notas transformada de laplace
TRANSCRIPT
-
7/26/2019 Notas Transformada de Laplace
1/83
-
7/26/2019 Notas Transformada de Laplace
2/83
-
7/26/2019 Notas Transformada de Laplace
3/83
-
7/26/2019 Notas Transformada de Laplace
4/83
-
7/26/2019 Notas Transformada de Laplace
5/83
s
t
V0
V0
-
7/26/2019 Notas Transformada de Laplace
6/83
-
7/26/2019 Notas Transformada de Laplace
7/83
cosh x= ex+ex
2 sinh x= exex
2 cos x= eix+eix
2 sin x= eixeix
2i
0 sin(t)e
tdt
0 cos(wt)e
tdt
0 sin
2(wt)etdt
0 cos
2(wt)etdt
0 sinh(t)e
2t
dt
0 cosh(t)e
2tdt
0
sin(wt)estdt
w
s
s
w
-
7/26/2019 Notas Transformada de Laplace
8/83
a0
estdt
a
s
a
s
|x|< 1
k=0 x
k = 11x
k=0 kx
k = x(1x)2
k=0 e
sk
k=0(1)kesk
k=0 kesk
s
F(s) = s26s+4
s33s2+2s
F(s) = s2+s2(s+1)3
F(s) = s(s+1)3
F(s) = 5s215s11
(s+1)(s2)3
F(s) = 3s22s1
(s3)(s2+1)
F(s) = s2+2s+3
(s2+2s+2)(s2+2s+5)
F(s) = 1s2(s2)
0
tetdt
0 t
2etdt
I=
0
|sin(t)| etdt
I=
k=0
(1)k k+1k
sin(t)etdt.
-
7/26/2019 Notas Transformada de Laplace
9/83
f(t)
L{f(t)}= 0
f(s)estdt
f(t)
L{f(t)} f(t) s
L{f(t)}= F(s) L{g(t)}= G(s) L{h(t)}= H(s)
f(t) = 1
L{1} =
0 1 est
dt
= lima
a0
estdt
= lima
1 esas
.
lima
1 esas
s > 0
L{1} = 1s
, s > 0.
f(t) = t
L{t} =0
testdt
= test
s
0
0
e
st
s
dt.
= test
s
0
+1
s
0
estdt.
-
7/26/2019 Notas Transformada de Laplace
10/83
test
s
0
lima
te
st
s
a
0
s > 0
1sL{1}
L{t} = 1sL{1}= 1
s2, s > 0.
f(t) = tn
tn1
t2
t3
L{t2} =0
t2estdt
= t2est
s
0
0
2t e
st
s
dt.
= 2
s
0
testdt=2
sL{t}= 2
s
1
s2 =
2
s3, s > 0
L{t3
} =
0 t
3
est
dt
= t3est
s
0
0
3t2 e
st
s
dt.
= 3
s
0
t2estdt=3
sL{t2}= 3
s
2
s3 =
3!
s4, s > 0
tn
L{tn}= n!sn+1
, s > 0.
f(t) = eat
L{eat} = 0
eatestdt
=
0
e(s+a)tdt
= e(s+a)t
s+ a
0
= 1
s + a, s + a > 0
f(t) = sin(wt)
L{sin(wt)} =0
sin(wt)estdt
=
sin(wt)est
s
0 1
s
0
(w cos(wt)) est dt
= w
s
0
cos(wt)estdt
= w
s
cos(wt)e
st
s
0
1s
0
(w sin(wt)) est dt
= w
s
1
s w
s
0
sin(wt)estdt
= w
s2 w
2
s2L{sin(wt)}.
-
7/26/2019 Notas Transformada de Laplace
11/83
L{sin(wt)}
L{sin(wt)}= ws2
w2
s2L{sin(wt)}.
L{sin(wt)}1 + w2s2
= ws2
,
L{sin(wt)}= ws2 + w2
, s > 0.
L{f(t)} f(t)
f(t) =
0, 0t45, t > 4.
ba f(t)dt=
xa f(t)dt+
bxf(t)dt
L{f(t)} = 0
f(t)estdt
=
40
f(t)estdt+
4
f(t)estdt
=
40
0 estdt+4
5estdt
= 5est
s
4
=5e4s
s
f(t) = cos(wt)
f(t)
f(t) =
0, 0t31, 3t5,0, t > 5.
f(t) = et2
f(t) = 1t
f(t) c c M > 0 T >0 |f(t)| M ect t > T
f(t) = t2 g(t) = sin(t) h(t) = et
|t2| et, t > 0,
|5 cos(t)| et, t > 2,|et| et, t > 0.
f
g
h
-
7/26/2019 Notas Transformada de Laplace
12/83
1
2
3
1
t
et
t2
1
2
3
4
12345
1 2 3
t
et
5 cos(t)
1
2
3
1
t
et
et
f(t) [0, ) c
f(t)
s > c
L{f(t)}=0
f(t)estdt=
T0
f(t)estdt+
T
f(t)estdt.
[0, T] T T c M >0 |f(t)| M ect t > T
T
f(t)estdt
T
|f(t)| estdt
T
Mectestdt
= M
T
e(sc)tdt
= M
c s e(sc)t
T
= M
s c e(sc)T, s > c.
0
f(t)estdt s > c
f(t) F(s) =L{f(t)}
lims
F(s) = 0.
F(s) =
0
f(t)estdt.
u= st
F(s) =1
s
0
fu
s
eudu.
-
7/26/2019 Notas Transformada de Laplace
13/83
f
M |f(t)|< M
|F(s)| Ms
0
eudu=M
s .
|F(s)| 0 s
lims
F(s) = 0.
F(s) =L{f(t)} f(t) f(t) =L1{F(s)}
F(s)
f(t)
F(s) t= 0
s > s0 F(s) =G(s)
0
f(t)estdt=
0
g(t)estdt
0
(f(t) g(t))estdt= 0
s > s0 f(t) g(t)
h(t) =
0, 0t 3,
0
h(t)estdt= 0 s > s0.
f(t) g(t)
h(t) =
1, 0t < ,0, t
0<
-
7/26/2019 Notas Transformada de Laplace
14/83
F(s) =L{f(t)} f(t) =L1{F(s)}
1
s 1
1
s2 t
1
sn, (n= 1, 2, 3,...)
tn1
(n 1)!
