A parameter identification technique for detection of spacer
locations in an assembly of two coaxial flexible tubes
J.K. Sinha a,*, P.M. Mujumdar b, R.I.K. Moorthy c
a
Reactor Engineering Di6ision, Bhabha Atomic Research Centre, Mumbai 400 085, India
Department of Aerospace Engineering, Indian Institute of Technology, Mumbai 400 076, India
c
Department of Mechanical Engineering, Indian Institute of Technology, Guwahati 781 001, India
b
Abstract
Assemblies of two horizontal coaxial flexible tubes with loosely held spacers to maintain the annular gap between
the coaxial tubes, are generally used in nuclear reactor for carrying hot fluid inside the inner tube with an insulating
gas filled annulus between the outer and inner tubes to reduce heat losses. The appropriate location of these spacers
is important for maintaining coaxiality and preventing contact between inner and outer tubes due to bending creep
of inner tube. Determination of spacer locations is therefore an important task. The conventional method of
inspection may be costly and time consuming. This paper presents a non-intrusive technique based on vibration
measurement, developed for the detection of such spacer spring locations in the assembly of the two coaxial tubes.
The technique is based on a parameter identification approach. It involves the identification of spacer locations by
updating the position parameters of the spacer in a Finite Element (FE) structural model through the optimization
of an error criterion based on the difference between measured and computed natural frequencies of the assembly of
the two coaxial tubes. A gradient-based method is used for optimization in the FE model updating problem. The
proposed technique has been validated by numerical simulation as well as on a laboratory scale experimental setup.
1. Introduction
Assemblies of two horizontal coaxial flexible
tubes with loosely held spacers to maintain the
annular gap between the coaxial tubes, are generally used in nuclear reactor for carrying hot fluid
inside the inner tube with an insulating gas filled
annulus between the outer and inner tubes to
reduce heat losses. Locating these spacers appropriately is important for maintaining coaxiality
and preventing contact between inner and outer
tubes due to bending creep of inner tube. Fig. 1 is
the schematic of a simple structural configuration
of such an assembly of the two coaxial tubes. The
location of these spacers may shift from their
design locations due to plant operation. Such
dislocated spacers may deteriorate the system
function. Hence, the identification of spacer loca-
140
tions in such an assembly of tubes is important.
The visual inspection of spacer locations in many
such assemblies may not be feasible due to complex constructional configuration of the structural
component. The other conventional technique for
such inspection may be costly and time
consuming.
Hence, a reliable and quicker method is required for detection of the spacer locations. The
change in the dynamic behavior of the tube assembly due to the shift of spacer locations can be
used for such detection. In fact, the use of such a
concept is very common in a similar class of
problems namely identification of crack or damage in structures. Doebling et al. (1998) gave an
excellent review of the studies in the domain of
crack identification. Such identification requires a
mathematical model (e.g. a Finite Element (FE)
model) and experimental modal parameters of the
structure. These studies, in general, use either the
change in mode shapes (Wolff and Richardson,
1989; Pandey et al., 1991; Fox, 1992; Kim et al.,
1992; Mayes, 1992; Srinivasan and Kot, 1992;
Ratcliffe, 1997) and/or the change in natural frequencies (Salawu, 1997) for detection of damage
location and damage size. The methods based on
mode shape changes require measurement of response at many locations over the structure. In
case of the spacer location problem in nuclear
reactor components considered here, such a measurement at a large number of locations for mode
shapes may not be possible. Only techniques
based on the change in the natural frequencies
(Salawu, 1997) can be used for the present problem. One of the methods in this domain is the
solution of the forward problem (Cawley and
Adams, 1979; Penny et al., 1993) which involves
prior computation of a set of natural frequencies
of various modes of vibration for each potential
damage location for crack identification. The solution of inverse problem, i.e. identification of
crack location for a given set of measured natural
frequencies could be then obtained by making
comparison with earlier computed data. However,
this would require a large computational effort.
Other methods are based on a direct solution of
inverse problem. These include non-iterative
methods (Gysin, 1986; Wang and Zhang, 1987;
Richardson and Mannan, 1991, 1992; Narkis,
1994) and iterative methods (Chen and Garba,
1980; Ricles and Kosmatka, 1992; Farhat and
Hemez, 1993) based on FE model updating (Imregun and Visser, 1991; Mottershead and Friswell,
1993; Friswell and Mottershead, 1995). Both forward and inverse methods are sensitive to the
initial baseline model of the structure. In a complex problem, initial modelling assumptions of the
baseline model may lead to erroneous identification. These errors could be minimized to some
extent by fine tuning the parameters of the baseline model itself through model updating methods
using a set of experimental modal data for a
known condition of the structure.
