Dynamics 2e 1943
Problem 9.17
For the silicon nanowire in Example 9.2, use the lumped mass
model shown, in which a point mass
m
is connected to a rod of
negligible mass and length
L
that is pinned at
O
, to determine the
natural frequency
!n
and frequency
f
of the nanowire. Use the
values given in Example 9.2 for the mass of the lumped mass, the
length of the massless rod, and the parameters used to determine
the spring constant
kD3EIcs=L3
. You may use either
ı
or
as the position variable in your solution. Assume that the
displacement of mis small so that it moves vertically.
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From Example 9.2, we know that the cross section of the Si nanowire is circular and that it has the following
The frequency of vibration goes down when all the mass is lumped at the end of the wire since it now has a
larger mass moment of inertia with respect to point
O
when compared with the rigid body model used in
Example 9.2.
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Dynamics 2e 1945
Problem 9.18
The small sphere
A
has mass
m
and is fixed at the end of the arm
OA
of negligible mass, which is pinned
at
O
. If the linear elastic spring has stiffness
k
, determine the equation of motion for small oscillations,
using
(a) the vertical position of the mass Aas the position coordinate,
(b) the angle formed by the arm OA with the horizontal as the position coordinate.
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Kinematic Equations. The needed kinematics relation ˛OA DR
.
Computation.
Substituting the force law and the kinematics equation into Eq. (1)), we obtain a differential
equation in instead of y:
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Dynamics 2e 1947
Problem 9.19
The uniform cylinder rolls without slipping on a flat surface. Let
k1D
k2Dkand rDR=2. Assume that the horizontal motion of Gis small.
Determine the equation of motion for the cylinder by writing its
Newton-Euler equations. Use the horizontal position of the mass center
Gas the degree of freedom.
Force Laws. For the spring forces, we can write
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Computation. Substituting Eqs. (2)–(4) into Eq. (1), we obtain the following equation of motion
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Dynamics 2e 1949
Problem 9.20
The uniform cylinder rolls without slipping on a flat surface. Let
k1D
k2Dkand rDR=2. Assume that the horizontal motion of Gis small.
Determine the equation of motion for the cylinder using the energy
method. Use the horizontal position of the mass center
G
as the degree
of freedom.
Solution
Referring to the FBD on the right, if
FA
is the force in the spring of constant
k1
and
FB
is the force in the spring of constant
k2
, we see that only
FA
and
FB
do work as the wheel undergoes small horizontal motion and rolls
without slipping.
Balance Principles.
Since both of those spring forces are conservative, we
can apply the energy method, which is
d
dt .T CV / D0
, to determine the
equation of motion of the cylinder. The kinetic energy of the cylinder is
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Computation. Using the expression for the mass moment of inertia and the kinematics equations derived
above, the kinetic energy can be written as
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Dynamics 2e 1951
Problem 9.21
Grandfather clocks keep time by advancing the hands a set amount per oscillation of the pendulum.
Therefore, the pendulum needs to have a very accurate period for the clock to keep time accurately. As a
fine adjustment of the pendulum’s period, many grandfather clocks have an adjustment nut on a bolt at
the bottom of the pendulum disk. Screwing this nut inward or outward changes the mass distribution of
the pendulum by moving the pendulum disk closer to or farther from the axis of rotation at
O
. Model the
pendulum as a uniform disk of radius
r
and mass
mp
at the end of a rod of negligible mass and length
Lr, and assume that the oscillations of are small. Let mpD0:7 kg and rD0:1 m.
If the pendulum disk is initially at a distance
LD0:85
m from the pin at
O
, how much would the
period of the pendulum change if the adjustment nut with a lead of
0:5 mm
was rotated four complete
rotations closer to the disk? In addition, how much time would the clock gain or lose in a
24
h period if
this were done?
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Kinematic Equations. Since we need to find the equation of motion, we let !pDP
.
Computation. Substituting the kinetic and potential energies into the energy method, we obtain
We are keeping more than four significant figures for
i
and
f
so that the change in period can be calculated to four significant
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