Numerical parametric study on interval shift variation in simo sstd technique for experimental modal analysis (ema)

IJRET: International Journal of Research in Engineering and Technology eISSN: 2319-1163 | pISSN: 2321-7308
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Volume: 03 Issue: 12 | Dec-2014, Available @ http://www.ijret.org 178
NUMERICAL PARAMETRIC STUDY ON INTERVAL SHIFT
VARIATION IN SIMO-SSTD TECHNIQUE FOR EXPERIMENTAL
MODAL ANALYSIS (EMA)
Ahmad Fathi Ghazali1
, Ahmad Azlan Mat Isa1
1
Faculty of Mechanical Engineering, Universiti Teknologi MARA, 40450 Shah Alam,Selangor D.E., Malaysia.
Abstract
SSTD (Single Station Time Domain) technique is one of the efficient methods used to extract modal parameters of a structure.
Instead of extracting the parameters in frequency domain by utilising Fast Fourier Transform (FFT) as used by some other
methods, this technique only relies on free decay responses to obtain the required parameters. In this paper, a parametric study
on time shift interval on algorithm method is investigated. The accuracy assessment is made using simulated analytical data with
known properties where percentage error between the results of simulated and this method can be calculated and compared.
Keywords: SSTD (Single Station Time Domain), FFT, EMA
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1. INTRODUCTION
One objective of doing modal analysis is to extract dynamic
characteristics of a system in terms of natural frequencies,
mode shapes and damping ratios. They are normally known
as modal parameters. The knowledge of these parameters
can feed various purposes including structural modification
and health monitoring, model updating, assessment of the
structural integrity and reliability to name a few [1, 2]. A
number of literature reported that the applications cover
wide range of engineering, science and technology fields
namely in aerospace, automotive, mechanical, civil, musical
instruments, medicine as well as bioengineering for example
[3-5].
One way of determining modal parameters is via theoretical
modal analysis which assumes the information of stiffness
matrix K, mass matrix M and damping matrix C to be
known. However, it is usually difficult to have a reliable
mathematical model that can accurately represent dynamic
behaviour of different components and of the complete
assembly. This include difficulties not only in modelling
joints, welding and other type of connections but also in
describing damping in an adequate manner [6].
Second approach is by experimental, popularly known as
‘experimental modal analysis (EMA)’. Due to inadequacy of
analytical modelling of structures, endeavours on
experimental or combined experimental method have
become centre of attention as seen by the diversity
developments of modal parameter identification techniques
since 1970’s. With the revolutionary advent of Fast Fourier
Transform (FFT) by Cooley and Tukey at that time as well,
EMA has grown steadily and become widespread [7]. The
techniques can be classified into two categories; frequency
and time domain.
The general idea behind the frequency domain methods is a
curve fitting process done on a set of function known as
Frequency Response Function (FRF). This function is
simply a ratio between the output responses to the input
excitation force. In practise, this is obtained by transforming
the raw time signal data to FRF using FFT algorithm found
in any signal processing analyser or in computer software
packages before it is curve fitted. Curve fitting is basically a
process of matching a predefined mathematical expression
to the measured data. Thus, the matching objective is to
minimize the error function defined between the analytical
expression and measured data before any process of
extracting modal parameters can be made. Clearly, the
accuracy will depend on the quality of measured FRF as
well as correct mathematical model used. Examples of this
identifications are Rational Fractional Polynomial (RFP) [8-
11], Ewins-Gleeson method [12] and Simultaneous
Frequency Domain (SFD) [3] .
A number of drawbacks have been reported such as aliasing,
leakage, requirement for large amount of data, sampling and
quantization error during measurement. Most of these
shortcomings were aided with the expansion of technology
in instruments precision, fast processing hardware, storage
capacity device as well as advancement in measurement
techniques. This is true except for leakage error. So far this
is the greatest challenge for frequency domain methods as
the error cannot be eliminated. Briefly, the Fourier
Transform process demanded sampled data consist of
complete representation of data for all time or contain
periodic repetition of the measured data. When this is not
the case as this is what exactly happen in practice, leakage
occur causing serious distortion of data. Although
windowing is applied to reduce the problem, the technique
could give harmful side effect on the accuracy of modal
parameter estimation especially damping [13, 14]. Other
problems associated with frequency domain methods are
listed in Ref [15].
Due to the aforementioned issues, the inertia of research
tends to shift back to the use of time domain. There are

