Mechanical Systems and Signal Processing 17 (5) 989-1001
APPLICATION OF PROPER ORTHOGONAL DECOMPOSITION TO
STRUCTURAL VIBRATION ANALYSIS
Sangbo Han*
Division of Mechanical Engineering
Kyungnam University
449 Wallyoung-dong,
Masan, 631-701
Korea
Tel) 82-551-249-2623
Fax) 82-551-249-2617
sbhan@kyungnam.ac.kr
Brian Feeny
Department of Mechanical Engineering
Michigan State University
2555 Engineering Building
East Lansing, MI 48824
Tel) 517-353-9451
Fax) 517-353-1750
feeny@msu.edu.
1
ABSTRACT
Mode shapes of a structure can be extracted without measuring a series of frequency response
functions by implementing proper orthogonal decomposition on the measured response data. If the
proper orthogonal decomposition is applied on the time responses of the structure, only one proper
orthogonal mode converges to the true normal modes of the structure. The degree of deviation of
the other extracted proper orthogonal modes from the true normal modes of the structure depends
on the spatial resolution, which is determined by the number of response measurement positions.
The suggested procedure of applying the proper orthogonal decomposition to the cross-spectral
density functions can extract all of the normal modes contained in the structural responses without
suffering from the limitation of the number of response measurement positions. Experimental data
of a homogeneous free-free beam and a non-homogeneous free-free beam were used to compare
the proper orthogonal modes and the structural normal modes, and the results strongly support the
applicability of the proper orthogonal decomposition to the structural vibration analysis.
1.
INTRODUCTION
Several efforts to extend the applicability of proper orthogonal decomposition (POD) as a modal
analysis tool have been reported since Lumley et al. introduced the idea to structural dynamics
community [1-3]. Since it is not required to measure the excitation forces as long as
simultaneously measured structural responses contain modes to be analyzed, POD is less restricted
to implement compared to the conventional modal testing, and therefore, the applicability of POD
can be maximized in the cases where it is difficult to assess or measure external excitation forces
to the structure.
However, there are several conditions satisfied with which the proper orthogonal modes (POMs)
from the POD can be considered as the normal modes of the structure. These are (1) the responses
contain every mode of the structure to be extracted, (2) the mass matrix of the structure is the
constant multiple of the identity matrix, or at least, the mass matrix of the structure corresponding
to the appropriate measurement coordinate system is given to perform a transformation on the
extracted POMs, (3) the natural frequencies of the structure are well separated to satisfy the
condition that the correlation of different modes converges to zero within the measured time
duration, (4) all the responses are measured simultaneously under the same excitation. Among
these restrictions, the most significant drawback of the POD is it requires the mass matrix of the
test structure should be known prior to extracting the mode shapes, which cannot be satisfied in
real situation.
From the structural vibration analysis point of view, one cannot extract all the information of the
structure with response measurement alone. In experimental modal analysis, there are techniques
to extract mode shapes of the structure using the responses alone. The so-called operational
deflection shape (ODS) can be extracted by comparing the responses of the structure with the
responses of the reference point along the structure when there is no information available on the
input forces [4-8]. On the other hands, to get the normal modes of the structure using the POD,
responses of the structures are measured simultaneously subjected with same input forces.
The POMs have been used as empirical modes in the analysis of fluids and structures for modal
projections of partial differential equations [1, 9-12]. POMS have been seen to approximate linear
modes in certain systems [2, 13-14]. Efforts to tie the POMs to linear normal modes in vibrating
systems include theoretical and numerical studies on discrete systems as well as experimental
2
study using the measured strain gage signals [15, 16]. But, until now, the application of POD to
structural vibration analysis has been limited to one-dimensional homogeneous structures.
Feeny and Kappagantu [17] and Kerschen and Golinval [18] showed that when the mass matrix of
the structure is proportional to the identity matrix and, at the same time, measured responses
contain all the modes of the structure, the proper orthogonal modes (POMs) converge to the mode
shapes of the structure. If the POD is applied on the raw time history data, experience suggests
that only one POM actually represents a true normal mode of the structure, corresponding to the
dominant mode in the responses [19]. In this case, the POMs can be considered as the normal
modes of corresponding lumped mass spring model of the structure that has the number of degree
of freedom equivalent to the number of response measurement positions.
This paper clarifies the applicability of the POD to distributed parameter systems by obtaining
experimental data of homogeneous and non-homogeneous beams, and propose a new method of
applying the POD on the frequency domain responses. It can be shown that this method can be
applied to extract normal modes of the structure with non-homogeneous mass distribution.
2. PROPER ORTHOGONAL DECOMPOSITION AND PROPER ORTHOGONAL MODES
Proper orthogonal decomposition (POD) is a procedure for simply extracting a basis for a modal
decomposition from an ensemble of signals measured from the vibrating structure [16]. In a
structural point of view, interpreting POD is based on the expansion theorem, by which responses
of a structure can be expressed as a linear sum of normal modes of that structure as follows.
L

