Model Updating: A Tool for Reliable Modelling, Design Modification and Diagnosis

Transcription

1 Articles Model Updating: A ool for Reliable Modelling, Design Modification and Diagnosis Jyoti K. Sinha * and Michael I. Friswell # ABSRAC Finite element model updating has become a mature area of research, that is a useful tool in the modelling of structures and is also able to diagnose faults. However, applying model updating techniques is not easy, and requires a good physical understanding of the structure. his physical understanding is required to enable parameters to be chosen that correct the underlying modelling errors, and also so that the response is sensitive to these parameters. In this way the number of parameters may be reduced, thereby significantly improving the conditioning of the estimation problem. his paper concentrates on the eigensensitivity approach to model updating, and demonstrates the use of this approach in modelling and also for health monitoring applications. he model of the GAREUR structure is updated, and the physical meaning of the updated model is tested by using design modifications. Fault detection and location often requires a large number of uncertain parameters since the location of the fault is unnown, resulting in ill-conditioned estimation equations that usually require regularization. Here fault location is used as a parameter and this is demonstrated by locating spacers and cracs in structures. Introduction Structural design and analysis generally requires a mathematical model representing the physical behaviour of the structure so that such a model may be used to predict responses for service loads, assess structural integrity and also to study the impact of suitable design modifications or system fault diagnosis. he Finite Element (FE) method (Zieniewicz and aylor, 1994, Coo et al., 1989) is the most appropriate tool for such numerical modeling in structural engineering today. his method can handle complex structural geometry, large complex assemblies of structural components and is also able to perform different types of analysis. Even with the great advances in the field of structural modeling, an initial FE model is often a poor reflection of the actual structure, particularly in the field of structural dynamics. his accuracy arises because of a number of simplifying assumptions and idealizations have to be made while constructing the FE model that generally depend on engineering udgement. o improve the correlation between the model and experimental data the FE model should be updated. Friswell and Mottershead (1995), Imregun and Visser (1991) and Mottershead and Friswell (1993) have given extensive reviews of the Jyoti K. Sinha and Michael I. Friswell, Department of Mechanical Engineering, University of Wales Swansea, Singleton Par, Swansea SA2 8PP, UK he Shoc and Vibration Digest, Vol. 34, No. 1, January Sage Publications various model updating methods that have been developed. A recent special issue of Mechanical Systems and Signal Processing highlighted recent developments (Mottershead and Friswell, 1998). Model updating may be performed either by direct methods or by sensitivity methods. he direct methods produce exact results, so that the model predictions match the experimental modal data. However, the resulting updated FE model may not provide any physical meaning, and this is the main reason why these methods have not been generally used in practice. he sensitivity methods overcome the limitations of the direct methods but require an iterative solution. In sensitivity methods the most important aspect is to define an error function between the computed and test data. he error could be defined either in the modal domain, involving the difference in natural frequencies alone or both natural frequencies and mode shapes, or in the frequency domain involving frequency response functions (FRFs). his error is generally minimized by the optimization of the error function, which is usually a highly non-linear function with respect to the updating parameters. In defining the error function, as well as in the construction of the sensitivity matrix, the correct pairing of computed modal data (natural frequencies or both natural frequencies and mode shapes) with the experimental modal data is essential. his is important because the pairing of computed and test data based on the sequential order of mode numbers may not always be correct. his correlation between the computed and the test data is generally established using the Modal Assurance Criterion (MAC) (Allemang and Brown, 1982). Another important tas in model updating is the selection of the parameters to be updated. he parameters should be chosen with the aim of correcting the recognized uncertainty in the model. Moreover the computed eigenvalues, eigenvectors and response of the FE model should be sensitive to the updating parameters. A number of such sensitivity methods were discussed by Friswell and Mottershead (1995). Ahmadian et al. (1998) and Ziaei-Rad and Imregum (1999) gave reviews of different regularization methods used in model updating. he obective of this paper is to illustrate the usefulness of model updating as a tool for reliable modelling, design optimization, structural modification and non-destructive diagnosis through examples. Of course such an illustration cannot be comprehensive, and here we concentrate on the vital role that parameterization plays in the estimation of physically meaningful updated models. he eigensensitivity approach, based on natural frequencies, is used in the

