Analysis of Swept-sine Runs During Modal Identification(2004)

8/10/2019 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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G. Gloth, M. Sinapius / Mechanical Systems and Signal Processing 18 (2004) 142114411422


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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

ARTICLE IN PRESS

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).

G. Gloth, M. Sinapius / Mechanical Systems and Signal Processing 18 (2004) 14211441 1423


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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)).

ARTICLE IN PRESS

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.



21/21

References

[1] P. Fargette, U. F.ullekrug, G. Gloth, B. Levadoux, P. Lubrina, H. Schaak, M. Sinapius, Tasks for improvements in

ground vibration testing of large aircraft, in: Proceedings of IFASD, CASA/AIAE, Madrid, 2001, pp. 121133.[2] G. Gloth, M. Degener, U. F .ullekrug, J. Gschwilm, M. Sinapius, P. Fargette, B. Levadoux, P. Lubrina, New

ground vibration testing techniques for large aircraft, Sound and Vibration 35 (11) (2001) 1418.

[3] P. Frachebourg, Comparison of excitation signals: sensitivity to nonlinearity and ability to linearize dynamic

behaviour, in: Proceedings of the 10th International Modal Analysis Seminar, Vol. 1, Leuven, 1985, pp. 110.

[4] ISO, Vibration and ShockExperimental Determination of Mechanical Mobility, Parts 15, iSO-7626/1-5.

[5] D. Ewins, Modal Testing: Theory, Practice and Application, 2nd Edition, Research Studies Press Ltd., Somerset,

UK, 2000.

[6] D. Bozich, Utilization of a digital computer for on-line acquisition and analysis of acoustic and vibration data,

Shock and Vibration Bulletin 35 (4) (1966) 151180.

[7] H. Irretier, Grundlagen der Schwingungstechnik, Vieweg, Braunschweig, 2000.

[8] R. Markert, An- und Auslaufvorg.ange linearer Schwinger im Frequenzbereich, Zeitschrift f.ur Angewandte

Mathematik und Mechanik 65 (4) (1985) T81T83.

[9] R. Markert, M. Seidler, Analytically based estimation of the maximum amplitude during the passage through

resonance, Applied Mechanics in the Americas 8 (ISBN 85-9000726-2-2) (1999) 15111514.

[10] R. Markert, Amplitudenabsch.atzung bei der instation.aren Resonanzdurchfahrt, in: Festschrift zum 60.

Geburtstag von Prof. Witfeld, Inst. f.ur Mechanik der UniBw Hamburg, UniBw Hamburg, 1996, pp. 1945.

[11] C. Harris (Ed.), Shock and Vibration Handbook, 4th Edition, McGraw-Hill, New York, 1995.

[12] D. Pollock, A Handbook of Time-Series Analysis, Signal Processing and Dynamics, Academic Press, New York,

1999.

[13] G. Gloth, M. Sinapius, Swept-sine excitation during modal identification of large aerospace structures,

Forschungsbericht DLR-FB 2002-18, DLR-Institut f.ur Aeroelastik, 2002.

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Analysis of Swept-sine Runs During Modal Identification(2004)

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