6 problem eigen

dwk 142

PROBLEMA EIGEN

PENDAHULUANPENDAHULUAN

dwk 143

0..........

0..........

0..........

2211

2222121

1212111

====++++++++++++

====++++++++++++

====++++++++++++

nnnnn

nn

nn

xqxqxq

xqxqxq

xqxqxq

.

.

.

.

.

.

.

.

.

.

.

.

====

0

.

.

0

0

.

.

..

...

...

..

..

2

1

21

22221

11211

nnnnn

n

n

x

x

x

qqq

qqq

qqq

[[[[ ]]]] 0

..

...

...

..

..

det

21

22221

11211

========

nnnn

n

n

qqq

qqq

qqq

Q

KARAKTERISTIK PERSAMAAN EIGENKARAKTERISTIK PERSAMAAN EIGEN

dwk 144

[[[[ ]]]] [[[[ ]]]] [[[[ ]]]]BAQ −−−−====

[[[[ ]]]]{{{{ }}}} [[[[ ]]]] [[[[ ]]]][[[[ ]]]]{{{{ }}}} {{{{ }}}}

[[[[ ]]]]

====

====−−−−====

)(...)(

...

...

.....

)(...)(

0

1

111

1

λλλλλλλλ

λλλλλλλλ

nnn

n

ff

ff

B

xBAxQ

where

I Tipe

[[[[ ]]]]{{{{ }}}} [[[[ ]]]] [[[[ ]]]][[[[ ]]]]{{{{ }}}} {{{{ }}}}[[[[ ]]]] constants. all are of elements where,

II Tipe

C

xCAxQ 02 ====−−−−==== λλλλ

[[[[ ]]]]{{{{ }}}} [[[[ ]]]] [[[[ ]]]][[[[ ]]]]{{{{ }}}} {{{{ }}}}03 ====−−−−==== xIAxQ λλλλ

III Tipe

EKSPANSI DETERMINAN LANGSUNGEKSPANSI DETERMINAN LANGSUNG

dwk 145

====

−−−−−−−−

−−−−−−−−−−−−

−−−−−−−−

0

0

0

210

131

012

3

2

1

x

x

x

λλλλ

λλλλ

λλλλ

[[[[ ]]]]

{{{{ }}}}

32

322

2

7148

25521010

2)156)(2(

)2(11)2)(3()2(

10

310

20

11)1(

21

13)2(

210

131

012

det

λλλλλλλλλλλλ

λλλλλλλλλλλλλλλλλλλλλλλλ

λλλλλλλλλλλλλλλλ


λλλλ

λλλλλλλλ

λλλλλλλλ

λλλλ

λλλλ

λλλλ

−−−−++++−−−−====

++++−−−−−−−−++++−−−−++++−−−−====

++++−−−−−−−−++++−−−−−−−−====

−−−−−−−−−−−−−−−−−−−−−−−−====

−−−−

−−−−−−−−++++

−−−−

−−−−−−−−−−−−−−−−

−−−−−−−−

−−−−−−−−−−−−====

−−−−−−−−

−−−−−−−−−−−−

−−−−−−−−

====Q

[[[[ ]]]] 08147det23 ====−−−−++++−−−−==== λλλλλλλλλλλλQ

SoalSoal TugasTugas EkspansiEkspansi DeterminanDeterminan LangsungLangsung

dwk 146

Tentukan nilai determinan dari matrik berikutdengan menggunakan metode ekspansi langsung !

[[[[ ]]]] 28

32det

λλλλ−−−−====A

[[[[ ]]]] λλλλ

λλλλ

−−−−

−−−−====

28

32det B

[[[[ ]]]]

λλλλ

λλλλ

λλλλ

−−−−−−−−

−−−−−−−−−−−−

−−−−−−−−

====

220

252

023

det C

57)

58)

59)

EKSPANSI DETERMINAN TIDAK LANGSUNGEKSPANSI DETERMINAN TIDAK LANGSUNG

dwk 147

[[[[ ]]]] )(det

21

22221

11211

λλλλf

qqq

qqq

qqq

Q

nnnn

n

n

========

L

MMM

L

L

∑∑∑∑====

====

++++++++++++++++====

n

i

ii

nn

af

aaaaf

0

2

210

)(

)(

λλλλλλλλ

λλλλλλλλλλλλλλλλ L

nnnnnnn

nn

nn

aaaaf

aaaaf

aaaaf

λλλλλλλλλλλλλλλλλλλλ



++++++++++++++++====→→→→

++++++++++++++++====→→→→

++++++++++++++++====→→→→

L

MM

L

L

2

210

1

2

1211011

0

2

0201000

)(

)(

)(

====

)(

)(

)(

1

1

1

1

0

1

0

11

00

nnnnn

n

n

f

f

f

a

a

a

λλλλ

λλλλ

λλλλ

λλλλλλλλ

λλλλλλλλ

λλλλλλλλ

MM

L

MMM

L

L

EkspansiEkspansi DeterminanDeterminan order 3 order 3 TidakTidak LangsungLangsung

