§7.3 translation and partial fractions partial fraction...
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Kidoguchi, Kenneth122 November, 2019 1
Rule 2 Quadratic Factor Partial Fractions
The portion of the partial fraction decomposition of the rational function
R(s) corresponding to the irreducible quadratic factor (s – a)2 of
multiplicity n is a sum of n partial fractions having the form
where A1, A2, A3, …., An , B1, B2, B3, …., Bn are constants.
1 1 2 22 22 2 22 2
n nn
A s BAs B A s B
s a b s a b s a b
Rule 1 Linear Factor Partial Fractions
The portion of the partial fraction decomposition of the rational function
R(s) corresponding to the linear factor s – a of multiplicity n is a sum of
n partial fractions having the form
where A1, A2, A3, …., and An are constants.
1 22
nn
AA A
s a s a s a
§7.3 Translation and Partial Fractions
Partial Fraction Decomposition Rules
22 November, 2019 2 Kidoguchi, Kenneth
2
1( )
2 5G s
s s
Find g(t) = L-1{G(s)} if:
§7.3 Translation and Partial Fractions
Inverse Laplace Transform – PFD Example
22 November, 2019 3 Kidoguchi, Kenneth
22 November, 2019 5 Kidoguchi, Kenneth
2 2( 5)s
5 sin( )te t L 2 2sin t
s
L
2 2
2
( 2) 3
s
s
2 cos(3 )te t L 2 2cos 3
3
st
s
L
( ) ( )ate f t F s a L ( ) ( )f t F sL
0
( ) ( )at st ate f t e dt e f t
L( )
0
Proof: ( ) ( ) s a tF s a f t e dt
If F(s) = L{f(t)} exists for s > c, the L{eat f(t)} exists for s > a + c, and
L{eat f(t)} = F(s – a)
Equivalently: L-1{F(s – a)} = eat f(t)
Thus the translation s s – a in the transform corresponds to
multiplication of the original function by eat.
§7.3 Translation and Partial Fractions
Theorem 1 Translation on the s-Axis
22 November, 2019 6 Kidoguchi, Kenneth
2
1( )
2 5G s
s s
Find g(t) = L-1{G(s)} if:
§7.3 Translation and Partial Fractions
Translation on the s-Axis, Another Example
22 November, 2019 8 Kidoguchi, Kenneth
2( )
4 8
sG s
s s
Find g(t) = L-1{G(s)} if:
§7.3 Translation and Partial Fractions
Translation on the s-Axis, Another Example
10 Kidoguchi, Kenneth22 November, 2019
k
m
x > 0x < 0
F(t)
PFD
22
1 10 10 50 10
29 43 25
s s
ss
22
60( )
4 3 25X s
s s
2
2
606 34 ( )
4s X s
ss
2
2
2( ) (0) (0) 6 ( ) (0) 34 ( ) 30
4X s sx v X s x X s
ss s
6 34 30sin 2x x x t L L L L
6 34 30sin 2 , (0) (0) 0x x x t x v
Consider the damped mass-spring system
whose motion is described by the IVP:
§7.3 Translation and Partial Fractions
Damped Oscillator Revisited
22 November, 2019 11 Kidoguchi, Kenneth
22
1 10 10 50 10( )
29 43 25
s sX s
ss
§7.3 Translation and Partial Fractions
Damped Oscillator Revisited (Cont)
22 November, 2019 13 Kidoguchi, Kenneth13
32 5( ) 5cos(5 ) 2sin(5 ) 5sin(2 ) 2cos(2 )
29 29
tx t e t t t t
§7.3 Translation and Partial Fractions
Damped Oscillator Revisited (Cont)
14 Kidoguchi, Kenneth22 November, 2019
k
m
x > 0x < 0
0 1 1
2 20 02 2
0
2sin sin
Ft t
m
F(t)
0
02 2
0
( ) cos cosF
x t t tm
0
2 2 2 22 2
00
F s s
s sm
0
2 2 2 2
0
( )F s
X sm s s
0If
2 2 0
0 2 2( )
F ss X s
m s
2 2 0
0 2 2( ) ( )
F ss X s X s
m s
2 0
0cos
Fx x t
m L L L
2 0
0cos
Fx x t
m
0Consider: cos , (0) (0) 0mx kx F t x x
§7.3 Translation and Partial Fractions
Beats and Resonance
22 November, 2019 15 Kidoguchi, Kenneth
k
m
x > 0x < 0
F(t)
0
0
0
( ) sin2
Fx t t t
m
00
22 2
00
2
2
F s
m s
0
22 2
0
F s
m s
0
2 2 2 2
0 0
( )F s
X sm s s
0If
2 2 0
0 2 2( )
F ss X s
m s
2 2 0
0 2 2( ) ( )
F ss X s X s
m s
2 0
0cos
Fx x t
m L L L
2 0
0cos
Fx x t
m
0Consider: cos , (0) (0) 0mx kx F t x x
§7.3 Translation and Partial Fractions
Beats and Resonance
22 November, 2019 16 Kidoguchi, Kenneth
k
m
x > 0x < 0
F(t)
2 2
2 50 55 45 55 45sin sin
2 20.1 55 45
sin(50 )sin(5 )
t t
t t
0 0 0
02 2
0
2( ) sin sin ,
2 2
Fx t t t
m
2 0
0cos , (0) (0) 0
Fx x t x v
m
0 00.1, 50, 45,and 55m F
Consider an undamped mass-spring system
that is initially at rest with:
§7.3 Translation and Partial Fractions
Beats – Example
22 November, 2019 17 Kidoguchi, Kenneth
sin(5 )envelope
x t
sin(5 )envelope
x t
( ) sin(50 )sin(5 )x t t t
§7.3 Translation and Partial Fractions
Beats – Example
22 November, 2019 18 Kidoguchi, Kenneth
( ) sin(50 )sin(5 )
( )
x t t t
dxv t
dt
§7.3 Translation and Partial Fractions
Beats – Example
22 November, 2019 19 Kidoguchi, Kenneth
k
m
x > 0x < 0
F(t)
( ) sin 50x t t t
0
0 0
0
( ) sin ,2
Fx t t t
m
2 0
0 0cos , (0) (0) 0
Fx x t x v
m
0 01, 100, and 50m F
Consider an undamped mass-spring system
that is initially at rest with:
§7.3 Translation and Partial Fractions
Resonance – Example
22 November, 2019 20 Kidoguchi, Kenneth
envelopex t
envelopex t
( ) sin 50x t t t
§7.3 Translation and Partial Fractions
Resonance – Example
22 November, 2019 21 Kidoguchi, Kenneth
( ) sin(50 )
( )
x t t t
dxv t
dt
§7.3 Translation and Partial Fractions
Resonance – Example