- Email: [email protected]
RESULTS OBTAINED BY MINIMISING NATURAL FREQUENCY AND MAC-VALUE ERRORS OF A PLATE MODEL
Mechanical Systems and Signal Processing (2003) 17(1), 55–64 doi:10.1006/mssp.2002.1539, available online at http://www.idealibrary.com on
RESULTS OBTAINED BY MINIMISING NATURAL FREQUENCYAND MAC-VALUE ERRORS OF A PLATE MODEL K. Bohle and C.-P. Fritzen Institute of Mechanics and Control, University of Siegen, Paul Bonatz Str. 9–11, Siegen 57 076, Germany. E-mail: [email protected] (Received 28 March 2002, accepted 1 October 2002) A plate model of the Garteur structure has been developed and updated. The cost function for the evaluation of the model’s improvement consisted of residuals between calculated and measured natural frequencies and mode shapes. A brief description of the FE-model is followed by the presentation of the different modules used for the updating procedure, namely a substructure sensitivity analysis for parameter selection as well as an error localisation technique to detect modelling errors. The performance of the model is verified by different measured datasets, including data obtained by a modified structure. # 2003 Published by Elsevier Science Ltd.
1. INITIAL MODEL
The FE-model of the Garteur test structure is made up of 121 four-node shell elements and 18 beam elements, which results in a model with about 1200 dof (Fig. 1). The modelling is done by using the Matlab FE-toolbox Matfem [1]. This offers the possibility to perform all the optimisation and analysis steps described in the following sections in the Matlab environment. The initial model is based on the geometric and material properties described in the technical documentation provided by the DLR [2]. Some simplifications have to be introduced concerning especially the joints between the wing and the fuselage due to the reduction of the model complexity. The connection is modelled by eight massless rigid beam elements and an intermediate steel layer between the wing and the fuselage (Fig. 2). The initial results comparing the model-generated modal-data to the originally measured data are shown in Tables 1 and 2. Modes 1–9 are the ‘active modes’ which are considered in the updating procedure. The ‘passive modes’ have not been used for updating.
2. MODEL ANALYSIS AND PARAMETER SELECTION
2.1. PARAMETRIC UPDATING To perform an updating, a set of design parameters has to be selected, which stay in physically meaningful bounds. In general one can select global, substructural or local material or geometric parameters. We concentrated on material parameters only due to the knowledge of the geometric properties of the structure, although it might be reasonable to define geometric parameters especially for simplified details in the model. The selection of the design parameters has been done with the aid of a sensitivity analysis and an error localisation algorithm which are described in the following sections. As 0888–3270/03/+$35.00/0
# 2003 Published by Elsevier Science Ltd.
56
K. BOHLE AND C.-P. FRITZEN
Figure 1. Garteur FE-model.
Figure 2. Detail of connection wing/fuselage.
Table 1 MAC-values of initial model Mode
Modepair
MAC (%)
MAC (%)
1 2 3 4 5 6 7 8 9
1;1 2;2 3;3 4;4 5;5 6;6 7;7 8;8 9;9
99.8 98.4 77.7 84.1 94.2 96.1 93.6 96.4 75.5
90.7
10 11 12 13 14
10;10 11;11 12;12 13;13 14;14
96.3 98.5 93.2 86.4 73.4
89.6
global parameter, the density of aluminium has been chosen, substructural parameters are the Young’s modulus and shear modulus of the wing, the fuselage and the vertical part of the tail and local parameters are the added masses on the wing tips and the Young’s modulus as well as the shear modulus of the steel plate connecting the wing to the fuselage.
57
PLATE MODEL
Table 2 Frequency deviation of initial model Mode
fmeas (Hz)
fmod (Hz)
Rel. diff. (%)
Rel. diff. (%)
1 2 3 4 5 6 7 8 9
6.4 16.1 33.1 33.5 35.6 48.4 49.4 55.1 63.0
5.6 16.6 34.3 34.3 38.6 44.3 52.6 58.9 65.6
12.64 2.94 3.29 2.32 7.54 9.18 6.10 6.45 3.92
6.04
10 11 12 13 14
66.5 102.9 130.5 141.4 151.3
73.5 104.9 127.7 131.1 146.7
9.55 1.87 2.20 7.87 3.05
4.91
Figure 3. Substructure sensitivities for mode 1.