F(s) =L{f(t)} f(t) =L1{F(s)}
a
a2
+s2
sin(at)
1
s+a e
at
s > s0
0
h3(t)estdt =
0
estdt 2
estdt
= est
s
0
est
s2
= 1
s e
s
s e
s
s +
e2s
s = 0, s > 0.
F(s) = 10100+s2 a= 10
L1
10
100 + s2
= sin(10t)
F(s) = 1
s30 n= 30
L1
1
s30
=
t29
29!.
F(s) f(t)
limt
f(t) =L,
-
7/26/2019 Notas Transformada de Laplace
15/83
lims0+
sF(s) = L.
sF(s) = s
0
f(t)estdt
= s a
0
f(t)estdt+ sa
f(t)estdt.
a
a
f(t) L limt
f(t) = L
s
a
f(t)estdt sa
Lestdt.
s Ls
esta
=Leas
eas 1 s0 lims
0+sF(s) = L.
F(s) f(t)
c
limt0+
f(t) =L,
lims
sF(s) = L.
sF(s) = s
0
f(t)estdt
= s a0 f(t)e
st
dt+ s ba f(t)e
st
dt+ sb f(t)e
st
dt.
s a
b
f(t) [a, b] |f(t)|< M s
ba
f(t)estdt
s ba
|f(t)|estdt
Ms ba
estdt
Ms
1
sest
b
a
=M(esa
esb).
b
c
M > 0 |f(t)|< M ect t > b s
b
f(t)estdt
sb
|f(t)|estdt
sb
M e(sc)tdt
Ms 1c s e
(sc)t
b
= M s
s c (e(sc)b).
-
7/26/2019 Notas Transformada de Laplace
16/83
a
f(t) L limt0
f(t) = L
s
a0
f(t)estdt s a0
Lestdt.
s L
s e
st
a
0=L 1 e
as
eas 0 s lims
sF(s) = L.
t
f(t)
c
k
a)
t
f(t)
c
k
b)
t
f(t)
c c + b
k
c)
t
f(t)
c
k
d)
t
f(t)
c 2c
k
e)
F(s) = ks esc
F(s) =
k
s (1 esc
)
F(s) = ks esc 1 esb
F(s) =kccsesc + 1s2 (e
sc 1)
F(s) =kc2cse
sc 2cse2sc + 1s2
2esc e2sc 1+ 2ks esc e2sc
-
7/26/2019 Notas Transformada de Laplace
17/83
f(t) = at
f(t) =
0, 0t
-
7/26/2019 Notas Transformada de Laplace
18/83
-
7/26/2019 Notas Transformada de Laplace
19/83
L{f(t) + g(t)}= L{f(t)} + L{g(t)}
L{f(t) + g(t)} =0
(f(t) + g(t)) estdt
=
0
f(t)estdt+
0
g(t)estdt
=
0
f(t)estdt +
0
g(t)estdt
= L{f(t)} + L{g(t)} .
f(t) = sin(wt)
eiwt = cos(wt) + i sin(wt)
eiwt = cos(wt) i sin(wt).
sin(wt) = eiwt eiwt
2i
cos(wt) = eiwt + eiwt
2
-
7/26/2019 Notas Transformada de Laplace
20/83
L{sin(wt)} = L
eiwt eiwt2i
= 1
2iL
eiwt
1
2iL
eiwt
= 1
2i
1
s iw 1s + iw
= 1
2i
s + iw (s iw)(s iw)(s+ iw)
= 1
2i
2iw
s2 + w2
= w
s2 + w2
f(t) = cos(wt)
f(t) = eat
ebt
f(t) = sinh(wt)
sinh(at) = eat eat
2
cosh(at) = eat + eat
2 .
L{sinh(at)} = Leat eat
2 = 1
2Leat 1
2Leat
= 1
2
1
s a 1
s + a
= 1
2
s+ a (s a)(s a)(s+ a)
= 1
2
2a
s2 a2
= a
s2 a2
f(t) =wt sin(wt)
L{wt sin(wt)} = wL{t} L {sin(wt)}=
w
s2 w
s2 + w2
= w(s2 + w2) ws2
s2(s2 + w2)
= w(s2 + w2) ws2
s2(s2 + w2) =
w3
s2(s2 + w2)
-
7/26/2019 Notas Transformada de Laplace
21/83
f(t) = cosh(at)
f(t) = 1cos(wt)
L1 {F(s) + G(s)}= f(t) + g(t).
L1 {F(s) + G(s)} = L1 {L{f(t)} + L{g(t)}}= L1 {L {f(t) + g(t)}}= f(t) + g(t).
F(s) = 2s + 4s2 1s1
L12
s+
4
s2 1
s 1 = 2L11
s+ 4L
1 1s2 L
1 1
s 1= 2 + 4t et
f(t) f(t) t0
L{f(t)}= sL{f(t)} f(0).
f(t) f(t)
L{f(t)} = 0
estf(t)dt
= estf(t)0 0
(sest)f(t)dt
= f(0) + s0
estf(t)dt
= f(0) + sL{f(t)}
f(t) f(t)
f(t) f(t) f(t)
L{f(t)} = sL{f(t)} f(0)= s (sL{f(t)} f(0)) f(0)= s2L{f(t)} sf(0) f(0).
f(t), f(t), , f(n1)(t) f(n)(t)
L{f(n)(t)}= snL{f(t)} sn1f(0) sn2f(0) f(n1)(0).
