Analysis of Swept-sine Runs During Modal Identification(2004)
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Mechanical Systems
and
Signal Processing
www.elsevier.com/locate/jnlabr/ymssp
Mechanical Systems and Signal Processing 18 (2004) 14211441
Analysis of swept-sine runs during modal identification
G. Gloth, M. Sinapius*
Institute of Aeroelasticity, German Aerospace Center (DLR), Bunsenstr. 10, 37073 G.ottingen, Germany
Received 27 February 2003; received in revised form 23 June 2003; accepted 24 June 2003
Abstract
Experimental modal analysis of large aerospace structures in Europe combine nowadays the benefits of
the very reliable but time-consuming phase resonance method and the application of phase separation
techniques evaluating frequency response functions (FRF). FRFs of a test structure can be determined by
a variety of means. Applied excitation signal waveforms include harmonic signals like stepped-sine
excitation, periodic signals like multi-sine excitation, transient signals like impulse and swept-sine
excitation, and stochastic signals like random. The current article focuses on slow swept-sine excitation
which is a good trade-off between magnitude of excitation level needed for large aircraft and testing time.
However, recent ground vibration tests (GVTs) brought up that reliable modal data from swept-sine test
runs depend on a proper data processing. The article elucidates the strategy of modal analysis based onswept-sine excitation. The standards for the application of slowly swept sinusoids defined by the
international organisation for standardisation in ISO 7626 part 2 are critically reviewed. The theoretical
background of swept-sine testing is expounded with particular emphasis to the transition through structural
resonances. The effect of different standard procedures of data processing like tracking filter, fast Fourier
transform (FFT), and data reduction via averaging are investigated with respect to their influence on the
FRFs and modal parameters. Particular emphasis is given to FRF distortions evoked by unsuitable data
processing. All data processing methods are investigated on a numerical example. Their practical usefulness
is demonstrated on test data taken from a recent GVT on a large aircraft. The revision of ISO 7626 part 2 is
suggested regarding the application of slow swept-sine excitation. Recommendations about the proper
FRF estimation from slow swept-sine excitation are given in order to enable the optimisation on these
applications for future modal survey tests of large aerospace structures.
r 2003 Elsevier Ltd. All rights reserved.
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*Corresponding author.
E-mail addresses: [email protected] (G. Gloth), [email protected] (M. Sinapius).
0888-3270/$ - see front matter r 2003 Elsevier Ltd. All rights reserved.
doi:10.1016/S0888-3270(03)00087-6
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1. Introduction
The European aircraft industry plans to extend their product offer towards high-capacity
aircraft. The prototypes of these aircraft are dynamically characterised by a high modal density inthe very low frequency range. This requires a high effort on the part of the experimental modal
analysis. On the other hand, the test time should be reduced to a minimum in order to reduce
costs. A new test strategy was proposed, developed, and applied during the ground vibration tests
(GVTs) of the recently built, new aircraft prototypes [1,2] in order to meet these requirements.
An essential part of the improved test strategy is the combination of the classical phase
resonance test method (sine dwell) with phase separation techniques which, in turn, are based on
the evaluation of measured frequency response functions (FRFs). Several excitation types (see
also [3]) were investigated with regard to their suitability for the modal identification of large
aircraft during a research GVT in 1999 [2]. The slow swept-sine excitation emerged from the
investigation as the most promising excitation signal. The tests also revealed that reliable FRFsdepend on the proper signal processing of the measured data.
In 1986, the international organisation for standardisation published guidelines with ISO 7626
part 2[4] for the application of slowly swept sinusoids. They are referred to in the standard issue
on modal testing [5], where the recommendation is given to check that progress through the
frequency range is sufficiently slow to check that the steady-state response conditions are attained
before measurements are made. If an excessive sweep rate is used, then distortionsof the FRF plot
are introduced, ....
This article investigates the source of these so-called FRF distortions. The standards on swept-
sine excitation are reviewed and the theoretical background of sweep excitation is given.
Furthermore, the methods of the correct estimation of FRFs are elucidated and compared with
those that produce FRF distortions. The energy of swept-sine waveforms is investigated and theeffect of swept-sine excitation on the FRF is studied using a single degree-of-freedom (sdof)
system. Finally, experimental data acquired during the vibration tests of large aerospace
structures are presented.
2. Review of the standards
The standard for the experimental determination of mobility is defined in the International
Standard ISO-7626[4]which was published in its first issue in 1986. Mobility is defined there as anFRF which is a phasor of the motion (acceleration, velocity, or displacement) at a structural point
due to a unit force excitation. The FRF is exclusively determined by the dynamics of the structure
which, in turn, are usually described by modal parameters. This implies linearity.
FRFs can be determined very well experimentally. Several waveforms are available to excite the
required structural motion. The most common types are harmonic excitation like discretely
stepped sine, periodic excitation like multi-sine, transient excitation like sinusoidal sweeps or
impact, and random excitation. They differ vastly in their spectral energy contents and test
duration. The best compromise between high energy input and short test duration for large
aerospace structures is the sinusoidal sweep.
