I. ��� ��

♦ ��(�� , Vibration) : ���� �(� , Period)� ���� ��.

♦ ���(or ����): ��� �� ���� ��( � !")#$% &'( )*+ , )*% -.'� /0 12'� �

3 - "456 ��0 78'9 :�'9 '�;?"

♦ ��+ <=> ?@- Galileo : A�B ��, ��CD

- Wallis & Sauveur : �E��FG HI��F

- Bernoulli : JK+ HIL+ M"NOPQ RS

- Euler & Bernoulli : T�U+ V�� -> WX Y�Z

- Raylegh : [+ �� ��

- Frahm : Propeller \ ]�% ^4 _`a ��, ��b��

- Stadola : c+ �� 78 ->d4e fg% >.

- Timoshenko & Minlin : cG !h% &i �� 78

- Lin & Rice , Crandall-Mark & Robson : jk����(��, lm, n� o[)

- FEM+ >. : p&+ Bqi �� 78

♦ ���+ r.Xs t Nuv: ��% ^4 ��w x7s y @z{ ^9, |z }.7~ P'� @z{ ^ .

- �* : 9�+ ��, �+ ��, c�, �� �� �.

- �w ���+ �� t 2H)+ �� : x���, ����(��), �{i o[, *�uo+ ��,

���

- �-�� : Chatter

- �� : �{i ��� ��

- Y� : Passive Vibration Control : Damper �

Active Vibration Control : Pusher, Magnetic Bearing �

- Wqi �� R4+ �{* ��, ;�R�, �� XQ�

- ��0 �.i ��+ �� �� 12

- ��. Shaker ]�, ;�� ]��

♦ ���+ �Euo � �� %��  ¡�'� uo(m , Ip)

� #$%��  ¡�'� uo(k, keq)

� %�� �¢ uo (damper , c)

♦ B}{(Degree of freedom): �+ �£ uo+ #$  ��'6 ��'� ¤ ¥ui ¦§� ��+ Joi+ F

¨)

♦©��78+ Nui ª "«� 9}��F t 9}¬d – 9}��F 78 , 9}��F t Stability

� �� r­ 78

22

21 )45cos(

2

1)]90cos([

2

1�� xkxkU +−= θ

Subjected to )0()0(,)0()0( XxXx == �� (1.1)

=> Depends on the I.C. & E.F.

C

J1 J2 J3

(Torsion only)

3 B}{

θ1 θ2 θ3

θ

1 B}{ ®��

C

k

x

1 B}{ ¯��

m

θ

x

2 B}{

♦ ��78 °±

• 1A� : &�X+ �R �� �²³´w µ¶ �·'9 F�>� 780 ¸7 Bqi �£ ¹´

0 9º'� ¹w �;»'¼� >°i ;�´0 ¸i F�> �½¾

=> �w @�� ¿�0 ;� ��>� �À� ¥u

=> �R 12%~+ Á ��

� ;»i i AÃI => M"

� M"��Ä �Å �· Æ[

� ¥u' Ç _M"v0 9ºi �R �½ 78

• 2A� : �?Y�Z }{

=> �È �?Y�Z� 1É �?Y�Z+ ±�?

=> 1A�+ B})*{��d ¥ui �� Y�Z´w ÊË+ R 2ÌÍ, D’Almbert+

PQ, %�� cÎ+ ÌÍ, Ï,Ð�Ì� Ñw �<�> PQ´��d Ò4� .

=> ¶� �È �?Y�Zw �È �²³Ó��d Ò4�� 1É �?Y�Zw 1É

Y�ZÓ��d Ò4� .

• 3A� : ÏÔϲ ÕÖ, ×Q% ÕÖ, F$780 ¸7 �?Y�Z0 Ø .

=> F�>Ó� 7  2y F ^0 @z 78� µ¶ Ùz� ��'6{ &�X+ �� �²³%~

F�>Ó� 7  ¿Á 2'( �� @z� µ¶ Ú) . => F$78� ¥u

• 4A� : ��% &i ]Û

=> Ò4� Õ#, É{, ;É{´w X8 Ü> t ]� ;»v% &'( XÛÝ ]Û �4s i .

♦©�½¾

• AH7Þ+ �½¾

� 1 D. O. F. �½

� 2 D. O. F. �½

� ß�� @z -> ®��� s� -> B}{ 3R

à

áH

nâ

ãv äg

�K åæ

ç

èâ t �K åæ

à

KsCs ç+ évÓ��d

x1

èâ t �K åæ

à

KsCs ç+ évÓ��d

x2

èâx1

KeCe ãv äg+ êvÓ��d

♦ ²ë¾ uo

• ìí+ ÌÍ(Hooke’s law)

îï% ;7�� /w µð� ñ4òQ� Nó(mg)� �&� ñ4ôQ� ²ë¾ /(fk)Ó� �õ4

ö ^ .

#G Ñ� K�+ x0� ²ë¾% ÷± îï0 ø7�% ùÏ ²ë¾+ Õ#G /�+ C��

[� Ñ .

# ,ðë+ M"ú<%~+ Y�Z  [� Ñ� �ûòü �  ìí+ ÌÍ(Hooke’s Law)�Ï

i .

2

2

1kxu = (1.2)

k : ²ë¾+ év(²ë¾+ êv0 ��)

x

y

k

0x

g

1x

2x

3x

• ²ë¾+ Hý

- @z 1 : ¯þ ²ë¾ (Springs in Parallel)

321

321

kkkk

kkkkw

eq

eq

++=⇒

++== δδδδ or

2

23

22

21

2

12

1

2

1

2

1

δ

δδδ

eqk

kkkE

=

++=

��>Ó� ¯þ ²ë¾% &7~

neq kkkkk +++= �321 (1.3)

- @z 2 : ¿þ ²ë¾ (Springs in Series)

21 k

w

k

w

k

w

eq

+==δ => 21

111

kkkeq

+=

��>Ó� ¿þ ²ë¾% &7

neq kkkk

1111

21

+++= �� (1.4)

k1 k2 k3

δ

k1δ k2δ k3δ

δ1

k1

k2

k1

k2

δ1

δ2

δ2

w

w

k2

k2

22δkw = 11δkw =

- ¡L� ²ë¾+ _�

� �w �&+ �� Y�% &i év

� \% &i _`a év

\% &7 32

4dJ

GJ

lM t πθ == , G : �A�F

l

Gd

l

GJMk t

t 32

4πθ

===

2 X Spring ¡ L

¿ þ �++=21

111

kkkeq

�++= 21 RRR

¯ þ �++= 21 kkkeq �++=21

111

RRR

l

h

d

D

�

Disk

�

J0 kt�

m

lEAk =l

x(t)

E = elastic modulus

A = cross-sectional area

l = length of bar

x(t) = deflection

¨R) ë���\+ _`a ²ë¾ �F eqk   2'Ï.

<ÁÂYÌ> ²ë¾+ HýÓ� \+ uo 1-2G 2-30 9ºy ¹.

Sol)

θθkT = , where l

GJStiffnessTorsionalk ==θ

122,112 )( θθkTT == 233,223 )( θθkTT ==

2312 θθθ += (¿þ 1�)

=> 3,22,1 )()()( θθθ k

T

k

T

k

T

eq

+= => 3,22,1 )(

1

)(

1

)(

1

θθθ kkk eq

+=

=>

23

423

423

12

412

412 )(

32

1

)(32

1

)(

1

l

dDG

l

dDGk eq −×+

−×= ππθ

d23D23d12D12

D23 = 0.2m

d23 = 0.15

d12D12d23D23

GJ

Tl=θ

¨R) í��+ �; ²ë¾ �F k  2'Ï.

<Á YÌ> �; #$%��

<;�> 1. �+ �K FA� ��.

2. ��å CDEB+ ��� Å�

B+ F¿Y� Õ# x% &7

²ë¾ k1+ Õ# : )90cos(1 θ−= �xx

²ë¾ k2+ Õ# : �45cos2 xx =

C

1.5m 1.5m45

10m

AF

D E

B

1000kg

11, kl

ml

k

102

2

=

wx

B

AF

3mw

ml

mNEmmA

steelmNE

mmAml

FA

FBFB

AB

ABAB

3

/1007.2,2100

)(/1007.2

2500,10

211

211

2

=×==

×=

==

�135cos2222ABFAABFAFB lllll −+=

FAFB

FAFBAB

ll

lll

2cos

222 −−=θ

�* #$ %�� (U )

22

21 )45cos(

2

1)]90cos([

2

1�� xkxkU +−= θ

AB

ABAB

FB

FBFB

l

EAk

l

EAk == 21 ,

2

2

1xkU eq=

�� 45cos)90(cos 22

21 kkkeq +−=∴ θ , !"²ë¾Ó�

45°θ

�45cosx)90cos( θ−�x

x

F

FBl

B

ABl

AFAl

θ�135

♦ ��(or ��)��

�� � �� ��� � � � ��

Newton’s 2nd law

∑∑ == θ���

���

IMamF , (1.5)

��� ���� �� ��� ����� ��� ��

• ��� ��

- � 1 : � !� �� "#$ %& ��

2

233

222

211

)()(

222)(

233

222

211

2233

222

211)(

2

1

2

1

)(2

1

2

1

2

1

2

1

l

lmlmlmm

EE

lmxmE

lmlmlmxmxmxmE

eq

ba

eqeqeqeqb

a

++=⇒

=

==

++=++=

θ

θ

��

����

o

Pivot point

1x� 2x� 3x�

3m2m

1m

1l

2l

3l

� !

3

22

11

3 xl

xl

xl

��

��

��

=

=

=

θθθ

o

Pivot point

eqx�

leqxl �� =θ

eqm

- � 2 : "�$ %& � '( ��

(1) )* +� �� eqm

20

220

2

2

20

2

2

1)(

2

1

2

1

,,2

12

1

2

1

R

Jmm

xmRxJxmTT

RxxxxmT

JxmT

eq

eqeq

eqeqeqeq

+=⇒

=+⇒=

===

+=

���

�����

��

θ

θ

(2) )* +� '( �� eqJ

20

220

2

2

2

1

2

1)(

2

1

,,2

1

mRJJ

JJRmTT

RxJT

eq

eqeq

eqeqeqeq

+=⇒

=+⇒=

===

θθθ

θθθθ

���

����

• �� �� ,-.

32

02

32,

12l

mlmIIl

mI GG =

+==

2

0

2

0 2]

2[

222

+=⇒

+=×+=×+= l

mIIl

mIll

mIl

ymII GGGG θθθθθ ������������

Rack, Mass m

R

/0

x�

Pinion, Mass moment of Inertia J0

O

l

!

m •1G

231415617891:;1<=>=?1@=ABC ;1DE@@FG H1IJ1K LM1NOP�1'(Q1RM1S1;1T

UV�1+�1��M1WJX

C41NWYZ1�[

)(

,

12

1121

slippingwithoutrollsxrr

l

r

x

xr

llx

r

x

cpcc

pp

pp

���

�

����

��

==

====

θ

θθθ

m

k2

No slip

*\]^

2lx2(t)

rcθc

Rigid link2 (mass m2)

Rigid link 1 (mass 1)

Rotates with pully about O

1lCylinder, mass mc

Pulley, mass moment of

Inertia Jpk1

x(t)

rp

θp

O

cyl

x

z

L

rc

222

3

1

4

1,

2

1mlmrImrI cxcz +==⇒

m

sphere

IG

rs 2

5

2sG mrI =

_41��1���

11

eq

eqn

peq

p

c

ppp

peq

eqeq

cp

cc

ppppp

ccpp

m

kk

r

xlkxkxk

r

lm

r

lm

r

lm

r

Jmm

xmT

rr

lxrm

r

lxm

r

x

r

lm

r

xJxm

JxmJJxmT

=+=∗

++++=⇒

=

++++=

++++=

ω

θθθ

,)(2

1

2

1

2

1)

2

1

3

1

2

1

))(2

(2

1)(

2

1))((

2

1)(

2

1

2

1

2

1

2

1

2

1

2

1

2

1

221

22

12

2

21

2

212

2

211

2

2

212

212

22

1122

2222

211

22

�

�����

�����

23141`ab1–1cd@@deFf1gFhiaA=jb1kB=AFl=h1mAFf>G1ndf1fdhBFf1afb4

111111122

32

32

22

)(2

1)(

2

1

2

12

1

2

1

θθθ

θ

���

��

mlJlmJ

xmJT

rr

r

+=+=

+=11111111aodEl1p

1111122

32 )(

2

1

2

1 θθ �� mlJIT ro +== 1111111111111111111aodEl1K

♦ �� ��(Damping Elements)- ���� ��� � �� (�� ����� ��) => ��� Sound� ��(�� �� �

! => "# $% �� &'()

- Damper : ��� �) ��� �*� +,- ./ 01

- ��23� 4� Damper 5�6 => 78��9 �:, �; ��

- Damper 9<= &' : � >? @ A6B C/ D�� )E

� ��0 FG HI JK) 4/ LM�N OP

- �� $% &$:

� Q6��(Viscous Damping) : 8R ST� U � � VW, ���B ��./

XR JK� YZ( xCFd �= )

� [\ ]/ ^����(Coulomb or Dry Friction Damping) : ��� U ��

� ����(Hysteretic Damping) : P_) �$` a bc de ��� f

g0/ ��

• Q6��0 1h

ij 1.1 ��� xcfc �= k b/ lmn opK

ij 1.1 ��� cf / qrs tB uv� f 8Kw��q.