-
7/26/2019 Notas Transformada de Laplace
22/83
f(t) = cos(t)
f(t) = cos(t), f(t) =sin(t) f(t) =cos(t).
f(t) =f(t).
s2F(s) sf(0) f(0) =F(s).
f(0) = cos(0) = 1 f(0) =sin(0) = 0
s2F(s) s=F(s),
F(s) = s
s2 + 1,
g(t) = t cos(t)
g(t) =t sin(t) + cos(t)
g(t) =t cos(t) sin(t) sin(t),
g(t) =g(t) 2 sin(t).
s2G(s) sg(0) g(0) =G(s) 2L{sin(t)}.
g(0) = 0 cos(0) = 0 g(0) =0 sin(0) + cos(0) = 1 L{sin(t)}= 1s2+1
s2G(s) 1 =G(s) 2s2 + 1
,
G(s) = s2 1(s2 + 1)2
,
y(t) + ay(t) + by(t) = f(t)
y(0) = y0
y(0) = y0
-
7/26/2019 Notas Transformada de Laplace
23/83
a
b
y0 y0
L{y(t)} + aL{y(t)} + bL{y(t)}=L{f(t)}.
s2L{y(t)} sy(0) y(0) + asL{y(t)} ay(0) + bL{y(t)}=L{f(t)}
Y(s) =F(s) + sy0+ y0+ ay0
(s2 + as+ b)
L{y(t)}= Y(s) L{f(t)}= F(s) y(t) =L1{Y(s)}
y(t) + y(t) = 2t
y(0) = 2
y(0) = 1
F{y(t)
}+
F{y(t)
}=
F{2t
}
s2L{y(t)} sy(0) y(0) + F{y(t)}= 2F{t}. L{y(t)}= Y(s) F{t}= 1s2
s2Y(s) sy(0) y(0) + Y(s) = 2s2
.
y(0) = 2 y(0) = 1
s2Y(s) 2s 1 + Y(s) = 2s2
.
Y(s)
Y(s)
s2 + 1
= 2
s2+ 2s + 1,
Y(s) = 2
s2 (s2 + 1)+
2s
(s2 + 1)+
1
(s2 + 1).
y(t) =L1
2
s2 (s2 + 1)
+ L1
2s
(s2 + 1)
+ L1
1
(s2 + 1)
.
L1 1
s2 (s2 + 1)
= t sin(t), w= 1 ,
L1
s
(s2 + 1)
= cos(t), w= 1
L1
1
(s2 + 1)
= sin(t), w= 1
y(t) = 2t 2sin(t) + 2 cos(t) + sin(t) = 2t + 2 cos(t) sin(t)
-
7/26/2019 Notas Transformada de Laplace
24/83
y(t) + 2y(t) + y(t) = et
y(0) = 1y(0) = 1
P(x) Q(x) P Q Q(x)
Q(x) = (a1x + b1)l1 (anx + bn)ln(c1x2 + d1x + e1)p1 (cmx2 + dmx + em)pm .
A1,1, . . . , An,l1ln B1,1, . . . , Bm,p1pn C1,1, . . . , C m,p1pn
P(x)
Q(x) =
l11k=0
A1,k(a1x + b1)l1k
+ +ln1k=0
An,k(anx + bn)lnk
+p11k=0
B1,kx + C1,k(c1x2 + d1x + e1)p1k
+ + pm1k=0
Bm,kx + Cm,k(cmx2 + dmx + em)pmk
.
F(s) = 1(s1)(s22s5)
A
B
C
F(s) = 1
(s 1)(s2 2s 5) = A
s 1+ B+ Cs
s2 2s 5=
A(s2 2s 5) + (B+ Cs)(s 1)(s 1)(s2 2s 5)
=
s2(A + C) + s(
2A + B
C)
5A
B
(s 1)(s2 2s 5) .
A +C = 0
2A + B C = 05A B = 1
A= B =16 C= 16
F(s) =1
6
1
s 1+ 1 + ss2 2s 5
.
F(s) =1
6
1
s 1+ s 1
(s 1)2 6
.
f(t) =1
6
et + et cosh(
6t)
.
F(s) = 3s2 2s 1(s 3)(s2 + 1) .
-
7/26/2019 Notas Transformada de Laplace
25/83
f(t) =8t + 2 sin(5t)
f(t) =3t2(t 2)
f(t) = (2t 3)3
f(t) = cos(t)2
f(t) = sin(t)3
f(t) = cosh(t)3
F(s) =8s2 + 10s2+25
F(s) =18s4 + 12s3
F(s) = 48s4 72s3 + 54s2 27s
F(s) = s2+2
s3+4s
F(s) = 6s4+10s2+9
F(s) = s(s27)
s410s2+9
F(s) = 2s8s2+4
F(s) = 2s
1s24 +
1s(s24)
f(t) = 2 cos(2t) 4 sin(2t)
f(t) = 2 cosh(2t)
sinh(2t)
L{ezt}= 1sz z C
L{e(a+ib)t}= s a+ ib(s a)2 + b2
-
7/26/2019 Notas Transformada de Laplace
26/83
L{eat cos(bt)}= s a(s a)2 + b2
L{eat sin(bt)}= b(s
a)2 + b2
y+ 2y= 1,y(0) = 3
y+ 3y= et,y(0) = 2
y+ 5y+ 6y= 0,y(0) = 1,y(0) = 0
y
+ 2y+ y= et,y(0) =1,y(0) = 1
y(t) = 52e2t + 12
y(t) = 14et + 74e
3t
y(t) = 3e2t 2e3t
y(t) = 12et(t2 2)
0
1
2
0 1 2 3 4
t
f(t)a)
t
f(t)b)
F(s) = 1s + ese2s
s2
F(s) = 2s +es+2e3se4s
s2
-
7/26/2019 Notas Transformada de Laplace
27/83
s
F(sa)
F(s)
F(s) f(t) eatf(t) F(s a)
Leatf(t)= F(s a), s > a
L1 {F(s a)}= eatf(t), s > a.
F(s a)
F(s a) =0
f(t)e(sa)tdt
=0
f(t)eatestdt
= Leatf(t)
f(t) = teat
L{t}= 1s2
=F(s),
f(t) = t
Leatt= F(s a) = 1(s a)2 , s > a
L
1
4a3(sin(at) cosh(at) cos(at) sinh(at))
=
1
s4 + 4a4,
L
1
asin(at)
=
1
s2 + a2,
-
7/26/2019 Notas Transformada de Laplace
28/83
L{cos(at)}= ss2 + a2
.
