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BepiColombo: sine test FEM correlation experiences
J.F. Mercer1, A.M. Kiley
2, G.S. Aglietti
1
1Surrey Space Centre,
University of Surrey, Guildford, Surrey, UK. GU2 7XH
2 Airbus Defence & Space,
Gunnels Wood Road, Stevenage, Herts. UK. SG1 2AS
e-mail: [email protected]
Abstract This paper reflects on some of the experiences of correlating the Finite Element Model (FEM) of ESA’s
BepiColombo spacecraft to sine sweep test measured data, in order to validate the model for subsequent
Coupled Loads Analysis (CLA). Post-model update procedures can take a significant number of man
hours to complete, without necessarily resulting in a final FEM which is notably more representative of
the real structure than the initial FEM. The long term research intention is to use the lessons learnt from
BepiColombo and other spacecraft correlations to work towards the containment of the FEM correlation
process, this paper addresses part of this on-going research effort. This is to be achieved through:
investigating the current techniques/algorithms for FEM test analysis correlation and validation;
identifying their limitations; thus ultimately developing a methodology which identifies where model
updates are or are not necessary to achieve a model which is validated for key participating modes in flight
predictions or during any sine test notching.
1 BepiColombo mission.
BepiColombo represents a joint ESA / JAXA corner stone mission in Space science and exploration of the
magnetospheric and surface mapping of the planet Mercury. The architecture consists of three separable
spacecraft modules in a stacked arrangement for launch. Two planetary orbiters are stacked on top of a
lower module whose function is to serve as propulsion spacecraft fulfilling the cruise phase to Mercury
after separation from an Ariane 5 launcher.
Figure 1 shows the Structural Thermal Model (STM) in readiness for longitudinal sine testing at the
ESTEC facility. The overall test specimen mass is 6400kg comprising 3900kg (stack structure), 2400kg
(test adapter) with the remaining mass allocated to a Force Measurement Device (FMD) and ancillaries.
2 Background
From a structural perspective, launch is one of the most challenging phases in the mission of a spacecraft.
The interaction of the spacecraft and the launch vehicle is an important aspect of this, however, for many
reasons, not least of which being the size of the launch vehicle, it is not possible to practically test the two
systems together. Thus, in order to simulate the launch experience, Coupled Loads Analyses (CLAs) are
835
Figure 1. BepiColombo STM stack in readiness for longitudinal sine test.
carried out which couple a Finite Element Model (FEM) of the spacecraft with a model of the launcher to
virtually predict flight loads. The process of qualification or “certification” to fly is therefore reliant on
predicted static and dynamic responses for which the Prime contractor of the system must show test
coverage or analytical coverage with elevated factors of safety. In summary, demonstration of FEM model
quality is key part of the process leading to final certification.
If there is to be confidence in the results of the CLA, it is necessary to first validate the spacecraft FEM
against test measured data. The analytical and experimental results are compared and the FEM updated
where necessary to improve its representation of the behaviour of the real structure at the modes of
interest. This correlation and model update process can take a considerable effort and time, and in certain
cases may result in no improved modal correlation on local modes. Therefore, containing this process and
ensuring meaningful targets and efficient processes is an important task with the potential to greatly aid in
improving industrial practices.
In terms of testing, dedicated modal survey tests on Spacecraft programs are nowadays generally rare with
dynamic correlation normally performed relative to dynamic acquisition from fixed base sine tests
(specimen clamped at launcher interface). For such fixed base sine tests, accelerometers generally capture
the response of the spacecraft at a few hundred locations, in contrast to the FEMs which tend to have
many hundreds of thousands of Degrees of Freedom (DOFs). For BepiColombo in excess of 500 channels
of data were acquired encompassing in excess of 400 accelerometer channels, with the remaining channels
dedicated to acquisition from strain gauges and 6-DOF load resultants at the specimen to electro-dynamic
shaker interface plane on a Force Measurement Device (FMD).
During definition of the accelerometer, strain gauge and load cell measurement point plan, the Prime
contractor focus is normally directed towards assignment of data acquisition at or on equipments/units or
appendages with the goal of showing coverage of the CLA on a unit-by-unit basis. Some account is often
taken to intuitively define a level of distribution of instrumentation that supports characterization of main
primary structure modes, local main appendage modes etc. but rarely are tools or processes applied to
confirm adequacy of the instrumentation to fulfil a robust modal characterization, the focus being manage
the sine test safely and to obtain discrete loads on equipments.