dwk 148

)(

210

131

012

]det[ λλλλ

λλλλ

λλλλ

λλλλ

fQ ====

−−−−−−−−

−−−−−−−−−−−−

−−−−−−−−

====

3

3

2

210

210

131

012

]det[ λλλλλλλλλλλλ

λλλλ

λλλλ

λλλλ

aaaaQ ++++++++++++====

−−−−−−−−

−−−−−−−−−−−−

−−−−−−−−

====

)6()6()6(40)(6

)5()5()5(12)(5

)3()3()3(2)(3

8

210

131

012

)(0

3

3

2

21033

3

3

2

21022

3

3

2

21011

000

aaaaf

aaaaf

aaaaf

af

++++++++++++====−−−−====→→→→====

++++++++++++====−−−−====→→→→====

++++++++++++========→→→→====

========

−−−−

−−−−−−−−

−−−−

====→→→→====

λλλλλλλλ

λλλλλλλλ

λλλλλλλλ

λλλλλλλλ

−−−−

−−−−

−−−−

====

48

20

6

216366

125255

2793

3

2

1

a

a

a

327148)( λλλλλλλλλλλλλλλλ −−−−++++−−−−====f

EkspansiEkspansi DeterminanDeterminan

order 3 order 3 TidakTidak

LangsungLangsung

dwk 149

EkspansiEkspansi DeterminanDeterminan order order 4 4 TidakTidak LangsungLangsung

dwk 150

[[[[ ]]]]

4

4

3

3

2

210

7130

4643

1254

0245

det


λλλλ

λλλλ

λλλλ

λλλλ

aaaaa

Q

++++++++++++++++====

−−−−−−−−−−−−

−−−−−−−−−−−−−−−−

−−−−−−−−−−−−−−−−

−−−−−−−−−−−−

====

)10()10()10()10(20)(10

)6()6()6()6(44)(6

)4()4()4()4(364)(4

)3()3()3()3(596)(3

860

7130

4643

1254

0245

)(0

4

4

3

3

2

21044

4

4

3

3

2

21033

4

4

3

3

2

21022

4

4

3

3

2

21011

000

aaaaaf

aaaaaf

aaaaaf

aaaaaf

af

++++++++++++++++========→→→→====

++++++++++++++++====−−−−====→→→→====

++++++++++++++++====−−−−====→→→→====

++++++++++++++++====−−−−====→→→→====

====−−−−====

−−−−−−−−

−−−−−−−−−−−−

−−−−−−−−−−−−

−−−−−−−−

====→→→→====

λλλλλλλλ

λλλλλλλλ

λλλλλλλλ

λλλλλλλλ

λλλλλλλλ

====

880

816

496

264

10000100010010

1296216366

25664164

812793

4

3

2

1

a

a

a

a

43223160212860)( λλλλλλλλλλλλλλλλλλλλ ++++−−−−++++−−−−−−−−====f

SoalSoal TugasTugas EkspansiEkspansi DeterminanDeterminan TidakTidak LangsungLangsung

dwk 151

Tentukan nilai determinan dari matrik berikut denganmenggunakan metode ekspansi tidak langsung !

[[[[ ]]]] 28

32det

λλλλ−−−−====A

[[[[ ]]]] λλλλ

λλλλ

−−−−

−−−−====

28

32det B

[[[[ ]]]]

λλλλ

λλλλ

λλλλ

−−−−−−−−

−−−−−−−−−−−−

−−−−−−−−

====

220

252

023

det C

60)

61)

62)

