- Email: [email protected]
RESULTS OBTAINED BY MINIMIZING NATURAL FREQUENCY AND MODE SHAPE ERRORS OF A BEAM MODEL
Mechanical Systems and Signal Processing (2003) 17(1), 21–27 doi:10.1006/mssp.2002.1535, available online at http://www.idealibrary.com on
RESULTS OBTAINED BY MINIMIZING NATURAL FREQUENCYAND MODE SHAPE ERRORS OF A BEAM MODEL D. G˛ge Institute of Aeroelasticity, German Aerospace Center (DLR), 37073 Go.ttingen, Germany. E-mail: [email protected]
and Michael Link Lightweight Structures and Structural Mechanics Laboratory, University of Kassel, 34109 Kassel, Germany. E-mail: [email protected]
(Received 28 March 2002, accepted after revisions 1 October 2002) The classical inverse sensitivity approach was applied. In the presented application, the deviations between analytical and measured natural frequencies and mode shapes were minimised to update user selected stiffness, mass and geometric parameters. The selection of the updating parameters was based on a sensitivity analysis. The capability of the updated model to predict modal data of a modified structure is demonstrated in order to check if the physical meaning of the updated parameters has not been altered during the updating process. # 2003 Elsevier Science Ltd. All rights reserved.
1. INTRODUCTION
In practice the degree of correlation between finite element (FE) model results and vibration test data (FRFs and modal data) is commonly used to assess the model quality. If the correlation is found to be unsatisfactory the need arises to modify the model. Modifications may be related not only to the model parameters like material parameters but also to model structure features like mesh density or element types. Computational model updating (CMU) methods have been developed and used successfully in the past with the aim of automatically updating the FE model parameters thereby reducing the discrepancies between the model predictions and the test results, however, with the assumption that the model structure was adequate. For the present application the inverse sensitivity approach was used where the deviations between analytical and measured natural frequencies and mode shapes are minimised [1]. A beam model was generated where ten parameters were used to minimise the deviations of nine measured natural frequencies and mode shapes.
2. MODELLING ASPECTS OF THE ANALYTICAL MODEL
The initial model generated for the GARTEUR SM-AG19 structure consists of 131 beam elements with 408 degrees of freedom, see Fig. 1. Special care was taken for modeling the joints. The joint between fuselage and vertical tail plane was linked by a rigid 0888–3270/03/+$35.00/0
# 2003 Elsevier Science Ltd. All rights reserved.
22
D. GO¨GE AND M. LINK Wing Screws
Beam Elements
X Fuselage
Rigid Elements
Detail Y
Detail X
Z VTP
Y
Beam Element Fuselage Rigid Element X Beam Element
Z X
Y
Figure 1. Finite element model of the GARTEUR SM-AG19 testbed with modelling aspects.
Table 1 Correlation between initial model and test (UWS) data from unmodified structure Mode
fexp (Hz)
fFE (Hz)
Df (%)
MAC (%)
1 2 3 4 5 6 7 8 9
6.55 16.61 34.88 35.36 36.71 50.09 50.72 56.44 65.14
5.98 16.06 33.19 33.25 37.34 48.00 49.85 57.98 64.46
8.70 3.25 4.84 5.97 1.74 4.17 1.72 2.73 1.03
97.98 96.55 69.86 70.10 85.22 97.40 95.61 99.31 92.17
10 11 12 13 14
69.64 105.47 134.68 145.87 155.82
71.03 101.04 128.61 140.41 151.30
2.00 4.20 4.51 3.74 2.90
96.98 98.10 75.80 95.70 95.51
body (offset) and a beam element, see Fig. 1. The joints fuselage/wing and vertical tail plane/horizontal tail plane were idealized by spring elements. Additional offsets for the wings were modeled using rigid elements to account for realistic kinematic constraints, Fig. 1. A detailed mass matrix was established taking into account all connecting bolts and accelerometer masses. The actual accelerometer positions, e.g. at the wing leading and tailing edges, were modeled by additional offset nodes. Table 1 shows the correlation of the initial finite element model and the test data measured by the University of Wales, Swansea (UWS). The frequency deviations for most mode shapes are relatively high. The average frequency deviations for the modes used for updating (so-called active frequencies) is: Df ini;act: ¼
9 X Dfini;i i¼1
9
¼ 3:79%:
ð1Þ
23
FREQUENCY AND MODE SHAPE ERRORS OF A BEAM MODEL
The mean for the dominant terms in the modal assurance criterion (MAC) matrix is: MAC ini;act: ¼
