8/9/2019 Dynamics and Vibration, Wahab,2008-chapter 8
http://slidepdf.com/reader/full/dynamics-and-vibration-wahab2008-chapter-8 1/46
8/9/2019 Dynamics and Vibration, Wahab,2008-chapter 8
http://slidepdf.com/reader/full/dynamics-and-vibration-wahab2008-chapter-8 2/46
Introduction
Two co-ordinates and two
euations o! motion
Two independent motions
k "
m"
m#
k #
x#
x"
Trailer
Suspension
$%le
Tyre
k r
y f yr
k f
yC
CG
C θ
k r
y f yr
k f
yC
CG
C θ
Chapter 8: Systems With Two Degrees Of Freedom (page 3!"
© John Wiley & Sons, Lt
8/9/2019 Dynamics and Vibration, Wahab,2008-chapter 8
http://slidepdf.com/reader/full/dynamics-and-vibration-wahab2008-chapter-8 3/46
#$uations of motion (page 3%"
k "
x"
m" m#k #
x#
f " f #
on!iguration
'sing (ewton)s #nd law
x=m F ..
x """∑ ###
..
x x=m F ∑
""""##""
..
x=m )+f x(x+k x-k −
###"##
..
x=m )+f x(x-k −
*ree body diagrams
+x
"" xk + "## x xk −
f "
+ "## x xk −
f #
© John Wiley & Sons, Lt
8/9/2019 Dynamics and Vibration, Wahab,2008-chapter 8
http://slidepdf.com/reader/full/dynamics-and-vibration-wahab2008-chapter-8 4/46
8/9/2019 Dynamics and Vibration, Wahab,2008-chapter 8
http://slidepdf.com/reader/full/dynamics-and-vibration-wahab2008-chapter-8 5/46
#$uations of motion
""""##""
..
x=m )+f x(x+k x-k −###"##
..
x=m )+f x(x-k −e-arranging
"##"#""" =f x-k )x+k +(k xm..
###"### =f x+k x-k xm..
$nd in matri% !orm
#
"
#
"
##
##"
#
"
#
"
=
+
f
f
x
x
k -k
-k +k k
x
x
m
m..
..
© John Wiley & Sons, Lt
8/9/2019 Dynamics and Vibration, Wahab,2008-chapter 8
http://slidepdf.com/reader/full/dynamics-and-vibration-wahab2008-chapter-8 6/46
#$uations of motion (page 3&"
a+ on!iguration
cθ
C y
k "
L"
k #
$
L#
yC f
C M θ
b+ *ree body diagram
$
L"
L#
mg
+" A A yk δ + +# B B yk δ +
yC f
C M θ
+x
+y
θ +
C
..
yC B B A A y=mmg+f y-k yk −++ ++- #" δ δ
&
..
##"" ++ θ δ δ θ C C B B A A =I +M L y-k L yk ++
'sing (ewton)s #nd law
C
..
y=m F ∑ y &
..
θ C C =I M ∑
© John Wiley & Sons, Lt
8/9/2019 Dynamics and Vibration, Wahab,2008-chapter 8
http://slidepdf.com/reader/full/dynamics-and-vibration-wahab2008-chapter-8 7/46
8/9/2019 Dynamics and Vibration, Wahab,2008-chapter 8
http://slidepdf.com/reader/full/dynamics-and-vibration-wahab2008-chapter-8 8/46
#$uations of motion
+x
+y
θ +
k "
L"
k #
$
L#
y A
y B
yC C θ
d+ 0e!ormed bar
" tan L
y y AC
C
−
=θ
tan" C C A -L=y y θ C C B +L=y y θ tan#
*or small 1ibration C C θ θ ≈tan
" C C A -L=y y θ C C B +L=y y θ #
Substituting in the euation o! motions
$nd in matri% !orm
© John Wiley & Sons, Lt
2##""#" yC C C c
..
f ) Lk L(k )yk +(k ym θ −−+
2
#
##
#
""##""&
..
C C C C M ) Lk L+(k )y Lk L(k I θ θ θ +−−
23
&
#
##
#
""""##
""###"
&
..c
..
