Problem 9.49
For identical systems, one with damping and the other without, would you expect the period of damped
vibration to be greater, less than, or equal to the period of undamped vibration? Explain your answer.
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Dynamics 2e 2001
Problem 9.50
A vibration test is performed on a structure, in which both the magnification factor
MF
and the phase
angle
are recorded as a function of excitation frequency
!0
. After the test, it is discovered that, for some
unfortunate reason, the recording of the magnification factor data is corrupted so that only the phase angle
data is available for analysis. Is it possible to determine the resonant frequency from the available data?
What can be inferred about the amount of damping in the system from the phase data?
Solution
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Problem 9.51
Suppose that the equation of motion of a damped forced harmonic
oscillator is given by
RxC2!nPxC!2
nxD.F0=m/ cos !0t
, where
x
is measured from the equilibrium position of the system. Obtain
the expression for the amplitude of the steady-state response of the
oscillator, and compare it with the expression presented in Eq. (9.72)
(which is for a system with equation of motion
RxC2!nPxC!2
nxD
.F0=m/ sin !0t).
Solution
To find the steady-state response for
cos !0t
forcing, we will follow the same procedure we used to find the
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Dynamics 2e 2003
This is easily seen to simplify to
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Problem 9.52
Differentiate Eq. (9.73) with respect to
!0=!n
, and set the result equal to zero to determine the frequency
!0
at which peaks in the
MF
curve occur as a function of
and
!n
. Use this result to show that the peak
always occurs at !0=!n1. Finally, determine the value of for which the MF has no peak.
Solution
Equation (9.73) is given by
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Dynamics 2e 2005
Problem 9.53
Calculate the response described by the equations listed below, in which
x
is measured in feet and time is
measured in seconds.
(a) 5RxC10 PxC100x D0, with x.0/ D0:1 and Px.0/ D 0:1
(b) 3RxC15 PxC12x D0, with x.0/ D0and Px.0/ D0:5
(c) RxC10 PxC25x D0, with x.0/ D0:15 and Px.0/ D0
(d) 25 RxC200 PxC1500x D0, with x.0/ D0:01 and Px.0/ D0
Solution
For each equation of motion, before finding the response, we will first need to determine the character of the
damping. That is, is the system overdamped, critically damped, or underdamped.
Part (a)
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Part (b)
Part (c)
For this system, mD1slug, cD10 lbs=ft, and kD25 lb=ft. Therefore the damping ratio is
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Dynamics 2e 2007
Part (d)
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Problem 9.54
Derive the equation of motion given in Eq. (1) of Example 9.9 for the
system in that example. The independent variable
y
is measured from
the equilibrium position of the system,
m
is the mass of the motor and
platform,
c
is the total damping coefficient of the dashpots,
k
is the total
constant of the linear elastic springs,
!r
is the angular velocity of the
unbalanced rotor,
“
is the distance of the eccentric mass from the rotor axis,
and
mu
is the eccentric mass. Note that
m
includes the eccentric mass so
that the nonrotating mass is equal to mmu.
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Dynamics 2e 2009
which becomes the equation of motion in Example 9.9, that is
r
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