9 X MACini;i i¼1
9
¼ 89:35%:
ð2Þ
The average frequency deviations for the modes not used in the updating procedure (so-called passive modes) amounts to: 14 X Dfini;i ¼ 3:47%: ð3Þ Df ini;pass: ¼ 5 i¼10 The mean for the dominant terms in the MAC matrix for the passive modes is MAC ini;pass: ¼
14 X MACini;i i¼10
5
¼ 92:42%:
ð4Þ
The average frequency deviations and the correlation via MAC is slightly better for the modes not used for updating than for the active modes. The reason for this is the bad correlation of the torsional wing modes 3 and 4, see Table 1. The fundamental modeling uncertainties in the finite element model were assumed to be in the joints, the mass distribution and in the flexural and torsional stiffnesses of the wing. Ten parameters were finally selected for the computational model updating, Table 2. The selection process was primarily but not only based on a sensitivity study. However, engineering judgement was also necessary, in particular, when lumping uncertain local parameters into global updating parameters. Earlier investigations on the GARTEUR SM-AG19 structure showed that the use of geometric parameters gave physically meaningful results [2, 3]. Three geometrical, six stiffness and one mass parameter were selected for updating. The horizontal offset of the wings tends to stiffen the wing/fuselage connection. Four modes in the active range are sensitive to the distance between the wings and the fuselage (vertical offset of wings). This parameter allows to compensate for inaccuracies in the complicated wing/fuselage connection. The fifth mode is most sensitive to the offset between the VTP and the fuselage and to the VTP bending stiffness. The remaining parameters for the wings were selected to compensate for the uncertainties in the thickness of the viscoelastic tape. Only one parameter was used for updating the wing bending stiffnesses about each principal axis because the wing bending modes did not exhibit significant non-symmetries. The non-symmetric behaviour of the torsional modes was more pronounced so that the
Table 2 Selected parameters for CMU and parameter changes after updating No. 1 2 3 4 5 6 7 8 9 10
Location and description
Parameter (p)
p change after CMU (%)
Offset VTP/fuselage Right and left wing bending stiffness Right and left wing bending stiffness Offset wings (vertical) Right wing torsional stiffness Left wing torsional stiffness Wing mass density Vertical tail plane (VTP) bending stiffness Fuselage bending stiffness Offset wings (horizontal)
l Imin Imax l Itor Itor r Imin Imin l
4.0 18.2 3.8 8.0 16.6 12.9 9.0 5.0 19.5 3.5
24
D. GO¨GE AND M. LINK
torsional stiffness of the left and right wing was updated separately. For the seventh mode the fuselage bending stiffness (Imin ) is the most sensitive parameter.
3. MODEL UPDATING RESULTS FOR THE INITIAL UNMODIFIED STRUCTURE
For updating only the first nine (active) eigenfrequencies and mode shapes were used. Modes 10–14 (passive) were only used to check the capability of the updated model to predict eigenfrequencies and modes shapes outside the active frequency range. Table 2 shows the parameter changes after computational model updating. The range of parameter changes is meaningful with respect to the possible modelling errors. The VTP/fuselage offset reduced from 0.075 to 0.072 m, the vertical offset of the wings changed from 0.12 to 0.1296 m and the horizontal offset of each wing increased from 0.04 to 0.0414 m. Table 3 shows the correlation between the updated model and the test data. The average frequency deviations for the updated modes and for the passive frequency range was reduced from Df ini;act: ¼ 3:79% and Df ini;pass: ¼ 3:47% to Df upd;act: ¼ 0:71% and Df upd;pass: ¼ 1:24% after updating. The mean for the dominant terms in the MAC increased for the active range from MAC ini;act: ¼ 89:35% to MAC upd;act: ¼ 96:21%: No big changes were found in the passive range with MAC upd;pass: ¼ 92:37% compared with MAC ini;pass: ¼ 92:42% of the initial model. The prediction of the analytical model is quite accurate. Figure 2 compares three measured and analytical FRFs calculated with the updated model. The updated model FRFs cover the measured FRFs quite well with regard to unavoidable uncertainties of the measurements. Experimental damping values have been used to calculate the response of the updated model.