2 28 he Shoc and Vibration Digest / January 2002 examples, although similar conclusions would be reached if mode shapes or frequency response functions were used. he Eigensensitivity Method A number of sensitivity methods have been used in model updating by many authors (Friswell and Mottershead, 1995), and is sufficiently mature to be implemented in several commercial codes. he obective function to be minimized in the following examples is the weighted sum of squares of the error in the eigenvalues, given by m 2 i1 J() θ Wε i ( λ ei λ ci ()) θ ( z e z c( θ)) Wε ( z e z c( θ)) (1) where λ ei and λ ci are the ith experimental and computed eigenvalues (natural frequency squared), the W εi are weighting factors that reflect the confidence level in the measurements, and m is the number of measured natural frequencies used for updating. hese eigenvalues and weighting factors are assembled into vectors, z [ λ, λ,..., λ ], z [ λ, λ,..., λ ] e e1 e2 em c c1 c2 m W ε ε ε ε diag( W 1, W 2,..., W m ). (2) Often only the measured eigenvalues are used since the measured mode shapes usually contain more noise than the natural frequencies, and the mode shapes are not very sensitive to parameter changes. However, the mode shapes are vital to ensure that the correct modes are paired. θ { θ} 1,{ θ} 2,...,{ θ} p is the vector of the p uncertain parameters to be estimated. he weighting matrix W ε must be positive definite, and is often diagonal with the reciprocals of variance of the corresponding measurements along the diagonal. he optimization given by equation (1) is non-linear and requires an iterative solution, usually by a gradient search technique. his 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 aylor series expansion of the computed eigenvalues as a function of the parameters, by retaining the first order terms. At the th iteration, δz where δz is the output error, S δθ, (3) δz z e z c,, (4) δθ is the perturbation in the updating parameters, δθ θ 1 θ (5) and θ is the current estimate of the parameters (that is at the th iteration). Note that the ouput vector in equation (4) is at the current parameter estimate, z z ( θ ). (6) c, c S is the sensitivity matrix, which is the first derivative of the eigenvalues with respect to the updating parameters, given by S z c θ λ ci or [ S ] i θ θ {} θ θ θ. (7) he eigenvalue derivatives with respect to the updating parameters can be obtained by the differentiation of the characteristic structural dynamic equation KλMφ 0 (8) where K and M are the stiffness and mass matrices and λ and φ are the eigenvalues and the normalized eigenvectors of the structural system. he elements of the sensitivity matrix can be obtained by differentiation of equation (8) with respect to each updating parameter and is (Fox and Kapoor, 1968) [ ] S i λ i K M φ i λ i φ {} θ {} θ {} θ where λ i and φ i are the ith eigenvalue and mass normalized eigenvector of the structure. If the errors in the eigenvectors are included in the penalty function then these derivatives may also be calculated (Nelson, 1976). Care should be exercised with repeated or very close eigenvalues (Friswell, 1996). he penalty function at the th iteration is approximated as (Friswell and Mottershead, 1995) J( δθ ) ( δ δθ ) ε( δ δθ ) i (9) z S W z S. (10) he solution of equation (10) is obtained by minimizing J with respect to δθ, which involves the differentiation of J with respect to each element of δθ and setting the result equal to zero. Finally this leads to the following equation for the estimation of updating parameters after each iteration: θ 1 θ S WεS S Wε( z e z, ). (11) Should the solution given by equation (11) be ill-conditioned, then various forms of regularisation may be applied (Friswell and Mottershead, 1995, Ahmadian et al., 1998, Friswell et al., 2001). he most popular approach is to weight deviations of the parameters from the initial model, on the basis that the initial model should be a reasonable approximation. In this paper physical insight is used to reduce the number of parameters so that equation (11) is always over-determined and well-conditioned. he GAREUR Structure he following example has been chosen to highlight the need for physical insight during the model updating process and to emphasize that this insight is vital to produce a better model of a structure, that may be used to test design modifications. he Structure and Materials Action Group (SM-AG19) of the Group for Aeronautical Research and 1 c

