A New Excitation Method:
Combining Burst Random Excitation with Cyclic Averaging
Allyn W. Phillips, PhD
Research Assistant Professor
Randall J. Allemang, PhD
Professor
Andrew T. Zucker
Research Assistant
Structural Dynamics Research Laboratory
Mechanical, Industrial and Nuclear Engineering
University of Cincinnati
Cincinnati, OH 45221-0072 USA
ABSTRACT
The use of controlled excitation is fundamental to the acquisition
of input-output data for the purpose of estimating modal
parameters. The characteristics of the excitation greatly influence
the quality of the resulting measurements. This paper first presents
an overview of the basic excitation methods, such as, random,
periodic random, pseudo-random, and burst random. The paper
then reviews the use of cyclic averaging with random excitation
methods. Finally, the paper develops a new excitation method that
is the result of combining cyclic averaging with burst random
excitation. This combination of methods results in an improved
signal-to-noise ratio (SNR) for lightly damped structures where
leakage is a significant problem. An experimental example is
given for a typical lightly damped, structural system.
Nomenclature
N avg = Number of averages.
N c = Number of cyclic averages.
N s = Number of spectral averages.
N i = Number of inputs.
N o = Number of outputs.
F max = Maximum frequency (Hz.).
Δ f = Frequency resolution (Hz.).
T = Observation period (Sec.).
1. Introduction
Single and multiple input estimation of frequency response
functions via shaker excitation has become the mainstay of most
mechanical structure measurements, particularly in the automotive
and aircraft industries. While there are appropriate occasions for
the use of deterministic excitation signals (sinusoids), the majority
of these measurements are made using broadband (random)
excitation signals. These signals work well for moderate to heavily
damped mechanical structures which exhibit linear characteristics.
When the mechanical structures are very lightly damped, care must
be taken to minimize the leakage error so that accurate frequency
response function (FRF) data can be estimated in the vicinity of the
modal frequencies of the system.
Historically, a number of random excitation signals have been
utilized, together with appropriate digital signal processing
techniques [1-5], to obtain accurate FRF data. The most common
random signal that is used in this situation is the burst random
signal, where the burst length of the random signal is adjusted to a
portion of the total observation time (T). The length of the burst
random signal is chosen based upon the measured signals being
completely observed transients in the observation time (T). This is
dependent upon the amount of system damping and the damping
characteristic provided by the shaker/amplifier operating in voltage
feedback mode. This works well in most cases but, in cases
involving very light damping, the burst length becomes so short
(10-20% of observation time, T) that the signal-to-noise ratio
(SNR) of the measurement starts to suffer. This situation can be
adequately addressed by the method presented in the following
sections.
2. Background
The background of this method involves a combination of the
traditional burst random excitation method with the cyclic
averaging technique. These concepts are reviewed in the following
sections.
2.1 Random Excitation Methods
Inputs which can be used to excite a system in order to determine
frequency response functions belong to one of two classifications,
random or deterministic [6-8]. Random signals are widely utilized
for general single-input and multiple-input shaker testing when
evaluating structures that are essentially linear. Signals of this
form can only be defined by their statistical properties over some
time period. Any subset of the total time period is unique and no
explicit mathematical relationship can be formulated to describe
the signal. Random signals can be further classified as stationary
or non-stationary. Stationary random signals are a special case
where the statistical properties of the random signals do not vary
with respect to translations with time. Finally, stationary random
signals can be classified as ergodic or non-ergodic. A stationary
random signal is ergodic when a time average on any particular
subset of the signal is the same for any arbitrary subset of the
random signal. All random signals which are commonly used as
input signals fall into the category of ergodic, stationary random
signals. Deterministic signals can be characterized directly by
mathematical formula and the characteristic of the excitation signal
can be computed for any instance in time. While this is true for the
theoretical signal sent to the exciter, it is only approximately true
for the actual excitation signal due to the amplifier/shaker/structure
interaction that is a function of the impedances of these electromechanical systems. Deterministic signals can, nevertheless, be
controlled more precisely and are frequently utilized in the
characterization of nonlinear systems for this reason. The random
classification of excitation signals is the only signal type discussed
in this paper.
The choice of input to be used to excite a system in order to
determine frequency response functions depends upon the
characteristics of the system, upon the characteristics of the
parameter estimation, and upon the expected utilization of the data.
