Download - Math Review – Complex Numbers (1)
2009 Fall ME451 - GGZ Page 1Week 1-2: Math Review and Laplace Transformation
Math Review Math Review –– Complex Numbers (1)Complex Numbers (1)
Complex number: ordered pair of two real numbers
Multiplication
1 and , where, −≅∈∈+= jRyxCjyxs
Conjugate: jyxss −== *
Addition:
)()( , ,212121222111
yyjxxssjyxsjyxs +++=++=+=
)()(1221212121
yxyxjyyxxss ++−=
22*yxss +=
2009 Fall ME451 - GGZ Page 2Week 1-2: Math Review and Laplace Transformation
Math Review Math Review –– Complex Numbers (2)Complex Numbers (2)
Euler’s identity
Phase:
j
eeeeje
jjjj
j
2sin ,
2cos where,sincos
θθθθθ θθθθ
−− −=
+=+=
Polar form:
θθθ sin,cos where, ryrxrejyxsj ===+=
Magnitude:22
yxr +=
)(
2
1
2
1
)(
21212211
21
2121
,
θθ
θθθθ
−
+
=
===
j
jjj
er
r
s
s
errssersers
x
y1tan
−=θ
x
y
Re
Im
rθ
2009 Fall ME451 - GGZ Page 3Week 1-2: Math Review and Laplace Transformation
Math Review Math Review –– LogarithmLogarithm
The logarithm of x to the base b is denoted by xb
log
The logarithm of 1000 to the base 10 is 3, i.e.,
Properties:
100010 that note ,31000log 3
10 ==
01log and 110log1010
==
yxy
x
yxxy
xyxxb
bbb
bbb
b
y
b
xb
loglog)(log
loglog)(log
)(log)(log and )(log
−=
+=
==
2009 Fall ME451 - GGZ Page 4Week 1-2: Math Review and Laplace Transformation
Math Review Math Review –– Matrix OperationsMatrix Operations
=
=
2221
1211
2221
1211 and
bb
bbB
aa
aaA
Determinant:12212211
det aaaaA −=
Multiplication: 2222122121221121
2212121121121111
++
++=
babababa
babababaAB
Inverse:
det
1121
1222
1
A
aa
aa
A
−
−
=−
2009 Fall ME451 - GGZ Page 5Week 1-2: Math Review and Laplace Transformation
Laplace TransformLaplace Transform
Definition:
∫∞
−==0
)()()]([ dtetfsFtfLst
The Laplace transform (LT) of f(t) is
LT: replace ODEs as linear input-output maps
solve ODEs w/ constant coefficients
t - domain
)(tf
s - domain
)(sF
LRe
Im
ts
2009 Fall ME451 - GGZ Page 6Week 1-2: Math Review and Laplace Transformation
Laplace Transform Laplace Transform –– Example (1)Example (1)
Example (1)
0 ,1)( ≥= ttf
ss
es
dtedtetfsFtfL
st
stst
110
1
)()()]([
0
0 0
=+=
−=
===
∞
−
∞ ∞
−−
∫ ∫
sLtfL
1]1[)]([ ==Q
∆≤
== ∆
→∆ 0;0
;lim)()(
1
0
tttf δ
11
lim
1lim
)()()()]([
0
00
0 0
=∆⋅∆
=
∆=
===
→∆
∆
−
→∆
∞ ∞
−−
∫
∫ ∫
dte
dtetdtetfsFtfL
st
stst δ
1)]([)]([ == tLtfL δQ
∆
1
∆ t
)(tδ
0→∆
2009 Fall ME451 - GGZ Page 7Week 1-2: Math Review and Laplace Transformation
Laplace Transform Laplace Transform –– Example (2)Example (2)
Example (2)
0 ,)( ≥= tttf
22
00
0 0 0 0 0
0 0
1)
10()00(
11
)(
)(
1 , )()()]([
sse
sse
s
t
dtt
vuvdt
t
udt
t
uvvdt
t
udt
t
uv
es
vtudttedtetfsFtfL
stst
ststst
=−−+−=−
⋅−
−−
=
∂
∂+
∂
∂=
∂
∂
∂
∂−
∂
∂=
−=====
∞
−
∞
−
∞ ∞ ∞ ∞ ∞
∞ ∞
−−−
∫ ∫ ∫ ∫ ∫
∫ ∫
2
1][)]([
stLtfL ==Q
2009 Fall ME451 - GGZ Page 8Week 1-2: Math Review and Laplace Transformation
Laplace Transform Laplace Transform –– Property (1)Property (1)
