FRF SUBSTRUCTURE SYNTHESIS : EVALUATION AND VALIDATION OF
DATA REDUCTION METHODS
Luigi Bregant’, Dirk Otte** and Paul Sas’
“LMS International
lnterleuvenlaan 66
B-3001 Leuven, Belgium
*KU Leuven, PMA Division
Celestijnenlaan 3008
B-3001 Leuven, Belgium
Abstract
This paper deals with dynamic substructuring techniques
using experimental FRFs. Despite a straightforward
fonulation. coupling predictions generally still fail to yield
acceptable results. An overview of previously performed
research is given. Next, the formulations are reviewed, in
view of their physical significance. A pseudo-inverse data
reduction technique, based on the singular value
decomposition is presented. The main part of the paper
then consists of the discussion of an extensive laboratory
case study. Apart from a number of caveats in using the
pseudo-inverse. the importance of not including rotational
degrees of freedom is also revealed.
Nomenclature
DOF : degree of freedom
FRF : frequency response function
[H,L,[Hs],:
FRF matrices at interface DOFs
tK& LHs1, :
row of FRFs between response i on A
P+&JH,l~
:
resp. 6 and all interface DOFs
column of FRFs between all interface
Hq,HBI,Hcg
:
DOFs and reference j on A resp. B
FRF on resp. A,B,C between i and j
Pl[Vl :
matrices of left and right singular vectors
+4}&1:
[+YJo;
:
i-th left, resp. right singular vector
diagonal matrix of singular values, resp
singular value
1. Introduction
Dynamic substructuring
techniques are methods that allow
to predict the dynamic behaviour of a structure based on
the dynamic behaviour of the composing substructures.
The theory of calculating the Frequency Response
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Functions (FRF) of a structure composed of several
substNctures
using the FRF’s available on the
SubstruCtures is well established. The basic formulation of
the method is very simple, Involving only elementary matrix
algebra. The method is attractive, for no parametric
(modal) modelling is required: the responses can be used
directly.
Initial applications of the method, also referred to as
impedance coupling, indicated however that the method
could only work well in ideal circumstances. This has
certainly deviated attention of the method to the methods
based on a modal formulation in the seventies. The
applicability of the modal approach is however limited to the
low frequency range where a reduced modal density allows
for accurate extraction of component modes.
The reasons of failure were, on the one hand, the
inconsistencies of measured data (noise, frequency shins),
the inabilfiy to measure correctly all DOFs of the
connection - specifically the full matffx of FRF’s between all
DOFs at which substructures are coupled and rotational
DOFs - , and one or more matrix inversions that could
become ill-conditioned for a larger number of DOFs.
Consistency of the data could largely be improved by
recent evolutions of data acquisition and processing
techniques: the advent of multichannel data acquisition
systems and the multiple input estimation of FRF’s. Such
methods have also enabled a better approach to the
measurements of the full matrix of FRF’s between all DOFs
of coupling.
Apart from this improvement, a lot ot research has been,
and still is, carried out for better conditioning the involved
matrix inversion. The actual fonulation onen depends on
the type of FRFs available and the set of constraint
equations that is used in the system assembly. Generally, it
requires some subsequent matrix inversions. A refined
algorithm requires only a single inversion of the sum of
FRF-matrices of the interface boundary DOFs [l]
A first research topic consisted of preprocessing of the
measured FRF data for each individual subsystem
(“smoothing”), before the actual coupling formulation was
applied [2]. A technique was used to smooth FRFs based
on a frequency band singular value analysis. The FRF
matrix is then reconstructed using a reduced number of
singular values. More successful was the symmatrisation
approach. The same principles -frequency band SVD- are
used on pairs of reciprocal FRFs.
For the actual coupling formulation, a pseudo-inverse
approach was used, by means of an SVD (at each spectral
line). Through a number of simulations, some difiiculties
were pointed out however, related to the pseudo-inverse
approach [3,4]. Data reduction, which is in this case
equivalent to singular value elimination. on a complex sum
of matrices may indeed eliminate strong contributions
appearing in both substructures but with opposite sign.
Nowhere else in literature, these diiicutties seem to have
been reported. It was concluded that some kind of
indicators were to be developed to assure that the singular
values to be eliminated do not contain significant structural
information.
This paper brings some of these ideas into practice,
considering an extensive case study on the coupling of two
laboratory aluminium structures using experimental FRFs.