In this paper, an investigation has been made to
determine the spacer positions in an assembly of
the two coaxial tubes by attacking the inverse
solution directly. The proposed technique also
makes use of the change in the natural frequencies
of the tube assembly. The proposed technique is
based on a parameter identification approach. It
requires the FE model of the assembly of the two
coaxial tubes for the computation of modal data
Fig. 1. Schematic of the assembly of two coaxial tubes.
141
(i.e. natural frequencies and mode shapes) for
given spacer locations in the tube assembly. An
FE model truly representative of the tube assembly with spacer locations in the design condition is
taken as the base line model. For off design
locations, the proposed technique involves updating of position parameters of the spacer in the FE
model through the optimization of an error function which is defined based on the difference
between measured and computed natural frequencies of the tube assembly. A gradient-based
method is used for optimization in the FE model
updating problem. The proposed technique has
been validated through the numerically generated
modal data for the tube assembly. This has also
been tested on a laboratory scale experimental
setup.
2. Overview of the model updating technique
The structural dynamic analysis of complex
structural configurations encountered in engineering practice is now-a-days usually carried out
using the FE method. A great deal of effort has to
be invested in making the FE model of the structure. The FE models require the material properties and physical dimensions of the structural
systems under consideration. Usually a number of
simplifying assumptions may have to be made
while constructing the FE models in practice.
Often such an FE model may not be fully reliable
because of the various idealizations made that
generally depend upon the understanding and the
engineering judgement of individuals involved in
the modelling. Thus, any parametric and or modelling errors/deviations may lead to a model
which may not be the true reflection of the ‘as
built’ structure, unless some level of validation is
carried out by making use of the measured modal
data from experimental tests. The parameters of
the FE model could be adjusted/updated to produce matching test behavior. This is known as
model updating.
Various methods have been developed for such
model updating requirements and Friswell and
Mottershead (1995), Imregun and Visser (1991)
and Mottershead and Friswell (1993) give excel-
lent reviews of these methods. The model updating could be done either by direct methods or by
sensitivity methods. The direct methods produce
exact results matching the experimental modal
data. However, the resulting updated FE model
may not provide any physical meaning or correlation. Hence, these have not been generally used in
practice. However, the sensitivity methods overcome the limitations of the direct methods but
require iterative solution. One such method
known as the Penalty Function method has been
used for the present purpose of spacer location
detection. The mathematical formulation using
this method is given later in Section 4.
In these methods the most important aspect is
to define an error function between the computed
and the test data. The error could be defined in
the modal domain in terms of the difference in
measured and computed natural frequencies or
both natural frequencies and mode shapes. Such
an error function is usually a highly non-linear
function with respect to the updating parameters.
The solution for the updating parameters is generally obtained by minimization of the error function through optimization technique. The iterative
solution of the non-linear optimization problem
by a gradient search technique requires the formulation and computation of the sensitivity matrix
of the error function with respect to the updating
parameters. Usually a local linearization is carried
out using a Taylor series expansion by retaining
the first order terms to compute the first order
sensitivity matrix. In defining the error function as
well as in the construction of the sensitivity matrix, the correct pairing up of computed modal
data (either natural frequencies or both natural
frequencies and mode shapes) with the experimental modal data is essential. This is important
because the pairing-up of computed and test data,
based on the sequential order of mode numbers,
may not be always correct. This correlation between the computed and the test data is generally
established using the Modal Assurance Criterion
(MAC; Allemang and Brown, 1982). Another important task in model updating is the selection of
the parameters to be updated. The parameters
should be chosen with the aim of correcting the
recognized uncertainty in the model. Moreover
142
Fig. 2. FE model of the assembly of two coaxial tubes.
the computed eigenvalues, eigenvectors, etc., of
the FE model should be sensitive to the updating
parameters. A number of such sensitivity methods
have been discussed by Friswell and Mottershead
(1995).