L
1
4a3(sin(at) cosh(at) cos(at) sinh(at))
= L
1
4a3
sin(at)
eat +eat
2 cos(at)e
at eat2
= L 18a3
eat (sin(at) cos(at)) +eat (sin(at) + cos(at))
= 1
8a3Leat (sin(at) cos(at)) + Leat (sin(at) + cos(at))
= 1
8a3
a
(s a)2 +a2 s a
(s a)2 +a2
+ a
(s+a)2 +a2+
s+a
(s+a)2 +a2
= 1
8a3
8a3
((s a)2 +a2) ((s+a)2 +a2)
=
1
s4 + 4a4
L
1
(n 1)! tn1eat
L
1
wsin(wt)eat
f(t) = et cosh(2t)
a= 2
L{cosh(2t)}= ss2 4=F(s).
s
Let cosh(2t)= F(s+ 1) = s + 1
(s+ 1)2
4, s >
1
L{sinh(2t) cos(t)}
L(4 + t2)et
F(s) = 1s22s3 F(s)
F(s) = 1
s2 2s 3= 1
(s 1)2 1 3= 1
(s 1)2 4 .
L12
sinh(2t)
= 1s2 4=G(s)
s
L1{F(s)}=L1{G(s 1)}= 12
et sinh(2t).
f(t) F(s)
F(s) = s(s+1)2+1
F(s) = s1(s22s+5)(s22s+10)
-
7/26/2019 Notas Transformada de Laplace
29/83
m
k
y(t)0
y(t)
t
ma=i
fi,
a = y (t)
ifi
k
f1 =ky(t)
f2 =y (t)
f3(t) =f(t)
my(t) = f1+ f2+ f3 =ky(t) y (t) + f(t),
y(t)
my(t) + y (t) + ky(t) = f(t).
y(0) =y0 y(0) =y0
mL{y(t)} + L{y(t)} + kL{y(t)}=L{f(t)}.
ms2
L{y(t)
} msy(0)
my(0) + s
L{y(t)
} y(0) + k
L{y(t)
}=
L{f(t)
}.
ms2Y(s) msy0 my0+ sY(s) y0+ kY(s) = F(s),
F(s) =L{f(t)} Y(s) =L{y(t)} Y(s)
Y(s) =F(s) + msy0+my0+ y0
ms2 + s + k .
y(t) =L1{Y(s)}
-
7/26/2019 Notas Transformada de Laplace
30/83
f(t) = 0 F(s) = 0
Y(s) =msy0+ my
0+ y0
ms2 + s + k .
2 0
2 >4mk F(s) = 0
Y(s) =msy0+ my
0+ y0
ms2 + s + k =
msy0+ my0+y0
m s + 2m2 2
4m+ k,
24m +k < 0
2 sinh(at) = eateat 2 cosh(at) =eat + eat
2 = 4mk F(s) = 0
Y(s) =msy0+ my0+ y0
ms2 + s + k =
msy0+ my0+y0s 2m
2 ,
= 0 F(s) = 0
Y(s) = msy0+ my
0
ms2 + k =
1
m
msy0+ my0
s2 + km,
= 2 m= 1 k= 5
y0 = 1 y0 =2 Y(s)
Y(s) = s
s2
+ 2s + 5
.
y(t) =L1
s
s2 + 2s + 5
.
ss2+2s+5
s
s2 + 2s + 5=
s
(s+ 1)2 + 4 =
s + 1
(s + 1)2 + 4 1
(s + 1)2 + 4.
-
7/26/2019 Notas Transformada de Laplace
31/83
y(t) = L1
s + 1
(s+ 1)2 + 22
1
2L1
2
(s+ 1)2 + 22
= et cos(2t) 12
et sin(2t).
et
cos(2t) 12
sin(2t)
= Aet cos(2t + ) = Aet (cos(2t)cos() sin(2t)sin()) .
A cos() = 1
A sin() = 1
2
A= 1 +1
4=
5
2
cos() = 2
5=
2
5
5 ,
0, 463648 26, 570
y(t) =
5
2 et cos(2t + 0, 463648).
y(t)
1/2
u(t) =
0, t < 01, t > 0.
t= a
u(t a) =
0, t < a1, t > a.
u(t) u(t
a) a >0
t = a
t = a
t = 0
g(t) =
0, t ,
-
7/26/2019 Notas Transformada de Laplace
32/83
t
y(t)
52
et
52
et
1
t
u(t)
1
t
u(t a)
a
1
t
u(t)
g(t) t= 0
lim0
g(t) = u(t), t= 0.
-
7/26/2019 Notas Transformada de Laplace
33/83
fp(t) =
0, t < a1, a < t < b0, t > b.,
a < b
1
t
a b
u(t)
fp(t) = u(t a) u(t b), a < b.
fp(t)f(t) f
t < a
f
t = a t = b u(t a)f(t) t= a u(t 1)sin(t)
1
1 2 3 4 5 6 7t
u(t 1) sin(t)
u(t 1) sin(t)
f(t)
f(t) =
0, t < 1
2, 1< t < 33, 3< t < 50, t > 5.
2(u(t 1) u(t 3)) 3(u(t 3) u(t 5))
f(t) = 2(u(t 1) u(t 3)) 3(u(t 3) u(t 5)).
f(t) = (u(t) u(2)) sin(t)
-
7/26/2019 Notas Transformada de Laplace
34/83
1
2
1
2
3
1 2 3 4 5 6 7t
f(t)
a0
L{u(t a)} =0
u(t a)estdt
=
a
estdt
= 1
s est
a
=eas
s .
a 0
L1 easF(s)= u(t a)f(t a), a > 0.
L{u(t a)f(t a)} =
0
u(t a)f(t a)estdt
=
a
0
u(t a)f(t a)estdt +a
u(t a)f(t a)estdt
=
a
f(t a)estdt,
u(ta) [0, a) (a, ) v= ta
a
f(t a)estdt=0
f(v)es(v+a)dv= eas0
f(v)esvdv.
