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 Ã (lc1 (lc1 à à (X1 (X2 à à (lc2 (lc2 à à à à (X1 (X2 à · à =à · à · · à à à · à à · Ã(lcm (lcm à 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. 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