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Sinusoidal sweeps are known as linear sweeps and logarithmic sweeps depending on their rule
on the change of frequencies (see Section 3). ISO-7626 sets the following standards and
recommendations for sinusoidal sweeps:
1. Any excitation waveform, the spectrum of which covers the frequency range of interest, can be
used provided that the excitation and response signals are processed properly. (In the ISO-
7626 section about Excitation.)
2. The frequency response function shall be computed using only those components of the
response and excitation signals corresponding to the excitation frequency. (In the ISO-7626
section about Processing.)
3. The sweep rate shall be chosen so that, in the frequency range within 710% of a resonance
frequency, the measured magnitude of the motion response of the structure is within 5% of the
true value. (In the ISO-7626 section about Control of excitation.)
Additionally, the standard points out, that tracking filters which are narrowband pass-band
filters are traditionally used for the data processing. The requirement on sweep velocity printed in
the standard is based on the assumption that a quasi-steady state response of the structure should
be achieved, i.e. the sweep must not be too fast. Recommended maximum sweep rates are given in
the standard. The maximum rate amax for linearly swept-sine excitation is recommended as
follows:
amaxo54f2rQ2
; given in Hz=min; 1
wherefr is the estimated resonance frequency and Q the dynamic amplification in the resonance
frequency. For logarithmically swept-sine excitation the maximum rate is recommended asfollows:
Smaxo77:6fr
Q2 ; given in oct=min: 2
The recommended sweep velocities are illustrated in Fig. 1. The frequency range plotted in the
figures covers the typical range of interest for large aircraft, i.e. from 1.0 to 20 Hz : The dampingvalues introduced in the figures are typical for large aerospace structures (z 0:33.0%). They are
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0
10
20
0
2
40
2
4
6
fr[Hz]
min=0.0028 oct/min
D [%]
S[oct/min]
0
10
20
0
2
40
50
100
fr[Hz]
min=0.0019 Hz/min
D [%]
a[Hz/min]
0.5 oct/min
5 Hz/min
Fig. 1. Recommended maximum sweep rates (left: logarithmic; right: linear).
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related to the factor Q by
Q 1=2 z: 3
Feasible minimum sweep rates, i.e. acceptable rates from a practical point of view, are includedas lines in the figures. For example, the recommended sweep rate for a logarithmic sweep for a
structure with its lowest resonance at 1 Hz and with a modal damping of 0.3% is 0 :0028 oct=min:Under these conditions, a typical sweep of 0.516 Hz; i.e. 5 octaves, will take (according toEq. (2)) 1785 min (nearly 30 h). This impressively demonstrates the inappropriateness of these
recommendations for large test structures having low eigenfrequencies and low damping values.
The recommended maximum sweep rates are referred to in ISO-7626 from [6]. The cited article
originates from 1966, i.e. 37 years ago. In those early days, the author tried to estimate the FRF
by evaluating the root-mean-square response amplitude; consequently, disregarding the correct
correspondence between the excitation frequency and response frequency. The author
investigated under which circumstances the assumption of a steady-state condition during
swept-sine excitation is acceptable.
The estimation of maximum acceptable sweep rates was based on three steps:
1. The duration Dt of a logarithmic sweep in the vicinity of the resonance fr is simplified
from Eq. (15) by using the relation between resonance amplification Q and the half-power
bandwidthDf:
Dt60 lnfr Df=2=fr Df=2
Sln2 4
which can be simplified using the relation for the half-power bandwidth:
Df fr=
Q:
5
Further simplification assuming small damping values leads to
Dt 60
SQ ln2: 6
2. The initial time period for reaching the steady-state condition of a lightly damped, sdof system
which is excited harmonically in its resonance or 2pfr by a unity force can be derived from
the solution of the equation of motion (Eq. (23)) [7]:
ut #ue2zort cosortj Q sinort 7
which, with the initial conditions ut 0 0 and ut 0 0; becomes
ut Q1e2zort sinort: 8
The expression 1e2zort in Eq. (8) determines the extent to which the steady-state
condition is achieved for a system excited from rest. It must be noted here that this assumption
is not valid during swept-sine excitation where the structure is continuously in motion.
3. The sweep time from the half-power point to the resonance Dt=2 should exceed the timerequired to reach a certain degree of the steady-state condition 0oko1:0:
1e2zortXk: 9
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These three steps finally yield an expression for the maximum sweep rate Smax by combining
Eqs. (6) and (9):
Smax 60p
ln2 ln1k
fr
Q2: 10
For a minimum 95% steady-state condition in amplitude k 0:95 Eq. (10) yieldsSmaxp90:7fr=Q
2; given in octaves per minute. Bozich [6] makes a similar estimation and givesthe recommendation of a maximum sweep rate ofSmaxp43:3fr=Q
2:It must be emphasised at this point that limits for sweep rates are only necessary if the FRFs are
derived from evaluating the root-mean-square response amplitude. This will be worked out in the
next sections.
