2/5/20151Chng 4BIN I LAPLACENi Dung1. nh ngha bin i Laplace2. Cc tnh cht ca bin i Laplace3. Cc nh l Heaviside 4. Bin i Laplace ca hm tun hon5. Bin i Laplace ngc6. ng dng php bin i Laplace2/5/20152nh Ngha Cho hm () tha mn cc iu kin Dirichlet vi 0. Bin i Laplace ca () l hm () sau: = () = 0

Trong , l mt bin phc Khi , bin i Laplace ngc ca hm () lhm (). K hiu: = 1()V D: , ,

: > ? p dng trc tip cng thc tm bin i Laplace:1 = 0

=

0= 1

= 0

=

0+ 0

=1

2eat= 0

= 0

() =

( )0=1 2/5/20153Tnh Cht Ca Bin i LAPLACE1. Tnh tuyn tnh2. Thay i t l thi gian3. Php dch min thi gian4. Php dch min S5. Vi phn min thi gian6. Tch phn min thi gian7. Vi phn min S8. Tch phn min S nh l Tch chp nh l Duhamel1) Tuyn tnh

1

1()

2

2()

1

1 +2

2

1

1() +2

2()2) Thay i t l thi gian 1

3) Dch min thi gian 0( 0)

0

0

0

( + 0)4) Dch min S

0

0Tnh Cht Ca Bin i LAPLACE ()2/5/201545) Vi phn min thi gian

(0)

=01

1

06) Tch phn min thi gian

0

Tnh Cht Ca Bin i LAPLACE ()7) Vi phn min S (1)

1

8) Tch phn min S

Tnh Cht Ca Bin i LAPLACE ()2/5/20155V d: , ?eat=1 ejat=1 = + +=

2 +2 +

2 +2Cng thc Euler & Tnh cht tuyn tnhejat = cos + sin ejat= cos +sincos =

2 + 2sin =

2 + 22/5/20156V d:

, ? p dng tnh cht tuyn tnh, ta c3 2

= 3 2

= 3 2 1 = 3 5 1 p dng tnh cht dch min S cos =

2 + 2 cos = +1 + 12 + 1V d:

, ? p dng tnh cht vi phn min S, ta co

=1

= 1=12osin =

2+2 t sin =

2 + 2=

2 + 2

2 +2 2=2

2 + 2 22/5/20157V d:

+? p dng tnh cht vi phn min S, ta c

= 1

1

= 1t 1

ln +1 1= 1

1

+1 1 + 1 1 = 1

12

2 1= 1

11 +1 1 1=

V d:

? p dng tnh cht tch phn min S, ta c

= 1

= 1

2 12 = 1 12(2 1)

= 112(2 1)= 4(

)2/5/20158Tnh Cht Ca Bin i LAPLACE ()nh l Tch chp = nh l Duhamel = 1 =

+0+ + 0+

Cng thc tch chp = 0

. . = 0

. . V d: .

+

? =. 2 + 2= .

2 +2.1 = . . p dng cng thc Duhamel = +0+ =1

2+2=sin =11 =

=cos()Ta c, g(0+) = g(0) = 0 =. cos

= .

0

cos . = 2. sin +. .

cos

2+22/5/20159Bin i LAPLACE Ca Hm Tun Hon Nu () l mt hm tun hon vi chu k T, tc l() =( +), trn on [0, ), v lin tc trntng on th = = 0

.

1 ; > 0V d: ?f(t) l hm lin tc, tun hon trong on [0, ) = 02.

1 2=11 2

0

2

=112

0

2=11212 +2

=12. 11+=1. 1+2/5/201510Bin i LAPLACE Ngcnh L Heaviside nh l 1: = 1

& > , :

t hoc l

tq(a): o hm ca q(s) ti aQ(a): a thc q(s) ti a v khng cha (s-a) H qu: = =1

t = =1

t; : ()V d:

++ +?Bc mu > bc t, v N0 ca mu: = 0, 1, 2Cch 1:

=32+6 +2 0 =2 1 =3 2 =6

0 =2 1 =1 2 =2Cch 2:

0 = +1 +2|=0 =2 =01 = +2|=1 =1 =12 = +1|=2 =2 =2 =0

0

0t +1

1

t +2

2

2t = 1 3 + 322/5/201511Bin i LAPLACE Ngcnh L Heaviside (tt) nh l 2: = 1

& > ,

: =

=011

1 !.