x ( , t )   r ( )qr (t )
(1)
r 1
where  r ( ) is the r th normal mode
q r (t ) is the r th natural modal coordinate
L is the number of modes to be considered.
The POD starts with the construction of ensemble matrix of measured response signals as
 x1 (t1 )
 x (t )
X    1 2
 

 x1 (t N )
x L (t1 ) 
x L (t 2 ) 
 q1  1T    qL  TL


 

x 2 (t N )  x L (t N )
x 2 (t1 ) 
x 2 (t 2 ) 


(2)
Here, L represents the number of measurement positions along the structure, chosen in this case
to be the same as the number of modes to be extracted, because the maximum number of POMs to
be extracted is the number of the response measurement along the structure. The vectors qr are
N  1 arrays of the time function q r (t ) sampled at times t  t1 , t2 ,t N , and the vectors  r represent
the normal modes discretized with L points along the structure. Thus, each column of the ensemble
matrix X  represents the time history of displacement (or velocity or acceleration) measured at
some point of the structure. The correlation matrix on which the POD is performed is defined as
R  
1
X T X 
N
(3)
Since the correlation matrix R is real and symmetric, its eigenvectors form an orthogonal basis,
and these eigenvectors are orthogonal, such that
3
 Ti   j
  ij
(4)
On the other hand, the normal modes of the structure have the orthogonality defined as
 Ti m  j
  ij
(5)
Here comes the limitation of the applicability of the POD on the real structure. Comparing Eqs. (4)
and (5), the mass matrix of the structure should be the constant multiple of the identity matrix in
order that the POMs represent the normal modes of the structure. If the mass matrix of the
structure is known, one can start the POD on the adjusted correlation matrix Rˆ  Rm to extract

POMs, which converge to the normal modes of the structure, even if the mass matrix is not the
constant multiple of the identity matrix. But as pointed out by Feeny and Kappagantu [17], the
resulting matrix is no longer symmetric. Despite the lack of symmetry, the eigenvalues and
eigenvectors of Rm are real if m is symmetric and positive definite. This can be seen by
examining the eigenvalue problem, Rmu  u . Letting m1/ 2 u  v , then m1/ 2 Rm1 / 2 v  v.
Thus, v and  are real, and therefore u  m1/ 2 v is real.
The relationship between the eigenvectors of the correlation matrix R , and the modal vectors of
the structure   j , can be found by post multiplying the correlation matrix R by a modal vector.
Thus,
R  j
1
X T X   j
N
1
q1  1T    qL  TL

N


 q  
T
1
T
1
(6)

   qL     j
T
L
Now, if the mass matrix of the structure in Eq. (5) is the constant multiple of the identity matrix, or
the modal vectors  j are mass normalized to satisfy the orthogonality conditions in Eq. (4), and
then Eq. (6) can be simplified as
R  j