3. Theoretical background
The effect of swept-sine excitation on the identification of the FRFs can be best studied in a
sdof system. Its dynamic behaviour can be described in general by the FRF of the system:
Ho Uo
Po; 11
where Uo and Po are the spectra of the response and excitation, respectively. The FRF is
exclusively determined by the dynamics of the structure.
3.1. Sweep excitation waveforms
The excitation force can be written for all kinds of sinusoidal excitation such as
pt #pt sin jt; jt ot: 12
Two types of sweeps are well known. Firstly, the linear sweep
ot osat; aoe os
T 13
between the start frequencyosand the end frequencyoewithin the time periodTis applied which
leads to a time-dependent excitation force of
pt #pt sin a
2t2 ost b
: 14
Secondly, the exponential (or logarithmic) sweep is very common. Its change in frequency isdefined by
ot os2St=60; 15
where os is the start frequency and S the sweep rate given in octaves per minute. Combining
Eqs. (12) and (15) yields the force function for the logarithmic sweep:
pt #pt sin 60os
Sln22St=60 1 b
: 16
In this case, the instantaneous sweep rate depends on the frequency.
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3.2. Spectral energy of swept-sine excitation
The spectral energy of swept-sine excitation within its start and end frequencies is determined
by two parameters. Firstly, the spectral distribution of the excitation energy is directly related tothe excitation amplitude. This enables the spectral control of the excitation energy by arbitrarily
setting the excitation amplitude #pt(Eq. (12)). Secondly, the spectral distribution of the excitation
energy is determined by the sweep rate.
An analytical expression can be found for the normalised Fourier spectrum (NFS) of the
excitation signal of a swept-sine run in the case of a linear sweep[8]. An infinite swept-sine run has
to be considered in order to be able to solve the corresponding Fourier integrals which finally
yields
NFSlino 1 i
4 ffiffiffiffiffiffi1
par wV3 iwV1 i1sgn aeio2=2a
#p; 17whereV1 and V3 are complex functions ofo;
V1 1i
2ffiffiffi
ap o; V3 1 i
2ffiffiffi
ap o 18
andwzis the complex error function. Ifa > 0 (sweep with increasing frequency), the oscillatorycharacteristic introduced by the error functions vanishes with increasing o: Ignoring theseoscillations by dropping the error functions leads to
NFSlino 1 i
4
ffiffiffiffiffiffi1
pa
r eio
2=2a#p 19
and the absolute value of the NFSis simply
jNFSlinoj
ffiffiffiffiffiffiffiffi1
2pa
r #p: 20
Fig. 2shows a comparison between the exact solution for an infinite sweep, the approximate
analytical solution, and the Fourier transform of a finite sweep. The oscillations at the start and
end frequencies are caused by the immediate start and end of the sweep which is equivalent to the
effect of a rectangular time window. Obviously, the approximation (Eq. (20)) provides the level of
the normalised spectrum. It does not depend on the length of the sweep: finite and infinite sweep
result in the same level of the spectrum.
No means have been found yet to analytically solve the Fourier integral in the case of alogarithmic sweep. But the final result of the linear sweep given in Eq. (20) can easily be extended
for the logarithmic case. The sweep rate can be considered constant for each time increment in the
Fourier integral that leads to the absolute value of the NFS given in Eq. (20). This value does not
depend on the length of the interval and thus can also be used for an infinitesimal short time
interval. Therefore, the sweep rate which is constant for the linear sweep depends on o in the case
of a logarithmic sweep:
a-ao Sln 2
60 o: 21
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As a result, the NFS for a logarithmic sweep is given by substituting Eq. (21) into Eq. (20):
jNFSlogoj
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1
2pao
s #p
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi60
2pSln 2
r ffiffiffiffi1
o
r #p: 22
Fig. 3shows the NFS of logarithmic sweeps for three different (logarithmic) sweep rates. The
faster the sweep, the lower the spectral energy.
It can be summarised as a conclusion from the energy considerations that the sweep rate is a
parameter which is able to control the excitation level as well as the force amplitude. This enables
the optimisation of excitation amplitude and testing time.
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0 1 2 3 4 5 6 70
2
4
6
8
[1/s]
nfs[N/Hz]
Analytical solutionFourier transformApproximate solution
Fig. 2. Normalised Fourier spectra of a linear sweep.
0 2 4 6 8 100
10
20
30
40
50
60
70
[1/s]
nfs[N/Hz]
Sweep rate SApproximate solutionSweep rate 0.5 SApproximate solutionSweep rate 2 SApproximate solution
Fig. 3. Normalised Fourier spectra of a logarithmic sweep.
