Unlock document.
This document is partially blurred.
Unlock all pages and 1 million more documents.
Get Access
Dynamics 2e 2039
Problem 9.68
A delicate instrument of mass
m
must be isolated from excessive
vibration of the ground, which is described by the function
u.t/ D
Asin !0t
. To do so, we need to design a vibration isolating mount,
modeled by the spring and dashpot system shown.
(a)
Find the equation of motion of the instrument and reduce it to
standard form.
(b) Find the steady-state response y.t/.
(c) Find the displacement transmissibility, i.e., the response ampli-
tude
D
divided by
A
, where
A
is the amplitude of the ground’s
vibration.
Solution
Part (a)
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.
and where we have again used mg Dkıst.
DmA!2
0sin !0t:
Because of the orthogonality of
sin !0t
and
cos !0t
, we can equate the coefficients of each to obtain the
following two equations for the unknowns Band C
Part (c)
To find the displacement transmissibility, we divide the amplitude of the response by the amplitude of the
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 2041
and then, since sin !0tand cos !0tare orthogonal, the response amplitude Dis given by
u
u
t"mA!2
0km!2
0
0
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 9.69
When the connecting rod shown is suspended from the knife-edge at point
O
and displaced
slightly so that it oscillates as a pendulum, its period of oscillation is
0:77
s. In addition, it
is known that the mass center
G
is located a distance
LD110 mm
from
O
and that the
mass of the connecting rod is
661
g. Using the energy method, determine the mass moment
of inertia of the connecting rod IG.
Solution
The FBD of the connecting rod when it is at an arbitrary angle is shown on the right.
We will use the energy method to obtain the equation of motion, which will then
allow us to determine IG.
Balance Principles.
Referring to the FBD on the right, the kinetic energy of the
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 2043
Problem 9.70
Derive the equation of motion for the system, in which the springs with
constants
k1
and
k2
connecting
m
to the wall are joined in series. Neglect
the mass of the small wheels, and assume that the attachment point
A
between the two springs has negligible mass. Hint: The force in the two
springs must be the same; use this fact, along with the fact that the total
deflection of the mass must equal the sum of the deflections of the springs,
to find an equivalent spring constant keq.
Solution
The FBD of the mass is shown on the right, where Fsis the force in the spring.
Balance Principles.
Referring to the FBD at the right and summing forces in the
xdirection, 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.
Problem 9.71
Revisit Example 9.2 and compute the natural frequency of the
silicon nanowire, using the energy method. Use a uniform Si
nanowire with a circular cross section that is
9:8 m
long and
330 nm
in diameter and with all its flexibility lumped in a torsional
spring at the base of the wire. In addition, use
D2330 kg=m3
for the density of silicon and
ED152 GPa
for its modulus of
elasticity.
Photo credit: “Bottom-up Assembly of Large-Area Nanowire Resonators Arrays,” Nature Technology, 3(2), 2008, pp. 88-92.
Solution
The FBD of the bar is shown at the right, where
Mt
is the restoring
moment due to the torsional spring,
measures the rotation of the
bar, and
ı
is the deflection of the end of the bar. We will ignore the
work done by gravity in our solution since the nanowire oscillates
about its static equilibrium position.
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 2045
Computation.
Substituting the kinetic energy, potential energy, and the kinematic equation into the energy
method, 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.
Problem 9.72
Structural health monitoring technology detects damage in civil, aerospace, and other structures. Structural
damage is usually comprised of cracking, delaminations, or loose fasteners, which result in the reduction of
stiffness. Many structural health monitoring methods are based on tracking changes in natural frequencies.
Modeling a structure as a one DOF harmonic oscillator, calculate the change in stiffness needed to cause a
3% reduction in the natural frequency of the structure being monitored.
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 2047
Problem 9.73
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 angular speed of the rotor for resonance to
occur. Express your answer in revolutions per minute.
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.