Also other sources of error, such as the inlluence of
rotational DOFs are also discussed.
The formulation that is used here requires the inversion of
a complex sum of two square FRF-matrices, related to the
interface DOFs in substructures A and B. The FRFs of the
assembled structure C are derived starting from the
compatibility condtions for the displacements and the
equilibrium conditions for the forces [l]. In the following, the
dependence of frequency is tacitly assumed.
If both DOFs i and j are on structure A,
(1)
(2)
in case DOF i is on the structure A and DOF j on the
structure B
Hq = tH~~([H~l,+[Hslss~‘~Hg)sI
symmetric, may tend to be singular at some frequencies. At
the resonance frequencies of the assembling structures, its
rank tends to be 1. On the other hand, the poles of the
assembled structure will coincide with minima of the
determinant and may cause rank deficiency. The latter is
clear from intuition. observing that this mattfx is really
independent of the particular set of DOFs I and j. so that
the poles of the coupled function should be described by
[H,L+[H,L. I f t h i s m a t r i x i s rankdeficient.
Is
inversion, required for the solution of either of the above
equations, may become ill conditioned at the frequencies
mentioned above, causing the solution to be heavily
influenced by minor pellurbations of the data.
2.2 Singular Value Analysis
By means of a linear least squares approach, the coupling
solution can be obtained by using the pseudo-inverse of
[HA], +[H,E, The pseudo-inverse is found by means of
= [uIWH
(5)
in which only the significant (“non-zero”) singular values
and the corresponding singular vectors are kept. The
singular values, considered. A certain number of the lower
singular values is then forced to zero.
Such data reduction can then be considered as a spatial
domain data reduction. reducing the original number of
interface DOFs to a number of effective or “principal
DOFs, the latter related to the physical DOFs by means of
the singular vectors.
In practice, whether important structural information at the
resonances is omined or not, is dependent on the position
of the singular values in which the minima of the
determinant of [HAL +[HBL (which equals the product
if both DOFs i and j are on structure B,
+[bI&Hdq
The kernel in these equations is the inversion of
[HAL +[HBh This matrix, which should in principle be
[H&+[HsE,
2.1 Basic Formulation
Hq = HsJ -tK&([H&
each of these
a singular value decomposition :
2. Theoretical Considerations
H,,=H,i-tH~~([H”Es+[HeEs~‘{H~>y
It is clear that for the interface DOFs,
formulations can be applied.
(3)
in case DOF i is on structure B and DOF j on structure B
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of the singular values) are reflected. Small singular values
can indeed be due to the sum of two small contributions
from A and 8, but also due to the sum of two large
opposite-sign
contributions.
The i-th singular value o,of this summed matrix can indeed
be considered as the sum of the contributions of the two
substructures:
.
the two substructures have to be easy to manufacture
and not to bulky
.
the two substructures have to be easy to measure
0; = CA; + Is&!;
The final test structure consists of two shaped aluminum
plate, each 3 mm thick, (fig. 2). The two substructures are
glued together at the three foreseen connection flaps to
form the coupled structure. The edges of the connection
flaps are tapered in order to increase the bonding surface.
The glue used to connect the two substructure is a very
stiff structural glue (3M Scotch weld 2216 B/A), this to
as much as possible. an infinitely r&id
simulate,
connection.
(7)
These two contributions can be obtained as follows,
starting from Eq. 5 :
Note that oli and og, are complex, but the imaginary parts
will have opposite sign.
2.3 Influence
of Rotational DOFs
To evaluate the influence of the rotational DOFs a simple
FE simulation was performed. Two square section bars of
different length were modeled by means of finite elements.
The first modes of the two structures and the FAFs at the
ends of the two bars were calculated.
The modes of the coupled bar and the FRFs at the
coupling DOFs were also calculated.
The FRFs of the coupled structure were then ‘predicted
using both translational and rotational DOF information (x,
y, 9). The match between the true FRFs and the ‘predicted’
FRFs is perfect.
The FRFs of the coupled structure were prediied
considering only the translational informations (x, y). In this
case, the predicted FRFs exhibit the peaks at lower
frequencies (fig. 1).
The obtained results seem to confirm the importance of the
rotational DOF information in the coupling process.
However, from experiences of other researchers lt seems
that the importance of the rotational DOF is structure
dependent and each case has to be investigated
separately.