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3.3. Transition through resonance
3.3.1. Phenomenology
During a swept-sine excitation, all resonances of the structure in the covered frequencyrange are passed with a certain sweep velocity. As a result, the maximum response for a resonance
is lower than the maximum response expected for a harmonic excitation with the correspond-
ing resonance frequency. Furthermore, there is a time delay for the maximum which thus
is not reached when the excitation frequency ft equals the resonance frequency fr; but a littlebit later instead when the excitation frequency is already higher than fr: Free vibrations ofthe structure are excited in the process (with frequency ffree fr) in addition to the forced
vibrations (with the swept-sine excitation frequency ft) and lead to a modulation of the response
signal.
Fig. 5 shows the response signal evoked by a linear sweep. The time axis is substituted by
the corresponding excitation frequency that is normalised with the resonance frequency ofthe mode. The delay of the maximum response which would correspond to fEfr in the case
of a harmonic excitation and the modulation of the signal can be observed. Fig. 4demonstrates
the effect of the sweep rate on the maximum response using the laboratory beam structure. It
shows six different runs in which the same frequency range was covered in different time
periods ranging from 6 to 90 s with a constant sweep rate. The same force amplitude was
applied in each run. Nevertheless, the maximum responses differ by more than a factor of
two.
3.3.2. Analytical solution for the linear sweep
The equation of motion for a sdof system can be written as
.u2zor uo2r u
pt
m ; 23
where z defines the damping, or is the eigenfrequency of the system. Eq. (23) can be solved
analytically in the case of a linear sweep if a constant force amplitude is applied (Eq. (14)).
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0 10 20 30 40 50 60
-3
-2
-1
0
1
2
3
t [s]
u(t)[g]
..
Fig. 4. Effect of different sweep rates on the response of a structure.
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The solution can be displayed in a format which resembles the solution of the stationary
case[9]:
ut #p
mjQtjsinjt bt: 24
The dynamic amplification Q is a constant in the case of harmonic excitation. However, for
transient processes Qt is time-dependent:
Qt jQtjeibt: 25
In the case of a linear sweep excitation, Qt can be expressed in the following way:
Qt Bwv1 wv2 C1ev2
1 C2ev2
2; 26
where
wv ev2
1 2iffiffiffi
pp Z v
0ez
2
dz" #
27
is the complex error function and
v1=2t 1 i
2ffiffiffi
ap atos izor7ior ffiffiffiffiffiffiffiffiffiffiffiffiffi1z2q
28
are complex-valued linear functions of time. B is a complex constant which depends on the
eigenfrequency, the sweep rate a; and the damping z:
Bi 1
4 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffio2rp
a1 z
2
s : 29
The exact solution given in Eq. (26) consists of four different terms of importance for the
behaviour of the solution close to the resonance frequency. The terms C1ev2
1C2ev2
2 describe
free vibrations which only influence the initial part of the sweep and which are usually damped out
when arriving at resonance. A more sophisticated discussion, given in [10], is required to estimate
that wv25wv1: Thus,
QtEBwv1: 30
The approximate analytical solution is compared with the exact solution in Fig. 5. The exact
solution was achieved by time integration of the equation of motion. The approximation is very
good near resonance. For comparison, the FRF is also given in the figure.
4. FRF estimation
The effect of swept-sine excitation on the FRF by means of different estimation methods is
investigated in this section. A typical corner case for large aerospace structures is taken as a
numerical example. The sdof system and its excitation is determined by
* eigenfrequency:fr 1 Hz* damping: z 1% andz 2%
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* logarithmic sweep* different sweep rates: S 0:252 oct=min* frequency range of excitation: f 0:71:4 Hz
4.1. Co-quad analyser, tracking filter, Hilbert transform
Co-quad analysers (also called vectormeters) evaluate the time-domain responses by relating
the instantaneous amplitudes in phase to the applied forces which yield real and imaginary parts
of the amplitude. The response utdue to a sinusoidal excitation is given in Eq. (24) in a general
form which can be rewritten as
ut R #utcos jt I #ut sin jt: 31
The real part R #utand imaginary part I #utof the response can be deduced from the response
multiplied with sinusoidal references which yields
ut sin jt 12I #ut 1
2I #ut sin 2jt 1
2R #ut cos 2jt; 32a
ut cos jt 12 R #ut 12R #utcos 2jt 12I #ut sin 2jt: 32b
The application of a low-pass filter which eliminates the second and third addend in Eq. (32)
directly yields the real and imaginary parts of the responses #ut and the applied forces #pt;
respectively. These complex envelopes are then assigned to the instantaneous frequency byapplication of Eq. (13) or Eq. (15). By this means they are used as an estimation *Uo and *Pofor the spectra of the forcePoand responsesUoimplying that the structure responded mono-
frequently with the instantaneous frequency of excitation. The FRF can finally be estimated by
HoER *Uo jI *Uo
R *Po jI *Po: 33
Eq. (33) can be extended for multiple input to
HoE *Uo *Po1 34
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0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3
-10
0
10
20
(t)/R
[-]
q(t)[-]
integration in time domainAnalytical solutionFrequency response function
Fig. 5. Time-domain integration and approximate analytical solution (a 0:002o2
r
; z 0:02).
