A Method for Identification of a Set of Optimal Measurement Points for
Experimental Modal Analysis
Thorsten Breitfeld
Institute for Statics and Dynamics of Aerospace Struct,ures
University of Stuttgart
Pfaffenwaldring 27
70550 Stuttgart, Germany
[email protected]
Abstract
[Ak] : Boolean selection matrix
Prior to a structural optimization using the FiniteElement-Method, it is often desirable to correlate measured mode shapes from an existing structure with
those calculated by a Finite-Element model using the
Modal-Assurance-Criterion
(MAC).
It can be shown that the elements of the MAC-matrix
are highly dependent upon the measurement points
used during a modal analysis, and thus an optimal set
of accelerometer locations should be identified prior to
performing the modal analysis.
In this paper such a set of points on the structure is
identified. Given the maximum number of available accelerometers and all possible application locations (i.e.
on the accessible surface of a structure) a set of measurement locations is found using combinatorial and optimization techniques. Potential aliasing of calculated
and measured mode shapes can now also be identified
prior to performing the modal analysis, and the best
attainable MAC-Matrix is calculated. The proposed
method is demonstrated on a full-scale FE-model of a
high-speed laser cutting robot.
Nomenclature
[M] : Mass matrix
[D] : Damping matrix
[K] : Stiffness matrix
{z} : Displacement vector
X
:
Eigenvalue
{a} : E i g e n v e c t o r
Z1, Z2 : Valuation function
k : Node number
no
:
Number of available accelerometers
n : Number of FE-Nodes
m : Number of calculated FE-modes
1
Introduction
The finite element method is being applied on an ever
growing scale in industry, since the cost of building and
testing prototypes during the product development cycle has become restrictive. ‘Thus, numerical simulations
have evolved into a cost-effective alternative to improve
the dynamic characteristics of structures.
If a first prototype exists which exhibits unsatisfactory
dynamic behaviour, a finite element model of it,s structure can be created. Parameters that govern the performance are identified with the help of this model and
and structural modifications can easily he implemented.
It is now possible to verify the effectiveness of proposed
modifications can by a recalculation of the modified finite element model.
In order to obtain a solid foundation for all further
simulation and optimization, the mathematical model
should represent the existing prototype as closely as
possible. In order to achieve a successful correlation the
experimental modal analysis of the prototype must, be
sure to capture all relevant dynamic effects. A significant source of error during it experimental modal analysis that makes correlation especially difficult is aliasing,
which hinders separation of the measured modes due to
large off-diagonal terms in the Modal Assurance Criterion (MAC) -Matrix[l]. Using the eigenmodes of the
non-correlated finite element model, all structural os-
1029
cillation modes can nevertheless be estimated and a
favourable distribution of accelerometer locations for
the experimental modal analysis which avoid aliasing
on the existing prototype can be found.
2 Aliasing Effects during Experimental Modal Analysis
During an experimental modal analysis, the responses
of a structure to an excitation are measured at no measurement poinbs. Various curve-fitting algorithms allow
the calculation of a maximum of no eigenvectors from
these rneaured structural responses. Figure 1 shows
that, due to to badly placed accelerometers (1,2), the
first two modes of the considered beam cannot be distinguished, since both measurement points exhibit the
same displacement for each eigenmode. This effect is
referred to as aliasing. Especially for very complex
struct,ures this effect is very difficult to avoid since an
overview of the eigenmodes is not easy to obtain and
thus oscillatory effects cannot immediately be uniquely
identified. This xi11 usually be the case for all considcred technical systems.
when damping is neglected. Unknowns in this equation
are the eigenvalues X and their corresponding real eigenvectors {G}, One important, characteristic of eigenvectors is their K- and M-orthogonality, which is also a
straight orthogonality if the mass-matrix is diagonal.
Thus, a criterion for distinguishing between eigenvalues
on the basis of this orthogonality condition was developed: The Modal Assurance Criterion [l]. This criterion calculates the normalized scalar product of two sets
of vectors {ax} and {+A} and arranges the resulting
scalars into the MAC-Matrix:
This matrix contains inform&ion about the orthogonality of the considered vector f&s. If a 1.0 is found in position [i,j] of the matrix, tha vector i is identical to the
vector j up to a scalar mulr.iple. A zero in the matrix
identifies orthogonal vectorzs.. If the investigated vector&s are all taken from the calculated eigenvectors
of the same finite element model, and all elements of
the vectors are included in the evalxxtion of the MAC,
the unity-matrix will result. If not all element~s of the
eigenvectors are included (t.runcated eigenvectors) offdiagonal values will result. These off-diagonal values
are the basis of the following considerations.
4 Measurement Point Selection
Figure 1: Aliasing due to badly placed measurement
points
3 An Orthogonality Criterion to
Discern Eigenmodes
The characteristics of an arbitrary linear vibrating system can be approximated with the help of mass, damping and stiffness matrices by applying the finite element
method. This results in the second-order differential
equation for free vibrations
[At]. {Z} + [II]. {i} + [K] {z) = 0.