3. FE model of the assembly of the two coaxial
tubes
For the purpose of the detection of spacer
locations in the tube assembly as shown in Fig. 1
by the proposed method, the dimensions (Length
(L), Diameter (d), and Thickness (t)) for the
Tube-A and Tube-B were assumed to be (6350,
100, and 5 mm) and (6350, 150, and 2 mm),
respectively. Material properties-modulus of elasticity (E) and density (r) for both the tubes were
chosen to be 0.21 × 1011 kg/m2 and 8000 kg/m3,
respectively. It was also assumed that two spacers
were used to maintain the annular gap between
the tubes and the boundary conditions assumed
to be fixed-fixed end conditions for both the
tubes. The mathematical model of the assembly
using the FE method has been constructed. The
FE model is shown in Fig. 2. The two-noded
simple Euler beam element has been used to
model the Tube-A and the Tube-B. A massless
spring element of stiffness 109 N/m has been used
to simulate the link between the Tube-A and
Tube-B as realized by the spacers. Each node of
the beam has been restricted to have two degrees
of freedom (DOF) — namely the bending displacement and bending rotation. Both Tube-A and
Tube-B are descritized into 127 beam elements. A
total of 254 beam elements (504 DOF—252 bending displacement and 252 bending rotation DOF)
have been used. The resulting size of the beam
elements used in the FE model for the Tube-A
and B was 50 mm so that the spacers could be
detected at the step of 50 mm. The positions of
spacer-1 and -2 are at X1 and X2 from the left end
(node 1) of the tubes assembly as shown in Fig. 2.
The spacers provide a link between the Tube-A
and Tube-B at their locations. Hence one end of
the spring element is connected to the node on the
Tube-A and another to the corresponding adjacent node on the Tube-B. It is also a well-known
fact that the spacers, in practice, shift from their
locations. To simulate this condition in the FE
model, the Tubes-A and -B are descritized such
that the nodes on both the tubes are at same
distance from the left end of the tubes assembly so
that spacers can be moved from one location to
another parallely. The constructed FE model
should be a true reflection of the tubes assembly
at the time of commissioning or at the time when
complete details of structure including the spacer
locations and corresponding dynamic characteristics are available. Such a model could be achieved
by updating the original baseline FE model by
using test modal data for known spacer locations,
through one of the sensitivity techniques itself.
Such an FE model has to be retained as a reference model so that any deviation in terms of
dynamic parameters in future could be used for
detecting spacer locations.
4. Mathematical formulation
The technique proposed in this paper is based
on the ‘Penalty Function method’ (Friswell and
Mottershead, 1995). The complete formulation
used for the specific application of spacer location
is brought out in this section. It has also been
assumed that the available vibration measurement
locations for the case of complex tube assembly
configuration are few and the measurement of
only natural frequencies is possible. Hence, the
definition of error function for this application
involves natural frequencies only.
143
The error function at the kth iteration of the
optimization is defined as (Friswell and Mottershead, 1995);
ok ={dZ}k −[S]k {du}k
(1)
where:
{u}Tk = [X1, X2]k is the position vector of the
locations of spacer-1 and -2, respectively at kth
iteration.
{Zc}Tk =[lc1, lc2, …, lcm ]k is the vector of the
eigenvalues (natural frequencies) of the 1st to
mth mode computed from FE model at the kth
iteration for location [X1, X2]k..
{Ze}T =[le1, le2,......, lem ] is the vector of the
eigenvalues of the 1st to mth mode measured
experimentally for some spacer locations which
are to be determined.
{du}k ={u}k + 1 −{u}k is the perturbation in
the updating parameters (spring locations).
{dZ}k = {Ze} −{Zc}k is the eigenvalue error
(natural frequency error).
[S]k =[#Zc/#u]k is the sensitivity matrix which
is the first derivative of eigenvalues with respect
to the spacer location.
=
({Zc} ({Zc}
(X1 (X2
n
k
Æ
Ç
à (lc1 (lc1 Ã
à (X1 (X2 Ã
à (lc2 (lc2 Ã
Ã
Ã
à (X1 (X2 Ã
· Ã
=Ã ·
à ·
· Ã
Ã
Ã
· Ã
à ·
Ã(lcm (lcm Ã
à (X
(X2 Ãk
È 1
É
(2)
{u}k + 1 = {u}k + [S]Tk [Wo ] [S]k− 1 [Sk ]T [Wo ]
×{{Ze}− {Zc}k }
(4)
The iteration process will continue till the problem converges to the required target, that is,
computed natural frequency data come close to
the experimentally measured natural frequency
data to the required accuracy.
