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Problem 21.63 The stepped disk weighs 20 lb and its
moment of inertia is I=0.6 slug-ft2. It rolls on the
horizontal surface. The disk is initially stationary with
the spring unstretched, and at t=0 a constant force
F=10 lb is applied as shown. Determine the position
of the center of the disk as a function of time. 8 in.
16 in.
16 lb/ft
8 lb-s/ft
F
Fx=−kS −cdx
dt +F+f.
x=8F
9k1−edt d
ωd
sin ωdt+cos ωdt
Problem 21.64* An electric motor is bolted to a metal
table. When the motor is on, it causes the tabletop to
vibrate horizontally. Assume that the legs of the table
behave like linear springs, and neglect damping. The
total weight of the motor and the tabletop is 150 lb.
When the motor is not turned on, the frequency of
horizontal vibration of the tabletop and motor is 5 Hz.
When the motor is running at 600 rpm, the amplitude of
the horizontal vibration is 0.01 in. What is the magnitude
of the oscillatory force exerted on the table by the motor
at its 600-rpm running speed?
768
Problem 21.65 The moments of inertia of gears A
and Bare IA=0.014 slug-ft2and IB=0.100 slug-ft2.
Gear Ais connected to a torsional spring with constant
k=2 ft-lb/rad. The system is in equilibrium at t=0
when it is subjected to an oscillatory force F(t) =
4 sin 3tlb. What is the downward displacement of the
5-lb weight as a function of time?
F(t)
5 lb
6 in
AB
10 in
3 in
Solve:
Fd=IB
RWRA=6.667 ft−1,
and M=W
g+IB
(RW)2+RB
RWRA2
IA=2.378 slug,
which is the equilibrium point. Make the transformation ˜xp=xp−
xpw. The equation of motion about the equilibrium point is
0=ωAh−12
770
Problem 21.66* A 1.5-kg cylinder is mounted on a
sting in a wind tunnel with the cylinder axis transverse
to the direction of flow. When there is no flow, a 10-N
vertical force applied to the cylinder causes it to deflect
0.15 mm. When air flows in the wind tunnel, vortices
subject the cylinder to alternating lateral forces. The
velocity of the air is 5 m/s, the distance between vortices
is 80 mm, and the magnitude of the lateral forces is
1 N. If you model the lateral forces by the oscillatory
function F(t) =(1.0)sin ω0tN, what is the amplitude
of the steady-state lateral motion of the sphere? 80 mm
5=0.016 s, from which the
period of a sinusoidal-like disturbance is τ=2(δt) =0.032 s, from
0=0.6667
[Note: This is a small deflection (113 microns) but the associated aero-
Problem 21.67 Show that the amplitude of the partic-
ular solution given by Eq. (21.31) is a maximum when
The maximum (or minimum) is found from
4[3ω2
0−ω2+2d2]ω0=˜ω0
Problem 21.68* A sonobuoy (sound-measuring
device) floats in a standing-wave tank. The device is
a cylinder of mass mand cross-sectional area A. The
water density is ρ, and the buoyancy force support-
ing the buoy equals the weight of the water that would
occupy the volume of the part of the cylinder below the
surface. When the water in the tank is stationary, the
buoy is in equilibrium in the vertical position shown at
the left. Waves are then generated in the tank, causing
the depth of the water at the sonobuoy’s position rela-
tive to its original depth to be d=d0sin ω0t. Let ybe
the sonobuoy’s vertical position relative to its original
position. Show that the sonobuoy’s vertical position is
governed by the equation
d2y
dt2+Aρg
my=Aρg
md0sin ω0t.
yd
Solution: The volume of the water displaced at equilibrium is
V=Ah where Ais the cross-sectional area, and his the equilibrium
immersion depth. The weight of water displaced is ρVg =ρgAh,so
Substitute:
d2y
772
Problem 21.69 Suppose that the mass of the sonobuoy
in Problem 21.68 is m=10 kg, its diameter is 125 mm,
and the water density is ρ=1025 kg/m3.Ifd=
0.1 sin 2tm, what is the magnitude of the steady-state
vertical vibrations of the sonobuoy?
yd
Problem 21.70 The mass weighs 50 lb. The spring con-
stant is k=200 lb/ft, and c=10 lb-s/ft. If the base is
subjected to an oscillatory displacement xiof amplitude
10 in. and frequency ωi=15 rad/s, what is the resulting
steady-state amplitude of the displacement of the mass
relative to the base?