Some other parameters have been tried but the mentioned ones turned out to be the most effective, so we focus on those parameters. 2.2. SENSITIVITY STUDY An eigenvalue/eigenvector-sensitivity study with respect to stiffness changes is performed with the goal to isolate parameters or sets of parameters on which certain modes are sensitive only. Those parameters can be updated with these sensitive modes while neglecting the others which are not affected. For this reason, the structure is divided into substructures. The calculation of the sensitivities is performed as described by Nelson [3] and Fox/Kapoor [4]. In Fig. 3, the eigenvalue sensitivity distribution of the first mode on stiffness changes in the parts of the wing which are near to the connection to the fuselage is shown (substructures 4–7). Figure 4 shows that mode 10 is sensitive only on changes in substructure 10, the vertical part of the tail. 2.3. ERROR LOCALISATION The error localisation algorithm applied to analyse the model errors is based on a sensitivity approach. It is originally used for damage localisation and described in
58
K. BOHLE AND C.-P. FRITZEN
Figure 4. Substructure sensitivities for mode 10.
Figure 5. Results of damage localisation algorithm for modes 1–5 (from left to right).
references [5, 6]. The interpretation of the algorithm’s indications can increase the physical understanding of the structure and can help to detect modelling uncertainties. Some results of the procedure are shown in Fig. 5. The erroneous elements are in dark colour. It turns out that the connection between the wing and the fuselage as well as the connection between the tail and the fuselage contain some errors. Also, the asymmetric behaviour of the real structure’s wings is pointed out, which can be seen especially in modes 3 and 4. This asymmetric behaviour is not expected when studying the technical drawings.
3. UPDATING PROCESS
For updating, a sequential quadratic programming algorithm has been used. The cost function which has to be minimised is defined as JðDpÞ ¼
n X i¼1
2
wMAC;i ½1 MACi ðDpÞ þ
n X i¼1
wfreq;i
fmod;i ðDpÞ fmeas;i 2 : fmeas;i
ð1Þ
Dp denotes the current vector of design parameters, wMAC and wfreq denote weighting factors specified by means of the sensitivity analysis or the damage localisation algorithm. n is the number of active modes, MACi is the criterion, fmod;i the model eigenfrequency, fmeas;i the measured eigenfrequency for the ith mode, respectively.
59
PLATE MODEL
4. RESULTS
4.1. ORIGINAL DATA SUPPLIED BY THE UNIVERSITY OF MANCHESTER AND THE DLR The updating has been performed in several iterations where the sensitive modes have been chosen to correct certain parameters. In the end, an average MAC value of about 96% has been reached for the active modes as well as for the passive modes (Table 3). The mean of frequency deviation (Table 4) is satisfactory for the active modes with less than 1%, the passive modes still have a larger shift with a maximum deviation of 9.6%. An outlier of the active modes is mode 5, which has a MAC value of 91% and a frequency shift of 4.4%. The resulting parameter values are listed in Table 5: Two examples of FRFs are illustrated in Figs 6 and 7. The correspondence of the calculated FRFs to the measured ones is very satisfactory.
Table 3 MAC values of updated model Mode
Modepair
MAC (%)
MAC (%)
1 2 3 4 5 6 7 8 9
1;1 2;2 3;3 4;4 5;5 6;6 7;7 8;8 9;9
99.9 94.5 97.6 98.1 91.4 98.0 97.7 99.5 87.6
96.0
10 11 12 13 14
10;10 11;11 12;12 13;13 14;14
97.4 99.5 98.5 94.6 94.4
96.9
Table 4 Frequency deviations of updated model Mode
fmeas (Hz)
fmod (Hz)
Rel. diff. (%)
Rel. diff (%)
1 2 3 4 5 6 7 8 9
6.4 16.1 33.1 33.5 35.6 48.4 49.4 55.1 63.0
6.4 16.1 33.0 33.6 37.2 48.4 49.4 56.3 62.8
0.1 0.1 0.3 0.2 4.4 0.0 0.0 2.1 0.3
0.7
10 11 12 13 14
66.5 102.9 130.5 141.4 151.3
66.3 98.5 126.3 127.8 145.1
0.3 4.2 3.2 9.6 4.1
4.3
60
K. BOHLE AND C.-P. FRITZEN
Table 5 Resulting parameter values No. 1 2 3 4 5 6 7 8 9 10 11
Section
Wing Tail Fuselage Left wing Right wing Steel plate
Parameter Density aluminium Young’s modulus Shear modulus Young’s modulus Shear modulus Young’s modulus Shear modulus Tip mass Tip mass Young’s modulus Shear modulus
Updated value 2738 kg/m 70.4 GPa 26.3 GPa 62.0 GPa 24.0 GPa 67.4 GPa 25.0 GPa 0.183 kg 0.180 kg 206 GPa 79.1 GPa
3
Figure 6. Comparison of FRF 12z212z (model: dashed).
