( Q J L Q H H U L Q J ' H V L J Q D Q G 5 D S L G 3 U R W R W \ S L Q J
/ H F W X U H
6 W U X F W X U D O 7 H V W L Q J
, Q V W U X F W R U V
3 U R I 2 O L Y L H U G H : H F N
- D Q X D U \
2 X W O L Q H
? 6 W U X F W X U D O 7 H V W L Q J
? : K \ W H V W L Q J L V L P S R U W D Q W
? 7 \ S H V R I 6 H Q V R U V 3 U R F H G X U H V ?
? 0 D V V 6 W D W L F ' L V S O D F H P H Q W ' \ Q D P L F V
? 7 H V W 3 U R W R F R O I R U
? ( [ S O D L Q S U R W R F R O
? 6 L J Q X S I R U W L P H V O R W V
' D W D $ F T X L V L W L R Q D Q G 3 U R F H V V L Q J
I R U 6 W U X F W X U D O 7 H V W L Q J
6 H Q V R U 2 Y H U Y L H Z
$ F F H O H U R P H W H U V / D V H U V H Q V R U V 6 W U D L Q * D J H V
) R U F H 7 U D Q V G X F H U V D Q G / R D G & H O O V * \ U R V F R S H V
6 H Q V R U & K D U D F W H U L V W L F V ' \ Q D P L F V
) 5 ) R I V H Q V R U V E D Q G Z L G W K U H V R O X W L R Q S O D F H P H Q W L V V X H V
' D W D $ F T X L V W L R Q 3 U R F H V V
( [ F L W D W L R Q 6 R X U F H V 1 R Q O L Q H D U L W \ $ Q W L $ O L D V ) L O W H U L Q J 6 L J Q D O
& R Q G L W L R Q L Q J
' D W D 3 R V W 3 U R F H V V L Q J
) ) 7 ' ) 7 & R P S X W L Q J 3 6 '
V D Q G D P S O L W X G H V S H F W U D
V W D W L V W L F D O Y D O X H V R I D V L J Q D O V X F K D V 5 0 6 F R Y D U L D Q F H H W F
, Q W U R G X F W L R Q W R 6 \ V W H P , G H Q W L I L F D W L R Q
( 7 ) ( ' \ Q D 0 R G 0 H D V X U H P H Q W 0 R G H O V
: K \ L V 6 W U X F W X U D O 7 H V W L Q J , P S R U W D Q W "
? 3 U R G X F W 4 X D O L I L F D W L R Q 7 H V W L Q J
( [ D P S O H * U R X Q G 9 L E U D W L R Q 7 H V W L Q J X Q G 9 L
? 3 H U I R U P D Q F H $ V V H V V P H Q W
? 6 \ V W H P , G H Q W L I L F D W L R Q
? ' H V L J Q 9 H U L I L F D W L R Q
? ' D P D J H $ V V H V V P H Q W
? $ H U R G \ Q D P L F ) O X W W H U 7 H V W L Q J
? 2 S H U D W L R Q D O 0 R Q L W R U L Q J
? 0 D W H U L D O ) D W L J X H 7 H V W L Q J
V W L P X O X V
U H V S R Q V H
5 H I K W W S Z Z Z D I P L O S K R W R V 0 D \ I K W P O
X W
[ W
) 5 D S W R U G X U L Q J J U R X Q G
Y L E U D W L R Q W H V W V D W ( G Z D U G V $ L U L U
) R U F H % D V H & D O L I L Q $ S U L O
' $ 4 G D W D D F T X L V L W L R Q
' 6 3 G L J L W D O V L J Q D O S U R F H V V L Q J
6 W U X F W X U D O
6 \ V W H P
' $ 4 ' 6 3
(Image is taken from U.S. Air Force Web site: http://www.af.mil.)
, 6 H Q V R U 2 Y H U Y L H Z
7 K L V 6 H Q V R U P R U S K R O R J \ L V X V H I X O I R U F O D V V L I L F D W L R Q
R I W \ S L F D O V H Q V R U V X V H G L Q V W U X F W X U D O G \ Q D P L F V
6 H Q V R U 0 R U S K R O R J \ 7 D E O H
7 \ S H
% D Q G Z L G W K
' H U L Y D W L Y H
5 H I H U H Q F H
4 X D Q W L W \
, P S H G D Q F H
/ L Q H D U 5 R W D W L R Q D O
/ R Z 0 H G L X P + L J K
3 R V L W L R Q 5 D W H $ F F H O H U D W L R Q
$ E V R O X W H 5 H O D W L Y H
) R U F H 7 R U T X H ' L V S O D F H P H Q W
/ R Z + L J K
( [ D P S O H X Q L D [ L D O V W U D L Q J D J H
1 H H G X Q L W V R I P H D V X U H P H Q W
> P @ > 1 P @ >μ V W U D L Q @ > U D G @ H W F ?
6 H Q V R U ( [ D P S O H V I R U 6 W U X F W X U D O ' \ Q D P L F V
( [ D P S O H I L [ H G I L [ H G
* R D O ( [ S O D L Q Z K D W W K H \ P H D V X U H D Q G K R Z W K H \ Z R U N
E H D P Z L W K F H Q W H U O R D G
V K D N H U
O D V H U V H Q V R U V
D F F H O H U R P H W H U V
O V
L Y H
V H Q V R U V
) L O
( )
Q
E
( ,
O P P
ω =
+
P
E
P
O
V W U D L Q J D J H V
J \ U R V F R S H V
O R D G F H O
J U R X Q G
L Q G X F W
H [ F L W D W L R Q
U V W I H [ L E O H P R G H I U H T X H Q F \
6 W U D L Q * D J H V
6 W U D L Q
? O
6 W U D L Q J D J H V P H D V X U H V W U D L Q G L I I H U H Q W L D O
G L V S O D F H P H Q W R Y H U D I L Q L W H D U H D Y L D D
ε =
F K D Q J H L Q H O H F W U L F D O U H V L V W D Q F H 5 Oρ >? @
O
R
& X U U H Q W 1 R P L Q D O O H Q J W K O
, =
9
L Q
R
R
O ρ
R
O
R : L W K D S S O L H G V W U D L Q ,
ε
=
9
L Q
O +? O ρ
R
, P S O H P H Q W D W L R Q
: K H D W V W R Q H E U L G J H
F L U F X L W
E R Q G W R W H V W
V W U D L Q J D J H V I H D W X U H
D U W L F O H
S R O \ L P L G H H Q F D S V X O D W H G
* 1 '
F R Q V W D Q W D Q J U L G V Z L W K
F R S S H U F R D W H G V R O G H U W D E V
9
0 I J
5 H I K W W S Z Z Z P H D V X U H P H Q W V J U R X S F R P
$ F F H O H U R P H W H U V
$ F F H O H U R P H W H U V P H D V X U H O L Q H D U
D F F H O H U D W L R Q L Q R Q H W Z R R U W K U H H
D [ H V : H G L V W L Q J X L V K
? V L Q J O H Y V P X O W L D [ L V D F F H O H U R P H W H U V
? ' & Y H U V X V Q R Q ' & D F F H O H U R P H W H U V
→ V ; V ? V [ ?