Q6 �9 �, xy h d$U z de� U deB 7Ew� 47 q{ deB JK v� !

|} a,

d~e= : A

vh

yu =

dt

dxv =

y

x

No slip condition

h

y

��$ 8R(�$8R)� �G2�(τ )/ JK 1�� YZUq.

gradientVelocityhvdyduWhere

dy

du

;=

= µτ (1.6)

!|�/ d~ ��e� Vg./ �G� ]/ ST�( cf )B

consantdampingviscoustheh

Ac

platemovingtheofareasurfacetheAWhere

cvh

AvAfc

:

:

µ

µτ

=

===

(1.7)

� N} ��) Y�$��e, �$: sEB }�=�� Y�$ ��( LM�� ��JK

( *v )� If �$: sEk =�.� ia �) ��H9/ qrs tq.

*v

c

dv

dfc =

�� ) d~ ��0(flat plate damper)

�m� +� !|�k ��m�0 �f �0) 30×30 mm �~k x�� 0.1mm�7 U u��

� d�.- !|�- Uq. � F d~��� Q6 smPa ⋅= 20η � �}� ��� 4q7   a �

�H9¡ ¢E.�.

Sol>

�£$ 8R� Vg./ �G�B cVAd

VAfc === ητ ⇒

d

Ac

η=

"0 mdmA 0001.0,108.103.02 232 =×=×= − �q.

¤� msNc /36.010

108.11020

4

33 ⋅=×××= −

−−

�� ) Y¥j Q6 ��0(Torsional viscous damper)

Y¥j ��0/ ��(clearance)¡ )�/ ¦¡ zo �§ �¨©� 16w� 47 Q68R� ��

� 4q. � Y¥j ��0/ �� ij�� �L mr 50.0= , ª� ml 75.0= , «*�¬�

mme 125.0= � w� 4q. M � �� ­®��7 θ ¡ ¯�°��7   a θ�tcM −= ±�� E

w/ Y¥j �� H9¡ ¢E.�.

Sol>

θθηη�r

e

dlrv

e

dAdF −=−=

�¨© ²� IU &³´/

θθηd

e

lrdFrdM �

3

−==

=µ.e

θπηθθηπ π��

e

lrd

e

lrdMM

32

0

2

0

3 2∫ ∫ −=−==

¤� ��H9/

smNe

lrct ⋅⋅=

×××××== −

−

3.2810125.0

50.075.0106223

333 ππη

Vy

d

d=2r

e M

θ

l

θ M

Oil

V dF

dθ

��) ¶�� ��0(Piston damper)

|L� D� 2o 1·k )�/ ª� L�

¶���� 16¸ ¹º»¼(Shock absorbers)

$½ 8R ��0) 4q. ¶�� |L�

d�7 Q6� η , ¾K) ρ �7   a

��H9¡ ¢E.�.

Sol>

U/ 1·k ¿f À®/ �} dÁJK, A/ ¶�� e=, a/ 1· e=�q.

ÂJuE½��cÃ

VAt

xA =∆

∆ = 1·k ¿U ÀÄ Ua2=

⇒

2

2

2

24

422

===

D

dV

D

dV

a

AVU

π

π

N} V) ¶�� JK�e

VD

d

D

Lp

2

22

32

=∆ η

1·� ÅM �q7 .e ¶��� ��./ ÆB

VD

dLp

dF

42

44

=∆= ηππ

cVF = ±�� Ew��/ ��H9/

4

4

=

D

dLc ηπ

� no�e

48

=

D

d

n

Lc

ηπ

x∆D

U UV Piston

L

d

p∆Fluid

♦���� (Harmonic motion)

• ����

��� � �� �� � ��� �, � ��� ������ �.

• ������ (Simple harmonic function)

�� �� 2� ���� ��! "#$% &'� �(� � ��) *�+ (,

�� �� ��� �.

-�+. “�/�� *0+ "#$1. 23� 45 06) 3$7 ��� ������ �� �.

Ex) Scotch yoke mechanism

tAAx ωθ sinsin ==

tAdt

dx ωω cos=

xtAdt

xd 222

2

sin ωωω −=−=

• ���� 89(Representation of harmonic motion)

����� 89$� 0, � :/� ω; <=$7 >�� A? op@A) %BC �,

@AD 89

- @A opX = ��� �E F+ GH$1

tAy ωsin= - �IF+ ( GHJ

tAx ωcos=

jtAitAX ˆsinˆcos ωω +=

K�. ji ˆ,ˆ 7 x, y F+ ( �0 @A

• L�M (cycle)

"NO PQ R7 ISPQ 06;&A 23T; U� 06VW ��, �X+ IS PQ

06; ��, ��. �Y 23T; U� 06VW ��, Z% �� ISPQ 06; ��$7

-�[ ��� -� L�M�� �.

\, op 1 revolution → tωθ = π2 rad :*0

• -] (Amplitude)

-�$7 ^[+. IS 06;&A _( *0) -� -]�� �.

\, 0 ` A

• -� �� (Period) : T or τ a= <=�� ( πθ 2= )+ bZ7 ��

\, @A op� :� π2 cd <=$7e bZ7 ��

ωπτ 2=

ω : f-�� or :-�� (Circular Frequency)

• -�� (frequency) or g-��

�0 ��h cycle � → -��� i�

π

ωτ 2

1 ==f , fπω 2=

♣ ω �0 : rad/s

f �0 : cycle/s or Hz

• ����(Synchronous motion)

j �P �� ��� kJ �l�) �W7 ��m

n) tAx ωsin11 = )sin(22 φω += tAx

• 0P:(Phase angle)

kJ -��) �Wo -�� �6$W p� � (Two synchronous motions) ��L� : *0q

(lead or lag)

• %r-��(Natural frequency)

s� t� uO �� vv; -�$�w xyB j �, s� uz {� -�$7 -��)

%r-��� �.

|%. n}r�s7 n ~ %r-��) ��.

• ��� (Beats)

-��� �V� �~ ����� �,� ��; o�o7 9P� ���� �.

tXtx ωsin)(1 = )sin()(2 ttXtx δωω += , δ is small

)2

cos()2

sin(2)()()( 21

tttXtxtxtx

δωδωω +=+=

⇒ rL$� ω -��) �W7 L?l! 7 -�� δω 0! 2xL�+. -]� �����

� ����. : Beat

n) ��-��� s %r-��+ ��C � ���� ��.

• ���(Octave)

-�� �0 _(�� _�� � �� ��� � ��� ��� ��� �.

n) 75-150 Hz, 150-300 Hz, 300-600 Hz

0P: φ

♦ ���� ��� 89

ibaX += θθ sincos iAAX +=

K�., 2122 )( baA +=

a

b1tan −=θ

��=~+.

���� +++=−+−=!4

)(

!2

)(1

!4!21cos

4242 θθθθθ ii

���� +++=−+−=!5

)(

!3

)(]

!5!3[sin

5353 θθθθθθθ iiiii

θθθθθθ ieii

ii =++++=+ ��!3

)(

!2

)(1sincos

32

θθθθθθ ieii

ii −=+−+−=− ��!3

)(

!2

)(1sincos

32

θθθ iAeiAX ±=±=∴ )sin(cos

<=@A : tiAeX ω=

1� ��� : XiAeiAedt

d

dt

xd titi ωω ωω === )(�

2� ��� : XAeiAedt

di

dt

xd titi 2222

2

)( ωωω ωω −===�

Displacement]Re[

cos)(tiAe

tAtxω

ω==

]Im[

sin)(tiAe

tAtxω

ω=

=

Velocity

)90cos(

sin

]Re[

�+=−=

tA

tA

Aei ti

ωωωω

ω ω

)90sin(

cos

]Im[

�+==

tA

tA

Aei ti

ωωωω

ω ω

Acceleration

)180cos(

cos

]Re[

2

2

2

�+=−=

−

tA

tA

Ae ti

ωωωω

ω ω

)180sin(

sin

]Im[

2

2

2

�+=

−=

−

tA

tA

Ae ti

ωωωω

ω ω

O

b

a

ibaX +=

x(Real)

y(imag.)

A

θ

• ���� �

)cos(,cos 2211 θωω +== tAXtAXFor

Ask : 21 XXX += ?

i) @A s�+.

)cos( αω += tAX

22

221 )sin()cos( θθ AAAA ++=

)cos

sin(tan

21

21

θθα

AA

A

+= −

ii) ��� s�

• ��(�

11111

θieAibaz =+= 2

2222θieAibaz =+=

K�., 2,1,22 =+= jbaA jij

2,1,tan 1 =

= − j

a

b

j

jjθ

)()()()( 21212211112111 bbiaaibaibaeAeAzz ii ±+±=+±+=±=± θθ

θiAezzz =+= 21

221

221 )()( bbaaA +++= ,

21

211tanaa

bb

++= −θ

]Re[ 11tieAX ω= , ]Re[ )(

22θω += tieAX

tiAtA ωω sincos 11 + )sin()cos( 11 θωθω +++ tiAtA

))sin(sin()cos(cos 2121 θωωθωω +++++ tAtAitAtA

αiAezzz =+= 21

2

12221

21

2221

21

}))sin(()sin(sin2)sin(

))cos(()cos(cos2)cos{(

θωθωωω

θωθωωω

++++

+++++=

tAttAAtA

tAttAAtAA

22

221

2221

21 )sin()cos(cos2 θθφ AAAAAAA ++=++=

1X

2X

αtωθ

O

ibaX +=

x(Re.)

y(Im.)

A

ωθsin2A

θcos2A)cos(2 αω +tA

++++= −

)cos(cos

)sin(sintan

21

211

θωωθωωα

tAtA

tAtA

]Re[]Re[ αiAezX ==

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∑∞

=++=

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tx ωω

K�., τπω 2= : �¬-��

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♣ �� v®¯°

±²*� : time , ³/*� : x(t)

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n tnaa

tx ω

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sin)(n

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=N

iix

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Original Function

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τ−

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��� 1.2������ ÂÃ

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velocityinitalvv

x =

= −

00

01tanωφ

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Phase=φ

x0

t�

*0

*0, x(t)

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Velocity

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A2ω

A2ω−

0

0

Aω

Aω−

0

−

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)sin()( φω += tAtx

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• 2D translational Equation : ∑= jextG FaM ,

• 2D rotational equation : jextGjkextG FrMH

dt

d,,)( ∑∑ ×+=

Where the angular momentum ωGG IH =

the angular acceleration ωαdt

d=

⇒ jextGjkextG FrMI ,, ∑∑ ×+=α

• If moments were taken about some arbitrary point A

the rotational equation becomes

jextAjkextGA

GG FrMamrI ,,)( ∑∑ ×+=×+α

• If A is a fixed point(pivot) , this eq. becomes

jextAjkextA FrMI ,, ∑∑ ×+=α

Where 2)(A

GGA rmII +=

kextM ,

x

y

dm j

A

AGr�

Mass center G

Angular acceleration

α

M: total mass

∫= dmrIG2

Angular velocity

ω

Ajr�

jextF ,

: mass moment

of inertia

♦ ��

���� : �� �� ��� �� �� �� ���� ��� ��.

���� : ��� �� ! "#� $% &' ���� �(� )) *+� ��

♦ ,�� -�.� ����• /01-�� 2/3� ��456

! 789:;< ��456= �>?@

0)()()()( =+−= txktxmortxktxm ������ (1.2)

II. 1 ���� ����

��� 1.1

1��>.� A

mm

B

lengthl =Gravity

g

Torsional

stiffnessk

J)(tθ

k

x(t)

/01-��

0=+ kxxm ��

CD EF

0=+ θθ km ��

G��

0)( =+ θθ lg��

m

0x0

+-Friction-freesurface

Resetposition

xy

-kx mg

N

• HI J 2KL= �MN ��456

∑ = xmF ����

O

0=+⇒−= kxxmkxxm ����

P

)()(2

2

stst xdt

dmxkmg δδ +=+−

2

2

)(dt

xdmkxkmg st =+− δ

0=+ kxxm ��

• D’Alembert � EQ

0=−− kxxm ��

0=+ kxxm ��

mkxRS T2 m

xm ��

k k k

stδm

mx

Static Eq. position

m

)( xk st +δ

)(2

2

stxdt

dm δ+

x

m

xm ��

kx

• UV� EQ(Principle of virtual work)

xkxforcespringthebyWV δ)(.. −=

xxmforceineriathebyWV δ)(.. ��−=

0=workTotal

0=+⇒ kxxm ��

• W� XYKL

ntconstaUT =+ T : ��W�, U : Z[(!\) W�

0)( =+UTdt

d

22

2

1,

2

1kxUxmT == �

0)2

1

2

1( 22 =+ kxxm

dt

d�

0=+⇒ kxxm ��

][ 456 : 00 222 =+⇒=+ nrm

kr ω

seigenvalueir n ;ω±=titi nn eCeCtx ωω −+=∴ 21)(

tite ti n ααω sincos ±=±

tAtAtx nn ωω sincos)( 21 +=⇒

0100 ,)0(,)0(:.. xAxxxxCI === ��

tAtxtx nnnn ωωωω cossin)( 20 +−=�

n

xA

ω0

2

�=⇒

tx

txtx nn

n ωω

ω sincos)( 00

�+=∴

m xm ��−kx

xδ

^_Aspects of the spring-mass system

1.

