engineering vibration [in korean]
TRANSCRIPT
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 ==
♦ ��, (harmonic analysis)- (&¡ -�J �� ����� ¢£ ���� ¤¡m ��T; 8�C � ¥7 ������.
- �� ��J : ¤¡ }[ -]� -��) �W¦ �Y ¤¡+ ($K K§ �W 0P ¨s;
��� ���.
- ����J ©Z+ ��(Fourier series)+ , �ªDT; 89«�.
• ©Z+ ��=~(Fourier series expansion)
c� )(tx � �� τ ����� $1
∑∞
=++=
1
0 )sincos(2
)(n
nn tnbtnaa
tx ωω
K�., τπω 2= : �¬-��
∫∫ ==τω
π
τπω
0
2
00 )(2
)( dttxdttxa
∫∫ ==τω
πω
τω
πω
0
2
0cos)(
2cos)( tdtntxtdtntxan
∫∫ ==τω
πω
τω
πω
0
2
0sin)(
2sin)( tdtntxtdtntxbn
c� n=1 : �¬
n=2,3,… : ��
∑∞
=
−+=1
0 )cos()(n
nn tnCCtx φω
K�., 20
0aC = , 2
122 )( nnn baC += , n
nn a
b1tan −=φ
♣ �� v®¯°
±²*� : time , ³/*� : x(t)
♣ �l� v®¯°
±²*� : Frequency , ³/*� : φ,, nnn Corba
��s�
(-] � s�)
♦ ´�� ! µ��• ´��(Even function) : )()( txtx =−
∑∞
=+=⇒
1
0 cos2
)(n
n tnaa
tx ω
• µ��(Odd function) : )()( txtx −=−
∑∞
=
=⇒1
sin)(n
n tnbtx ω
♦ �Hi =~(Half range expansion)
♣ �Hi =~2¶m 4 �· �T;� �� 0 +. τ VW �7 x(t)) ¸7e L¹«�.
♦s� �6s�• x(t)� s�) º$� 0 D¡SQ; 8�� ¹�$W p� »¼ �6s�T; Coefficient) º�.
•, � : Simpson½¶
1. �¬�� ¾ → 0� π2 �(1 cycle)T; ¾
2. =��(τ )+ b¿ ´�~ À��T; NÀ¡
�� f$7 s� _%�7 M
nM 4≥ 3. (Á� x(t)) s�
4. ∑=
=N
iix
Na
10
2
τπ i
N
iin
tnx
Na
2cos
2
1∑
=
= , τπ i
N
iin
tnx
Nb
2sin
2
1∑
=
=
τt
x(t)
Original Function
τt
x(t)
τ−
µ��; =~
τt
x(t)
τ−
´��; =~
��� 1.2������ ÂÃ
)sin()(2
202
0 φωω
ω ++= tv
tx
velocityinitalvv
x =
= −
00
01tanωφ
Periodωπ2
_( /�
Phase=φ
x0
t�
*0
*0, x(t)
-]
A
��� 1.3
������+. *0, /�, �/� ¨s
Velocity
)cos()( φωω += tAtx�
t
t
t
Acceleration
)sin()( 2 φωω +−= tAtx��
A2ω
A2ω−
0
0
Aω
Aω−
0
−
Displacement
)sin()( φω += tAtx
♦ �� ���(Nomograph): ��� �� �� ��� � ��, ��, ���� �� ���� � !".
�# 1.9 $%&' ��� () *+ �����(nomograph)
- �,-� : ./' f � 0'1.
,-� : ��� 0'1
232� +1% 4$� 5 : �� 5
232� -1% 4$� 5 : 67 ��� 5
- 8 ��� 9:5; �< () =0 ��, ��, ���� ��� >? @�A ' B�C 8D
��� �����(nomograph)E *F.
- ��' ω , G�H IJ� K7LM �� ��� NOPQ G�H IJ� K71�� NO8 R
2 ST� UVW�; X� YZ [\� 1]^�+� 0_�� ^)&.
`a 1.2.2
�# 1.7� ���� .Y� 22� �� () b�� 2-8 Hz8F. �����c� 2-8 Hz� _d
e� 'f5; ��eg ��, ��, ���� N+� h@e? iF. 2-8 Hz� ./' b���
je� =0 ���� 1g(9.8m/s2 ), ek ��� 400mm/s, *7A S, lm ��� 30mm, *
7PQ � nH ��W�o �# 1.9� �* 4+5� �_ ��i pq r� *7iF.
