=
k
m
q = det(A) = \frac{k}{m}
q
=
d
e
t
(
A
)
=
m
k
Δ
=
p
2
−
4
q
=
c
2
m
2
−
4
k
m
=
c
2
−
4
m
k
m
2
\Delta = p^2 - 4q = \frac{c^2}{m^2} - \frac{4k}{m} = \frac{c^2 - 4mk}{m^2}
Δ
=
p
2
−
4
q
=
m
2
c
2
−
m
4
k
=
m
2
c
2
−
4
m
k
No Damping:
c
=
0
c = 0
c
=
0
p
=
0
p = 0
p
=
0
q
=
k
m
>
0
q = \frac{k}{m} > 0
q
=
m
k
>
0
Δ
=
−
4
k
m
<
0
\Delta = -\frac{4k}{m} < 0
Δ
=
−
m
4
k
<
0
Center.
Under Damping:
c
2
<
4
m
k
c^2 < 4mk
c
2
<
4
m
k
p
=
−
c
m
<
0
p = -\frac{c}{m} < 0
p
=
−
m
c
<
0
q
=
k
m
>
0
q = \frac{k}{m} > 0
q
=
m
k
>
0
Δ
=
−
c
2
−
4
m
k
m
2
<
0
\Delta = -\frac{c^2 - 4mk}{m^2} < 0
Δ
=
−
m
2
c
2
−
4
m
k
<
0
Stable spiral.
Critical Damping:
c
2
=
4
m
k
c^2 = 4mk
c
2
=
4
m
k
p
=
−
c
m
<
0
p = -\frac{c}{m} < 0
p
=
−
m
c
<
0
q
=
k
m
>
0
q = \frac{k}{m} > 0
q
=
m
k
>
0
Δ
=
0
\Delta = 0
Δ
=
0
Stable degenerated node.
Over Damping:
c
2
>
4
m
k
c^2 > 4mk
c
2
>
4
m
k
p
=
−
c
m
<
0
p = -\frac{c}{m} < 0
p
=
−
m
c
<
0
q
=
k
m
>
0
q = \frac{k}{m} > 0
q
=
m
k
>
0
Δ
=
−
c
2
−
4
m
k
m
2
>
0
\Delta = -\frac{c^2 - 4mk}{m^2} > 0
Δ
=
−
m
2
c
2
−
4
m
k
>
0
Stable spiral.