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several advantages using this domain. Unlike FRF, time
domain methods manipulate only time signal of response
instead the need to measure excitation force. Thus, less
information is needed and no issue of leakage will take
place. It is also reported that the methods are able to
differentiate closed mode which has been a problem among
frequency domain methods [16].
However, this type of methods has its own challenges. This
includes high sensitivity towards noise and the ‘invisibility’
of structural modes in the time signal. The existence of noise
may pave to computational modes in the calculation which
they are not the genuine vibration mode but nevertheless
‘identified’. Some examples of this type technique are
Ibrahim Time Domain (ITD) [16-18], Single Station Time
Domain (SSTD) [19], Complex Exponential (CE) and
Eigensystem Realization Algorithm (ERA).
2. THEORETICAL BACKGROUND
This section outlined the SSTD algorithm including parts
where different shift intervals are used in the study. More
detailed explanation regarding the algorithm can be referred
to the paper written by Zaghlool [19].
2.1 SSTD Algorithm
SSTD method is a variant of Ibrahim Time Domain (ITD)
method as both techniques share the same way on how the
algorithm works. The difference is that ITD is a Single-
Input-Multiple-Output (SIMO) type while SSTD is a Single-
Input-Single-Output (SISO) type [16]. In order to make
SSTD as a SIMO type, the algorithm will include least
square method as explained by Mohanty [20].
Considering at time tj, a system with N structural modes can
be expressed as a summation of the individual responses
specific at point i and time j of each mode r:


N
r
ts
irji
jr
eptx
2
1
)(
(1)
where pir is the eigenvector at point i for mode r. In this
algorithm, N refers to the number of identification order
which will identify converged modal parameters value
instead of defining N as number of structural modes. In
addition, the total number of converged modal parameters
will be the number of structural mode for that system. From
(1), the response measured at L instances of time can be
written as follow:
LNNNLN 

2222
][][][ ΛPX
(2)
Another set of equation that is exactly similar to (2) but is
shifted by ∆t1 is expressed as below







N
r
ts
ir
N
r
tts
irji
jr
jr
ep
epttx
2
1
2
1
)(
1
ˆ
)( 1

(3)
1
ˆ ts
irir
r
epp 

(4)
This can also be written in symbolic form as below:
LNNNLN 

2222
][]ˆ[]ˆ[ ΛPX
(5)
A square matrix [As] of order 2N is defined as below:
NNNNNN 222222
][][]ˆ[

 PAP s
(6)
Multiply (2) by [As] yield
LNNNNNLNNN 

22222222
][][][][][ ΛPAXA ss
(7)
Substitute (6) into (7) leads to
LNNNLNNN 

222222
][]ˆ[][][ ΛPXAs
(8)
Insert (5) into (8) gives
LNLNNN 

2222
]ˆ[][][ XXAs
(9)
As stated earlier, Equation (9) use a single output signal
with respect to a single input. Hence, a least square
approach is used in order to get a SIMO-SSTD type as
Matrix [As] is independent of the location of measurements
and remain valid for any SISO combination [20].
Transforming Equation (9) for n responses due to single
input can be outlined as below:
nLN
LNLNLN
nLN
LNLNLNNN












2
222
2
22222
ˆˆˆ][ n21n21s XXXXXXA 
(10)
where [Xn] refer to matrix with point i and time j as
explained in (1). In more compact form,
nLNnLNNN 

2222
]ˆ[][][ YYAs
(11)
In order to find Matrix [As], pseudo inverse technique is
used to give an expression known as Double Least-Squares

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(DLS). It is claimed that the equation below result better
estimates of the damping factors [21].
 11
)]][)([]][ˆ([)]ˆ][)([]ˆ][ˆ([
2
1
][ 
 TTTT
YYYYYYYYAs
(12)
From (6), the eigenvalues are deduced from matrix [As] as
below:
Nre
ep
ep
ep
ep
ep
ep
ts
N
tNs
r
ts
r
ts
r
N
tNs
r
ts
r
ts
r
NN
r
r
r
r
r
r
r
2...2,1][ 1
1
1
1
1
1
1
12
)12(
1
0
12
)12(
1
0
22











































sA
(13)
Rearranging (13) to have a standard eigenvalue problem as
follow:
0















 1
1
1
1
)12(
1
0
][
tNs
r
ts
r
ts
r
ts
r
r
r
r
ep
ep
ep
e





IAs
(14)
For underdamped case:
2
1 rrrr
rrr
i
ibas
 

(15)
The relationship between the eigenvalues (which is in
complex form, βr + iγr) and natural frequency are as follow:



2
2
)ln(
rr
rr
r
iba
t
i
f




(16)
For damping ratio:
r
r
r
rrr
a
a






(17)
Modal Confidence Factor (MCF) is then applied to
differentiate between genuine structural modes and
computational modes for each N. If pij is the ith
element of
the jth
identified eigenvector or jth
mode at the actual
measurement, then the same measurement delayed at
∆t3=10∆t1 is expected to be:
ij
t
ij pep j 3
 (18)
if this identified eigenvector is a structural mode [16, 22].
Thus, MCF can be defined as the ratio between left and right
hand side of (18) as below:
ijij
t
ij
t
ij
ij
t
ij
ij
ij
t
ij
ppefor
pe
p
pepfor
p
pe
MCF
j
j
j
j


3
3
3
3




(19)
and should be near unity for genuine structural mode. The
selected measurements from MCF filter are then listed and
grouped. Average measurements for each group will be
calculated to obtain the final result for natural frequency and
damping ratio.
2.2 Modified SSTD Algorithm
The idea of both SSTD and ITD methods is to reorder all the
output or time signal responses to become a standard
eigenvalue problem and solve it to get the dynamics
characteristics. One of the parameter involve in the process
is the shift interval. Mathematically they are described from
equation (2) and (5). As the objective of this paper is to
observe the impact of varying different shift interval
towards the accuracy of natural frequency and damping
ratio, following cases are considered for both matrices [X]
and [X^] where the time interval is shifted with:
1) multiple by 1 (default settings in the algorithm)
2) multiple by 2
3) multiple by 3
4) multiple by 4
5) multiple by 5
3. ANALYSIS AND PROCEDURES
Matlab was used in the analysis where five simulated data
with known properties were generated. Unlike previous
work, these five simulated data will be tested in six different
situations corrupted with distinct value of ‘white Gaussian
noise’. In other words, there were six different situations
where each situation consists of five simulated data sets.
The reason for using five simulated data for each situation is
to simulate the response of a system at five various locations
when an input was subjected at one of these points. An

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analogy for this is as depicted in
Fig 1 where a beam with five accelerometers at these points
is subjected with an input at each point separately. When an
input applied at one point results five responses for each
points, this means that there are 25 responses when the input
is subjected at all points. It is set up that for each data set to
have five degree of freedom. The following table
summarizes the details of the simulated data:
Table 1: Data collection parameters
Parameter Value
Sample rate 2048 Hz
Number of samples 2048
Frequency resolution 1 Hz
Nyquist frequency 1024 Hz
Table 2: Properties of the simulated data set
Mod Resid Natural Acceptab Dampi Acceptab
e ue Frequen
cy
le
Percenta
ge Error
(%)
ng
Ratio
(%)
le
Percenta
ge Error
(%)
1 10 128.0 2.344 0.300 16.667
2 20 256.0 1.172 0.250 20.000
3 30 512.0 0.586 0.139 35.971
4 39 768.0 0.391 0.093 53.763
5 50.7 806.4 0.372 0.046 108.696
Table 3: List of SNR
Situation SNR
1 0.1
2 1
3 10
4 30
5 60
6 100
From previous work, the acceptable value used for MCF
was greater than 0.9 and the acceptable number of
identifications model, N is around 70 to 80. Thus N = 80 is
used throughout this study. It is also important to note that
the acceptable percentage error for natural frequency and
damping ratio is different. Moreover the percentage may
vary for each mode for both parameters. For the purpose of
this research, more restricted limits than previous work were
defined where for natural frequency, deviation as much as
±3Hz from the true value is used as the limit. For damping
ratio, maximum data deviation of ±0.05 will be applied. Any
value greater than these is considered erroneous. Table 2
shows a list of acceptable percentage error at each mode for
both parameters.
Further analysis will look upon the impact of following
aspects towards the results:
1) The effect of shifting time interval
2) The effect of noise
3) Relationship between shifting time interval and
quantity of selected data together with its impacts
4) The effect of data distributions towards the results

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Fig 1: Five simulated data are analogous as a beam producing five responses when an input is applied at specific location.
4. RESULTS AND DISCUSSIONS
4.1 Results for Natural Frequency
(a)