!Trong (s) l thng s ca p(s) v tt c cc h s ca q(s) ngoitr

V d:

+

+? i vi h s +2 = 2, L Heaviside 1

1 = 2 2

2 =213

2 =22 i vi h s +13 = 1, L Heaviside 2 =

+ 2; =2 + 22; =4 + 23;1 = 1; 1 = 2; 1 = 4;

2 =

131 !+

132 !

1!+ 10!

3131 !

=42+21

22

=2+212

2

=1 +2 =22+2+2 12

2

2/5/201512Bin i LAPLACE Ngcnh L Heaviside (tt) nh l 3: = 1

& > , () + 2 + 2 =

cos +

sinVi

l phn o &

l phn thc ca biu thc (a+ib) l thngs ca p(s) v tt c cc h s ca q(s) ngoi tr + 2 + 2V d:

+

++? i vi h s +22 = 2, L Heaviside 2 1 =

22 1 0!

00!+22 1 1!

11!

2=350 15 2 =

2 + 2 +10

=2 + 10

2 + 2 + 1022 = 15

2 =3502/5/201513V d:

+

++? 2 +2 +10 = +12 +32 = 1, = 3, L Heaviside 3 2 =

cos +

sin =

+ 22 =1+3 =13 950

= 1350;

= 950 2 = 3950cos3 + 1350sin3 =1 +2 =35015 2 +

3 950cos3 +1350sin 3WD Gii PTVPTT H S HngGii pt sau: 3 10 =2; 0 =1;0 =2 p dng tnh cht vi phn min t, ta c() = () = 0 = 1() = 2 0 0 =2 2 3 10 = 2 } 3 10{ = 2 2 2 3 1 10 =2

2/5/201514WD Gii PTVPTT H S Hng Rt gn

2 3 10 =

2 +2

=

2 +2 5 + 2 p dng nh l Heaviside 1, ta c Bc mu > bc t, v N0 ca mu: = 0, 5, 2

0 = 5 + 2|=0 = 105 = + 2|=5 = 352 = 5|=2 = 14

0 = 2 + 2|=0 = 25 = 2 + 2|=5 = 222 = 2 + 2|=2 = 8 = 15+2235

5 +47

2WD Phn Gii Mch inPhng php 1Phng php 22/5/201515V d: Tm () ? Bit t < 0, K 1; v t 0, K 2Phng Php 11. Lp phng trnh vi phn2. PTVP PT Ton T3. Gii pt ton t4. Chuyn n0pt ton t sang n0 thi gian2/5/201516Phng Php 1 Xt < 0, mch t 0= 0

0= 5PP1: 1/ Lp PTVP Xt 0, 2 p dng nh lut Kirchoff 2 + +1

=011

+ +1

=02 Ta c, 0= 0+= 0 &

0=

0+= 51 +5 =0 =52/5/201517PP1: 2/ PTVP PT Ton T () = () = 0 = () = 2 0 0 = 2 +52 +2 +5 +1

=0 =5 +2+1

=5

2 + +1=5 +122+322PP1:3/ Gii PT Ton T4/ Chuyn N0 p dng nh l Heaviside 3, ta c = 5 + = 12+32= 5 =

cos +

sin = 12

325 sin32

= 103

12

sin32

2/5/201518Phng Php 21. Laplace ha mch in2. p dng nh lut Ohm PT Mch in3. Gii pt mch in ( c Laplace ha)4. Suy ra nghimPP2: Laplace Ha Mch in i vi R i vi L i vi C2/5/201519Phng Php 2 Xt < 0, mch t 0= 0

0= 5PP2: 1/ Laplace Ha Mch in Xt 0, 2 p dng nh lut Kirchoff 2 + 0 +

+0

=0 =0 0

+ +1

=5

2 + +1