Linearity
)()()]([)]([)]()([ sbGsaFtgbLtfaLtbgtafL +=+=+
)()(
)()(
)]()([)]()([
)(
0
)(
0
0
sbGsaF
dtetgbdtetfa
dtetbgtaftbgtafL
sG
st
sF
st
st
+=
+=
+=+
∫∫
∫∞
−
∞
−
∞
−
4342143421
Proof:
2009 Fall ME451 - GGZ Page 9Week 1-2: Math Review and Laplace Transformation
Laplace Transform Laplace Transform –– Property (2)Property (2)
“s” Shifting property
)(lim)( where),()]([ then ),()]([ If sFasFasFtfeLsFtfLass
at
−→=−−==
)(
)(
)()]([
0
)(
0
asF
dtetf
dtetfetfeL
tas
statat
−=
=
=
∫
∫∞
−−
∞
−
Proof:
2009 Fall ME451 - GGZ Page 10Week 1-2: Math Review and Laplace Transformation
Laplace Transform Laplace Transform –– Property (3)Property (3)
“t” Shifting property
)()]()([
then ,function stepunit is )( and )()]([ If
sFeatuatfL
tusFtfL
as−=−−
=
( )
)(
)()()(
)()()()()]()([
00
)(
0
sFe
atdefedeuf
dteatuatfdteatuatfatuatfL
as
sasas
a
stst
−
∞
−−
∞
+−
∞
−
∞
−
=
−===
−−=−−=−−
∫∫
∫∫
ττττττ ττ
Proof:
<
≥=−
at
atatu
,0
,1)(
a
1
)( atu −
t
2009 Fall ME451 - GGZ Page 11Week 1-2: Math Review and Laplace Transformation
Laplace Transform Laplace Transform –– Property (4)Property (4)
Transforms of Derivatives
∂
∂=−==
t
tftffssFtfLsFtfL
)()( )0()()]([ then ,)()]([ If &&
Proof:
( )
)0()()()(
)(
)(
)( , )()]([
)(
00
0 0 0 0 0
00
fssFdtetfsetf
dtt
vuvdt
t
udt
t
uvvdt
t
udt
t
uv
tfveudtt
vudtetftfL
sF
stst
stst
−=
−−=
∂
∂+
∂
∂=
∂
∂
∂
∂−
∂
∂=
==∂
∂==
∫
∫ ∫ ∫ ∫ ∫
∫∫
∞
−∞
−
∞ ∞ ∞ ∞ ∞
−
∞∞
−
43421
&&
)0()0()()]([ 2fsfsFstfL &&& −−=
2009 Fall ME451 - GGZ Page 12Week 1-2: Math Review and Laplace Transformation
Laplace Transform Laplace Transform –– Property (5)Property (5)
Transforms of Integrals
)(1
])([ then ,)()]([ If0
sFs
dfLsFtfL
t
== ∫ ττ
Proof:
.0)0( and )()( then,,)()(Let 0
=== ∫ gtftgdftg
t
&ττ
)()0()]([)]([)]([)(
sderivative of s transformUsing
)(
ssGgtgLstgLtfLsF
sG
=−===321
&
)(1
])([)(
Therefore
t
0
sFs
dfLs G == ∫ ττ
2009 Fall ME451 - GGZ Page 13Week 1-2: Math Review and Laplace Transformation
Laplace Transform Laplace Transform –– Example (3)Example (3)
Linearity ttf 21)( +=
)].([G(s) find ,)( If 2tgLetg
t ==
2
11
]1[)]1([)(
2
2
2
−==
==
−→
−→
ss
LeLsG
ss
ss
t
2
1][
Therefore
2t
−=
se L
22
212
1][2]1[]21[)]([
s
s
sstLLtLtfL
+=+=+=+=
“s” Shifting properties:
2009 Fall ME451 - GGZ Page 14Week 1-2: Math Review and Laplace Transformation
Laplace Transform Laplace Transform –– Example (4)Example (4)
Then, .3)( and ,)( ,3Let −=−== tatfttfa
).3()62()( where
)]([)( Find
−−=
=
tuttg
tgLsG
2
3 12
)]3()3[(2)]3()3(2[)]([)(
se
tutLtutLtgLsG
s−=
−−=−−==
3
3s-
2
2G(s)
Therefore,
es
=
“t” Shifting properties:
6 93−
6−
3
6)(tg
t
2009 Fall ME451 - GGZ Page 15Week 1-2: Math Review and Laplace Transformation
Laplace Transform Laplace Transform –– Example (5)Example (5)
)0()0()]([)]([ using ][sin 2fsftfLstfLwtL &&& −−=
wtwtf
wfwtwtf
fwttf
sin)(
)0( ,cos)(