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which requires the application ofnx linear-independent force vectors for nx simultaneous inputs.
The linear independence can be checked by the condition criterion cHo of Hadamard of the
matrix %Po built by nx different sweeps:
cHo det %PoQnx
n1
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPnxl1
%Pnlo %P
nloq
: 35
The procedure using co-quad analysers is valid only as long as the assignment of the complex
envelopes to the instantaneous frequency can be used as an approximation for the spectra Uo
and Po: This is acceptable only in very special cases.Tracking filters work on a similar principle which again implies that the structure responds
mono-frequently with the instantaneous frequency of excitation.
The Hilbert transform ofut;
Hut 1
p
Z N
N
ut
ttdt 36
can also be used to evaluate the envelope of the time-dependent functions ft andut [11]. The
amplitude of FRFs can be derived from the envelopes after transforming the time axis into the
frequency domain by means of Eq. (13) or Eq. (15). This procedure is affected by the same
constraints as the previous methods.
The FRF estimation by means of vectormeters, tracking filters, or Hilbert transform leads to a
distortion of the identified FRF. This is depicted in Fig. 6 for a damping value ofz 2% and
different sweep velocities. In the figure, the absolute values of the FRFs are shown in the graph on
the left. The real and imaginary parts are depicted in the waterfall plot on the right. Even for avery slow sweep of 0:1 oct=min; which hardly makes sense from a technical point of view, thedistortion still is considerable. It leads to a shift of the eigenfrequency and a lower peak value
which results in a higher damping value identified in modal analysis.
The amount of distortion depends on the damping of the structure. A smaller damping leads to
a more pronounced distortion of the estimated FRF. Finally, the recommendation of the ISO
standard is investigated for a sweep velocity of 0:5 oct=min which leads from Eq. (2) to a
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0.9 1 1.1 1.2 1.30
0.1
0.2
0.3
0.4
0.50.6
f [Hz]
Hexact
S = 0.25 oct/min
S = 2 oct/minS 0.9 1 1.1 1.2 1.3
0
2-0.5
0
0.5
f [Hz]
0.9 1 1.1 1.2 1.3 2-1
-0.5
0
0.5
S
f [Hz]
Himag
Hreal
S
Fig. 6. Sweep evaluation by means of vectormeters.
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minimum recommended frequency of 4 Hz for a damping ofz 2%: Fig. 7 illustrates that theestimated FRF is still distorted mainly in the frequency range near the eigenfrequency. Moreover,
the shift of the FRF depends on whether the sweep is moving upwards or downwards.
These examples illustrate that FRF estimation by means of vectormeters, tracking filters and
Hilbert transform is not recommendable for swept-sine excitation.
4.2. Fourier transform
A straightforward FRF calculation method is the Fourier transform:
Uo 1
2p
Z N
N
uteiot dt; Po 1
2p
Z N
N
pteiot dt 37
of the responses ut and forces pt; respectively. The FRF can then be directly calculated fromEq. (11). In practical application, the measured responses and forces are discrete but non-periodic
sequences in time. Thus, the Fourier transform usually is performed as discrete Fourier transform
(DFT)[12]. The main disadvantage of this procedure arises from the high number of samples nsrequired for the acquisition of a slow sweep which, for a linear sweep (Eq. (13)), amounts to
ns Tuomax
p
38
where uX1 is a factor of oversampling recommended for a better amplitude resolution. For a
logarithmic sweep (Eq. (15)), the number of samples is
ns 60uomaxlnomax=omin
2p ln2S : 39
For example, a typical sweep from 2 to 32 Hz with a rate of 0 :5 oct=min produces 122 880samples, assuming that an oversampling of u 4 is applied. In most practical cases, the fast
Fourier transforms (FFTs) are not applicable since the number of samples is not a power of two.
Of course, a number of samples being a power of two may be obtained by zero padding of the
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3 3.5 4 4.5 50
0.02
0.04
f [Hz]
|H|
3 3.5 4 4.5 5
-2
0
2
f [Hz]
angle
Hexact
Fig. 7. Sweep evaluation by means of vectormeters f 4 Hz; S 0:5 oct=min:
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measured time sequences. However, this may lead to even larger FFT sizes for long sweep runs.Moreover, the high number of samples, which is not necessary for the resolution of the FRF in the
frequency domain, can lead to noisy FRF spoiling the subsequent modal analysis.
The FRF estimation based on the Fourier transform yields FRFs which do not differ from the
exact solution as shown inFig. 8.