3 Sinha and Friswell / MODEL UPDAING 29 Figure 2. he FE Model of GAREUR Structure (Common FE Nodes and Measurement Locations Indicated, Rigid Lin Between Nodes 45 and 60) Figure 1. he GAREUR Structure echnology in EURope (GAREUR) was initiated in 1995 with the purpose of comparing a number of current measurement and identification techniques applied to a common structure (Degener and Hermes, 1996; Degener, 1997; Balmes, 1997, 1998). he testbed, shown in Figure 1, was designed and manufactured by ONERA and has now been accepted as a benchmar structure by the woring group on Finite Element (FE) Model Updating of COS Action F3 on Structural Dynamics (Lin and Graetsch, 1999; Keye, 1999). he testbed represents a typical aircraft design with fuselage, wings and tail, and is 1.5 m long with a 2 m wingspan. Realistic damping levels are achieved by the application of a viscoelastic tape bonded to the upper surface of the wings and covered by a thin aluminum constraining layer. An FE model of the GAREUR structure was developed using beam elements, lumped masses and rigid offsets, using the MALAB based Structural Dynamics oolbox (Balmes, 2000). Euler Bernoulli beam elements were used for modelling the fuselage, wings and tail. Rigid lins were used between the wing and fuselage, wing and drum and lumped masses for wing tip masses. Figure 2 shows the FE model. A total of 80 beam elements were used to give a total of 486 degrees of freedom. Modal tests (Ewins, 1984) were also conducted on the structure. An accelerometer was fixed at location 112 in the z-direction to measure the response and a roving force from an instrumented hammer was used to excite the structure. able 1 compares the measured natural frequencies with those computed from the initial FE model. It is clear from able 1 that the computed natural frequencies deviate significantly from the measured ones even for such a relatively simple structure. For a complex structure this deviation can be much greater. he mechanical oints in the GAREUR structure are potential sources of error in the modeling and these errors should be corrected by model updating. An equivalent beam model with offsets (Ahmadian et al., 1996, Mottershead et al., 2000) is used to represent the oint between the fuselage wings, fuselage vertical tail and the horizontal tail-vertical tail, as shown in Figure 3. he damping layer on the wings also imparts an added stiffness to wings that is difficult to quantify. By consideration of the experimental Figure 3. he Updating Parameters mode shapes and the eigensensitivities, the following six parameters were chosen for updating: 1. the flexural rigidity (EI z ) w of the wings, 2. the torsional rigidity (GI t ) w of the wings, 3. the beam bending offset, (a z ) w, which has the effect of stiffening the wings at the oint with the fuselage, 4. the torsional rigidity of the vertical tail, (GI t ) vt, 5. the beam bending offset (a b ) vt, which has the effect of stiffening the fuselage vertical tail oint, and 6. the beam bending offset (a b ) ht vt, which has the effect of stiffening the horizontal vertical tail oint. hese parameters are similar to an earlier study by Mares et al. (2000). Using the above six updating parameters, model updating was carried out. he first 11 modes of the structure correlated well with the predictions from the initial FE model and these were used to define the penalty function. Figure 4 shows the convergence history of the six updating parameters and able 2 gives their initial and final values. he bending off-set, (a z ) w, for the wings at the fuselage oint remains at zero during model updating. his is liely to be because the effect of this parameter is similar to that of the wing flexural rigidity, (EI z ) w. he computed natural frequencies from the updated (final) FE model are also given in able 1 for comparison with the experimentally identified natural frequencies. he maximum initial error of 11.3% in the frequencies is reduced to the order of 2.1%. he predictions from the updated model of the GAREUR structure are close to the measured data. he quality of the model may be tested by comparing natural frequencies that were not used in the updating. However, the updated models are often used for the prediction of faults in structures or for design modifications, and comparing these higher