4. PREDICTION OF THE MODAL DATA OF THE MODIFIED STRUCTURE
In order to see if the updated model is capable of predicting the affects of structural modifications, experimental data from two modified structures were used, as described by Link and Friswell in the preceding summary of the working group 1 contributions. At first, a lumped mass at the tail was introduced in the initial unupdated FE model, Table 4.
Table 3 Correlation between updated model and test (UWS) data from unmodified structure Mode
fexp (Hz)
fFE (Hz)
Df (%)
MAC (%)
1 2 3 4 5 6 7 8 9
6.55 16.60 34.88 35.36 36.71 50.09 50.72 56.44 65.14
6.47 16.90 34.91 35.40 36.97 50.40 50.47 56.69 65.77
1.22 1.75 0.09 0.11 0.71 0.62 0.49 0.44 0.97
97.99 96.70 96.84 95.55 94.68 97.12 94.64 99.41 92.92
10 11 12 13 14
69.64 105.47 134.68 145.87 155.82
70.73 105.12 129.98 146.46 156.44
1.57 0.33 3.49 0.40 0.40
97.00 99.13 76.98 95.28 93.45
25
FREQUENCY AND MODE SHAPE ERRORS OF A BEAM MODEL
2
m/s /N
REF 12z DOF 12−z
100
0
10
20
30
40 50 60 REF 12z DOF 101−z
70
80
90
100
70
80
90
100
70
80
90
100
2
m/s /N
Test Data from UWS
100
Updated FE Model
10−2 0
10
20
30
40
50
60
2
m/s /N
REF 12z DOF 5−z
100
0
10
20
30
40
50 Hz
60
Figure 2. Test and updated model FRFs, excitation at ref 12 z, unmodified structure.
Table 4 Correlation between modified initial model and test (IC London) data with tail modification (mod.1) Mode
fexp (Hz)
fFE (Hz)
Df (%)
MAC (%)
1 2 3 4 5 6 7 8 9
6.54 13.94 32.36 35.09 35.52 38.10 48.68 50.17 56.46
5.98 13.52 31.90 33.24 33.56 37.60 47.71 48.04 56.40
8.56 3.01 1.42 5.27 5.52 1.31 1.99 4.25 0.11
99.66 99.18 83.92 97.48 78.67 99.31 81.66 97.02 87.97
10 11
58.16 78.51
58.04 78.62
0.21 0.14
96.53 96.70
The average frequency deviation for all 11 modes is relatively high (Dfini;mod:1 ¼ 2:89%). The mean of the dominant terms in the MAC is MAC ini;mod:1 ¼ 92:55%: In a second step modification 1 was introduced in the updated FE model, Table 5. Comparing the average frequency deviation for the updated model (Df upd;mod:1 ¼ 0:84%) with the corresponding value from the initial model a drastic reduction can be seen. Also the mean for the dominant terms in the MAC (MAC upd;mod:1 ¼ 95:67%) was improved. Using experimental data from modification 2 (added lumped mass to the wing tip) leads to the same satisfactory results. Table 6 shows the frequency deviations and MAC values of the initial unupdated FE model with modification 2 and of the test data from
26
D. GO¨GE AND M. LINK
Table 5 Correlation between modified updated model and test (IC London) data with tail modification (mod.1) Mode
fexp (Hz)
fFE (Hz)
Df (%)
MAC (%)
1 2 3 4 5 6 7 8 9
6.54 13.94 32.36 35.09 35.52 38.10 48.68 50.17 56.46
6.47 14.09 31.73 35.04 35.55 37.50 48.10 50.47 56.68
1.07 1.08 1.95 0.14 0.08 1.57 1.19 0.60 0.39
99.66 99.41 99.24 78.97 99.42 97.05 92.47 98.57 93.50
10 11
58.16 78.51
58.11 77.64
0.09 1.11
97.71 96.39
Table 6 Correlation between modified initial model and test (UWS) data with wing modification (mod.2) Mode
fexp (Hz)
fFE (Hz)
Df (%)
MAC (%)
1 2 3 4 5 6 7 8 9
6.31 16.43 27.28 35.36 36.46 48.92 50.01 54.34 64.54
5.70 15.82 26.45 33.22 37.13 47.86 48.05 55.37 63.91
9.67 3.71 3.04 6.05 1.84 2.17 3.92 1.90 0.98
79.10 96.09 97.54 84.04 75.47 (39.18) 73.16 95.28 92.30
10 11
69.55 105.09
71.01 100.49
2.10 4.38
95.40 91.04
structural modification 2. As before, the average frequency deviation is relatively high (Df ini;mod:2 ¼ 3:61%). Also the mean of the MAC is quite low (MAC ini;mod:2 ¼ 83:51%). When introducing the second modification in the updated FE model, the results are improved in a drastic manner, Table 7. The average frequency deviation reduced to Df upd;mod:2 ¼ 1:05% and the average MAC value increased to MAC upd;mod:2 ¼ 93:27%:
5. CONCLUSIONS