4.1. Computational implementation
To implement the above formulation for the
detection of spacer locations, the computed modal
data using the above FE model for the known
spacer locations and the construction of the sensitivity matrix is required along with the experimental modal data for some unknown spacer
locations which are to be found.
4.1.1. Estimation of sensiti6ity matrix
The construction of the sensitivity matrix, [S],
requires the derivative of the eigenvalues with
respect to updating parameter (i.e. #l/#u). This
can be obtained by the differentiation of the
following characteristic structural dynamic
equation:
[K− lM] {F}= 0
The penalty function (J) at kth iteration is
formed as;
Jk (du)=o Tk Wo ok
deviation) of the corresponding measurements
(Friswell and Mottershead, 1995).
The vector of spacer locations, {u}, can be
obtained by minimizing ‘J’ with respect to ‘du‘
which involves the differentiation of ‘J’ with respect to each element of ‘du‘ and setting the result
equal to zero. The solution so obtained at each
step is, in fact, a weighted least square solution.
Finally this leads to the following equation for the
vector of spacer locations after each iteration.
(3)
where ‘Wo ’ is the positive diagonal weighting matrix which reflects the confidence level in the measurements. It is generally taken as the reciprocals
of the variance (i.e. the squares of the standard
(5)
where [K] and [M] are the Stiffness and Mass
matrices and {F} is the normalized eigenvector of
the structural system and l is the eigenvalue.
Mathematically, each element of the sensitivity
matrix can be written as (Friswell and Mottershead, 1995):
Si, j =
n
(lci
(K
(M
= {Fi }T
− lci
{Fi }
(uj
(uj
(uj
(6)
144
where lci and {FI} are the ith eigenvalue and
eigenvector of the structural system and uj is the
jth updating parameter (spacer location).
In general, when the stiffness and mass matrices
of the structural system are continuous functions
of the updating parameters, Eq. (6) can be directly used to construct the sensitivity matrix.
This is so for example when the parameters are
physical, material properties and boundary stiffnesses of the structural system. In such cases, the
updating parameters are generally value dependent and their position in the system stiffness and
mass matrices during and after model updating
does not change. Most updating problem studied
earlier in literature are of this type, for example,
the various works reported earlier (Friswell and
Mottershead, 1995).
However, the present thrust of study is different. In the present problem the value of the spring
representing the spacer in the FE model has an
appropriately high fixed value. The shift in spacer
location is reflected as a change in the nodes to
which the springs are attached. This is further
reflected in the FE model in the form of discrete
(step) jumps in the value of those elements of the
stiffness matrix, which correspond to the direct
stiffness at the nodes to which the springs are
attached. In short, this is equivalent to a change
in the model itself, for each pair of spacer locations. Thus it is not possible to compute the
sensitivity matrix directly using Eq. (6) in the
present case.
Thus, an alternate method based on the basic
definition of derivative has been used to derive the
eigenvalue derivative for the present problem,
which is:
−lci,k − 1
(lci
l’
)k = ci,k − 1
uj,k −uj,k − 1
(uj
(Si, j )k = (
(7)
where, uj,k is the jth updating parameter (or
spacer location) at kth iteration; lci,k − 1 is the ith
eigenvalue at (k −1)th iteration; and l%ci,k is the
ith eigenvalue at kth iteration for ‘uj,k ’, such that
the other updating parameters (spacer location)
remaining same as at (k −1)th iteration. This is
nothing but the eigenvalue for incremental change
in the jth updating parameter.
A similar concept has been used earlier for
eigenvector derivatives for continuous updating
parameter problem by Suther et al. (1988).
4.1.2. Iterati6e process
The iterative process starts with an initial guess
for the spacer locations. Using Eq. (7) for the
eigenvalues derivative the sensitivity matrix is
constructed as in Eq. (2) and the iteration is
carried out as per Eq. (4). The computed locations
at each iteration by Eq. (4) may not coincide with
the nodes of FE model. One way to proceed
further is that the FE model of the tubes assembly
be re-descritized in the vicinity of the updated/
computed location of the spacers at every iterations. But this practice would definitely be time
consuming and requires more computational effort without gaining much benefit. Alternatively,
one can forcibly shift the updated locations of the
spacer to the nearest nodes at the end of each
iteration. The latter scheme was adopted for the
examples being illustrated subsequently. The iteration process will continue till the problem converges to the required target, that is, computed
natural frequency data come close to the experimentally measured natural frequency data to the
required accuracy.