Circular Natural Freq. : mk

n =ω

st

mgk

δ=

gfgfg st

nn

stn

stn

δπτδπδω 21

,2

1, ====

2.

)cos()cos()(

)2

cos()sin()(

22 πφωωφωω

πφωωφωω

+−=−−=

+−=−=

tAtAtx

tAtAtx

nnnn

nnnn

��

�

3.

�V 00 =x �@

tx

tx

tx nn

nn

ωω

πωω

sin)2

cos()( 00 ��=−=

�V 00 =x� �@

txtx nωcos)( 0�=4.

A

xttAtx nn =−→−= )cos()cos()( φωφω

A

y

A

xttwinAtx

nnnn −=−=−→−−=

ωφωφωω �

� )sin()()(

m

k

x

stδ

A

A

x

y

A

nAω

x

x�

122

=

+

⇒

A

y

A

x 1

22

=

+

⇒

nA

x

A

x

ω�

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AJ)

`���ab c?d.( eQf� RSD ��= T2)

�� m� g# ��hQ x.

21

21

2121

21

)(4)

44()

22(2

)(2

kk

kkw

kkw

k

w

k

w

xxx

+=+=+=

+=

)(4 21

21

kk

kk

x

wk eq +

==

�� 456 (/01-��) : 0=+ xkxm eq��

Hzkk

kkf

sradkkm

kk

m

k

nn

eqn

)(44

1

2

/)(4

21

21

21

21

+==

+==⇒

ππω

ω

mO

Pulley 2

Pulley 1

2k

1k

11xk

1x w

w

22 xk

2x

m

m

w

w=mg xm ��)(2 21 xxx +=

11

11

2

2

k

wx

wxk

=

=

22

22

2

2

k

wx

wxk

=

=

AJ 1.1.1

!Ui φ /01-��. jk ��= l & ��m! x0 nN Un opq= ��rs.

6 (1.9) ��jt x(0)=x0b �uv= Xwd.

sol) t=0b 6 (1.9) nx?@

+== −

0

0120

20

2

tansinsin)0(v

xvxAx

ωω

ωφ

piyiz� {|D }~D Z�� va� 5�:;

< x(0)� �� s�D �s.

020

20

2

020

20

2

)0( xvx

xvxx =

+

+=

ωω

ωω

AJ 1.1.2

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� � �R~?

sol ) ��a, ��, �[ �� �. mk=ω �s. �d�

mNsradkgmk /2960)/)(300( 22 === πω

φ0xω

0v

20

20

2 vxw +

�>� 1.4

jk��� � � ��

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��:, �b

1)( 21 −==+= − jmkeaeatx tjtj ωωω

: �a �  , w�� a1D a2 ¡¢a Ua�s. £�:, �b

)sin()( φω += tAtx

: �a �  , w�� A� ¤ ¥a�� Ua�s. ¦�:, �b

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� ©5� ªs. i Ua�� yiva� �6D «V¬(Euler)� �69:;<

22

)(

tan

212

211

212211

2

1122

21

jAAa

jAAa

jaaAaaA

A

AAAA

+=+=

−=+=

=+= −φ

= ­= a �s.

AJ 1.2.1

78D �� /01� �� 10mmmz��= & ®

¯� °n �(= c?wd.

Sol)

`���a sf

THzfsradm

k0476.0

1,21

2,/132

102.49

8.8573

=====×

== − πωω �s.

m!� °n�� AJ 1.1.1� A� �±?² /01 �� �� ³> �= ´�,

mmxvx

Atx 10)( 0

20

20

2

==+

==ω

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)132cos(10)2132sin(10)( tttx =+= π �s.

kgm 3102.49 −×=

mNk 8.857=

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kf

J

k tn

tn π

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842

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1 224222

0 ===×== πρπρ

- ��� �(torsional pendulum) : ��� ]ef-H3�

• 0000 & θθθθ �� == == tt � �g1 �1 ttt nn

n ωωθωθθ sincos)( 0

0

�

+=

l

h

d

D

Q

Disk

Q

J0 ktQ

hX 1.5.1)

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sol)

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2

422102 /9087.4

5.02

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mkgm

mmN

lJ

GJ

J

k p =⋅××===

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hX) Natural Frequency of a wind Turbine

�� 2.14�[ t1uv w" p=�� x; Uy� *U1 z{� |{M �$. ''� FG&

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���Y : *U� FGH3./0v ��� �3� �� 1���*[ .�f �$.

�� : *U� ' |{1 m"� l"� FG" mn �?t* �$.

�" >

θ��00 IM =∑

)3

1(2, 2

00 mlIIkt ==− θθ ��

l2

m

d

l l

2

4

2 32 l

Gd

l

GJkt

π==

22

4

22

4

0 64

3

2

3

32

00

lml

Gd

ml

G

l

d

I

kn

kI

t

t

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=+

hX ) ,��� ��p

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�&' Q����� θθ ≅sin "$�

00 =+ θθ wdJ ��

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mgd

J

wdeq

eqn

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00

=⇒===ω

200 mkJ = ; k0 : O�� �� % � H3 �L(radius of gyration)

d

k

md

mk

md

Jleq

20

200 === : ,��� �� N� m"

G�� �� % � H3 �L� KG# �R

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+=

2

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xAx γγγ =+

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22 )(2

1))( (

2

1x

g

AlT �

γ== °�FGp&�

- ��!�� �#R tXtx ωcos)( =

tTTtuu nn ωω 2max

2max sin,cos ==

22

max2

max 2

1, X

g

lATXAu nωγγ ==

maxmax Tu =

l

gn

2=ω

sradrev /1060

2300min/300

2

6001 ππ =×==×=�³p(�¾;¿4�±©�

srad /95.3

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22

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x

hX ) nω � �� ]ef FG� Á¨

]ef 4¥� () )/( lxy

]ef 4¥� °� )/( lxy �

2))((2

1x

l

ydy

l

mdT s

s �= , ms : ]ef FG

)( )( sm TTT !��¼g]ef�!��¼gFG� +=

++=

+= ∫ =

22

2

22

0

2

32

1

2

1

)()(2

1

2

1

xm

xm

l

xydy

l

mxm

s

l

y

s

��

��

)¡�¼gP �� =u

2

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1kx=

��!�T* ���R tXtx nωcos)( =

2max

22max

2

1

)3

(2

1

kXu

Xm

mT n

=

+= ω

maxmax uT =

3

mm

kn

+=ω

m

l

y

dy

x

hX) %ÂÃ� ��p

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�pM q�#.

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���Y > �Î� �3� �� ÂÃM 1���* �K�#.

�� > 1. %ÂÃ1 Ï� FG"$.

2. �Î& �?NR� �$.

sol )

L: 1) %ÂÃ� ��p� �Î� FG� �K�g Ð� �

3

3

l

EIPk ==

δ

44444 1096600)96120(64

)(64

×⋅=−=−= ππio ddI

concreteforinlbfEinl )/(104,)(360012300 6×==×=( )( )

( ) inlbk /67.15453600

109660010433

46

=×⋅××=∴

srad

gM

kM

k

c

n /1106

12174.3267.15455

2

≈×

××===ω

E. I.M

P

EI

Pl

3

3

=δ

L: 2) %ÂÃ� ��p� �Î� FG� �KÑ �

BN� Ï���" �dÑ � ÒÓÔ Õ� �½ (>&

( )

( )323

max

3

max2

3max

2

32

33

2)3(

6)(

xlxl

y

EI

Plyxlx

l

yxl

EI

Pxxy

−=

=⇐−=−=

- � lx = � ��FG eqm � �$� �R

{ } dxxyl

mymT

l

eqbeam2

0

2max )(

2

1

2

1�∫==

( )dxxllxxll

y

l

ml

∫ +−

=

0

65422

3max 69

22

1 �

+−=

75

9

42

1 777

7

2max l

lll

ym�

2max140

33

2

1ym �

=

lxatmmeq ==⇒140

33

Rayleigh Energy MethodM "d�R

( ) 2max2

1yMmTTT eqMbeam �+=+=

2max3

2max

3

2

1

2

1y

l

EIyku ��

==

j? tYy nωcosmax = "R

maxmax uT =

E. I.mM

P

maxy=δ

l

x

y(x)

=> �;� Á¨" �K�

Ð& ÖX

3

3

l

EIPk ==

δ

( ) 23

22 3

2

1

2

1Y

l

EIYMm neq

=+ ω

mM

lEI

mM

k

eqn

140

33

33

+=

+=⇒ ω

♦ ���� ����(Free vibration with viscous Damping)

�� �� xcF �−= (c : �� �)

� �-�����-���� ���� ��� 1����� ���

• !�"#$

∑ = xmF ��

xmkxxc ��� =−−⇒0=++⇒ kxxcxm ��� (2.1)

• %�"#$ : 02 =++ kcrmr

m

k

m

c

m

c

m

mkccr −

±−=−±−=

22

2,1 222

4 (2.2)

• &� �� � ' ��((Critical damping constant & the damping ratio)

: &� �� � (cc)) $ (2.2)* +,+- 0�� ./) �� �.

02

2

=−

m

k

m

cc

nc mmkm

kmc ω222 ===⇒ (2.3)

&* ��01 ��((Damping ratio) ζ ) &� �� �0 2� �� �* (

cc

c=ζ (2.4)

$ (2.1)- 34 5

0=++ xm

kx

m

cx ���

2,22

nnnc

m

k

m

m

m

c

m

c ωζωωζζ ====

02 2 =++⇒ xxx nn ωζω ��� (2.5)

mO

Static Eq. positin.

x

m

kx xc�

x

1��� 678901 ��:;�

�< $ (2.2))

nr ωζζ )1( 22,1 −±−= (2.6)

�� ��&( 0≠ζ )- �=� 3> ?@A !�� �BC3.

• D�� E) FG ���(Underdamped System, mk

mccc c <<< 2,,1ζ )

nn irir ωζζωζζ )1(,)1( 22

21 −−−=−+−=

titi nn ecectx ωζζωζζ )1(2

)1(1

22

)( −−−−+− +=

titi nn ecectx ωζζωζζ )1(2

)1(1

22

)( −−−−+− +=

)(22 1

21

1titit nnn ecece ωζωζζω −−−− +=

[ ]tccitcce nntn ωζωζζω 2

212

21 1sin)(1cos)( −−+−+= −

[ ]tBtAe nntn ωζωζζω 22 1sin1cos −+−= −

( )φωζζω +−= − tXe ntn 21sin

( )φωζζω −−= − teX ntn 2

0 1cos

H�1, ),(),,(),,( 00 φφ XXBA ) I� JK0 *L M#C3.

I�JK� 0000 , xxxx tt �� == == � ,

01)0( xcx ==

[ ]tBtxetx nntn ωζωζζω 22

0 1sin1cos)( −+−= −

[ ][ ]tBtxe

tBtxetx

nnnt

nnt

n

n

n

ωζωζωζ

ωζωζζωζω

ζω

220

2

220

1cos1sin1

1sin1cos)(

−+−−−⋅+

−+−−=−

−�

n

nnn

xxBxBxx

ωζζωωζζω

2

000

20

11)0(

−

+=⇒=−+−=�

��

−

−

++−=∴ − txx

txetx n

n

nn

tn ωζωζ

ζωωζζω 2

2

0020 1sin

11cos)(

� (210)

( ) ( )AB

BABAXX 1

0122

0 tan,tan, −− ==+== φφ

$ (2.10)� NOP) !�Q R ST� nωζ 21− - U) ��JV!���W X� tne ζω−0 *

L1 �Y� A�Z�� �[�3.

\ �� ����� (Frequency of damped vibration)

n

dnd ω

ωωζω 21−=

)10(11 2

2

2

2

<<=+

⇒−=

ζζ

ωωζ

ωω

n

d

n

d

\ ���� 4�]01 FG��* ^_@ ����0 `�1 ab� cQ FG��01.