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(b)
(c)
(d)

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(e)
(f)
Fig 2: Natural frequency percentage error for all cases (a) SNR 0.1 (b) SNR 1 (c) SNR 10 (d) SNR 30 (e) SNR 60 (f) SNR 100
The discussions start by looking at the effect of shifting the
time interval that occurs at the strongest signal, SNR 100.
This is to ensure that the observation on the effect of time
interval shift is not affected by the presence of noise. The
comparison will made relative to Case 1 as it is the default
settings in SSTD algorithm.
From Fig. 2-f, it can be seen that the results for all cases
reside within the limit of acceptability as outlined in Table
2. Case 5 have the highest percentage error among all cases
and there is one mode undetected by this case which is
Mode 4. For the remaining cases, the results were much
closed to each other. If taking the results for Case 1 as a
point of reference for comparison, the results for Case 2, 3
and 4 fluctuated around the results for Case 1. For example,
the results for Case 2 is higher than Case 1 for Mode 1,
lower at Mode 2, slightly higher at Mode 3 before becoming
lower back at Mode 4 and apparently same for Mode 5. The
same pattern is also experienced by Case 3 and Case 4.
The results also stressed out that Case 1 has the most
consistent results in keeping low percentage for each mode
for this situation. This is illustrated with a downward trend
in the graphs from one mode to another. Cases such as Case
3 and 4 has upward trend after experiencing downward
direction. If considering all cases, a small conclusion can be
drawn where in general, the larger the gap in time interval,
the higher the percentage errors will be. The ultimate errors
were depicted by the results of Case 5. These findings
confirmed and validated the pattern of graph for each case in
the previous research work where it is similar to this study.
The following discussions will take account the presence of
noise in SNR 0.1 until SNR 100. In a nutshell, the majority
of the results for all cases meet the limits defined. From Fig.
2, it can be seen that some of the cases did not detect some
of the modes in the data. This is obviously shown in
situation SNR 0.1 where Case 2, Case 4 and Case 5 did not
detect mode 4, mode 3 and mode 4 respectively. However
the issue is reduced when the value of SNR is increasing

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except for Case 5 as stated earlier. This phenomenon
highlights an attention where the amount of noise contained
in the data do gave impact towards the results especially for
cases that have shifted time interval.
(a)
(b)
(c)

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(d)
(e)
(f)
Fig 3: Natural frequency percentage error for Case 1 and Case 3 (a) SNR 0.1 (b) SNR 1 (c) SNR 10 (d) SNR 30 (e) SNR 60 (f)
SNR 100
However, there is an exception for Case 3 as cases that
always stay within the limits for all situations even when the
signal is highly polluted with noise at SNR 0.1 are Case 1
and 3. If comparing these two cases only, Fig. 3 exhibits that
Case 1 has lower percentage error than Case 3 for most of
the times. This is in line with earlier statement mentioning

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about higher percentage error if the gap for time interval is
increased. Hence, Case 1 and Case 3 are said to be robust
with consistent low percentage errors for each mode for all
situations including high presence of noise condition in
comparison to other cases with the results for Case 1 better
than Case 3 for almost all the time.
Also, it is essential to notify that most of the modes that
clearly have a bit higher in percentage error relative to other
modes as shown in Fig. 1 are Mode 3 and Mode 4 yielded
by Case 2 and Case 4. This is obviously seen in situation
from SNR 0.1 to SNR 60 for respective cases. Although the
figures were small and indicated that the difference is not
significant, this marked an important point in next
discussions. Finally, among all cases, Case 5 gave the most
inaccurate result in terms of acceptability of percentage
error and detection of mode.
4.2 Results for Damping Ratio
On the contrary, the percentage error results for damping
ratio are higher than the results for natural frequency. These
are expected as one of the reasons is due to mathematical
situation where small deviation in numerator with respect to
small value of denominator could sensitively lead to big
difference (See Table 2). The interest lies more on this
parameter as it has long been the subject of research since
Rayleigh introduced the well-known ‘viscous damping’
(a)
(b)

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(c)
(d)
(e)

_______________________________________________________________________________________
(f)
Fig 4 Damping ratio percentage error for all cases (a) SNR 0.1 (b) SNR 1 (c) SNR 10 (d) SNR 30 (e) SNR 60 (f) SNR 100
Fig 5 Damping ratio percentage error for Case 1, Case 2, Case 3 and Case 4 for SNR 100
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0 1 2 3 4 5 6
%
Mode
Case 1
Case 2
Case 3
Case 4

