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Dynamics 2e 2049
Problem 9.74
When the electric motor is resting on the beam, the static de-
flection of the beam is
ısD15 mm
. The motor is not perfectly
balanced, so when it is operating the unbalanced mass is equiva-
lent to a mass
muD200
g at a distance of
"D150 mm
from the
axis of the rotor. The combined mass of the motor and sprung
mass of the beam is mcD40 kg.
Determine the amplitude of steady-state vibration of the mo-
tor if the rotor is spinning at !rD150 rpm.
of McGraw-Hill, and must be surrendered upon request of McGraw-Hill. Any duplication or distribution, either in print or electronic form, without the
permission of McGraw-Hill, is prohibited.
Canceling terms and rearranging, we obtain the equation of motion as
of McGraw-Hill, and must be surrendered upon request of McGraw-Hill. Any duplication or distribution, either in print or electronic form, without the
permission of McGraw-Hill, is prohibited.
Dynamics 2e 2051
Problem 9.75
When the electric motor is resting on the beam, the static de-
flection of the beam is
ısD15 mm
. The motor is not perfectly
balanced, so when it is operating the unbalanced mass is equiva-
lent to a mass
muD200
g at a distance of
"D150 mm
from the
axis of the rotor. The combined mass of the motor and sprung
mass of the beam is mcD40 kg.
It is determined that the largest allowable vibration amplitude
of the motor is 10 mm. Determine the angular speed of the rotor
at which this will occur.
of McGraw-Hill, and must be surrendered upon request of McGraw-Hill. Any duplication or distribution, either in print or electronic form, without the
permission of McGraw-Hill, is prohibited.
Computation. Solving Eq. (1) for Ryand substituting the result into Eq. (2), we obtain
of McGraw-Hill, and must be surrendered upon request of McGraw-Hill. Any duplication or distribution, either in print or electronic form, without the
permission of McGraw-Hill, is prohibited.
Dynamics 2e 2053
Problem 9.76
The harmonic oscillator shown has a mass
mD5kg
, a spring with
constant
kD4000 N=m
, and a dashpot with a damping coefficient
cD20 Ns=m
. Calculate the amplitude
F0
of the sinusoidal excita-
tion force that is necessary to produce a steady-state vibration with a
velocity amplitude of
10 m=s
at resonance. What is the corresponding
amplitude of the acceleration?
of McGraw-Hill, and must be surrendered upon request of McGraw-Hill. Any duplication or distribution, either in print or electronic form, without the
permission of McGraw-Hill, is prohibited.
where we used
cD20 Ns=m
,
kD4000 N=m
, and
mD5kg
. Next, letting
!d
denote the damped natural
frequency and recalling that
!dD!np12
, since we are interested in finding
F0
at resonance, we must
of McGraw-Hill, and must be surrendered upon request of McGraw-Hill. Any duplication or distribution, either in print or electronic form, without the
permission of McGraw-Hill, is prohibited.
Dynamics 2e 2055
Problem 9.77
Modeling the beam as a uniform thin bar, ignoring the inertia of the pulleys, assuming that the system is
in static equilibrium when the bar is horizontal, and assuming that the cord is inextensible and does not
go slack, determine the linearized equation of motion of the system. In addition, determine the system’s
natural frequency of vibration. Treat the parameters shown in the figure as known.
Solution
of McGraw-Hill, and must be surrendered upon request of McGraw-Hill. Any duplication or distribution, either in print or electronic form, without the
permission of McGraw-Hill, is prohibited.
of McGraw-Hill, and must be surrendered upon request of McGraw-Hill. Any duplication or distribution, either in print or electronic form, without the
permission of McGraw-Hill, is prohibited.
Dynamics 2e 2057
Problem 9.78
Revisit Example 9.9 and obtain the expression for the force transmitted
to the floor, using the expression for the steady-state response of the
unbalanced motor.
of McGraw-Hill, and must be surrendered upon request of McGraw-Hill. Any duplication or distribution, either in print or electronic form, without the
permission of McGraw-Hill, is prohibited.
Problem 9.79
The system shown is released from rest when both
springs are unstretched and
xD0
. Neglecting the
inertia of the pulley
P
and assuming that the disk
rolls without slip, derive the equation of motion of
the system in terms of
x
. Assume that point
G
is
both the mass center of the disk and its geometric
center. Treat the quantities
k1
,
k2
,
c
,
m1
,
m2
, and
IG
as known, where
IG
is the mass moment of iner-
tia of the disk. Finally, assuming that the system is
underdamped, derive an expression for the damped
natural frequency of the system.
of McGraw-Hill, and must be surrendered upon request of McGraw-Hill. Any duplication or distribution, either in print or electronic form, without the
permission of McGraw-Hill, is prohibited.