(1)
The solution of this equation is represented by a linear combination of harmonic functions, leading to the
rigenvalue problem
0 WI + WI) {@I = 0
(2)
Since only a limited number of locations can be fitted with accelerometers during an experimental modal
analysis, a subset of points must be identified that significantly contribute to the preservation of the orthogonality of the eigenvectors and thus avoid aliasing. Selection of these points is accomplished using results of
a finite element calculation.
The number of potential measurement points to be located is usually limited by t,he number of available sensors no. Another restriction of the selection process
is the accessibility of points for application of the accelerometers and other hardware. The overall goal
would be to identify those ILO points of all n possible
(finite element) points that would lead to the lowest
possible off-diagonal terms in the MAC-matrix. It is
important to note at this point that the MAC-value of
a pair of different vectors can be made worse (higher)
even if a measurement point is added. Figure 2 shows
two different eigenvectors that become more difficult to
distinguish once an accelerometer is placed at point P
the MAC-value becomes larger.
Valuation functions that couple the selection quality
1030
Figure 2: Increase of MAC-Value due to added measurement point.
to the size of the off-diagonal terms are defined for the
selection process. The first of these functions is
m--l
Z1
m
= c c MACij
(4)
i=l j=i+,
which weighs all upper off-diagonal terms of the [m x m]
MAC-matrix equally. Due to the symmetry of the
MAC-matrix, only values above the diagonal are considered for the valuation. All values on the diagonal of
the matrix are unity During an experimental modal
analysis with closely spaced modes, eigenfrequencies are
often especially difficult to distinguish. If the eigenvectors of the finite element calculation are arranged in
order of frequency, a second valuation function
m--l
Z,
L:
m
E c MAC&.
Ii -jl
(5)
identified measurement point are very likely to have
a similar negative effect on the MAC-value. If this is
the case, they will not cont.ribute any new information
during a experimental modal analysis. Thus, all neighboring points are investigated by the selection process
and removed from the analysis if they are found to
exhibit similar characteristics as the identified measurement point. The definition of being ‘geometrically
close’ as well as defining the maximum number of points
to be removed from the selection process in each iteration are a function of mesh size and can be specified by
the user. The implemented software assists in finding
these parameters, since they affect the runtime of the
selection process.
The iteration process of
l
l
l
identification of a node with a large (negative) effect on the MAC-value of two vectors,
definition of this point as a potential measurement
point for these eigenvectors and
removal of this and all eligible neighboring points
from the selection process
is repeated for each eigenvector pair until the desired
number of no points is reached for each pair. Finally,
all results of all vector pairs are analyzed and a global
set of measurement points for all vectors is found.
i=l j=i+1
5 Sample Large Scale FE-Model
can be defined. Here, the MAC-values of modes with
neighboring frequencies are weighed more since it is important that they be clearly identified.
The valuation functions judge the effect of removing
node k when calculating the MAC-value of two vectors.
Afrer all vector pairs are investigated, the functions ax
used to select those measurement points which lead to
an overall optimal MAC-matrix for all possible pairings
as described below. If a boolean selection matrix [Ak] is
used to simulate the removal of node k, the MAC-value
at at position [i, j] of the matrix is given by
An internal list keeps track of the change in the MACvalue for the two involved vectors after removal of node
k. The node that worsens (increases) the MAC-value
of eigenvectors i and j the most is considered important for distinguishing these modes and is marked as
a measurement point. It, is removed from the selection process. Nodes that are geometrically close to the
I~-
~
L
Figure 3: FE-Model of the laser-cutting center
After verifications using academic examples [2], the
applicability of the proposed method to large finite
1031
element-models is demonstrated using a laser-cutting
center (See Fig.3). The structure was modelled using
7716 elements with 6872 nodes leading to a total of
40944 DOFs. The first 9 calculated eigenvectors are
used in the selection process.
are emphasized.) Fig.4 shows that some measurement
points were placed inside the structure in order to measure and identify internal vibrational behaviour.
5.1 All Nodes as Potential Measurement Points
200 measurement points are to be distributed across
the structure, allowing all 6872 FE-nodes to be potential measurement points. Figure 4 shows the computed
distribution. Adjacent pictures show the structure from
the same viewpoint with the elements blanked out on
the right-hand side. The corresponding MAC-matrices
prior to and after the selection process are given below:
MAC(6872).
:t.
.; :__
-,
..,
::
. . . . .: _
.:
:
i’
i..’ .
, ; :_ :
I.‘:.
.
.
.
:
100 =
Figure 4: Placement of 200 measurement
6872 nodes
5.2
MAC(200,)
.,.
:
:,
:
:.;
.;. :
._ ;. *
‘;‘:,,;
”
_:_ . ..i.L.
points using
Using Only Surface Nodes as Possible Measurement Points
Since placing acceleromet~ers
inside a structure is not
always possible, using only accessible points on the surface of a structure as potential measurement points
is examined next. As expected, the MAC-matrix
MAC(3550) formed using only surface nodes has large
off-diagonal values even before reduction (mode pairs
[9/2], (9/4] and [6/7]). R,educing these 3350 nodes to
200 measurement points nevertheless leads to a satisfactory MAC-matrix MAC(2002).
zz
I
0
3
0
Before reduction the matrix (MAC(6872)) has
small off-diagonal elements (< 9) with the exception of
the MAC values for the eigenvector pair 4 and 9. These
two modes show very similar oscillating behavior, differing only in the vibration of the laser-cone. Since this
part of the structure was modelled using only beamelements and lumped masses, a better resolution is not
to be expected here. To better investigate these two
modes, the FEmodel would have to be improved in
this area.
