- Email: [email protected]
RESULTS OBTAINED BY MINIMISING NATURAL-FREQUENCY ERRORS AND USING PHYSICAL REASONING
Mechanical Systems and Signal Processing (2003) 17(1), 39–46 doi:10.1006/mssp.2002.1537, available online at http://www.idealibrary.com on
RESULTS OBTAINED BY MINIMISING NATURAL-FREQUENCY ERRORS AND USING PHYSICAL REASONING C. Mares and J. E. Mottershead Department of Engineering, Mechanical Engineering Division, University of Liverpool, Brownlow Hill, Liverpool L69 3GH, UK. E-mail: [email protected]
and M. I. Friswell Department of Mechanical Engineering, University of Wales Swansea, Swansea SA2 8PP, UK (Received 28 March 2002, accepted 1 October 2002) A finite element model of the GARTEUR SM-AG19 test structure is updated. The philosophy applied in selecting the updating parameters is based on physical understanding, scrutiny of the mode shapes and sensitivity calculations. The ‘corrected’ model is shown not only to accurately reproduce test data used in the updating process, but also to provide results in agreement with test data (not used for updating) from the modified structure. # 2003 Published by Elsevier Science Ltd.
1. FINITE ELEMENT MODEL
The test structure consists of aluminium beams of rectangular cross-section as described by Link and Friswell [1] in first article of this special issue. The modelling uncertainty, which we aim to reduce by updating, is undoubtedly concentrated at the joints and, to a lesser extent, at the constrained viscoelastic layer that runs over the length of the wings. In the physical structure, the joint between the fuselage and the wings is achieved by screwed connections through a small plate which is sandwiched between the two. In several modes this joint is at a vibration node, in which case its representation in the finite element model is not critical. In other modes however the joint is strained considerably and the form of the model in the region of the joint may then be important. This joint is the main one that influences the choice of beams or plates for the finite element model. The wings and fuselage are long slender structures which might suggest the use of beam elements. However, they are not connected at the neutral axes. Therefore, locally at the joint, there will almost certainly be deformations that cannot be properly represented by the assumption of plane section remaining plane. If plate elements were to be used then a fine mesh would be needed to achieve the same accuracy that can be readily obtained from a small number of Hermitian beams. We choose to construct a finite element model consisting of 126 beams, 26 rigid elements and five lumped masses. The complete model has 534 degrees-of-freedom. An equivalent beam-model is used to represent the joint between the fuselage and wings as shown in Fig. 1. The fuselage imparts an added stiffness to the wings at the joint and similarly the wings have a stiffening effect on the fuselage as they pass over it, though to a 0888–3270/03/+$35.00/0
# 2003 Published by Elsevier Science Ltd.
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C. MARES ET AL.
wing offset
θ1 wing
wing fuselage connection
fuselage offset
θ2
fuselage
Figure 1. Wing-fuselage model.
Tail plane.
generic element θ θ
Tail fin.
3 4
bending offset θ5 Figure 2. Tail model.
lesser extent. Beam offsets, shown with lengths exaggerated in Fig. 1, are used. The damping and constraining layers on the wings are each represented by beams with nodes offset to the neutral axis of the main wing-beam. Rigid elements are used to connect the winglets at the wing-tips. Offsets, as in Fig. 2, are used at the connection between the fin and the tail-plane and between the fin and the fuselage. A point mass (and a point inertia) at the tail-plane/tail-fin joint represents the mass of two steel strips used in the construction of the joint. The physical accelerometer positions such as on the wing leading and trailing edges are located using offset nodes.