4 30 he Shoc and Vibration Digest / January 2002 able 1. Measured and Computed Natural Frequencies (Hz) of the GAREUR Structure Mode No Data Initial FE model Final FE model Error Initial (%) Error Final (%) Description Wing two node bending Global fuselage rotation First anti-symmetric wing torsion First symmetric wing torsion Wing three node bending Wing four node bending Second fuselage rotation mode Symmetric in plane bending of wing Wing five node bending ail torsion Fuselage two node bending able 2. he Changes in the Updating Parameters Parameters (EI z ) w (GI t ) w (a z ) w (GI t ) vt (a b ) ht vt (a b ) vt Initial FE model Final FE model mm mm are listed in able 3. he natural frequencies predicted by the initial FE model corresponding to measured ones are also listed in able 3 for comparison. he predicted natural frequencies from the initial model deviate significantly from the measured frequencies and the mode order is not correct. his brings out the lac of reliability of the initial FE model even for a relative simple structure such as the GAREUR testbed. However, the prediction by the final FE model is very encouraging. he error in the predicted natural frequencies from the final FE model are within an error of approximately 4% and show good mode shape correlation. he asymmetry introduced into the GAREUR structure eliminates all uncertainty about the updated model, and separates the close modes. Model Updating as a Diagnostic ool Figure 4. he Convergence of the Updating Parameters modal frequencies does not guarantee the physical meaning of the model. For the GAREUR structure the predictive capacity of the updated model was tested using two structural modifications. he first modification consisted of replacing the cylinder of mass 150 g at measurement location 112 on the wing with one having a mass of 725 g. he second modification was to add a cylinder of mass 925 g to location 301 on the outside edge of the tail. able 3 lists the experimentally identified modes for both modifications. hese modifications were designed such that the mode ordering changes considerably, and the close modes at approximately 35 Hz separate. A lumped mass is added to the updated model to model each of the modifications of the GAREUR structure. he computed natural frequencies for both sets of modifications Non-intrusive and non-destructive damage or crac detection for mechanical structures using experimentally measured vibration data has been an area of active research for decades. he basic principle of most of the research studies is to use test modal data of structures before any crac has formed as base-line data, and all subsequent tests are compared to it. Deviation in the test results from the base-line data is then used to estimate crac size and location (Cawley and Adams, 1979). Doebling et al., (1998) and Salawu (1997) gave reviews of the research on crac and damage detection and location in structures. he estimation of crac size and location generally requires a mathematical model (typically an FE model) along with experimental modal data from the structure. hese estimation methods are predominately based on the change in natural frequencies, in mode shapes or in dynamically measured flexibility, and typically use a sensitivity based approach. Another class of crac detection methods has evolved from the direct model updating approach and modify the structural model

5 able 3. Measured and Computed Natural Frequencies (Hz) of the Modified GAREUR Structure Sinha and Friswell / MODEL UPDAING 31 Wing Modification ail Modification Prediction Prediction Final mental Initial Final Initial Error Error Experi- Error Error Mode Data FE Initial Final Data Initial Final FE Model FE Model FE Model Model (%) (%) (%) (%) g mass at location 112 replaced by 725 g 925 g mass added at location 301 matrices (mass and stiffness matrices) (Doebling et al., 1998; Kaou and Zimmerman, 1994). Many research studies on crac estimation have been reported that explicitly use model updating methods (for example, Ricles and Kosmata, 1992; Farhat and Hemez, 1993; Hemez and Farhat, 1995). he methods of model updating may also be used as non-destructive and non-intrusive tools for the diagnosis of various faults in structural systems. Once again the correct choice of updating parameters is vital to ensure a meaningful diagnosis. For health monitoring purposes using the element stiffness is quite reasonable, since damage, for example from cracs, is often local. Providing the element size is small compared to the wavelength of the lower mode shapes of interest, then the element stiffness represents an average of the stiffness variation over the element, and accurate modelling of the damage mechanism is not required. However, the elements need to be small enough to enable the accurate localization of the damage. he maor difficulty with this approach is that the location of the damage is not nown a priori, and every element stiffness is a candidate parameter, leading to a large number of parameters. In model updating, the number of parameters may be reduced by only including those parameters that are liely to be in error, as demonstrated earlier. Regularization may be useful, where extra constraints are applied to the parameter estimation problem to ensure a unique solution. Although using parametric models can reduce the number of parameters considerably, for damage location there will still be a large number of parameters. Most