The measurement time for the sweep response decreases for a higher sweep rate which, in turn,
yields a lower frequency resolution for the chosen parameters. This can be improved by acquiring
longer time sequences which may include the decay after the end of the swept-sine excitation. For
long sweep runs up to high frequencies it becomes more and more difficult to achieve a number of
samples without zero padding, which is a power of two as required for the FFT. A data reduction
is helpful in this case to avoid the time-consuming DFT and to smooth the FRFs.
4.3. Data reduction via averaging
Data reduction in the frequency domain is possible by partitioning the time series, transforming
these partitioned time blocks, and averaging them. In order to minimise leakage due to the non-
periodic time sequences, weighting of the time sequences with suitable time windows like the
Hanning window is very common. This, in turn, requires an appropriate overlapping of the time
sequences in order to avoid a loss of information. Two alternative ways of averaging are possible.
4.3.1. Cross-power spectra evaluation
The averaging of power spectra (XPOW) is a very common procedure. Expanding Eq. (11) bythe conjugate complex of the Fourier transformed time sequence of the force pt; yields in thecase of multiple responses an expression for the FRFs:
fHog f %Supog
%Sppo; 40
wheref %Supog are the averaged cross-power spectra
f %Supog 1
N
XNl1
fSupogl 1
N
XNl1
fUogPol 41a
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0.9 1 1.1 1.2 1.30
0.1
0.2
0.3
0.40.5
0.6
f [Hz]
|H|
0.9 1 1.1 1.2 1.3 1.4
0
2-0.5
0
0.5
S
f[Hz]
0.9 1 1.1 1.2 1.3 1.4
0
2-1
-0.5
0
0.5
f[Hz]
Hreal
Himag
S
Fig. 8. Sweep evaluation by means of FFT.
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and %Sppo is the averaged auto-power spectrum of the input force:
%Sppo 1
N
XNl1
fSppogl 1
N
XNl1
PoPol: 41b
In the case of multiple input, the matrix of FRFs can be calculated from
Ho %Supo %Sppo1 42
which requires the averaging of the responses ofnxindependent force patterns for nxsimultaneous
inputs.The effect of data reduction by means of the cross-power spectra averaging is investigated in
Fig. 9. A data reduction by a factor of two is utilised for the investigation which yields different
length of time partitions for each sweep velocity.Fig. 9reveals a bad effect of the data reduction
on the identification of FRFs. The reduction yields an additional damping, which is listed in
Table 1. In this case, the FRF distortion is caused by the truncation of the decay of the resonance.
This is illustrated inFig. 10which shows the sweep response and the pure decay on the left. On the
right, the estimated FRFs which are derived from this response are plotted for different reduction
grades expressed by means of the time Tof the time partition. The related length of the time
frames of data evaluation are added in the figure. Additionally, the remaining decay amplitude is
given in percentage for each curve calculated by
Adecay e2pfrzT 43
which indicates the dependency of the truncation effect on the eigenfrequency and the
modal damping. The latter effect is evaluated from the FRFs and is listed in Table 1. It is
considerable even for a small truncation of the decay part of the response. The truncation error
for the damping is lower than 5% for time windows where the remaining decay amplitude Adecayis
lower than 0.001%. This leads to the following recommendations to avoid severe truncation
effects:
1. The time partitions should be as large as possible.
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0.9 1 1.1 1.2 1.30
0.1
0.2
0.3
0.40.5
0.6
f [Hz]
|H|
Hexact
S=0.25 oct/min
S=2 oct/min
S
0.9 1 1.1 1.2 1.3
0
2-0.5
0
0.5
S
f [Hz]
Hreal
0.9 1 1.1 1.2 1.3
0
2-1
-0.5
0
0.5
S
f [Hz]
Himag
Fig. 9. Sweep evaluation by means of cross-power spectra average.
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2. The minimum length of the time frame should satisfy the condition
Tmin > 2
frz: 44
Eq. (44) contains the unknown parameters fr and z of the lowest resonance which have to be
estimated for the initial step of the data reduction. If the modal analysis of the FRFs yields
parameters that violate Eq. (44), the FRF estimation has to be repeated with a larger time frame
for the response partition. However, there is no way to secure the applicability of a time window
because the identified dampingthat may be estimated larger than the actual valuecould fulfil
Eq. (44) whereas the true damping value would not. It should be noted that the fulfilment ofEq. (44) yields only 4 points in the half-power bandwidth of the resonance which is not very much
for a reliable modal analysis.
4.3.2. Peak reference hold evaluation
An alternative is the utilisation of the peak reference hold(PRH) averaging technique which is
defined by
%Uol %Uol if jUREFoljpj %UREFolj;
Uol if jUREFolj> j %UREFolj:
( 45
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Table 1
Damping valuesvarying reduction grades
Reduction grade PRH XPOW
Exact solution 1.00%
T180 s 1.05% 1.03%
T90 s 1.35% 1.28%
T45 s 1.95% 1.60%
T22:5 s 3.55% 3.00%
0 50 100 150
-1
0
1
t [s]
u
0 50 100 150-1
0
1
t [s]
u
0.9 1 1.1 1.2 1.3 1.40
0.2
0.4
0.6
0.8
1
1.2
f [Hz]
|H|
Hexact
Heval
: T=180 s; Adecay
= 0.0012 %
Heval: T=90 s; Adecay= 0.35 %
Heval
: T=45 s; Adecay
= 5.9 %
Heval
: T= 22.5 s; Adecay
= 24.3 %
Fig. 10. The effect of different reduction grades.