-
7/26/2019 Notas Transformada de Laplace
35/83
L{u(t a)f(t a)}= easL{f(t)}= easF(s).
f(t) = 1
L{u(t a)}= eass
, a > 0
a= 0
L{t2}= 2s3
e3s 2s3
L1
e3s2
s3
= u(t 3)(t 3)2.
G(s) =e2s ss2+4
G(s) =es 1s2 3e3s 1s2
F(s) = es 1
(s + 1)2 1 .
1(s+1)21
L1
1
(s + 1)2 1
= et sinh(t).
L1
es 1
(s + 1)2 1
= u(t 1)L1
1
(s+ 1)2 1
= u(t 1)e(t1) sinh(t 1).
F(s) =e2s(s 1)s2 2s + 5 .
F(s) f(t) t0
f()d 1sF(s)
L t
0
f()d
=
1
sF(s),
L1
1
sF(s)
=
t0
f()d.
-
7/26/2019 Notas Transformada de Laplace
36/83
g(t) =t0
f()d g(t) = f(t)
L{g(t)}= sL{g(t)} g(0).
g(0) = 0
L t
0
f()d = L {g(t)}=
1
sL{g(t)}
= 1
sL{f(t)}
= 1
sF(s).
V0
V(t)
V(t) = V0(u(t a) u(t b)) .
t= a V0 t= a t= b
CR
V
i(t)
Ri(t) + 1
C
q(t) = V(t),
q(t) 1Cq(t) C Ri(t)
R
q(t) =t0
i()d i(t)
Ri(t) + 1
C
t0
i()d =V0(u(t a) u(t b)) .
L{i(t)} + 1sRC
L{i(t)}= V0Rs
eas ebs ,
-
7/26/2019 Notas Transformada de Laplace
37/83
-
7/26/2019 Notas Transformada de Laplace
38/83
1
2
3
11 2 3 4 51
t
i(t)
1
2
3
11 2 3 4 51
t
q(t)
d1 = 0 d2 = 1CR
L1
1
s
s+ 1CR
= CR
1 e tCR
.
t
q(t) = CV0 u(t a)1 e ta
CR u(t b)1 e tb
CR .
q(t) =
0, t < a
CV0
1 e taCR
, a < t < b,
CV0
e
tb
CR e taCR
, t > b,
a= 0.5 b= 2 R= 1 C= 1 V0 = 3V
-
7/26/2019 Notas Transformada de Laplace
39/83
f(t) = u(t
1) + 3u(t
3)
4u(t
5)
f(t) = tu(t) + (t2 t)u(t 1) + (6 t t2)u(t 2) + (t 6)u(t 6)
1
2
3
4
1 2 3 4 5 6
t
f(t)
1
2
3
4
1 2 3 4 5 6
t
f(t)
f(t) = 1s
es + 3e3s 4e5s
f(t) = 1s2 +2s3 +
1s2
es +
2s3 5s2 e2s + 1s2 e6s
-
7/26/2019 Notas Transformada de Laplace
40/83
1
2
1
2
1 2 3 4 5 6t
f(t)
1
2
1
2
1 2 3 4 5 6
t
f(t)
1
1 2 3 4 5 6
t
f(t)
1
2
3
4
1 2 3 4 5 6
t
f(t)
f(t) = u(t 1) 3u(t 3) + 2u(t 5)
f(t) = tu(t) + (1 t)u(t 1) + 32 t+ 92u(t 3) + 32 t 112 u(t 5)
f(t) =
n=0(1)nu(t n)
-
7/26/2019 Notas Transformada de Laplace
41/83
f(t) =
n=0 u(t 2n)
F(s) = 1s
es 3e3s + 2e5s
F(s) = 1s2
1 es 32e3s + 32e5s + 2se5s
F(s) = 1s(1+es)
F(s) = 1s(1e2s)
F(s)
F(s) = 2s
s2 + 2s + 5
F(s) = 2s 2
(s2 2s + 5)(s2 2s + 10)
F(s) = 2e4s
s2 4
F(s) = 2 e2s
(s 3)(s 1)
F(s) = 2s
s2 + 2s + 5es
F(s) =
2s
2
(s2 2s + 5)(s2 2s + 10)e3s
f(t) = (2 cos(2t) + sin(2t)) et
f(t) = et
5(cos(2t) sin(3t))
f(t) = u(t 4) sinh(2(t 4))
f(t) = u(t 2)
e3(t2) e(t2)
f(t) = u(t 1) (2 cos(2(t 1)) + sin(2(t 1)))
f(t) = u(t 3)
5 (cos(2(t 3)) sin(3(t 3)))
y+ 5y+ 6y= 12et,y(0) = 1,y(0) =1
-
7/26/2019 Notas Transformada de Laplace
42/83
y+ 5y+ 6y= 6u(t 3),y(0) = 1,y(0) =1
y(t) =2et
+ e
t
+ 2e3t
y(t) = u(t)
2e2t e3t+ u(t 3) 1 3e2t+6 2e3t+9
f(t) = tu(t 1)
f(t) = t2u(t 1)
F(s) = 1s2 +
1s e
s
F(s) = 2s3 +
2s2 +
1s
es
F(t) = F0sin
km t
my(t) + ky(t) = F(t)y(0) = 0y(0) = 0
.
y(t) = F0
2k
sin
km
t
km
t cos
km
t
F(t) = 2(u(t 1) u(t 2))
y(t) 3y(t) + 2y(t) = F(t)y(0) = 0y(0) = 0
.
y(t) = (1
2e(t1) + e2(t1))u(t
1)
(1
2e(t2) + e2(t2))u(t
2)
-
7/26/2019 Notas Transformada de Laplace
43/83
[, ]
(t) = 1
2(u(t + ) u(t )) =
0, t .
t= a
(t a) = 12
(u(t (a )) u(t (a+ ))) =
0, t < a 12 , a < t < a + 0, t > a + .
