04_SEMD3_Ch3_LaplaceText - 1 Standard notation in dynamics...

L l T f Laplace Transforms
1. Standard notation in dynamics and control
(shorthand notation) = how you communicate
with other engineers
r 3
with other engineers.
2. Converts differential calculus mathematics to
hapte
algebraic operations, very useful for:
Initial Value Problems (IVPs).
Ch
3. Advantageous for block diagram analysis = a
“systems” way to analyze more complex systems.
(study subsystems then combine to large system) (study subsystems, then combine to large system)
r 3
hapte
For “simple”
Ch
For simple
Y(s) -- maybe
a short cut !
Laplace Transform Definition
-
0
[ ]= ( ) ( ) ( ) st e F t dt f s f t
∞
=
∫
L
Two derivations
(“a” is a constant, f ’ is derivative (“a” is a constant, f ’ is derivative wrt wrt time): time):
-st st
a a a
[ ]= e dt e 0 a a
∞
∞
−
⎡ ⎤
⎡ ⎤ = − = − − =
⎣ ⎦
⎢ ⎥
∫
L
3
0
0
-st -(b+s)t (b s)t -bt -bt
[ ]
s s s
1 1
[ ]= e dt e dt -e
b b
e e
∞ ∞
∞
− +
⎣ ⎦
⎡ ⎤
= = =
⎣ ⎦
∫ ∫
L
hapte
0
0 0
-st
b+s s+b
df df
and a theorem ... [f ] e dt ( ) ( 0)
dt dt
sF s f t
∞
⎡ ⎤
′
= = = − =
⎢ ⎥
⎣ ⎦
∫
L L
Ch
0
Very convenient for zero initial condition f(0) = 0
An example is when f = the “error” in a process control IVP An example is when = the error in a process control IVP,
i.e at t = 0 the process is at a “nominal design” steady-state.
Other Transforms
2
where we assign ( )
d f dg df
g f t
⎡ ⎤
⎡ ⎤
′
⎢ ⎥
⎢ ⎥
L L
[ ]
2
where we assign
( ) - (0)
dt dt dt
sG s g
= = =
⎣ ⎦
⎣ ⎦
=
2
( ) - (0) (0)
( ) (0) (0)
s sF s f f
s F s sf f
′ = −
′
= − −
n
r 3
etc. for higher orders
-
[ ( )] sin
2
j t j t e e
t
j
ω ω
ω
+ −
⎡ ⎤
=
⎢ ⎥
⎣ ⎦
L L
n
d f
dt
- j t j t e e ω ω + ⎡ ⎤ +
hapte
1 1 1
⎛ ⎞
+
⎜ ⎟
2 2 s
ω
ω
=
+
cos [ ( )]
2
t ω =
⎢ ⎥
⎣ ⎦
L L
Ch
Note:
2 2 2 2
2
1
2
s j s j
s j s j
ω ω
ω ω
=
+ −
⎝ ⎠