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(a)
(b)
(c)
(d)

_______________________________________________________________________________________
(e)
(f)
Fig 6 Damping ratio percentage error for Case 1 and Case 3 (a) SNR 0.1 (b) SNR 1 (c) SNR 10 (d) SNR 30 (e) SNR 60 (f) SNR
100
concept in 1877. However, the understanding of this
parameter is still primitive until today in contrast to inertia
and stiffness forces. This sparks an impetus to extensively
conduct more research in fulfilling better understanding
towards this parameter [23-26].
Similarly, the discussion begins by looking at the strongest
signal, SNR 100. Generally, Fig. 4 portrayed that all cases
have percentage errors within the limits introduced. An
exception goes to Case 5 as it did not identify Mode 4 as it
also did not detect for natural frequency parameter. Plus,
there is also an obvious, over-exceeding result for Mode 2.
These evidences indirectly make Case 5 to have the poorest
result among all.
For the rest of the cases, findings showed that increasing the
shift in time interval produced better results in comparison
to Case 1 for certain mode. In previous work, Case 4 was
proven to have the most consistent result in keeping low
percentage error for most of the modes. Whereas in this
study, by assessing the results at each mode individually
highlights Case 2 and Case 4 having promising potentials
where they do have better results at specific mode in
comparison to Case 1. The results for Case 3 were also
improved especially for higher modes. Hence, from this
observation, the statement ‘larger gap in time interval yield
larger percentage error’ only hold true for Case 5.
However, if the analysis considers the presence of noise
starting from SNR 0.1 to SNR 100, some of these promising
cases turned out to be very erroneous from the boundary
agreed. Surprisingly in certain situations, the results were
worse than Case 5 as illustrated in Fig. 4. Good examples
were Case 2 and Case 4 themselves seen in each situation
except for SNR 100. The over-exceeding errors are reduced
as the amount of noise is also reduced. Some failure
detection of mode in natural frequency also followed
consecutively in damping ratio. Again, the phenomenon

_______________________________________________________________________________________
highlights the impact of noise towards damping ratio results
especially for cases that have shifted time interval.
By inspecting closely at the results, it appears that only Case
1 and Case 3 reside within the limits defined for all
situations except for SNR 0.1 and SNR 1. For these
situations, the results only for Mode 1 and Mode 2 for both
cases were drastically surpassed the limits. Fig 6 depicts that
the erroneous results for Mode 1 were very high by merely
triple in relative to the rest of the modes including Mode 2.
It is suspected that this is primarily due to significant
amount of noise in the signal. In general, the consistency on
keeping low percentage for damping ratio achieved by Case
1 and Case 3 for each mode in almost all situations makes
them robust among all cases. Again, comparing both cases
only, Case 1 provided lower percentage error results than
Case 3 for most of the times.
An important note to point out is the contrast results
observed for the strongest signal situation, SNR 100 from
the expected. It is stated earlier that the results improved as
the amount of noise is reduced. This can be observed in both
natural frequency as well as damping ratio parameter.
However, for this situation, the percentage error is increased
once again instead of decreasing relative to the results for
SNR 60 in both cases. Moreover, the increment is higher
than the results for SNR 10. Although this happen, the
results were still within the limit. Nevertheless, if
comparison is made with the previous work, approximately
the same range of figures is obtained in both studies. A
suggestion for repeating numerical experiment for both SNR
60 and SNR 100 is recommended.
Another observation that can be peeked is that for Case 2
and 4, the over-exceeding result mostly occurred for Case 2
and Case 4 are Mode 3 and Mode 4. The errors were
obvious where it can rise up to five times greater than
percentage error for the rest of the mode. The same pattern
is also observed in natural frequency results. The difference
is that the figures were small and thus, the phenomena may
not as obvious as in damping ratio results.
4.3 Number of Data
One way of looking the factors that give effects towards the
results is to observe the number of data selected by the
algorithm for each case. These data is then used to calculate
the average result for both natural frequency and damping
ratio parameters. Following discussions will imply the same
previous format where it begins with SNR 100.
Fig. 6 depicts that Case 2 produced the largest number of
data for almost every mode. This is then followed by Case 1,
Case 3 or Case 4 and finally Case 5. It was mentioned
earlier that Case 5 did not detect one mode for this situation
which is Mode 4. The diagram spells out the reason where
none of data were selected at that mode after MCF filter is
applied in the algorithm. From these findings, a number of
deductions can be made:
1) In general, the overall trend illustrated that as the shift
in time interval is increased, the number of data
selected will decrease. This evidence is showed by
the ranking of cases according to the data quantity
possessed by each case. The ultimate least number of
data held by Case 5 agreed with this statement
although the largest number of data was initially
occupied by Case 2 instead of Case 1.
2) The number of data is an indicator in assessing the
accuracy of results but the quantity is not the sole
indicator for this purpose. As a proof, the least
percentage errors for damping ratio among all cases
were held by Case 4 with 59 data. Whereas, Case 1
had the largest with 68 data but occupied second in
terms of lowest percentage errors. This can be clearly
seen in next example.
(a)

