2 2 - 36 -

2.9 (method of undetermined coefficients) , , , , ,

( )xr

(1) ( )xrbyyay =++ (A) (basic rule) (1) 2.1 , , (1) .

( )xrpy py

(B) (modification rule) (1) ,

py

py x ( ) . 2x

(C) (sum rule) 2.1 , .

( )xrpy

2.1

( )xr py xx Ceke

kx ( ) 0111,1,0 KxKxKxKn nnnnn ++++= "" xMxK

xkxk

sincos

sincos

+

( )xMxKexkexke x

x

x

sincossincos

+

2 2 - 37 -

1 (A) 284 xyy =+ i2404 22 ===+ xBxAyh 2sin2cos +=

0122 KxKxKyp ++= 122 KxKyp +=

22Kyp =

( ) 201222 8424 xKxKxKKyy pp =+++=+ 4 042048 0212 =+== KKKK 102 012 === KKK

122sin2cos 2 ++=+= xxBxAyyy ph

2 (B) xeyyy =+ 23 ( )( ) 2,1021232 ===+ xxh ececy 221 +=

xp Cxey = xxp CxeCey += xxx

p CxeCeCey ++=

( ) xxxxxxppp eCxeCxeCeCxeCeyyy =+++=+ 23223 1=C

xxxph xeececyyy +=+= 221

3 (B) ( ) ( ) 10,10,2 ===++ yyeyyy x ( ) 10112 22 ==+=++

( ) xh exccy += 21

xp eCxy = 2 xxp eCxCxey = 22

xxxxp eCxCxeCxeCey += 2222

( ) xxxxxxxppp eeCxeCxCxeeCxCxeCeyyy =+++=++ 222 22422

2 2

- 38 -

2112 == CC

( ) xxph exexccyyy ++=+= 221 21 ( ) 10 1 == cy

( ) xxxx exxeexccecy ++= 2212 21

( ) 010 212 === cccy xxx exexey

=+= 121

21 22

4 (C) ( ) ( ) 1.600,2.00,4sin554cos4025.152 5.0 ==+=++ yyxxeyyy x ( ) ii 212104152 22 ==+=++=++ ( )xBxAey xh 2sin2cos += xMxKCey xp 4sin4cos5.0 ++= xMxKCey xp 4cos44sin45.0 5.0 +=

xMxKCey xp 4sin164cos1625.0 5.0 =

( ) ( )

xxexKMxKMCexMxKCexKxMCe

xMxKCeyyy

x

x

x

x

xppp

4sin554cos4025.14sin8114cos11825.64sin54cos554sin84cos8

4sin164cos1625.052

5.0

5.0

5.0

5.0

5.0

+=

++=

+++

++

=++

2.025.125.6 == CC 40118 = KM 0=K 55811 = KM 5=M

( ) xexBxAeyyy xxph 4sin52.02sin2cos 5.0 +++=+=

( ) 02.02.00 ==+= AAyxexBey xx 4sin52.02sin 5.0 ++=

xBexBey xx 02cos22sin ++= xe x 4cos201. 5.0 +( ) 201.60201.020 ==++= BBy

xexey xx 4sin52.02sin20 5.0 ++=

2 2

- 39 -

2.11 : , - (1) m 0=++ kyycy ( )trkyycym =++ , : (input), (driving force) ( )tr : (output), (response) ( )ty (2) tFkyycym cos0=++ ( )0,00 >> F (3) ( ) tbtatyp sincos += ( ) tbtatyp cossin +=

( ) tbtatyp sincos 22 = (2)

( ){ } ( ){ } tFtbmkcatcbamktkbtkatactbc

tbmtamkycymy ppp

cossincossincossincos

sincos

022

22

=+++=

++

+

=++

(4) ( ) ( ) 020

2

=+

=+

bmkcaFcbamk

Cramer rule

( )( ) 2222

20

2

2

20

0cmk

mkF

mkccmk

mkcF

a

+

=

=

2 2

- 40 -

( ) 222202

2

02

0

cmkcF

mkccmk

cFmk

b

+=

=

2.5 ( )00 >=mk

(5) ( )( ) 222220222

00

cm

mFa

+

= , ( ) 22222020

cm

cFb

+=

(2) (6) ( ) ( ) ( )tytyty ph += 1. . 0=c

mk=2 02

(3) (5)

(7) ( ) ( ) tk

Ft

mkm

Ftm

Ftyp

cos

1

cos

1

cos2

0

02

0

022

0

0

=

=

=

2.5 (4*) (p.88) (8) ( ) ( ) ( ) tm FtCty coscos 220 00 +=

(natural frequency)