The characterization of the system is primarily concerned with the
linearity of the system. As long as the system is linear, all input
forms should give the same expected value. Naturally, though, all
real systems have some degree of nonlinearity. Deterministic input
signals result in frequency response functions that are dependent
upon the signal level and type. A set of frequency response
functions for different signal levels can be used to document the
nonlinear characteristics of the system. Random input signals, in
the presence of nonlinearities, result in a frequency response
function that represents the best linear representation of the
nonlinear characteristics for a given level of random signal input.
For small nonlinearities, use of a random input will not differ
greatly from the use of a deterministic input.
The characterization of the parameter estimation is primarily
concerned with the type of mathematical model being used to
represent the frequency response function. Generally, the model is
a linear summation based upon the modal parameters of the
system. Unless the mathematical representation of all
nonlinearities is known, the parameter estimation process cannot
properly weight the frequency response function data to include
nonlinear effects. For this reason, random input signals are
prevalently used to obtain the best linear estimate of the frequency
response function when a parameter estimation process using a
linear model is to be utilized.
The expected utilization of the data is concerned with the degree of
detailed information required by any post-processing task. For
experimental modal analysis, this can range from implicit modal
vectors needed for trouble-shooting to explicit modal vectors used
in an orthogonality check. As more detail is required, input
signals, both random and deterministic, will need to match the
system characteristics and parameter estimation characteristics
more closely. In all possible uses of frequency response function
data, the conflicting requirements of the need for accuracy,
equipment availability, testing time, and testing cost will normally
reduce the possible choices of input signal.
With respect to the reduction of the variance and bias errors of the
frequency response function, random or deterministic signals can
be utilized most effectively if the signals are periodic with respect
to the sample period or totally observable with respect to the
sample period. If either of these criteria are satisfied, regardless of
signal type, the predominant bias error, leakage, will be minimized.
If these criteria are not satisfied, the leakage error may become
significant. In either case, the variance error will be a function of
the signal-to-noise ratio and the amount of averaging.
Many signals are appropriate for use in experimental modal
analysis. Some of the most commonly used random signals, used
with single and multiple input shaker testing, are described in the
following sections.
Pure Random - The pure random signal is an ergodic, stationary
random signal which has a Gaussian probability distribution. In
general, the frequency content of the signal contains all frequencies
(not just integer multiples of the FFT frequency increment) but
may be filtered to include only information in a frequency band of
interest. The measured input spectrum of the pure random signal
will be altered by any impedance mismatch between the system
and the exciter.
PseudoRandom - The pseudorandom signal is an ergodic,
stationary random signal consisting only of integer multiples of the
FFT frequency increment. The frequency spectrum of this signal
has a constant amplitude with random phase. If sufficient time is
allowed in the measurement procedure for any transient response
to the initiation of the signal to decay, the resultant input and
response histories are periodic with respect to the sample period.
The number of averages used in the measurement procedure is
only a function of the reduction of the variance error. In a noise
free environment, only one average may be necessary.
Periodic Random - The periodic random signal is an ergodic,
stationary random signal consisting only of integer multiples of the
FFT frequency increment. The frequency spectrum of this signal
has random amplitude and random phase distribution. Since a
single history will not contain information at all frequencies, a
number of histories must be involved in the measurement process.
For each average, an input history is created with random
amplitude and random phase. The system is excited with this input
in a repetitive cycle until the transient response to the change in
excitation signal decays. The input and response histories should
then be periodic with respect to the observation time (T) and are
recorded as one average in the total process. With each new
average, a new history, uncorrelated with previous input signals, is
generated so that the resulting measurement will be completely
randomized.
Burst Random (Random Transient) - The burst random signal is
neither a completely transient deterministic signal nor a completely
ergodic, stationary random signal but contains properties of both
signal types. The frequency spectrum of this signal has random
amplitude and random phase distribution and contains energy
throughout the frequency spectrum. The difference between this
signal and the random signal is that the random transient history is
truncated to zero after some percentage of the observation time (T).
Normally, an acceptable percentage is fifty to eighty percent. The
measurement procedure duplicates the random procedure but
without the need to utilize a window to reduce the leakage
problem. The point that the input history is truncated is chosen so
that the response history decays to zero within the observation time
(T). For light to moderately damped systems, the response history
will decay to zero very quickly due to the damping provided by the
exciter/amplifier system trying to maintain the input at zero
(voltage feedback).