(t a) = 1 > 0 (t a) a > 0
= 1 = 12 = 14
= 18 = 112
(t a) = lim0
(t a).
(t a) (t a)
(t)
f(t) = 1
e
t2
2
-
7/26/2019 Notas Transformada de Laplace
44/83
t
(t a) a= 1 = 1
a
t
(t a) a= 1 = 12
a
t
(t a) a= 1 = 14
a
t
(t a) a= 1 = 18
a
t
(t a) a= 1 = 112
a
t
(t a) a= 1
a
t= a
(t a) =
0, t=a, t= a
(t a)dt= 1
f(t) t= a
(t a)f(t)dt= f(a).
-
7/26/2019 Notas Transformada de Laplace
45/83
V0
t= a [a, a+] f(t) f(a)
(t a)f(t)dt =
(t a)f(t)dt
(t
a)f(a)dt
= f(a)
(t a)dt= f(a).
(t a) = lim0
1
2(u(t (a )) u(t (a + ))) .
lim0
1
2(u((t a) + )) u((t a) ))) = d
dtu(t a)
(t a) = ddt
u(t a).
L{(t a)}=0
(t a)estdt= eas.
V0
V(t)
V(t) = V0(u(t a) u(t b)) .
i(t)
Li(t) + Ri(t) + 1
Cq(t) = V0(u(t a) u(t b)) ,
q(t) 1Cq(t) C Ri(t)
R
Li(t) L i(0) = 0 q(0) = 0
dq(t)dt =i(t)
Li(t) + Ri(t) + 1
Ci(t) = V0((t a) (t b)) ,
i(0) = 0 i(0) = 0
Ls2I(s) + RsI(s) + 1
CI(s) =V0
eas ebs ,
-
7/26/2019 Notas Transformada de Laplace
46/83
L
CR
V
I(s) = V0
eas
ebs
Ls2 + Rs + 1C
= 1
L
V0
eas ebss+ R2L
2 R2L2 + 1LC=
V0L
eas
s + R2L2
+ e
bss + R2L
2+
= 1
LC
R
2L
2.
V0 = 10V
a= 1
b= 5
>0 L= 1 C= 110 R= 2
I(s) = 10
es
(s+ 1)2
+ 9 e
5s
(s+ 1)2
+ 9
.
i(t) =10
3
u(t 1)e(t1) sin(3(t 1)) u(t 5)e(t5) sin (3(t 5))
.
-
7/26/2019 Notas Transformada de Laplace
47/83
V0
1
2
3
1
2
1 2 3 4 5 6 7 81t
i(t)
1
2
3
4
5
11 2 3 4 5 6 7 81
t
i(t)
= 0 L= 1 C= 1 R= 2
I(s) = 10
es
(s+ 1)2
e5s
(s+ 1)2
.
i(t) = 10
u(t 1)e(t1)(t 1) u(t 5)e(t5)(t 5)
.
-
7/26/2019 Notas Transformada de Laplace
48/83
1 2 3 4 5 6 7 81
t
i(t)
L
x
x = L
W(x) x[0, L]
k(x) (x) (x)
x
k(x) =d(x)
dx .
dy(x)
dx = tan((x))
(x) tan((x))(x) dy(x)
dx =(x),
d2y(x)
dx2 =k(x).
k(x) = M(x)EI E I
M(x)
d2y(x)dx2
= M(x)EI
.
M(x) V(x)
d
dxM(x) = V(x)
d
dxV(x) = W(x).
-
7/26/2019 Notas Transformada de Laplace
49/83
-
7/26/2019 Notas Transformada de Laplace
50/83
1
2
1 1 2 3 4 5 x
y(x)
c(t) +1
c(t) =x(t), t > 0
c(t) x(t) t
c0 T
x(t) =c0((t) + (t T) + (t 2T) + (t 3T) + ) c(0) = 0 c(t)
sC(s) +1
C(s) = c0
1 + esT + e2sT + e3sT + = c0
n=0
esT
n.
C(s)
s+ 1
= c0
n=0
esT
n
C(s) =
c0s+ 1
n=0
esT
n .
s
c(t) = c0
e
t
+ etT
u(t T) + e t2T u(t 2T) + e t3T u(t 3T) +
= c0e t
1 + e
T
u(t T) + e 2T u(t 2T) + e 3T u(t 3T) +
c0 = 1 = 1 T = 1
-
7/26/2019 Notas Transformada de Laplace
51/83
1
2
3
1 2 3 4 5
t
i(t)
c0
limtnT
c(t) = limtnT
c0e
t
1 + e
T
+ e2T + + e (n1)T
=
c0enT
1 + e
T
+ e
2T
+ + e(n1)T
=
c0
e
nT
+ e(n1)T
+ e(n2)T
+ + eT
limtnT+
c(t) = limtnT+
c0e
t
1 + e
T
+ e2T + + e (n1)T + enT
=
c0enT
1 + e
T
+ e2T + + e (n1)T + enT
=
c0
e
nT
+ e(n1)T
+ e(n2)T
+ + eT + 1
,
c0
t = 0 c(0) = c0 c(0) = 0
y(t) + y(t) = (t)
y(0) = 0
sY(s) y(0) + Y(s) = 1
Y(s) = 1s+1
y(t) = et.
y(0) = 1= 0
t > 0 y(t)
y(t) = u(t)et.
y(0)
y(t) + y(t) = u(t)u(t)y(0) = 0
-
7/26/2019 Notas Transformada de Laplace
52/83
u(t) u(t )
(t) 0+ Y(s)
Y(s) =
1
s(s+ 1)
1
es
=1
s 1
s+ 11
es
,
y(t) =1 et
u(t) u(t ) 1 e
(t)
.
y(t) =
0, t0,
1et
, 0< t,e1
et, t.
>0
et = 1 + t+t2
2 +
t3
3!+ . . .1 + t
y(t)
0, t0,t , 0< t,
et, t. 0 y(t)
y(t) = 0.2 = 0.1 = 0.05
etu(t)
1
0 1 2
[0, ] f g f g
(f g)(t) = t0
f()g(t )d.