Figure 7. Comparison of FRF 12z2111z (model: dashed).
Initial value 2800 kg/m3 72 GPa 27 GPa 72 GPa 27 GPa 72 GPa 27 GPa 0.2 kg 0.2 kg 210 GPa 81 GPa
61
PLATE MODEL
4.2. NEW DATA SUPPLIED BY THE IMPERIAL COLLEGE LONDON In this section, the same initial model as before has been used for updating. The frequencies of the new measurements tend to higher values, which might be explained by an ageing of the damping layer applied to the wings. The updating procedure has been performed considering the dataset 1 of the unmodified structure and the resulting model has then been compared to the other dataset as well as to the modified datasets after modelling the added masses without further updating. The MAC values (Table 6) of dataset 1 are comparable to those obtained from the original data, still mode 5 has the lowermost MAC with 91%. The largest frequency deviation (Table 7) is reduced to 2.1%, but the average of the frequency deviations stays at about 1%. The parameters which changed most were the Young’s modulus (72.3 GPa) and the shear modulus (27.4 GPa) of the wing, the Young’s modulus (211.5 GPa) and the shear modulus (27.4 GPa) of the steel plate and especially the two wing tip masses (0.172 kg left and 0.167 kg right wing). This can be seen as physically reasonable due to the different method of data acquisition by means of a hammer test. The original data has been obtained by a shaker excitation. The MAC values between the updated model and the dataset 2 (Table 10) are significantly lower (89%) than with dataset 1 although it is the same model. In dataset 2, there were more measured degrees of freedom, so the decreasing MAC values can indicate that the model is not well suited to explain the overall movement of the structure or that existing errors accumulate and lead to lower values in the end. Another reason for the decreasing MAC values could be that the measurement locations are not exactly matched by model nodes and the differences in the eigenvectors are due to this distance. The modification at the tail causes decreasing MAC values (Table 8) especially at mode 6 where the tail has a large movement. It is affected most with a MAC of 65%. However, the frequency predictions (Table 9) of the model are satisfying with a mean deviation of 1.8% and a maximum outlier of 3.6% for mode 9. A similar conclusion can be drawn for the prediction of the modification at the wing mass. The predicted frequency deviation (Table 13) is in mean 1.2%. Although the mean MAC value (Table 12) is slightly lower compared to the mean MAC of the unmodified dataset 2, the third mode of the modified model has an even higher MAC than the unmodified model. 4.2.1. Dataset 1 (unmodified structure) (Tables 6 and 7)
Table 6 MAC values of updated model Mode 1 2 3 4 5 6 7 8 9 10
Modepair
MAC (%)
MAC (%)
1;1 2;2 3;4 4;3 5;5 6;6 7;7 8;8 9;9 10;10
99.4 95.5 95.8 95.6 91.1 93.3 94.1 98.4 97.6 96.8
95.8
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K. BOHLE AND C.-P. FRITZEN
Table 7 Frequency deviations of updated model Mode
fmeas (Hz)
fmod (Hz)
Rel. diff. (%)
Rel. diff. (%)
6.5 16.6 34.9 35.3 36.5 49.8 50.6 56.4 65.0 69.8
6.5 16.5 35.6 35.2 36.5 49.2 51.5 56.0 65.6 69.5
0.7 0.3 2.1 0.3 0.0 1.3 1.6 0.7 1.0 0.4
0.8
1 2 3 4 5 6 7 8 9 10