G [
[ W
G W
J H Q H U D O O \ Q H J O H F W L Q L W L D O F R Q G L W L R Q V
P D J
$ F F H O H U R P H W H U
U H V S R Q V H W R
E D V H P R W L R Q
G % G H F
U R O O R I I
5 H F R U G H G Y R O W D J H
ω
= . [ W 9 W
+ 9
R X W D
& D Q P H D V X U H O L Q H D U F H Q W U L I X J D O D Q G J U D Y L W D W L R Q D O D F F H O H U D W L R Q
8 V H F D X W L R Q Z K H Q G R X E O H L Q W H J U D W L Q J D F F H O H U D W L R Q W R J H W S R V L W L R Q G U L I W
6 L Q J O H $ [ L V
&
$ F F H O H U R P H W H U P X V W E H
D O L J Q H G Z L W K V H Q V L Q J D [ L V
&
( [ D P S O H . L V W O H U 3 L H ] R E H D P
Q R W U H V S R Q V L Y H D W ' &
( [ D P S O H 6 X P P L W F D S D F L W L Y H
0 D Q X I D F W X U H U V . L V W O H U 9 L E U R P H W H U 6 X P P L W
D F F H O H U R P H W H U ' & F D S D E O H
D
/ D V H U ' L V S O D F H P H Q W 6 H Q V R U V
5 H F R U G V G L V S O D F H P H Q W G L U H F W O Y L D V O D Q W U D Q J H D Q W U D Q J H P H D V X U H P H Q W
W D U J H W
[ W ; V→
9
R X W
[ > P @
> 9 ' & @
' L V W D Q F H [ L V U H F R U G H G Y L D W U L D Q J X O D W L R Q
E H W Z H H Q W K H O D V H U G L R G H H P L W W H U W K H
W D U J H W D Q G W K H U H F H L Y H U
[
V H Q V R U
&
/
U D Q J H
7 \ S L F D O 6 H W W L Q J V
, μ P P V
, , μ P P V
, , , μ P P V
5 H V R O X W L R Q W U D G H R I I
V S D W L D O Y V W H P S R U D O
S R V L W L R Q V H Q V L W L Y H G H Y L F H 3 6 '
$ G Y D Q W D J H V
9 L E U R P H W H U V L Q F O X G H D G Y D Q F H G S U R F H V V L Q J
F R Q W D F W I U H H P H D V X U H P H Q W
D Q G V F D Q Q L Q J F D S D E L O L W L H V
' L V D G Y D Q W D J H V
0 D Q X I D F W X U H U V . H \ H Q F H 0 7 , , Q V W U X P H Q W V
Q H H G U H I O H F W L Y H I O D W W D U J H W
O L P L W H G U H V R O X W L R Q a μ P
) R U F H 7 U D Q V G X F H U V / R D G & H O O V
) R U F H 7 U D Q V G X F H U V / R D G & H O O V D U H F D S D E O H R I P H D V X U L Q J
8 S W R ' 2 ) R I I R U F H R Q W K U H H R U W K R J R Q D O D [ H V
D Q G W K H P R P H Q W W R U T X H D E R X W H D F K D [ L V W R
F R P S O H W H O \ G H I L Q H W K H O R D G L Q J D W W K H V H Q V R U
V O R F D W L R Q
7 K H K L J K V W L I I Q H V V D O V R U H V X O W V L Q D K L J K U H V R Q D Q W I U H T X H Q F \
D O O R Z L Q J D F F X U D W H V H Q V R U U H V S R Q V H W R U D S L G I R U F H F K D Q J H V
/ R D G F H O O V D U H H O H F W U R P H F K D Q L F D O W U D Q V G X F H U V
W K D W W U D Q V O D W H I R U F H R U Z H L J K W L Q W R Y R O W D J H
7 K H \ X V X D O O \ F R Q W D L Q V W U D L Q J D J H V L Q W H U Q D O O \
0 D Q X I D F W X U H U V - 5 7 U D Q V G X F H U 7 H F K Q L T X H V , Q F
) [
) ]
) \
0 [
0 ]
0 \
2 W K H U 6 H Q V R U V
? ) L E H U 2 S W L F V W U D L Q V H Q V R U V % U D J J * U D W L Q J V
L Q J
V S D F L Q J Z K
,
λ
U H I O H F W L R Q
L
O
L
L L
L Q S X W
W U D Q V P L W W H G
R S W L F D O I E H U
% U R D G V S H F W U X P L Q S X W O L J K W V U H I H F W H G R Q O \ D W D
V S H F L I F Z D Y H O H Q J W K G H W H U P L Q H G E \ W K H J U D W
F K Y D U L H V Z W K V W U D L Q
? 5 L Q J / D V H U * \ U R V F R S H V 6 D J Q D F ( I I H F W
? 3 9 ' ) R U 3 = 7 V H Q V R U V
, , 6 H Q V R U & K D U D F W H U L V W L F V ' \ Q D P L F V
* R D O ( [ S O D L Q S H U I R U P D Q F H F K D U D F W H U L V W L F V D W W U L E X W H V R I U H D O V H Q V R U V
: K H Q F K R R V L Q J D V H Q V R U I R U D
S D U W L F X O D U D S S O L F D W L R Q Z H P X V W
V S H F L I \ W K H I R O O R Z L Q J U H T X L U H P H Q W V
6 H Q V R U 3 H U I R U P D Q F H 5 H T X L U H P H Q W V
? ' \ Q D P L F 5 D Q J H D Q G 6 S D Q
? $ F F X U D F \ D Q G 5 H V R O X W L R Q
? $ E V R O X W H R U 5 H O D W L Y H P H D V X U H P H Q W
? 6 H Q V R U 7 L P H & R Q V W D Q W
? % D Q G Z L G W K
? / L Q H D U L W \
? , P S H G D Q F H
? 5 H O L D E L O L W \ 0 7 % )
& R Q V W U D L Q W V
3 R Z H U 9 ' & + ] $ & + ] $ &
& R V W : H L J K W 9 R O X P H ( 0 , + H D W
6
;
V D W X U D W L R Q O L P L W
G \ Q D P L F
U D Q J H
O L Q H D U
Q R Q O L Q H D U
6 ;
<
& D O L E U D W L R Q L V W K H S U R F H V V R I
R E W D L Q L Q J W K H 6 ; U H O D W L R Q V K L S
I R U D Q D F W X D O V H Q V R U , Q W K H S K \ V L F D O
Z R U O G 6 G H S H Q G V R Q W K L Q J V R W K H U W K D Q ;
& R Q V L G H U P R G L I \ L Q J L Q S X W < H J 7 H P S
( J / R D G F H O O F D O L E U D W L R Q G D W D
; P D V V N J ?
