Problem 9.55
A module with sensitive electronics is mounted on a panel that vibrates due to excitation from a nearby
diesel generator. To prevent fatigue failure, the module is placed on vibration-absorbing mounts. The
displacement of the panel is measured to be
yp.t/ Dy0sin !0t
, where
y0D0:001
m,
!0D300 rad=s
,
and the time
t
is measured in seconds. Letting the mass of the electronic module be
mD0:5 kg
, calculate
the amplitude of the vibration of the module if the equivalent stiffness and damping coefficients for all the
mounts combined are kD10;000 N=m and cD40 Ns=m, respectively.
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Dynamics 2e 2011
and then substituting it into the equation of motion in ´, we obtain
Since tan is given by Eq. (3), we can write cos as
„ ƒ‚ …
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cos sin.!0t /;
where we have let tan DDsin =.y0CDcos /. Therefore, the amplitude of vibration is given by
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Dynamics 2e 2013
Problem 9.56
A hard drive arm undergoes flow-induced vibration caused by
the vortices of air produced by a platter that rotates at
!0D
10;000 rpm
. The arm has length
LD0:037
m and mass
mD
0:00075 kg
, and it is made from aluminum with a modulus of
elasticity
ED70 GPa
. In addition, assume that the cross section
of the arm has an area moment of inertia
Ics D8:5 1014 m4
.
Following the steps in Example 9.2 on p. 672, the arm can be mod-
eled as a rigid rod that is pinned at one end and is restrained by a
torsional spring with equivalent spring constant
ktD3EIcs=L
.
In addition to the torsional spring, assume that the arm’s motion is
affected by a torsional damper with torsional damping coefficient
ct
. Assuming that the damping ratio is
D0:02
and that the vor–
tices produce an aerodynamic force with the same frequency as
the rotation of the platter, determine the amplitude of the aerody-
namic force needed to cause a steady-state vibration amplitude of
0:0001
m at the tip of the arm. Assume that the aerodynamic force
is applied at the midpoint
B
of the hard drive. What vibration
amplitude will result if the same excitation is applied to a hard
drive arm assembly with the damping ratio of 0.05?
Solution
The FBD of the arm is shown at the right, where we have
neglected the weight of the arm.
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Defining the following equivalent quantities
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Dynamics 2e 2015
Problem 9.57
The mechanism consists of a disk
D
pinned at
G
, which is both the
geometric center of the disk and its mass center. The outer circumfer-
ence of the disk has radius
roD0:1
m and is connected to an element
consisting of a linear spring with stiffness
k1D100 N=m
in parallel
with a dashpot with damping coefficient
cD50 Ns=m
. The disk
has a hub of radius
riD0:05
m that is connected to a linear spring
with constant
k2D350 N=m
. Knowing that for
D0
the disk is in
static equilibrium and that the mass moment of inertia of the disk is
IGD0:001 kgm2
, derive the linearized equation of motion of the
disk in terms of
. In addition, calculate the resulting vibrational
motion if the system is released from rest with an initial angular
displacement iD0:05 rad.
Solution
The FBD of the disk is shown on the right, where
Fc
is the force due to the
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and the solution is given by
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Dynamics 2e 2017
Problem 9.58
A box of mass
0:75 kg
is thrown on a scale, causing both the scale
and the box to move vertically downward with an initial speed
of
0:5 m=s
. Before the box lands on the scale, the scale is in
equilibrium. The total mass of the scale’s moving platform and
the box is
mD1:25 kg
. Modeling the platform’s support as a
spring and dashpot with stiffness
kD1000 N=m
and damping
coefficient
cD70:7 Ns=m
, find the response of the scale. Hint:
Place the origin of the
y
axis at the position of the platform
corresponding to the equilibrium configuration of the platform
and box together.
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permission of McGraw-Hill, is prohibited.