4.2.2. Modification of the tail (Tables 8 and 9) Table 8 Predicted MAC values with tail modification Mode 1 2 3 4 5 6 7 8 9 10
Modepair
MAC (%)
MAC (%)
1;1 2;2 3;3 4;4 5;5 6;6 7;7 8;8 9;9 10;10
99.2 92.8 92.6 94.9 93.7 64.8 83.4 96.3 85.2 85.5
88.8
Table 9 Predicted frequency deviations with tail modification Mode 1 2 3 4 5 6 7 8 9 10
fmeas (Hz)
fmod (Hz)
Rel. diff. (%)
Rel. diff. (%)
6.5 13.9 32.4 35.1 35.5 38.1 48.7 50.2 56.5 58.2
6.6 14.3 32.1 35.7 36.1 39.0 49.0 51.0 54.4 59.2
0.6 2.7 0.8 1.7 1.7 2.5 0.7 1.6 3.6 1.7
1.8
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PLATE MODEL
4.2.3. Dataset 2 (unmodified structure) (Tables 10 and 11) Table 10 Predicted MAC values of updated model for second set Mode 1 2 3 4 5 6 7 8 9 10
Modepair
MAC (%)
MAC (%)
1;1 2;2 3;4 4;3 5;5 6;6 7;7 8;8 9;9 10;10
97.9 92.9 82.9 83.1 92.3 93.6 82.1 79.8 87.8 97.0
88.9
Table 11 Predicted frequency deviations of updated model for second dataset Mode 1 2 3 4 5 6 7 8 9 10
fmeas (Hz)
fmod (Hz)
Rel. diff. (%)
Rel. diff. (%)
6.5 16.6 34.9 35.3 36.5 49.8 50.6 56.4 65.0 69.8
6.5 16.5 35.6 35.2 36.5 49.2 51.5 56.0 65.6 69.5
0.7 0.3 2.1 0.3 0.0 1.3 1.6 0.7 1.0 0.4
0.8
4.2.4. Modification of the wing mass (Tables 12 and 13) Table 12 Predicted MAC values with wing mass modification Mode 1 2 3 4 5 6 7 8 9 10
Modepair
MAC (%)
MAC (%)
1;1 2;2 3;3 4;4 5;5 6;6 7;7 8;8 9;9 10;10
96.7 92.3 86.8 83.3 89.8 84.4 79.8 76.1 87.3 95.7
87.2
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K. BOHLE AND C.-P. FRITZEN
Table 13 Predicted frequency deviations with wing mass modification Mode 1 2 3 4 5 6 7 8 9 10
fmeas (Hz)
fmod (Hz)
Rel. diff. (%)
Rel. diff. (%)
6.3 16.4 27.3 35.4 36.5 48.9 50.0 54.3 64.5 69.5
6.2 16.1 27.6 36.0 36.3 48.8 49.3 54.8 65.8 69.2
1.2 1.8 1.0 1.9 0.4 0.3 1.3 0.9 2.0 0.5
1.2
5. CONCLUSIONS
The different modules used for model updating yielded satisfactory results both in the improvement of the initial model and the prediction of modifications in the structure. The error localisation technique worked well to identify erroneous locations and/or to investigate uncertain regions of the structure. A better physical understanding of the crucial parts of the structure and the insight which details may have been simplified too much in modelling are the contributions of this technique. The substructure sensitivity analysis supported the tuning of the weighting factors in the cost function. The most sensitive target values can be privileged to update a certain parameter.
REFERENCES 1. M. Link 1997 Matfem User’s Guide. University of Kassel. 2. M. Degener and M. Hermes 1996 Ground Vibration Test and Finite Element Analysis of the GARTEUR SM-AG19 Testbed. DLR Institut fu. r Aeroelastik. 3. R. B. Nelson 1976, AIAA Journal 14, 1201–1205. Simplified calculation of eigenvector derivatives. 4. R. L. Fox and M. P. Kapoor 1968 AIAA Journal 6, 2426–2429. Rates of change of eigenvalues and eigenvectors. 5. C.-P. Fritzen and K. Bohle 1999 In Structural Health Monitoring 2000, F.-K. Chang (ed.), pp. 901–911, Technomic Publishing Co., Stanford, CA. Parameter selection strategies in model-based damage detection. 6. C.-P. Fritzen and K. Bohle, 1999 Proceedings of Identification in Engineering Systems IES ’99, 492–505. Model-based health monitoring of structures}application to the I40-highway-bridge.