6 Y R O W D J H P 9
P
F N
[ W
[
E
W
( [ D P S O
P
D
E
V ; V F V N
* V
V ; V P V F
+
= =
++
6 H Q V R U ) U H T X H Q F \ 5 H V S R Q V H ) X Q F W L R Q
H $ F F H O H U R P H W H U
P J
N H 1
F 1 V P
V N
0
D J Q
L
W
X
G H
> G %
O
, G H D O $ F F H O H U R P H W H U ) 5 ) I U R P % D V H 0 R W L R Q
E D Q G Z L G W K
7 \ S L F D O O \ V S H F L I \ E D Q G Z L G W K D V I R O O R Z V
) U H T X H Q F \ > + ] @
( [ D P S O H . L V W O H U % $ F F H O H U R P H W H U
1 R W H % D Q G Z L G W K R I V H Q V R U V K R X O G
) U H T X H Q F \ 5 H V S R Q V H + ]
E H D W O H D V W W L P H V K L J K H U W K D Q
K L J K H V W I U H T X H Q F \ R I V L J Q D O V W
6 H Q V R U 7 L P H & R Q V W D Q W
+ R Z T X L F N O \ G R H V W K H V H Q V R U U H V S R Q G W R L Q S X W F K D Q J H V "
) L U V W 2 U G H U , Q V W U X P H Q W V
6 H F R Q G 2 U G H U , Q V W U X P H Q W V
D
G
G + D
G + D \ E X
+ D \ = E X
D
=
R R
R R
G W
G W G W
' L Y L G L Q J E \ D
R
J L Y H V
( V V H Q W L D O S D U D P H W H U V D U H
D G \ E
R
+= X
R
D G W D
.
E
R
ω
D
ζ
D
Q
R
N N
D
R
Q
D
D D
R
τ .
V W D W L F
Q D W X U D O
V H Q V L W L Y L W \
I U H T X H Q F \
G D P S L Q J U D W L R
, Q V G R P D L Q
. . < V < V
=
, Q V G R P D L Q
=
τ V +
8 V ω ω
8 V
V + ζ V +
Q Q Q
7 L P H F R Q V W D Q W K H U H L V
τ ζ= ω
τ W L P H F R Q V W D Q W
Q
. V W D W L F V H Q V L W L Y L W \
7 L P H I R U D H R X W S X W F K D Q J H
6 H Q V R U 5 D Q J H 5 H V R O X W L R Q
U D Q J H V S D Q
U H V R O X W L R Q
' \ Q D P L F U D Q J H U D W L R R I O D U J H V W W R
V P D O O H V W G \ Q D P L F L Q S X W D V H Q V R U P H D V X U H V
G %
O R J 1
/ R Z R S H U D W L Q J O L P L W
+ L J K R S H U D W L Q J O L P L W
V D W X U D W L R Q
+ \ V W H U H V L V G X H W R V G X H W R
L Q W H U Q D O V H Q V R U
I U L F W L R Q
L Q S X W
R X W S X W
W K U H V K R O G
5 H V R O X W L R Q V P D O O H V W L Q S X W L Q F U H P H Q W W K D W J L Y H V 7 K U H V K R O G V P D O O H V W
U L V H W R D P H D V X U D E O H R X W S X W F K D Q J H 5 H V R O X W L R Q P H D V X U D E O H L Q S X W
D Q G D F F X U D F \ D U H 1 2 7 W K H V D P H W K L Q J
$ F F X U D F \
0 H D V X U H P H Q W W K H R U \ H V V H Q W L D O O \ H U U R U W K H R U \
7 R W D O H U U R U U D Q G R P H U U R U V V \ V W H P D W L F H U U R U V
$ F F X U D F \ O D F N R I H U U R U V
% , $ 6
, 0 3 5 ( & , 6 , 2 1
? 7 H P S H U D W X U H I O X F W X D W L R Q V
? , Q Y D V L Y H Q H V V R I V H Q V R U
? H [ W H U Q D O Y L E U D W L R Q V
? 6 S D W L D O D Q G W H P S R U D O D Y H U D J L Q J
? H O H F W U R Q L F Q R L V H D P S O L I L H U
? K X P D Q E L D V
±3σσ X Q F H U W D L Q W \ O L P L W V
? S D U D O O D [ H U U R U V
? I U L F W L R Q P D J Q H W L F I R U F H V K \ V W H U H V L V
E L D V
σ D F F X U D F \ T X R W H G D V
3 ? με ′
με
3 U R E D E O H H U U R U D F F X U D F \
5 H D G L Q J
% H V W
7 U X H
( V W L P D W H
9 D O X H X Q N Q R Z Q H σ 3 ? με ′
S
1 R W H , Q V W U X P H Q W V W D Q G D U G X V H G I R U F D O L E U D W L R Q V K R X O G E H a W L P H V P R U H
D F F X U D W H W K D Q W K H V H Q V R U L W V H O I 1 D W L R Q D O 6 W D Q G D U G V 3 U D F W L F H
/ L Q H D U L W \
3 , Q G H S H Q G H Q W / L Q H D U L W \ ′
8 V X D O O \ K D Y H O D U J H V W H U U R U V D W
I X O O V F D O H G H I O H F W L R Q R I V H Q V R U
$ S H U F H Q W R I U H D G L Q J R U
% S H U F H Q W R I I X O O V F D O H Z K L F K H Y H U
L V J U H D W H U
% R I I X O O V F D O H
$ R I U H D G L Q J
2 X W S X W
G 6 a P 9 o
?
G ;
?
N J
?
?