��� de�3.

fg 2.1 FG���* L.

n

d

ωω

1

ζ

��0 hi dω * jV

• &� ���(Critically damped system , mk

mccc c === 2,,1ζ )

nc

m

cr ω−=−=

22,1

tnetcctx ω−+= )()( 21

I�JK� 0000 , xxxx tt �� == == k ^_0

01 xc =

[ ] tn

n

n

ttn

t

n

nn

n

etxxxtx

xxc

xcxx

ecetcxtx

etcxtx

ω

ωω

ω

ωω

ωω

−

−−

−

++=

+=⇒=+−=

++−=

+=

)()(

)0(

)()(

)()(

000

002

020

220

20

�

�

��

�

(S�Z ∞→t k< 0)(..0 →→− txeie tnω (Ml [m)

• n���(Overdamped system, mk

mccc c >== 2,,1ζ )

212

2

21

0)1(

0)1(rr

r

r

n

n >

<−−−=

<−+−=

ωζζ

ωζζ

tt nn ecectx ωζζωζζ )1(2

)1(1

22

)( −−−−+− +=

I�JK 0000 , xxxx tt �� == == k ^_0

201021)0( cxcxccx −=⇒=+=

tn

tn

nn ecectx ωζζωζζ ωζζωζζ )1(22

)1(21

22

)1()1()( −−−−+− −−−+−+−=�

[ ] 02

022

2

02

22

20

02

22

1

)1()1()1(

)1()1)((

)1()1()0(

xxc

xccx

xccx

nnn

nn

nn

ωζζωζζωζζ

ωζζωζζ

ωζζωζζ

−+−−=−+−−−−−⇒

=−−−+−+−−⇒

=−−−+−+−=

�

�

��

12

)1(2

02

2−

−−−−−=

ζωωζζ

n

n xc

�

12

)1(

12

)1(12

2

002

2

002

02

202

−

+−+=

−

+−−+−=−=

ζωωζζ

ζωωζζζω

n

n

n

nn

xx

xxxcxc

�

�

=> �0 FnC I� JK0 o�p� !�� (S�Z�� 21 & rr @ qr >�s� 4�� A

t0 hu A�Z�� v�w3.

fg 2.2

\ Aspects of viscous damping vibration system

1. x[ 6 01* r+ r1n r2* yZ

2. &� ���* :zZ *{(Physical meaning of critically damped system)

- (S� !�- �) | }b� ~[��

- @� �i 4�0 Overshootp� #A 89�

- 2�* S� x��0 �� : �� � ��� ��p� �i4�0 #A(Overdampingk ^_

4�� A� �)

3.

\ 1����* ����Q 86 n ���01 ���� � � `3.

nω

nω

nζω−

12

>ζfor

s

nrr ω−== 21

10 << ζ

0=ζ

11

>ζfor

s

0Real

Imag.

• 2��[�(logarithmic decrement, δ )

���������*��Y���[�)���������)�r��Y*�(0����2��������

fg����

−=−+=

⇒+=+=)cos()cos(

2211

12112 φωφω

πωωω

πτtt

ttttt

dd

dd

dd

)cos(

)cos(

20

10

2

1

2

1

φωφω

ζω

ζω

−−= −

−

teX

teX

x

x

dt

dt

n

n

1

1

1

)(2

1 τζωτζω

ζωn

dn

n

ee

e

x

xt

t

==⇒ +−

−

m

ce

x

x

dn

ndndn

2

2

1

2lnln

22

1 ⋅=−

====ω

πωζ

πζωτζωδ τζω

( ) 222 21

2

δπδζ

ζπζδ

+=⇒

−=⇒

πδζπζδζ2

21 ≈⇒≈⇒<<if

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dτ dτ

x1

x2

t3t2t1t4

\ 1

1

3

2

2

1

1

1

+

−

+

⋅⋅=m

m

m

m

m x

x

x

x

x

x

x

x

x

x�

dndndn eee τζωτζωτζω�=

)( dnme τζω=

1

1)(

1

1 ln1

)(lnln++

=⇒===m

dnm

m x

x

mmme

x

xdn δδτζωτζω

• ����* [´ 0µA

\�������01�4�0�2��0µA�jV�� dtdw /

22

−=−==

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dt

dw���¶·�0µA@�4�0�hu�"¸

tXx dωsin= u�@#� ����S�¹3�"¸�)�0µA)

( )∫∫ ⋅=

=∆

=

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ωωω2

0

222

0

2

cos tdtcXdtdt

dxcw dddt

d

22

0

2

2

2cos1Xctd

tXc dd ωπω

π=′′+= ∫

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w∆ )� dω *�¿��3�

\���·� �·���������À�<�

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0

22

2

0

2

2

0

cos

cossin

XctdtXk

tdttkX

dtFvw

dddd

ddd

t

d

d

d

ωπωωω

ωωω

ωπ

ωπ

ωπ

=⋅⋅+

⋅⋅⋅=

=∆

∫∫∫=

Â;�*�0µA)�~2�89�0µA )2

1( 2kx �E)�~2�!��0µA� )

2

1

2

1( 222

max dmXmv ω=

m

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�3����@�ÃQ��� �r��Q�¬*��k�

)(422

22

2

1 22

2

ntconstam

c

Xm

Xc

w

w

dd

d πζδω

π

ω

ωπα ≅=

==∆=

���� :α (��Ä®ª�¤¡¦Å¦¡�§ ��¦©Æ�¡ � ¡¦¢Ç°

ζπ

πη 22

2 ≅∆=∆

=w

w

w

w

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•�������(Ëg�

·�������Ì�Í)

θ�td cT −=

·�!�"#$Q

00 =++ θθθ tt kcJ ���

0

2

00

2

,1

22

02

Jk

Jk

c

J

c

c

cwhere

tnnd

t

t

n

t

tc

t

nn

=−=

===

=++⇒

ωωζω

ωζ

θωθζωθ ���

μ°�ÏШ¡Ñ� Òª¨Ò¤¥�¨Å� ��¨¢¨¥¡Ç¡Ó¤

�

4037.0

1

27726.24ln

5.0

1ln

12

5.1

1

=⇒−

===

=

ζζ

πζδx

x

m

( ))/(4338.3

sec24037.01

2

1

2222

sradn

nndd

=⇒

=−

=−

==

ωω

πζω

πω

πτ

ª¢¦ÅÅ©¤ªª��Ñ

)/(2652.23584338.3200 22 mNmk n =×== ω

Ô ��¦©Æ�¡¨©ª¢ ©¢��¡

)/(4981.55454.13734037.0

)/(54.13734338.320022

msNcc

msNmc

c

nc

−=×==−=××==

ζω

nddt wheretXetx n ωζωωζω 21sin)( −== −

­®¢°*�~2�~[�0�2L

0)cossin( =+−= − ttXedt

dxdddn

tn ωωωζωζω

�� tfiniteforeAs tn 0≠−ζω

�� 0cossin =+−⇒ tt dddn ωωωζω

m

k/2 k/2c

x

m=200kg

0)0(,0)0( xxx == �

?,4

1sec,2 15.1 =⇒== ckxxdτ

~2j8 = 250 mm => x0=?

������ζ

ζζωωω

21tan

−==

n

dd t

2

22

2

21

11

1

tan1

tansin ζ

ζζ

ζζ

ωωω −±=

−+

−

±=+

±=t

tt

d

dd

ζωζω

ζωζω

−=−−=

=−=⇒

tt

tt

dd

dd

cos&1sin

cos&1sin

2

2

[ ]tttXedt

xdddddndn

tn ωωωωζωωωζζω sincos2sin 2222

2

−−= −

� ζωζω =−= tandtWhen dd cos1sin 2

01 222

2

<−−= − ζωζωn

tnXedt

xd

)(1sin 2 txofaximommtosCorrespondtd ζω −=⇒

ζωζω −=−−= ttWhen dd cos&1sin 2

01 222

2

>−−= − ζωζωn

tnXedt

xd

)(1sin 2 txofnimommitosCorrespondtd ζω −−=⇒

~2®E)�~[°�Õ-�«n�)�Ö×-�3>n�Ø��r

tncetx ζω−=)(

~2�0�2L

d

tω

ζ 21

max

1sin −=

−

maxsinmaxmax tXece dtt nn ωζωζω −− =

21 ζ−=⇒ Xc

tneXx ζωζ −−= 21 1

~[�0�2L

d

tω

ζ )1(sin 21

min

−−=

−

minsinmin tXece dtt nmnn ωζωζω −− =

21 ζ−−=⇒ Xc

tneXx ζωζ −−−= 21 1

~2�-�U)� 1t

(sec)3678.04037.01sin

21sin

1sin

2121

1

21

=−=−

=⇒

−=−−

πτπ

ζ

ζω

d

d

t

t

~2�-�«n�)�Ö×

455.0

4337.01250

1

3678.04338.34337.02

21

=⇒−=

−=××−

−

X

eX

eXx tnζωζ

I����

tXetx dtn ωζω sin)( −= ���Â;�j8�Ö×

)cossin()( ttXetx dddntn ωωωζωζω +−= −

�

¢ºÙ 01

)/(4294.1455.02

00 smXXXxxd

dt =×=⋅=⋅==== ππτπω��

μ°���LÚ

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ª¨Ó�°

sradm

kn /4721.4

500

1000 ===ω

É°� )/(1.44724721.450022 msNmc nc −=××== ωtnetcctx ω−+= )()( 21

[ ] tn

netxxXtx ωω −++= )()( 000 �

)(tx @�~2���)�4�� *t

[ ] tnn

tn

nn etxxXexxtx ωω ωωω −− ++−+= )()()( 00000 ���

+

−=⇒00

01*

XX

Xt

nn ωω �

��ä¼01�tntextxx ω−== 00 )(,0 �

n

tω1

* =∴

nn

t

e

Xe

Xetxttxx n

ωωω 010*

0max **)(��

� ===== −−

�°� smexX n /8626.47183.2721.44.0max0 =××== ω�

�°� (sec)8258.01.0 24721.4

202 =⇒⋅= − tetX t�

♦ �� ��(Hysterisis Damping) ����• �� ��(Hysteritic damping)

: � � � � ����� �� ����� ��� �� � ��

• !"#� $%�� plot&

xckxF �+=��' ω , �( X) *� +&,�- ./

tXtx ωsin)( =

22

22 )sin(

cossin)(

xXckx

tXXckx

tcXtkXtF

−±=

−±=

+=∴

ω

ωω

ωω

Area : ��0$ �1 2 ,� Cycle�3 !"#� $%�

22

0)cos)(cossin( cXdttXtcXtkXFdxw πωωωωωωω

π=+==∆ ∫∫ (1)

• �� ��� � -4� �� 5� 6-� ��$7 Hysterisis 89:- ;<3.

- � =-� =(loading-unloading)� 1 ��> �?� @A$%�� 89:B� C� DE �FG

H3.

- I (1)- �J K �� L �� �� M') N"

ck

Fx(t)

Xcω−

Xcω22 xXc −ω

kx

Xx

-X

F

x

• AOF .� : �� ��$ �2 1 ��> $%� @AP ��'$ QR L ST �(� UV$

WX

- ntconstadampinghysteresisthehWhere

cwh

:

=

�Y- I (1)$ SZ �

• [\ ]^(Complex stiffness)

;_ tiXex ω= ��

xick

eXickxF ti

)(

)(

ωω ω

+=+=

M� [\]^

k

hik

k

hikihk =⇒+=+=+ ββ )1()1(

)( xorε

)( Forσ � =

� =

2hXwArea π=∆=

ck

Fx(t)

hk

F(t)x(t)

xihkF )( +=

• M� 4`(Response of the system)

- 2 ��>a @A $%�22 XkhXw βππ ==∆

- �� ��$7� w∆ � b0 �c$ d� +&,�$ �ef

- P&Q : 1/2 ��> g

4442

25.0

25.0

22+≠ =−−=−∴ jjjj

Qp

kXXkXkkXEE

βπβπ

25.0

2 )2()2( ++=−⇒ jj XX πβπβ

πβπβ

−+=⇒

+ 2

2

5.0j

j

X

X

�� L πβπβ

−+=⇒

+

+

2

2

1

5.0

j

j

X

X

.12

22

2

2

1

5.0

5.01

constX

X

X

X

X

X

j

j

j

j

j

j =+≅−

+−=−+=⋅=∴

+

+

++

πβπβ

πβπβπβπβ

- h� i^ ��W eqζ

k

h

k

heqeq 22

2 ==⇒=≅≅ βζππβπζδ

- h� i^ M' eqc

ωωβββζ hk

mkmkcc eqceq ===⋅=⋅=2

2

j k� !!

1) l+ ��M$ S1 h� i^ ��W) l � !mP +& ���� no$; �p.

2) q0 1rP M� ��' ω� d� +&,� 23. �s $7 �tuv.

♦ ���� (Stability condition)

∑ = θ��00 IM

θθθθ sin2

cos)sin(23

2

mgl

lklml +−=��

�� ��

02

23

22

=−+ θθθ mglkl

ml��

02

3122

2

=

−+ θθml

wlkl��

�� 1) 02

3122

2

>−ml

wlkl ��

tAtAt nn ωωθ sincos)( 21 +=

2

2

2

312

ml

wlkln

−=ω => ��� ��

�� 2) 22 2312 mlwlkl =− ��

21)(0 CtCt +=⇒= θθ��

�� �� 0000 , θθθθ �� == == tt �

00)( θθθ += tt � : � ��� ��� 0θ� �� ��

00 )(,0 θθθ == t� : ���� with 0θθ = (Maginally stable)

mg

l

2

lG

kk

mg

θ

θsinklθsinkl

θsinl

θcosl

�� 3) 22 2312 mlwlkl <− ��tt eBeBt ααθ −+= 21)(

2

2

2

123

ml

klwl −=α

���� 0000 , θθθθ �� == == tt �

( ) ( )[ ]tt eet αα θαθθαθα

θ −−++= 00002

1)( ��

� !�� �� : Unstable

"# $%& < '"# $%&

♦ Rayleigh Energy Method

'() *�+�, �-� ./01�

�2��3 �45

maxmax11 00 uTuT =⇒+=+ => nω : Raylegh Energy method

2211 uTuT +=+

�!67�89 :;<= >?

@AB C��� BD= >?

