PROBLEM 19.132
A loaded railroad car weighing 30,000 lb is rolling at a constant velocity v0 when it couples with a spring and
dashpot bumper system (Figure 1). The recorded displacement-time curve of the loaded railroad car after
coupling is as shown (Figure 2). Determine (a) the damping constant, (b) the spring constant. (Hint: Use the
definition of logarithmic decrement given in Problem 19.129.)
SOLUTION
30,000
W
PROBLEM 19.132 (Continued)
PROBLEM 19.133
A torsional pendulum has a centroidal mass moment of inertia of 0.3 kg-m2 and when given an initial twist
and released is found to have a frequency of oscillation of 200 rpm. Knowing that when this pendulum is
immersed in oil and given the same initial condition it is found to have a frequency of oscillation of 180 rpm,
determine the damping constant for the oil.
SOLUTION
P
R
Th
e
co
n
b
a
r
b
a
r
S
O
R
OBLEM 1
e
barrel of a
f
n
stant
110
0
c=
r
rel into firin
g
r
rel to move b
O
LUTION
9.134
f
ield gun wei
g
0
lb s/ft.⋅
Det
e
g
position in
t
ack two-third
s
g
hs 1500 lb
a
e
rmine (a) the
t
he shortest
p
s
of the way f
r
a
nd is returne
d
constant k w
h
p
ossible time
w
r
om its maxi
m
d
into firing
p
h
ich should
b
w
ithout any
o
m
um-recoil po
p
osition after
r
b
e used for th
e
o
scillation, (b
)
sition to its fi
r
r
ecoil by a re
e
recuperator
t
)
the time ne
e
r
ing position.
cuperato
r
of
t
o return the
e
ded for the
l
c
y
t
m
n
t
t
n
6
e
e
6
e
PROB
L
A 2-kg
b
dashpot
w
in equilib
the bloc
k
decreme
n
equilibri
u
L
EM 19.13
5
b
lock is supp
o
w
ith a coeffici
e
rium when it
k
an upward
v
n
t, (b) the m
a
u
m after two c
y
5
o
rted by a s
p
e
nt of viscous
is struck fro
m
v
elocity of 0.
a
ximum up
w
y
cles. (Hint:
S
p
ring of cons
t
damping
c
=
m
below by a
h
4 m/s. Deter
m
w
ard displace
m
S
ee Probs. 19.
t
ant k = 128
0.6 N s/m.
=
⋅
T
h
ammer whic
h
m
ine (a) the
m
ent of the
b
128 and 19.1
2
N/m and a
T
he block is
h
imparts to
logarithmic
b
lock from
2
9.)
s
m
g
S
O
O
LUTION
PROBL
E
A 4-kg bl
o
which is a
t
and is at
t
Knowing
t
b
locks wil
l
E
M 19.136
o
ck A is drop
p
t
rest. Block
B
t
ached to a
t
hat there is
l
move after t
h
p
ed from a
h
B
is supported
dashpot of
no rebound,
h
e impact.
h
eight of 800
by a spring
o
damping co
e
determine th
e
mm onto a 9
o
f constant
k
=
e
fficient
c=
e
maximum
d
-kg block B
1500 N/m
=
230 N s/m.⋅
d
istance the
a
n
u
o
r
t
)
t
PROBLEM 19.136 (Continued)
2
2
kc
⎛⎞
S
O
Fr
o
O
LUTION
o
m the given
d
PRO
B
A 0.9-
k
shown
coeffic
i
connec
t
spring
d
d
at
a
B
LEM 19.1
3
k
g block B is
from two sp
r
i
ent
7.5
N
c=
t
ing A and B
d
uring the res
u
3
7
connected b
y
r
ings, each o
f
N
s/m.⋅
Kno
w
is cut, deter
m
u
lting motion
.
y
a cord to a
f
constant
k
=
w
ing that th
e
m
ine the mi
n
.
2.4-kg block
180 N/m
=
, a
n
e
system is
n
imum tensio
n
A which is s
u
n
d a dashpot
at rest wh
e
n
that will o
c
u
spended as
of damping
e
n the cord
c
cur in each
o
i
o
PROBLEM 19.138
A 0.9-kg block B is connected by a cord to a 2.4-kg block A which is suspended as
shown from two springs, each of constant k = 180 N/m, and a dashpot of damping
coefficient 60 N s/m.c
=
⋅ Knowing that the system is at rest when the cord connecting
A and B is cut, determine the velocity of block A after 0.1 s.
SOLUTION
PROBLEM 19.139
A machine element weighing 800 lb is supported by two springs, each having a
constant of 200 lb/in. A periodic force of maximum value 30 lb is applied to the
element with a frequency of 2.5 cycles per second. Knowing that the coefficient
of damping is 8 lb s/in.,⋅ determine the amplitude of the steady-state vibration of
the element.