3 R L Q W D W Z K L F K $ R I U H D G L Q J
% R I I X O O V F D O H L Q S X W
) X O O V F D O H
L
, Q F U H D V L Q J Y D O X H V
L W
6 W D W L F 6 H Q V L W Y L W \
V O R S H R I F D O L E U D W L R Q F X U Y H
O H D V W V T X D U H V I
, Q S X W
G H F U H D V L Q J Y D O X H V
1 H H G W R F R Q V L G H U W K H G \ Q D P L F V
R I W K H V W U X F W X U H W R E H W H V W H G
E H I R U H F K R R V L Q J Z K H U H W R S O D F H
V H Q V R U V
3 O D F H P H Q W , V V X H V
( [ D P S O H 7 3 ) 6 & , $ U F K L W H F W X U H
a
,
o a
o
T =
?
?? ? ?
?
?
T +
?
?
Φ β
X ?
?
X
7 ?
=
β a Φ
?
$ %
\ = o T +
[]
X
?
N
0 R G H D W + ] X Q R E V H U Y D E O H
&
'
L I V H Q V R U S O D F H G D W W K L V 3 Q R G H ′
2 E V H U Y D E L O L W \ G H W H U P L Q H G
2 E V H U Y D E L O L W \ J U D P L D Q
E \ S U R G X F W R I P R G H V K D S H P D W U L [
Φ D Q G R X W S X W L Q I O X H Q F H F R H I I L F L H Q W V β
\
7
+
7
: → $ :+ : $ & &=
R R R
2 W K H U F R Q V L G H U D W L R Q V
? 3 R O H ] H U R S D W W H U Q L I V H Q V R U X V H G I R U F R Q W U R O F R O O R F D W H G V H Q V R U D F W X D W R U S D L U
? 3 O D F H P H Q W F R Q V W U D L Q W V Y R O X P H Z L U L Q J V X U I D F H S U R S H U W L H V H W F ?
, Q Y D V L D E L O L W \ , P S H G D Q F H
+ R Z G R H V W K H P H D V X U H P H Q W V H Q V R U L Q I O X H Q F H W K H S K \ V L F V R I W K H V \ V W H P "
5 H P H P E H U + H L V H Q E H U J ? V X Q F H U W D L Q W \ S U L Q F L S O H
?? ≥ = [ S
V H Q V R U
[
? T T
L
= 3 > : @
L
3 R Z H U
T =
N
)
N
V
G U D L Q
3 =
L J L
, P S H G D Q F H F K D U D F W H U L ] H V 3 O R D G L Q J ′ H I I H F W R I V H Q V R U R Q W K H V \ V W H P
6 H Q V R U H [ W U D F W V S R Z H U H Q H U J \ ! & R Q V L G H U 3 L P S H G D Q F H ′ D Q G 3 D G P L W W D Q F H ′
( I I R U W Y D U L D E O H
) R U F H
: D Q W W R * H Q H U D O L ] H G
=
J L
T
T
L
) O R Z Y D U L D E O H
9 H O R F L W \ P H D V X U H )
L Q S X W L P S H G D Q F H
L
( U U R U G X H W R P H D V X U H P H Q W T =
T
=
J L
∝ N
V
L P
= =
J L
+
L X
J R
3 X Q G L V W X U E H G ′
3 P H D V X U H G ′
/ R D G & H O O + L J K , P S H G D Q F H N
V
O D U J H Y V 6 W U D L Q * D J H / R Z , P S H G D Q F H N
V
V P D O O
& R Q F O X V L R Q , P S H G D Q F H R I V H Q V R U V O H D G W R H U U R U V W K D W P X V W E H P R G H O H G L Q
L P S H G D Q F H G \ Q D P L F V
D K L J K D F F X U D F \ P H D V X U H P H Q W F K D L Q , H L Q F O X G H V H Q V R U
, , , ' D W D $ F T X L V L W L R Q 3 U R F H V V
' L V S O D \
6 V L J Q D O Y D U L D E O H
7 \ S L F D O V H W X S
;
7 U D Q V G X F H U
$ P S O L I L H U
) L O W H U
$ ' & R Q Y H U W H U
' 6 3
6
7 U D Q V G X F H U 3 W U D Q V I R U P V ′ W K H S K \ V L F D O
Y D U L D E O H ; W R D P H D V X U H G V L J Q D O
$ P S O L I L H U L V X V H G W R L Q F U H D V H W K H
P H D V X U H P H Q W V L J Q D O V W U H Q J W K
) L O W H U L V X V H G W R U H M H F W X Q Z D Q W H G
Q R L V H I U R P W K H P H D V X U H P H Q W V L J Q D O
$ Q D O R J W R ' L J L W D O F R Q Y H U W H U V D P S O H V
W K H F R Q W L Q X R X V P H D V X U H P H Q W V L J Q D O
L Q W L P H D Q G L Q D P S O L W X G H
* R D O ( [ S O D L Q W K H P H D V X U H P H Q W F K D L Q
( [ F L W D W L R Q
( [ F L W D W L R Q V R X U F H S U R Y L G H V S R Z H U W R
W K H V W U X F W X U D O V \ V W H P V X F K W K D W D G \ Q D P L F
U H V S R Q V H L V R E V H U Y D E O H L Q W K H I L U V W S O D F H
' L J L W D O 6 L J Q D O 3 U R F H V V L Q J W X U Q V U D Z G L J L W D O
V H Q V R U G D W D L Q W R X V H I X O G \ Q D P L F V L Q I R U P D W L R Q
; S K \ V L F D O Y D U L D E O H
6 \ V W H P ( [ F L W D W L R Q 7 \ S H V
2 Y H U Y L H Z R I ( [ F L W D W L R Q 7 \ S H V
7 \ S H $ , P S X O V L Y H ( [ F L W D W L R Q
7 \ S H
'
7 \ S H
&
7
\ S H
%
7 \ S H
$
, P S X O V H + D P P H U V
7 \ S H % % U R D G E D Q G 1 R L V H
( O H F W U R P H F K D Q L F D O 6 K D N H U V
7 \ S H & 3 H U L R G L F 6 L J Q D O V
1 D U U R Z E D Q G ( [ F L W D W L R Q
7 \ S H ' ( Q Y L U R Q P H Q W D O
6 O H Z L Q J : L Q G * X V W V 5 R D G
7 H V W W U D F N : D Y H V
7 L P H > V H F @
( [ F L W D W L R Q 6 R X U F H V
) W ) Wδ X W==
: L G H E D Q G H [ F L W D W L R Q D W Y D U L R X V H Q H U J \ O H Y H O V
R
F D Q E H D S S O L H G W R D V W U X F W X U H X V L Q J L P S X O V H I R U F H
K D P P H U V 7 K H \ J H Q H U D W H D Q H D U O \ S H U I H F W L P S X O V H
W
* ω
G = X τ K W?ττ F R Q Y R O X W L R Q L Q W H J U D O
X W \ W
\ W
3
?∞
ω + ω
, P S X O V H 5 H V S R Q V H K W
< = 8
* = + ωω
ω
N
) ?