1
 1 q1T q j     L qTL q j
N

(7)
With the condition that the correlation between the principal coordinates qi q j approaches zero
T
with sufficiently long time period when i  j , Eq. (7) becomes
R  j
  j   j
(8)
where the eigenvalue
j 
1
qTj q j
N
(9)
is the mean squared value of the time series q j . Therefore, the eigenvectors of the correlation
matrix R are the modal vectors of the structure and the corresponding eigenvalues are the mean
square value of the signal corresponding to that mode as long as the mass matrix of the structure is
the constant multiple of the identity matrix and the correlation between modes are negligible.
If the mass matrix of the structure is not constant multiple of the identity matrix, then the
simplification in Eq. (7) does not hold, and the modal vectors and POMs have no physical
4
relationships. In other words, each modal vector of the structure  i and the corresponding POM
 i
has different shapes depending on the mass distribution of the structure. This difference
between the orthogonality conditions of the normal modes and the POMs is the main obstacle in
utilizing the simplicity of the POD in conjunction with the structural vibration analysis. Even though
the correlation matrix contains information about the normal modes of the structure, the POD
scheme tries to extract successive vectors that satisfy the orthogonality conditions of Eq. (4).
Therefore, except the first extracted POM that has the largest power among the modal components,
all other POMs deviate from the modal vector  i as the mode goes up. In practice, one can
consider the POMs as the normal modes of the uniform lumped-mass-spring system that
approximates the actual behavior of the structure.
To overcome this crucial restriction, we first consider the case where the responses of the
structure consist of only one particular mode, say i th mode. Modal parameter extraction method
using the SDOF modal analysis technique is based on the fact that we may obtain an estimate for
the modal constants of the mode being analyzed by assuming that the total response in this
resonant region is attributed to a single term in the general FRF series [12]. The same can be
applied to the responses of the structure measured simultaneously under the same excitation force,
that is, the response of the structure excited in one particular mode is the mode itself. Then,
structural responses measured at different positions are given as
x  i A sin  i t
(10)
And the ensemble matrix and the correlation matrix are given as
 A sin  i t1 
 A sin  t 
T
i 2 
X   
 i



 A sin  i t N 
R 
1
X T X    i  1
N
N
(11)
N

k 1

 A 2 sin 2  i t k  i
T
(12)
1 N

Since the term   A 2 sin 2  i t k  is simply the mean squared value of the harmonic signal, the
 N k 1

correlation matrix of the response signals with only one modal component becomes
R  qi rms 2  i  Ti
(13)
The rank of this L L matrix of  i  i is only one; its eigenvector corresponding to non-zero
eigenvalue is  i itself. In this case, extracted POM corresponds to the normal mode of the
T
structure. Therefore, if one filters out all the measured signals such as to contain only one
particular mode to construct the ensemble matrix, then no matter what the mass matrix of the
structure is, the POM extracted from those signals converges to that particular mode of the
structure. Note that this filtered signal should not be confused with the measured signal under
single harmonic excitation, in which case the response does not necessarily contain any particular
modal component of the structure.
According to Eq. (2) and (3), each element of the correlation matrix is simply a cross correlation
function between individual ensemble signal with time delay   0 . The relationship between the
two-sided cross-spectral density function and the cross correlation function is given as
5
R xy ( ) 
1 
1
j 2f
*
df  0 G xy
( f )e  j 2f df
0 G xy ( f )e
2
2

0

(14)

  C xy ( f ) cos 2f  Q xy ( f ) sin 2f df
where G xy ( f ) is two-sided cross-spectral density function between signal x and y
*
G xy
( f ) is the complex conjugate of G xy ( f )
C xy ( f ) is the real part of G xy ( f )
Q xy ( f ) is the imaginary part of G xy ( f )
Therefore, each element of the correlation matrix corresponds to the following quantity.