SOLUTION
[1 (1.130) ] [2(0.1390)(1.130)]
m
m
PROBLEM 19.140
In Problem 19.139, determine the required value of the coefficient of damping if
the amplitude of the steady-state vibration of the element is to be 0.15 in.
PROBLEM 19.139 A machine element weighing 800 lb is supported by two
springs, each having a constant of 200 lb/in. A periodic force of maximum value
30 lb is applied to the element with a frequency of 2.5 cycles per second.
Knowing that the coefficient of damping is 8 lb s/in.,⋅ determine the amplitude
of the steady-state vibration of the element.
SOLUTION
P
R
In
t
w
h
S
O
R
OBLEM 1
t
he case of t
h
h
ich the magn
i
O
LUTION
9.141
h
e forced vibr
a
i
fication facto
r
a
tion of a sys
t
r
will always
d
t
em, determi
n
d
ecrease as th
n
e the range o
f
e frequency r
a
f
values of th
e
a
tio
/
fn
ωω
in
c
e
damping fa
c
c
reases.
c
tor
/
c
cc
for
r
1
o
o
s
s
r
PROBLEM 19.142
Show that for a small value of the damping factor /,
c
cc the maximum amplitude of a forced vibration
occurs when
f
n
ω
ω
≈ and that the corresponding value of the magnification factor is 1
2(/ ).
c
cc
SOLUTION
From Eq. (19.53′):
1
m
x
S
O
O
LUTION
PR
A c
o
des
c
on
a
det
e
rpm
axe
s
equ
i
is 0.
spri
n
OBLEM 1
9
o
unte
r
-rotatin
g
c
ribing circles
a
machine el
e
e
rmine some
o
a stroboscop
s
of rotation
i
librium. Kno
w
6 in. and that
n
g constant k,
9
.143
g
eccentric
m
of 6-in. radi
u
e
ment to ind
u
o
f the dynami
e shows the
e
and the ele
m
w
ing that the
the total mas
s
(b) the damp
i
m
ass exciter c
o
u
s at the sam
e
u
ce a steady
–
c characterist
i
e
ccentric mas
s
m
ent to be
p
amplitude of
s
of the syste
m
i
ng factor
/
c
cc
o
nsisting of t
w
e
speed but in
–
state vibrati
o
i
cs of the ele
m
s
es to be exa
c
p
assing thro
u
the motion o
f
m
is 300 lb, d
e
.
c
w
o rotating 1
4
opposite sen
s
o
n of the ele
m
m
ent. At a sp
c
tly under the
i
u
gh its positi
o
f
the element
a
e
termine (a) t
h
4
-oz masses
s
es is placed
m
ent and to
eed of 1200
i
r respective
o
n of static
a
t that speed
h
e combined
r
a
d
c
h
a
h
e
a
s
4
i
e
e
4
e
PROBLEM 19.143 (Continued)
PROBLEM 19.144
A 36-lb motor is bolted to a light horizontal beam which has a static
deflection of 0.075 in. due to the weight of the motor. Knowing that the
unbalance of the rotor is equivalent to a weight of 0.64 oz located 6.25
in. from the axis of rotation, determine the amplitude of the vibration of
the motor at a speed of 900 rpm, assuming (a) that no damping is
present, (b) that the damping factor /c
cc is equal to 0.055.
SOLUTION
m
PROBLEM 19.145
A 45-kg motor is bolted to a light horizontal beam which has a static
deflection of 6 mm due to the weight of the motor. The unbalance of the
motor is equivalent to a mass of 110 g located 75 mm from the axis of
rotation. Knowing that the amplitude of the vibration of the motor is
0.25 mm at a speed of 300 rpm, determine (a) the damping factor /,
c
cc
(b) the coefficient of damping c.
SOLUTION
From Eq. (19.53′)
m
P
k
c
c
⎝⎠
Then 0.1269
c
c=
1
2
2
−
PROBLEM 19.146
The unbalance of the rotor of a 180-kg motor is equivalent to a mass of 85 g
located 150 mm from the axis of rotation. The pad which is placed between
the motor and the foundation is equivalent to a spring of constant
k = 7.5 kN/m in parallel with a dashpot of constant c. Knowing that the
magnitude of the maximum acceleration of the motor is 9 mm/s2 at a speed
of 100 rpm, determine the damping factor /.
c
cc
SOLUTION
2
23 1
10
π
−−
⎛⎞
PROBLEM 19.147
A machine element is supported by springs and is connected to a dashpot as
shown. Show that if a periodic force of magnitude P = Pm sin
ω
ft is applied to
the element, the amplitude of the fluctuating force transmitted to the foundation is
()
()
()
()
()
2
22
22
12
12
f
cn
ff
ncn
c
c
mm
c
c
FP
ω
ω
ωω
ωω
⎡⎤
+⎢⎥
⎣⎦
=
⎡
⎤⎡ ⎤
−+
⎢
⎥⎢ ⎥
⎣
⎦⎣ ⎦
n
m
ω
=
PROBLEM 19.147 (Continued)