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In this paper, the focus is on the modal analysis and comparison of FEM eigen-values and eigen-vectors
with the test measured natural frequencies and mode shapes. Particular focus is directed towards showing,
even with a relatively high instrumentation count such as on BepiColombo, quality of modal “metrics”
may be heavily influenced by the selection of model reduction technique. Here comparisons are made
primarily by using Modal Assurance Criteria (MAC) [1] and Normalised Cross-Orthogonality (NCO) [2]
checks. For MAC the mode shapes and eigen-vectors are compared, resulting in a value which indicates
how closely matched the vectors are, with a perfect match yielding a 1. It is generally considered, and
indicated in ECSS-E-ST-32-11C [3], that target modes achieving a MAC of at least 0.9 indicates a good
correlation. The NCO check is also an ECSS [3] requirement, and works similarly to MAC, but with the
mass matrix employed to weight the relative importance of the DOFs being considered. An ideal result of
perfectly matched mode shapes which are orthogonal to the mass matrix will yield a diagonal matrix, and
for mass normalised modes this becomes an identity matrix. Off diagonal values <0.1 and leading
diagonal terms >0.9 are deemed to indicate a good correlation.
When performing the MAC check, the vectors describing the mode shapes must be of equal vector order
and the mass matrix dimensions must match this for the NCO check. It is therefore necessary to either
expand the experimental data to a DOF count matching the FEM or reduce the analytical results [4] to a
DOF count equal to that of the test acquisition count. A potential problem with using the FEM to expand
the test data is that any errors in the FEM, which is still to be validated, may corrupt the experimental data
and therefore undermine the following correlation and update process. It is therefore generally considered
that the modal reduction of the FEM to the test measured DOFs is the preferred approach [5].
The reduced FEM is commonly known as the Test Analysis Model (TAM), and there are many methods
currently available for performing these reductions. There have already been several studies (e.g. [6] &
[7]) comparing the various methods including; static (or Guyan) reduction [8], improved reduction system
(IRS) [9], System Equivalent Reduction Expansion Process (SEREP) [10] and Craig-Bampton [11]. Here
the focus is on; the Guyan reduction method, which has some historical basis and availability within FE
tools such as MSC-NASTRAN, and SEREP which has been identified as suitable for generating
spacecraft TAMs ([12] & [13]).
This paper concentrates mainly on the ability of the TAM to represent the behaviour of the full model. In
practice other factors are also extremely important, such as the robustness of the TAM for NCO checks.
This is a crucial aspect as both the FEM and the test mode shapes are inevitably affected by some level of
“errors” (i.e. differences between the analytical or experimental mode shape and the true mode shape of
the physical structure). These discrepancies can be difficult to identify and accurately quantify as they
result from a wide variety of sources. For example, for the experimental mode shapes, typical sources of
errors can be: inaccuracies in the accelerometers sensitivities, non-ideal boundary conditions etc. For the
analytical mode shapes, approximations of the geometry and properties of the structure and possibly
errors/inaccuracies in production of the FEM are the most likely causes of errors. The quality of the TAM
is also influenced by the sensitivity of the reduction method to sensor placement locations as highlighted
previous.
Another important aspect of ensuring that the correlation process is meaningful is confirming that the
experimental deflected shapes, which are used as targets in MAC and NCO checks, represent true mode
shapes for comparison with the mode shapes found through analytical modal analysis. This is an
important issue to explore as the data collected during the fixed base sine test of the structure is a
representation of how the structure behaves when excited at a given frequency; this is known as the
Operating Deflected Shape (ODS). When the structure is excited at a resonant frequency, which is well
separated and uncoupled from any other modes, the ODS will be essentially the same as the mode shape
with the deflection dominated exclusively by that mode. It is, however, often the case that modes are
closely spaced and the ODS at a resonant frequency is “contaminated” by other nearby modes and is not
equivalent to the mode shape. This issue has led to the development of many algorithms and software,
such as Frequency-Domain Direct Parameter Identification (FDDPI) [14], Least Squares Complex
Exponential (LSCE) [15] or PolyMAX [16] methods for estimating and extracting modal parameters and
mode shapes from test data.