0)0( ,sin)(Let
2−=
==
==
&&
&&
{ {wtftf
ffswtLswtwL )0()0(]sin[]sin[ Thus,0)(
2
)(
2 &32143421
&&
−−=−
22][sin
Therefore,
ws
wwtL
+=
Calculating
wwtLswtLw −=− ][sin][sin 22
wwtLws =+ ][sin)( 22
2009 Fall ME451 - GGZ Page 16Week 1-2: Math Review and Laplace Transformation
Cosine
Sine
Exponential
nth order ramp
Unit ramp
Unit step
Unit impulse
Description
Laplace Transform Laplace Transform –– LT Table to rememberLT Table to remember
)(tf
t
s
1)(tu
)(tδ
)]([)( tfLsF =
1
2
1
s
nt 1
!+n
s
n
ateas −
1
wtsin22
ws
w
+
wtcos 22ws
s
+
2009 Fall ME451 - GGZ Page 17Week 1-2: Math Review and Laplace Transformation
Convolution
Transforms of integrals
Transforms of
derivatives
“s” shifting properties
Linearity
Laplace Transform Laplace Transform –– LT PropertiesLT Properties
)]([)]([)]()([ tgbLtfaLtbgtafL +=+
)0()()]([ fssFtfL −=&
)]([)()]()([ tfLesFeatuatfLasas −− ==−−
ass
attfLasFtfeL
−==−= )]([)()]([
)0()0()()]([2
fsfsFstfL &&& −−=
M
∫ ==
t
tfLs
sFs
dfL0
)]([1
)(1
])([ ττ
∫∫ −=−=−
tt
dtfgdtgfsGsFL00
1 )()()()()]()([ ττττττ
“t” shifting properties
2009 Fall ME451 - GGZ Page 18Week 1-2: Math Review and Laplace Transformation
Inverse Laplace TransformInverse Laplace Transform
)]([)( Find )],([)(Given 1sFLtftfLsF
−==
t - domain
)(tf
s - domain
)(sF
1−L
Approach:
etc. ,properties tables,using term-by- term)]([Obtain (2)
functionscommon of s transform theselike"look " )( Make (1)
1sFL
sF
−
2009 Fall ME451 - GGZ Page 19Week 1-2: Math Review and Laplace Transformation
Inverse Laplace Transform Inverse Laplace Transform ––Example (1)Example (1)
)]([)( Find ,6
1)(Given 1
23sYLty
sss
ssY −=
−+
+=
3
152
2
103
61-
6
1)(
nextit discuss willWe
23
444 3444 21 +
−
+−
+=−+
+=
ssssss
ssY
ttee
sL
sL
sLty
32
111
15
2
10
3
6
1
]3
1[
15
2]
2
1[
10
3]
1[
6
1)(
−
−−−
−+−
=
+−
−+
−=
2009 Fall ME451 - GGZ Page 20Week 1-2: Math Review and Laplace Transformation
Inverse Laplace Transform Inverse Laplace Transform –– PFE Case (1)PFE Case (1)
CA, Bs
C
s
B
s
A
sss
ssY and , Find ,
32)3)(2(
1)(
++
−+=
+−
+=
)2()3()3)(2(1 −++++−=+⇒ sCssBsssAs
PFE: Partial Fraction ExpansionsA way to expand general rational functions into forms that
appears in the LT table
Case 1: real and distinct roots
15
2 )5)(3(2;3
10
3 )5)(2(3;2
6
1 )3)(2(1;0
−=⇒−−=−−=
=⇒==
−=⇒−==
CCs
BBs
AAs
2009 Fall ME451 - GGZ Page 21Week 1-2: Math Review and Laplace Transformation
Inverse Laplace Transform Inverse Laplace Transform –– PFE Case (1)PFE Case (1)
tt eetysss
sY 32
15
2
10
3
6
1)(
3
15/2
2
10/36/1)( −−+−=⇒
+
−+
−+
−=
AsCBAsCBA
CsCsBsBsAAsAs
sCssBsssAs
6)23()(
236
)2()3()3)(2(1
2
222
−−++++=
−+++−+=
−++++−=+
Case 1: real and distinct roots (cont’d)
unknowns 3 with equations 3
61:
231:
0:
0
1
2
−=
−+=
++=
As
CBAs
CBAs
Alternative approach:
We have
2009 Fall ME451 - GGZ Page 22Week 1-2: Math Review and Laplace Transformation
