8/9/2019 Forced Vibration of a Sdof System
1/79
Forced Vibration of a SDOF System
S. Ilanko
2007
8/9/2019 Forced Vibration of a Sdof System
2/79
Simple Harmonic Excitation
m
State 2 (at time t during vibration)
State (e!uilibrium" springs unstressed)
k T
c T
F sin t
m
8/9/2019 Forced Vibration of a Sdof System
3/79
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
8/9/2019 Forced Vibration of a Sdof System
4/79
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
8/9/2019 Forced Vibration of a Sdof System
5/79
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
8/9/2019 Forced Vibration of a Sdof System
6/79
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!"
8/9/2019 Forced Vibration of a Sdof System
7/79
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!"
8/9/2019 Forced Vibration of a Sdof System
8/79
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
8/9/2019 Forced Vibration of a Sdof System
9/79
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
8/9/2019 Forced Vibration of a Sdof System
10/79
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
8/9/2019 Forced Vibration of a Sdof System
11/79
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
8/9/2019 Forced Vibration of a Sdof System
12/79
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
8/9/2019 Forced Vibration of a Sdof System
13/79
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
8/9/2019 Forced Vibration of a Sdof System
14/79
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
8/9/2019 Forced Vibration of a Sdof System
15/79
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
8/9/2019 Forced Vibration of a Sdof System
16/79
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
8/9/2019 Forced Vibration of a Sdof System
17/79
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"$
8/9/2019 Forced Vibration of a Sdof System
18/79
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
8/9/2019 Forced Vibration of a Sdof System
19/79
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
8/9/2019 Forced Vibration of a Sdof System
20/79
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
)(
)(
8/9/2019 Forced Vibration of a Sdof System
21/79
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
8/9/2019 Forced Vibration of a Sdof System
22/79
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
8/9/2019 Forced Vibration of a Sdof System
23/79
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
8/9/2019 Forced Vibration of a Sdof System
24/79
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
8/9/2019 Forced Vibration of a Sdof System
25/79
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
8/9/2019 Forced Vibration of a Sdof System
26/79
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$rite%
( )222
2
)()(
cossin)(
+
=
cmk
tctmkFu
8/9/2019 Forced Vibration of a Sdof System
27/79
Simple Harmonic Excitation
m
State 2 (at time t during vibration)
State (e!uilibrium" springs unstressed)
k T
c T
F sin t
m
tFumucku =++ sintBtAu += cossin
&gain re$rite%
( )222
2
222
)()(
cossin)(
)()( +
+
=
cmk
tctmk
cmk
Fu
( )
( )22222
)()(
cossin)(
+
=
cmk
tctmkFu
8/9/2019 Forced Vibration of a Sdof System
28/79
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
8/9/2019 Forced Vibration of a Sdof System
29/79
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
8/9/2019 Forced Vibration of a Sdof System
30/79
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
8/9/2019 Forced Vibration of a Sdof System
31/79
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
8/9/2019 Forced Vibration of a Sdof System
32/79
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
8/9/2019 Forced Vibration of a Sdof System
33/79
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
8/9/2019 Forced Vibration of a Sdof System
34/79