DYNAMICS OF AEROSPACE STRUCTURES 837
3 Theory
3.1 Modal reduction methods
The dynamics of a system can be defined by the following equation:
(1)
where M, C and K are the mass, damping and stiffness matrices respectively, u is the physical
displacements and F is the applied forces.
3.1.1 Guyan reduction
In order to perform the Guyan, or static, reduction equation (1) can be written in the following, partitioned
form, with the damping neglected as is common practice in this context:
[
] { } [
] { } {
} (2)
Here the subscripts M and S are used for the master and slave DOFs respectively, where the masters are
the DOFs of interest to be retained and the slaves are those to be eliminated in the reduction. In an
NASTRAN context, the “masters” may be considered the problem A-Set definition.
For the static reduction, the inertias of the slave DOFs are assumed small and thus are neglected, allowing
the second line of equation (2) to be simplified to:
(3)
This allows for the slave displacements to be defined in terms of the master displacements, and this
transformation may then be substituted back into equation (2) as follows:
[
] [
] { } [
] [
] { } { } (4)
The slave displacements, uS, have been eliminated from the equation. The reduced TAM can now be
defined by pre-multiplying by the transformation matrix:
[
]
[
] [
] (5)
It is important to note that this method gives exact results for static problems. As the solution is derived
only from the static stiffness of the system, there may be discrepancies when this is applied to dynamic
analyses. The level of inaccuracy in the TAM is heavily dependent on the selected DOFs as the amount of
inertia associated with the neglected slave DOFs will affect the accuracy of the final result.
838 PROCEEDINGS OF ISMA2014 INCLUDING USD2014
3.1.2 SEREP reduction
The SEREP reduction method is derived by first defining the displacement of master DOFs as follows:
(6)
The target mode shapes, partitioned to the required DOFs, are contained in the modal matrix, , which is
multiplied by appropriate modal coordinates contained in the vector .
It is then possible to rearrange the above equation:
(7)
Often the number of master DOFs is significantly greater than the number of modes being considered, the
following procedure can be performed to obtain the pseudo inverse of the modal matrix:
( )
(8)
where the superscript P represents the pseudo-inverse of the reduced modal matrix. It is therefore possible
to re-define the displacements as:
(9)
If equation (9) is substituted into equation (1), and the resulting equation is then pre-multiplied by the
transpose of the reduced modal matrix and its pseudo-inverse, the result is:
(10)
With the mode shapes mass-normalised, which is commonly performed automatically by FE software, the
following holds true:
(11)
And so, looking back to equation (10) it can be seen that the TAM can be represented in the following
manner and is therefore determined from the mode shapes alone:
(12)
Note that no manipulation of the full FEM mass matrix has been required to obtain MTAM_SEREP. If
required, the reduced stiffness matrix may be derived in the same way. It should be noted that this method
of reduction inherently yields modes exactly matching the partitioned full FEM modes.
DYNAMICS OF AEROSPACE STRUCTURES 839
3.2 Experimental modal analysis
The ODS is the deflection of a structure when excited at a given frequency, such as that measured during
testing. This deflection results from a combination of mode shapes when the excitation occurs between
modes. At or near a resonance peak, when the ODS is dominated by a single mode and other modes have
a negligible contribution, the ODS may be approximately equal to the mode shape, as illustrated in Figure
2.
Figure 2. Excitation between modes 1 and 2 (left) and close to mode 1 (right) (Source: [17])
Figure 3. Curve fitting FRF measurements (Source: [18])
There are many algorithms and software packages available for extracting mode shapes from Frequency
Response Functions (FRFs). A simple method, however, is to use the peaks in the imaginary part of
(acceleration/force) FRFs to identify the resonant frequencies and to estimate the mode shapes as the
ODS’ at those frequencies, as depicted by Figure 3. This method can work well when there is only light
modal coupling, however a more complex curve fitting method may be required in order to obtain a good
representation of the true mode shapes when there is heavy modal coupling [18].