2. UPDATING PHILOSOPHY AND PARAMETERISATION
The choice of updating parameters is an important aspect of the finite element model updating process [2–4]. In this article sensitive parameters are only chosen for updating if they can be justified on the basis of engineering understanding of the test structure. The parameterisation of joints in an aluminium space frame [5] is an example of how this
41
GARTEUR TEST STRUCTURE
Table 1 Table of sensitivities for the updating parameters Mode f1 f2 f3 f4 f5 f6 f7 f8 f9 f10 f11 f12 f13 f14
y1
y2
y3
y4
y5
y6
y7
2.27e+02 7.87e+02 1.38e+03 2.02e+03 1.11e+03 1.59e+04 1.06e+04 2.17e+04 1.07e+03 2.37e+04 1.20e+03 1.24e+05 6.37e+05 5.65e+05
0.0 3.55e+00 2.98e+01 0.0 2.58e+02 0.0 1.10e+04 0.0 6.11e+02 1.02e+03 8.29e+00 3.11e+01 8.51e+01 9.61e+01
0.0 1.13e+02 9.81e+02 0.0 1.30e+03 0.0 1.24e+03 0.0 9.10e+00 1.14e+04 8.18e+03 2.73e+03 0.0 3.77e+02
0.0 0.0 0.0 0.0 9.88e01 0.0 9.46e+02 0.0 4.76e+04 4.94e+02 5.33e+02 6.66e+02 0.0 1.83e+02
0.0 1.69e+03 7.55e+03 2.85e01 1.01e+04 5.51e01 3.08e+02 3.81e+01 1.11e+02 6.71e+03 6.05e+04 1.42e+05 8.51e02 7.80e+04
1.42e+03 7.64e+03 7.54e+03 4.03e+02 1.22e+04 8.24e+04 6.14e+03 7.89e+01 2.15e+03 9.82e+04 1.26e+05 1.76e+05 0.0 6.36e+04
8.92e01 1.90e+01 2.28e+04 3.97e+02 1.71e+04 7.74e+02 6.31e+02 4.99e+02 1.91e+01 6.03e+01 6.90e+03 1.37e+04 1.32e+05 1.57e+05
Figure 3. Finite element mode 7:
philosophy can be applied to obtain an updated model with physical meaning. The parameters chosen in the exercise reported here have been used consistently in a series of studies [6,7] using the original DLR measurements [1]. In [7] the test structure is used as an example of model updating by a method which is robust to measurement and parameter uncertainties. Here, we discuss each of the chosen updating parameters in turn. The first-order sensitivities of the first 14 finite-element modes to changes in the parameters are shown in Table 1. The wing-offset, y1 ; used to represent local stiffening (in bending) of the wings by the fuselage is the single parameter most responsible for the deviation between the finite element predictions and test modes. The use of offset parameters in model updating is described by Mottershead et al. [8]. Table 1 shows parameter y1 to be sensitive to all the modes, and in numerous updating exercises it was found to converge consistently to the same value. The torsional rigidity, y2 ; of the wing-fuselage connection is the controlling parameter for the seventh mode, characterised by in-plane wing bending with the two wing-tips in anti-phase as shown in Fig. 3.
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C. MARES ET AL.
Two generic-element eigenvalues, y3 and y4 ; for the substructure stiffness matrix of the three elements at the tail-plane/tail-fin joint (including the offsets and joint mass described in Section 1) are important for the prediction of many modes with activity at the tail. y3 represents the stiffness of the first in-plane bending eigenvector of the joint separately from other ‘modes’ of the substructure stiffness matrix. y4 is the torsional ‘mode’ of the joint, which is important to the ninth and tenth mode, as well as to the seventh. Generic elements are described by Gladwell and Ahmadian [9] and Ahmadian et al. [10]. The tail bending offset, y5 ; at the connection with the fuselage affects modes 5; 11; 12 and 14 particularly strongly. The flexural and torsional rigidities (y6 and y7 ) of the wings are of course very sensitive because the wings are the most active components in all of the lower modes. Their use as updating parameters seems reasonable mainly because of uncertainty in the thickness of the viscoelastic and constraining layers. y6 and y7 are penalised in updating to prevent them taking unjustifiably large values. Updating was carried out using the first natural frequencies obtained by UWS and shown in column 7 of Table 1 [1]. It proved necessary to include 10 modes, rather than nine as specified in the ‘rules’ of the benchmark study, for reasons that will be elaborated in the following section. The measured modes were used only for the purpose of mode pairing. This leads to a 10 7 over-determined system of updating equations.
3. MODEL UPDATING RESULTS
The initial and updated finite element natural frequencies are given in Table 2. The initial and updated mode-shape correlations (MAC) are provided in Tables 3 and 4. The errors in natural frequency predictions of the first eight modes, from the initial model, are all positive, the largest being over 6% at the seventh mode, probably due to overestimation of most of the joint stiffnesses. Modes 3; 4 and 5 are close modes with significant cross coupling in the finite element model. Modes 9 and 10 are another set of close modes which change places to produce small diagonal and large off-diagonal terms in the MAC array of Table 3.