Rxy (0)  0 Cxy ( f )df
(15)
This means that the whole area under the real part of the cross-spectral density function gives the
cross correlation function of time delay   0 .
The filtering scheme extracting particular mode from the response signals can be more easily done
with the cross-spectral density functions than with time history data when using a FFT signal
analyzer. Another advantage of using cross spectral density function over a time response signal is
that one can significantly reduce the random error of the measured signals by taking the average of
individual ensemble signals [20, 21]. Furthermore, by looking at the cross-spectral density
functions, one can easily identify the natural frequencies of the structure, which cannot be obtained
directly from the POD applied on the time signal data.
3.
EXPERIMENTS
A 12.7 X 12.7 X 1500 mm uniform steel beam was prepared to compare its normal modes and
experimentally extracted POMs. The boundary conditions of the beam are chosen to be free-free,
because it is the easiest to set up and its experimental data contains the least error involved in the
realization of boundary conditions [20]. The beam was hung horizontally with two soft strings
attached at the ceiling and the vibration responses in the horizontal direction were measured under
the impact in the same direction.
Six equally spaced accelerometers were attached along the length of the beam to measure the
acceleration response of the beam. Ideally, all accelerometers should have not only the same
physical dimensions but also the same electronic characteristics including sensitivities and time
constants to ensure precise measurement of both amplitude and phase differences of each response
along the beam. But, it is not common for an academic laboratory to have six or more identical
accelerometers. And furthermore, even the accelerometers with the same serial number from the
same manufacturer can have slightly different sensitivities and time constants depending on the
signal conditioning circuits and amplifiers used. The accelerometers used in this experiment were
from two different manufacturers. Each accelerometer was carefully calibrated using a standard
back-to-back method to find its sensitivity [22]. The dimensions, model numbers, manufactures of
the accelerometers used in the experiment along with the calibration results are given in Table 1.
Beam responses were measured by impacting the beam with a hammer at one free end, which can
activate all the modes of the beam within the range of frequencies of interest. The number of
modes of the beam to be activated can be controlled by either adjusting the duration of the impact
or adjusting the Nyquist frequency of the measurement. All the accelerometer signals were
simultaneously fed into the B&K 2035 FFT signal analyzer with 8-channel input module. Therefore,
each measured signal represents the response of the beam subjected to a single external
6
disturbance. B&K 2035 FFT is a signal analyzer that captures the time domain signals with 2048
sampling points and converts them into various time and frequency domain functions with 801
discretized points. The responses of the beam were captured for one second. As such the sampling
time accommodates up to 7th mode of the beam in the frequency domain. In addition, as in the
conventional modal analysis, cross spectral density functions of these measured signals were
calculated by taking linear average of 8 independent measurements. The locations of the
accelerometers along the beam are sketched in Fig. 1.
To construct a beam with non-homogeneous mass distribution, an additional mass of 0.5 kg is
attached at the accelerometer position #5.
Table 1. Physical dimensions and sensitivities of the accelerometers used in the experiment
Figure 1: Setup for the response measurements of the test beam.
4. RESULTS AND DISCUSSIONS
Figure 2 shows the correlation values between two principal coordinates in various phase angles
and frequency ratios. The frequency ratio is defined as the ratio of the frequency of the higher
mode to the frequency of the lower mode. When the sampling time duration corresponds only one
period of the lower mode signal as in case (a), it seems that the correlation term is not negligible
for closely adjacent modes. But, when the sampling time duration is expanded to 5 periods of the
