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Measurement of Structural Damping and Equivalent Mass of a
Vibrating Beam
1. Introduction:In this experiment the frequency Response
Function (FRF) will be used to:
a) determine the damping ratio, and equivalent damping
coefficient of a vibratingbeam
b) estimate the equivalent mass of a vibrating beamc) determine
the equivalent stiffness coefficient of a vibrating beam
Consider a vibrating beam with two layers of damping material
(plastic strips)as shown in figure
1.
Figure 1
This beam can be represented by a simple single degree of
freedom (SDOF) system. The damped
SDOF model and its free body diagram are shown in figure 2.
FBD
Figure 2
The equation of motion for this model is given by
(1)Where m, C, and kare the equivalent mass, damping and
stiffness coefficients respectively.
Note that from the previous experiment the equivalent mass and
stiffness can be measured or
determined from equations 2, and 3.
m=mtip+0.24 mbeam (2) (3)
E, I, L,mbeam
mtip
Damping Material
m
Ck
m
ky
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2. Estimating mass of a system based on F requency Response
Function (F RF)Consider a damped system subjected to a harmonic
force f(t) as shown in figure 3.
Figure 3The equation of motion for this system is given by
F0 cos (4)As shown in figure 3 for a harmonic force the response
of this system is also a harmonic
function. Assume a solution of x . The amplitude X, and phase
angle , aredetermined based on this assumed solution and its
derivatives.
(5) and (6)x, , and are substituted into equation 4 to get
(8)Equation 7 could be modified to represent FRF. Note that the
amplitude of the acceleration
function in equation 6 is . Therefore equation 7 can be written
as OR
(9)
In fact equation 9 represents the FRF that was generated by the
Real Time Analyzer (RTA) in
the previous experiment (see figure 4). Expanding the terms in
the denominator results in
(10)
Assume that the damping coefficient C is very small and is much
greater than n. In this case
the only dominant term in the denominator is and the remaining
terms in the denominatorcould be ignored.
C
k
m
C
F0 cos
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Equation 10 can be now simplified to
OR (11)
Equation 11 shows that mass of a system can be estimated from
its FRF. That is at large values
of, the plot becomes asymptotic to a value of 1/m (see figure
4). Note also the unit of is , which is the correct unit for
1/m.
Figure 4
3. Use Frequency Response Function (FRF) to determine damping
ratio The FRF can be also used to determine damping ratio of a
mechanical system. Equation 7 can be
written in a non dimensional form as
(12)
Where isF0/k, and damping ratio, as well as frequency ratio rare
defined by
The plot of versus frequency ratio r for a given damping ratio
is shown in figure 5. Notice
that the two plots in figures 4, and 5 are very similar.
Bandwidth
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Figure 5
There are two methods of finding damping ratio using FRF plot.
In the first method the
maximum value (Q) (or) is used, and in the second method the
half powerpoints 1, and 2 are used to determine the damping
ratio.
a)Determine damping ratio using the quality factor QAs is shown
in figure 5 for n (r=1), the ratio of or Q reaches its maximum
value.Therefore in equation 12, for a r=1, Q is given by
(13)However in this experiment FRF (
is measured, and ) should be corrected in equation 13.
There is a relationship between these two ratios. As it is shown
in equation 6, .
Therefore damping ratio can be evaluated by (14)
= =
Bandwidth
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b)Determine damping ratio usingmethod the half power points 1,
and2.As it is shown in figure 5, the half power points are obtained
by finding the intersections of
line and the given FRF. Note that from equation 13, . The
intersection points r1 (),
and r2 () are determined by
(15)The two roots of equation 15 are given by
(16)
For small damping ratios (
equations 16 can be simplified. Assume that
to get
Equations 17, and 18 are subtracted to get
(19)The damping ratio in equation 19 is then given by.