`a 1.2.3
�stu vs8w� IJ mNk /400= � ��x� �_ yyP� 1kg� G�z, �{xe|
F. 8}8 ��A S =0��� 2.4mm 8F. vs8w� ~% t��� ST� G�o 1.4kg z
, ��ek =0��� 3.4mm8F. 8D }o �# 1.7� ������ BY� Y� pq� _
de��?
_) lm��� vs8w� �� 2.4mm , sradmk /201 ==ω , Hzrad
sradf 18.3
2
/201 ==
π8
k .Y� o mmttx )20(sin4.2)( = 8F. ���, ��M ���� ��
smmtttx /)20(cos48)20)(cos4.2(20)( ==�
22 /)20(sin960)20)(sin4.2()20()( smmtttx −=−=��
8F. 8}o ./' 3.18 Hz, ���� 2.4mm, �� �� 48mm, ��� �� 960mm� 5�, 7
�iF.
8 �stu ��� F� lm�o v s8w� ~% t���� �_ G�8 1.4kgz, ��
S 6Y�F. 8 4�� ./'� sradmk /9.162 ==ω , Hzrad
sradf 69.2
2/9.16
2 ==π
8k
��i G�� �_ =0 ��� 3.4mm� Pk ��, ��, ����
mmttx )9.16(sin4.3)( =smmtttx /)9.16(cos46.57)9.16)(cos4.3(9.16)( ==�
22 /)9.16(sin074.971)9.16)(sin4.3()9.16()( smmtttx −=−=��
8F. 8}o ./' 2.69 Hz, ���� 3.4mm, �� �� 57.46mm, ��� �� 971.074mm� 5
�, 7�iF.
�� � �� ��� 5�� �_ �� �% pqo ./'� 2.69�� 3.18 Hz�y� b��
� ��� ��; ��ek BF. 8}8 ��c\� t���; ~y �o c\�� t���;
�� �o c\, �&� �� vs8w� ��b�� _d*F. ��� �# 1.9� ���� 7
�i pqo �� �k �2� Y<WC 8D 4��� �� �* ��� �# 1.10� �� �
��� ��eF.
�# 1.10 `a 1.2.3 �� h@* s8w� ��b�� ���� �����
• 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�
Steady space representatin of phase plane ),( xx � plane
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
���� ��, ���, ��\ �8�+ 1��> /01-��.: {�1 l a �s. �
��� ��� � 300kg9: �5��` ��a �rad/s�s. ��, ���, ��\� �[
� � �R~?
sol ) ��a, ��, �[ �� �. mk=ω �s. �d�
mNsradkgmk /2960)/)(300( 22 === πω
φ0xω
0v
20
20
2 vxw +
�>� 1.4
jk��� � � ��
0� +� ��jt= � 0=+ kxxm �� � � � �� ��N 4K9: � a �s.
��:, �b
1)( 21 −==+= − jmkeaeatx tjtj ωωω
: �a � , w�� a1D a2 ¡¢a Ua�s. £�:, �b
)sin()( φω += tAtx
: �a � , w�� A� ¤ ¥a�� Ua�s. ¦�:, �b
tAtAtx ωω sincos)( 21 +=
: � a � , w�� A1D A2 ¥a�= � Ua�s. §¨� Uaf� ��jt �
� ©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
ω
³>®¯� °n�� Aω =1320mm/s�², ³> ®¯� °n�� A2ω =174.24*10-3mm/s2�s.
µN ��³> 0v =0�� &' !U� 2)0(tan 01 πωφ == − x �s. �d� � .� ®¯�
)132cos(10)2132sin(10)( tttx =+= π �s.
kgm 3102.49 −×=
mNk 8.857=
♦ ��� ����� ��- � � ��� ��� ���� �� �� �� �� � � � !�� ��� �"# �$.
- % � !�& ' ()* +�, ,- ./01 23 45� ���� �� �6�7� 8 91 :
;� <=> ./0� �� ?@A$.