lower mode as in case (2), the correlation term rapidly decays out with increased value of the
frequency ratio. Figure 2 supports the validity of the derivation of Eq. (7) to (8), and at the same
time, addresses another restriction the POD scheme has, that is, normal modes of structures with
closely spaced natural frequencies can not be extracted correctly by the POD.
Figure 2. Correlation of two signals with various phase shifts and frequency ratios.
Figure 3 shows the POMs extracted from the experimental data of the homogeneous beam. The
correlation matrix was constructed using six simultaneously measured acceleration responses, and
each acceleration signal represents one ensemble signal without averaging or any signal processing.
The comparison of six POMs extracted from this acceleration data with the lowest six normal
modes of the beam revealed a very unsatisfactory result. Examining the power spectral density
functions of six acceleration responses of the beam that are shown in Fig. 4, it was found that the
order of the power of each mode is quite reverse to the case of the displacement signals. Unlike
typical displacement signals, the signal power of the 7th mode is the largest among the modes
contained in the acceleration responses. Since the eigenvalues of the correlation matrix are simply
the mean squared values of the ensemble data in the direction of the individual mode extracted, the
order of the POMs and the order of the normal modes of the structure can be quite different,
especially when the natural frequencies of the structure are very high. This indicates that auxiliary
information is needed for the important task of identifying which POM corresponds to which normal
modes of the actual structure [15].
Figure 3. Comparison of lowest theoretical mode shapes and the POMs extracted from only the
single measurement of acceleration responses.
Figure 4. Time histories and PSD functions of the acceleration response of the test beam.
The result of comparing the POMs with the normal modes of the beam in the reverse order from
the 7th mode is shown in Fig. 5. As can be seen in this figure, the first two POMs nicely match with
the 7th and 6th modes of the beam. The poor agreement between the POMs and the normal modes
of the beam except the first two modes are due to the lack of spatial resolution. Since the POD tries
7
to extract eigenvectors that satisfy the orthogonality conditions of Eq. (4), the POMs extracted
successively will deviate from the true normal modes of the structure except the first POM which is
also a discretized approximation of the true normal mode. This addresses the inconvenience of the
practical aspect of the POD, in which one cannot extract all the modes of the structure from the
correlation matrix constructed with the measured time history data. From the result of Fig. 5, it can
be said that the POD scheme can extract even a very high mode of the structure as long as the
modal contribution of that mode to the response is the most significant. But, it should also be
mentioned here that even though higher modes of the structure can be extracted using the POD, it
is practically absurd to express fluctuating higher modes with limited number of node points.
Figure 5. Comparison of theoretical mode shapes and the POM extracted from the single
measurement of acceleration responses in the reverse order from the 7th mode.
To emphasize the fact that the POD process can extract only one true normal mode that has the
maximum modal contribution, the response signals of the beam were low pass filtered below 500 Hz
to contain modes lower than the 5th mode. The first POM extracted from these signals is compared
with the 5th theoretical normal mode of the beam in Fig. 6. It can be seen that the POM is much
closer to the theoretical mode than the one given in Fig. 5.
Figure 6. Comparison of 5th theoretical mode shape and the first POM extracted from the low
pass filtered response signals.
Since lower modes of the structure are usually required in the preliminary stage of structural
vibration analysis, it is necessary to extract lower modes of the beam effectively from the
experimental data. It is possible to apply the POD on the measured response data after converting
the acceleration responses of the test beam into the displacement responses. There are two