_______________________________________________________________________________________
(b)
(c)
(d)

_______________________________________________________________________________________
(e)
(f)
Fig 7: Number of data for all cases (a) SNR 0.1 (b) SNR 1 (c) SNR 10 (d) SNR 30 (e) SNR 60 (f) SNR 100
Glancing through situations corrupted with noise starting
from SNR 0.1 to SNR 100 drew a number of attentions. It
can be seen how the noise give significant impact towards
number of data of each cases especially with shifted time
interval. An exception is for Case 3 as together with Case 1,
they produced greatest number of data consistently for each
mode for all situations. Comparing both cases, Case 1 have
larger number of data than Case 3 almost all the time. Thus,
this is an elucidation for why Case 1 had lower percentage
errors than Case 3 for most of the time as discussed
previously. This indirectly also unlock one of the reason
behind the robustness of Case 1 and Case 3 in surviving
acceptable percentage errors for both parameter although the
situation is corrupted extremely with noise.
The factor ‘number of data’ may also explain the erroneous
result experienced by Case 2 and Case 4 at Mode 3 and
Mode 4. This is clearly seen due to insufficient number of
data at those modes. However, the effect is more observable
for damping ratio instead of natural frequency although both
gained same pattern of locations for the errors to occur. This
factor also gave answer to the lack in results yielded by Case
5. Another important notification is that the statement does
not valid for situations SNR 60. As a result, an alternative
explanation is required to remedy the problem.
4.1 Data Distributions
4.4.1 Introduction to Data Distributions Analysis
As mentioned earlier, number of data can play as an
indicator in assessing the accuracy of results but the factor is
not the primary indicator for this purpose. For example in
situation SNR 60, Case 2 has sufficient number of data for
all modes where the numbers surpassed the quantity of data
for Case 1 except for Mode 4. At a glance, the majority of
good results normally possessed approximately 50 data or
higher referring only to Case 1 and Case 3 for all situations.
These cases were selected as reference due to their
consistencies in providing lower percentage error for each
mode. With this definition, the difference in quantity for