+−
−+
C
yC
C C M
f y
Lk Lk Lk Lk
Lk Lk k k y
I
m
θ θ θ
8/9/2019 Dynamics and Vibration, Wahab,2008-chapter 8
http://slidepdf.com/reader/full/dynamics-and-vibration-wahab2008-chapter-8 9/46
#$uations of motion (page 3&'"
$ spring-suspended mass system )+y(x=-k P iiii θ θ sincos
x=m +f θ P ..
x
i=
ii∑4
"
cos y=m +f θ P ..
y
i=
ii∑4
"
sin
'sing (ewton)s #nd law
) =f θ θ +yθ (xk + xm xii
i=
ii
..
sincoscos4
"
#∑
) =f θ +yθ θ (xk + ym yi
i=
iii
..#
4
"
sincossin∑
© John Wiley & Sons, Lt
k "
mk 3
k #
#θ
4θ
"θ f
x
f y
x
y
P "#θ
4θ
"θ f
x
f y
P #
P 4
x
y
8/9/2019 Dynamics and Vibration, Wahab,2008-chapter 8
http://slidepdf.com/reader/full/dynamics-and-vibration-wahab2008-chapter-8 10/46
#$uations of motion
=
+
∑= f
f
y
x
θ θ θ
θ θ θ k
y
x
m
m
y
x
iii
iii
i
i..
..
#
#4
" sincossin
cossincos
$nd in matri% !orm
The general matri% !orm +S D=F D M
..
F
F
D
D
S S
S S
D
D
M
M ..
..
=
+
#
"
#
"
###"
"#""
#
"
##
""
© John Wiley & Sons, Lt
P "#θ
4θ
"θ f
x
f y
P #
P 4
x
y
8/9/2019 Dynamics and Vibration, Wahab,2008-chapter 8
http://slidepdf.com/reader/full/dynamics-and-vibration-wahab2008-chapter-8 11/46
#$uations of motion
+S D=F D M ..
5n case o! !ree 1ibration
+S D= D M ..
D
D
S S
S S
D
D
M
M ..
..
=
+
#
"
###"
"#""
#
"
##
""
The displacement time response solutions +sin"" ψ ω += t D D m +sin## ψ ω += t D D m
Where Dm" and Dm# are the ma%imum 1alues or amplitudes+
0i!!erentiation twice
n)damped free *i+ration (page 3&,"
© John Wiley & Sons, Lt
+sin"
#..
" ψ ω ω +−= t D D m+sin#
#..
# ψ ω ω +−= t D D m
8/9/2019 Dynamics and Vibration, Wahab,2008-chapter 8
http://slidepdf.com/reader/full/dynamics-and-vibration-wahab2008-chapter-8 12/46
#$uations of motion
=
−
−
D
D
M S S
S M S
m
m
#
"
##
#
###"
"#""
#
""
ω
ω Substituting in the euation o! motions
*or non-tri1ial solutions
##
#
###"
"#""
#
"" =−
− M S S
S M S
ω ω *rom which
+- #
"###""
#
""####""
6
##"" =−++ S S S S M S M M M ω ω
Sol1ing using the uadratic !ormula
© John Wiley & Sons, Lt
8/9/2019 Dynamics and Vibration, Wahab,2008-chapter 8
http://slidepdf.com/reader/full/dynamics-and-vibration-wahab2008-chapter-8 13/46
#$uations of motion
!!c"-"
#62
##
#," −±ω Where
!=M M # "=-(M S M S )# c=S S S"" ## "" ## ## "" "" ## "#
#+ −
#
"###""
#
"#####""
# 66 S M M )S M S !c=(M -" +−The term is always positi1e
=
−
−
D
D
M S S
S M S
m
m
#
"
##
#
###"
"#""
#
""
ω
ω
The amplitude ratios
S-SS-
#"
###"##
""
#
"""
"#
#
""
M S M D
D=r m
m ω ω
−=−
= S-S
S- 2#"
######
""
#
#""
"#
#
"#
M S M D
D=r m
m ω ω
−=−
© John Wiley & Sons, Lt