This damping, provided by the
exciter/amplifier system, is often overlooked in the analysis of the
characteristics of this signal type. Since this measured input,
although not part of the generated signal, includes the variation of
the input during the decay of the response history, the input and
response histories are totally observable within the sample period
and the system damping that will be computed from the measured
FRF data is unaffected.
2.2 Cyclic Signal Averaging
The cyclic classification of signal averaging involves the added
constraint that the digitization is coherent between sample
functions [9-10]. This means that the exact time between each
sample function is used to enhance the signal averaging process.
Rather than trying to keep track of elapsed time between sample
functions, the normal procedure is to allow no time to elapse
between successive sample functions. This process can be
described as a comb digital filter in the frequency domain with the
teeth of the comb at frequency increments dependent upon the
periodic nature of the sampling with respect to the event measured.
The result is an attenuation of the spectrum between the teeth not
possible with other forms of averaging.
This form of signal averaging is very useful for filtering periodic
components from a noisy signal since the teeth of the filter are
positioned at harmonics of the frequency of the sampling reference
signal. This is of particular importance in applications where it is
desirable to extract signals connected with various rotating
members. This same form of signal averaging is particularly
useful for reducing leakage during frequency response
measurements and also has been used for evoked response
measurements in biomedical studies.
A very common application of cyclic signal averaging is in the
area of analysis of rotating structures. In such an application, the
peaks of the comb filter are positioned to match the fundamental
and harmonic frequencies of a particular rotating shaft or
component. This is particularly powerful, since in one
measurement it is possible to enhance all of the possible
frequencies generated by the rotating member from a given data
signal. With a zoom Fourier transform type of approach, one shaft
frequency at a time can be examined depending upon the zoom
power necessary to extract the shaft frequencies from the
surrounding noise.
The application of cyclic averaging to the estimation of frequency
response functions can be easily observed by noting the effects of
cyclic averaging on a single frequency sinusoid. Figures 1 and 2
represent the cyclic averaging of a sinusoid that is periodic with
respect to the observation time period T . Figures 3 and 4 represent
the cyclic averaging of a sinusoid that is aperiodic with respect to
the observation time period T . By comparing Figure 2 to Figure 4,
the attenuation of the nonperiodic signal can be clearly observed.
2.2.1 Theory of Cyclic Averaging
In the application of
cyclic averaging to frequency response function estimates, the
corresponding fundamental and harmonic frequencies that are
enhanced are the frequencies that occur at the integer multiples of
Δ f . In this case, the spectra between each Δ f is reduced with an
associated reduction of the bias error called leakage.
The first observation to be noted is the relationship between the
Fourier transform of a history and the Fourier transform of a time
shifted history. In the averaging case, each history will be of some
finite time length T which is the observation period of the data.
Note that this time period of observation T determines the
fundamental frequency resolution Δ f of the spectra via the
1
Rayleigh Criteria (Δ f = ).
T
The Fourier transform of a history is given by:
+∞
X(ω ) =
∫∞ x(t) e
− jω t
(1)
dt
−
Using the time shift theorem of the Fourier transform, the Fourier
transform of the same history that has been shifted in time by an
amount t 0 is:
X(ω ) e− j ω t0 =
+∞
∫∞ x(t + t ) e
0
− jω t
dt
(2)
−
For the case of a discrete Fourier transform, each frequency in the
spectra is assumed to be an integer multiple of the fundamental
1
frequency Δ f = . Making this substitution in Equation (2)
T
2π
(ω = n
with n as an integer) yields:
T
X(ω ) e
−j n
2π
t
T 0
+∞
=
∫∞ x(t + t ) e
0
− jω t
dt
(3)
−
Note that in Equation (3), the correction for the cases where
t 0 = N T with N is an integer will be a unit magnitude with zero
phase. Therefore, if each history that is cyclic averaged occurs at a
time shift, with respect to the initial average, that is an integer
multiple of the observation period T , then the correction due to the
time shift does not effect the frequency domain characteristics of
the averaged result. All further discussion will assume that the
time shift t 0 will be an integer multiple of the basic observation
period T .