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PRH averaging is applied on time partitions and yields spectra %Uoand %Powhich can be used
for the FRF estimation (Eq. (11)).
Independent force patterns are required for multiple inputs. The linear independence can be
checked by the condition criterion that is defined in Eq. (35).Data reduction via PRH averaging leads to similar drawbacks like the cross-power spectra
averaging if the decays of resonances are truncated (see Table 1) [13]. Consequently, the same
recommendations are valid as those given for the FRF evaluation based on XPOW averaging.
4.4. Data reduction via reduced Fourier transform
An alternative to averaging is the performance of the Fourier transform for selected frequencies
for which the FRF is required. In practice, a maximum acceptable frequency resolution Df is
selected and the DFT is evaluated for all multiples of this Df:If the following resolution is chosen:
Df 12sDt
; 46
where Dtis the sampling rate of the sweep ofns samples, the reduced DFT (RDFT) can simply be
performed by means of a sum of FFTs:
UjDf 2
ns
Xns=2sm0
X2s1n0
um2s nDtei2pjDfnDt 47
which significantly reduces the calculation effort.
A second way of a reduced Fourier transform is the evaluation of the Fourier integral at
discrete frequencies with a resolution which depends on the number of spectral lines Nwithin the
half-power bandwidth of all resonances being passed during the sweep. The frequency resolutionis determined by the increment
Df 2zf=N 48
which results in a reduction to nf spectral lines:
nf lnoe=os
ln12z=N 49
within the start frequencyos and the end frequency oe of a sweep run. This reduced DFT results
in an unequally spaced frequencies for the FRF evaluation. This method is referred to as LDFT.
Table 2compares the calculation effort and the amount of data reduction for the different data
reduction methods. The example is based on a typical sweep from 2 to 32 Hz ; acquired with131 072 samples. A data reduction to 2048 spectral lines is chosen for the RDFT, the LDFT is
based on a damping assumption of 1% combined with 5 data points within the half-power
bandwidth of all resonances.
5. Experimental results of large aircraft
Typical swept-sine test data acquired during the GVT of a four-engine aircraft are presented
here in order to investigate the effect of the different evaluation methods on real test data. The
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aircraft is dynamically characterised by a high modal density E4 modes=Hz in the low-frequency range. A photo of a typical GVT test set-up is shown inFig. 11.
The sweep test run investigated here is defined by the following parameters:
* frequency band from 0.5 to 32 Hz* logarithmic sweep, velocity 0:5 oct=min* constant force amplitude* total length 947 s; 75 776 samples* two simultaneous shakers, symmetric and antisymmetric excitation
One FRF calculated by different methods is presented here, and some identified modal
parameters are compared. The FRFs are scaled with the values of the fundamental mode.
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Fig. 11. GVT set-up of a four engine aircraft.
Table 2
Calculation effort for different Fourier transformation methods
Example
Method Number of operations Number of operations Spectral lines
DFT n2s 17 179 869 184 65 536
FFT nslog2ns 2 228 224 65 536
RDFT nslog2nf 1 572 864 2048
LDFT lnoe=os=ln12z=N 136 414 513 1040
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The FRF derived from the FFT of the time sequence without any data reduction is shown in
Fig. 12. The plot reveals a noisy FRF which makes a modal analysis difficult, especially in the
higher frequency range. The mode indicator function (shown in the right ofFig. 12for the low-
frequency part of the sweep), which is calculated from the driving point FRFs, emphasises this
experience. The modal parameters derived from the FRFs differ from the results of the
appropriated mode measured by means of the phase resonance method only slightly by 0.4% for
the fundamental eigenfrequency, by 6.9% in the related modal damping, and by 3:7% in thegeneralised mass.
Averaging is a standard procedure for noise reduction and, moreover, to reduce the amount ofdata. A data reduction by means of PRH averaging is shown in Fig. 13for different grades of
reductionR: Obviously, the FRFs are smoother than those calculated from the complete FFT.A global comparison of the different reduction grades (Fig. 13, left side) shows no significant
differences. However, truncation errors can be discovered in the low-frequency range which is
depicted on the right part ofFig. 13.
The FRFs estimated by using four different reduction grades are modally evaluated. The results
are displayed inTable 3. A high data reduction to 2.7% of the total length of the sweep leads to
a small error in the eigenfrequency, but to a considerable error in the modal damping. The
generalised mass is nearly unaffected by the truncation. The effects decrease for the second mode.