5. Numerical examples
For the case of the tube assembly, the exercise
for detection of spacer locations has been carried
out using first three natural frequencies only.
More number of natural frequencies could also be
used depending upon the type of structural system
and how many numbers of natural frequency can
actually be measured non-intrusively. For the case
chosen, it has also been observed from the analysis that the value of first few natural frequencies
generally appear in order irrespective of the spacers locations. So, the pairing up of computed and
experimental natural frequencies for defining,
{Ze}− {Zc}k, the error function based on MAC is
not required in the present example.
For illustration of the effectiveness of the proposed approach, some exercises were carried out
to identify the locations of both the spacers in the
145
tube assembly for different sets of target data
starting from the same initial guess. The target
data is nothing but the set of measured natural
frequencies of the tube assembly corresponding to
the unknown spacer locations. The initial guess
data is also a set of natural frequencies corresponding to initially assumed spacer locations in
the tube assembly, which are computed from the
FE model. However, in the absence of actual
measured data, numerically simulated data using
the FE model for known spacer locations is hence
forth treated as target data for validation of the
proposed method. It was observed that the spacer
locations were successfully detected. These results
are listed in Table 1. These are discussed below.
5.1. Initial guess for the iteration
For the illustrative examples, it was assumed to
start with that the spacers-1 and -2 were at X1 =
2450 mm (node 50-178) and X2 =4350 mm (node
88-216), respectively, from the left end of the tube
assembly (Fig. 2 and Table 1).
5.2. Target data
The following three numerically simulated
target data were used. These data are also listed in
Table 1.
Target data-1: 18.703 Hz, 49.609 Hz, 72.284 Hz
for X1 =2950 mm and X2 =3950 mm
Target data-2: 18.725 Hz, 50.755 Hz, 80.128 Hz
for X1 =2350 mm and X2 =3850 mm
Target data-3: 18.685 Hz, 49.141 Hz, 71.073 Hz
for X1 = 2850 mm and X2 = 3350 mm
5.3. Results and discussions
Assuming that the effective travel distance of
the spacers in the tubes assembly is approximately
70% of the length of the Tube-A, that is, approximately 4.50 m. This length consists of around 90
number of nodes at a step of 50 mm for the
Tube-A. Obviously this will results in large number of iterations due to various combination of
two spacers position in the tubes assembly if one
decides to detect the springs by manually shifting
the springs one by one to another pair of nodes of
FE model of the tubes assembly (i.e. by brute
force method) and comparing the computed natural frequencies with the target data. The total
number of iterations required would be around
3060.
Let us consider the example of target data-1
(Table 1). The target locations for spacers-1 and
-2 are only away by ten and eight nodes, respectively, from initial starting guess locations. To
arrive at these locations from the initial locations
by a brute force method, many number of solutions are required. An optimized solution to such
a problem must not only be quicker but also
reliable. This has indeed achieved by using the
model updating technique as outlined in Section
4. In fact, the target locations of spacers-1 and -2
in the tubes assembly for the target data-1 is
detected at only the fourteenth iteration from the
Table 1
Detection of spacer positions in the FE model of the two coaxial tubes assembly
Parameters
Initial guess
Target data-1
Target data-2
Target data-3
Spacer positions
X1, mm
X2, mm
2450 (Node 50-178)a
4350 (Node 88-216)
2950 (Node 60-188)
3950 (Node 80-208)
2350 (Node 48-176)
3850 (Node 78-206)
2850 (Node 58-186)
3350 (Node 68-196)
Frequency
First Hz
Second Hz
Third Hz
18.721
50.919
80.098
No. of iterations to achieve target
a
Tube-A–Tube-B node numbers.
18.703
49.609
72.284
18.725
50.755
80.128
18.685
49.141
71.073
14
06
19
146
This method has further been applied and
tested on a simple laboratory scale experimental
set-up, which is as discussed in the following
section.
6. Testing of the approach on a laboratory setup
Fig. 3. Iteration history for detection of spacer locations in the
FE model of the two coaxial tubes assembly.
initial guess by the proposed technique, as can be
seen in Table 1. The iteration history for detection
of spacer locations in the tube assembly is shown
in Fig. 3. The results of the other two examples,
target data-2 and -3, are also listed in Table 1. In
these examples the spacer locations were detected
at the sixth and nineteenth iteration, respectively,
when the same initial guess was used.