The signal averaging algorithm for histories averaged with a
boxcar or uniform window is:
x(t) =
where:
1
Nc
Nc − 1
Σ
i=0
x i (t)
(4)
•
x(t) = Cyclic averaged time history
1
For the case where x(t) is continuous over the time period N c T ,
the complex Fourier coefficients of the cyclic averaged time history
become:
0.8
0.6
0.4
0.2
0
−0.2
−0.4
T
1
T
Ck =
∫ x(t) e
− jω t
(5)
dt
0
−0.6
−0.8
−1
0
0.5
1
1.5
2
2.5
Time Periods (T)
3
3.5
4
Ck =
Figure 1. Contiguous Time Records (Periodic Signal)
1
T
T
∫
0
1
Nc
Nc − 1
T
Nc − 1
Σ
i=0
x i (t) e−
jω t
dt
(6)
x i (t) e−
jω t
dt
(7)
Finally:
1
0.8
Ck =
0.6
0.4
1
Nc T
0.2
∫ Σ
0
i=0
0
Since x(t) is a continuous function, the sum of the integrals can be
replaced with an integral evaluated from 0 to N c T over the
original function x(t). Therefore:
−0.2
−0.4
−0.6
−0.8
−1
0
0.1
0.2
0.3
0.4
0.5
0.6
Time Periods (T)
0.7
0.8
0.9
1
Ck =
Figure 2. Averaged Time Records (Periodic Signal)
1
Nc T
Nc T
∫
x(t) e−
jω t
dt
(8)
0
The above equation indicates that the Fourier coefficients of the
1
cyclic averaged history (which are spaced at Δ f = ) are the same
T
Fourier coefficients from the original history (which are spaced at
1
Δf =
).
Nc T
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
2
2.5
Time Periods (T)
3
3.5
4
The results of cyclic averaging of a general random signal with the
application of a uniform window are shown in Figures 5 and 6.
Likewise, the results of cyclic averaging of a general random signal
with the application of a Hann window are shown in Figures 7 and
8.
Figure 3. Contiguous Time Records (Non-Periodic Signal)
1
3. New Method
0.8
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
0
0.1
0.2
0.3
0.4
0.5
0.6
Time Periods (T)
0.7
0.8
0.9
Figure 4. Averaged Time Records (Non-Periodic Signal)
•
x i (t) = Time history, average i
•
N c = Number of cyclic averages
1
The new method developed in this paper is simply a combination
of burst random excitation and cyclic averaging. Burst random
excitation is useful whenever a leakage problem exists in making a
frequency response function measurement on a lightly damped
mechanical system. The limitation of burst random excitation is
that, if the burst length is shortened too far (10-20 percent of the
observation time block, T) in order to minimize the leakage
problem, the signal-to-noise ratio (SNR) of the excitation signal
deteriorates. Cyclic averaging allows the burst random signal to
exist over a large portion of the contiguous observation time and
yet permits the burst length of random signal to be adjusted to any
time length needed. Note that this combination of techniques
allows leakage to be reduced in two ways while maintaining a
reasonable SNR: 1) the burst length can be adjusted as needed and
following figures. Figure 9 shows four contiguous observation
times for a burst random signal measured at the load cell and a
typical response transducer. Figure 10 is the cyclic average of
these four blocks. This becomes the first average for any spectral
averages that will be accumulated. Note that in this case, even
though the burst length is chosen so that the excitation signal will
be zero after 20 percent of the fourth block of the four contiguous
averages, the response signal, shown in Figure 9, still has not
decayed to zero by the end of the fourth block.
0.1
0.05
0
−0.05
−0.1
−0.15
0
0.5
1
1.5
2
2.5
Time Periods (T)
3
3.5
4
4. Structural Example
Figure 5. Contiguous Time Records (Random Signal)
The following example represents a single measurement on an Hframe test structure in a test lab environment. The test results are
representative of all data taken on the H-frame structure. This Hframe test structure is very lightly damped and has been the subject
of many previous studies.