In this paper the classical inverse sensitivity approach was used to update stiffness, mass and also geometric parameters of a finite element model of the GARTEUR SM-AG19 test structure. The updated model results correlate well with the test data not only in the active
FREQUENCY AND MODE SHAPE ERRORS OF A BEAM MODEL
27
Table 7 Correlation between modified updated model and test (UWS) data with wing modification (mod.2) Mode
fexp (Hz)
fFE (Hz)
Df (%)
MAC (%)
1 2 3 4 5 6 7 8 9
6.31 16.43 27.28 35.36 36.46 48.92 50.01 54.34 64.54
6.17 16.64 27.77 35.36 36.76 48.26 50.43 54.39 65.20
2.22 1.28 1.80 0.00 0.82 1.35 0.84 0.09 1.02
78.96 96.03 97.42 91.21 90.62 97.41 96.93 96.91 93.02
10 11
69.55 105.09
70.71 104.61
1.67 0.46
95.45 92.04
but also in the passive frequency range. Excellent results were achieved in predicting the modal changes due to structural modifications so that it can be concluded that the final model is physically meaningful and satisfies the requirements for validated models with regard to industrial applications.
REFERENCES 1. M. Link 1999 Modal Analysis and Testing, pp. 281–304. Dordrecht: Kluwer Academic Publications, J. M. M. Silva and N. M. M. Maia (eds.), Updating of analytical models}basic procedures and extensions. 2. M. Link and T. Graetsch 1999 Proceedings of the 2nd International Conference on Identification in Engineering Systems, 48–62. Assessment of model updating results in the presence of model structure and parameterisation errors. 3. C. Mares, J. E. Mottershead and M. I. Friswell 2000 Proceedings of the 25th International Seminar on Modal Analysis, 635–640. Selection and updating of parameters for the GARTEUR SM-AG19 testbed.
RESULTS OBTAINED BY MINIMIZING NATURAL FREQUENCYAND MODE SHAPE ERRORS OF A BEAM MODEL D. G˛ge Institute of Aeroelasticity, German Aerospace Center (DLR), 37073 Go.ttingen, Germany. E-mail: [email protected]
and Michael Link Lightweight Structures and Structural Mechanics Laboratory, University of Kassel, 34109 Kassel, Germany. E-mail: [email protected]
(Received 28 March 2002, accepted after revisions 1 October 2002) The classical inverse sensitivity approach was applied. In the presented application, the deviations between analytical and measured natural frequencies and mode shapes were minimised to update user selected stiffness, mass and geometric parameters. The selection of the updating parameters was based on a sensitivity analysis. The capability of the updated model to predict modal data of a modified structure is demonstrated in order to check if the physical meaning of the updated parameters has not been altered during the updating process. # 2003 Elsevier Science Ltd. All rights reserved.
1. INTRODUCTION
In practice the degree of correlation between finite element (FE) model results and vibration test data (FRFs and modal data) is commonly used to assess the model quality. If the correlation is found to be unsatisfactory the need arises to modify the model. Modifications may be related not only to the model parameters like material parameters but also to model structure features like mesh density or element types. Computational model updating (CMU) methods have been developed and used successfully in the past with the aim of automatically updating the FE model parameters thereby reducing the discrepancies between the model predictions and the test results, however, with the assumption that the model structure was adequate. For the present application the inverse sensitivity approach was used where the deviations between analytical and measured natural frequencies and mode shapes are minimised [1]. A beam model was generated where ten parameters were used to minimise the deviations of nine measured natural frequencies and mode shapes.