8/9/2019 Dynamics and Vibration, Wahab,2008-chapter 8
http://slidepdf.com/reader/full/dynamics-and-vibration-wahab2008-chapter-8 14/46
#-amp.e 8/,: Free *i+ration I (page 3&0"
"7 8(/m
x"
# 8g " 8g
x#
#7 8(/mTwo masses are eual to #
8g and " 8g
The two springs) constants are eual
to "7 8(/m and #7 8(/m alculate the natural !reuencies and the mode
shape amplitude ratios
S$%&ti$'
=
=
#
"
##
""
m
m
M
M M M ""2m"2# 8g, M ##2m#2" 8g
© John Wiley & Sons, Lt
8/9/2019 Dynamics and Vibration, Wahab,2008-chapter 8
http://slidepdf.com/reader/full/dynamics-and-vibration-wahab2008-chapter-8 15/46
#-amp.e 8/,: Free *i+ration I
"7 8(/m
x"
# 8g " 8g
x#
#7 8(/m
=
##
##"
###"
#"""
k -k
-k +k k
S S
S S S=
S ""
2k "
3k #
26 (/m, S "#
2S #"
2-k #
2-#7 (/m,
S ##2k #2#7 (/m
9#
"###""
7
""####""##"" ":7.4";# ×−×−=+= =S S # c=S )S M S # "=-(M M !=M
rad/s<4.7#672rad/s,#".777<2 ,##
":7.4#6+";";2
#"
9#77#
#,"
ω ω ω
×
×××−×±×
π ω
#" π
ω #
# f "2 24.64 =>, f
#2 2"."" =>
© John Wiley & Sons, Lt
8/9/2019 Dynamics and Vibration, Wahab,2008-chapter 8
http://slidepdf.com/reader/full/dynamics-and-vibration-wahab2008-chapter-8 16/46
#-amp.e 8/,: Free *i+ration I
The amplitude ratios are?
phase+-anti .<"6-2
#+7#67.<46
#7
S
S- 2
phase+-in .9"62#+777<.#"6
#7- 2
#
""
#
#""
"##
#
""
#
"""
"#"
×−
=
−
×−=
−
M
r
M S
S r
ω
ω
x" x#
.9"6 "
x" x#
-.<"6"
© John Wiley & Sons, Lt
8/9/2019 Dynamics and Vibration, Wahab,2008-chapter 8
http://slidepdf.com/reader/full/dynamics-and-vibration-wahab2008-chapter-8 17/46
#-amp.e 8/0: Free *i+ration II (page 3&%"
$ racing car has a mass o! #4 8g
The sti!!ness o! the !ront and rear
wheel/suspension are both eual to
":< (/m The mass moment o! inertia o! the
car is 67 8g.m#
0etermine the !reuencies and mode shape amplitude
ratios o! the car
© John Wiley & Sons, Lt
*igure @9.#-"
.# m
A
".: m # m .4 m
8/9/2019 Dynamics and Vibration, Wahab,2008-chapter 8
http://slidepdf.com/reader/full/dynamics-and-vibration-wahab2008-chapter-8 18/46
#-amp.e 8/0: Free *i+ration II
S$%&ti$'
=
=
C I
m
M
M M
##
""
M ""
2m"
2#4 8g, M ##
2 I
267 8g.m#
S ""2k "3k #247# (/m, S "#2S #"2k # L#-k " L"27#9 (, S ##2k " L"
#3k # L##2"#"#<6 (.m,
© John Wiley & Sons, Lt
+−
−+=
#
##
#
""""##
""###"
###"
#"""
Lk Lk Lk Lk
Lk Lk k k
S S
S S S=
*igure @9.#-"
.# m
A
".: m # m .4 m
8/9/2019 Dynamics and Vibration, Wahab,2008-chapter 8
http://slidepdf.com/reader/full/dynamics-and-vibration-wahab2008-chapter-8 19/46
#-amp.e 8/0: Free *i+ration II
"#6.6
"#"6.6;4"7
;#
"###""
:
""####""##""
×−
×−=+=
=S S c=S