3. Experimental Case Study
3.1Measurementset-up
A test structure was designed in order to fulfill the following
requirements :
.
the structure has to consist of two substructures
.
the two substructure have to be different, in order to
study the most general coupling conditions
.
the two substructure have to be easily coupled and
separated (1 necessary, more than once; the coupled
structure must always have the same dynamic
characteristics)
.
the two substructure need to have middle-high modal
density in the mid-frequency range
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Other techniques to connect the two substructure, life bolts
or rivets. have been taken into account but were not
retained due to the tikficukies these connection systems
might bring in the measurement phase (i. 8. non lineadties,
rattling, ..) and due to the problems that might arise for
obtaining constant connection’s characteristics (i.e. bok
torque, time loosening...).
The optimization of the test structure was performed using
finite elements calculations. The final dimensions of the two
pieces were decided on the base of comparisons of FRFs
of the different substructures. An equal number of points,
sufficient to describe the deformations of the connection
areas was selected on the two substructures and the
relative FRFs were measured. The same FRFs were also
measured for the coupled system in order to have a base
for the evaluation of the obtained results. The msasunng
grid ls shown in fig. 3.
Roving hammer excitation was used in all the FRFs
measurements, the choice of this kind of excitation was
dictated by the need of direct transfer functions in all the
considered DOFs. The use of a shaker wouki have
requested a much longer set-up preparation time.
To take in account the mass loading effect of the
accelerometers attached to the structure under test dummy
masses were used.
For each measurement point, only the vibration orthogonal
to the plane was measured, since the vibration amplitude in
the plane of the plate is negligible compared to the first
one.
For each connection flap (fig. 3) the two edge points and
the middle points were measured; this selection yields a full
coupling matrix of 81 elements (9 by 9).
The acquisition parameters used during the measurements
WBW
Spectral lines:
Central frequency:
Bandwidth:
Trigger IeveI:
Pretrlgger:
600
400Hz
600Hz
20%
3%
Force window:
Exponential decay:
N”
averages:
H estimator:
5%
5%
5
H”
During the measurements, the structures were suspended
by soft rubber bands, in order to simulate the free-free
boundary conditions.
3.2 Quality of the measurements
The test structure is very lightly damped and as a
consequence the peaks of the FFtFs are very sharp at the
natural frequencies. Since there are no sources of noise or
other contaminating effects the obtained functions are also
very smooth, furthermore the position of the natural
frequencies was found very stable for all the measured
response function (fig. 3: three different direct FRFs, at the
middle flap).
The coherence functions reaches always vary high values,
giving confidence on the good quality of the data. The
reciprocity check, performed on different points of the two
substructures, gives also a good match (fig. 4).
The mass loading effect of the accelerometers has been
tested too. Below 600 Hz, no evident dterences were
perceptible when comparing of the FRFs obtained with or
without dummy masses. In order to be more confident in
the measurements all of them were performed with dummy
masses anyhow.
Numerical techniques used to smooth the FRFs and to
improve the symmetry of the FRF matrices are not taken
in account here. In fact the measured FRFs are practically
noise-free and the reciprocity check indicate that those
matrices are already symmetric.
3.3 Singular Value Analysis
A singular value decomposition was performed on the
coupling DOF matrix [/-/,1,+[He],. The peaks in the
highest singular value give an indication on the damped
natural frequencies of the substructures, furthermore for
each frequency line, the comparison between the different
singular values gives information about the number and the
relative importance of the principal responses of the
system.
shows the higher frequency band in more detail (200400
Hz). It is clear that the prediction using the direct inverse
reveals some spurious peaks due to the illconditioned
inversion. On the other hand, eliminating the singular value
below a certain threshold introduces some spurious dips
and chopped-off resonance peaks. There is also a clear
negative frequency shift of the resonance peaks,
comparable to what happened in the simulation (Fig.1).
Finally, the amplitude estimation is generally satisfying.
The chopping of the peaks is apparently due to the
disregarding of small singular values bearing important
structural information as the sum of opposite s@n
contributions. This can clearly be seen in Fig. 6, showing
the smallest singular value and the ampliiudes of both
partial contributions. This phenomenon implies that the
singular values, candkiate to being forced to zero should at
each frequency be checked whether they consist of large
or small partial conttfbutions. In the latter case however,
one would be obliged also to keep the noise and illcondiiioning effects.