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
8/9/2019 Forced Vibration of a Sdof System
35/79
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
8/9/2019 Forced Vibration of a Sdof System
36/79
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
8/9/2019 Forced Vibration of a Sdof System
37/79
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
8/9/2019 Forced Vibration of a Sdof System
38/79
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
8/9/2019 Forced Vibration of a Sdof System
39/79
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
8/9/2019 Forced Vibration of a Sdof System
40/79
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
8/9/2019 Forced Vibration of a Sdof System
41/79
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
8/9/2019 Forced Vibration of a Sdof System
42/79
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 $actor&
k
Fu =
)(tan
2
=
mk
c
8/9/2019 Forced Vibration of a Sdof System
43/79
Simple Harmonic Excitation
State 2 (at time t during vibration)k T
c T
F sin t
m
222
)'())'((
%
kckmu
uR
+
==
phase change
+et us si"pli$y the $or"ula $orR
yna"ic *agni$ication $actor&
k
Fu =
)(tan
2
=
mk
c
8/9/2019 Forced Vibration of a Sdof System
44/79
Simple Harmonic Excitation
State 2 (at time t during vibration)k T
c T
F sin t
m
222
)'())'((
%
kckmu
uR
+
==
phase change
+et us si"pli$y the $or"ula $orR
e kno! that k - m2
k
Fu =
)(tan
2
=
mk
c
8/9/2019 Forced Vibration of a Sdof System
45/79
Simple Harmonic Excitation
State 2 (at time t during vibration)k T
c T
F sin t
m
222
)'())'((
%
kcu
uR
+
==
phase change
+et us si"pli$y the $or"ula $orR
e kno! that k - m2
k
Fu =
)(tan
2
=
mk
c
8/9/2019 Forced Vibration of a Sdof System
46/79
Simple Harmonic Excitation
State 2 (at time t during vibration)k T
c T
F sin t
m
222
)'())'((
%
kcu
uR
+
==
phase change
+et us si"pli$y the $or"ula $orR
e also kno! that c - cc=(2m)
k
Fu =
)(tan
2
=
mk
c
8/9/2019 Forced Vibration of a Sdof System
47/79
Simple Harmonic Excitation
State 2 (at time t during vibration)k T
c T
F sin t
m
222
)'())'((
%
kcu
uR
+
==
phase change
+et us si"pli$y the $or"ula $orR
e also kno! that c - cc=(2m)
k
Fu =
)(tan
2
=
mk
c
8/9/2019 Forced Vibration of a Sdof System
48/79
Simple Harmonic Excitation
State 2 (at time t during vibration)k T
c T
F sin t
m
222
)'2())'((
%
kmu
uR
+
==
phase change
+et us si"pli$y the $or"ula $orR
nce again use k - m2
k
Fu =
)(tan
2
=
mk
c
8/9/2019 Forced Vibration of a Sdof System
49/79
Simple Harmonic Excitation
State 2 (at time t during vibration)k T
c T
F sin t
m
222
)'2())'((
%
+
==
u
uR
phase change
+et us si"pli$y the $or"ula $orR
nce again use k - m2
k
Fu =
)(tan
2
=
mk
c
8/9/2019 Forced Vibration of a Sdof System
50/79
Simple Harmonic Excitation
State 2 (at time t during vibration)k T
c T
F sin t
m
222
)2()(
%
+
==
u
uR
phase change
+et us si"pli$y the $or"ula $orR
In ter"s o$ the $re/uency ratio - '
k
Fu =
)(tan
2
=
mk
c
8/9/2019 Forced Vibration of a Sdof System
51/79
Rs
1esonance occurs !henRis "ai"u"
R
8/9/2019 Forced Vibration of a Sdof System
52/79
Rs
1esonance occurs not !hen =
R
8/9/2019 Forced Vibration of a Sdof System
53/79
Rs
1esonance occurs not !hen = d
R
8/9/2019 Forced Vibration of a Sdof System
54/79
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 $re/uency ratio - '
k
Fu =
)(tan
2
=
mk
c
8/9/2019 Forced Vibration of a Sdof System
55/79
Finding resonancefre!uency
02 =
d
dR
( )( )
( ) ( )( )( )( )
( ) ( )( ) 2'322222
2'3222
22
2
22
4)(2
2
4)(2
2
+
+=
+
+=
d
dR
( ) 04)(2 22 =+
22 =
222 )2()(
%
+==
uuR
8/9/2019 Forced Vibration of a Sdof System
56/79
Finding resonancefre!uency
222 )2()(
%
+==
u
uR
02 =
d
dR
( )( )
( ) ( )( )( )( )
( ) ( )( ) 2'322222
2'3222
22
2
22
4)(2
2
4)(2
2
+
+=
+
+=
d
dR
( ) 04)(2 22 =+ 22 =
5eak occurs only i$ 2'
Top Related