regularization techniques rely on minimum norm type solutions that will tend to spread the identified damage over a large number of parameters (Friswell and Penny, 1997). he maor difference between model updating and damage/error location is that in damage location only a limited number of parameters are liely to be in error. If only these parameters are chosen then the updating would be over-determined. Unfortunately we do not now which parameters might be in error and this must be determined. Indeed determining which parameters are in error may be thought of as a form of regularization nown as subset selection (Friswell et al., 1997). In damage location statistical methods and performance measures have been used that wor on a similar principle (Friswell et al., 1994). Only a limited number of sites are assumed to be damaged, and the model updated based on the reduced number of parameters. his process is repeated for all possible combinations of damage site, and possibly even damage mechanisms. he results from all the updated models are compared and the one that best matches the measured data is chosen. he approach advocated in this paper is different. Here the damage or fault is parameterized using the position of the fault as a parameter. hus a small number of parameters may be used, and an over-determined estimation problem generated. Estimation of Support Stiffnesses and Locations Many flexible mechanical systems such as fuel pins, heat exchanger tubes, control rods and various instrumented and shrouded tubes used in nuclear power plants and other engineering industries are beam-lie components with a number of intermediate supports along their length. In many cases these intermediate supports are firmly fixed. However, in some cases they may be loosely coupled and may move from their original locations during operation, for example because of flow induced vibration. he movement of the supports may or may not affect the support stiffnesses depending upon the structural configuration. Undetected, such dislocated supports may deteriorate the system performance and consequently eopardize the safety of the structure

6 32 he Shoc and Vibration Digest / January 2002 Figure 5. Schematic of wo Beams with Interconnected Intermediate Supports Figure 6. Modeling of the th Support Spring or plant. Visual inspection of such support locations in the structural system is not always possible if the structural configuration is complex. A non-intrusive and non-destructive method for the detection of support stiffnesses and their locations in the structural system was developed by Sinha et al. (2000, 2001) and Sinha and Friswell (2001). he identification technique is based on model updating using only natural frequencies in the penalty function. wo beams with a number of massless intermediate spring supports is considered, as shown schematically in Figure 5. It is assumed that the support springs are able to move along the beam length. Euler Bernoulli beam elements were used to model both beams, labelled A and B, and only bending in a single plane is considered. Each node has two degrees of freedom, namely the translational displacement and bending rotation. he supports are modeled as springs (of stiffness spring ), that may be placed within the beam elements of the FE model (see Figure 6). he stiffness matrix for the spring is then obtained from the relative displacement of its ends, which is approximated using the beam shape functions. he stiffness matrix for only one support (i.e. the th support of spring stiffness, spring, ) within the eth element between beams A and B can be written as (Sinha and Friswell, 2001) K S spring, N( ξ ) N( ξ ) N( ξ ) N( ξ ) (12) N( ξ ) N( ξ ) N( ξ ) N( ξ ) where N( ξ) [ N e1( ξ) N e2( ξ) N e3( ξ) N e4( ξ)] and the N ei () ξ are the standard Euler Bernoulli shape functions (Petyt, 1990). he size of the stiffness matrix K S for the th spring is (8 8) corresponding to the degrees of freedom of the eth element of both beams A and B. Similarly, the stiffness matrix K S can be constructed for other supports. hese element matrices for the supports are then added to the global matrices for the beams. A typical experimental example for the detection of both the stiffness and location of a support is presented here. Sinha and Friswell (2001) should be consulted for further examples and detail. Both the positions and the stiffnesses of the supports are chosen as updating parameters. he vector of updating parameters is then defined as θ [ x ], where x x 1, x 2,..., x p and [ spring, 1, spring, 2,..., spring, p ] are the vectors of locations (measured from one end of the beams) and Figure 7. Laboratory Setup stiffnesses of the p supports. he first m eigenvalues (natural frequency squared) are measured and used in the penalty function. An initial estimate of the location and stiffness of the support was made and the iterative process continued until the solution converged. he example