2. MODELLING ASPECTS OF THE ANALYTICAL MODEL
The initial model generated for the GARTEUR SM-AG19 structure consists of 131 beam elements with 408 degrees of freedom, see Fig. 1. Special care was taken for modeling the joints. The joint between fuselage and vertical tail plane was linked by a rigid 0888–3270/03/+$35.00/0
# 2003 Elsevier Science Ltd. All rights reserved.
22
D. GO¨GE AND M. LINK Wing Screws
Beam Elements
X Fuselage
Rigid Elements
Detail Y
Detail X
Z VTP
Y
Beam Element Fuselage Rigid Element X Beam Element
Z X
Y
Figure 1. Finite element model of the GARTEUR SM-AG19 testbed with modelling aspects.
Table 1 Correlation between initial model and test (UWS) data from unmodified structure Mode
fexp (Hz)
fFE (Hz)
Df (%)
MAC (%)
1 2 3 4 5 6 7 8 9
6.55 16.61 34.88 35.36 36.71 50.09 50.72 56.44 65.14
5.98 16.06 33.19 33.25 37.34 48.00 49.85 57.98 64.46
8.70 3.25 4.84 5.97 1.74 4.17 1.72 2.73 1.03
97.98 96.55 69.86 70.10 85.22 97.40 95.61 99.31 92.17
10 11 12 13 14
69.64 105.47 134.68 145.87 155.82
71.03 101.04 128.61 140.41 151.30
2.00 4.20 4.51 3.74 2.90
96.98 98.10 75.80 95.70 95.51
body (offset) and a beam element, see Fig. 1. The joints fuselage/wing and vertical tail plane/horizontal tail plane were idealized by spring elements. Additional offsets for the wings were modeled using rigid elements to account for realistic kinematic constraints, Fig. 1. A detailed mass matrix was established taking into account all connecting bolts and accelerometer masses. The actual accelerometer positions, e.g. at the wing leading and tailing edges, were modeled by additional offset nodes. Table 1 shows the correlation of the initial finite element model and the test data measured by the University of Wales, Swansea (UWS). The frequency deviations for most mode shapes are relatively high. The average frequency deviations for the modes used for updating (so-called active frequencies) is: Df ini;act: ¼
9 X Dfini;i i¼1
9
¼ 3:79%:
ð1Þ
23
FREQUENCY AND MODE SHAPE ERRORS OF A BEAM MODEL
The mean for the dominant terms in the modal assurance criterion (MAC) matrix is: MAC ini;act: ¼
9 X MACini;i i¼1
9
¼ 89:35%:
ð2Þ
The average frequency deviations for the modes not used in the updating procedure (so-called passive modes) amounts to: 14 X Dfini;i ¼ 3:47%: ð3Þ Df ini;pass: ¼ 5 i¼10 The mean for the dominant terms in the MAC matrix for the passive modes is MAC ini;pass: ¼
14 X MACini;i i¼10
5
¼ 92:42%:
ð4Þ
The average frequency deviations and the correlation via MAC is slightly better for the modes not used for updating than for the active modes. The reason for this is the bad correlation of the torsional wing modes 3 and 4, see Table 1. The fundamental modeling uncertainties in the finite element model were assumed to be in the joints, the mass distribution and in the flexural and torsional stiffnesses of the wing. Ten parameters were finally selected for the computational model updating, Table 2. The selection process was primarily but not only based on a sensitivity study. However, engineering judgement was also necessary, in particular, when lumping uncertain local parameters into global updating parameters. Earlier investigations on the GARTEUR SM-AG19 structure showed that the use of geometric parameters gave physically meaningful results [2, 3]. Three geometrical, six stiffness and one mass parameter were selected for updating. The horizontal offset of the wings tends to stiffen the wing/fuselage connection. Four modes in the active range are sensitive to the distance between the wings and the fuselage (vertical offset of wings). This parameter allows to compensate for inaccuracies in the complicated wing/fuselage connection. The fifth mode is most sensitive to the offset between the VTP and the fuselage and to the VTP bending stiffness. The remaining parameters for the wings were selected to compensate for the uncertainties in the thickness of the viscoelastic tape. Only one parameter was used for updating the wing bending stiffnesses about each principal axis because the wing bending modes did not exhibit significant non-symmetries. The non-symmetric behaviour of the torsional modes was more pronounced so that the
Table 2 Selected parameters for CMU and parameter changes after updating No. 1 2 3 4 5 6 7 8 9 10
Location and description
Parameter (p)
p change after CMU (%)
Offset VTP/fuselage Right and left wing bending stiffness Right and left wing bending stiffness Offset wings (vertical) Right wing torsional stiffness Left wing torsional stiffness Wing mass density Vertical tail plane (VTP) bending stiffness Fuselage bending stiffness Offset wings (horizontal)
l Imin Imax l Itor Itor r Imin Imin l
4.0 18.2 3.8 8.0 16.6 12.9 9.0 5.0 19.5 3.5
24
D. GO¨GE AND M. LINK
torsional stiffness of the left and right wing was updated separately. For the seventh mode the fuselage bending stiffness (Imin ) is the most sensitive parameter.