20

2

2 2

- 41 -

(7) py

(9) kFa 00 = 2

0

1

1

=

(resonance) 0 =

(resonance factor) 57 , (2)

tmFy

mky 00 cos=+

(10) tmFyy 00

20 cos =+

tBtAyh 00 sincos +=

( )tbtatyp 00 sincos +=

( )tbtattbtayp 000000 cossinsincos +++=

( )tbtattbtatbtayp

02

002

0

00000000

sincos

cossincossin

+

++=

( )( )

tmFtbta

tbtat

tbtattbtayy pp

00

0000

02

002

0

02

002

000002

0

coscos2sin2

sincos

sincoscos2sin2

=+=

++

++=+

0

0

2,0

mFba ==

(11) ttmFyp 0

0

0 sin2

= 58

2 2

- 42 -

0

) ( ) ( ) 00,00 == yy ( ) ( ) tm FtBtAty cossincos 220 000 ++= ( ) ( ) ( )220 0220 0 00 ==+= m

FAm

FAy

( ) ( ) tm FtBtAty sincossin 220 00000 += ( ) 000 0 === BBy

(12) ( ) ( )( ) ( 00220 0 coscos = ttmFty )

BABABABABABA

sinsincoscos)cos(sinsincoscos)cos(

=+

+=

BABABA sinsin2)cos()cos( =+

22abBbaAbBA

aBA

=

+==+

=

2sin

2sin2coscos abbaba +=

( ) ( )

+

= ttm

Fty2

sin2

sin2 00220

0

59

2 2

- 43 -

2.

0>c

( ) ( ) ( )tth ececty + += 21 024

02

2

>

=>=mmkc

mc

t ( ) ( )h etccty += 21 ( ) ( )tBtAety th ** sincos += ( )042

42

22* >=

=

mc

mk

mcmk

ph yyy += (transient solution)

py (steady-state solution)

( )

py

(3) (13) ( ) ( ) = tCtyp cos*

( ) ( )( ) ( )( )( ){ } ( ) ( ) 2222202 02222202

20

222222

02

2220

2220

220

2

222220

2

0

2

222220

2

220022*

cm

F

cm

F

cm

cFmF

cm

cF

cm

mFbaC

+=

+=

+

+=

++

+

=+=

( )220tan

==

mc

ab

( ) 0* =

ddC , ,

0)(2 22202

=+ cm

2

2

2

2

2

22

02

22

22

2 mcmk

mcmk

mc

=

== (14) (16)

1,1 == km , () 60

5 - 44 -

5 Laplace (3) 1 : () 2 : 3 : sine cosine

5.1 Laplace, , ,

( )f t : t 0 ( ) ( )F s e f t dtst= 0 ( )F s : Laplace(Laplace transform) ( )f t ( )f (1) ( ) =sF ( ) ( ) = 0 dttfef st Laplace(Laplace transformation) ( )f t ( )F s (inverse transform) , ( )f t ( )F s ( )1 F =( )f t ( )1 F

5 - 47 -

Laplace 5.1

5.1 , .

5.1 Laplace ( )f t ( )f ( )f t ( )f ( )f t ( )f

( )( )

( )( )( ) 22

221

221

2232

222

22

sin1216

cos111positive

5

sinh10!,2,1,0

4

cosh9!23

sin812

cos7111

+

+

+=

+

+

+

+

ast

ase

asast

sa

at

asaat

sn

nt

assat

st

st

st

sst

s

at

a

a

n

n

5.9 .

(1), (2), (3) (4) . (4) . 1 0 . n = 0 1! = . n

( ) ( )t e t dt s e t s n e t dt n s e t dtn st n st n st n st+ +

+

= =

+ =

+ 1 10 10

0 0

1 1 1 1 n ( ) ( )t n s ns

ns

nn n

+

+ +=

+=

+11 2

1 1! !

(4) (5) ( ) n + =1 n! (, n ) .

5 - 57 -

5.3 Unit Step, Delta Functions

: sL ( s) -

3 : Y(s) Y(s) y(t)

- (Heaviside ) u ( t-a) - ( 110, 111)

u ( t-a)= { 0 t

5 - 58 -

f ( t-a)u( t-a)=L-1{e-asF(s)} (3*)

e-asF(s)= e-as

0e - s f ()d=

0e - s (+ a)f ()d

+a= t = t-a, d=dt, =0 t=a

e-asF(s)=

ae- stf ( t-a)dt=

0e- stf ( t-a)u( t-a)dt

( )

L{u ( t-a)}=

0e- stu ( t-a)dt

=a

0e- st0dt+

ae- st1dt=- 1s e

- st |

a= e

-as

s (s> 0)

a = 0 ,

L{u ( t)}= 1s

( u(t-a) = 1u(t-a), y(t)=1 )

