PROBLEM 19.161 (Continued)
P
osition 2 2
2
2
0
2
C
Cm
T
Vmgh
mg
θ
=
=
=
PROBLEM 19.162
The block shown is depressed 1.2 in. from its equilibrium position and released.
Knowing that after 10 cycles the maximum displacement of the block is 0.5 in.,
determine (a) the damping factor c/c, (b) the value of the coefficient of viscous damping.
(Hint: See Problems 19.129 and 19.130.)
SOLUTION
From Problems 19.130 and 19.129:
2
1ln
c
c
c
n
x
π
⎛⎞
⎛⎞ =
c
PROBLEM 19.162 (Continued)
k
PRO
B
An 0.8
–
consta
n
accord
i
the ma
b
ecom
e
B
LEM 19.1
6
–
lb ball is con
n
n
t
5lb/ft.k=
i
ng to the rel
a
ximum allo
w
e
slack.
6
3
n
ected to a pa
Knowing t
h
a
tion
m
δ
δ
=
w
able circular
ddle by mean
at the pad
sin ,
f
t
ω
whe
r
frequency
ω
s
of an elastic
d
le is move
d
r
e
8 in.,
m
δ
=
f
ω
if the cor
d
cord AB of
d
vertically
determine
d
is not to
1
e
1
PROBLEM 19.163 (Continued)
f
f
PROBLEM 19.164
A 3-kg slender rod AB is bolted to a 5-kg uniform disk. A dashpot of
damping coefficient 9N s/mc
=
⋅ is attached to the disk as shown.
Determine (a) the differential equation of motion for small oscillations,
(b) the damping factor /.
c
cc
SOLUTION
disk
22
AB d AB AB
Wx Fr I I m
θθ θ
−−= ++
⎜⎟
⎝⎠
PROBLEM 19.164 (Continued)
l
cc
c
SO
Sm
a
LUTION
a
ll angles:
sin ,
c
δ
δ
δ
=
=
=
f
f
f
θ
θ
≈
PROBLE
M
A 4-lb unif
o
and is conn
e
equation of
rod will for
m
0.9 in. dow
n
c
os 1
θ
≈
M
19.165
o
rm rod is sup
e
cted to a das
h
motion for s
m
m
with the ho
r
n
and released
.
ported by a p
h
pot at B. Det
e
all oscillatio
izontal 5 s aft
.
i
n at O and a
e
rmine (a) the
ns, (b) the a
n
er end B has
b
spring at A,
differential
n
gle that the
b
een pushed
L
−
α
=
−
PROBLEM 19.165 (Continued)
44
==
0.07246 0.3375 1.25 0
θθθ
+
+=
(b) Substituting t
e
λ
into the above differential equation,
00
0
(0) 0 2.329 sin 3.439 cos
3.439
tan 2.329
0.9755 rad
0.05 0.06039 rad
sin (0.9755)
θ
θφ θφ
φ
φ
θ
==− +
=
=
==
PROBLEM 19.165 (Continued)
t
−
S
O
O
LUTION
PROBL
E
A 400-kg
a dashpot
Knowing
t
at a dista
n
800 rpm
foundatio
n
E
M 19.166
motor suppo
r
of constant
t
hat the unbal
a
n
ce of 100 m
m
(a) the amp
n
, (b) the amp
l
r
ted by four s
p
6500 N sc
=
⋅
a
nce of the ro
m
from the ax
litude of th
e
l
itude of the
v
p
rings, each o
/m
is constr
a
tor is equival
e
is of rotation
,
e
fluctuating
v
ertical motio
n
f constant 15
ined to mo
e
nt to a 23-g
m
determine f
force transm
of the motor
0
kN/m, and
v
e vertically.
m
ass located
o
r a speed of
itted to the
.
7
PROBLEM 19.166 (Continued)
m
PROBLEM 19.167
The compressor shown has a mass of 250 kg and operates at 2000 rpm. At this operating condition the force
transmitted to the ground is excessively high and is found to be 2
f
mr
ω
where mr is the unbalance and
ω
f is the
forcing frequency. To fix this problem, it is proposed to isolate the compressor by mounting it on a square
concrete block separated from the rest of the floor as shown. The density of concrete is 2400 kg/m3 and the
spring constant for the soil is found to be 80 × 106 N/m. The geometry of the compressor leads to choosing a
block that is 1.5 m by 1.5 m. Determine the depth h that will reduce the force transmitted to the ground by 75%.
SOLUTION
f
ω
ω
f
PROBLEM 19.167 (Continued)
2
S
O
O
LUTION
PROBLE
A small bal
l
a tightly str
e
on a horiz
o
displaceme
n
cord and re
cord to re
m
equation of
period of vi
b
M 19.168
l
of mass m a
t
e
tched elastic
o
ntal plane.
T
n
t in a direc
t
leased. Assu
m
m
ain constant,
motion of t
h
b
ration.
tached at the
cord of lengt
he ball is gi
ion perpendi
ing the tens
(a) write th
e
h
e ball, (b) d
e
midpoint of
h
l can slide
v
en a small
c
ular to the
i
on T in the
e
differential
e
termine the
PROBLEM 19.169
A certain vibrometer used to measure vibration amplitudes consists
essentially of a box containing a slender rod to which a mass m is attached;
the natural frequency of the mass-rod system is known to be 5 Hz. When
the box is rigidly attached to the casing of a motor rotating at 600 rpm, the
mass is observed to vibrate with an amplitude of 0.06 m. relative to the box.
Determine the amplitude of the vertical motion of the motor.
SOLUTION
44
m
S
O
O
LUTION
PROBLE
M
If either a
acceleration
the string is
location of t
h
difficulty c
a
and B are pl
and the dist
then adjuste
Show that t
h
that g
=
4
π
2
l
M
19.170
simple or a
of gravity g,
not truly we
h
e mass cent
e
a
n be eliminat
aced so that t
h
a
nce l is me
a
d so that the
p
h
e period
τ
o
b
l
/
τ
2
.
compound
p
difficulties a
r
ightless, whil
e
r is difficult t
o
ed by using
a
h
ey are obvio
u
a
sured with g
r
p
eriod of osc
i
b
tained is eq
u
p
endulum is
u
r
e encountere
e in the case
o
establish. I
n
a
reversible, o
u
sly not at th
e
r
eat precision
.
i
llation
τ
is th
u
al to that of
a
u
sed to dete
r
d. In the cas
e
of the comp
o
n
the case of a
r Kater, pend
u
e
same distanc
.
The positio
n
e same when
a
true simple
mine experi
of the simpl
und pendulu
compound p
lum. Two k
e from the m
n
of a counte
r
either knife
pendulum of
m
entally the
e
pendulum,
m
, the exact
e
ndulum, the
n
ife edges A
a
ss center G,
r
weight D is
e
dge is used.
length l and
d
r
v
p
p
PROBLEM 19.170 (Continued)
π
2
τ