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5 10 15 20 250
0.5
1
1.5
2
2.5
3
f
|H|
2 4 6 8 10 12 140
200
400
600
800
1000
f [Hz]
MIF
Fig. 12. FRF based on FFT.
5 10 15 20 250
0.5
1
1.5
2
2.5
3
f
|H|
2.7 % (high)5.4 %10.8 %21.6 % (low)
0.5 1 1.5 2 2.5 30
0.2
0.4
0.6
0.8
1
f
|H|
2.7 % (high)5.4 %10.8 %21.6 % (low)
Fig. 13. FRFs based on PRH average.
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The deviation shown inTable 3is related to the modal parameters extracted from the complete
FFT of the responses.
Data reduction by means of cross-power spectra averaging is shown inFig. 14for four different
grades of reduction. Again, truncation errors can be discovered in the low-frequency range.The FRFs estimated by using four different reduction grades are modally evaluated. The results
are listed inTable 3. A considerable data reduction to 2.7% of the total length of the sweep run
leads to correct eigenfrequencies, but also to a significant error in the modal damping. The
generalised mass again is nearly unaffected by the truncation. The effects decrease for higher
frequencies, e.g. for the second mode. The deviations shown in Table 3are related to the modal
parameters extracted from the complete FFT of the responses.
A data reduction by means of RDFT is shown inFig. 15for three different reduction grades.
No significant differences are visible although the FRFs are more noisy than those derived from
averaging especially in the high-frequency range. The detail of the FRF in the low-frequency
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Table 3
Influence of data reduction on the modal parameters in percentage
Frequency Damping Generalised mass
Reduction to % PRH XPOW RDFT PRH XPOW RDFT PRH XPOW RDFT
Mode 1
2.7 1.7 0.0 0.2 145.5 211.7 2.6 1:3 6:2 4.55.4 0.3 0.0 0.0 105.2 83.1 0.0 6:4 3:6 2:5
10.8 0.0 0.0 0.0 41.6 29.9 1.3 2.3 1:7 1:421.6 0.0 0.0 15.6 26.0 0:9 0:9
Mode 2
2.7 1.0 0.1 0.1 79.8 79.8 4.3 0.5 2:7 7:05.4 0.2 0.1 0.1 39.4 28.7 1.1 0:6 1:1 0:3
10.8 0.1 0.1 0.0 11.7 8.5 1.1 1:5 0:5 0:8
21.6 0.1 0.1 2.1 3.2 0:2 0:2
5 10 15 20 250
0.5
1
1.5
2
2.5
3
f
|H|
2.7 % (high)5.4 %10.8 %21.6 % (low)
0.5 1 1.5 2 2.5 30
0.2
0.4
0.6
0.8
1
f
|H|
2.7 % (high)5.4 %10.8 %21.6 % (low)
Fig. 14. FRF based on cross-power spectra averaging.
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range is plotted in the right ofFig. 15. The FRF derived from the complete FFT is added in theplot. The three different reduction grades are modally evaluated. The results are tabled inTable 3.
A high data reduction does not lead to any significant error in the eigenfrequency, modal
damping, and generalised mass. The deviations shown in Table 3are again related to the modal
parameters extracted from the complete FFT of the responses.
In conclusion, the low-frequency range should be analysed without any data reduction whereas
the higher frequency range should preferably be analysed with data reduction in order to reduce
noise.
6. Summary and conclusion
The European aircraft industry is constantly calling for a reduction of the testing time of
prototypes without diminishing the accuracy of the data. As a consequence, substantial changes in
the testing strategy have been introduced in past ground vibration tests of European aircraft. This
test strategy is mainly based on the combination of the benefits of the very reliable but time-
consuming phase resonance method and the use of phase separation techniques on data stemming
from swept-sine excitation.
This article reviews the standards for the application of swept-sine excitation defined by the
international organisation for standardisation (ISO). The theoretical background of swept-sine
testing is expounded with particular emphasis on the transition through structural resonances.The effect of different standard procedures of data processing is investigated. Particular care is
needed for the data reduction of long swept-sine runs which may affect the identification of the
modal damping values in the low-frequency range. Recommendations on the proper estimation of
frequency response functions are given in this article. On the other hand, the article shows that no
restrictions for the sweep rate are needed as long as a proper data processing is applied to the
measured time-domain data.
As a conclusion it is suggested to adapt the guidelines for the application of slowly swept
sinusoids published in ISO 7626 part 2 [4] to the actual capabilities of contemporary data
processing.
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5 10 15 20 250
0.5
1
1.5
2
2.5
3
f
|H|
2.7 % (high)5.4 %10.8 % (low)
0.5 1 1.5 2 2.5 30
0.2
0.4
0.6
0.8
1
f
|H|
2.7 % (high)5.4 %10.8 % (low)
Fig. 15. FRF based on RDFT.
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