Q R Q R L V H
% U R D G E D Q G
R
7 K H Q R L V H I U H H U H V S R Q V H W R D Q L G H D O L P S X O V H F R Q W D L Q V
D O O W K H L Q I R U P D W L R Q D E R X W W K H / 7 , V \ V W H P G \ Q D P L F V
6 K D N H U F D Q E H G U L Y H Q E \ S H U L R G L F R U
E U R D G E D Q G U D Q G R P F X U U H Q W I U R P D V L J Q D O J H Q H U D W R U
5 H F R U G
+
6
\ 6
X X
6 = * Mω 6 * Mω ωω
\ \ X X
D Q G V R O Y H
, Q S X W 3 6 '
I R U *
2 X W S X W 3 6 '
( [ D P S
( [ F L W D W L R Q $ P S O L W X G H 1 R Q / L Q H D U L W \
O H I U R P 0 2 ' ( ( [ S H U L P H Q W L Q μ G \ Q D P L F V W R U V L R Q P R G H
3 O R W V & R X U W H V \
0 L W F K , Q J K D P
G D
P S L Q
J
& R Q F O X V L R Q / L Q H D U L W \ L V R Q O \
S U H V H U Y H G I R U U H O D W L Y H O \ V P D O O D P S O L W X G H
H [ F L W D W L R Q J H R P H W U L F D O R U P D W H U L D O
Q R Q O L Q H D U L W \ I U L F W L R Q V W L F W L R Q H W F ?
3 = 7 ( [ F L W D W L R Q
( [ F L W D W L R Q D P S O L W X G H V H O H F W L R Q L V D W U D G H R I I E H W Z H H Q L Q W U R G X F L Q J Q R Q O L Q H D U L W \
X S S H U E R X Q G D Q G D F K L H Y L Q J J R R G V L J Q D O W R Q R L V H U D W L R 6 1 5 O R Z H U E R X Q G
Q R Q O L Q H D U
I
U
H T X
H Q F \
X W
.
7 H V W
$ U W L F O H
Q W
6 L J Q D O & R Q G L W L R Q L Q J D Q G 1 R L V H
: K H Q Z H D P S O L I \ W K H V L J Q D O
Z H L Q W U R G X F H P H D V X U H P H Q W
Q R L V H Q W Z K L F K F R U U X S W V W K H
\ W
P H D V X U H P H Q W \ W E \ V R P H
D P R X Q W
& R Q V L G H U 6 L J Q D O W R 1 R L V H 5 D W L R
3 R Z H U & R Q W H Q W L Q 6 L J Q D O
+∞
6
\ ω
Gω
6 1 5
3 R Z H U & R Q W H Q W L Q 1 R L V H
3
6 ω
?∞
Q Q
< V + 1 V = . * V 8 V
6 ω
\
6
Q Q
. * Mω +
/ R R N D W 3 6 ' ? V
=
6 ω 6
\ \ X X
6 R O Y H I R U V \ V W H P G \ Q D P L F V
Y L D & U R V V F R U U H O D W L R Q X \
6
' H F U H D V H Q R L V H H I I H F W E \
1 R L V H
ω
X ? , Q F U H D V L Q J 6
X X
O L P L W Q R Q O L Q H D U L W \
F R Q W U L E X W L R Q
* Mω ?
6
? , Q F U H D V L Q J . D O V R L Q F U H D V H V 6
Q Qω
X X
? ' H F U H D V L Q J 6 Q Q E H V W R S W L R Q
4 X D O L W \ H V W L P D W H Y L D
&
\ X
ω
F R K H U H Q F H I X Q F W L R Q
$ ' 4 X D Q W L ] D W L R Q
7 L P H T X D Q W L ] D W L R Q I
V
I =
V
? 7
[ W
6 D P S O L Q J ) U H T X H Q F \
D Q D O R J V L J Q D O
G L J L W D O V L J Q D O
( [ D P S O H E L W V
E L W I R U V L J Q
O H Y H O V
W
N
=?? 7 N
[ N ?
[ W $ '
) L J X U H F R X U W H V \ $ O L V V D & O D Z V R Q
R U L J L Q D O D Q G T X D Q W L ] H G D F W X D W R U V L J Q D O V ;
6 L J Q
D
O $ P S O L W X G
H
> E
L W V
@
W
? 7
Q R L V H V L J Q D O
W L P H > V H F @
$ P S O L W X G H T X D Q W L ] D W L R Q E L W V U D Q J H
1 \ T X L V W 7 K H R U H P , Q R U G H U W R U H F R Y H U D V L J Q D O [ W H [ D F W O \ L W L V Q H F H V V D U \ W R
V D P S O H W K H V L J Q D O D W D U D W H J U H D W H U W K D Q W Z L F H W K H K L J K H V W I U H T X H Q F \ S U H V H Q W
H V W 5 X O H R I W K X P E R I 6 D P S O H W L P H V I D V W H U W K D Q K L J K H V W I U H T X H Q F \ R I L Q W H U 6 D P S O H W L P H V I D V W H U W K D Q J K H V W I U H T X H Q F \ R I L Q W H U H V W
$ Q W L $ O L D V L Q J ) L O W H U L Q J
$ '
V
I
Iπ
ω ≤
[ W
[ N ?