NILAI EIGENNILAI EIGEN

dwk 152

08147)(

0

0

0

210

131

012

23

3

2

1

====−−−−++++−−−−====

====

−−−−−−−−

−−−−−−−−−−−−

−−−−−−−−


λλλλ

λλλλ

λλλλ

f

x

x

x

VektorVektor Eigen Eigen PertamaPertama

dwk 153

11 ====λλλλ

====

−−−−

−−−−−−−−

−−−−

⇒⇒⇒⇒

====

−−−−−−−−

−−−−−−−−−−−−

−−−−−−−−

0

0

0

110

121

011

0

0

0

1210

1131

0112

3

2

1

3

2

1

x

x

x

x

x

x

Metode Eliminasi Gauss-jordan

−−−−

−−−−

⇒⇒⇒⇒

++++

++++

→→→→

−−−−

−−−−

−−−−

⇒⇒⇒⇒++++→→→→

−−−−

−−−−−−−−

−−−−

⇒⇒⇒⇒

0

0

0

000

110

101

0

0

0

110

110

011

0

0

0

110

121

011

RR

R

RR

R

RR

R

R

R

R

32

2

21

3

21

1

3

2

1

====

−−−−

−−−−

0

0

0

110

101

3

2

1

x

x

x

0

0

32

31

====−−−−

====−−−−

xx

xx

{{{{ }}}}

====

====

====

1

1

1

3

3

3

3

3

2

1

1 x

x

x

x

x

x

x

X

VektorVektor Eigen Eigen KeduaKedua

dwk 154

22 ====λλλλ

====

−−−−

−−−−−−−−

−−−−

⇒⇒⇒⇒

====

−−−−−−−−

−−−−−−−−−−−−

−−−−−−−−

0

0

0

010

111

010

0

0

0

2210

1231

0122

3

2

1

3

2

1

x

x

x

x

x

x

{{{{ }}}}

−−−−

====

−−−−

====

====⇒⇒⇒⇒

1

0

1

0 3

3

3

3

2

1

2 x

x

x

x

x

x

x

VektorVektor Eigen Eigen KetigaKetiga

dwk 155

43 ====λλλλ

{{{{ }}}}

−−−−====

−−−−====

====⇒⇒⇒⇒

1

2

1

2 3

3

3

3

3

2

1

3 x

x

x

x

x

x

x

x

====

−−−−−−−−

−−−−−−−−−−−−

−−−−−−−−

⇒⇒⇒⇒

====

−−−−−−−−

−−−−−−−−−−−−

−−−−−−−−

0

0

0

210

111

012

0

0

0

4210

1431

0142

3

2

1

3

2

1

x

x

x

x

x

x

SoalSoal TugasTugas NilaiNilai Eigen Dan Eigen Dan VektorVektor EigenEigen

dwk 156

Evaluasi nilai Eigen dan vektor Eigen darikarakteristik persamaan yang diperoleh pada soal

berikut !

[[[[ ]]]] 28

32det

λλλλ−−−−====A

[[[[ ]]]] λλλλ

λλλλ

−−−−

−−−−====

28

32det B

[[[[ ]]]]

λλλλ

λλλλ

λλλλ

−−−−−−−−

−−−−−−−−−−−−

−−−−−−−−

====

220

252

023

det C

63)

64)

65)

APLIKASI GETARAN MEKANISAPLIKASI GETARAN MEKANIS

dwk 157

k2=2kk1=kP(t)=0m2=2m

x2

m1=m

x1

AplikasiAplikasi GetaranGetaran MekanisMekanis

dwk 158

====

++++

−−−−

−−−−

0

0

20

0

22

23

2

1

2

1

x

x

m

m

x

x

kk

kk

&&

&&

(((( )))) (((( ))))

(((( )))) (((( ))))

====

−−−−

−−−−

−−−−→→→→

++++−−−−====⇒⇒⇒⇒++++====

++++−−−−====⇒⇒⇒⇒++++====

0

0

20

01

22

23

sinsin

sinsin

2

12

2

1

2

2222

2

1111

A

Am

A

Ak

tAxtAx

tAxtAx

ωωωω

θθθθωωωωωωωωθθθθωωωω

θθθθωωωωωωωωθθθθωωωω

&&

&&

(((( ))))

0)(

00

1112211

1211

====++++−−−−−−−−

====++++−−−−⇒⇒⇒⇒====←←←←++++ ∑∑∑∑xmxxkxk

FFFF Issx

&&

(((( ))))

0)(

00

22122

222

====++++−−−−

====++++⇒⇒⇒⇒====←←←←++++ ∑∑∑∑xmxxk

FFF Isx

&&


dwk 159

[[[[ ]]]]

eigenvalue second

eigenvalue first

⇒⇒⇒⇒====→→→→====++++====→→→→

⇒⇒⇒⇒====→→→→====−−−−====→→→→

====⇒⇒⇒⇒++++−−−−====

====

−−−−−−−−

−−−−−−−−

mk

mk

,

,

kmQ

A

A

93217320.332

51802679.032

14det

0

0

222

23

22

11

22

2

1

ωωωωλλλλ

ωωωωλλλλ

ωωωωλλλλλλλλλλλλ

λλλλ

λλλλ

{{{{ }}}}

====

====

⇒⇒⇒⇒

====⇒⇒⇒⇒

1

732,0

732,0

2679,0

1

21

1

V

AA

reigenvecto First λλλλ

{{{{ }}}}

−−−−

====

−−−−====

⇒⇒⇒⇒

====⇒⇒⇒⇒

1

732.2

732.2

732.3

2

21

2

V

AA

reigenvecto Second λλλλ


dwk 160

k2=2kk1=k

P(t)=0m2

x2

m1

x1

0.73 1.0


dwk 161

-2.73 1.0

k2=2kk1=k

P(t)=0m2

x2

m1

x1

Soal Tugas 66: Aplikasi Nilai Awal

dwk 162

2m2k

mk

m3k

x1 x2 x3

Evaluasi pola vibrasi dari sistem mekanis massa-pegasyang mempunyai 3 derajat kebebasan seperti pada

gambar berikut. Normalisasi vektor Eigennya sehingga

nilai elemen vektor terbesar sama dengan 1,0.

Data & Analytics

6 problem eigen