# )S M S # "=-(M M !=M
rad/s":.479;2rad/s,"#.#;2
,;4"7#
"#6.6;4"76+"#"6.6"#"6.62
#"
;#::#
#,"
ω ω
ω ×
×××−×±×
#.:<=>2#
".;<=>,2#
##
""
π
ω
π
ω == f f
© John Wiley & Sons, Lt
-#24<
#"m/rad-"".69"92#4+#;."#47#
7#9
S
S- 2 4#
""
#
"""
"#"
π
ω ×××−
−=− M r mm/degree anti-phas
#.:24<
#"m/rad."7692
#4+479;.":47#
7#9
S
S- 2 4
#
""
#
#""
"##
π
ω ××
×−
−=
− M r mm/degree in-phase+
-# mm
"o
cθ
C y
Bibration mode "
+x
+y
θ +
#.: mm
"o
cθ
C y
Bibration mode #
8/9/2019 Dynamics and Vibration, Wahab,2008-chapter 8
http://slidepdf.com/reader/full/dynamics-and-vibration-wahab2008-chapter-8 20/46
#-amp.e 8/3: Free *i+ration III (page 3&&"
x"
m"
x#
k m#
$ semide!inite or unrestrained system
two masses m" and m# are attached to
each other with a spring o! sti!!ness k
0etermine the two natural !reuencies o! the system
S$%&ti$'
M ""2m", M ##2m#
=
k -k
-k k
S S
S S S=
###"
#""" S ""2k , S "#2S #"2-k , S ##2 k
© John Wiley & Sons, Lt
8/9/2019 Dynamics and Vibration, Wahab,2008-chapter 8
http://slidepdf.com/reader/full/dynamics-and-vibration-wahab2008-chapter-8 21/46
8/9/2019 Dynamics and Vibration, Wahab,2008-chapter 8
http://slidepdf.com/reader/full/dynamics-and-vibration-wahab2008-chapter-8 22/46
8/9/2019 Dynamics and Vibration, Wahab,2008-chapter 8
http://slidepdf.com/reader/full/dynamics-and-vibration-wahab2008-chapter-8 23/46
Torsiona. 1i+ration
k " #G" #I P "
"", θ
"
##,( θ
k # #G
# #I
P #
L" L#
#
*igure 9.:? Torsional two degrees o! !reedom system
""θ k
"
+ "## θ θ −k
31e
#(
#(
Where the torsional sti!!ness
"
"""
L
I Gk
*=
#
###
L
I Gk
*=
#
"
#
"
##
##"
#
"
#
"
=
+
k -k
-k +k k
..
..
θ
θ
θ
θ
5n matri% !orm
$nd !or !ree 1ibration
#
"
##
##"
#
"
#
"
=
+
k -k
-k +k k
..
..
θ
θ
θ
θ
© John Wiley & Sons, Lt
8/9/2019 Dynamics and Vibration, Wahab,2008-chapter 8
http://slidepdf.com/reader/full/dynamics-and-vibration-wahab2008-chapter-8 24/46
#-amp.e 8/2: Free torsiona. *i+ration (page 38"
.7 m.7 m
# 8g
" 8g
Two dis8s o! masses " 8g and
# 8g, ha1ing radii o! 7 cm
and " cm
Counted on a steel solid sha!t o!diameter "7 mm and length " m
0etermine the two torsional natural !reuencies o!
the system and the corresponding amplitudes ratios
© John Wiley & Sons, Lt
8/9/2019 Dynamics and Vibration, Wahab,2008-chapter 8
http://slidepdf.com/reader/full/dynamics-and-vibration-wahab2008-chapter-8 25/46
#-amp.e 8/2: Free torsiona. *i+ration
.7 m.7 m
# 8g
" 8g
S$%&ti$'
The polar moment o! inertia o!
the sha!t
6;
66
m"-;:.64#
-"7.-
4#
−×=
×==
π π D
I *
The dis8s) mass moments o! inertia
4##
""" "#7."
#
7."