>
@
/ 3 )
6 L
O
) V + ]
>
O
9 ( 5 < , 0 3 2 5 7 $ 1 7
) L O W H U L Q J V K R X O G E H G R Q H R Q W K H
D Q D O R J V L J Q D O H J W K R U G H U % X W W H U Z R U W K
6 D P S O H V L J Q D O [ W
$ P S O L W X G H V @
I U H T X H Q F L H V >
J
Q
D O
D Q D O R J
V D P S H G
: D Q W I + ]
1 R $ Q W L $ O L D V L Q J
) U H T X H Q F \ + ] @
$ Q W L $ O L D V L Q J I L O W H U Q G R U G H U Z L W K + ] F R U Q H U I U H T X H Q F \
+ ] V L J Q D O L V 3 I R G H G ′ G R Z Q
I U R P W K H 1 \ T X L V W I U H T X H Q F \ + ] W R + ]
6 R O X W L R Q 8 V H D O R Z S D V V I L O W H U / 3 ) Z K L F K D Y R L G V V L J Q D O 6 P D O O
F R U U X S W L R Q E \ I U H T X H Q F \ F R P S R Q H Q W V D E R Y H W K H 1 \ T X L V W I U H T X H Q F \ D W W H Q X D W L R Q
, 9 ' D W D 3 R V W 3 U R F H V V L Q J
* R D O ( [ S O D L Q Z K D W Z H G R D I W H U G D W D L V R E W D L Q H G
6 W D W L R Q D U \ S U R F H V V H V
$ Q D O \ ] H L Q I U H T X H Q F \ G R P D L Q
( > [ @ ( > [
@ ? D U H W L P H L Q Y D U L D Q W
$ Q D O \ ] H L Q W L P H G R P D L Q
7 U D Q V L H Q W S U R F H V V H V
7
V
3 H U F H Q W R Y H U V K R R W H W F
, P S X O V H U H V S R Q V H
) R X U L H U W U D Q V I R U P R I K W ! + ω
7 K H ) ) 7 ) D V W ) R X U L H U 7 U D Q V I R U P L V W K H Z R U N K R U V H R I ' 6 3
' L J L W D O 6 L J Q D O 3 U R F H V V L Q J
0 H W U L F V I R U V W H D G \ V W D W H S U R F H V V H V
6 L
J Q D
O [
> P
@
[
6 W H D G \ 6 W D W H 3 U R F H V V ( [ D P S O H
0 H D Q 9 D O X H
7
μ = ( [
7
[
> @=
3
[ W G W
? 7
F H Q W U D O W H Q G H Q F \ P H W U L F
1
' L V F U H W H
μ
[
=
1
|
[
N
N=
9 D U L D Q F H
> σ = ( [? μ @=
[ [
( [ > @? μ
[
G L V S H U V L R Q P H W U L F
> @
5 0 6
> @ ( [
L
( T X D O W R σ
[
7 L P H V H F
5 R R W P H D Q V T X D U H 5 D U H O \ L Q W H U H V W H G
L Q K J K H U P R P H Q W V
R Q O \ I R U ] H U R P H D Q
0 H W U L F V I R U W U D Q V L H Q W S U R F H V V H V
3 2
7 V
7 S
W L P H G R P D L Q
( Y D O X D W H L Q
[
6 W H S 5 H V S R Q V H V L J Q D O H [ D P S O H
( [ D P S O H 6 W H S 5 H V S R Q V H
R I W H Q X V H G W R H Y D O X D W H
S H U I R U P D Q F H R I D F R Q W U R O O H G
V W U X F W X U D O V \ V W H P
πα
7 =
3 H D N 7 L P H
S
ω
Q
6 H W W O L Q J 7 L P H 7 =
V
ζω
Q
3 H U F H Q W 2 Y H U V K R R W
3 2 = H [ S
(
?πζα
)
' D P S L Q J
α =
W H U P
?ζ
ζ
7 L P H W > V H F @
a
?
§ ·
o
, I D V V X P H R Q H G R P L Q D Q W
[ W = [
?
?
H
?ζω
Q
W
V L Q ωα W + W D Q
?
αζ
?
1
?
?
S D L U R I F R P S O H [ S R O H V
R
?
α
Q ¨
6 L J Q D O [
W
> P @
0 H W U L F V I R U L P S X O V H U H V S R Q V H G H F D \
I U R P , Q L W L D O & R Q G L W L R Q V
6 L J Q D O 6 W U
H
Q J W K
( [ D P S O H ' H F D \ S U R F H V V
2 V F L O O D W R U \ ( [ S R Q H Q W L D O ' H F D \
R U L P S X O V H U H V S R Q V H
W
H
?
[ W , Q L W L D O & R Q G L W L R Q , P S X O V H U H V S R Q V H W L P H G R P D L Q
+∞
= X τ K W?ττ [ W
3
G
?∞
F R Q Y R O X W L R Q R S H U D W L R Q
/ D S O D F H G R P D L Q
= + V 8 V < V
P X O W L S O L F D W L R Q
, P
5 D W H R I G H F D \
G H S H Q G H G R Q
S R O H V
7 L P H > V H F @
5 H
[ W = H [ S ? W + H [ S ? + M W
) ) 7 D Q G ' ) 7
ω
Q
=
|
F H
) R X U L H U V H U L H V
[ W
Q
L W
' L V F U H W H ) R X U L H U 7 U D Q V I R U P
$ S S U R [ L P D W H V W K H K H
1?
F R Q W L Q X R X V ) R X U L H U W U D Q V I R U P L
7
? L π I W
? L W
;
N
=
|
[ H
N U
; =
3
[ W H
ω
G Wω
U
U= N 1
7
U ? 1
N
I
N
= W = U N D Q G U D U H L Q W H J H U V
U
7 1
1 R I G D W D S R L Q W V
1 R W H
7 W L P H O H Q J W K R I G D W D
;
N
D U H F R P S O H [
7 K H 3 R Z H U 6 S H F W U D O ' H Q V L W \ 3 6 ' ) X Q F W L R Q
J L Y H V W K H I U H T X H Q F \ F R Q W H Q W R I W K H S R Z H U L Q :
N
=
? 7
;
N
? ;
+
W K H V L J Q D O
1
N
) ) 7 I D V W H U D O J R U L W K P L I 1 S R Z H U R I ?