���-�

�8�-�

������(Harmonically Excited vibration)

♦����� �

)(0)( φω += tieFtF

)cos(0 φω += tF

)sin(0 φω += tF

♦��� � ����� ��

tFkxxm ωcos0=+��

m

kn =ω

tctctx nnF ωω sincos)( 21 +=

Assume tBtAtxp ωω sincos)( +=

tBtAtxp ωωωω cossin)( +−=�

tBtAtxp ωωωω sincos)( 22 −−=��

02 FkAmA =+− ω , 0=B

2

0

ωmk

FA

−=

tmk

Ftxp ω

ωcos)(

20

−=

tmk

Ftctctx nn ω

ωωω cossincos)(

20

21 −++=⇒

Initial condition : 0)0( Xtx == , 0)0( Xtx �� ==

m

k

tFtF ωcos)( 0=

m

kx

tFtF ωcos)( 0=

20

01 ωmk

Fxc

−−= ,

n

xc

ω0

2

�=

tmk

Ft

xt

mk

Fxtx n

nn ω

ωω

ωω

ωcossincos)(

200

20

0

−+

+

−−=∴

�

♣ ����(Amplification Factor) , ����(Magnification Factor) , ���(Amplitude Ratio)

: ��� �� ��� �� ��� �� �

• ����( stδ ) = kF0

• ����( X ) = 2

0

2

0

20

11

rk

Fk

F

mk

F

n

−=

−

=−

ωωω

, ��� r : ��� �

21

1

rX

st −=δ

! I ) 10 <<nω

ω " #,

21

1

rX

st −=δ : px $ %& �'(

1

)(r ωω=

st

Xδ

tmk

F ωω

cos2

0

−

tF ωcos0

! II ) 1>nω

ω " #,

)180sin(cos

1

)(2

�+=

−

= tXttx

n

stp ωω

ωω

δ

tFF ωcos0=

0→→∞→ Xnω

ω

! III) 1=nω

ω " #, → )�(Resonance)

−

−+

+=

20

0

1

coscossincos)(

n

nstn

nn

ttt

xtxtx

ωω

ωωδωω

ω�

nωω → , 0

0(*)→

⇒ By L’hospitals’s rule

tttt

d

d

ttd

dtt

nn

nn

n

n

n

nnn

ωω

ωω

ω

ωω

ω

ωωω

ωω

ωωωωωωωω

sin22

sin

1

)cos(cos

1

coscos

222 limlimlim =

=

−

−=

−

−←←←

tt

tx

txtx nstn

nn

n ωδωωω

ω sin2

sincos)( 00 +

+=

� ��� ��* +,� ,-� .��/0 ��.

�180 � '(1

*

♦ 23��

>

−

+−

<

−

+−

=1cos

1

)cos(

1cos

1

)cos(

)(

2

2

n

n

st

n

n

st

forttA

forttA

tx

ωωω

ωω

δφω

ωωω

ωω

δφω

♦45* 6((Beating phenomenon)

• �� ���� +78� 9:���$ ��-; "<-= >=? @! �AB !

“45* 6(”* CD

• ��� � ����� ���� E��F* 000 == xx � G9 -H

)cos(cos)(2

0

ttmF

tx nn

ωωωω

−−

=

)2

sin2

sin2[2

0

ttmF

nn

n

ωωωωωω

−⋅+−

= (1)

•�� ��� ω� 9:���IJ K, LJ9 -H

εωω 2=−n , ��� ε & L& M�

ωωωωω 2≈+⇒≈ nn

εωωωωωωω 4))((22 =+−=−∴ nnn (2)

(2)N (1)� �O-H

ttmF

tx ωεεω

sinsin2

)(0

=

⇒ ε * LJH

�� : tmF

εεω

sin2

0

PPQ R-9 S T� επ2 & UJ.

⇒ tεsin V.* � W*X� *Y �Z tεsin V.& �[ W*X� �\

•45* T�(period of beating) : dτ

ωω

πεπτ

−==

nd

2

2

2

•45* ��� : ωωεω −== nb 2

•45* 6( V.

]^ ) _`N ==- a

)(2083.0)5.0)(20(12

1

1243

3

inbh

I ===

inlbl

EIk /0.1200

100

)2083.0)(1030)(192(1963

6

3=×==

inslbmlbF /4.386/150,50 20 −==

)(1504.0)832.62)(4.386/150(1200

5022

0 inmk

FX −=

−=

−=

ω

20 in

5 in

steelforpsiE )(1030 6×=

♦ ���� �� �� ��

tFkxxcxm ωcos0=++ ���

tBtAtxp ωω sincos)( +=

tBtAtxp ωωωω cossin)( +−=�

tBtAtxp ωωωω sincos)( 22 −−=��

tF

tkBtkAtBctActBmtAm

ωωωωωωωωωωω

cos

sincoscossinsincos

0

22

=

+++−−−

−=⇒=−+−

=++−

Bc

mkAAckBBm

FBckAAm

ωωωω

ωω2

2

02

0

0

22 )(FB

c

mk =

−⇒

ωω

222

02

2220

)()(

)(,

)()( ωωω

ωωω

cmk

FmkA

cmk

FcB

+−−=

+−=

)cos(

sin)()(

cos)()(

)()(

2220

2220

2

φω

ωωω

ωωωω

ω

−=+−

++−

−=∴

tX

tcmk

Fct

cmk

Fmktxp

Where, 2

1222

0

])()[( ωω cmk

FX

+−= ,

−= −

21tan

ωωφmk

c

k

tFF ωcos0=

���� �� ��� ��

or 2

1222

0

21222

0

])2()1[(]))(2())(1[( rr

kF

kF

X

nn

ζωωζ

ωω +−

=+−

=

��� 2

1222 ])2()1[(

1

rr

XM

st ζδ +−== , ��� )

1

2(tan

21

r

r

−= − ζφ

���� r� � X � φ !�

♣ )(st

XM δ= ��� � r" ��� ζ !�� #$ %&

1. 0=ζ (����)� �' , 1→r , ∞→M

2. 0>ζ (����)� �' , () � ����* �+�(M)� �,�-..

3. /0 %1 2 r� �' , ↓→↑ Mζ4. 0=r � �' , /3 ζ � �' 1=M 4..

5. ��5 6 �+� �,� 7�" 7� 89�* :;< =>?..

6. 0, →∞→ Mr

7. 2

10 << r � �' , 221 ζ−=r @, 221 ζωω −= n �* maxM AB ..

nωω = C. nd ωζωω 21−== C. DE 2

{ } 0)2)(2(2)2)(1(2])2()1[(

1

2

1 2

23222

=+−−+−

−= ζζζ

rrrrrdr

dM

08)1(4 22 =+−− ζrrr

221 ζ−=⇒ r

8. M2 F�2E (When 221 ζ−=r )

22

12222max

max12

1

)]21(4)211[(

1

ζζζζζδ −=

−++−=

=

st

XM

⇒ ��G HI�J5 KL�M 4N. @, maxX � O1LP ζ G K ..

Q� ζ RST UJP F� �+� VW X

nωω = ( 1=r )�*

)ζδ ωω 2

11 =

=

==

nst

r

XM

9. 2

1=ζ � �' , 0,0 ==dr

dMr @, 0=ω ; 1LY Q� Z[��?

2

1>ζ � �' , ↓→↑ Mζ

♣ ��� %&

1. 0=ζ (��� �\])� �',

2. 0>ζ � 10 << r

�� 900 <<φ x F� ^_

3. 0>ζ � 1>r

�� 18090 <<φ x F� `a

4. 0>ζ � 1=r ; �90=φ

��b

5. 0>ζ � 1>>r � �' ; �180→φ6. ���E ω , nω , ζ � cLde

♦ fg �� (Total Response)

)cos()cos()( 00 φωφωζω −+−= − tXteXtx dtn

Where 21 ζωω −= nd

kmFX &,,,& 0 ζωφ ← conditionsinitialX ←00 &φ

♦ %&�(Quality Factor)� �h+(Bandwidth)• Q Factor(Q �) Q� %&�(Quality factor)

�� iE jk ( 05.0<ζ )� �Ll 7��* �+�

QXX

nstst

==

≅

⇒

=ζδδ ωω 2

1

max

→ m�6 n o&��G VWL�M pN

; q) 7��* �+� rs 't L� Radio ��u5

=φ)10(0 << r� : Fx & ���

)1(180 >r� : Fx & h��

F(real)

x(Imag.)

• v��o(Half power points, 21 & RR )

w+� 2

Q5 x/d� y(

nωω

)

��� z�{� ��( ω∆ )= 2maxXcωπ

→ half point : 22

max QXX ==

• �h+(Bandwidth)

v��o 1R " 2R � '|L� ��� b ( 12 ωω − ) ← n

Rωω1

1 = , n

Rωω2

2 =

222 )2()1(

1

22

1

2 rr

Q

ζζ +−==

0)81()42( 2224 =−+−−⇒ ζζrr

2222

2221 1221,1221 ζζζζζζ ++−=+−−=⇒ rr

ζ i� jk

2

222

22

2

121

21 21,21

≈+==−≈

==

nn

RrRrωωζζ

ωω

Where 21

21 ,RR

ωωωω ==

ζωωωωωωω 4)())(( 221

221212

21

22 ≈+=−+=− nRR

nωωω 212 =+ 4}5

�h+ nb ζωωωω 2)( 12 ≅−=

• Q� �h+" �

b

nQωω

ζ≈≅

2

1

♦ tieFtF ω0)( = � �� ���� �

�� ��� ��� ��� tieFtF ω0)( = � ����

tieFkxxcxm ω0=++ ���

��� )(tx p � tip Xex ω= , ti

p Xeix ωω=� , tip Xex ωω 2−=�� � ����

{ } 02 )( FXicmk =+− ωω

ωω icmk

FX

+−=

)( 20

+−

−+−

−=222222

2

0 )()()()( ωωω

ωωω

cmk

ci

cmk

mkF

=+=−

−

−−

a

bebaiba

iba

i 122 tan, φφ

[ ]

φ

ωωie

cmk

FX

21

222

0

)()( +−=

Where )(tan2

1

ωωφmk

c

−= −

���� � , px

[ ]

)(

21

222

0

)()()( φω

ωω−

+−= ti

p ecmk

Ftx

• � !� �� "#� �(complex frequency response)

)(21

12

0

ωζ

iHrirF

kX =+−

=

• $%��(Magnification factor, M)

[ ] 2

1222

0 )2()1(

1)(

rrF

kxiHM

ζω

+−===

• �&

φωω ieiHiH −= )()( , )1

2(tan

21

r

r

−= − ζφ

)(0 )()( φωω −== tip eiH

k

Ftx

'( tFtF ωcos)( 0=

[ ]

)cos()()(

)(2

1222

0 φωωω

−+−

= tcmk

Ftxp

])(Re[])(Re[ )(00 φωω ωω −== titi eiHk

FeiH

k

F

'( tFtF ωsin)( 0=

[ ]

)sin()()(

)(2

1222

0 φωωω

−+−

= tcmk

Ftxp

])(Im[ )(0 φωω −= tieiHk

F

♦ ��� ����� � � �� ��

(Response of a damped system under the harmonic motion of the base)

0)()( =−+−+ yxkyxcxm ����

if tYty ωsin)( = tYctkYkxxcxm ωωω cossin +=++ ���

[ ] [ ] 2

1222

1

21

222

1

)()(

)cos(

)()(

)sin()(

ωω

φωω

ωω

φω

cmk

tcY

cmk

tkYtxp

+−

−++−

−=⇒

)cos()( 21 φφω −−=⇒ tXtxp

)cos()()(

)(21

2/1

222

22

φφωωω

ω −−

+−

+= tcmk

ckY

• �����(Displacement Transmissibility)

“�� )(txp � ��� ���� )(ty � ��� �� �

2/1

222

22/1

222

22

)2()1()2(1

)()()(

+−

+=

+−

+=rr

r

cmk

ck

Y

X

ζζ

ωωω

)12

(tan)(tan2

12

11 r

r

mk

c

−=

−= −− ζ

ωωφ , )

2

1(tan)(tan 11

2 rc

k

ζωφ −− ==

m

k c

x

tYty ωsin)( =

ym

x

)( yxc �� −)( yxk −

♦ ���(Transmitted Force)

• ��(base)� ���� , F

xmyxcyxkF ���� −=−+−= )()(

)cos()cos( 21212 φφωφφωω −−=−−= tFtXmF T

• ���(Transmissiblity)

2/1

222

22

)2()1()2(1

+−

+=rr

rr

kY

FT

ζζ

��� ��! � ��" � r# ζ � $� ��

• %���

- &'� ��� �� %��� , z yxz −=

- tYmymkzzczm ωω sin2=−=++ �����

)sin(])()[(

)sin()(

2/1222

2

φωωω

φωω −=+−

−= tzcmk

tYmtz

2/1222

2

2/1222

2

])2()1[(])()[( rr

rY

cmk

YmZ

ζωωω

+−=

+−=

k c

base

)( yxc �� −)( yxk −

&'� �� )(tx ( �) �%

���� �� *� +�,

YZ / � r# ζ � �� plot

Ex) -./0 �1� 2'(Vehicle moving on a rough road)