#
−×=×
==r m
8g.m#
".#
".#
#
####
# =×
==r m
8g.m#
© John Wiley & Sons, Lt
8/9/2019 Dynamics and Vibration, Wahab,2008-chapter 8
http://slidepdf.com/reader/full/dynamics-and-vibration-wahab2008-chapter-8 26/46
#-amp.e 8/2: Free torsiona. *i+ration
.7 m
.7 m
# 8g
" 8g
The torsional sti!!ness o! the
sha!t is constant
49.:<77.-
"-;:.6"-::;;
#" =×××
===−
L
GI k k *
(.m
=
=
#
"
##
""
M M M
M ""
2 "2 4
"-#7." −× 8g.m#, M ##
2 #2." 8g.m#
=
##
##"
###"
#"""
k -k
-k +k k
S S
S S
S=
S ""
2k "3k
#2"74.:< (.m, S
"#2S
#"2-k
#2-:<7.49 (.m, S
##2k
#2:<7.49 (/m
© John Wiley & Sons, Lt
8/9/2019 Dynamics and Vibration, Wahab,2008-chapter 8
http://slidepdf.com/reader/full/dynamics-and-vibration-wahab2008-chapter-8 27/46
8/9/2019 Dynamics and Vibration, Wahab,2008-chapter 8
http://slidepdf.com/reader/full/dynamics-and-vibration-wahab2008-chapter-8 28/46
8/9/2019 Dynamics and Vibration, Wahab,2008-chapter 8
http://slidepdf.com/reader/full/dynamics-and-vibration-wahab2008-chapter-8 29/46
n)damped forced *i+rations
k "
x"
m" m#
k #
x#
f " f #
on!iguration
The steady state solutions
t A D ω sin"" = sin## t A D ω =
0i!!erentiating twice with
respect to time
t A D ω ω sin"
#"
..
−= sin#
##
..
t A D ω ω −=
Substituting in the euation o! motions
=
−
−
F
F
A
A
M S S
S M S
$
$
#
"
#
"
##
#
###"
"#""
#
""
ω
ω
Sol1ing !or D AE SS-
S-S"
#
"
""
#
""#"
"###
#
##
#
"
−
−=
F
F
M
M
C A
A
$
$
ω
ω
© John Wiley & Sons, Lt
8/9/2019 Dynamics and Vibration, Wahab,2008-chapter 8
http://slidepdf.com/reader/full/dynamics-and-vibration-wahab2008-chapter-8 30/46
8/9/2019 Dynamics and Vibration, Wahab,2008-chapter 8
http://slidepdf.com/reader/full/dynamics-and-vibration-wahab2008-chapter-8 31/46
#-amp.e 8/,': Forced *i+ration I (page 32'"
".7 m .7 m
$
*igure F9."-"
The bar has a length o! # m and
a mass o! " 8g
k A2< (/m and k B2; (/m
a+ alculate the two natural !reuencies o! the bar.b+ 0etermine the amplitude ratios in mm/degree+.
c+ 5! a harmonic !orce o! magnitude P 2# ( and !orcing
!reuency o! " rad/s is applied at A, determine the
absolute 1alues o! the two amplitudes.
© John Wiley & Sons, Lt
8/9/2019 Dynamics and Vibration, Wahab,2008-chapter 8
http://slidepdf.com/reader/full/dynamics-and-vibration-wahab2008-chapter-8 32/46
#-amp.e 8/,': Forced *i+ration I
".7 m .7 m
$
*igure F9."-"
S$%&ti$'
=
=
C I
m
M
M M
-
-
-
-
##
""
+−
−+=
#
##
#
""##""
##""#"
###"
#"""
Lk Lk Lk Lk
Lk Lk k k
S S
S S S=
M ""
2" 8g, M ##
2 I 2mL#/"#24.44 8g.m#
× ×"- ; .72"7 (/m,S ""
2<3;2"7 (/m # S "#
2S #"
2<
× "#3; .7#29#7 (/m.S ##
2 < ×
© John Wiley & Sons, Lt
8/9/2019 Dynamics and Vibration, Wahab,2008-chapter 8
http://slidepdf.com/reader/full/dynamics-and-vibration-wahab2008-chapter-8 33/46
8/9/2019 Dynamics and Vibration, Wahab,2008-chapter 8
