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Problem 9.38
Revisit Example 9.6 and discuss whether it is possible to obtain the
equation of motion of the system via the energy method.
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Dynamics 2e 1981
Problem 9.39
A fatigue-testing machine for electronic components consists of a
platform with an unbalanced motor. Assume that the rotor in the
motor spins at
!rD3000 rpm
, the mass of the platform is
mpD
20 kg
, the mass of the motor is
mmD15 kg
, the unbalanced mass is
muD0:5 kg
, and the equivalent stiffness of the platform suspension
is
keq DnksD5106N=m
, where
n
is the number of springs. For
the testing machine, the distance
"
between the spin axis of the rotor
and the location at which
mu
is placed can be varied to obtain the
desired vibration level. Calculate the range of values of
"
that would
provide amplitudes of the particular solution ranging from
0:1 mm
to
2mm.
Solution
This system is identical to that found in Example 9.6. Therefore, using
kD5106N=m
,
mmD15 kg
, and
mpD20 kg, we know that the square of the natural frequency of the system is given by
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Problem 9.40
At time
tD0
, a forced harmonic oscillator occupies position
x.0/ D
0:1
m and has a velocity
Px.0/ D0
. The mass of the oscillator is
mD10 kg
, and the stiffness of the spring is
kD1000 N=m
. Calculate
the motion of the system if the forcing function is
F .t/ DF0sin !0t
,
with F0D10 N and !0D200 rad=s.
Solution
As we saw in Eq. (9.31) on p. 681, the equation of motion for this system is given by
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Dynamics 2e 1983
Problem 9.41
The forced harmonic oscillator shown has a mass
mD10 kg
. In
addition, the harmonic excitation is such that
F0D150
N and
!0D200 rad=s
. If all sources of friction can be neglected, determine
the spring constant ksuch that the magnification factor MF D5.
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. Substituting the force law, the kinematic equation, and F .t/ DF0sin !0tinto Eq. (1), we
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Dynamics 2e 1985
Problem 9.42
A ring of mass
m
is attached by two linear elastic cords with elastic
constant
k
and unstretched length
L0< L
to a support, as shown.
Assuming that the pretension in the cords is large, so that the cords’
deflection due to the ring’s weight can be neglected, find the linearized
equation of motion for the case where
F .t/ DF0sin !0t
and
w.t/ D
0
(i.e., the support is stationary). In addition, find the response of the
system for y.0/ D0and Py.0/ D0.
In addition, since we want the equation of motion, we let ayD Ry.
Computation.
Substituting the kinematics equations and the force law into the balance law in Eq. (1), we
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permission of McGraw-Hill, is prohibited.
where we obtained the last approximation by noting that
yL
and so
.y=L/2
and higher powers can be
ignored. Therefore the linearized equation of motion becomes
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Dynamics 2e 1987
Problem 9.43
A ring of mass
m
is attached by two linear elastic cords with
elastic constant
k
and unstretched length
L0< L
to a support, as
shown. Assuming that the pretension in the cords is large, so that
the cords’ deflection due to the ring’s weight can be neglected,
find the linearized equation of motion for the case where
F .t/ D0
and
s.t/ Ds0sin !t
. In addition, find the response of the system
for y.0/ D0and Py.0/ D0.
In addition, the acceleration of the ring can be found using
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permission of McGraw-Hill, is prohibited.
Looking at the second term in parentheses, we note that using a Taylor series expansion about
yD0
(this is
sometimes called a binomial expansion) we can write it as
1
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Dynamics 2e 1989
Problem 9.44
Modeling the beam as a rigid uniform
thin bar, ignoring the inertia of the pul-
leys, assuming that the system is in
static equilibrium when the bar is hor-
izontal, and assuming that the cord is
inextensible and does not go slack, de-
termine the linearized equation of mo-
tion of the system in terms of
x
, which
is the position of
A
. Finally, determine
an expression for the amplitude of the
steady-state vibration of block A.
2FsLmBgL
2DIO˛B;(2)
where
T
is the cord tension,
˛B
is the angular acceleration of the bar,
Fs
is the force in the spring, and
IOD1
3mBL2is the mass moment of inertia of the bar Babout point O.
Force Laws. The force laws are
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permission of McGraw-Hill, is prohibited.