-
7/26/2019 Notas Transformada de Laplace
53/83
f(t) = et g(t) = cos(t) f g
(f g)(t) = t0
e cos(t )d
= 1
2e (cos(t ) sin(t ))
t
0
= 1
2
et cos(t) + sin(t) .
e cos(t )d= 12e (cos(t ) sin(t )) +
f g= g f
F(s) =L{f(t)} G(s) =L{g(t)}
L{(f g)(t)}= F(s)G(s).
L1
{F(s)G(s)
}= (f
g)(t).
F(s) =L{f(t)}=0
f(t)estdt
G(s) =L{g()}=0
g()esd.
F(s)G(s) =
0
f(t)estdt
0
g()esd
=
0
f(t)
0
g()es(t+)ddt
t
v= t+
F(s)G(s) =
0
f(t)
t
g(v t)esvdvdt
v [t, ] t [0, ] t [0, v] v [0, ]
F(s)G(s) =0
v
0
f(t)g(v t)esvdtdv
=
0
v0
f(t)g(v t)dt
esvdv
=
0
(f g)esvdv= L{f g}
-
7/26/2019 Notas Transformada de Laplace
54/83
s(s1)(s2+1)
s
(s 1)(s2 + 1) = 1
s 1s
s2 + 1,
L1 1
s1= et L1 ss2+1= cos(t) L1
1
s 1s
s2 + 1
=
t0
e cos(t )d.
L1
1
s 1s
s2 + 1
=
1
2
et cos(t) + sin(t) .
y(t) = 4 + 9
t0
y()(t )d.
f(t) = y(t) g(t) = t
L{y(t)}= 4s
+ 9L{y(t)}L{t}
Y(s) = 4
s+ 9Y(s)
1
s2.
Y(s) = s2
s2 94
s = 4s
s2 9 .
y(t) = 4 cosh(3t)
y(t) = t +
t0
y()sin(t )d
du(t)
dt =(u(t) uamb) + q(t)
u(t) uamb q(t) u(0) = 50 uamb= 50 = 4
u(t) q(t) = 100(t 1) u(t)
-
7/26/2019 Notas Transformada de Laplace
55/83
q(t)
dq(t)
dt =(ua u(t)).
ua
f(t) g(t)
f(t) =
t2
, 0t < 2t2 ,
t
-
7/26/2019 Notas Transformada de Laplace
56/83
-
7/26/2019 Notas Transformada de Laplace
57/83
f(t) T
L{f(t)}= 11 esT
T0
f(t)estdt.
f(t)
L{f(t)} =0
f(t)estdt
= T
0
f(t)estdt+ 2T
T
f(t)estdt+ 4T
3T
f(t)estdt+
=
n=0
(n+1)TnT
f(t)estdt.
=t nT
L{f(t)} =n=0
T0
f(+ nT)es(+nT)d
=
n=0
esnT T0
f(+ nT)esd.
f() =f(+ nT)
L{f(t)} = n=0
esnT T
0
f()esd
=
T0
f()esd
n=0
esT
n
=
T0
f()esd
1
1 esT
= 1
1 esT T0
f()esd,
-
7/26/2019 Notas Transformada de Laplace
58/83
esT
f(t) = cos(wt) cos(wt)estdt=
est (w sin(wt) s cos(wt))s2 + w2
+
L{cos(wt)} = 11 es 2w
2w
0
cos(wt)estdt
= 1
1 es 2w
est (w sin(wt) s cos(wt))s2 + w2
2w
0
= 1
1 es 2ws ses 2w
s2 + w2
= s
s2 + w2.
f(t) 2a
0
1
1
t
f(t)
a 2a 3a 4a 5a 6a
T= 2a
L{f(t)} = 11 e2sa
2a0
f(t)estdt
= 1
1 e2sa a
0
estdt 2aa
estdt
= 1
1 e2sa
1 2eas + e2ass
= 1
(1 esa)(1 + esa)
(1 eas)2s
= 1
s
1 eas1 + esa
.
eas
2
L{f(t)} = 1s
eas
2 e as2eas
2 + eas
2 )
= 1
s
sinhas2
cosh
as2
=
1
stanh
as2
.
-
7/26/2019 Notas Transformada de Laplace
59/83
g(t) 2a
0
1
t
g(t)
2a 4a 6a
g(t) [0, a] 1a [a, 2a] 1a
1
a
L{g(t)}= 1sL{g(t)} +1
sg(0).
L{ } = 1as
L{ } +1s
,
L{g(t)} = 1
as
1
stanhas
2
= 1
as2tanh
as2
.
h(t)
h(t) =
sin(wt), 0< t < w0, w < t 0
L
2
t(1 cos(wt))
= ln
s2 + w2
s2
.
L(1
cos(wt)) =
1
s s
s2 + w2.
L
2
t(1 cos(wt))
= 2
s
1
v v
v2 + w2
dv
= 2
ln(v) 1
2ln(v2 + w2)
s
= (2 l n(v) ln(v b))|s= ln
v2
v2 + w2
s
= ln s2
s2 +w2
= ln
s2 + w2
s2
limv
ln
v2
v2 + w2
= ln
limv
v2
v2 + w2
= ln ( 1)
= 0.
ln(x) x >0
L
1
tsin(wt)
= arctan
ws
.
L{sin(wt)}= ws2 + w2
.
-
7/26/2019 Notas Transformada de Laplace
68/83
L
1
tsin(wt)
=
s
w
v2 + w2
dv
= arctan v
w
s
= limv arctan v
w arctan sw
=
2 arctan
sw
tan2
= cot()
L
1
tsin(wt)
= cot1
sw
.
= cot1sw
cot = sw
1tan =
sw tan =
ws
L1t
sin(wt)
= arctan
ws
L{ (t)}= 1s
cot1(s),
(t)
(t) =
t0
sin(x)
x dx.
L{ (t)}=L t
0
sin(x)
x dx
=
1
sL
sin(t)
t
.