3. MODEL UPDATING RESULTS FOR THE INITIAL UNMODIFIED STRUCTURE
For updating only the first nine (active) eigenfrequencies and mode shapes were used. Modes 10–14 (passive) were only used to check the capability of the updated model to predict eigenfrequencies and modes shapes outside the active frequency range. Table 2 shows the parameter changes after computational model updating. The range of parameter changes is meaningful with respect to the possible modelling errors. The VTP/fuselage offset reduced from 0.075 to 0.072 m, the vertical offset of the wings changed from 0.12 to 0.1296 m and the horizontal offset of each wing increased from 0.04 to 0.0414 m. Table 3 shows the correlation between the updated model and the test data. The average frequency deviations for the updated modes and for the passive frequency range was reduced from Df ini;act: ¼ 3:79% and Df ini;pass: ¼ 3:47% to Df upd;act: ¼ 0:71% and Df upd;pass: ¼ 1:24% after updating. The mean for the dominant terms in the MAC increased for the active range from MAC ini;act: ¼ 89:35% to MAC upd;act: ¼ 96:21%: No big changes were found in the passive range with MAC upd;pass: ¼ 92:37% compared with MAC ini;pass: ¼ 92:42% of the initial model. The prediction of the analytical model is quite accurate. Figure 2 compares three measured and analytical FRFs calculated with the updated model. The updated model FRFs cover the measured FRFs quite well with regard to unavoidable uncertainties of the measurements. Experimental damping values have been used to calculate the response of the updated model.
4. PREDICTION OF THE MODAL DATA OF THE MODIFIED STRUCTURE
In order to see if the updated model is capable of predicting the affects of structural modifications, experimental data from two modified structures were used, as described by Link and Friswell in the preceding summary of the working group 1 contributions. At first, a lumped mass at the tail was introduced in the initial unupdated FE model, Table 4.
Table 3 Correlation between updated model and test (UWS) data from unmodified structure Mode
fexp (Hz)
fFE (Hz)
Df (%)
MAC (%)
1 2 3 4 5 6 7 8 9
6.55 16.60 34.88 35.36 36.71 50.09 50.72 56.44 65.14
6.47 16.90 34.91 35.40 36.97 50.40 50.47 56.69 65.77
1.22 1.75 0.09 0.11 0.71 0.62 0.49 0.44 0.97
97.99 96.70 96.84 95.55 94.68 97.12 94.64 99.41 92.92
10 11 12 13 14
69.64 105.47 134.68 145.87 155.82
70.73 105.12 129.98 146.46 156.44
1.57 0.33 3.49 0.40 0.40
97.00 99.13 76.98 95.28 93.45
25
FREQUENCY AND MODE SHAPE ERRORS OF A BEAM MODEL
2
m/s /N
REF 12z DOF 12−z
100
0
10
20
30
40 50 60 REF 12z DOF 101−z
70
80
90
100
70
80
90
100
70
80
90
100
2
m/s /N
Test Data from UWS
100
Updated FE Model
10−2 0
10
20
30
40
50
60
2
m/s /N
REF 12z DOF 5−z
100
0
10
20
30
40
50 Hz
60
Figure 2. Test and updated model FRFs, excitation at ref 12 z, unmodified structure.