RESULTS OBTAINED BY MINIMISING NATURAL FREQUENCYAND MAC-VALUE ERRORS OF A PLATE MODEL K. Bohle and C.-P. Fritzen Institute of Mechanics and Control, University of Siegen, Paul Bonatz Str. 9–11, Siegen 57 076, Germany. E-mail: [email protected] (Received 28 March 2002, accepted 1 October 2002) A plate model of the Garteur structure has been developed and updated. The cost function for the evaluation of the model’s improvement consisted of residuals between calculated and measured natural frequencies and mode shapes. A brief description of the FE-model is followed by the presentation of the different modules used for the updating procedure, namely a substructure sensitivity analysis for parameter selection as well as an error localisation technique to detect modelling errors. The performance of the model is verified by different measured datasets, including data obtained by a modified structure. # 2003 Published by Elsevier Science Ltd.
1. INITIAL MODEL
The FE-model of the Garteur test structure is made up of 121 four-node shell elements and 18 beam elements, which results in a model with about 1200 dof (Fig. 1). The modelling is done by using the Matlab FE-toolbox Matfem [1]. This offers the possibility to perform all the optimisation and analysis steps described in the following sections in the Matlab environment. The initial model is based on the geometric and material properties described in the technical documentation provided by the DLR [2]. Some simplifications have to be introduced concerning especially the joints between the wing and the fuselage due to the reduction of the model complexity. The connection is modelled by eight massless rigid beam elements and an intermediate steel layer between the wing and the fuselage (Fig. 2). The initial results comparing the model-generated modal-data to the originally measured data are shown in Tables 1 and 2. Modes 1–9 are the ‘active modes’ which are considered in the updating procedure. The ‘passive modes’ have not been used for updating.
2. MODEL ANALYSIS AND PARAMETER SELECTION
2.1. PARAMETRIC UPDATING To perform an updating, a set of design parameters has to be selected, which stay in physically meaningful bounds. In general one can select global, substructural or local material or geometric parameters. We concentrated on material parameters only due to the knowledge of the geometric properties of the structure, although it might be reasonable to define geometric parameters especially for simplified details in the model. The selection of the design parameters has been done with the aid of a sensitivity analysis and an error localisation algorithm which are described in the following sections. As 0888–3270/03/+$35.00/0
# 2003 Published by Elsevier Science Ltd.
56
K. BOHLE AND C.-P. FRITZEN
Figure 1. Garteur FE-model.
Figure 2. Detail of connection wing/fuselage.
Table 1 MAC-values of initial model Mode
Modepair
MAC (%)
MAC (%)
1 2 3 4 5 6 7 8 9
1;1 2;2 3;3 4;4 5;5 6;6 7;7 8;8 9;9
99.8 98.4 77.7 84.1 94.2 96.1 93.6 96.4 75.5
90.7
10 11 12 13 14
10;10 11;11 12;12 13;13 14;14
96.3 98.5 93.2 86.4 73.4
89.6
global parameter, the density of aluminium has been chosen, substructural parameters are the Young’s modulus and shear modulus of the wing, the fuselage and the vertical part of the tail and local parameters are the added masses on the wing tips and the Young’s modulus as well as the shear modulus of the steel plate connecting the wing to the fuselage.
57
PLATE MODEL
Table 2 Frequency deviation of initial model Mode
fmeas (Hz)
fmod (Hz)
Rel. diff. (%)
Rel. diff. (%)
1 2 3 4 5 6 7 8 9
6.4 16.1 33.1 33.5 35.6 48.4 49.4 55.1 63.0
5.6 16.6 34.3 34.3 38.6 44.3 52.6 58.9 65.6
12.64 2.94 3.29 2.32 7.54 9.18 6.10 6.45 3.92
6.04
10 11 12 13 14
66.5 102.9 130.5 141.4 151.3
73.5 104.9 127.7 131.1 146.7
9.55 1.87 2.20 7.87 3.05
4.91
Figure 3. Substructure sensitivities for mode 1.