4 BepiColombo correlation experiences
4.1 Requirements.
The requirements for subsequent correlation were assigned by BepiColombo Prime recognizing the
current move towards application of specific “metrics” from [3].
Target requirements were assigned in advance and are summarized in the Table 1.
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Ref. Requirement Target value Comment
1. Freq. domain for correlation 0-100Hz
2. Eigen-freqs. ±3% (1st lateral bending modes)
±5% (remaining “principal
modes”)
3. Damping Average of selected measurement
locations for given mode.
Computed within error of test
freq. resolution using ½
Power Point method or
alternative.
Transfer Function based upon
Pilot Average.
4. Freq. responses / base FMD
loads
No metric Comparative only (FEM v. Test)
5. Test Analysis Model (TAM) MAC >0.9
NCO( j , j) >0.9,
NCO ( i , j) <0.1
Freq domain above.
Applicable to principal
modes.
Table 1. Pre-correlation targets.
With reference to Table 1 the following comments are relevant:
Definition of “principal mode”: this term was introduced with the aim of containing the scope of
the modal update exercise. Prime definition for a “Principal Mode” or “Principal Notch” is a
structural mode generating an active auto-notch or a mode which was predicted, on forward
extrapolation of low level sine data, to generate a notch or generate significant interface load as
seen by the Force Measurement Device (FMD).
Generally the commercial cost in terms of effort associated with evaluating such correlation
metrics does not constitute the major cost of the FEM correlation activity. The major cost is
associated with FEM update where such update measures are rational and not an arbitrary tuned
amendment. In this context it therefore follows that by containing the scope of the number of
modes to be updated then this introduces a constraint on the scope of update effort.
Target by definition means non-contractually binding. An example of non-compliance relates to
the measure assigned to the normalised cross-orthogonality check (NCO –see later) which
assesses mode shape orthogonality with respect to an analytical mass matrix. Reduced mass
matrix production was limited to static (Guyan) reduction based upon tool availability of specific
MSC-NASTRAN DMAPS. Prior to application of this NCO measure, a pre-check was performed
evaluating the orthogonality of the reduced Guyan eigen-vectors with respect to the partitioned set
at test measurement locations in the full FEM. The aim of this pre-check was to evaluate quality
of the TAM reduced mass matrix. The consequence of this check was that in spite some ~400
accelerometer points these master DOFs were poor for adequate TAM production. Further post-
test study assessing TAM production quality is discussed below.
0-100Hz domain corresponds with sine test range upper limit.
4.2 Comparison of Guyan and SEREP reduction methods
In order to investigate the quality of TAM generated by the Guyan and SEREP reduction processes, a
modal analysis has been performed on the BepiColombo FEM. In the context here, the “quality
assessment” is respect to a consistent model providing a common basis for production of the reduced mass
matrices. The assessment provided here would normally be performed in advance of Test versus FEM
mode shape correlation as a pre-check on TAM quality.
For the purposes of this investigation, the modes of interest have been selected based solely on the modal
effective mass of each mode in the translational directions, as shown in Table 2. The modal effective
DYNAMICS OF AEROSPACE STRUCTURES 841
mass is a measure of the percentage mass of the structure involved in each mode and is often used to give
an indication of significant modes for correlation [19].
Table 2. Modal Effective Masses of Selected Target Modes
The modal assurance criteria (MAC) is commonly one of the first comparisons made to give an indication
of the degree of correlation between two modal vectors.
( )
(13)
The subscripts i and j denote, commonly, the test and FEM modes respectively but also could represent
the correlation between two differing analysis bases.
The normalised cross-orthogonality (NCO) check is also a means of assessing the level of correlation
between two modal vectors, however it makes use of the system mass matrix to weight the relative
importance of the selected DOFs. The NCO it is given by the following equation:
( )
( )(
) (14)
With reference to equations (13) and (14), the context of this paper “i” , “j” are the correlation
(orthogonality) of eigen-vectors resulting from the reduced model (“j”) and those of the full FEM with
vector components partitioned to the test accelerometer locations (“i”). For the Guyan reduction (see
previous) the retained DOFs are those corresponding with test accelerometer positions. For such reduction
correspondence of eigen-values is dependent upon the master (A-SET) selection hence MAC assessments
were performed to determine which reduced mode matched most closely the corresponding selected full
mode, partitioned to the test accelerometer DOFs. For SEREP reduction the reduced problem eigen-
values match the full model from which the reduction was produced, hence for the SEREP method, the
selected modes are used to derive the reduced modes and thus only the required modes are present and no
mode matching is required.