1. 1 ,

f ( t)= { 2 (0< t< )0 (< t< 2)sin t ( t> 2) , F(s)=?f ( t) =2u ( t)-2u ( t-)+u ( t-2)sin t

=2u ( t)-2u ( t-)+u ( t-2)sin ( t-2)

F(s)= 2s -2e - s

s +e -2s

s 2+1

5 - 59 -

2. 1 ,

F(s)= 2s 2

- 2e-2s

s 2- 4e

-2s

s +se -s

s 2+1

L-1( 1s 2

)= t, L-1( e-2s

s 2)=( t-2)u ( t-2)

L-1( e-2s

s )=u ( t-2), L-1(e-s s

s 2+1)= cos ( t-)u ( t-)

f ( t) =2t-2( t-2) u ( t-2)-4 u ( t-2)+ cos ( t-) u ( t-)=2t-2t u ( t-2)- cos t u ( t-)

f ( t)= { 2t (0< t< t)0 (2< t< )- cos t ( t> ) , Dirac

- ( ) * * * hard landing*

- f(t) a ta+k (impulse) * a ta+k f(t) * : k ( zero)

(5) f k ( t-a)= { 1/k ata+k0 otherwise ( 117), I k

(6) I k=

0f k(t-a)dt=

a+ k

a

1k dt=1

5 - 60 -

- f k(t-a)

f k ( t-a)=1k [u( t-a)-u( t-a-k)]

- f k(t-a)

L{f k ( t-a)}=1ks [e

-as-e -(a+k) s]= e-as 1-e- ks

ks

- Dirac Delta

( t-a)= limk0

f k(t-a)= { t=a0 ta ,

0( t-a)dt=1

- Dirac Delta

L{( t-a)}= limk0

e-as 1-e- ks

ks = e-as ( )

L{( t)}=1

4 - y''+3y'+2y= r( t), y(0)=0, y'(0)=0, r( t)=( t-1),

s 2Y+3sY+2Y= e- s, Y=F(s)e- s

F(s)= 1(s+2)(s+1) =1

s+1 -1

s+2

f ( t)=L-1(F(s))= e- t-e-2t

y( t)=L-1(F(s) e- s )= f ( t-1)u ( t-1)

y ( t)={ 0 (0t< 1)e-( t-1)-e-2( t-1) ( t> 1)

5 - 61 -

5.4 Differentation and Integration( )

F(s)

F(s)=L(f)=

0e- stf ( t) dt

F(s) s

F'(s)= dFds =-

0e- stt f ( t) dt

L{ t f ( t) }=-F '(s) or L-1{F '(s)}= t f ( t)

1

L{ f ( t)t }=

sF( s)ds, L-1{

sF( s)ds }= f ( t)t

()

sF( s)ds=

s [

0e - s

tf ( t)dt ] ds

=

0 [

sf ( t) e - s

td s ] dt=

0f ( t) [

se - s

td s ] dt

se - s

td s=- 1t [e- s t]s = 1t e

- s t

sF( s)ds=

0e - st f ( t)t dt

5 - 62 -

Ex 2. F(s)= ln (1+ 2

s 2 ), f ( t)=?,

G(s) =F'( s)= 11+2/s 2

(-2) 2

s 3=- 2

2

s(s 2+2)=- 2s +

2ss 2+2

g( t)=L-1{G(s)}=-2+2cost

g( t)=L-1{G(s)}=L-1{F '(s)}=- tf ( t)

f ( t)= 2t (1- cost)

(7) L(t y ')=- dds [sY(s)-y(0)]=-Y(s) - sdY(s)ds

(8) L(t y '')=- dds [ s2Y(s)- sy(0)-y'(0)]=-2sY(s) - s 2 dY(s)ds +y(0)

5.5 (Convolution)

L-1{F(s)}= f( t), L-1{G(s)}=g( t) , H(s)=F(s) G(s) f( t) g( t) , h( t)= (f*g)( t) , f g (convolution) .

- ( convolution )h( t)= (f*g)( t) = f( t)*g( t)

= t

0f(t-)g()d =

t

0f()g( t-)d

5 - 63 -

1. H(s)= 1(s 2+1)2

h( t)=?,

H( s)= 1(s 2+1)2

= 1s 2+1

1s 2+1

=F( s)G(s),

f ( t)=g( t)= sin t

h( t)= f ( t)*g( t)=t

0sin ()sin ( t-)d

sin ()sin ( t-)= 12 [- cos t+ cos (2- t)],

h( t)= 12

t

0[- cos t+ cos (2- t)]d=- 12 tcos t+

12 sin t

2. H(s)= 1s 3

h( t)=?