Some other parameters have been tried but the mentioned ones turned out to be the most effective, so we focus on those parameters. 2.2. SENSITIVITY STUDY An eigenvalue/eigenvector-sensitivity study with respect to stiffness changes is performed with the goal to isolate parameters or sets of parameters on which certain modes are sensitive only. Those parameters can be updated with these sensitive modes while neglecting the others which are not affected. For this reason, the structure is divided into substructures. The calculation of the sensitivities is performed as described by Nelson [3] and Fox/Kapoor [4]. In Fig. 3, the eigenvalue sensitivity distribution of the first mode on stiffness changes in the parts of the wing which are near to the connection to the fuselage is shown (substructures 4–7). Figure 4 shows that mode 10 is sensitive only on changes in substructure 10, the vertical part of the tail. 2.3. ERROR LOCALISATION The error localisation algorithm applied to analyse the model errors is based on a sensitivity approach. It is originally used for damage localisation and described in
58
K. BOHLE AND C.-P. FRITZEN
Figure 4. Substructure sensitivities for mode 10.
Figure 5. Results of damage localisation algorithm for modes 1–5 (from left to right).
references [5, 6]. The interpretation of the algorithm’s indications can increase the physical understanding of the structure and can help to detect modelling uncertainties. Some results of the procedure are shown in Fig. 5. The erroneous elements are in dark colour. It turns out that the connection between the wing and the fuselage as well as the connection between the tail and the fuselage contain some errors. Also, the asymmetric behaviour of the real structure’s wings is pointed out, which can be seen especially in modes 3 and 4. This asymmetric behaviour is not expected when studying the technical drawings.
3. UPDATING PROCESS
For updating, a sequential quadratic programming algorithm has been used. The cost function which has to be minimised is defined as JðDpÞ ¼
n X i¼1
2
wMAC;i ½1 MACi ðDpÞ þ
n X i¼1
wfreq;i
fmod;i ðDpÞ fmeas;i 2 : fmeas;i
ð1Þ
Dp denotes the current vector of design parameters, wMAC and wfreq denote weighting factors specified by means of the sensitivity analysis or the damage localisation algorithm. n is the number of active modes, MACi is the criterion, fmod;i the model eigenfrequency, fmeas;i the measured eigenfrequency for the ith mode, respectively.
59
PLATE MODEL
4. RESULTS
4.1. ORIGINAL DATA SUPPLIED BY THE UNIVERSITY OF MANCHESTER AND THE DLR The updating has been performed in several iterations where the sensitive modes have been chosen to correct certain parameters. In the end, an average MAC value of about 96% has been reached for the active modes as well as for the passive modes (Table 3). The mean of frequency deviation (Table 4) is satisfactory for the active modes with less than 1%, the passive modes still have a larger shift with a maximum deviation of 9.6%. An outlier of the active modes is mode 5, which has a MAC value of 91% and a frequency shift of 4.4%. The resulting parameter values are listed in Table 5: Two examples of FRFs are illustrated in Figs 6 and 7. The correspondence of the calculated FRFs to the measured ones is very satisfactory.
Table 3 MAC values of updated model Mode
Modepair
MAC (%)
MAC (%)
1 2 3 4 5 6 7 8 9
1;1 2;2 3;3 4;4 5;5 6;6 7;7 8;8 9;9
99.9 94.5 97.6 98.1 91.4 98.0 97.7 99.5 87.6
96.0
10 11 12 13 14
10;10 11;11 12;12 13;13 14;14
97.4 99.5 98.5 94.6 94.4
96.9
Table 4 Frequency deviations of updated model Mode
fmeas (Hz)
fmod (Hz)
Rel. diff. (%)
Rel. diff (%)
1 2 3 4 5 6 7 8 9
6.4 16.1 33.1 33.5 35.6 48.4 49.4 55.1 63.0
6.4 16.1 33.0 33.6 37.2 48.4 49.4 56.3 62.8
0.1 0.1 0.3 0.2 4.4 0.0 0.0 2.1 0.3
0.7
10 11 12 13 14
66.5 102.9 130.5 141.4 151.3
66.3 98.5 126.3 127.8 145.1
0.3 4.2 3.2 9.6 4.1
4.3
60
K. BOHLE AND C.-P. FRITZEN
Table 5 Resulting parameter values No. 1 2 3 4 5 6 7 8 9 10 11
Section
Wing Tail Fuselage Left wing Right wing Steel plate
Parameter Density aluminium Young’s modulus Shear modulus Young’s modulus Shear modulus Young’s modulus Shear modulus Tip mass Tip mass Young’s modulus Shear modulus
Updated value 2738 kg/m 70.4 GPa 26.3 GPa 62.0 GPa 24.0 GPa 67.4 GPa 25.0 GPa 0.183 kg 0.180 kg 206 GPa 79.1 GPa
3
Figure 6. Comparison of FRF 12z212z (model: dashed).
