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Forced Vibration of a SDOF System
S. Ilanko
2007
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Simple Harmonic Excitation
m
State 2 (at time t during vibration)
State (e!uilibrium" springs unstressed)
k T
c T
F sin t
m
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Simple Harmonic Excitation
m
State 2 (at time t during vibration)
State (e!uilibrium" springs unstressed)
k T
c T
F sin t
m
T
T mF sin
t
u
FREEBODY DIAGRAM
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Simple Harmonic Excitation
u
m
State 2 (at time t during vibration)
State (e!uilibrium" springs unstressed)
k T
c T
F sin t
m
T
T mF sin
t
umtFTT =+ sin
Newtons Second Law
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Simple Harmonic Excitation
u
m
State 2 (at time t during vibration)
State (e!uilibrium" springs unstressed)
k T
c T
F sin t
m
T
T mF sin
t
umtFTT =+ sin
umtFucku =+ sin
Constitutive equations
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Simple Harmonic Excitation
u
m
State 2 (at time t during vibration)
State (e!uilibrium" springs unstressed)
k T
c T
F sin t
m
T
T mF sin
t
umtFTT =+ sin
umtFucku =+ sin
tFumucku =++ sin
Rearrane!"
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Simple Harmonic Excitation
m
State 2 (at time t during vibration)
State (e!uilibrium" springs unstressed)
k T
c T
F sin t
m
tFumucku =++ sin
Now it is #at$s!"
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Simple Harmonic Excitation
m
State 2 (at time t during vibration)
State (e!uilibrium" springs unstressed)
k T
c T
F sin t
m
tFumucku =++ sin
Try combination of sin tand cos t for u
tBtAu += cossin
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Simple Harmonic Excitation
m
State 2 (at time t during vibration)
State (e!uilibrium" springs unstressed)
k T
c T
F sin t
m
tFumucku =++ sin
Calculus
tBtAu += cossin
tBtAu = sincos
tBtAu = cossin 22
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Simple Harmonic Excitation
m
State 2 (at time t during vibration)
State (e!uilibrium" springs unstressed)
k T
c T
F sin t
m
tFumucku =++ sin
Substitute
tBtAu += cossin
tBtAu = sincos
tBtAu = cossin 22
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Simple Harmonic Excitation
m
State 2 (at time t during vibration)
State (e!uilibrium" springs unstressed)
k T
c T
F sin t
m
tFumucku =++ sintBtAu += cossin
tBtAu = sincos
tBtAu = cossin 22
k (A sin t + B cos t)+ c (A cos t - B sin t)
m 2(A sin t + B cos t) = Fsin t
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Simple Harmonic Excitation
m
State 2 (at time t during vibration)
State (e!uilibrium" springs unstressed)
k T
c T
F sin t
m
tFumucku =++ sin Group sin, cos terms
tBtAu += cossin
tBtAu = sincos
tBtAu = cossin 22
k (A sin t + B cos t)+ c (A cos t - B sin t)
m 2(A sin t + B cos t) = Fsin t
(kA-c m 2A - F)sin t +(k B+c m 2B)cos t = 0
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Simple Harmonic Excitation
m
State 2 (at time t during vibration)
State (e!uilibrium" springs unstressed)
k T
c T
F sin t
m
tFumucku =++ sin Let us mae somespace!!tBtAu += cossin
tBtAu = sincos
tBtAu = cossin 22
k (A sin t + B cos t)+ c (A cos t - B sin t)
m 2(A sin t + B cos t) = Fsin t
(kA-c m 2A - F)sin t +(k B+c m 2B)cos t = 0
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Simple Harmonic Excitation
m
State 2 (at time t during vibration)
State (e!uilibrium" springs unstressed)
k T
c T
F sin t
m
tFumucku =++ sin Let us mae somespace!!tBtAu += cossin
tBtAu = cossin 22
k (A sin t + B cos t)+ c (A cos t - B sin t)
m 2(A sin t + B cos t) = Fsin t
(kA-c m 2A - F)sin t +(k B+c m 2B)cos t = 0