H(s)= 1s 1s 2

=F(s)G(s)

f( t)=1, g( t)= t ,

h( t)= f ( t)*g( t)=t

01( t-)d= [ t- 12 2] t0 = t

2

2

3. H(s)= 1s 2(s-a )

h( t)=?,

H(s)== 1s 2

1s-a =F(s)G(s)

, f ( t)= t, g( t)= eat

h( t)= f ( t)*g( t)=t

0e a( t-)d= eat

t

0e -ad= 1

a 2(eat-at-1)

- ( t)

t

0( t-) f()d = f( t) *f= f* = f

, 1*f f ( t) ( (0)=),

5 - 64 -

(2) y''+ay'+by= r( t)

, (s 2+as+b)Y( s)= (s+a)y(0)+y'(0)+R( s) (3) Y(s)= [ (s+a)y(0)+y'(0)]Q(s)+R(s)Q (s), Q( s)= 1

s 2+as+b

y(0) = y'(0) = 0 , Y(s)=R( s)Q(s)

y( t) ,

(4) y( t) = r(t)*q( t) = t

0r() q( t-) d =

t

0q() r ( t-) d

4. 5.3 4

y''+3y'+2y= r( t), y(0)=0, y'(0)=0

r( t)=

0, 0< t< 11, 1< t< 20, 2< t

,

Q( s) = 1s 2+3s+2

= 1(s+1)(s+2) = 1

s+1 -1

s+2

q( t)=e- t-e-2t

t < 1 , r( t) = 0 y( t) = r(t)*q( t) = 0*q( t) = 0

1 < t < 2 , r( t) = 1

y( t) = r(t)*q( t) = t

0r() q( t-) d =

1

0r() q ( t-) d +

t

1r() q ( t-) d

5 - 65 -

= 0 + t

11 q ( t-) d

= t

1(e-( t-)-e-2( t-)) d

= [e-( t-)- 12 e-2( t-)]= t

=1

= ( (1- 12 ) - (e-( t-1) - 12 e-2( t-1))= 12 - e

-( t-1) + 12 e-2( t-1)

2 < t ,

y( t) = r(t)*q( t) = t

0r() q( t-) d

= 1

0r() q ( t-) d +

2

1r() q ( t-) d+

t

2r() q ( t-) d

= 0 + 2

11 q ( t-) d + 0

= 2

1(e-( t-)-e-2( t-)) d

= [e-( t-)- 12 e-2( t-)]= 2

=1

= ( (e-( t-2) - 12 e-2( t-2)) - (e-( t-1) - 12 e-2( t-1)))= e-( t-2) - e-( t-1) - 12 [e

-2( t-2)) - e-2( t-1))]

5.

y( t) = t+t

0y() sin ( t-)d

1 : y( t)= t+y * sin t 2 :

Y(s) = 1s 2

+Y(s) 1s 2+1

Y(s)= s2+1s 4

= 1s 2

+ 1s 4

3 :

y(t) = t+ 16 t3

5 - 66 -

5.6 (Partial Fraction),

-

Y(s)= F(s)G(s)

G(s)

Case I. : G(s)= s-a ( )

Ex 1. y''+y'-6y=1, y(0)=0, y'(0)=1

s 2Y-sy(0)-y'(0)+sY-y(0)-6Y= 1s

(s 2+s-6)Y=1+ 1s

Y( s)= s+1s( s-2)( s+3) =A 1s +

A 2s-2 +

A 3s+3

A 1, A 2, A 3=?

s+1=A 1( s-2)(s+3)+A 2s(s+3)+A 3s(s-2)

s=0 1=A 1(-2)(3), A 1=-1/6

s=2 3=A 225, A 2=3/10

s=-3 -2=A 3(-3) (-5), A3=-2/15

y(t) =- 16 +310 e

2t- 215 e-3t

Case II. : G(s) = (s-a)m ( )

e.g.) G( s)= ( s-a) 3 A 1(s-a) + A 2

(s-a) 2 +

A 3(s-a) 3

5 - 67 -

Ex 2. y''-3y'+2y=4t, y(0)=1, y'(0)=-1

s 2Y-sy(0)-y'(0)-3sY+3y(0)+2Y= 4s 2

(s 2-3 s+2)Y= 4s 2

+s-4= 4-s3-4s 2

s 2

Y= 4+ s3-4s 2

s 2(s-1) ( s-2)=

A 1s 2

+A 2s +

A 3s-1 +

A 4s-2

A 1, A 2, A 3, A 4=?