• � B" ��C -> � �� $D B� E FG H3 ./0 J0� -I" JK �1 L:M �
K��. � ��N� �� -I� OP'� Q# �R ->�� �� ��� "ST*4U
VW� X 2YZ�[
000 =+⇒−= θθθθ tt kJkJ ����
00 2
1
J
kf
J
k tn
tn π
ω =⇒=
• \]^+R
- � NR" ->" _`R �� ab� Spring constantM cd
- -I� EH3 ./0 (FG)
g
wDD
g
wDhDhDmrJ
842
1
324)
4(
2
1
2
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)
�i* jk@ lL" 0.5cm"� m"� 2mn �� B� EH3 ./0 J=0.5kg m2n -I"
B� oJK �� � ��pM q�r#. N, �i� PN�p1 G=8*1010 N/m2"$.
sol)
( ) ( )[ ]( )( ) ( )22
2
422102 /9087.4
5.02
32/105.0/108srad
mkgm
mmN
lJ
GJ
J
k p =⋅××===
−πω
s#[ ��p1 srad /2156.2=ω "$.
hX) Natural Frequency of a wind Turbine
�� 2.14�[ t1uv w" p=�� x; Uy� *U1 z{� |{M �$. ''� FG&
m"� m"1 l"$. �}� �~& m"� l2"� lL" dn ���� gg�@ �$. " ��
��� �pM ���#.
�@ �� : *UM � p=� x; Uy" �� gg�@ �$.
���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
ππω
θθ
===
=+
hX ) ,��� ��p
���[ ��gg�@ �1 � 1 � �;� ��r ���� ��T* ��� C$. "v
w& �M ,��#� �$. " �� ��pM q�r#.
�@ �� : �;��[ ��� OM ��T* ��1 �
q�1 � :'� (angulat oscillation)� ��p
���Y : 1 �� ����* "a�
�">
() Q� ��[ � � �� w� �� ,- Torque1 wdsinQ"$.
0sin0 =+ θθ wdJ ��
�&' Q����� θθ ≅sin "$�
00 =+ θθ wdJ ��
md
Jl
l
g
J
mgd
J
wdeq
eqn
0
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
2220 dkk G +=
OAdGAGAOGdd
kleq G =+=+=
+=
2
OA
gn =ω : % � �� gg�" A 91 On~� H��" ��p� ?�
A� : ����(Center of percussion)
1. ¡ : ¢�£M OP ��, ¤}4¥ ����
2. ¦q �" : ����T* §� 6� => l' �¨T* �;" �©.
3. Izod �� \ª : �� 6�|" �� �� �«� �© => �� ¬8 (> �g
4. ���[� ����.
Q m
l
gn =⇒ ω
hX )
p&�� ®k� ��p� 600rpm� ¯°��[ ±©�� ²; (� �³pt$ 3.5´ µ
� �� )� ²;H� ¶±m"1 ·¸n�?
²;(� �³p = (�¹p*¯°�)/2
u = aº� »�� )¡�¼g
+ ��� »�� )¡�¼g
= 2
2)(
2)( xA
xAx
xAx γγγ =+
A : p&�� NR½
γ : p&� ��
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
10 == π
�p1»�¾;��
)(243.09
81.929
22
mll
g =×=⇒=
��Àx
l
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
2
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
_Ä ��� %ÂÃ1 Å"� 300ft"� 8ft� ¿L 10ft� ÆL� �g1 ÇÈ> NR� �
ÉÃ}0* jk@Ê �$. %� ËÌ� ÂÃ� ��� 6*105 lb "$. %ÂÃ� Í�� �
�pM q�#.
�@ �� > %ÂÃ� �� 2.10
���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
\��*�����q��<0) ��� ¡¢�£ ��¤¥ ������¢¦�¤�§¨� ¦©�¢¤ª¢ ��«L��*�¬�01��
n��®��S�������¯#�����)�r��Y°��¯#�H�±²Z���³�3�
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
−=−==
dt
dxccvFv
dt
dw���¶·�0µA@�4�0�hu�"¸
tXx dωsin= u�@#� ����S�¹3�"¸�)�0µA)
( )∫∫ ⋅=
=∆
=
πωπ
ωωω2
0
222
0
2
cos tdtcXdtdt
dxcw dddt
d
22
0
2
2
2cos1Xctd
tXc dd ωπω
π=′′+= ∫
º»�"¸�0µA)�!���Y*�¼½0�(¾���S������n��Y0�2�H��"¸�0µA
w∆ )� dω *�¿��3�
\���·� �·���������À�<�
�Á�JV�!��u��@#� �� tXtx dωsin)( =
( )
( ) 22
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
xckxF �−−=⇒
�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
���� :η ȱ���®É�¥ §¦ © ¹3�"¸�)�0µAÊ�Â;�j7�0µA*�(°
•�������(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
fg
S���JK�»���ºÛÙÙÑÆ����ѺÉÙÙÙÙ�®ÜÝ�°
��������������S��x�����&������ �·������
��������������Þ�¬z���Ù�ß�
³À�c»��É°����*�&�����������°���*�I��Þ�����
�����������°�����I��89�Fà�89�Ù�É� 0�áD�â)|�ãz)�4��
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