Table 4 Correlation between modified initial model and test (IC London) data with tail modification (mod.1) Mode
fexp (Hz)
fFE (Hz)
Df (%)
MAC (%)
1 2 3 4 5 6 7 8 9
6.54 13.94 32.36 35.09 35.52 38.10 48.68 50.17 56.46
5.98 13.52 31.90 33.24 33.56 37.60 47.71 48.04 56.40
8.56 3.01 1.42 5.27 5.52 1.31 1.99 4.25 0.11
99.66 99.18 83.92 97.48 78.67 99.31 81.66 97.02 87.97
10 11
58.16 78.51
58.04 78.62
0.21 0.14
96.53 96.70
The average frequency deviation for all 11 modes is relatively high (Dfini;mod:1 ¼ 2:89%). The mean of the dominant terms in the MAC is MAC ini;mod:1 ¼ 92:55%: In a second step modification 1 was introduced in the updated FE model, Table 5. Comparing the average frequency deviation for the updated model (Df upd;mod:1 ¼ 0:84%) with the corresponding value from the initial model a drastic reduction can be seen. Also the mean for the dominant terms in the MAC (MAC upd;mod:1 ¼ 95:67%) was improved. Using experimental data from modification 2 (added lumped mass to the wing tip) leads to the same satisfactory results. Table 6 shows the frequency deviations and MAC values of the initial unupdated FE model with modification 2 and of the test data from
26
D. GO¨GE AND M. LINK
Table 5 Correlation between modified updated model and test (IC London) data with tail modification (mod.1) Mode
fexp (Hz)
fFE (Hz)
Df (%)
MAC (%)
1 2 3 4 5 6 7 8 9
6.54 13.94 32.36 35.09 35.52 38.10 48.68 50.17 56.46
6.47 14.09 31.73 35.04 35.55 37.50 48.10 50.47 56.68
1.07 1.08 1.95 0.14 0.08 1.57 1.19 0.60 0.39
99.66 99.41 99.24 78.97 99.42 97.05 92.47 98.57 93.50
10 11
58.16 78.51
58.11 77.64
0.09 1.11
97.71 96.39
Table 6 Correlation between modified initial model and test (UWS) data with wing modification (mod.2) Mode
fexp (Hz)
fFE (Hz)
Df (%)
MAC (%)
1 2 3 4 5 6 7 8 9
6.31 16.43 27.28 35.36 36.46 48.92 50.01 54.34 64.54
5.70 15.82 26.45 33.22 37.13 47.86 48.05 55.37 63.91
9.67 3.71 3.04 6.05 1.84 2.17 3.92 1.90 0.98
79.10 96.09 97.54 84.04 75.47 (39.18) 73.16 95.28 92.30
10 11
69.55 105.09
71.01 100.49
2.10 4.38
95.40 91.04
structural modification 2. As before, the average frequency deviation is relatively high (Df ini;mod:2 ¼ 3:61%). Also the mean of the MAC is quite low (MAC ini;mod:2 ¼ 83:51%). When introducing the second modification in the updated FE model, the results are improved in a drastic manner, Table 7. The average frequency deviation reduced to Df upd;mod:2 ¼ 1:05% and the average MAC value increased to MAC upd;mod:2 ¼ 93:27%:
5. CONCLUSIONS
In this paper the classical inverse sensitivity approach was used to update stiffness, mass and also geometric parameters of a finite element model of the GARTEUR SM-AG19 test structure. The updated model results correlate well with the test data not only in the active
FREQUENCY AND MODE SHAPE ERRORS OF A BEAM MODEL
27
Table 7 Correlation between modified updated model and test (UWS) data with wing modification (mod.2) Mode
fexp (Hz)
fFE (Hz)
Df (%)
MAC (%)
1 2 3 4 5 6 7 8 9
6.31 16.43 27.28 35.36 36.46 48.92 50.01 54.34 64.54
6.17 16.64 27.77 35.36 36.76 48.26 50.43 54.39 65.20
2.22 1.28 1.80 0.00 0.82 1.35 0.84 0.09 1.02
78.96 96.03 97.42 91.21 90.62 97.41 96.93 96.91 93.02
10 11
69.55 105.09
70.71 104.61
1.67 0.46
95.45 92.04
but also in the passive frequency range. Excellent results were achieved in predicting the modal changes due to structural modifications so that it can be concluded that the final model is physically meaningful and satisfies the requirements for validated models with regard to industrial applications.
REFERENCES 1. M. Link 1999 Modal Analysis and Testing, pp. 281–304. Dordrecht: Kluwer Academic Publications, J. M. M. Silva and N. M. M. Maia (eds.), Updating of analytical models}basic procedures and extensions. 2. M. Link and T. Graetsch 1999 Proceedings of the 2nd International Conference on Identification in Engineering Systems, 48–62. Assessment of model updating results in the presence of model structure and parameterisation errors. 3. C. Mares, J. E. Mottershead and M. I. Friswell 2000 Proceedings of the 25th International Seminar on Modal Analysis, 635–640. Selection and updating of parameters for the GARTEUR SM-AG19 testbed.