Figure 7. Comparison of FRF 12z2111z (model: dashed).
Initial value 2800 kg/m3 72 GPa 27 GPa 72 GPa 27 GPa 72 GPa 27 GPa 0.2 kg 0.2 kg 210 GPa 81 GPa
61
PLATE MODEL
4.2. NEW DATA SUPPLIED BY THE IMPERIAL COLLEGE LONDON In this section, the same initial model as before has been used for updating. The frequencies of the new measurements tend to higher values, which might be explained by an ageing of the damping layer applied to the wings. The updating procedure has been performed considering the dataset 1 of the unmodified structure and the resulting model has then been compared to the other dataset as well as to the modified datasets after modelling the added masses without further updating. The MAC values (Table 6) of dataset 1 are comparable to those obtained from the original data, still mode 5 has the lowermost MAC with 91%. The largest frequency deviation (Table 7) is reduced to 2.1%, but the average of the frequency deviations stays at about 1%. The parameters which changed most were the Young’s modulus (72.3 GPa) and the shear modulus (27.4 GPa) of the wing, the Young’s modulus (211.5 GPa) and the shear modulus (27.4 GPa) of the steel plate and especially the two wing tip masses (0.172 kg left and 0.167 kg right wing). This can be seen as physically reasonable due to the different method of data acquisition by means of a hammer test. The original data has been obtained by a shaker excitation. The MAC values between the updated model and the dataset 2 (Table 10) are significantly lower (89%) than with dataset 1 although it is the same model. In dataset 2, there were more measured degrees of freedom, so the decreasing MAC values can indicate that the model is not well suited to explain the overall movement of the structure or that existing errors accumulate and lead to lower values in the end. Another reason for the decreasing MAC values could be that the measurement locations are not exactly matched by model nodes and the differences in the eigenvectors are due to this distance. The modification at the tail causes decreasing MAC values (Table 8) especially at mode 6 where the tail has a large movement. It is affected most with a MAC of 65%. However, the frequency predictions (Table 9) of the model are satisfying with a mean deviation of 1.8% and a maximum outlier of 3.6% for mode 9. A similar conclusion can be drawn for the prediction of the modification at the wing mass. The predicted frequency deviation (Table 13) is in mean 1.2%. Although the mean MAC value (Table 12) is slightly lower compared to the mean MAC of the unmodified dataset 2, the third mode of the modified model has an even higher MAC than the unmodified model. 4.2.1. Dataset 1 (unmodified structure) (Tables 6 and 7)
Table 6 MAC values of updated model Mode 1 2 3 4 5 6 7 8 9 10
Modepair
MAC (%)
MAC (%)
1;1 2;2 3;4 4;3 5;5 6;6 7;7 8;8 9;9 10;10
99.4 95.5 95.8 95.6 91.1 93.3 94.1 98.4 97.6 96.8
95.8
62
K. BOHLE AND C.-P. FRITZEN
Table 7 Frequency deviations of updated model Mode
fmeas (Hz)
fmod (Hz)
Rel. diff. (%)
Rel. diff. (%)
6.5 16.6 34.9 35.3 36.5 49.8 50.6 56.4 65.0 69.8
6.5 16.5 35.6 35.2 36.5 49.2 51.5 56.0 65.6 69.5
0.7 0.3 2.1 0.3 0.0 1.3 1.6 0.7 1.0 0.4
0.8
1 2 3 4 5 6 7 8 9 10