$ P S O L W X G H 6 S H F W U D D Q G 3 6 '
( [ D P S O H S U R F H V V L Q J R I / D V H U G L V S O D F H P H Q W V H Q V R U G D W D I U R P 6 6 / W H V W E H G
> @
6
L
J Q D O
>
P
@
7 L L J Q D O
> @
L >
P
@
L
L L H V
> @
3 6 '
>
P
@
L
> @
@
O L
L L
L L J Q D O
L
L
L
L > @
L > P
[
7 L P H V H F
P H ' R P D L Q 6
[
) U H T X H Q F \ + ]
$
P
S
O
W
X G H
$ P S O W X G H 6 S H F W U X P
[
' R P Q D Q W ) U H T X H Q F
[
) U H T X H Q F \ + ]
+
]
3 R Z H U 6 S H F W U D O ' H Q V W \
[
) U H T X H Q F \ + ]
5
0
6
>
P
& X P X D W Y H 5 0 6
6 8 0 0 $ 5 < 2 ) 5 ( 6 8 / 7 6
0 ( $ 1 R I W P H V J Q D O H
5 0 6 R I W P H V H
5 0 6 R I 3 6 ' Z W K I I W P H
5 0 6 R I $ 6 Z W K I I W P H
5 0 6 R I F X P X O D W Y H 5 0 6 H
' R P Q D Q W ) U H T X H Q F \ + ]
' R P Q D Q W 0 D J Q L W X G H H
0 $ 7 / $ % F R G L Q J
7 L P H G R P D L Q
* L Y H Q V L J Q D O [ D Q G W L P H Y H F W R U W 1 V D P S O H V G W F R Q V W
dt=t(2)-t(1); % sampling time interval dt
fmax=(1/(2*dt)); % upper frequency bound [Hz] Nyquist
T=max(t); % time sample size [sec]
N=length(t); % length of time vector
x_mean=mean(x); % mean of signal
D V V X P H ] H U R P H D Q
x_rms=std(x); % standard deviation of signal
$ P S O L W X G H 6 S H F W U X P
V K R X O G P D W F K
X_k = abs(fft(x)); % computes periodogram of x
AS_fft = (2/N)*X_k; % compute amplitude spectrum
k=[0:N-1]; % indices for FT frequency points
f_fft=k*(1/(N*dt)); % correct scaling for frequency vector
f_fft=f_fft(1:round(N/2)); % only left half of fft is retained
AS_fft=AS_fft(1:round(N/2)); % only left half of AS is retained
3 R Z H U 6 S H F W U D O ' H Q V L W \
PSD_fft=(2*dt/N)*X_k.^2; % computes one-sided PSD in Hertz
PSD_fft=PSD_fft(1:length(f_fft)); % set to length of freq vector
rms_psd=sqrt(abs(trapz(f_fft,PSD_fft))); % compute RMS of PSD
9 6 \ V W H P , G H Q W L I L F D W L R Q
* R D O ( [ S O D L Q H [ D P S O H R I G D W D X V D J H D I W H U S U R F H V V L Q J
* R D O & U H D W H D P D W K H P D W L F D O P R G H O R I W K H V \ V W H P E D V H G R Q L Q S X W
D Q G R X W S X W P H D V X U H P H Q W V D O R Q H
L
X
L
W L ? Q
V
( V W L P D W R U
, Q S X W W P H K L V W R U L H V
2 X W S X W W L P H K L V W R U L H V
7 U D Q V I H U I X Q F W L R Q V
7 U D Q V I H U P D W U L [ *
[
M
W M ? P
6 W D W H V S D F H V \ V W H P
=
Q W
Z K L W H
T $ T+ % X
"
Q R L V H
L Q S X W
*
\ X
\ & T+ ' X =
\ W
R X W S X W
: D Q W W R R E W D L Q * I U R P
X W
\ X
*
\ X
L V W K H D F W X D O S O D Q W Z H D U H W U \ L Q J W R
< Mω = * Mω 8 Mω + 9 Mω
L G H Q W L I \ L Q W K H S U H V H Q F H R I Q R L V H
( P S L U L F D O 7 U D Q V I H U ) X Q F W L R Q
( V W L P D W H ( 7 ) (
?
N
?
* Mω =
< Mω
* Mω = * Mω +
8 M
1 Mω
N O
N O
ω
8 Mω
N O
O
O
2 E W D L Q D Q H V W L P D W H R I ( V W L P D W H G 7 )
7 U X H 7 )
1 R L V H
W K H W U D Q V I H U I X Q F W L R Q
I U R P W K H , W K L Q S X W W R
: K D W D U H W K H F R Q V H T X H Q F H V R I Q H J O H F W L Q J
W K H M W K R X W S X W
W K H F R Q W U L E X W L R Q V E \ W K H Q R L V H W H U P "
> > >
& R P S X W H
6
\ X
= ( < V 8
V @ 6
8 8
= ( 8 8
@ 6
< <
= ( < <
@
4 X D O L W \ $ V V H V V P H Q W R I W U D Q V I H U I X Q F W L R Q H V W L P D W H Y L D W K H F R K H U H Q F H I X Q F W L R Q F R K H U H Q F H I X Q F W L R Q
& →
, P S O L H V V P D O O Q R L V H 6 Q Q a
6
\ X
\ X
& =
\ X , P S O L H V O D U J H Q R L V H
\ \ X
6 6
X X
& →
3 R R U ( V W L P D W H
7 \ S L F D O O \ Z H
Z D Q W & !
\ X
$ Y H U D J L Q J
< W
7
7 7
G
?
G
G
1
L L
L
G
1
L
L
V
1
* V
8 V
1
=
=
?
=
|
|
7
1 G
G D W D V X E G L Y L G H G L Q 1 S D U W V
< V 8
G
( 7 ) ( Z L W K D Y H U D J H V
L @
L
L @
( U U R U L P S U R Y H V Z L W K
% L D V > @
G
( * * 1? ∝
V W D W H V \ V W H P
0
D J Q
W
X G H
>
G
%
( 7 ) ( Z W K Q R D Y H U D J L Q J
( 7 ) (
7 U X H 6 \ V W H P
0
D
J Q
W
X
G H
>
G %
( 7 ) ( D Y J
7 U X H 6 \ V W H P
) U H T X H Q F \ > + ] @
) U H T X H Q F \ > + ] @
( [ D P S O H / L Q H D U / H D V W 6 T X D U H V
E V E % M
* V
ωθ
?
++
0 R G H O 6 \ Q W K H V L V 0 H W K R G V
Q?
Q R
==
3 R O \ Q R P L D O I R U P
V
Q
+ D
Q?
+ + D $
(
Mωθ
)
Q? R
: H Z D Q W W R R E W D L Q D Q H V W L P D W H R I W K H S R O \ Q R P L D O F R H I I L F L H Q W R I * V W W R R E W D L Q D Q H V W L P D H R I W K H S R O \ L O F R H I I L F L H Q W R I * V
7
θ =
[
D
R
D
D
Q?
E
R
E
Q?
]
N
1
a
% Mωθ
o
' H I L Q H D F R V W I X Q F W L R Q - =
|
N=
?
?
*
?
Mω ?
1
N
N
$ Mωθ
?
?
- L V T X D G U D W L F L Q θ F D Q D S S O \ D J U D G L H Q W V H D U F K W H F K Q L T X H W R P L Q L P L ] H F R V W -
? -
6 L P S O H P H W K R G E X W W Z R P D M R U S U R E O H P V
6 H D U F K I R U
?θ
=→θ
R S W L P
? 6 H Q V L W L Y H W R R U G H U Q
? 0 D W F K H V S R O H V Z H O O E X W Q R W ] H U R V
2 W K H U 0 H W K R G V $ 5 ; O R J D U L W K P L F 1 / / 6 ) 2 5 6 (
6 W D W H 6 S D F H 0 H D V X U H P H Q W 0 R G H O V
0 H D V X U H P H Q W P R G H O V R E W D L Q H G I R U 0 , 7
2 5 , * , 1 6 W H V W E H G V W D W H P R G H O
*
L
P
E D O
!