The schematic of the laboratory scale experimental setup is shown in Fig. 4(a). The set-up
consists of two tubes (namely Tube-1 and Tube-2)
made of steel which are inter-connected by a
simple rubber band. The detail of dimensions and
the boundary conditions of both the tubes are
also marked in the figure. The setup shown in Fig.
4(a) is not a coaxial arrangement of two tubes,
however the method suggested for detection of
spacer locations can be tested as well. The same is
presented here.
The experimental modal test was carried out on
the set-up, using the impulse–response method
(Ewins, 1984). It was also assumed that the spring
action of the rubber band was linear for the small
level of excitation given for the modal test. The
Fig. 4. Laboratory experimental setup.
147
Fig. 5. Measured FRFs of the tube-1 of the experimental setup
(Acceleration response at node 82 due to an impulse at node
1).
modal test was conducted by using impulsive
force at one end of the Tube-1 and the response
was measured at the another end, based on the
assumption that only these two locations were
available for measurement. It is because such
limitations could be the case for many more structural systems at site. This modal test was conducted for the two different locations (i.e. 656.5
and 746.5 mm from one end) of the rubber band.
The measured FRFs (inertance) for these two
tests are shown in Fig. 5. The identified natural
frequencies are listed in Table 2 as the test data-1
and -2.
The FE model of the test set-up was also
developed using simple beam elements for the
tubes and the translational spring element for the
rubber band. The FE model is shown in Fig. 4(b).
In fact, the FE model was initially updated using
the test modal data-1 as listed in Table 2 by the
model updating technique itself to obtain a tuned
FE model, truly representative of the setup, so
that the analysis for the detection of spring location could be carried out reliably. The value of the
rubber band stiffness was also obtained by the
model updating. The details are given in Sinha
(1998).
In the initial FE model it was assumed that the
rubber band was at the position of 508.44 mm
from the left end of the Tube-1. This data is listed
in Table 2 as the initial guess. The detection of the
spring location has been carried out using the
approach evolved by considering the position (X1)
of the spring as an updating parameter. As can be
seen from Table 2, the target spring location for
both the cases has been detected at the third and
fourth iteration only. It is once again observed
that much fewer iterations compared to the iterations required by brute force method is sufficient
for position parameter identification of spring location. This is because the number of iterations
required by brute force method to detect the
spring for these cases is to be at least equal to the
difference between the target and starting nodes.
This is estimated to be 7 and 11, respectively
(Table 2), for both cases, which is definitely more
than double the iterations required in the present
method. The position of the spring for both cases
has been detected at very small error of the order
of 0.0366 and 0.672% from their target locations,
respectively. The detection would have been even
better by using finer elements in the FE models. It
is also important to note that when the iterative
processes were deliberately continued even after
achieving the target for both cases, it was observed that the computed location was shuttling
between the target node and the adjacent node
and not diverging. The histories of these iterations
are also listed in Table 3 and graphically shown in
Fig. 6. Hence, the example of this simple setup
also qualifies the evolved technique for detection
of the spring locations as a fast, reliable and
non-intrusive technique.
148
Table 2
Detection of spacer position in the experimental setup
Parameters
Test data-1
(a)
Initial guess
(b)
% Diff.
(a&b)
Updated
data (c)
Spacer position X1,
mm
656.5
508.44
(Node 2588)a
−22.55
656.74
+0.036
(Node 32-95)
Frequency
First Hz
Second Hz
Third Hz
20.938
28.750
39.375
17.757
28.695
37.029
No. of iterations to achieve target
a
Tube-1–Tube-2 node numbers.
−15.19
−0.191
−5.958
20.513
28.658
39.554
3
% Error
(a&c)
−2.030
−0.320
+0.455
Test data-2
(d)
Initial guess
(e)
746.50
508.44
−31.89
(Node 25-88)
21.875
29.375
39.375
–
17.757
28.695
37.029
% Diff.