All expected FErnodes, except those mentioned above,
can be distinguished clearly using the 200 measurement points found by the selection process as can be
seen in MAC(2001). (The largest off-diagonal elements
1032
MAC(2002). lOO=
- 100
0
0
1
0
0
1
3
1
0
-Fl
0
01
:
0
100
m
0
0
0
0
9
100
To put the above results into perspective, 50 meaurement points were selected ‘by hand’ for a experimental modal analysis. Their distribution is shown in
fig.7, the matching MAC-matrix MAC(502) is given
below. High off-diagonal elements result from these
hand-picked and well distributed measurement points
even though knowledge of the resulting eigenvectors was
used in the selection.
MAC(502).
100 =
By removing some measurement points, the offdiagonal matrix elements have improved in some posit,ions, making the corresponding eigenvectors easier to
distinguish (See also Fig.2). Fig. 5 shows the positions
of the selected measurement points.
J
1
‘_’ ,.
::.
..’
“.,m!..j ,_,, _ -. .“i..L_,
.’ . :‘.
.:: , *: ..-“.
,.;, _’ , < ..__ . ,..
._ , ._. .
“..._::
.,;,.a
I, ..:.
:... ‘:
..*
:,.
.
.
..:.:..
;:,_-
a....
i:
:::
,:
i:..
.
..i
:.__
‘..
__.
..
_; 1.‘.. . . .
,.:.
:!_I.
.’
‘.
lifEi++
MAC(6872)
_
:
;’
;;.
Finally, table 5.2 summarizes the valuation functions
Z1 and Z, of all examples.
MAC(3550)
MAC(200,)
‘:$
..&
I
Table 1: Valuation functions
Figure 5: 200 measurement points selected from 3550
surface nodes
Reducing all 3550 surface nodes to 50 measurement
points leads to the MAC-matrix MAC(501) and the
distribution of accelerometers shown in fig.6. Once
again, the eigenvector
pairs [4/9] as well as [6/7] have
the highest MAC-values, and this effect can be taken
into consideration during the subsequent experimental
6 Concluding Remarks
The algorithm described abow identifies measurement
points for an experimental modal analysis from a set of
given possible measurement points utilizing information
supplied by a dynamic analysis of an existing finite element model. The computed measurement points lead
to a sub-optimal MAC-matrix and can be used to identify and differentiate all modes predicted by the finite
element analysis. Using these measurement points, the
probability of missing modes during an experimental
modal analysis is significantly reduced, especially when
dealing with complex structures whose dynamic behavior is generally not easy to predict.
Since the boundary conditions of a finite element, model
often do not represent reality in a sufficient, manner
and model-uncertainties like coupling, connections and
joints are usually present to some extent in every real
1033
structure, the identified measurement points can be
augmented by ‘hand-picked’ measurement locations defined by the experience of the test engineer.
In summary it is important to note that:
l
The MAC-matrix computed from the input data
(all possible measurement points) prior to reduction identifies modes that will be difficult to distinguish in the subsequent experimental modal analysis, for example internal modes and modes with
similar dynamic characteristics.
l
Pretest assessment of the best possible MACmatrix aids the experimental engineer in the interpretation of his results.
l
Frequency separation aids the discernibility of
modes. If eigenvectors are ordered by frequency,
comparatively high MAC-values far from the diagonal are not crucial, since the modes are separated
by a large frequency band.
. Adding a node as a measurement point can decrease the quality of the MAC-matrix, but in reality improves the quality of the experimental modal
analysis by redundant measurement point.
_:
,:
., :
,I .
.:_ . .
I
_.
.:
:. .:
.
I’
1.
:’
,.
.
!.
.
:
I
L..
Figure 6: 50 measurement points selected from 3550
surface nodes
. Using only ‘hand-picked’ measurement points one
might miss detecting an unexpected mode or unexpected dynamic behavior in an experimental modal
analysis.
. Augmenting the calculated measurement points
by ‘hand-picked’ measurement points is advantageous, since modelling errors and incorrect boundary conditions may cause the iinite element calculation to not calculate all relevant modes.
.
.
.
. . .
References
.__
:
(11 R.J. ALLEMANG, D.L. BR O W N: A Correlation Coe&ient for Modal Vector Analysis. Proc. of the
International Modal Analysis Conference, Orlando,
November 1982.
.
.
.
j
[2] KUNZMANN, G.: Enittlung ULJ~ optimalen Messstellen fiir eine Modale Analyse unter Verwendung
eines FE-Modells. Studienarbeit, Institut fiir Statik
und Dynamik, Universitiit Stuttgart, Dez. 1993.
:
:
:
..”
Figure 7: 50 ‘hand-picked’ measurement points
1034
k