34 56 (a)� -. /0 �1� �7 "89:� ��; " <� 2'� =>� ?@A BCD

<E. 2'� &'F 1200kgCD, GHI,� JKL%"� 400kN/m, � � ζ =0.5 CE. M�2� N

�C 100km/hr ) O 2'� ��� ��A P�Q. RSF �� Y=0.05m, T� 6m UGVW� �

��E.

sol> H� TX" sradT

f /09.29

36006

100010022

2 =

××

=== πππω

3600

)1000)(100(

6

11 ×=T

DY��" )/(2574.18 sradm

kn ==ω

��" � 593.12574.18

09.29 ===n

rωω

��� 8493.0)2()1(

)2(1222

2

=+−

+=rr

r

Y

X

ζζ

∴ 2'� ��� �� (X) )(0425.005.08493.08493.0 mYX =×==

Ex) Z[ \\��� ],! ��(Machine on Resilient Foundation)

^_H 3000 N ^-� ��H Z[\\� �� \\`a <E. ��� ^_ � \\�� Ub

cdF 7.5cm efgE. \\�� ��H �� �� DY��" �� 0.25cm� ����A hA

O ��� 1cm� ��� ���� iC $j`kE. CO (1)\\�� � �", (2)��� ��! �

b � ��, (3)��� ��� $� %���� ��A P�Q.

sol> (1) 122.30681.9

3000 ==m

mNkk st

st /40000==⇒=δωωδ

l� ( nωω = , m r=1)

1291.0)2(

)2(1

025.0

01.02

2

=⇒+== ζζ

ζY

X

mNsmkcc r /9032 === ζζ

(2) r = 1

NkXYkrr

rrkYFT 400)01.0)(40000(

)2(

)2(1

)2()1(

)2(1 21

2

221

222

22 ===

+=

+−

+=ζ

ζζ

ζ

(3) r = 1 ) O %���

00968.01291.02

0025.0

2)2()1( 222

2

=×

==+−

=ζζY

rr

rYz

01.0=X , 025.0=Y , YXZ −≠= 00968.0 → x, y, z� �%2 On

♦ �� ����� � �� ��

(Response of a damped system under rotating unbalance)

• ������ ���� �� � ��� ���� ���(Unbalance)

- �� �� �� : M

- �� � !"�� �# $%& ω� ��'� ()�� 2/m

→$$ �)* 22ωme � �� M+ ,�

- -�./� 01

- -2 ./� tem ωω sin2

2

⇒ tmekxxcxM ωω sin2=++ ���

])()(Im[)sin()( )(2 φωωωωφω −=−= ti

n

eiHM

metXtx

Where M

kn =ω

)()()()(

2

222

2

ωωω

ωωω

iHM

me

cmk

meX

n

=+−

=

)(tan2

1

ωωφMk

c

−= −

tme ωω sin2

M

xc�kx

x

)()2()1(

2

222

2

ωζ

iHrrr

r

me

MX =+−

=⇒

)12

(tan2

1

r

r

−= − ζφ

• ζ 3� meMX r� 45

1. �6 0�� 78, nωω = 9:�� �6� � ;"+ <='> ?�@.

← A� 9:�� �87 ζ B C � D,

2. ↑ω : 1→meMX , � ;"� E7

3. meMX F 3�,

0=

me

MX

dr

d ⇒

21

1

ζ−=r

G H� 1=r (A�H) IJK� LM

Ex) NO7P -Q(Francis water turbin)

NO7P -Q RST� UV, WX YZ� �[� \@. ] YZ�� ^� A�� ��_ B`�

� YaC !-� C� bJ@. ��� ��� 250kg]C ���� (me)� 5 kg-mm]@. ���c C

#_ d] �e!" fG� 5 mm]@. -Q� 600 – 6000rpm %& gh�� 8�@. ��� i

jk� l�m� C#n� \@C ,#o - \@. -Q pq 8� %&�� ���, C#_� rs

t&u k 2e+ v#'w. �� E7ox'@C ,#@.

sol> 0≈c ]� yz� �e !" F �6�

)1()( 2

2

2

2

rk

me

Mk

meX

−=

−= ω

ωω

sradrpm /20600 π= , sradrpm /2006000 π=

sradkk

m

kn /0625.0

250===ω

For rpm600=ω

2425

2

2

23

1004.1010

20

]004.0

)20(1[

)20)(105(005.0 π

ππ

ππ ×=⇒

−=

−

×=−

kk

kk

For rpm6000=ω

2627

2

2

23

1004.1010

200

]004.0

)200(1[

)200)(105(005.0 π

ππ

ππ ×=⇒

−=

−

×=−

kk

kk

0 {� YZ��D|

↑=n

rωω

: X(��k ���6) ↓

1>>⇒nω

ω (Needed)

nω⇒ ] 8W} 'C, ~ m

kn =ω �� k, 8W} @.

)/(1004.10 24 mNk π×=⇒ � ��

mmdmE

kld

l

EIk 127.0106.2

3

643 443

43

=⇒×==⇒= −

π

���� �� ��� ��

♦ �� �� �� ��

• ��� �� ��� : �� ��� ��� ���� ���� �

(using Fourier !�)

→ "# �� : ���� $ ���� ��% �� & '�. ( by Superposition )

♦ ( �� (non-periodic force) �� ��

1. Fourier ���� ���% )*+, -.

2. Convolution ��% /0�, -.

3. Laplace 12% /0�, -.

4. F(t)3 �45 678�� 9/ : ; �<��� =>

5. ?�-@8% �<��� ��

♦ Convolution Integral% /0�, -.

• ( �� ���� AB :

- ���� C�, ����� 7 DE 1�F ���G �@ �7H �=

� ; IJ

• KL� :

- “MN 7 t�O PQ R S F� T0U V G S% KL�GE 5W.”

- KLX

12 xmxmtFF �� −=∆=

∫∆+

=tt

tdtFF

- C�� 1� YZ KLX f

1lim0

=== ∫∆+

→∆FdtFdtf

tt

tt

- For a finite Fdt as dt → 0,

∞→F ([\�� �], ^_)

• KLX `5 ��

- t=0 YZ KLX% a, 1bcde f�gh– ijk lXe

0=++ kxxcxm ���

+

+= − txx

txetx dd

nd

tn ωωζωωζω sincos)( 00

0

�

sI.C’for

00 x(0)x ,)0( x �� == x

m

k== nn

,2m

c ω

ωζ

2

2n 2m

k 1

−=−=

m

cd ζωω

- YZ KLXG �=m� " lXG @m nB opWq

( or t 0 t -0< �V 0 x x == � )

KLX – ?�X re

0)0()0(1 xmtxmtxmf ��� =−−===

- s���

m

1 x 0)(tx , 0 x )0( 00 ====== ��tx

F(t)

F

t

1=∆tF

t∆

k c

F(t)

m

- Response with unit impulse at t = 0

tm

etgtx d

d

tn

ωω

ζω

sin)()(−

==

- KLX� C�� 1G tuF v�� FEq

)(F sinF

x(t) mF

x0 tgtm

ed

d

tn

==⇒=−

ωω

ζω

�

- v�� 7 t = τ KLX F� �=�Wq

mF

)(tx == τ�

KLXG �=l Vwm x = 0Eq

)F x(t) g(t −=

)(sinF

x(t))(

τωω

τζω

−=⇒−−

tm

ed

d

tn

t

)()(

tg tx

=

d

2

ωπ

τ∆τ

F(t)F

t

~FF =∆τ

0t

)(tx)�g(tF

~−

0

• �� �� ��� ��

v�� x� )(tF

: C�� 1��, �y� KLXz� {.

- 7 τ S F(τ)� ∆τ� MN |7�O e T05Wq t = τ KLX = F(τ)∆τ - 7 t G KLX �5 e� ��, ∆x(t)

)( )F( x(t) τττ −∆=∆ tg

- "#��N " 7 }~ T0�, ��� KLXz �5 �� ��� {

∑ ∆−≅ τττ )()F( x(t) tg

- ∆τ →0

∫ −=t

0)()( x(t) τττ dtgF

∫ −=⇒ −−t

0

)(

d

)(sin)(1

x(t) ττωτω

τζω dteym d

tn��

; Duhamel Integral (or Convolution Integral)

��G ��5 AB� tuq �<�� -. G0

• �s �� ��� ��

ymkzzczm ����� −=++

∫ −−=⇒ −−t

dt

d

dtezytz n

0

)( )(sin)(1

)( ττωω

τζω��

ττ ∆+τ

F(t)

t

τ∆

0

)(τF

• •

< v�� �� �� >

m

k c

y

Ex) ��� T0�, eY ��� ( Step force on a Compacting Machine )

1bcde� �A��, ���� t� �� (a)� �G ��W. �Ti� ��

G �=l V lX m ( mN �i�, mmY, ��m, [l� lX% ��5W)

T0�, SN �� (b) d � ���G eY ���(step force)�� Gn�

� � oW. GV e� ��% '�E.

sol> 0F F(t) = % �mF Duhamel integral% /0.

∫ −= −−t

0

)(

d

0 )(sinF

x(t) ττωω

τζω dtem d

tn

( ) ( )

t

dn

ddnt tte

mn

0

22)(

d

0 )()(sinF

=

−−

+−+−=

τ

τζω

ωζωτωτωζω

ω

−

−−= − )cos(

1

11

F

2

0 φωζ

ζω tek d

tn ,

−=

2

1-

1 tan

ζζφ

[ ] cos-1kF

x(t)

) system undamped ei ( 0

n0 t

if

ω

ζ

=

⋅=→�� (d)

H� ��G (ghe |7��� T05Wq

k2F x 0

max = � @� 1Z� 2�

Ex) Time-Delayed Step Force on a compacting Machine

t� ��� �G d � ���% a% V Z �� (a)� ���� ��% '�E.

≤≤≤

= t tF

tt0 0 F(t)

00

0

) t- t ( t 0→ in x(t) of the previous Example.

{ }

−−

−−= −− φω

ζζω )(cos

1

11

k

F x(t) 0

)(

2

0 0 tte dttn

[ ])(cos1k

F x(t)

system. undampedan

00 tt

if

n −−= ω

F(t)

0

0F

0tt

≤≤≤= t t 0

tt0 F F(t)0

00

Ex) Rectangular Pulse Load on the

�� (a)� ���� 00 tt ≤≤ �OH �@5 S% a% V � ��% '�E.

= —

( ) ( )

−−−

−−

−−

−=⇒ −−− φωζ

φωζ

ζωζω )(cos1

11cos

1

11

k

F x(t) 0

)(

22

0 0 ttete dtt

dt nn

{ }[ ]φωφωζ

ζωζω

−−+−−−

= −−

)(cos)cos(11k

F x(t) 02

0 0 ttete

dt

d

tn

n

,

−=

2

1-

1 tan

ζζφ

(ghe� �Q

[ ]ttt nn ωω cos)(cosk

F x(t) 0

0 −−=

(1) 2

t n0

τ> : �` 1Z� �� �7 � ��

(2) 2

t n0

τ< : �` 1Z� �� �� �7� tt 0> ��

t0

0F

0t

)(tF

nτ nτ2

2 t n

0

τ>

2 t n

0

τ<

0F

)(tF

0F

)(tF

0tt

EX) �A��� 1�, ���% a, ���

�� ?��� ��� �A��� 1�, S �� (b)G T0�, �Q �� (a)

d � ���� ��% '��E.

Sol> By Duhamel’s integral

∫ −= −−t

dt dte n

0

)(

d

)(sinm

F x(t) ττωτ

ωδ τζω

∫∫ −−•−−−= −−−− t

dtt

dt dtedtet nn

0

)(

d0

)(

d

))((sinm

tF - ))((sin)(

m

F ττω

ωδττωτ

ωδ τζωτζω

−−+−=⇒ − ttet d

dn

ndd

n

t

n

n ωωω

ωζωωωζ

ωζδ ζω sincos

22

k

F x(t)

2

222

For an undamped system

[ ]tt nnn

ωωωδ

sink

F x(t) −=

Ex) Blast Load on a Building Frame. ( (ghe )

�[ j�vG t� ��(a)� �G 1bcde (ghe� �A��pW. �� (b)

� �N �$  i� � �, �Ti� �� `5 j�v� ��% '�E.

(a) (b)

≤

≤≤

=⇒

t, 0

t,0 t

-1F )F(

0

00

0

τ

τττ

Sol>

∫ −=t

n dtF0

n

)(sin)(m

1 x(t) ττωτ

ω

(1) Response during 0 t t 0 ≤≤

[ ] )(sincoscossin)1(m

F x(t)

00

2n

0 ∫ −−=t

nnnnn dttt

τωτωωτωωτω

∫∫ ⋅−⋅−=t

nnn

t

nnn dt

tF

dt

t0

0

0

00

0 )(sin)1(cosk

-)(cos)1(sink

F τωτωτωτωτωτω

Note that

⋅+=⋅

⋅+=⋅

∫

∫

τωω

τωττωτωτ

τωω

τωττωτωτ

nn

nnn

nn

nnn

sds

sd

sin1

cos- )(sin

cos1

sin )(cos

−++−−

+−−=∴

tt

tt

ttt

tt

tt

t

ttt

k

nn

nnn

nn

nnnn

ωω

ωωω

ωω

ωωωω

sin1

cos1coscos

1cos

1sinsinsin

F x(t)

00

000

0

+−−=⇒ t

tt

t

t

k nn

n ωω

ω sin1

cos1F

x(t)00

0

x(t)

F(t)

m

2

k

2

k

t

0F

)(tF

0t

(2) Response during 0 t t >

+−−= ∫∫

t

t

t

n ddtt 0

0

0)(sin)1(m

F x(t)

00n

0 τττωτω

[ ]tttttt nnnnn ωωωωω

ωcos)sin(sin)cos1(

k

F x(t) 000

0n

0 −−−=⇒

♦ �� i¡¢£(Response Spectrum)

; “¤@5 ���� `= e� Fc ���(¥N Fc �)� 1� D¦ 1bcd

e� �` ��( �` 1Z, §d, �§d, ¨)� 1�©n% )*+, ��j”.