(
1
&
:
K
H H
O
!
'
3
/
:
K H
H O
!
(
1
&
: K
H
H
O
!
*
U
R
* L
P
E
D
O
!
'
3
/
* L
P
E
D O
!
*
U
R
G D W D
I L W
6 R I W Z D U H X V H G ' \ Q D 0 R G E \ 0 L G p 7 H F K Q R O R J \ & R U S
6 X P P D U \
8 S I U R Q W Z R U N E H I R U H D F W X D O W H V W L Q J G D W D D F T X L V L W L R Q L V F R Q V L G H U D E O H
? : K D W D P , W U \ L Q J W R P H D V X U H D Q G Z K \ "
? 6 H Q V R U V H O H F W L R Q D Q G S O D F H P H Q W G H F L V L R Q V Q H H G W R E H P D G H
? : K L F K E D Q G Z L G W K D P , L Q W H U H V W H G L Q "
? + R Z G R , H [ F L W H W K H V \ V W H P F D X W L R Q I R U Q R Q O L Q H D U L W \ "
7 K H W R S L F R I V L J Q D O F R Q G L W L R Q L Q J L V F U X F L D O D Q G D I I H F W V U H V X O W V
? ' R , Q H H G W R D P S O L I \ W K H Q D W L Y H V H Q V R U V L J Q D O "
? : K D W D U H W K H H V W L P D W H V I R U Q R L V H O H Y H O V "
? : K D W L V P \ V D P S O L Q J U D W H ? 7 D Q G V D P S O H O H Q J W K 7 1 \ T X L V W / H D N D J H "
? 1 H H G W R F R Q V L G H U / H D N D J H $ O L D V L Q J D Q G $ Y H U D J L Q J
' D W D S U R F H V V L Q J W H F K Q L T X H V D U H S R Z H U I X O D Q G G L Y H U V H
? ) ) 7 D Q G ' ) 7 P R V W L P S R U W D Q W W U \ W R K D Y H A 1 S R L Q W V I R U V S H H G
? 1 R L V H F R Q V L G H U D W L R Q V K R Z J R R G L V P \ P H D V X U H P H Q W " ! F R K H U H Q F H
7 H V W 3 U R W R F R O
( Q J L Q H H U L Q J ' H V L J Q D Q G 5 D S L G 3 U R W R W \ S L Q J , $ 3
0 D V V D F K X V H W W V , Q V W L W X W H R I 7 H F K Q R O R J \
7 H V W 3 U R W R F R O
7 H D P 1 X P E H U B B B B B B B B B B B B B
7 H D P 0 H P E H U V B B B B B B B B B B B B B B B B B B B B B B B B B B
B B B B B B B B B B B B B B B B B B B B B B B B B B
3 U R G X F W B B B B B B B B B B B B B B B B B B B B B B B B B B 9 H U V L R Q B B B B B B B B B B B B B B B B
0 D Q X I D F W X U L Q J & R V W B B B B B B B B B B B B B B > P L Q @ 0 D Q X I D F W X U L Q J 7 L P H D F W X D O
B B B B B B B B B B B B B B > @ 2 P D [ H V W L P D W H
6 W U X F W X U D O 3 H U I R U P D Q F H
/ R D G & D V H ) B B B B B B B ) B B B B B B B B ) B B B B B B B B > O E V @
' L V S O D F H P H Q W I R U N 1 R O R D G ] H U R U H D G L Q J B B B B B B B B B > 9 R O W V @
0 H D V X U H P H Q W V B B B B B B B B B B B B B B B B B B B B > 9 R O W V @ $ Y H U D J H B B B B B B B B > 9 R O W V @
& D O L E U D W L R Q ) D F W R U B B B B B B B B B > P P 9 R O W V @ ) L [ W X U H ) O H [ L E L O L W \ B B B B B B B B B B B > P P @
δ
B B B B B B B B > 9 R O W V @
B B B B B B B B B B > P P 9 R O W V @ B B B B B B > P P @ B B B B B B B B B > P P @
' L V S O D F H P H Q W V D G G O H 1 R / R D G V U H D G L Q J B B B B B B B B B > 9 R O W V @
0 H D V X U H P H Q W V B B B B B B B B B B B B B B B B B B B B > 9 R O W V @ $ Y H U D J H B B B B B B B B > 9 R O W V @
& D O L E U D W L R Q ) D F W R U B B B B B B B B B > P P 9 R O W V @ ) L [ W X U H ) O H [ L E L O L W \ B B B B B B B B B B B > P P @
δ
B B B B B B B B > 9 R O W V @
B B B B B B B B B B > P P 9 R O W V @ B B B B B B > P P @ B B B B B B B B B > P P @
1 D W X U D O ) U H T X H Q F \ ( V W L P D W H B B B B B B B B B B B B B B B B B B B B B B B B > + ] @ 2 V F L O O R V F R S H
0 D V V B B B B B B B B B B B B B B B B B B B B > J U D P V @ D Y J B B B B B B > J U D P V @
6 W D I I 6 L J Q D W X U H B B B B B B B B B B B B B B B B B B B B B B B B B B
美国麻省理工大学：Engineering Design and Rapid Prototyping：l7_str_test
| 课件名称： | 美国麻省理工大学：Engineering Design and Rapid Prototyping |
| 课件分类： | 航空航天 |
| 课件类型： | 教学课件 |
| 文件大小： | 6.61MB |
| 下载次数： | 0 |
| 评论次数： | 0 |
| 用户评分： | 0 |
- 1. 美国麻省理工大学：Engineering Design and Rapid Prototyping：l11_biography
- 2. 美国麻省理工大学：Engineering Design and Rapid Prototyping：l1_intro
- 3. 美国麻省理工大学：Engineering Design and Rapid Prototyping：l3_cad
- 4. 美国麻省理工大学：Engineering Design and Rapid Prototyping：l4_cae
- 5. 美国麻省理工大学：Engineering Design and Rapid Prototyping：l5_mfg
- 6. 美国麻省理工大学：Engineering Design and Rapid Prototyping：l6_nucast_slides
- 7. 美国麻省理工大学：Engineering Design and Rapid Prototyping：l7_str_test
- 8. 美国麻省理工大学：Engineering Design and Rapid Prototyping：l8_optimize