Figure 4. Natural Frequencies (Hz) for Reduced vs Full FEM
842 PROCEEDINGS OF ISMA2014 INCLUDING USD2014
Figure 5. MAC comparing reduced and partitioned full FEM mode shapes. Left: Guyan, Right: SEREP
Comparisons were made between the natural frequencies of the full (partitioned) target modes and the
reduced modes selected as “best match” to these targets based on MAC comparisons. Figure 4 illustrates
the reduced model natural frequencies plotted against those calculated from the full FEM. Note that for
the SEREP natural frequencies the points all lie on the line, indicating a perfect match. For the Guyan
reduction, some points deviate from the line indicating a difference between the reduced and full model
results.
It should be noted that when using MAC to determine the “best match” reduced mode for each target full
(partitioned) mode, it was found that in some cases the same reduced mode gave the highest MAC for
more than one of the target modes. As a result, some reduced modes have been repeated in order that the
best match is given for each target mode in turn.
The MAC for the reduced modes compared with the full partitioned modes is given above in Figure 5 for
both Guyan and SEREP reductions.
Figure 6 shows the NCO results obtained using the Guyan and SEREP reduced TAMs. The reduced and
full modes are again compared. The full modes are then compared with themselves, using both TAMs to
give an indication of the TAM quality. As a final check the reduced modes are compared with themselves
to test the orthogonality to the reduced mass matrices. It should be noted that the repeated reduced modes,
mentioned previously, have resulted in some unity off-diagonal values in Figure 6 (c), however this result
still indicates full orthogonality of the Guyan reduced modes with the Guyan mass matrix.
The MAC and NCO comparisons of the partitioned full and Guyan reduced modes revealed a poor match
in modes other than the first few fundamental modes. This is a notable finding as it demonstrates the
errors introduced as a result of the reduction process, which means the reduced model is not representative
of the full model. Thus, any comparisons between the reduced FEM and test modes are not necessarily
indicative of how representative of the real spacecraft the full FEM actually is even with reduction to ~400
test instrumentation DOFs.
In contrast, the SEREP method has also been implemented to reduce the FEM to the test measured DOFs.
Here the reduced model matches exactly the full model results for natural frequencies and mode shapes.
This is an inherent aspect of the SEREP process and means the reduced model is representative of the full
model, making comparisons between reduced model and test results potentially more meaningful.
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(a) Partition of Full Modes vs Reduced Modes. Left: Guyan, Right: SEREP
(b) Partition of Full Modes vs Partition of Full Modes. Left: Guyan, Right: SEREP
(c) Reduced vs Reduced Modes. Left: Guyan, Right: SEREP
Figure 6. Cross-orthogonality (NCO) results using reduced TAMs
4.3 Comparison of ODS and mode shapes
When vibration testing is performed the resulting behaviour of the spacecraft is in the form of ODS’,
where an ODS is the deflected shape of a structure at a given excitation frequency. When the correlation
process is conducted, the test results are compared with the FEM pure natural modes. It is therefore
important to confirm that the test data is processed such that the vectors used in the correlation are
representative of the corresponding mode shapes.
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A simple method is to use the peaks in the imaginary part of the test acceleration FRFs to identify the
resonant frequencies and to extract the ODS at those frequencies. In order to examine the validity of this
method, a frequency response analysis has been performed on the FEM and the resulting ODS’ compared
to the FEM mode shapes. Here only x-direction excitation has been considered (spacecraft lateral). The
effective modal masses of the 21 target modes selected previously were again examined and it was found
that 8 of these modes were dominated mainly by x-direction deflections, as such these modes have been
used for comparison with the ODS. Figure 7 shows the MAC comparison of these FEM mode shapes
compared to their closest matching ODS from the FEM frequency response analysis.
The actual BepiColombo test data, processed as described above, was then compared to both the FEM
mode shapes and the FEM ODS, with the resulting MAC comparisons given in Figure 8.