To improve the resufts (especially the frequency shifts) and
to have a bener insight in the coupling formulation two
different experiment enlarging the number of considered
interface DOFs have been tried out:
.
increasing the number of measurements points
(considering the same measuring directions of the
original set up)
.
increasing the number of directions in which the
measurements are performed (using the same number
of points of the original set up)
For the first experiment two extra points for each
connection area have been measured bringing the total
number of coupling points to 15. The interface FRF
matrices then have 225 (15 by 15) elements instead of the
original 81 (9 by 9). These 15 by 15 matrices are used in
the reconstmction and the results compared with the ones
obtained with the original 9 by 9 matrices. No major
improvements were experienced.
The second experiment was not successful: for the
extremely difficult measuring conditions, it was not possible
to acquire the desired FRFs. The few good FRF measured
in the plane of the plate are in any case very low compared
to the ones measured in direction orthogonal to the plane
(25-50 dB lower).
IA the present case, the smallest singular (a*) value is
relatively small compared with the highest one (o,),
between 1000 and 10000 times lower for all the considered
matrices (fig. 5: o,, 0,. oT,, o,, oJ.
Taking into account the earlier FEM simulation, it was
concluded that the frequency shifts are due to the fact that
no rotational DOFs were measured.
3.4 Coupling Prediction
4. Conclusions
Fig. 6 shows reconstructed FRFs (direct inverse and
pseudo-inverse for two different thresholds) in comparison
with the measured FRF on the coupled structure. Fig. 7
An experimental case study on two aluminium plates has
been presented. Despite a generally reasonable prediction
1595
qua order of magnitude, two major diiicuities could be
observed :
.
the data reduction method using the pseudo-inverse
embeds some serious caveats, as the matrix to be
inverted fs a complex sum of two FRF matrices. This
observation perfectly confirms the simulation msuits
described in [4].
.
not measuring FRFs between rotational DOFs causes
a negative frequency shift of the predicted resonance
frequencies.
Figures
Further research on how to deal with both these numerical
aspects and measuring aspects is therefore imposed. The
inversion problem can be dealt with using a total least
squares approach; previous simulations [4] were shown to
be promising. At higher frequencies, the exact resonance
frequencies are not important anymore and one is more
interested in global amplitude levels. A frequency averaging
approach [4], also based on singular value analysis. was
proven successful for simulations. Also here, experimental
validation should be canted out.
Acknowledgement
The presented research work was performed within the
framework of the Human Capital and Mobility Project
MONICA (Methods, On Noise Identification Control
Activities) funded by the European Commission.
Fig. 1 : FEM simulation :_ : FRF coupled structure: :
coupling prediction
References
[l] Van Loon P. “Modal Parameters of Mechanical
Structures”, Ph.D. Dissertation, Dept. of Mech. Eng.,
KU Leuven, Leuven, Belgium, 1974.
[Z] Leuridan J., De Vis D., Grangier H., Aquilina R..
“Coupling of Structures Using Measured FRF’s :
Some Improved Techniques”, Proc. of the 13th tnt.
Seminar on Modal Analysis, Leuven. Belgium, 19-23,
1938, paper No. l-19.
[3] Otte D., Leuriian J.. Grangier H.. Aquilina R.,
“Prediction of the Dynamics of Structural Assemblies
Using Measured FRFdata: Some Improved Data
Enhancement Techniques”, ‘Proc. 9-th Int. Modal
Analysts Conference, Florence (ltaly), Apr-15-19,
1991, pp.909-915. ‘Prcc. Florence Modal Analysis
Conference, Florence (Italy), Sept-10-12.
1991,
pp.325334.
[4] Otte D. “Development and Evaluation of Singular Value
Analysis Methodologies for Studying Muitivadate Noise
and Vibration Problems”, Doctoral Dissertation,
Katholieke Universiteit Leuven, Belgium, 1994.
1596
Rg. 2 : Test structure
/-a+uzMONICA
t!
F :g.
Fig. 3 : Test st~cture : measurement grid
6 : Coupling prediction : -- : measured FRF, --- : direct
ilwerse; . . . : pseudo-inverse
r....
:
I
I”
a. I”
r.., “.” .,..,. I..
/-a+MONfCA
p+%~ONfCA
‘ig. 7 : Coupling prediction (200-400 Hz)
ig. 4 : Measured FRFs : reciprocity check
;:
,
I,
p+zMONlCA
pm+&
/m
1.1
MONKA
Fii. 8 : Smallest singular value (---) and contributions from
A and B (---; . ..)
Fig. 5 Singular values of [HAL, +[&L, t ,3,5,7,g
1597