0.05
0.04
0.03
0.02
0.01
0
−0.01
−0.02
−0.03
−0.04
−0.05
0
0.1
0.2
0.3
0.4
0.5
0.6
Time Periods (T)
0.7
0.8
0.9
1
Figure 6. Averaged Time Records (Random Signal)
0.08
0.06
Nine representative cases were measured on this structure. The
configuration of the test involved two shaker locations (inputs) and
eight response accelerometers (outputs). The frequency response
function measurements presented in the following discussion were
calculated utilizing both the H 1 and H v algorithm approaches. The
H 1 data is shown in the plots. The digital signal processing
characteristics of each case are shown in Table 1. In all cases, the
total data acquired and hence the total test time are the same. In
this case 96 data ensembles are used simply changing the relative
number of cyclic and spectral averages.
0.04
0.02
0
−0.02
−0.04
−0.06
−0.08
−0.1
0
0.5
1
1.5
2
2.5
Time Periods (T)
3
3.5
4
Figure 7. Contiguous Time Records with Hann Window (Random
Signal)
Cases 2, 3, and 4 are typical attempts to reduce the leakage by use
of Burst Random excitation. The bursts lengths of 80%, 50% and
20%, respectively, span the range of practical burst excitation
record lengths. The increase in coherence indicates that while
leakage has been reduced, it has not been eliminated by reducing
the burst length. Further reduction in burst length would seriously
degrade the SNR, as well as, cause the test technique to approach
impact testing with all its attendant problems. A different
testing/averaging technique is required to eliminate leakage.
0.03
0.02
0.01
0
−0.01
−0.02
−0.03
−0.04
0
Case 1 is considered a baseline case since this a very popular
method for making a FRF measurement. However, it is clear that
in this measurement situation, there is a significant drop in the
multiple coherence function at frequencies consistent with the
peaks in the FRF measurement. This characteristic drop in
multiple (or ordinary) coherence is often an indication of a leakage
problem. This can be confirmed if a leakage reduction method
reduces or eliminates the problem when the measurement is
repeated. In all subsequent cases, the test configuration was not
altered in any way - data was acquired simply using different burst
random, cyclic averaging combinations.
0.1
0.2
0.3
0.4
0.5
0.6
Time Periods (T)
0.7
0.8
0.9
1
Figure 8. Averaged Time Records with Hann Window (Random
Signal)
2) the number of cyclic averages can be altered.
This new method is demonstrated in the time domain in the
Cases 5 and 7 show the use of cyclic averaging with Random
excitation. The reduction in leakage for these cases is comparable
to the use of 80% Burst Random.
Cases 6 and 8 show the use of cyclic averaging with Burst Random
excitation. For both cases, the burst length was chosen such that
the burst continued 20% into the last block. The reduction in
leakage is significant when compared to the previous cases, but is
total record length. This allows 1.2 blocks for the response to
decay. In this case, the leakage is virtually eliminated. This
reduction is not possible when using only burst excitation or cyclic
averaging alone.
Input Time Data
3
1
Input:
Phase (Deg)
Amplitude
2
0
−1
1 Output: 5
180
0
0
10
−2
−3
1
−1
10
0
0.5
1
1.5
2
2.5
3
3.5
4
0.8
Output Time Data
0.1
0.6
−3
Coherence
Magnitude
−2
10
10
0.4
Amplitude
0.05
−4
10
0.2
0
−5
10
0
50
100
150
−0.05
200
250
300
Frequency (Hertz)
350
400
450
0
500
Figure 11. Case 1: Random Excitation with Hann Window
−0.1
0
0.5
1
1.5
2
2.5
Time Periods (T)
3
3.5
4
Time
Records
(Burst
Phase (Deg)
Input:
Figure 9. Input/Output: Contiguous
Random Signal)
1 Output: 5
180
0
0
10
1
Cyclic Averaged Input Time Data
−1
10
6
0.8
4
2
0
10
0.6
−3
10
0.4
−2
−4
10
0.2
−4
−6
−5
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Amplitude
0.1
0
−0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
Time Periods (T)
0
50
100
150
200
250
300
Frequency (Hertz)
350
400
450
Figure 12. Case 2: Burst Random Excitation (80 %)
Cyclic Averaged Output Time Data
0.2
−0.2
10
0.7
0.8
0.9
1
Figure 10. InputOutput: Averaged Time Records (Burst Random
Signal)
not yet eliminated.