was a laboratory scale experiment comprising of two tubes made of steel which are inter-connected by a rubber band (Sinha, 1998). Figure 7(a) shows a schematic of the setup and gives details of the dimensions and the boundary conditions of both of the tubes. A modal test was carried out using impact excitation (Ewins, 1984). It was assumed that the spring action of the rubber band was linear for the small levels of excitation used in the test. Modal tests were conducted for two different locations of the rubber band (656.5 and mm from one end), and the identified natural frequencies are listed in able 4. Figure 7(b) shows the FE model of the setup that was developed using lumped mass Euler Bernoulli beam elements for both of the tubes and a spring element for the rubber band. he location and stiffness of the spring has been carried out using the position (x 1 ) and stiffness ( spring,1 )ofthe spring as the updating parameters. he initial estimate of the spring location was 508 mm and the estimated stiffness was 2 N/m. he first three natural frequencies were used to define the penalty function. able 4 shows the estimated location and stiffness of the support on convergence. he

7 able 4. he Estimation of the Stiffness and Location of a Spring Between wo ubes Sinha and Friswell / MODEL UPDAING 33 Parameters Initial Initial Error Final Error est Data Estimated Value Value (%) (%) Case 1 Support Location (mm) x Support Stiffness (N/m) spring, Natural Frequency (Hz) Case 2 Support Location (mm) x Support Stiffness (N/m) spring, Natural Frequency (Hz) positions of the spring for both cases were identified with errors of 5.11% and 0.39% of their target locations, whereas the estimated spring stiffnesses had errors of 20.01% and 2.65% from the target stiffness of N/m, that was obtained from static tests on the elastic band. One reason for high error in the spring stiffness was that the spring stiffness was relatively small, and the difference in the natural frequency for stiffness values of N/m and N/m was very small. However, it was gratifying that the spring was located accurately, and the estimate of location seemed more robust to errors that the estimate of stiffness. he Estimation of Crac Size and Location in Beam Structures An example of the estimation of the crac size and location in a beam using model updating is now discussed. Emphasis will be placed on producing a model of the damaged structure that is suitable for model updating. he problem of a simple beam with multi-cracs along the beam length is considered, as shown schematically in Figure 8(a). It is assumed that the depth of cracs in the beam is uniform across the width and they do not change the beam mass but reduce the beam stiffness. he Euler Bernoulli beam element was used to model the beam and only bending in a single plane is considered. Each node has two degrees of freedom, namely the translational displacement and bending rotation. he cracs are modelled such that they can be placed within the beam elements of the FE model. his modelling approach is once again such that the system stiffness matrix is a continuous function of the crac depth and its location, similar to the spring support model discussed earlier. If the th crac is located at x, within the eth element of beam, as shown in Figure 8(b), then the stiffness matrix of the eth element of the beam can be written as Ke, cracke Kc (13) where K e is the element stiffness of the eth element with no crac and K c is the effect of the th crac. Obviously some Figure 8. he Craced Beam and its FE Model: (a) Beam with multiple cracs; (b) FE Modelling of th Crac Assuming Linear Variation of Local Flexibility (EI) from Uncrac-Crac-Uncrac Depth along Line of the structure adacent (either side) of the crac will not contribute significantly to the stiffness. Chritides and Barr (1984) derived an expression for the stiffness reduction due to a crac, although this was not local to the crac in that the stiffness reduction extended the whole length of the beam. Sinha, Friswell and Edwards (2002) approximated the variation of local flexibility around the crac by a triangular reduction in flexural rigidity, shown by the line in Figure 8(b). he length of the stiffness reduction, l c,is 1.5 times the beam thicness, d. his model is equivalent to the low frequency dynamics of the craced beam, and approximates the solution of Christides and Barr (1984) near to the crac. he value of flexural rigidity at point 2 in Figure 8(b) is obtained from the crac depth d c as I c = 1/12w(d d c ) 3. Similar element stiffness matrices are constructed for the other cracs, and these are assembled into the global stiffness matrix for the beam. Once again, the crac identification is performed using model updating. he updating parameters are the crac depths and their locations, and iteration continues until convergence. he proposed approach to the estimation of crac depth and location was demonstrated using a cantilever beam of length 1832 mm and cross-section mm. he material density and the Young's modulus of the beam were 2600