Inverse Laplace Transform Inverse Laplace Transform –– PFE Case (2)PFE Case (2)
323 )2()2(2)2(
3)(
++
++
+=
+
+=
s
C
s
B
s
A
s
ssY
0 20
1 2Let
)2(21
1 2Let
)2()2(3 2
=⇒=⇒∂
∂
=⇒−=
++=⇒∂
∂
=⇒−=
++++=+⇒
AA s
Bs
BsA s
Cs
CsBsAs
Case 2: real and repeated roots
32 )2(
1
)2(
1)(
++
+=
sssY
2009 Fall ME451 - GGZ Page 23Week 1-2: Math Review and Laplace Transformation
Inverse Laplace Transform Inverse Laplace Transform –– PFE Case (2)PFE Case (2)
tt
ssss
ssss
et
tety
tLtL
sssssY
2
2
2
2
2
2
2
3
2
232
2)(
]2
[][
11
)2(
1
)2(
1)(
−−
+=+=
+=+=
+=
+=
+=+
++
=
Case 2: real and repeated roots (cont’d)
2009 Fall ME451 - GGZ Page 24Week 1-2: Math Review and Laplace Transformation
Inverse Laplace Transform Inverse Laplace Transform –– PFE Case (3)PFE Case (3)
)( bias ±=
]2cos[]2[cos2
)(1
1
22teLtL
s
ssY
t
ss
ss
−
+=
+=
==+
=
Case 3: complex conjugate roots
iss
s
ss
ssY 21 :Roots
2)1(
1
52
1)(
222±−=⇒
++
+=
++
+=
tetyt 2cos)(
Therefore
−=
2009 Fall ME451 - GGZ Page 25Week 1-2: Math Review and Laplace Transformation
Inverse Laplace Transform Inverse Laplace Transform –– Example (2)Example (2)
22222
2
21)4)(1(
1)(
+
++
+++=
++
+=
s
EDs
s
C
s
B
s
A
sss
ssF
)1()()4()4)(1()4)(1(1222222 ++++++++++=+ ssEDssCsssBssAss
20/3,4163
)](24[4
)12)(4)(2(32
5/2521
4/1410
−=−=⇒−=−=−⇒
++−−=
+−+=−⇒=
=⇒=⇒−=
=⇒=⇒=
EDEDED
EDiDE
iEiDis
CCs
BBs
4/1 440 0
])1(2)[()1(]2)4(2[
)]1(2)4[()]1(2)4()4)(1[(2
2232
2222
−=−=⇒+=⇒=
+++++++++
++++++++++=⇒∂
∂
BABAs
sssEDssDssssC
sssBssssssAss
)2sin2
12cos(
20
3
5
2
4
1
4
1)(
2
1
20
3
1
1
5
21
4
11
4
1)(
222ttettf
s
s
ssssF
t +−+++−=⇒+
+−+
+++−= −
2009 Fall ME451 - GGZ Page 26Week 1-2: Math Review and Laplace Transformation
Laplace Transform Laplace Transform –– ConvolutionConvolution
∫∫ −=−=
==
−
−−
tt
dttfgdttgfsGsFL
tgsGLtfsFL
00
1
11
)()()()()]()([
),()]([ and )()]([ If
ττττ
Example:
44 344 214342143421tttysGsF
sssssssY
3sin8
1sin
8
3)(
22
)(
2
)(
222 9
8/3
1
8/3
9
3
1
1
)9)(1(
3)(
−=
+−
+=
+
+=
++=
tsGLttsFLtf 3sin)]([)(g and sin)]([)( 11 ==== −−
tt
dtdtgfty
tt
3sin8
1sin
8
3
)(3sinsin)()()(00
−==
−=−= ∫∫
L
ττττττ
2009 Fall ME451 - GGZ Page 27Week 1-2: Math Review and Laplace Transformation
Laplace Transform Laplace Transform –– Solving Solving ODEsODEs
ODE in
domain t
AE in
domain s
Solution
)(ty
Direct solution
Solve for
)(sY
L
1−L)(sY
2009 Fall ME451 - GGZ Page 28Week 1-2: Math Review and Laplace Transformation
Laplace Transform Laplace Transform –– Solving Solving ODEsODEs ExampleExample
0)0()0( ,21)(4)( ==++−=+ −yyettyty
t &&&
44 344 214434421
&
&
&
Solution Particular
22
2
Solution sHomogeneou
2
2
2
2
2
2
)4)(1(
1
4
)0()0()(
)1(
1)0()0()()4(
1
211)(4)0()0()( :
++
++
+
+=
+
+++=+
+++−=+−−
sss
s
s
ysysY
ss
sysysYs
ssssYysysYsL
0
)2sin2
12cos(
20
3
5
2
4
1
4
1)( ttetty
t +−+++−= −
222 2
1
20
3
1
1
5
21
4
11
4
1)(
+
+−+
+++−=
s
s
ssssY
ILT Example (2)