methods, in general, to convert the acceleration signal into the displacement signal. One is directly
integrating the acceleration signal in time domain. The other is dividing the Fourier transformed
acceleration signal by the scale factor of   and taking its inverse Fourier transform. It turned
out both methods produced a significant amount of errors depending on the sampling resolution in
time and frequency domain to digitize the response signals. It is well known that to have better
resolution in time domain, one has to compromise with the coarse resolution in frequency domain
and visa versa with given number of sampling. Therefore, with a fixed resolution in the time and
frequency domains, converting high frequency signals in time domain and converting low frequency
signals in frequency domain will produce biased errors. An effective way to convert the
acceleration signal into a displacement signal without significant errors should be studied further.
2
The required processing on the signals to provide the condition specified in Eq. (13) involves
filtering out the rest of the modal components to ensure the signals contain only one desired mode
to be extracted. This signal process may seem irritating especially when filtering circuits are not
available. But, when the signals are stored in frequency domain functions, it is much easier to apply
the band pass filter. According to Eq. (15), each element of the correlation matrix can be calculated
by integrating the real part of the cross-spectral density functions. Simply taking only an
appropriate portion (usually 5 or 7 spectral lines around the resonant frequency) of the entire
cross-spectral density functions and summing the ordinate values gives the filyered elements of the
correlation matrix. This one-by-one process of signal filtering can be considered as the trade off
to the conventional modal analysis in which frequency response functions (FRFs) not responses
alone are to be measured.
To improve the quality of the POMs, all the acceleration signals were averaged in the frequency
domain to reduce the random errors. Then correlation matrices were constructed with filtered
signals to contain only one specific modal component. The resulting POMs are considered as a
reasonable representation of the theoretical normal modes of the beam up to sixth mode as can be
seen in Fig. 7. The results in Fig. 7 state that the suggested POD procedure of constructing the
8
correlation matrix using the cross-spectral density functions can extract all of the normal modes of
the structure without suffering from the spatial resolution due to the limited number of response
measurement positions.
Figure 7. Comparison of theoretical mode shapes and the POMs extracted from the
averaged and signal processed acceleration responses.
To find out whether this mode-by-mode scheme works for the responses of structure with nonhomogeneous mass distribution, the beam with an additional mass attached on the #5 accelerometer
position was tested. All the measurement conditions were identical to the case of the homogeneous
beam. The POMs of the non-homogeneous beam are compared with the normal modes estimated
from the finite element model as well as the normal modes of the homogeneous beam in Fig. 8,
which suggests that the procedure is satisfactory for the structure with non-homogeneous mass
distribution, too.
The experiments conducted in this paper were limited to one-dimensional structures. For general 3
dimensional structures, one can apply the same procedure to find the corresponding mode shape of
that particular part of the structure.
Figure 8. Comparison of theoretical mode shapes and the POMs of non-homogeneous
beam.
5. CONCLUSIONS
When simultaneously measured vibration responses along the structure are available, normal modes
of the structure can be extracted without measuring the full series of FRFs as long as the response
signals contain all the modes to be extracted.
If the POD is applied to the time responses of the structure, experience indicates that only one
POM converges to the true normal modes of the structure. The degree of deviation of the other
POMs from the true normal modes of the structure depends on the spatial resolution, which is
determined by the number of response measurement positions.
The suggested procedure to apply the POD on the cross-spectral density functions can be used to