_______________________________________________________________________________________
Mode 4 is considered small as Case 1 obtained 95 data while
Case 2 with 82 data. However, Fig. 4 and Fig. 6 show that
the percentage errors for damping ratio at Mode 3 and Mode
4 are 72.3% and 53.8% respectively whereas for Case 1, the
results are 0.009% and 0.003%. The huge difference
demands an alternative elucidation in fulfilling the missing
understanding to this problem.
Other way of looking at this is by inspecting the pattern of
data distributions and their closeness to the theoretical value.
For this purpose, the following analysis would concentrate
on situation SNR 60. The parameter investigated for this
analysis only focusing on damping ratio as the impact of
data quantity has less significance towards natural
frequency. The term ‘closeness’ to the theoretical value will
use the definition of acceptable percentage error where
maximum deviation allowed from true values is ±0.05. It is
essential to not be confused between these two distinct terms
as the limit for the percentage errors is related to the mean
damping ratio while the term ‘closeness’ applied to all data
that will contribute to the final value of mean damping ratio.
Data residing outside this definition will be labelled as
‘outliers’.
4.4.2 Analysis of Data Distribution Results
The following table listed the sample standard deviation and
the mean at each mode for all cases for SNR 60:
Table 4: List of sample standard deviations for each mode
and all case for SNR 60
Mo
de
Sample Standard Deviation
Case 1 Case 2 Case 3 Case 4 Case 5
1 9.65645
E-06
0.00647
871
0.00326
8874
0.04642
859
0.23585
5401
2 1.1991E
-05
0.00434
3426
0.00115
2304
0.00608
9391
0.36542
1001
3 3.47831
E-06
0.16454
0102
0.00076
094
0.24799
8069
0.11000
5963
4 5.41418
E-06
0.08151
4118
3.89237
E-05
0.13870
4117
0.12389
807
5 0.00867
5979
0.00317
5876
0.01124
0715
0.00883
3524
0.02930
2872
Table 5: List of sample mean for each mode and all case for
SNR 60
Mo
de
Sample Mean
Case 1 Case 2 Case 3 Case 4 Case 5
1 0.29999
8346
0.30090
6747
0.29948
6286
0.31530
4064
0.26493
8887
2 0.24999
2192
0.24921
3822
0.24965
8943
0.24826
8181
-
2.50940
2018
3 0.13889
016
0.03781
1737
0.13900
2891
-
0.19160
8597
0.07430
8167
4 0.09259
4922
0.04281
2912
0.09260
0173
-
0.09159
4876
-
0.68868
5268
5 0.04632
0023
0.04669
0393
0.04543
6313
0.04507
0485
0.05029
8969
From the Fig. 8, Table 4 and Table 5, the results depict that
broad dispersion of data at certain mode lead to erroneous
result for damping ratio. Case 2 as well as Case 4 at Mode 3
and Mode 4 exhibited that wide distribution of data
influenced the average value of damping ratio. Monitoring
closely at Fig. 8-b for Case 2 indicated that around 50 data
were closed to the theoretical value, 0.139. However, with
the presence of a number of outliers had affected the final
average value to 0.038. Similar pattern is also observed for
Mode 4. These evidences showed that as the dissemination
of data is wider, the more it affects the value for mean
damping ratio which will also give impact towards
percentage errors. This explains why Case 2 has high
percentage errors for both modes although the quantities of
data are 89 and 82 respectively.
It can also be seen that the dispersion of data for Case 4 is
larger than Case 2. For Mode 3 and 4, Table 4 marked 0.165
and 0.082 for Case 2 whereas 0.248 and 0.139 for Case 4.
The vast spread of data for Case 4 at both modes had also
influence the value for mean damping ratio as well as
percentage error as shown in Fig. 8 –g,h. Thus, the above
statement is also valid for Case 4.
Switching to Case 5, the over-exceeding errors mostly
occurred at Mode 2 and Mode 4. It is stated beforehand that
the cause for this is due to insufficient number of data. The
dissemination pattern for this condition illustrates that most
of these points are very far from theoretical values by range
of 0.6 to 3.5 thus affecting the mean damping ratio. The
analysis using data distribution for Case 5 is not relevance
as not much information can be extracted from this amount
of quantity.

_______________________________________________________________________________________
(a)
(b)
(c)

_______________________________________________________________________________________
(d)
(e)
(f)

_______________________________________________________________________________________
(g)
(h)
(i)

_______________________________________________________________________________________
Fig 8: Data distribution for Case 2 and Case 4, SNR 60 (a) Mode 1 Case 2 (b) Mode 2 Case 2 (c) Mode 3 Case 2 (d) Mode 4 Case
2 (e) Mode 5 Case 2 (f) Mode 1 Case 4 (g) Mode 2 Case 4 (h) Mode 3 Case 4 (i) Mode 4 Case 4 (j) Mode 5 Case 4
In comparison to Case 1 and Case 3, the distributions for all
data at each mode are closed to each other and the
theoretical value. Thus, there are no outliers as they meet the
qualification of definitions although looking at the Graph
seemed to illustrate some outliers.
An important note to realize is the source of these outliers.
The primary role of MCF is to initially differentiate between
structural and computational mode where in this study,
value greater than 0.9 is chosen. The role is only effectively
success for the default settings, Case 1 and also Case 3
where almost no outliers identified. For the rest of the cases,
the quantity of these outliers is increased starting from Case
2 then Case 4. A number of deductions can be outlined from
these phenomena:
1) As the shift in time interval is increased, the number
of computational mode/outliers with value of MCF
greater than 0.9 is also increased along with
decreasing quantity of selected data and this will
occur at certain mode. The presence of these outliers
will influence the mean damping ratio and thus
affecting the percentage errors. An exception is for
Case 3 as no outliers identified in Graph. However,
there is compromise between these two factors;
number of outliers and quantity of data. A solid
example is Case 5 where it has greatest reduction in
number of data among all cases as obviously seen
for Mode 2 and Mode 4 but these data are all
outliers.
2) As the shift in time interval is increased, the error in
value for computational mode/outliers that occur at
certain mode is also increased. Quoting example of
Case 5 again, the values of these outliers were very
erroneous in comparison to the values of outliers
from Case 4. This is also the same as the error in
outliers for Case 4 is higher than Case 2.
3) As stated beforehand that in general, numbers of
data can act as an indicator in assessing the accuracy
of results but the factor is not the primary indicator
for this purpose. This is due to that the factor is
connected to other factor which is the amount of
computational modes/outliers exist depending on
how much shift in time interval for that case. More
vitally, if all the data are all considered within the
‘closeness’ limit and have minimum 50 data but still
having high percentage error, the cause rely on the
pattern of dispersion among the data or from
theoretical value if known. However, logically this
should not happen as the ‘closeness’ limits
introduced should ensure that the percentage error is
in line with it.
(a)

