This preview shows pages 1–3. Sign up to view the full content.
B. Balachandran
ENME 361
Fall 2008
Solutions for Quiz #6
(October 23, 2008; Duration: 15 minutes)
1)
Provide the definition of resonance.
Solution
:
For a linear vibratory system with a single degree of freedom, the condition when the frequency
of excitation
ω
equals the natural frequency
ω
n
of the system; that is,
ω
=
ω
n
, is called resonance.
2) Consider the following differential equation governing the vibrations of a single degreeof
freedom system subjected to a harmonic excitation:
2
2
jt
o
dx
d
x
mc
k
x
F
e
dt
dt
ω
++
=
Assume a solution of the form
x
(
t
) =
X
o
e
j
t
and show that the velocity is given by
()
() /
2
( )
o
F
xt
H
e
k
ωθ
π
−Ω
+
=Ω
±
where
1
2
2
2
2
12
( )
,
( )
tan
,
and
cos
sin
1
jx
n
He
x
j
x
ζ
θ
−
Ω
Ω=
=
+
+
Ω
Solution
:
After substituting
x
(
t
) =
X
o
e
j
t
into the given equation and dividing throughout by the mass
m
,
we get
2
2
00
0
2
22
2
0
0
2
2
2
o
nn
o
o
dXe
F
X
ee
dt
dt
m
F
jj
X
e
e
m
F
jX
m
ωω
ζω
ωζ
=
=
−+
+
=
which we solve for
X
o
and get
0
2
2
2
2
n
FF
X
mj
==
⎛⎞
⎜⎟
⎝⎠
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document=
()
0
2
1
12
F
k
j
ζ
−Ω+
Ω
=
( )
2
0
2
2
2
j
F
This is the end of the preview. Sign up
to
access the rest of the document.
 Spring '11
 yoo

Click to edit the document details