978-0073380308 Chapter 9 Solution Manual Part 13

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)

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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

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Dynamics 2e 2041

and then, since sin !0tand cos !0tare orthogonal, the response amplitude Dis given by

u

t“mA!2

0km!2

0

0

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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

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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

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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.

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Dynamics 2e 2045

Computation.

Substituting the kinetic energy, potential energy, and the kinematic equation into the energy

method, we obtain

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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.

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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

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permission of McGraw-Hill, is prohibited.