c
x
k
xi
m
W=128.7(rad/s)2,
and (see Eq. (21.38),
Problem 21.71 The mass in Fig. 21.21 is 100 kg. The
spring constant is k=4 N/m, and c=24 N-s/m. The
base is subjected to an oscillatory displacement of fre-
quency ωi=0.2 rad/s. The steady-state amplitude of the
displacement of the mass relative to the base is measured
and determined to be 200 mm. What is the amplitude of
the displacement of the base? (See Example 21.7.)
c
x
k
xi
m
d2x
dt2+2ddx
dt +ω2x=a(t)
2=k
a(t) =−d2xi
dt2=xiω2
isin(ωit−φ).
Ep=0.2=ω2
ixi
(ω2−ω2
i)2+4d2ω2
i
0.8333 =0.24 m
Problem 21.72 A team of engineering students builds
the simple seismograph shown. The coordinate ximea-
sures the local horizontal ground motion. The coordi-
nate xmeasures the position of the mass relative to
the frame of the seismograph. The spring is unstretched
when x=0. The mass m=1 kg, k=10 N/m, and c=
2 N-s/m. Suppose that the seismograph is initially sta-
tionary and that at t=0 it is subjected to an oscil-
latory ground motion xi=10 sin 2tmm. What is the
amplitude of the steady-state response of the mass? (See
xix
k
c
m
TOP VIEW
774
Problem 21.73 In Problem 21.72, determine the posi-
tion xof the mass relative to the base as a function of
time. (See Example 21.7.)
Solution: From Example 21.7 and the solution to Problem 21.70,
the canonical form of the equation of motion is
Apsin ωit+Bpcos ωit, where
Ap=(ω2−ω2
i)ω2
ixi
(ω2−ω2
i)2+4d2ω2
i=4.615.
ixi
Solve:
i)ω2
ixi
ωd=√ω2−d2=3 rad/s. The complete solution is x=xh+xp.
Problem 21.74 The coordinate xmeasures the dis-
placement of the mass relative to the position in which
the spring is unstretched. The mass is given the initial
conditions
t=0
x=0.1m,
dx
dt =0.
(a) Determine the position of the mass as a function
of time.
(b) Draw graphs of the position and velocity of the
mass as functions of time for the first 5 s of motion.
90 N/m
x
10 kg
776
Problem 21.75 When t=0, the mass in Prob-
lem 21.74 is in the position in which the spring is
unstretched and has a velocity of 0.3 m/s to the right.
Determine the position of the mass as functions of time
and the amplitude of vibration
(a) by expressing the solution in the form given by
Eq. (21.8) and
(b) by expressing the solution in the form given by
Eq. (21.9)
90 N/m
x
10 kg
Solution: From Eq. (21.5), the canonical form of the equation of
Problem 21.76 A homogenous disk of mass mand
radius Rrotates about a fixed shaft and is attached to
a torsional spring with constant k. (The torsional spring
exerts a restoring moment of magnitude kθ, where θis
the angle of rotation of the disk relative to its position in
which the spring is unstretched.) Show that the period
of rotational vibrations of the disk is
τ=πR2m
k.
k
R
Solution: From the equation of angular motion, the equation of
Problem 21.77 Assigned to determine the moments of
inertia of astronaut candidates, an engineer attaches a
horizontal platform to a vertical steel bar. The moment
of inertia of the platform about Lis 7.5 kg-m2, and
the frequency of torsional oscillations of the unloaded
platform is 1 Hz. With an astronaut candidate in the
position shown, the frequency of torsional oscillations
is 0.520 Hz. What is the candidate’s moment of inertia
about L?
L
778
Problem 21.78 The 22-kg platen Prests on four roller
bearings that can be modeled as 1-kg homogenous
cylinders with 30-mm radii. The spring constant is k=
900 N/m. What is the frequency of horizontal vibrations
of the platen relative to its equilibrium position?
kP
2.
Since the system is conservative, T+V=const. Substitute the
kinematic relations and reduce:
Problem 21.79 At t=0, the platen described in
Problem 21.78 is 0.1 m to the left of its equilibrium
position and is moving to the right at 2 m/s. What are
the platen’s position and velocity at t=4s?
kP
780