8 34 he Shoc and Vibration Digest / January 2002 able 5. he Estimation of the Crac Depth and Location for a Cantilever Beam (Initial Estimates, x 1 = 400 mm and d c1 = 2 mm) Crac depth, d c1, (mm) Natural Frequency (Hz) 4.00 Case 1 Case 2 Case 3 Estimated ( 0.99%) 3.65 ( 8.75%) Estimated (+0.58%) ( 11.71%) Crac Location, x 1, (mm) Estimated (+4.83%) (+29.08%) the detection is small. he crac location estimation is more accurate (error less than 5%) compared to the crac depth (error within 30%). he inaccuracy in the crac depth estimates may be due to the difficulties in measuring the crac depth, or possibly because the natural frequencies are more sensitive to the location. Conclusion Figure 9. Craced Cantilever Beam and its FE Model: (a) Schematic of Setup; (b) FE Model g/m 3 and GN/m 2 respectively. Figure 9(a) shows a schematic of the test setup. Once again, the modal parameters of the beam were obtained by impact tests conducted on the beam (Ewins, 1984). he modal test was conducted on the beam without a crac and with a crac at a single location and different crac depths. he experimentally identified natural frequencies are listed in able 5. An FE model of the beam was developed using Euler Bernoulli beam elements and the clamped end of the beam was simulated using translational and rotational springs, t and θ. he FE model is shown in Figure 9(b). he boundary stiffnesses of the beam were fine tuned so that the computed natural frequencies closely matched the experimental results for the beam without a crac. hese boundary stiffnesses were estimated as t = 26.5 MN/m and θ = 150 Nm/rad. Using this FE model, the detection of crac depth and location was performed. here were two updating parameters, namely the crac location (x 1 ) and depth (d c1 ). he first four measured natural frequencies were considered as the target data for this exercise and the weighting matrix was taen as the inverse of the target (measured) eigenvalues. able 5 gives the results of the crac detection and shows that the crac detection is quite effective and the error in he application of model updating using the eigenvalue sensitivity approach has been demonstrated through examples in design modification and health monitoring. Of critical importance is the choice of updating parameters. Choosing a good set of updating parameters means the updated model has physical meaning and can be used with confidence for design modifications and structural optimization. his was shown dramatically for the GAREUR example, where the updated model was able to predict changes due to significant mass modifications. he use of the model updating method as a non-intrusive and non-destructive diagnostic technique has also been demonstrated by the detection of crac and support locations and their stiffnesses. Here the location of the crac or support was used, and the reduced number of parameters enabled an over-determined set of equations to be generated. he location of the crac or support was estimated more accurately than the stiffness. In practice accurate location is more important, since once a fault is located more labour intensive local methods may be employed to determine the extent of the damage. hese applications have shown that model updating, when applied correctly, can be profitably exploited in modelling, design modification and fault diagnosis. o effectively apply model updating requires a physical understanding of the dynamics of a structure and the liely modelling errors. However, applying model updating techniques also improves this understanding. Such understanding is vital to improve the modelling of a structure in the design process, and leads to more accurate predictions when the structure is modified. It is doubtful whether model updating will ever be completely automated, and indeed such automation is probably not desirable. he understanding and insight gained by the engineer trying to update a model, through many iterations and parameter choices, should not be under-estimated.

9 Sinha and Friswell / MODEL UPDAING 35 Acnowledgements Jyoti K. Sinha acnowledges the support of the Department of Science and echnology of India through the award of a BOYSCAS Fellowship. JKS also acnowledges the parent organization B.A.R.C., India for consistent support and encouragement. Michael Friswell acnowledges the support of the EPSRC through the award of an Advanced Fellowship. he authors acnowledge Dr. Liu Wenie of the Imperial College of Science, echnology and Medicine in London, UK, who provided some of the measured data on the GAREUR structure. Notes * Permanent address: Scientific Officer, Vibration Laboratory, Reactor Engineering Division, Bhabha Atomic Research Centre, Mumbai , India. vilred@magnum.barc.ernet.in. # Corresponding author: m.i.friswell@swansea.ac.u. 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