Figure 7 shows that, although there is a reasonable match for the first couple of x-direction dominated
modes, overall the FEM ODS are not representative of the FEM mode shapes. Figure 8 reveals that the
test results generally give a better match to the FEM ODS than the FEM mode shapes, confirming that the
ODS at resonance frequency is not necessarily representative of the corresponding pure mode shape.
Figure 7. MAC results of FEM modes and FEM ODS
Figure 8. MAC results of experimental ODS compared with FEM modes (left) and FEM ODS (right)
4.4 Modelling practices & assumptions affecting correlation.
A key focus of the FEM update process in terms of modal correlation improvement is the update of model
parameters which are rational and do not represent arbitrary tuning. The process of correlation and update
is complicated further by identification of modelling practice which can be construed as poor assumptions
or based upon “poor modelling practice” which means automated or semi-automated optimisation routines
can be difficult to apply at the front end of update. Three examples, worthy of note, relating to modelling
practice which were found to affect dynamic correlation are provided below.
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DYNAMICS OF AEROSPACE STRUCTURES 845
Non-structural mass assumptions in dynamic models:
A “classical” dilemma faced by the engineer when modelling structures is when mass should be added
discretely or should be smeared in non-structural mass (NSM) means. During the sine test campaign a
significant notch was introduced to maintain structural responses on a pressurant tank mass dummy
mounted centrally on one of the panels within the centre stack module. The mode is shown schematically
in Figure 9 (a) and is characterized by offset flexure of a backing panel with the tank operating in kind of
torsional mode.
Outboard of the tank is a lattice panel arrangement (Figure 9 (b)) which supports high temperature multi-
layer insulation (MLI [blanket]). This blanket is effectively a low mass thin membrane only discretely
coupled to the supporting lattice. Classically such idealizations are implemented as smeared NSM which
in turn loads the local structure with mass, conservative for local stress analysis, but non-representative in
terms of dynamic idealization. Such a blanket membrane will have an “infinite” number of modes near
“zero eigen-frequency” hence negligible effective mass in the structural frequency domain. As a
consequence of this modelling approach, a high modal density of lattice modes is introduced into the
FEM, which were not apparent on test instrumentation, these modes spanned the tank torsion mode and
incorrectly coupled with it. Linked to the initial poor correlation was the tank idealization as a lumped
mass but with omission of a prescribed c of g inertia. This practice of omission represents a lack of
interrogation of a higher order mode and consideration if self-inertia of the tank was important. Since the
model involves rotation of the body about a point of flexure on the panel, omission of self-inertia is
acceptable if the (mass * distance 2) about the panel flexural axis dominates relative to the c of inertia
when this is not the case then such omission is clearly erroneous.
A key point to the discussion here is the interaction of these two erroneous or incorrect model
implementations. Identification of rational approach through a semi-automated update or mode shape
optimsation tool could be difficult and the example here highlights one area where manual scrutiny is
probably the only means of correction.
(a) (b)
Figure 9. Helium pressurant tank “torsion” mode.
Internal panel boundary conditions in terms of panel cleat RBE2s:
Invariably boundary conditions in whatever form, often represent a potential source for correlation
discrepancy. During post-test correlation of BepiColombo by far the most common source of model error
could be attributed to the area of rigidity implemented by the depended DOFs on RBE2 entries within the
model. At certain floor locations cleat features were represented as “RBE2 spiders” coupling the floor to
the primary structure central cylinder or closed perimeter panels. RBE2 dependent DOFs at the primary
structure union were in certain cases coupled to single NASTRAN GRID entries hence introducing point
flexibilities. As a consequence certain floor eigen-frequencies were found to be much lower than test as a
result of inherent boundary flexibility. For one floor in the central spacecraft model, such correction was
important since in a spacecraft axial sense this floor, together with a centrally mounted hydrazine tank of
~300kg, couples with the axial behavior of the module at the top of the stack thus generating the main
spacecraft axial mode and elevated base loads a result of such feature coupling.
The idealization of RBE2’s as previously described is deemed a “poor modelling” approach which
potentially needs manual identification or specific addressment in a semi-automated correlation tool which
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recognizes the need to address modal sensitivity subject to equations of constraint dependent freedoms
spacial coverage.
