PROBLEM 19.148
A 91-kg machine element supported by four springs, each of constant k = 175 N/m,
is subjected to a periodic force of frequency 0.8 Hz and amplitude 89 N. Determine
the amplitude of the fluctuating force transmitted to the foundation if (a) a dashpot
with a coefficient of damping c = 365 N ⋅ s/m is connected to the machine element
and to the ground, (b) the dashpot is removed.
SOLUTION
()
()
()
22
2
2
2
12
1 1 (1.8123) 2.2844
ff
ncn
c
c
f
n
ωω
ωω
ω
ω
⎡⎤
⎡
⎤
−+
⎢⎥
⎢
⎥
⎣
⎦
⎣⎦
⎛⎞
−=− =−
⎜⎟
⎝⎠
PROBLEM 19.148 (Continued)
c
()
n
ω
PROBLEM 19.149
A simplified model of a washing machine is shown.
A bundle of wet clothes forms a weight wb of 20 lb in
the machine and causes a rotating unbalance. The
rotating mass is 40 lb (including mb) and the radius of
the washer basket e is 9 in. Knowing the washer has
an equivalent spring constant k = 70 lb/ft and
damping ratio
ζ
= c/cc = 0.05 and during the spin
cycle the drum rotates at 250 rpm, determine the
amplitude of the motion and the magnitude of the
force transmitted to the sides of the washing machine.
SOLUTION
π
mf
fm f
PROBLEM 19.149 (Continued)
m
P
m
PROBLEM 19.150*
For a steady-state vibration with damping under a harmonic force, show that the mechanical energy dissipated
per cycle by the dashpot is 2,
mf
Ecx
π
ω
= where c is the coefficient of damping, m
x
is the amplitude of the
motion, and
f
ω
is the circular frequency of the harmonic force.
ω
S
O
O
LUTION
PR
O
The
sim
p
equ
a
syst
e
of a
m
amp
l
O
BLEM 19
suspension
p
lified spring-
a
a
tion defining
e
m moves at
a
m
plitude
m
δ
a
l
itude of the
v
.151*
of an auto
m
a
nd-dashpot s
the vertical
d
a
speed
v
ove
r
a
nd wave len
g
v
ertical displa
c
m
obile can
b
ystem shown
.
d
isplacement
r
a road with
g
th L. (b) De
r
c
ement of the
m
b
e approxim
a
.
(a) Write th
e
of the mass
a sinusoidal
c
r
ive an expre
s
m
ass m.
a
ted by the
e
differential
m when the
c
ross section
s
sion for the
f
u
n
T
T
f
2
f
s
s
PROBLEM 19.151* (Continued)
k
r
r
a
u
u
d
F
B
F
d
ω
x
PROBLE
M
Two blocks
A
of the same
c
is connected
coefficient o
P
sin
m
f
P
ω
=
and
B
x
of th
e
M
19.152*
A
and B, eac
h
c
onstant k. Bl
o
to the grou
n
f damping
c
.
f
t
Write the
d
e
two
b
locks f
r
h
of mass m, a
o
cks A and B
a
n
d by two da
c
. Block A i
s
d
ifferential e
q
r
om their equ
i
re supported
a
a
re connected
shpots, each
s
subjected
t
q
uations defin
i
i
librium positi
a
s shown by
t
by a dashpot,
dashpot havi
n
t
o a force o
f
i
ng the displ
a
ons.
t
hree springs
and block B
n
g the same
f
magnitude
a
cements
A
x
m
n
m
a
n
f
a
e
f
e
o
o
a
a
q
q
c
d
p
PR
O
Expr
for
w
S is
c
O
BLEM 19
.
ess in terms
o
w
hich oscillati
o
c
losed.
.
153
o
f L, C, and E
o
ns will take
p
the range of
v
p
lace in the c
i
v
alues of the
r
i
rcuit shown
w
r
esistance R
w
hen switch
m
o
o
l
l
e
P
R
Co
de
t
(1
−
S
O
u
e
s
n
h
e
n
h
a
=
a
d
a
a
R
OBLEM 1
nsider the cir
c
t
ermine (a) th
e
1/ )
e
−
times i
t
O
LUTION
9.154
c
uit of Probl
e
e
final value
o
t
s final value.
(
e
m 19.153 wh
o
f the current
(
The desired
v
en the capaci
t
in the circuit,
v
alue of t is k
n
t
or C is remo
v
(b) the time
t
n
own as the ti
m
v
ed. If switc
h
t
at which the
m
e constant o
h
S is closed
a
current will
h
f the circuit.)
a
t time
t
=
0,
h
ave reached
e
e
e
c
c
M
d
o
o
n
m
a
a
PROB
L
L
EM 19.154
(Continue
d)
S
O
O
LUTION
PRO
B
Draw t
h
corresp
o
B
LEM 19.1
5
h
e electrical
a
o
nding to the
f
5
5
a
nalogue of t
h
f
ree bodies m
h
e mechanical
and A.)
system sho
w
w
n. (Hint: Dr
a
a
w the loops
r
m
p
s
m
t
c
o
d
s
m
o
o
PROB
L
Draw th
e
correspo
L
EM 19.15
e
electrical a
n
nding to the
fr
6
n
alogue of th
e
fr
ee bodies m
a
e
mechanical
a
nd A.)
system show
n
n
. (Hint: Dra
w
w
the loops
S
O
O
LUTION
P
R
W
r
of
t
R
OBLEM 1
r
ite the differ
e
t
he Point A, (
b
9.157
e
ntial equatio
n
b
) the charges
n
s defining (
a
on the capac
i
a
) the displac
e
i
tors of the el
e
e
ments of the
e
ctrical analo
g
mass m and
g
ue.
l
PROBLEM 19.157 (Continued)
PROBLEM 19.158
Write the differential equations defining (a) the displacements of the masses m1
and m2, (b) the charges on the capacitors of the electrical analogue.
22
2
2
dt C
dt
PROBLEM 19.159
An automobile wheel-and-tire assembly of total weight 47 lb is attached to a mounting
plate of negligible weight which is suspended from a steel wire. The torsional spring
constant of the wire is known to be 0.40 lb in./rad.K
=
⋅ The wheel is rotated through
90° about the vertical and then released. Knowing that the period of oscillation is
observed to be 30 s, determine the centroidal mass moment of inertia and the
centroidal radius of gyration of the wheel-and-tire assembly.
=
S
O
P
R
T
h
re
m
c
o
O
LUTION
R
OBLEM
1
h
e period of v
m
oved, the p
e
o
nstant of the
s
1
9.160
ibration of th
e
e
riod is obse
r
s
pring.
2
2
W
m
=
e
system sho
w
r
ved to be 0.
5
3
0
A
W
τ
+=
w
n is observe
d
5
s. Determin
e
0
.6 s
d
to be 0.6 s.
A
e
(a) the wei
g
A
fter cylinde
r
g
ht of cylind
e
r
B has been
e
r A, (b) the
t
n
m
n
n
S
O
O
LUTION
PROB
Disks A
C is att
a
b
etwee
n
system.
LEM 19.1
6
and B weigh
a
ched to the
n
the disks,
d
6
1
30 lb and 12
l
rim of disk
B
d
etermine th
e
l
b, respective
l
B
. Assuming
e
period of
s
l
y, and a smal
that no slip
p
s
mall oscillat
i
l 5-lb block
p
ing occurs
i
ons of the
a
s