(d&e)
−18.83
−2.315
−5.958
Updated
data (f)
% Error
(d&f)
741.48
−0.672
(Node 36-99)
21.337
28.385
40.155
4
−2.459
−3.370
+1.981
149
updating technique. The baseline FE model
should be a true reflection of the tube assembly at
the time of commissioning or at the time when
complete details of the dynamic characteristics are
available. The validation and the advantages of
the suggested approach have been presented
through a few numerical examples of different
spacer locations in the tube assembly. The potential of the suggested approach has also been
brought out successfully on a simple laboratory
test setup. Hence, it has been illustrated that the
spacer locations could be detected quickly, reliably
and non-intrusively, that is without disturbing the
normal setup. In the illustrated examples, only one
and two spacers are used. However, the proposed
technique could be extended for more than two
spacers also.
7. Conclusion
Detection of spacer locations in the assembly of
two coaxial tubes used in nuclear reactor is important from safety consideration. The conventional
method for such requirement requires an extended
shut down of the reactor. The proposed technique
making use of an FE model along with the Penalty
Function technique for model updating seems to
be ideally suited for the required objective of the
detection of the spacer locations in the assembly of
two coaxial tubes. The suggested approach uses
the natural frequency characteristics of the tube
assembly for the detection of the spacer locations.
The methodology presented has made use of the
baseline FE model along with the test modal data
of the tube assembly in the gradient-based model
Table 3
Iteration history for detection of spacer position in the experimental setup
Iteration no.
For Test Data-1. Target Location = 656.5 mm. Test
Frequency; 20.938 Hz; 28.750 Hz; 39.375 Hz
For Test Data-2. Target Location =746.5 mm. Test
Frequency; 21.875 Hz; 29.375 Hz; 39.375 Hz
Updated location
Updated frequency
Updated location
Updated frequency
17.757
28.695
37.029
17.246
28.590
36.666
20.230
28.720
39.303
20.513
28.658
39.554
20.230
28.720
39.303
20.513
28.658
39.554
20.230
28.720
39.303
20.513
28.658
39.554
508.444 mm (Node 25-88)
17.757
28.695
37.029
17.246
28.590
36.666
20.764
28.589
39.759
21.174
28.450
40.054
21.337
28.385
40.155
21.472
28.327
40.231
21.337
28.385
40.155
21.472
28.327
40.231
0 (Initial guess) 508.444 mm (Node 25-88)
1
487.259 mm (Node 24-87)
2
635.555 mm (Node 31-94)
3
656.741 mm (Node 32-95)
4
635.555 mm (Node 31-94)
5
656.741 mm (Node 32-95)
6
635.555 mm (Node 31-94)
7
656.741 mm (Node 32-95)
Hz;
Hz;
Hz
Hz;
Hz;
Hz
Hz;
Hz;
Hz
Hz;
Hz;
Hz
Hz;
Hz;
Hz
Hz;
Hz;
Hz
Hz;
Hz;
Hz
Hz;
Hz;
Hz
487.259 mm (Node 24-87)
677.926 mm (Node 33-96)
720.296 mm (Node 35-98)
741.481 mm (Node 36-99)
762.666 mm (Node 37-100)
741.481 mm (Node 36-99)
762.666 mm (Node 37-100)
Hz;
Hz;
Hz
Hz;
Hz;
Hz
Hz;
Hz;
Hz
Hz;
Hz;
Hz
Hz;
Hz;
Hz
Hz;
Hz;
Hz
Hz;
Hz;
Hz
Hz;
Hz;
Hz
150
Fig. 6. Iteration history for detection of spacer location between two tubes in the FE model of the experimental setup.
The discussion and comments made so far are
totally based on the results of numerically simulated examples and a simple laboratory experimental setup. However, the real test for the
suggested approach would be in the implementation of the technique on the real life problems
where the complexities involved could be more.
Some of these complexities are:
1. Difficulties in numerical modelling of the
structure for baseline model.
2. Quality of experimental modal results that depends upon the number of locations available
for external excitation and response measurements, optimization of these measurement locations, etc. It may happen that only few
locations are accessible for vibration measurements due to complex geometry of the
structure.
The above difficulties are more or less common
for all real life problems but approach to overcome these could be different depending upon the
type of problem being tackled. For many real life
problems, the proposed technique based on a
simple optimized gradient-based FE model updating algorithm can be used as such provided the
baseline FE model of the structure is as close as to
in-situ behavior. Such a baseline FE model could
be constructed by more sensitive model updating
methods using experimental inputs. In a complex
practical problem, a further question, which
arises, is about the certainty of correlating the
changes in frequencies to changes in the selected
updating parameters. Additional indicative information may be needed to confirm this correlation.