Propellant tank fluid torsional de-coupling:
Although tank slosh representation was not a key element of FEM correlation, for post-STM FEM update
to the flight configuration, it proved necessary to dynamically compensate for dynamic characteristics of
certain fluid filled tanks. Certain Tank models are sub-contractor supplied, in Craig-Bampton form, and
are subject to U.S. ITAR restrictions. Available model updates were limited hence the possibility to
incorporate new FEM updates was restricted. Within the Craig-Bampton tank model lateral slosh /
pendulum modes were incorporated but torsionally the fluid mass was effectively smeared on the Tank
shell thus no-torsional sloshing mode featured. In the absence of an update, to compensate for non-
representative mid-frequency range torsion modes of the Tank on the spacecraft platform support
structure, the Tank was suspended in its torsional DOF on a spring of low stiffness at the tanks lower polar
mount to introduce such a low frequency slosh mode. Omitting a higher torsion mode resulting from shell
inertia only, with no fluid, was not deemed important as such modes would be expected to be well outside
of the sine test range.
4.5 Overview of attained correlation.
In terms of overall assessment and with reference to Table 1, the targets of requirement targets “2” and
“4” were generally achieved with “1”, the domain, respected by implementation of test modal dampings
“3” derived from qualification level inputs.
With regards to Table 1 item “5” subject to poor pre-check correspondence of TAM mode orthogonality
with respect to full model partitioned vectors (*), the NCO check using a Guyan based mass matrix was
not assessed in correlation effort outside of the first and second order cantilever modes on each axis.
Subsequent study has been presented in this paper in the context of the reduced mass matrix production. (*) Partition to test measurement point locations.
In terms of MACs based upon ODS’:
Good correlation was achieved for the first order cantilever modes on each axis. MACs >0.9 were
attained.
Of equal importance the second cantilever types modes on each lateral axis were also reasonably
well correlated. These modes are critical to the sine test as the modes drive critical notches on the
inter-module separation interfaces (a four point cup/cone plus low shock NEA). Baseline MACs
of >0.6 were computed which improved to ~0.8 when a limited number of non-critical DOFs were
eliminated from the vector assessment.
For modes other than the above, MACs were significantly lower than targets.
Figure 10 compiles frequency response correlation at the FMD located on the Shaker – Test Adapter
interface plane, for one lateral axis and the longitudinal axis sine cases. The base force discrepancy
annotated “A” in Figure 10(c) is due to the lower spacecraft module tanks. This stack module houses five
propellant tanks of significant mass. In the FEM the elevated base force is generated by all tanks being
responsive in the FEM, this response characteristic was not exhibit during the longitudinal sine test.
In general good correspondence between Test and FEM 6-DOF base loads at the FMD is shown,
indicating indirectly, relatively good FEM representation of major effective masses.
5 Conclusions
In order to carry out the required cross-orthogonality checks it was necessary to reduce the BepiColombo
FEM to the DOFs from the test measured points. This paper has assessed influence of the selection of the
reduction method, namely comparison of Guyan static reduction versus SEREP. The Guyan reduction has
been performed for modes selected as important based on modal effective mass from the full FEM modal
DYNAMICS OF AEROSPACE STRUCTURES 847
analysis. It was found that the natural frequencies of the reduced modes deviated by up to 28% from those
of the full FEM. The MAC and NCO comparisons of the reduced and partitioned full modes revealed a
varying quality of match for the Guyan reductions. The SEREP method on the other hand inherently
produces exact matches and perfect MAC and NCO results when reduced and full models are compared.
Although these results highlight the benefits of the SEREP method, further work is required to explore the
influence of sensor placement techniques on the reductions, as well as investigating the robustness of the
methods.
It has been found that there is often a poor match between the FEM mode shape and the FEM ODS. It
was also shown that the test results matched the FEM ODS more closely than the FEM mode shapes.
(a) Lateral Sine X. FMD forces. (left = test, right =FEM)
(b) Lateral Sine X. FMD moments. (left = test, right =FEM)
(c) Longitudinal Sine Z. FMD forces. (left = test, right =FEM)
(d) Longitudinal Sine Z. FMD moments. (left = test, right =FEM)
Figure 10. Test Specimen FMD load resultants.
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