_______________________________________________________________________________________
(b)
(c)
(d)

_______________________________________________________________________________________
(e)
(f)
Fig 9: Time elapsed for all cases in each situation (a) SNR 0.1 (b) SNR 1 (c) SNR 10 (d) SNR 30 (e) SNR 60 (f) SNR 100
From these deductions, it is understandable that the reason
affecting the level of broadness for data spread relies on the
quantity and the values of these outliers that exist in the
data. This give an answer on why at Mode 4, Case 1 has
better results than Case 2 although both cases have the
largest amount of data for that mode relative to other cases
with 95 and 82 data respectively.
4.5 Time Elapsed
Fig. 9 exhibited the time taken for each case to calculate all
the modal parameters. It can be seen that the time taken is
proportional to the size of shift in time interval in general. In
other words, this indicates that Case 1 has the longest time
taken followed by Case 2, Case 3, Case 4 and finally Case 5.
The reduction in time for Case 5 is almost double from the
default settings, Case 1. Thus, Case 3 can be used as
alternative for large application of modal analysis project if
time saving is the main concern as it also robust with
slightly higher percentage error than Case 1.
5. CONCLUSION
In this paper, a numerical parametric study regarding the
effect of time shift interval in SSTD algorithm for EMA had
been carried out successfully. The results have shown that
the default setting in SSTD algorithm which is Case 1 is
robust and produce accurate result among all cases although
in situations with high presence of noise. Case 3 is also
robust as Case 1 but slightly less accurate. Following list
stress out a number of vital conclusions deduced from the
overall research work:
1) As the shift in time interval is increased, the
misleading value for computational mode/outliers that
occur at certain mode is also increased.
2) As the shift in time interval is increased, the quantity
of computational mode/outliers with value of MCF
greater than 0.9 is also increased in line with
decreasing quantity of selected data. Results show that
this will occur at certain mode only.

_______________________________________________________________________________________
3) The cause that influences the level of broadness for
data spread relies on the quantity and the values of
these outliers that exist in the data. Thus, it can also be
stated that as dissemination of data is wider, the more
it affects the value for mean damping ratio which will
also give impact towards percentage errors.
4) In general, as the shift in time interval is increased, the
number of data selected is decreased. This would lead
to higher percentage error for both parameters.
5) The number of data is an indicator in assessing the
accuracy of results but the quantity is not the sole
indicator for this purpose.
6) Modes that have few numbers of data will yield poor
results for damping ratio. However, this has less effect
on natural frequency.
7) In general, as the amount of noise contained in the data
is increased, the accuracy for natural frequency and
damping ratio as well as quantity of data selected are
decreased. For same amount of noise, the statement is
true as the shift in time interval is increased. An
exception is only for Case 3 where the results are
robust even for high presence of noise.
ACKNOWLEDGEMENTS
The authors acknowledge with gratitude the great support
from the management of Research Management Institute
(RMI), UiTM amd financial support from the Minister of
higher Education Malaysia (MOHE) for the granted grant of
Fundamental Research Grant (FRGS).
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BIOGRAPHIES
Ahmad Fathi Ghazali is a postgraduate student at at
Mechanical Engineering Faculty, MARA university of
Technology, 40450 Shah Alam, Malaysia
Ahmad Azlan Mat Isa is a lecturer at Mechanical
Engineering Faculty, MARA University of Technology,
40450 Shah Alam, Malaysia

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