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Simple Harmonic Excitation
m
State 2 (at time t during vibration)
State (e!uilibrium" springs unstressed)
k T
c T
F sin t
m
tFumucku =++ sin Let us mae somespace!!tBtAu += cossin
k (A sin t + B cos t)+ c (A cos t - B sin t)
m 2(A sin t + B cos t) = Fsin t
(kA-c m 2A - F)sin t +(k B+c m 2B)cos t = 0
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Simple Harmonic Excitation
m
State 2 (at time t during vibration)
State (e!uilibrium" springs unstressed)
k T
c T
F sin t
m
tFumucku =++ sin T"is is true for anytime ttBtAu += cossin
(kA-c m 2A - F)sin t +(k B+c m 2B)cos t = 0
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Simple Harmonic Excitation
m
State 2 (at time t during vibration)
State (e!uilibrium" springs unstressed)
k T
c T
F sin t
m
tFumucku =++ sin T"is is true for anytime ttBtAu += cossin
(kA-c m 2A - F)sin t +(k B+c m 2B)cos t = 0
T"erefore, bot" t"ese
terms must #anis"$
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Simple Harmonic Excitation
m
State 2 (at time t during vibration)
State (e!uilibrium" springs unstressed)
k T
c T
F sin t
m
tFumucku =++ sintBtAu += cossin
(kA-c m 2A - F)sin t +(k B+c m 2B)cos t = 0
Consider t"is first
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Simple Harmonic Excitation
m
State 2 (at time t during vibration)
State (e!uilibrium" springs unstressed)
k T
c T
F sin t
m
tFumucku =++ sintBtAu += cossin
(kA-c m 2A - F)sin t +(k B+c m 2B)cos t = 0
(k B+c m 2B)= % &i#es)( 2
=
mk
AcB
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Simple Harmonic Excitation
m
State 2 (at time t during vibration)
State (e!uilibrium" springs unstressed)
k T
c T
F sin t
m
tFumucku =++ sintBtAu += cossin
(kA-c m 2A - F)sin t +(k B+c m 2B)cos t = 0
(k A-c m 2A)= ' and)( 2
=
mk
AcB &i#e
FAm
mk
AckA =
+
2
2
2
)(
)(
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Simple Harmonic Excitation
m
State 2 (at time t during vibration)
State (e!uilibrium" springs unstressed)
k T
c T
F sin t
m
tFumucku =++ sintBtAu += cossin
(kA-c m 2A - F)sin t +(k B+c m 2B)cos t = 0
(k A-c m 2A)= ' and)( 2
=
mk
AcB &i#e
FAm
mk
AckA =
+
2
2
2
)(
)(
222
2
)()()(+
=cmk
mkFA
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Simple Harmonic Excitation
m
State 2 (at time t during vibration)
State (e!uilibrium" springs unstressed)
k T
c T
F sin t
m
tFumucku =++ sintBtAu += cossin
)( 2
=
mk
AcB
FAm
mk
AckA =
+
2
2
2
)(
)(
222
2
)()()(+
=cmk
mkFA
222 )()( +
=
cmk
FcB
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Simple Harmonic Excitation
m
State 2 (at time t during vibration)
State (e!uilibrium" springs unstressed)
k T
c T
F sin t
m
tFumucku =++ sintBtAu += cossin
222
2
)()()(+
=cmk
mkFA
222 )()( +
=
cmk
FcB
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Simple Harmonic Excitation
m
State 2 (at time t during vibration)
State (e!uilibrium" springs unstressed)
k T
c T
F sin t
m
tFumucku =++ sintBtAu += cossin
222
2
)()(
)(
+
=
cmk
mkFA222 )()( +
=
cmk
FcB
Substitute
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Simple Harmonic Excitation
m
State 2 (at time t during vibration)
State (e!uilibrium" springs unstressed)
k T
c T
F sin t
m
tFumucku =++ sintBtAu += cossin
222
2
)()(
)(
+
=
cmk
mkFA222 )()( +
=
cmk
FcB
( )222
2
)()(
cossin)(
+
=
cmk
tctmkFu
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Simple Harmonic Excitation
m
State 2 (at time t during vibration)
State (e!uilibrium" springs unstressed)
k T
c T
F sin t
m
tFumucku =++ sintBtAu += cossin
( )
( )22222
)()(
cossin)(
+
=
cmk
tctmkFu
#ust re$rite%
( )222
2
)()(
cossin)(
+
=
cmk
tctmkFu
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Simple Harmonic Excitation
m
State 2 (at time t during vibration)