Case 9 shows the use of cyclic averaging with Burst Random
excitation, but this time the burst length is chosen as 80% of the
0
500
Coherence
Magnitude
Amplitude
−2
1 Output: 5
Input:
Phase (Deg)
0
0
10
1 Output: 5
180
0
0
10
1
1
−1
−1
10
10
0.8
0.8
0.6
−3
10
10
0.6
−3
10
0.4
0.4
−4
−4
10
10
0.2
−5
10
0
50
100
150
200
250
300
Frequency (Hertz)
350
400
450
10
0
50
100
150
200
250
300
Frequency (Hertz)
350
400
450
0
500
Figure 16. Case 6: Burst Random Excitation (80 %), 4 Cyclic
Averages
1 Output: 5
180
Input:
0
Phase (Deg)
Phase (Deg)
Input:
0.2
−5
0
500
Figure 13. Case 3: Burst Random Excitation (50 %)
0
10
1
1 Output: 5
180
0
0
10
1
−1
10
0.8
−1
10
0.8
0.6
−3
10
0.4
−2
Magnitude
10
Coherence
−2
Magnitude
Coherence
10
Magnitude
−2
Coherence
Magnitude
−2
10
0.6
−3
Coherence
Phase (Deg)
Input:
180
10
0.4
−4
10
0.2
−4
10
−5
10
0
50
100
150
200
250
300
Frequency (Hertz)
350
400
450
0.2
0
500
−5
10
0
50
100
150
200
250
300
Frequency (Hertz)
350
400
450
0
500
Figure 14. Case 4: Burst Random Excitation (20 %)
Figure 17. Case 7: Random Excitation with Hann Window, 6
Cyclic Averages
1 Output: 5
0
Input:
Phase (Deg)
Phase (Deg)
Input:
180
0
10
1
−1
10
1 Output: 5
180
0
0
10
0.8
1
−2
10
−1
−5
0.2
10
0.6
−3
Coherence
0.4
0.8
−2
Magnitude
−4
10
Coherence
Magnitude
10
0.6
−3
10
10
10
−4
−6
10
0.4
0
50
100
150
200
250
300
Frequency (Hertz)
350
400
450
0
500
10
0.2
−5
Figure 15. Case 5: Random Excitation with Hann Window, 4
Cyclic Averages
10
0
50
100
150
200
250
300
Frequency (Hertz)
350
400
450
0
500
Figure 18. Case 8: Burst Random Excitation (86 %), 6 Cyclic
Averages
Phase (Deg)
Input:
6. References
1 Output: 5
180
0
0
10
[1]
Bendat, J.S.; Piersol, A.G., Random Data: Analysis and
Measurement Procedures, John Wiley and Sons, Inc.,
New York, 1971, 407 pp.
[2]
Bendat, J. S., Piersol, A. G., Engineering Applications of
Correlation and Spectral Analysis, John Wiley and Sons,
Inc., New York, 1980, 302 pp.
[3]
Otnes, R.K., Enochson, L., Digital Time Series Analysis,
John Wiley and Sons, Inc., New York, 1972, 467 pp.
[4]
Hsu, H.P., Fourier Analysis, Simon and Schuster, 1970,
274 pp.
[5]
Potter, R.W., "Compilation of Time Windows and Line
Shapes for Fourier Analysis", Hewlett-Packard Company,
1972, 26 pp.
[6]
Halvorsen, W.G., Brown, D.L., "Impulse Technique for
Structural Frequency Response Testing", Sound and
Vibration Magazine, November, 1977, pp. 8-21.
[7]
Brown, D.L., Carbon, G., Zimmerman, R.D., "Survey of
Excitation Techniques Applicable to the Testing of
Automotive Structures", SAE Paper No. 770029, 1977.
[8]
Van Karsen, C., "A Survey of Excitation Techniques for
Frequency Response Function Measurement", Master of
Science Thesis, University of Cincinnati, 1987, 81 pp.
[9]
Allemang, R.J., Phillips, A.W., "Cyclic Averaging for
Frequency Response Function Estimation", Proceedings,
International Modal Analysis Conference, pp. 415-422,
1996.
[10]
Allemang, R. J., Brown, D.L., "A Correlation Coefficient
for Modal Vector Analysis", Proceedings, International
Modal Analysis Conference, pp.110-116, 1982.
