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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 1991
Problem 9.45
For the system in Prob. 9.44, deter-
mine the maximum forcing frequency
!0
for steady-state motion, such that
the cord does not go slack.
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.
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.
Canceling an Land noting that 1
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Dynamics 2e 1993
Problem 9.46
One of the propellers on the Beech King Air 200 is unbalanced, such that the unbalanced weight
Wu
is a distance
R
from the spin axis of the propeller. The propellers spin at a constant rate
!p
, and the
weight of each engine is
We
(this includes the mass of the propeller). Assume that the wing is a uniform
beam that is cantilevered at
A
, has weight
Ww
and bending stiffness
EI
, and whose mass center is at
G
. Treat the engine as a point mass and evaluate your answers for
WuD3oz
,
WeD450 lb
,
RD5:1 ft
,
!pD2000 rpm
,
EI D1:13 1011 lbin:2
,
dD8:7 ft
, and
hD10:9 ft
. In addition, ignore the angular
motion of the wing in computing the angular velocity and angular acceleration of the propeller, and ignore
the time-dependent inertia term in the final equation of motion.
Neglect the mass of the wing, and model the wing as was done in Example 9.2. Determine the
resonance frequency of the system, and find the MF for the given parameters.
gd2
have treated the engine as a point mass.
Force Laws. The force law for the torsional stiffness is given by
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.
Kinematic Equations. Relating the acceleration of the unbalanced mass to that of the engine, 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.
To find the
MF
, we need to find a particular solution
p
to the equation of motion in Eq. (4), and then use
its amplitude to determine the
MF
. The most direct way to find a particular solution is to assume a solution
of the form pDDsin !ptand then substitute it into the equation of motion. Doing so, we obtain
where we have used the expression for !nfound above and !pD2000 rpm D2000
30 rad=s.
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 9.47
One of the propellers on the Beech King Air 200 is unbalanced, such that the unbalanced weight
Wu
is a distance
R
from the spin axis of the propeller. The propellers spin at a constant rate
!p
, and the
weight of each engine is
We
(this includes the mass of the propeller). Assume that the wing is a uniform
beam that is cantilevered at
A
, has weight
Ww
and bending stiffness
EI
, and whose mass center is at
G
. Treat the engine as a point mass and evaluate your answers for
WuD3oz
,
WeD450 lb
,
RD5:1 ft
,
!pD2000 rpm
,
EI D1:13 1011 lbin:2
,
dD8:7 ft
, and
hD10:9 ft
. In addition, ignore the angular
motion of the wing in computing the angular velocity and angular acceleration of the propeller, and ignore
the time-dependent inertia term in the final equation of motion.
Let the mass of the wing be
mwD350 lb=g
, and model the wing as was done in Example 9.2.
Determine the resonance frequency of the system, and find the MF for the given parameters.
gd2C1
12
g.2h/2CWw
gh2DWe
gd2C4
3
gh2;
where we have assumed that the wing is a uniform beam of length 2h.
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 1997
Force Laws. The force law for the torsional stiffness is given by
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 therefore, the inertia term is dominated by the constant part, which means that we can approximate the
gsin !ptC3EI
dDsin !ptD Wu
gdR!2
which, upon canceling sin !ptand solving for D, gives
p
p=.3gEI /
Therefore,
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Dynamics 2e 1999
Problem 9.48
In the design of a MacPherson strut suspension, what would you choose for
the damping ratio
? Explain your answer in terms of automotive ride and
comfort.
shock
absorber
and spring
steering
link
car frame
lower
control arm
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permission of McGraw-Hill, is prohibited.