http://slidepdf.com/reader/full/dynamics-and-vibration-wahab2008-chapter-8 34/46
".7 m .7 m
$
(+t "-sin#-
#-amp.e 8/,': Forced *i+ration I
The amplitude ratios
phase+-in /mm7#.#4<
#"m/rad."66#2
"+;4:."7"7
"7
S
S- 2
phase+-anti /mm#7.64<
#"m/rad-#.4<2
"+;:;."""7
"7
S
S- 2
4
#
""
#
#""
"##
4
#
""
#
"""
"#"
°=×××−
−=
−
°−=×××−
−=
−
π
ω
π
ω
M r
M r
'sing F o"2# (, F o#2
mm66m66.44.##4444
444.;944
"#"7+""4#7+"444.44
#+444.4+"9#7
S-+
-#6
#
"##
##""
#
""####""
6
##""
#"#"##
#
##
"
===
+×−×
××−
=++−== S S S M S M M M
F -S )F M (S
A$$
C ω ω
ω
© John Wiley & Sons, Lt
8/9/2019 Dynamics and Vibration, Wahab,2008-chapter 8
http://slidepdf.com/reader/full/dynamics-and-vibration-wahab2008-chapter-8 35/46
#-amp.e 8/,': Forced *i+ration I
°−===
+×−×
×−=
++−==
::.rad"464.##.##4444
:
"#"7+""4#7+"444.44
#"7
S-+
#6"##
##""
#
""####""
6
##""
"#"#""
#
""#
S S S M S M M M
F -S )F M -(S A $$
C ω ω
ω θ
*igure F9."-#
".7 m .7 m
$
(+t "-sin#-
© John Wiley & Sons, Lt
8/9/2019 Dynamics and Vibration, Wahab,2008-chapter 8
http://slidepdf.com/reader/full/dynamics-and-vibration-wahab2008-chapter-8 36/46
#-amp.e 8/,,: Forced *i+ration II (page 320"
"7 8(/m
x"
# 8g " 8g
x#
*igure @9.""
#7 8(/m
" (
Two-degree o! !reedom system
=armonic loading o! magnitudes
F o"2" ( and F $#2
0raw the steady-state amplitudes 1ersus
ω cur1e response spectrum+
S$%&ti$'
9#
"###""
7
""####""##"" ":7.4";# ×−×−=+= =S S # c=S )S M S # "=-(M M !=M
'sing F o"2# (, F o#2
© John Wiley & Sons, Lt
8/9/2019 Dynamics and Vibration, Wahab,2008-chapter 8
http://slidepdf.com/reader/full/dynamics-and-vibration-wahab2008-chapter-8 37/46
#-amp.e 8/,,: Forced *i+ration II
"7 8(/m
x"
# 8g " 8g
x#
*igure @9.""
#7 8(/m
" (
'sing F o"2# (, F o#2
9#76
<
9#76#
9#76
#<
9#76
#
"
":7.4";-#
"#.7-
":7.4";-#
"#7+-2
":7.4";-#
"-"#.7
":7.4";-#
"+"-#72
×+×
×=
×+×
×
×+×
×=
×+×
×
ω ω ω ω
ω ω
ω
ω ω
ω
A
A
*igure @9.""-4
rad/s+ω
" A # A
ω rad/s
© John Wiley & Sons, Lt
8/9/2019 Dynamics and Vibration, Wahab,2008-chapter 8
http://slidepdf.com/reader/full/dynamics-and-vibration-wahab2008-chapter-8 38/46
1i+ration +sor+ers (page 32!"
k "
x"
m" m#
k #
x#
f " f #
on!iguration
The two amplitudes
#"###
###""
#""
#"#"###
##"
-
)-S M - )(S M -(S
F -S )F M (S = A
$$
ω ω
ω
2$#"######""#""
"#"#""#
""
# )-S M - )(S M -(S
F -S )F M -(S $$
ω ω
ω
$$ F F =" -# =$ F 'sing the !orce amplitudes as
""" m M = ### m M =#"# k S −=### k S =The mass and stiffness elements
###
##"
##"
###"
-
)-k m- )(k m-k (k
)F m(k = A $
ω ω
ω
+ 2$
###
##"
##"
##
)-k m- )(k m-k (k
F k $
ω ω +
© John Wiley & Sons, Lt
t F f $ ω sin=
8/9/2019 Dynamics and Vibration, Wahab,2008-chapter 8
http://slidepdf.com/reader/full/dynamics-and-vibration-wahab2008-chapter-8 39/46
1i+ration +sor+ers
Bibration absorption is achie1ed when
the amplitude o! the mass m" is >ero
-
-#