Further studies on a practical scale are required to
confirm robustness of the presented technique.
References
Allemang, R.J., Brown, D.L., 1982. A Correlation Coefficient
for Modal Vector Analysis. 1st IMAC, Florida, pp. 110–
116.
Chen, J.C., Garba, J.A., 1980. Analytical model improvement
using modal test results. AIAA J. 18 (6), 684 – 690.
Cawley, P., Adams, R.D., 1979. The locations of defects in
structures from measurements of natural frequencies. J.
Strain Anal. 14 (2), 49 – 57.
Doebling, S.W., Farrar, C.R., Prime, M.B., 1998. A summary
review of vibration-based damage identification methods.
Shock Vibration Digest 30 (2), 91 – 105.
Ewins, D.J., 1984. Modal Testing: Theory and Practice. Research Studies Press.
Farhat, C., Hemez, F.M., 1993. Updating of FE dynamic
models using an element by element sensitivity methodology. AIAA J. 31 (9), 1702 – 1711.
151
Fox, C.H.J., 1992. The Location of Defects in Structures: A
Comparison of the Use of Natural frequency and Mode
Shape Data. 10th IMAC, pp. 522–528.
Friswell, M.I., Mottershead, J.E., 1995. Finite Element Model
Updating in Structural Dynamics. Kluwer.
Gysin, H.P., 1986. Critical Application of an Error Matrix
Method for Location of Finite Element Modelling Inaccuracies. 4th IMAC, pp. 1339–1351.
Imregun, M., Visser, W.J., 1991. A review of modal updating
techniques. Shock Vibration Digest 23 (1), 9–20.
Kim, J.H., Jeon H.S., Lee, C.W., 1992. Application of the
Modal Assurance Criteria for Detecting and Locating
Structural Faults. 10th IMAC, pp. 536–540.
Mayes, R., 1992. Error Localization using Mode Shapes — An
Application to a Two Link Robot Arm. 10th IMAC, pp.
886 – 891.
Mottershead, J.E., Friswell, M.I., 1993. Model updating in
structural dynamics: a survey. J. Sound Vibration 167 (2),
347 – 375.
Narkis, Y., 1994. Identification of crack location in vibrating
simply supported beams. J. Sound Vibration 172 (4), 549 –
558.
Pandey, A.K., Biswas, M., Samman, M.M., 1991. Damage
detection from change in curvature mode shapes. J. Sound
Vibration 145 (2), 321–332.
Penny, J.E.T., Wilson, D.A.L., Friswell, M.I., 1993. Damage
Location in Structures Using Vibration Data. 11th IMAC,
pp. 861 – 867.
Ratcliffe, C.P., 1997. Damage detection using a modified
laplacian operator on mode shape data. J. Sound Vibration
204 (3), 505 – 517.
Richardson, M., Mannan, M.A., 1991. Determination of
Modal Sensitivity Function for Location of Structural
Faults. 9th IMAC, pp. 670 – 676.
Richardson, M.H., Mannan, M.A., 1992. Remote Detection
and Location of Structural Faults using Modal Parameters. 10th IMAC, pp. 502 – 507.
Ricles, J.M., Kosmatka, J.B., 1992. Damage detection in
elastic structures using vibratory residual forces and
weighted sensitivity. AIAA J. 30, 2310 – 2316.
Salawu, O.S., 1997. Detection of structural damage through
changes in frequencies: a review. Eng. Structures 19 (9),
718 – 723.
Sinha, J.K., 1998. Model Updating in Structural Dynamics
Through Gradient Search Techniques, M. Tech. Dissertation, Department of Aerospace Engineering, Indian Institute of Technology Bombay, India.
Srinivasan, M.G., Kot, C.A., 1992. Effects of Damage on the
Modal Parameters of a Cylindrical Shell. 10th IMAC, pp.
529 – 535.
Suther, T.R., Camarda, C.J., Walsh, J.L., Adelmans, H.M.,
1988. Comparison of several methods for calculating vibration mode shape derivatives. AIAA J. 26 (12), 1506 –1511.
Wang, W., Zhang, A., 1987. Sensitivity Analysis in Fault
Vibration Diagnosis of Structures. 5th IMAC, pp.496–501.
Wolff, T., Richardson, M., 1989. Fault Detection in Structures
from Changes in their Modal Parameters. 7th IMAC, pp.
87 – 94.