L{ (t)}= 1s
arctan
1
s
=
1
scot1 (s) .
s
-
7/26/2019 Notas Transformada de Laplace
69/83
-
7/26/2019 Notas Transformada de Laplace
70/83
-
7/26/2019 Notas Transformada de Laplace
71/83
40i
+
110
5 i1
1
10 i2
2
i1 i2
i1 i2 i1(0) =i2(0) = 0
di1(t)
dt + 5i1(t) + 40i(t) = 110
2di2(t)
dt + 10i2(t) + 40i(t) = 110.
i(t) =i1(t) + i2(t)
di1(t)dt
+ 45i1(t) + 40i2(t) = 110
2di2(t)
dt + 40i1(t) + 50i2(t) = 110,
di1(t)
dt + 45i1(t) + 40i2(t) = 110
di2(t)
dt + 20i1(t) + 25i2(t) = 55.
sI1(s) i1(0) + 45I1(s) + 40I2(s) = 110s
sI2(s) i2(0) + 20I1(s) + 25I2(s) = 55s
.
(s+ 45) I1(s) + 40I2(s) = 110
s
20I1(s) + (s + 25) I2(s) = 55
s .
-
7/26/2019 Notas Transformada de Laplace
72/83
(s + 45) 40
20 (s + 25)
I1(s)I2(s)
=
110s
55s
I1(s)I2(s)
=
1
(s+ 25)(s + 45) 800
(s + 25) 4020 (s + 45)
110s55s
I1(s) = 1
s2 + 70s + 325
110
s (s+ 25) 2200
s
=
1
(s + 5)(s+ 65)
110 +
550
s
I2(s) = 1
s2 + 70s + 325
2200
s +
55
s(s + 45)
=
1
(s+ 5)(s+ 65)
55 +
275
s
.
I1(s) = 2I2(s) I2(s)
I2
= 1
(s+ 5)(s + 65)55s + 275
s= 55
(s+ 5)(s + 65)s + 5
s= 55
s(s+ 65).
i2(t) = 55
65
1 e65t= 11
13
1 e65t .
i1(t) = 2i2(t)
i1(t) =22
13
1 e65t .
k1 k2 m1 m2
m1x1(t) = k1x1(t) + k2[x2(t) x1(t)] + f1(t)m2x2(t) = k2[x2(t) x1(t)] + f2(t),
x1 x2 f1 f2
m1(s2X1(s) x1(0) sx1(0)) = (k1+ k2)X1(s) + k2X2(s) + F1(s)
m2(s2X2(s) x2(0) sx2(0)) = k2X2(s) + k2X1(s) + F2(s)
m1s2 + k1+ k2X1(s) k2X2(s) = F1(s) + m1x1(0) + sm1x1(0)k2X1(s) + m2s2 + k2X2(s) = F2(s) + m2x2(0) + sm2x2(0)
m1s2 + k1+ k2 k2
k2 m2s2 + k2
X1(s)X2(s)
=
F1(s) + m1x1(0) + sm1x1(0)F2(s) + m2x2(0) + sm2x2(0)
X1(s)X2(s)
=
1
P(s)
m2s2 + k2 k2
k2 m1s2 + k1+ k2
F1(s) + m1x1(0) + sm1x1(0)F2(s) + m2x2(0) + sm2x2(0)
,
-
7/26/2019 Notas Transformada de Laplace
73/83
-
7/26/2019 Notas Transformada de Laplace
74/83
t
x(t)
t
y(t)
RST
R
S
T
/l
x(t) y(t) z(t)
x(t) = x(t)y(t) = x(t) y (t)z(t) = y (t),
x(0) = 1, y(0) =z(0) = 0.
x(t) y(t) z(t)
= 1 = 2
sX(s) x(0) = X(s)sY(s) y(0) = X(s) Y(s)sZ(s) z(0) = Y(s).
X(s) = x(0)
s + =
1
s+ 1.
-
7/26/2019 Notas Transformada de Laplace
75/83
Y(s) =X(s)
s+ =
x(0)
(s+ )(s + )=
1
(s + 1)(s+ 2).
Z(s) =
Y(s)
s =
2
s(s+ 1)(s + 2) .
x(t) y(t) z(t)
x(t) =L1 {X(s)}= et
y(t) =L1 {Y(s)}=e2t + et,
a= 2 b=1
z(t) =L1 {Z(s)} = 2L1Y(s)
s = 2
t
0
y()d
= 2
t0
e2 + e d= 2
e2t 1
2 et 1
= e2t 2et + 1,
x(t) y(t) z(t)
t
x(t) y(t)
z(t)
-
7/26/2019 Notas Transformada de Laplace
76/83
-
7/26/2019 Notas Transformada de Laplace
77/83
(k) =
0
exxk1dx, (k >0)
I(x) =m=0
(1)mm!(m+ + 1)
x2
2m+
0
J0(x) = 1 x2
22(1!)2+
x4
24(2!)2 x
6
26(3!)2+
(t) =
t0
sin(x)
x dx
= 0.57721566490153286060651209008240243104215933593992...
-
7/26/2019 Notas Transformada de Laplace
78/83
-
7/26/2019 Notas Transformada de Laplace
79/83
-
7/26/2019 Notas Transformada de Laplace
80/83
L{f(t) +g (t)}= L{f(t)} +L{g(t)}
Lf(t)= sL{f(t)} f(0)Lf(t)= s2L{f(t)} sf(0) f(0)
s Leatf(t)= F(s a) t L {u(t a)f(t a)}= easF(s)
L{u(t a)}= eas
s
L
t
0
f()d
=
F(s)
s
L{(t a)}= eas
L{(f g)(t)}= F(s)G(s),
(f
g)(t) =
t
0
f()g(t
)d
L{f(t)}= 11 esT
T
0
es
f()d
L{tf(t)}=dF(s)ds
Lf(t)
t
=
s
F(s)ds
-
7/26/2019 Notas Transformada de Laplace
81/83
-
7/26/2019 Notas Transformada de Laplace
82/83
-
7/26/2019 Notas Transformada de Laplace
83/83