; 6ª Fc «�, ¬, Fc � DE d

; m�� ry5 ­e G0.

♦ Laplace Transformation.

∫∞ −==

0)( L(x(t) X(s) dttxe st

• Laplace Transform� ��� �� �� ��

1. Derive the eq. of motion

2. Transform each term of the equation, using known I.C’s.

3. Solve for the transformed response of the system

4. Obtain the desired solution (response) by usging inverse Laplace transformations.

• ��� �� Laplace ��

∫∫∞ −∞−∞ − +==

000)(s )(

dt

dxdttxetxedt

dt

dxeL ststst

x(0)- (s) dt

dxL sX=

⇒

∫∞ −=

0 2

2

2

2

dt

xddt

dt

xdeL st

{ } (0)x-x(0)- (s) L 2��� Xsx =

• { } F(s) )(L =xf

• ���� 1.D.O.F �� ���� Laplace ��

{ } { }f(t)L L =++ kxxcxm ���

( ) c)x(0)(ms (0)xm F(s) X(s)L +++=++ ���� kxxcxm

�� �� ���

• �� �� ! (generalized Impedence), Z(s)

0 (0)x 0, For x(t) == �

kcsms ++== 2

X(S)

F(s) Z(s)

• Transfer function or Admittance , Y(s)

kcsms ++

==2

1

F(s)

X(s) Y(s) ,

Z(s)

1 Y(s) =

⇒ "# $%� &' )i( )H(i ωω Yk ⋅= ⇒ )()( X(s) sFsY=

• Response 0 (0)x 0, for x(0) x(t) == �

{ } { }Y(s)F(s) L X(s) L x(t) -1-1 ==

• Response 00 x 0)(tx , x ) 0 tfor x( x(t) �� ====

02202222 2s

1

2s

2s

)2m(s

F(s) X(s) x

sx

ss nnnn

n

nn

�

ωζωωζωζω

ωζω +++

++++

++=

{ }

+++

++++=

22

1-022

1-0

1-1-

2s

1L

2s

2sL)()(L x(t)L

nnnn

n

sx

sxsYsF

ωζωωζωζω

�

∫ ⋅=⋅t

0

1- Theoryn Convolutio : d )-y(t )( } Y(s) )( {L τττfsF

∫ −=t

0 d)(-

d

)d-(tsin e )( m

1 n ττωτ

ωτζω tf

)(cos where, )sin( -1

1

2s

2s L 1

111222n1- ζφφω

ζωζωζω ζω −− =+=

+++

tes d

t

nn

n

tess d

t

nn

n ωωωζω

ζω sin 1

2

1 L

d22

1- −=

++

)sin( )(1

sin )sin( 1

)( )(t

0

0112

0 τωτω

ωω

φωζ

τζωζωζω −+++−

=∴ −−−− ∫ tefm

tex

tex

tx dt

dd

t

dd

t nnn�

T(s)F(s)

X(s)

Ex) Response of a Compacting Machine for Rectangular Pulse Load

Using Laplace transformation

>≤≤

=0

00

t t0

t t 0 )(F

tF

s

eFtFsF

st )1( )}({L )(

00

−−==

022022220

2

1

2

2

)2(

)1( )(

0

xs

xss

s

ssms

eFsX

nnnn

n

nn

st

�

ωζωωζωζω

ωζω +++

++++

++−=

−

++

++

++

+

++

−

++

=

⋅

−

12

12

12

12

12

1

2

2200

2

220

2

220

2

220

0

nn

nn

nn

n

nn

st

n

nn

n

ss

xx

ss

sx

sss

e

m

F

sss

m

F

ωζ

ωωω

ζ

ωζ

ωω

ωζ

ωω

ωζ

ωω

�

{ } { }

{ }

−

−

++

−−

−

+−−

−−

+−

−−=

−−

−−−

)1sin(1

2 1sin

1 -

)(1sin1

1 - 1sin1

1 )(

2

2200

12

2

2

20

102

220

12

2201

texx

tex

tte

m

Ft

e

m

FtxL

ntn

nnn

tn

n

n

t

n

n

t

n

n

n

nn

ζωζ

ωωω

ζφζωζ

ωω

φζωζω

φζωζω

ζωζω

ζωζω

�

[ ])1sin(

1

2)1sin(

1 -

)1sin()1sin(1

)(

2

2

001

2

2

0

12)(

12

22

0 0

texx

tex

tetem

Ftx

nt

n

nn

t

ntt

nt

n

nn

nn

ζωζω

ζωφζωζ

φζωφζωζω

ζωζω

ζωζω

−−

++−−−

+−++−−−

=⇒

−−

−−−

�

For the undamped system

[ ] sinx

cosxcos)(cos

sin)2

sin( - 2

)(sin)2

sin( )(

n

000

0

0002

0

tttttk

F

tx

txtttm

Ftx

nnnn

nn

nnn

n

ωω

ωωω

ωω

πωπωπωω

�

�

++−−=

+−

+−++−=

2 ���� ��� (Two Degree of Freedom Systems)

• 2 ���� System :

“ � �� � � � 2�� ����� ��� ��� ”

• ���� �� � ��� !" :

�� ��� = �� #$% × & #$� �'� �()� %

• 2 ���� System → 2�� *+, � -./

→ 0� 1% 2�

• normal mode ( or principal mode or natural mode )

�2 3 �45 � 6 7�� 0� 1%8� ��9: 1.

“ �� 1; ��9� 0� 1%<= �>? @, 7 1AB;<� �.� CD

; �9EF ; G�� .H ModeI �J. ”

• K�: L� �45 � 6 ��1� MN� 2�� .H OPQ� RSTJ.

• UV WX5< �� YZ WX �� MN� �15N [\ ]^%<= _`

(WX �15� �� 0�1%8� � �a �b @ c1; _` )

7��� 1A; de

• �� ���( general ized Coordinates )

“ �-./; *+E>= & -./< Of ��� gh. ”

• ] ���

“ & �-./; i9 � �9� ��j gh �k lm� ��� Wn

o � �-./ p*+�q =: ���r: s % t�k jf ��� ”

♦ YZ 1� � -./

• �� �\ K�� �C t<= &&� u(�v:Vw #$ 1m N 2m � �v�

.� � ��� )(1 tx , )(2 tx < ex ��

2232122321222

1221212212111

)(k)(c

)(k)(c

Fxkxkxcxcxm

Fxkxkxcxcxm

=++−++−=+++−++

����

����

Vector) (Force Matrix) (Stiffness Matrix) (Damping Matrix) (

F K C M

0

0

2

1

2

1

322

221

2

1

322

221

2

1

2

1

Mass

F

F

x

x

kkk

kkk

x

x

ccc

ccc

x

x

m

m

=

+−

−++

+−

−++

�

�

��

��

FkxxM x c =++⇒ ���

�y�zw y�zw {�zw

• TkkM , c c , M TT ===

• 0c 22 == kif

Eq @ : p*+ -./( 1m N 2m � =: |} ~v9 ��)

, c , kM : e&��

1m

1x

1F

2m

2x

2F11xk

11xc �

23xk

23xc �

)( 122 xxk −)( 122 xxc �� −

@

1k

1c

2k

2c

3k

3c1m 2m

)(1 tx

)(1 tF

)(2 tx

)(2 tF

♦ p���� �� 1 x�

• �� 1 : 0 F F 21 ==

• p��� : 0C C C 321 ===

00

0

2

1

322

221

2

1

2

1 =

+−

−++

∴

x

x

kkk

kkk

x

x

m

m

��

��

1m N 2m � �� 1% �a �� ��& �� WX� �J0 �. 6

)cos(X )( 11 φω += ttx

)cos(X )( 22 φω += ttx

0)cos()cos(0

0

2

1

322

221

2

1

22

21 =+

+−

−+++

−

− φωφωω

ωt

X

X

kkk

kkkt

X

X

m

m

02

1

22322

22

121 =

−+−

−−+⇒

X

X

mkkk

kmkk

ωω

0 X =A

0XX 21 == (no vibration, trivial solution) ⇒ For a non-trivial solution

0kk

kk2

2322

22

121 =−+−

−−+ω

ωmk

km

0))((})(){()(m 223221

2132221

421 =−++++++−⇒ kkkkkmkkmkkm ωω

2

1

21

223221

2

21

132221

21

13222122

21

))((4-

)()(

2

1

)()(

2

1,

−++

+++

+++

=⇒

mm

kkkkk

mm

mkkmkk

mm

mkkmkk

�

ωω

������ ��� 1ω ��� 2ω a�[�@�/�����h�

�

�

�

�

prob. Eigenvalue A Matrix �

: 1% -./

or l+ -./�

�

• 21 and ωω : �� 0�1% ; Eigenvalue.

• 21ω N 2

2 ω < e� � .H OPQ (2)(1) X ,X��

X X X X

X X X X (2)2

(2)1212

(1)2

(1)1211

����������

������

ωω

/�� �;�:, 1%p X

X r ,

X

X(2)1

)2(2

2(1)1

)1(2

1 ==r

)(m-

)(m-

X

X r

)(m-

)(m-

X

X

32222

2

2

21221

(2)1

)2(2

2

32212

2

2

21211

(1)1

)1(2

1

kk

k

k

kk

kk

k

k

kkr

++=++==

++=++==

ωω

ωω

- Mode Vector of the system

X

X

)2(1

)2(1

)2(

)2(1)2(

)1(1

)1(1

)1(

)1(1(1)

=

=

=

=∴

rX

X

X

X

rX

X

X

X

- ��1� x �� �. ( .H Mode or 0� Mode )

� ������������ � , ,

Mode 2nd )cos(cos

)cos( x

Mode 1)cos(

)cos(

)(

)( (t)x

21)2(

2)2(

1

22)2(

1

22)2(

1)2(

2

)2(1)2(

11)1(

11

11)1(

1)1(

2

)1(1(1)

φφ

φωφω

φωφω

XX

trX

tX

x

x

sttXr

tX

tx

tx

=

++

=

=

=

++

=

=

�

�

L�W� >

• &&� #$< 7��� L�W� �� � 2� �h%

• l.� L�W� � i�� Mode ( � =1 , 2 )j 1

0)0(x ,)0( x

0)0(x ,�� ���)0( x

2)(

112

1)(

11

====

=====

tXrt

tXti

i

�

�

• K�� ��� L�W� � 7 Mode� �nE> h� �1(��x).

)cos()cos()(x)( x )( x

)cos()cos()(x)( x )( x

)(x)(x )(

22)2(

1211)1(

11)2(

2)1(

22

22)2(

111)1(

1)2(

1)1(

11

(2)(1)

φωφωφωφω+++=+=

+++=+=⇒

+=

tXrtXrttt

tXtXttt

tttx���

L�W�

)0(x)0(x (0),)0( x

)0(x)0(x , (0))0( x

2222

1111

��

��

========

txt

txt

2)2(

11)1(

11 coscos)0(x φφ XX += , 2)1(

121)1(

111 sinsin)0(x φωφω XX −−=�

2)2(

121)1(

112 coscos)0( x φφ XrXr += , 2)2(

1221)1(

1112 sinsin)0(x φωφω XrXr −−=�

{ } { }[ ]212

1)1(

1

2

1)1(

1)1(

1 sincosX +++=⇒ φφ XX

{ } { } 2

1

21

22122

21212

)0(x)0(xr-)0(x)0(xr

rr

1

++−−

=ω

��

{ } { } 2

1

22

22112

21112

)1(1

)0(x)0(xr)0(x)0(xr-

rr

1X

−++−

=ω

��

−+=

= −−

)]0()0([

)0(x)0(xr-tan

cos

sintan

2121

2121

1)1(

1

1)1(

111 xxrX

X

ωφφφ

�� (5.1)

+−−=

= −−

)]0()0([

)0(x)0(xrtan

cos

sintan

2111

2111

2)2(

1

2)2(

112 xxrX

X

ωφφφ

��

Ex) Frequencies of Spring-Mass Systems

n=1

0kx-2kxx m

0kx-2kxx m

122

211

=+=+

��

��

2 1,i ),cos()( xAssume i =+= φωtXt i

• l+-./ ; 02m-

2m-2

2

=+−

−+kk

kk

ωω

034 2242 =+−⇒ kkmm ωω

[ ]

m

k

m

mkmkkm =

−−=⇒

2

1

2

2

12222

1 2

12164ω

[ ]

m

k

m

mkmkkm 3

2

12164

2

1

2

2

12222

2 =

−+=ω

12m-

2m-

X

X r

12m-

2m-

X

X

22

22

(2)1

)2(2

2

21

21

(1)1

)1(2

1

−=+

=+==

=+

=+==

k

k

k

k

k

k

k

kr

ωω

ωω

• Normal Modes

+

+

==

1)1(

1

1)1(

1(1)

cos

cos

)(x mode 1

φ

φ

tm

kX

tm

kX

tst�

,

+−

+

==

2)2(

1

2)2(

1(2)

3cos

cos

)(x mode 2

φ

φ

tm

kX

tm

kX

tnd�

k

mm =1

mm =2

nk

k

1x

2x

• Interpretation

- �� 1� Mode: 1b @ :

; 7 #$� 1A; [J. (8C Spring�  ;� ¡� �.)