extract all of the normal modes contained in the structural responses without suffering from the
limitation of the number of response measurement positions. The frequency domain signal process
has additional advantage of identifying the natural frequencies of the structure without conducting
the conventional modal decomposition.
The validity of the application of the POD to non-homogeneous structures was also verified in an
example, and the result supports that this idea can be extended to a more general structure. In
practice, the main obstacle of applying the POD is the availability of sufficiently large numbers of
transducers to measure the responses simultaneously. The POD scheme can be applied to any
structures, provided responses are measured simultaneously in the unitary direction by
decomposing the structure into substructures.
CKNOWLEDGEMENTS
Part of this work is supported by the Kyungnam University Research Fund, 2002, and Author BF is
grateful for support from the National Science Foundation (CMS-0099603).
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9
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‘ Optimal’ Modal Reduction of a System with Frictional Excitation.
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Anniversary Volum,. 159-143. Solitons, Chaos and Modal Interactions in Periodic Structures.
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Table 1. Physical dimensions and sensitivities of the accelerometers used in
the experiment
Manufacturer
PCB
PCB
PCB
PCB
PCB
IMV
Model
Number
352B22
338B01
302A07
351B11
303A03
VP4200
Dimension (mm)
Base Dia. X height
9.14 X 3.6
6.35 X 12.5
12.7 X 33.0
7.9 X 10.9
7.14 X 12.2
12.7 X 25.4
Weight
(gr)
0.5
2.2
25.0
2.0
1.9
32.6
12
Sensitivity
(mV/g)
9.97
10.06
10.0
5.35
11.1
9.51
NO. 1
NO. 2
NO. 3
NO. 4
NO. 5
ACCELEROMETER
NO. 6
IMPACT HAMMER
Figure 1. Setup for the response measurements of the test beam.
13
Correlation Value
0.5
phase=0 degree
phase=45 degree
phase=90 degree
phase=135 degree
0
-0.5
1
1.5
2
Frequency ratio
2.5
3
(a) Sampling time is 1 period of the lower frequency signal
Correlation Value
0.5
phase=0 degree
phase=45 degree
phase=90 degree
phase=135 degree
0
-0.5
1
1.5
2
Frequency ratio
2.5
3
(b) Sampling time is 5 periods of the lower frequency signal
Figure 2. Correlation of two harmonic signals with various phase shifts and
frequency ratios.
14
1
1
0
0
-1
-1
0
0.5
1
1.5
1
1
0
0
-1
0
0.5
1
1.5
-1
1
1
0
0
-1
0
0.5
1
1.5
0
0.5
1
1.5
0
0.5
1
1.5
-1
0
0.5
1
1.5
Figure 3. Comparison of lowest theoretical mode shapes and the POMs
extracted from only the measurement of acceleration responses.
 : Theoretical mode shapes
* : POMs
15
4
g2/Hz
g2/Hz
1
0.5
1
g2/Hz
g
0.5
0
-20
20 0
g
1
0
-20
20 0
0
-20
20 0
g
0.5
0.5
1
g2/Hz
g
-20
20 0
0
-20
20 0
g
g2/Hz
0
0.5
1
g2/Hz
g
20
0
-20
0
0.5
TIME (SEC.)
1
10
2
10
0
104
10 0
200
400
600
800
200
400
600
800
200
400
600
800
200
400
600
800
200
400
600
800
200
400
600
FREQUENCY (Hz)
800
2
10
0
10
4
10 0
2
10
0
104
10 0
2
10
0
10
4
10 0
2
10
0
104
10 0
2
10
0
10
0
Figure 4. Time histories and PSD functions of the acceleration response of the
test beam.
16
1
1
0
0
-1
-1
0
0.5
1
1.5
1
1
0
0
-1
0
0.5
1
1.5
-1
1
1
0
0
-1
0
0.5
1
1.5
0
0.5
1
1.5
0
0.5
1
1.5
-1
0
0.5
1
1.5
Figure 5. Comparison of theoretical mode shapes and the POM extracted
from the time histories of acceleration responses in the reverse
order from the 7th mode.
 : Theoretical mode shapes
* : POMs
17
1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
0
0.5
1
1.5
Figure 6. Comparison of 5th theoretical mode shape and the first POM
extracted from the low pass filtered response signals.
 : Theoretical 5th mode shape of the beam
* : First POM extracted from the low pass filtered signals
18
1
1
0
0
-1
0
0.5
1
1.5
-1
1
1
0
0
-1
0
0.5
1
1.5
-1
1
1
0
0
-1
0
0.5
1
1.5
-1
0
0.5
1
1.5
0
0.5
1
1.5
0
0.5
1
1.5
Figure 7. Comparison of theoretical mode shapes and the POMs extracted
from the averaged and signal processed acceleration responses.
 : Theoretical mode shapes
* : POMs
19
1
1
0
0
-1
-1
0
0.5
1
1.5
1
1
0
0
-1
0
0.5
1
1.5
0
0.5
1
1.5
0
0.5
1
1.5
-1
0
0.5
1
1.5
1
1
0
0
-1
-1
0
0.5
1
1.5
Figure 8. Comparison of theoretical mode shapes and the POMs of
non-homogeneous beam.
 : Mode shapes of the non-homogeneous beam
* : POMs
-- : Theoretical mode shapes of the homogeneous beam
20