State (e!uilibrium" springs unstressed)
k T
c T
F sin t
m
tFumucku =++ sintBtAu += cossin
&gain re$rite%
( )222
2
222
)()(
cossin)(
)()( +
+
=
cmk
tctmk
cmk
Fu
( )
( )22222
)()(
cossin)(
+
=
cmk
tctmkFu
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Simple Harmonic Excitation
m
State 2 (at time t during vibration)
State (e!uilibrium" springs unstressed)
k T
c T
F sin t
m
tFumucku =++ sintBtAu += cossin
'o$ te tricy*
( )222
2
222
)()(
cossin)(
)()( +
+
=
cmk
tctmk
cmk
Fu
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Simple Harmonic Excitation
m
State 2 (at time t during vibration)
State (e!uilibrium" springs unstressed)
k T
c T
F sin t
m
tFumucku =++ sintBtAu += cossin
'o$ te tricy*
trigonometry
( )222
2
222
)()(
cossin)(
)()( +
+
=
cmk
tctmk
cmk
Fu
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Simple Harmonic Excitation
m
State 2 (at time t during vibration)
State (e!uilibrium" springs unstressed)
k T
c T
F sin t
m
tFumucku =++ sintBtAu += cossin
)sin()()( 222
+
= tcmk
Fu
( )222
2
222
)()(
cossin)(
)()( +
+
=
cmk
tctmk
cmk
Fu
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Simple Harmonic Excitation
m
State 2 (at time t during vibration)
State (e!uilibrium" springs unstressed)
k T
c T
F sin t
m
tFumucku =++ sintBtAu += cossin
)sin()()( 222
+
= tcmk
Fu
222
2
222 )()(
)(cos;
)()(sin
+
=
+
=
cmk
mk
cmk
c
( )222
2
222
)()(
cossin)(
)()( +
+
=
cmk
tctmk
cmk
Fu
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Simple Harmonic Excitation
m
State 2 (at time t during vibration)
State (e!uilibrium" springs unstressed)
k T
c T
F sin t
m
tFumucku =++ sintBtAu += cossin
)sin(
)()(222
+
= t
cmk
Fu
222
2
222 )()(
)(cos;
)()(sin
+
=
+
=
cmk
mk
cmk
c
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Simple Harmonic Excitation
m
State 2 (at time t during vibration)
State (e!uilibrium" springs unstressed)
k T
c T
F sin t
m
tFumucku =++ sintBtAu += cossin
)sin(
)()(222
+
= t
cmk
Fu Or*+
222
2
222 )()(
)(cos;
)()(sin
+
=
+
=
cmk
mk
cmk
c
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Simple Harmonic Excitation
m
State 2 (at time t during vibration)
State (e!uilibrium" springs unstressed)
k T
c T
F sin t
m
tFumucku =++ sintBtAu += cossin
)sin(
)()(222
+
= t
cmk
Fu
)(tan
2
=
mk
c
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Simple Harmonic Excitation
m
State 2 (at time t during vibration)
State (e!uilibrium" springs unstressed)
k T
c T
F sin t
m
tFumucku =++ sintBtAu += cossin
)sin(
)()(222
+
= t
cmk
Fu
phase change)(
tan2
=
mk
c
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Simple Harmonic Excitation
State 2 (at time t during vibration)k T
c T
F sin t
m
)sin(
)()(222
+
= t
cmk
Fu
phase change
This is the steady state response
)(tan
2
=
mk
c
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Simple Harmonic Excitation
State 2 (at time t during vibration)k T
c T
F sin t
m
)sin(
)()(222
+
= t
cmk
Fu
phase change
This is the steady state response
Force dependent unlike transientresponse, it doesnt decay !ith
ti"e
)(tan
2
=
mk
c
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Simple Harmonic Excitation
State 2 (at time t during vibration)k T
c T
F sin t
m
)sin(
)()(222
+
= t
cmk
Fu
phase change
This is the steady state response
#"plitude o$ steady state response
!%
)(tan
2
=
mk
c
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Simple Harmonic Excitation
State 2 (at time t during vibration)k T
c T
F sin t
m
222
)()(
%
+
=
cmk
Fu
phase change
This is the steady state response
#"plitude o$ steady state response
)(tan
2
=
mk
c
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Simple Harmonic Excitation
State 2 (at time t during vibration)k T
c T
F sin t
m
222
)()(
%