###
###
#"#
#"
##
#" =⇒=
+ )m(k
)-k m- )(k m-k (k
)F m(k = A $ ω
ω ω
ω
*rom which ##
#
m
k
=ω
x"
m"
x#
k "/# k "/#
x"
m"
m#
x#
k "/#
t F f $ ω sin=
k "/#
k #
when A"2
#
#
k
F A $=
© John Wiley & Sons, Lt
t F f $ ω sin=
8/9/2019 Dynamics and Vibration, Wahab,2008-chapter 8
http://slidepdf.com/reader/full/dynamics-and-vibration-wahab2008-chapter-8 40/46
1i+ration +sor+ers
x"
m"
x#
k "/# k "/#
x"
m"
m#
x#
k "/#
t F f $ ω sin=
k "/#
k #
rad/s+
Griginal system without
1ibration absorber+
Codi!ied system with1ibration absorber+
" A
ω
© John Wiley & Sons, Lt
8/9/2019 Dynamics and Vibration, Wahab,2008-chapter 8
http://slidepdf.com/reader/full/dynamics-and-vibration-wahab2008-chapter-8 41/46
8/9/2019 Dynamics and Vibration, Wahab,2008-chapter 8
http://slidepdf.com/reader/full/dynamics-and-vibration-wahab2008-chapter-8 42/46
#-amp.e 8/,0: 1i+ration a+sor+ers I
Cotor
x"
k #
x#
m#
S$%&ti$'
'sing S0G*
<:9.<9<
#<7" =×=
π ω rad/s
"<:9.<9 "
"
""
k
m
k =⇒=ω
*rom which
<" "<44.6 ×=k (/m
© John Wiley & Sons, Lt
8/9/2019 Dynamics and Vibration, Wahab,2008-chapter 8
http://slidepdf.com/reader/full/dynamics-and-vibration-wahab2008-chapter-8 43/46
#-amp.e 8/,0: 1i+ration a+sor+ers I
94.#"7"------
-<:9.<9 #
#
#
#
## =⇒=⇒= mmm
k ω 8g
<"<44.7 × M ""
2m"2" 8g, M
##2m
#2#"7.94 8g, S
""2k
"3k
#2 (/m,
<" <"N/m, S ##2k #2 (/mS "#2S #"2-k #2-
The two natural !reuencies o! the modi!ied system
"##
"###""
;
""####""
7
##"" "<44.6"#"7:.#""79.# ×−×−=+×= =S S # c=S )S M S # "=-(M M !=M
rad/s97.<;;:2rad/s,76.<62 ,""79.##
"<44.6""79.#6+"#"7:.#"#"7:.#2 #"7
"#7#;;
#
#," ω ω ω ××
××××−×±×
π ω
#" π
ω #
#
f "2 29.<=>, f
#2 2"4.<6 =>
© John Wiley & Sons, Lt
8/9/2019 Dynamics and Vibration, Wahab,2008-chapter 8
http://slidepdf.com/reader/full/dynamics-and-vibration-wahab2008-chapter-8 44/46
#-amp.e 8/,!: 1i+ration a+sor+ers III (page !''"
x"
"< tons
m#
x#
k #
k /#
$ bridge is modelled as a singledegree o! !reedom
=as a mass o! "<, tons and
sti!!ness o! k 2#", 8(/m
$ harmonic !orce o! ; (
0esign an undamped 1ibration
absorber so that its amplitude does
not e%ceed #.7 cm
© John Wiley & Sons, Lt
8/9/2019 Dynamics and Vibration, Wahab,2008-chapter 8
http://slidepdf.com/reader/full/dynamics-and-vibration-wahab2008-chapter-8 45/46
#-amp.e 8/,!: 1i+ration a+sor+ers III
x"
"< tons
m#
x#
k #
k /#
'sing S0G*
rad/s67<6.""
""<
"#"<
9
=
×
×==
m
k ω
The amplitude A#
##
#
;----#7.-
k k
F A $ =⇒=
*rom which4<----# =k N/m=360 kN/m
S$%&ti$'
© John Wiley & Sons, Lt
8/9/2019 Dynamics and Vibration, Wahab,2008-chapter 8
http://slidepdf.com/reader/full/dynamics-and-vibration-wahab2008-chapter-8 46/46
#-amp.e 8/,!: 1i+ration a+sor+ers III
x"
"< tons
m#
x#
k #
k /#
*rom which
The suppressed e%citation !reuency
#
#
#
## 4<+#9#4."
mm
k =×⇒= π ω
#:74# =m kg
© John Wiley & Sons, Lt