4.2.2. Modification of the tail (Tables 8 and 9) Table 8 Predicted MAC values with tail modification Mode 1 2 3 4 5 6 7 8 9 10
Modepair
MAC (%)
MAC (%)
1;1 2;2 3;3 4;4 5;5 6;6 7;7 8;8 9;9 10;10
99.2 92.8 92.6 94.9 93.7 64.8 83.4 96.3 85.2 85.5
88.8
Table 9 Predicted frequency deviations with tail modification Mode 1 2 3 4 5 6 7 8 9 10
fmeas (Hz)
fmod (Hz)
Rel. diff. (%)
Rel. diff. (%)
6.5 13.9 32.4 35.1 35.5 38.1 48.7 50.2 56.5 58.2
6.6 14.3 32.1 35.7 36.1 39.0 49.0 51.0 54.4 59.2
0.6 2.7 0.8 1.7 1.7 2.5 0.7 1.6 3.6 1.7
1.8
63
PLATE MODEL
4.2.3. Dataset 2 (unmodified structure) (Tables 10 and 11) Table 10 Predicted MAC values of updated model for second set Mode 1 2 3 4 5 6 7 8 9 10
Modepair
MAC (%)
MAC (%)
1;1 2;2 3;4 4;3 5;5 6;6 7;7 8;8 9;9 10;10
97.9 92.9 82.9 83.1 92.3 93.6 82.1 79.8 87.8 97.0
88.9
Table 11 Predicted frequency deviations of updated model for second dataset Mode 1 2 3 4 5 6 7 8 9 10
fmeas (Hz)
fmod (Hz)
Rel. diff. (%)
Rel. diff. (%)
6.5 16.6 34.9 35.3 36.5 49.8 50.6 56.4 65.0 69.8
6.5 16.5 35.6 35.2 36.5 49.2 51.5 56.0 65.6 69.5
0.7 0.3 2.1 0.3 0.0 1.3 1.6 0.7 1.0 0.4
0.8
4.2.4. Modification of the wing mass (Tables 12 and 13) Table 12 Predicted MAC values with wing mass modification Mode 1 2 3 4 5 6 7 8 9 10
Modepair
MAC (%)
MAC (%)
1;1 2;2 3;3 4;4 5;5 6;6 7;7 8;8 9;9 10;10
96.7 92.3 86.8 83.3 89.8 84.4 79.8 76.1 87.3 95.7
87.2
64
K. BOHLE AND C.-P. FRITZEN
Table 13 Predicted frequency deviations with wing mass modification Mode 1 2 3 4 5 6 7 8 9 10
fmeas (Hz)
fmod (Hz)
Rel. diff. (%)
Rel. diff. (%)
6.3 16.4 27.3 35.4 36.5 48.9 50.0 54.3 64.5 69.5
6.2 16.1 27.6 36.0 36.3 48.8 49.3 54.8 65.8 69.2
1.2 1.8 1.0 1.9 0.4 0.3 1.3 0.9 2.0 0.5
1.2
5. CONCLUSIONS
The different modules used for model updating yielded satisfactory results both in the improvement of the initial model and the prediction of modifications in the structure. The error localisation technique worked well to identify erroneous locations and/or to investigate uncertain regions of the structure. A better physical understanding of the crucial parts of the structure and the insight which details may have been simplified too much in modelling are the contributions of this technique. The substructure sensitivity analysis supported the tuning of the weighting factors in the cost function. The most sensitive target values can be privileged to update a certain parameter.
REFERENCES 1. M. Link 1997 Matfem User’s Guide. University of Kassel. 2. M. Degener and M. Hermes 1996 Ground Vibration Test and Finite Element Analysis of the GARTEUR SM-AG19 Testbed. DLR Institut fu. r Aeroelastik. 3. R. B. Nelson 1976, AIAA Journal 14, 1201–1205. Simplified calculation of eigenvector derivatives. 4. R. L. Fox and M. P. Kapoor 1968 AIAA Journal 6, 2426–2429. Rates of change of eigenvalues and eigenvectors. 5. C.-P. Fritzen and K. Bohle 1999 In Structural Health Monitoring 2000, F.-K. Chang (ed.), pp. 901–911, Technomic Publishing Co., Stanford, CA. Parameter selection strategies in model-based damage detection. 6. C.-P. Fritzen and K. Bohle, 1999 Proceedings of Identification in Engineering Systems IES ’99, 492–505. Model-based health monitoring of structures}application to the I40-highway-bridge.