; 1m N 2m � �� ���� 0°

- �� 1� Mode: 1b @ :

; 7 #$� {� ¢�� �. 0 -}\ �e (��� 180°); Node point at the center of the middle spring (Of �C t< ex £¤;9 ��J.)

• �� �< G� ��x

)3

cos()cos( )( x

)3

cos()cos( )( x

2)2(

11)1(

12

2)2(

11)1(

11

φφ

φφ

+−+=

+++=

tm

kXt

m

kXt

tm

kXt

m

kXt

Ex) Initial Condition to Excite Special Mode

(1) 1� .H OPQa

(2) 2� .H OPQjr: 1 ��� L�W�(¥� ¦Z� §¨< ex)

Sol>

K�� L�W�< e� #$� �. system) for the 1 ,1( 21 −== rr

)3

cos()cos( )( x

)3

cos()cos( )( x

2)2(

11)1(

12

2)2(

11)1(

11

φφ

φφ

+−+=

+++=

tm

kXt

m

kXt

tm

kXt

m

kXt

K�� L�W�< e � ) 1 1 with 5.1 ( 21 −== randr�

[ ] [ ] 2

1

221

221

)1(1 )0()0()0()0(

2

1X

+++−= xx

m

kxx ��

[ ] [ ] 2

1

221

221

)2(1 )0()0(

3)0()0(

2

1X

−++−−= xx

m

kxx ��

[ ]

−+= −

)]0()0([k

)0(x)0(xm-tan

21

2111

xx

��φ ,

[ ]

+−+= −

)]0()0([3k

)0(x)0(xmtan

21

2112

xx

��φ

(1) �� 1� .H Mode

)0(x)0(x & )0(x)0( x 0

cos

cos

)(x

2121)2(

1

1)1(

1

1)1(

1(1)

��

�

==→=⇒

+

+

=

X

tm

kX

tm

kX

t

φ

φ

(2) �� 2� .H Mode

)0(x)0(x & )0(x)0( x 0

3cos

3cos

)(x

2121)1(

1

2)2(

1

2)2(

1(2)

��

�

−=−=→=⇒

+−

+

=

X

tm

kX

tm

kX

t

φ

φ

♦ ����

21223t222

12212t111

)(k

)(k

ttt

ttt

MkkJ

MkkJ

=−++

=−++

θθθθθθ

��

��

♦Coordinate Coupling and Principal Coordinates

• n d.o.f system → n coordinates to solve the motion of system.

• �� ��� �� �� ���� ���� ��

• ��� : ���� ��� �� ��� � .

• Example

- �!� "# $%# ��

; &'( )*+ ,-� .� /�0 $1

; 12"3 45"( 67)*80 "�

; &'� 9%� Spring80 ��

1θ

11θtk)( 122 θθ −tk

2θ

23θtk

1l 2l1k 2k

A BG01Jm

1tM

1tk 2tk 3tk

1J 1J

2tM

1θ 2θ

; �� :;( � �<# ���(=>?@A @B�<)

CDC ),( 21 xx EC�!CFG �CH9%CIJ

CKC ) ,( θx ECLMN �COJCP 3CQRS

CTC ) ,( 11 θx EC9%CF �CIJC 1x +CQRC 1θCUC ) ,( θy ECLMN �CVW80CXYCZ �C �#C[C\ �CIJC )(ty 3CQR )(tθ

• �� ]-� ^_

- (x, S)���( ;B# �� ` a

0)()(k

0)()(k

222111100

2221

=++−−⇒=

=++−+⇒=

∑∑

llxkllxJJM

lxklxxmxmF

θθθθ

θθ����

����

=

+−−−−+

+

⇒

0

0

)()(

)()(

0

02

222

112211

221121

0 θθx

lklklklk

lklkkkx

J

m��

��

k 2211 lklif = : bc

k 2211 lklif ≠ : �! AB� d+��e f�+ QR� g-��

(h, ij �kl�� QR+ �m ij ��)

: elastic coupling (n- ]-) o� static coupling

- (y, S)���( ;B# �� ` a

θx

)(ty

θ)(tθ

θ′− 1ly θ′+ 22 ly

A B

eGC ⋅

GC ⋅P

A′

B′′

1l′

2l

)( 11 θ′− lyk )( 22 θ′+ lyk

f�+ QR� "# ��` a :

��������

0

0)()(

l)l(yl)l-(yk

)l(y)l-(yk

222

211221

112221

222111

2211

=

′+′′+′−′−′+

+

−′′+−′′=

−′+−′−=

θθ

θθθ

θθθ

y

lklklklk

lklkkky

Jme

mem

ymekJ

mekym

p

p

��

��

����

����

, k 2211′=′⇒ lklif �� �p ]-(dynamic coupling) or ,-(inertial) ]-q� �J.

CCCCr �!@ y`s80 l���� #tk ��� 7u74� vB�� ,-u ym ��

CCCC � wxy eym �� z{� S`s80 QR

|}��0 S`s80� ��e ~ �me 80 � y`s80 ��

• 2 ���� ���� �-

1. �� ` a([- ��( .� ����)

=

+

+

0

0

2

1

2221

1211

1

1

2221

1211

2

1

2221

1211

x

x

kk

kk

x

x

cc

cc

x

x

mm

mm

�

�

��

��

- /- �� ≠ D : n- ]- or p ]-

- �� �� ≠ D : �� ]- or �� ]-

- )* �� ≠ D : )* ]- or ,- ]-

2. ���� ��� ,��@ ��� ���-"0 ��

: ���� �Y# �0 ��

3. ]-� �)e ��� ��� �� ��, �� �� �-@ ��.

- 1��� o� �� ��� (pricipal Coord. Or natural Coord.)

: p@A �p80 ]-�� �� �� ` a ( decoupled motion)� 1�

��� 21 & qq

- 1���( ;B�k �]- �� ` a� �� i ��, @� ` a� bc

p80 � i �� �[@ ��.

Ex ) Spring – )*�� 1���.

Sol>

)3

cos()cos()(

)3

cos()cos()(

2)2(

11)1(

12

2)2(

11)1(

11

φφ

φφ

+−+=

+++=

tm

kXt

m

kXtx

tm

kXt

m

kXtx

bc� ���( ),( 21 qq � ��� 21 & qq � ��(

)3

cos()(

)cos()(

2)2(

12

1)1(

11

φ

φ

+=

+=

tm

kXtq

tm

kXtq

 �k "¡�� ��` ae

03

0

22

11

=

+

=

+

qm

kq

qm

kq

��

��

03

0

0

10

01

2

1

2

1 =

+

⇒

q

q

m

km

k

q

q

��

��

21 & qq : 1��� ← p@A �p80 ]-�� ��

)()()( 211 tqtqtx += , )()()( 211 tqtqtx −=

[ ] [ ]

)()(2

1)(q ,)()(

2

1)(q 212211 txtxttxtxt +=+=∴

k

mm =1

mm =2

nk

k

1x

2x

Ex) ��¢� £��(pitch)��i, l� �� �� ��i ¤ QR 74� �

(node point)( d ��.

Sol> Derive the motion of eq. for (x, θ) Coord.

θθθ

���

����

022112

222

110

221121

0)()(

0)()(

JMxlklklklkJ

xmFlklkxkkxm

=⇐=+−+++

=⇐=+−+++

∑∑

����� "# ¥¦(

)cos()(

)cos()(

θωθθω+Θ=+=

tt

tXtx0 �

=

Θ

++−+−

+−++−0

0

)()(

)()(222

211

202211

2211212 X

lklkJlklk

lklkkkm

ωω

Substituting known data into the above eq.

=

Θ

+−

+−0

0

)67500810(15000

15000)400001000(2

2 X

ωω

⇒ �- ` a

sradsrad /4341.9 ,/8593.5

0750.249991.8

21

24

==⇒=+−ωω

ωω

fk

θ

GC ⋅

2l1l

�§�

x

rk

mkNk

mkNkmlml

1.1m r kgm

r

f

/22

/185.10.1

1000

2

1

====

==

QR!¨

.30610X -2.6461X

3061.0 , 6461.2

(2)(2)(1)(1)

)2(

)2(

)1(

)1(

Θ=Θ=

=Θ

−=Θ

XX

)1(Θ−GC ⋅

)1(X

6461.2

)2(ΘGC ⋅

306.0

♦�� �� �� ( Forced Vibration Analysis )

• �� �� �� 2 ����� �� ���

=

+

+

2

1

2

1

2221

1211

2

1

2221

1211

2

1

2221

1211

F

F

x

x

kk

kk

x

x

cc

cc

x

x

mm

mm

�

�

��

��

and a harmonic forcing function

2 1,j ,)( 0 == tijj eFtF ω , ��� ω� �� ���

Assume the steady state solution )(tx j be a harmonic

2 1, j , )( == tijj eXtx ω

���� 1X � 2X � ω �� parameter! �"

=

++−++−++−++−

20

10

2

1

2222122

1212122

1212122

1111112

)()(

)()(

F

F

X

X

kcimkcim

kcimkcim

ωωωωωωωω

define the mechanical impedence )( ωiZrs as

1,2 s r, , )( 2 =++−= rsrsrsrs kcimiZ ωωω

Then [ ] 0)( FXiZ��

=ω (#$

where,

[ ]

=

=

=

20

100

2

1

2212

1211

F

FF

X

XX

Impedence of :)()(

)()()(

�

�

�

MatrixiZiZ

iZiZXiZ

ωωωω

ω

Solve (#$%& [ ] 01)( FiZX�� −= ω

Where [ ]

−

−−

=−

)()(

)()(

)()()(

1)(

1112

1222

2122211

1

ωωωω

ωωωω

iZiZ

iZiZ

iZiZiZiZ

)()()(

)()()(

)()()(

)()()(

2122211

201110122

2122211

201210221

ωωωωωω

ωωωωωω

iZiZiZ

FiZFiZiX

iZiZiZ

FiZFiZiX

−+−=

−−=⇒

Ex) Steady-State Response of Spring – Mass System

'( 1m ) tFF ωcos101 = � *�+ ,

�� �--. /0 123.

4 ��� /056 783

Sol>

�� ���

=

−

−+

0

cos

2

2

0

0 0

2

1

2

1 tF

x

x

kk

kk

x

x

m

m ω��

��

Assume the solution , 2 1, j , cos)( == tXtx jj ω

k- )( , 2)()( 122

2211 =+== ωωωω ZkmZZ

))(3(

)2(

)2(

)2()(X

2210

2

22210

2

1 kmkm

Fkm

kkm

Fkm

+−+−+−

=−+−

+−=

ωωω

ωωω

))(3()2(

)(X22

10222

102 kmkm

kF

kkm

kF

+−+−=

−+−=

ωωωω

9� ��� 3

, 21 m

k

m

k == ωω � ��

m

k

m

k

k

)(1 tx

)(2 tx

tFtF ωcos)( 101 =

−

−

−

=2

1

2

1

2

1

2

10

2

1

1

1

2

)(X

ωω

ωω

ωω

ωω

ω

k

F

−

−

=

2

1

2

1

2

1

2

102

1

)(X

ωω

ωω

ωω

ω

k

F

• 2:� ;� <=

• >?@ '( 1m � ��A 0�� BC2� DEF ���G H"I

J Apply to design a dynamic Vibration absorber.

1x � �K LA� MF N�� OPQ.(� R�S ��! �T /P)

0 13

2

2 3 4

10

1

F

kX

1ωω1ω 2ω

•

0 13

1

2 3

10

2

F

kX

1ωω1ω 2ω

••

2m

)1m(��

2k

1k

)(1 txtF ωsin0

)(2 tx

�U��

♦ Semi-define System • �V��(Semidefinite System): W1X(unrestrained system) or YZ�(degenerate system)

0)( 2111 =−+ xxkxm ��

0)( 1222 =−+ xxkxm ��

<Z��A3 *�

2 1, j , )cos()( 1 =+= φωtXtx jj

0)(-m 212

1 =−+ kXXkω 2

221 )(-m XkkX ++− ω

[ ] 0)( 212

212 =+−⇒ mmkmm ωω

21

2121

)( and 0

mm

mmk +==⇒ ωω

• �F�� : “9����[� 2\* 0� G ]� �”

&&&&•&H=0 J ^ '(OA! _` -.��A ab.( Rigid Body Motion)

• 2ω ! )1(1X )1(

2X � M-A �c : def [g! hiA "I

1m 2mk

)(1 tx )(2 tx