1
−1
10
0.8
10
0.6
−3
Coherence
Magnitude
−2
10
0.4
−4
10
0.2
−5
10
0
50
100
150
200
250
300
Frequency (Hertz)
350
400
450
0
500
Figure 19. Case 9: Burst Random Excitation (80 %), 6 Cyclic
Averages
Table 2 is a summary of the FRF magnitude/multiple coherence for
several peak frequencies for each case. Comparing the relative
magnitude of the FRF for Cases 1 and 4 to Case 9, the error in
magnitude is severe for the random excitation and significant for
the burst random (20%). These two cases are representative of
typical measurement conditions and the burst random is the best
non-cyclic average result. The random excitation result averages
approximately 70% magnitude error. The burst random, while
much better showing 5% or less error for most peaks, exhibit about
30% magnitude error for the 77.5 Hz peak. When estimating
modal parameters, the frequency and mode shape would probably
be estimated reasonably, however, the damping and modal scaling
would be distorted (over estimating damping and under estimating
modal scaling). Using these results for model prediction or FE
correction would bias the predicted results.
5. Conclusions
The most important conclusion that can be drawn from the results
of this measurement exercise on a lightly damped mechanical
system is that accurate data is an indirect function of measurement
time or number of averages but is a direct function of measurement
technique. The leakage problem associated with utilizing fast
Fourier transform (FFT) methodology to estimate frequency
response functions on a mechanical system with light damping is a
serious problem that can be managed with proper measurement
techniques, like cyclic averaging and burst random excitation. It is
also important to note that while ordinary/multiple coherence can
indicate a variety of input/output problems, a drop in the
ordinary/multiple coherence function, at the same frequency as a
lightly damped peak in the frequency response function, is often a
direct indicator of a leakage problem. Frequently, comparisons are
made between results obtained with narrowband (sinusoid)
excitation and broadband (random) excitation when the
ordinary/multiple coherence function clearly indicates a potential
leakage problem. It is important that good measurement technique
be an integral part of such comparisons.
Case
Excitation
Window
Blocksize
Case 1
Case 2
Case 3
Case 4
Case 5
Case 6
Case 7
Case 8
Case 9
Random
Burst Random (80%)
Burst Random (50%)
Burst Random (20%)
Random
Burst Random (80%)
Random
Burst Random (86%)
Burst Random (80%)
Hann
Uniform
Uniform
Uniform
Hann
Uniform
Hann
Uniform
Uniform
2048
2048
2048
2048
2048
2048
2048
2048
2048
F max
Nc
Ns
N avg
512 Hz.
512 Hz.
512 Hz.
512 Hz.
512 Hz.
512 Hz.
512 Hz.
512 Hz.
512 Hz.
1
1
1
1
4
4
6
6
6
96
96
96
96
24
24
16
16
16
96
96
96
96
96
96
96
96
96
TABLE 1. Test Cases - DSP Parameters
Case
Case 1
Case 2
Case 3
Case 4
Case 5
Case 6
Case 7
Case 8
Case 9
24 Hz. Peak
FRF
Coh.
0.0198
0.0542
0.0677
0.0659
0.0472
0.0678
0.0525
0.0673
0.0688
0.3593
0.8966
0.9771
0.9631
0.8072
0.9981
0.9029
0.9965
0.9950
55.5 Hz. Peak
FRF
Coh.
0.0046
0.0115
0.0147
0.0174
0.0146
0.0174
0.0158
0.0170
0.0173
0.2797
0.8620
0.9623
0.9917
0.8428
0.9887
0.9266
0.9957
0.9977
77.5 Hz. Peak
FRF
Coh.
0.0042
0.0089
0.0117
0.0138
0.0154
0.0187
0.0151
0.0191
0.0200
0.2451
0.8606
0.9595
0.9833
0.8086
0.9851
0.8341
0.9823
0.9967
162.5 Hz. Peak
FRF
Coh.
0.0566
0.0689
0.0702
0.0718
0.0703
0.0694
0.0629
0.0695
0.0701
TABLE 2. Test Cases - Measurement Results
0.7305
0.9978
0.9988
0.9991
0.9787
0.9988
0.9938
0.9988
0.9990
169.5 Hz. Peak
FRF
Coh.
0.0262
0.0681
0.0686
0.0668
0.0583
0.0693
0.0626
0.0687
0.0706
0.4317
0.9889
0.9989
0.9992
0.9599
0.9987
0.9740
0.9987
0.9990