+
=
cmk
Fu
phase change
This is the steady state response
Static displace"ent&kFu =
)(tan
2
=
mk
c
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Simple Harmonic Excitation
State 2 (at time t during vibration)k T
c T
F sin t
m
222222 )'())'(()'())'((%
kckm
u
kckmk
Fu
+
=
+
=
phase change
This is the steady state response
Static displace"ent&kFu =
)(tan
2
=
mk
c
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Simple Harmonic Excitation
State 2 (at time t during vibration)k T
c T
F sin t
m
222
)'())'((
%
kckmu
uR
+
==
phase change
This is the steady state response
yna"ic *agni$ication $actor&
k
Fu =
)(tan
2
=
mk
c
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Simple Harmonic Excitation
State 2 (at time t during vibration)k T
c T
F sin t
m
222
)'())'((
%
kckmu
uR
+
==
phase change
+et us si"pli$y the $or"ula $orR
yna"ic *agni$ication $actor&
k
Fu =
)(tan
2
=
mk
c
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Simple Harmonic Excitation
State 2 (at time t during vibration)k T
c T
F sin t
m
222
)'())'((
%
kckmu
uR
+
==
phase change
+et us si"pli$y the $or"ula $orR
e kno! that k - m2
k
Fu =
)(tan
2
=
mk
c
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Simple Harmonic Excitation
State 2 (at time t during vibration)k T
c T
F sin t
m
222
)'())'((
%
kcu
uR
+
==
phase change
+et us si"pli$y the $or"ula $orR
e kno! that k - m2
k
Fu =
)(tan
2
=
mk
c
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Simple Harmonic Excitation
State 2 (at time t during vibration)k T
c T
F sin t
m
222
)'())'((
%
kcu
uR
+
==
phase change
+et us si"pli$y the $or"ula $orR
e also kno! that c - cc=(2m)
k
Fu =
)(tan
2
=
mk
c
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Simple Harmonic Excitation
State 2 (at time t during vibration)k T
c T
F sin t
m
222
)'())'((
%
kcu
uR
+
==
phase change
+et us si"pli$y the $or"ula $orR
e also kno! that c - cc=(2m)
k
Fu =
)(tan
2
=
mk
c
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Simple Harmonic Excitation
State 2 (at time t during vibration)k T
c T
F sin t
m
222
)'2())'((
%
kmu
uR
+
==
phase change
+et us si"pli$y the $or"ula $orR
nce again use k - m2
k
Fu =
)(tan
2
=
mk
c
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Simple Harmonic Excitation
State 2 (at time t during vibration)k T
c T
F sin t
m
222
)'2())'((
%
+
==
u
uR
phase change
+et us si"pli$y the $or"ula $orR
nce again use k - m2
k
Fu =
)(tan
2
=
mk
c
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Simple Harmonic Excitation
State 2 (at time t during vibration)k T
c T
F sin t
m
222
)2()(
%
+
==
u
uR
phase change
+et us si"pli$y the $or"ula $orR
In ter"s o$ the $re/uency ratio - '
k
Fu =
)(tan
2
=
mk
c
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Rs
1esonance occurs !henRis "ai"u"
R
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Rs
1esonance occurs not !hen =
R
-
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Rs
1esonance occurs not !hen = d
R
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Simple Harmonic Excitation
State 2 (at time t during vibration)k T
c T
F sin t
m
222
)2()(
%
+
==
u
uR
phase change
To $ind the resonance $re/uency, $ind !hen dR/d2
=0
In ter"s o$ the $re/uency ratio - '
k
Fu =
)(tan
2
=
mk
c
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Finding resonancefre!uency
02 =
d
dR
( )( )
( ) ( )( )( )( )
( ) ( )( ) 2'322222
2'3222
22
2
22
4)(2
2
4)(2
2
+
+=
+
+=
d
dR
( ) 04)(2 22 =+
22 =
222 )2()(
%
+==
uuR
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Finding resonancefre!uency
222 )2()(
%
+==
u
uR
02 =
d
dR
( )( )
( ) ( )( )( )( )
( ) ( )( ) 2'322222
2'3222
22
2
22
4)(2
2
4)(2
2
+
+=
+
+=
d
dR
( ) 04)(2 22 =+ 22 =
5eak occurs only i$ 2'