4+ s 3-4s 2=A 1(s-1) ( s-2)+A 2s( s-1) ( s-2)+A 3s 2(s-2)+A 4s 2(s-1)

s=1 1=+A 3(1) (-1), A 3=-1

s=2 -4=+A 441, A 4=-1

s=0 4=A 1(-1) (-2), A 1=2

3s 2-8s=A 1(2s-3)+A 2(s-1) ( s-2)+ s(k( s))

s=0 0=A 1(-3)+A 2(-1) (-2)=-6+2A 2, A 2=3

y(t) =2t+3-e 2t-e t

Case III. : G(s)= ( s-a)(a- a) () ( a=+ i, a=- i )

e.g.) As+B(s-a) (s- a)

or As+B(s-)2+2

5 - 68 -

Ex 3. y''+2y'+2y= r( t), y(0)=1, y'(0)=-5 , r( t)=10 sin 2t [1-u( t-)] = 10 sin 2t -10 u( t-)sin 2( t-)

,

[s 2Y(s)-sy(0)-y'(0)]+2[Y( s)-y(0)]+2Y (s)=10 2s 2+4

(1-e-s )

(s 2+2s+2)Y(s) = sy(0)+y'(0)+2y(0)+10 2s 2+4

(1-e-s )

= s-5+2+10 2s 2+4

(1-e-s)

= s-3+ 20s 2+4

- 20e-s

s 2+4

Y( s) = s-3s 2+2s+2

+ 20( s 2+4)( s 2+2s+2)

- 20e-s

(s 2+4)( s 2+2s+2)

L-1{ s-3s 2+2s+2 }=L-1{ (s+1)-4(s+1)2+1 } =L-1{ ss+1 } s s+1- 4L-1{ 1s+1 } s s+1

=e- t cost - 4e- tsint

L-1{ 20(s 2+4)(s 2+2s+2) }= As+Bs 2+4 + Cs+Ds 2+2s+2

(As+B)(s 2+2s+2) = (Cs+D)( s 2+4)

s 3, s 2, s 1, s 0 A=-2, B=-2, C=2, D=6

L-1{ 20(s 2+4)(s 2+2s+2) } =L-1{ -2s-2s 2+4 + 2s+6s 2+2s+2 }

=-2L-1{ ss 2+4 }-L-1{ 2s 2+4 }+L-1{ 2(s+1)+4(s+1)2+1 }

=-2 cos2t- sin2t+L-1{2 ss 2+1 +4 1s 2+1 } s s+1=-2 cos2t-sin2t+e- t(2cos t+4sin t)

5 - 69 -

-L-1{ 20e-s

(s 2+4)(s 2+2s+2) } = -[-2 cos2( t-)- sin2( t-) +e-( t-) (2cos ( t-)+4sin ( t-)) ] u( t-)

= -[-2 cos2t- sin2t+e-( t-)(-2 cos t-4 sin t)]

= [2cos2t + sin2t + e-( t-)(2 cos t+4 sin t)] u( t-)

y( t) = + + = [e- t cost - 4e- tsint]+[-2 cos2t- sin2t+e- t (2cos t+4sin t)]

+[2cos2t + sin2t + e-( t-)(2 cos t+4 sin t)] u( t-)

= -2 cos2t - sin2t + 3e- tcost

+ [2cos2t + sin2t + e-( t-)(2 cos t+4 sin t)] u( t-)

,

y( t)= { -2 cos2t - sin2t + 3e- tcost , 0< t<

e- t [ (3+2e )cos t+4e sint] , < t

Case IV. : G(s) = [ (s-a)(a- a)] 2 ()

e.g.) As+B[ (s-a) (s- a)] 2

+ As+B(s-a) (s- a)

Ex 3. y''+w 0y = Ksinw 0t, y(0)=0, y'(0)=0

s 2Y(s) + w20Y(s) =Kw 0

s 2+w20

Y( s) = Kw 0(s 2+w20)

2 = Kw 0

w 0s 2+w20

w 0

s 2+w20 = Kw 0

F( s)G( s)

5 - 70 -

f( t)= sinw 0t , g( t)= sinw 0t,

y( t)= Kw 0f( t) * g( t) = Kw 0

t

0sinw 0 sinw 0(t-) d

= Kw 0t

0

12 [ cos {w 0-w 0 (t-)}- cos {w 0+w 0 (t-)}]d

= K2w 0 [t

0cos (2w 0 -w 0t) d-

t

0cosw 0t d]

= K2w 0 [ 12w 0 ( sinw 0t+ sinw 0t ) - t cosw 0t ]= K

2w20( sinw 0t - w 0t cosw 0t)