Table 2 Measured natural frequencies and finite element predictions (Hz) Mode no. f1 f2 f3 f4 f5 f6 f7 f8 f9 f10 f11 f12 f13 f14
Exper Model
FE Model initial
FE Model final
Error initial (%)
Error final (%)
6.55 16.61 34.88 35.36 36.71 50.09 50.72 56.44 65.14 69.64 105.47 134.68 145.87 155.82
6.61 16.95 35.98 36.45 37.22 50.88 53.94 59.30 64.48 66.94 105.71 133.09 146.33 158.68
6.54 16.77 34.95 35.29 36.35 49.91 50.73 56.42 65.16 69.63 101.22 130.04 143.25 153.18
0.88 2.05 3.15 3.09 1.40 1.58 6.34 5.07 1.01 3.87 0.22 1.18 0.31 1.84
0.07 0.97 0.20 0.20 0.97 0.35 0.02 0.02 0.03 0.02 4.03 3.45 1.79 1.69
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GARTEUR TEST STRUCTURE
Table 3 Initial MAC Exp. mode no. FE mode no. MAC
1 1 98
2 2 96
3 3 63
4 4 70
5 5 81
6 6 97
7 7 95
8 8 99
9 10 97
10 9 78
11 11 98
12 12 82
13 13 93
14 14 92
8 8 99
9 9 93
10 10 97
11 11 97
12 12 86
13 13 93
14 14 91
Table 4 Final MAC Exp. mode no. FE mode no. MAC
1 1 98
2 2 96
3 3 69
4 4 70
5 5 88
6 6 97
7 7 95
Table 5 Updated parameters y1 0.342
y2 0.538
y3 0.876
y4 1.957
y5 0.814
y6 1.081
y7 0.935
The convergence of the seven updating parameters is complete after about three iterations and their final values are given in Table 5. Except for the torsional eigenvalue of the generic element at the tail-plane/tail-fin joint, all the joint stiffnesses are reduced (yi 51; i ¼ 1; . . . ; 5; i=4) as is the torsional rigidity of the wings. The tail joint is stiffened by two steel strips of rectangular cross-section placed in the corners of the joint as shown in the top right-hand detail of Fig. 2 [1]. The steel strips impart considerable torsional rigidity to the joint over and above that of the finite element model, which includes only the mass of the two reinforcing strips. Torsion in the tail joint is the one exception to the general observation that the joints are over stiffened in the initial model. Modes 9; 10 and 12 are seen from Table 2 to be sensitive to y4 which explains the negative initial errors in the fifth column of Table 2. The converged value of y4 was different depending upon whether nine or ten modes were included in the objective function. The reason for including 10 modes was that the error after convergence was significantly less than when using nine modes. The flexural rigidity y6 is increased slightly. The torsional rigidity of the beam between the wing and fuselage y2 is substantially reduced, but this is an equivalent parameter having a numerical value that cannot be attributed any physically meaning. It was assigned an initial value in the range where mode 7 is sensitive. The seventh mode fails to converge if y2 is not selected as an updating parameter. The greatest stiffness reduction occurs in the wing-offset over the fuselage y1 : This indicates that the stiffening effect of the fuselage is considerably over-estimated in the initial model. y1 is the dominant updating parameter which, in numerous updating tests (using different candidate parameters), consistently converges to a normalised value close to 0.34. If y1 is set to 0:34 without changing the other parameters, modes 9 and 10; which had been interchanged, are swapped back to be in the same modal order as in the test. The rms error in the first 10 natural frequencies predicted by the updated model is 0:46%, and as can be seen in Table 2 the predictions of individual frequencies do not exceed 1% in the same range. Outside the range of the first 10 natural frequencies the
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C. MARES ET AL.
predictions are less accurate, the 11th mode being predicted with an error that just exceeds 4%. The loss of accuracy in predicting modes 11–14 can be attributed to the fact that the mode shapes become more complicated at higher frequencies so that new parameters describing different joint behaviour become sensitive. An example of this can be seen in the sensitivities of the modes to the bending offset at the tail-fuselage joint y5 ; which take large values for the eleventh and twelfth natural frequencies outside the range of data used in the objective function. In this case, however, y5 must be included to obtain convergence of the fifth mode. It is seen from Table 4 that after updating some improvement in mode correlation is also achieved. As discussed above, the interchange in mode order between modes 9 and 10 is corrected, although the cross-coupling between the close modes 3 and 5 remains largely unchanged, because torsion cannot be observed fully by a single row of accelerometers along the wings (all three modes are sensitive to y7 ). There is also the possibility that non-proportional viscoelastic damping of the wings in torsion couples modes 3–5. The best way to form a judgement on the ‘quality’ of the updated model is not by inspecting the out-of-range predictions but by considering its performance in determining the natural frequencies and modes of the structure following a physical modification, which is the subject of the following section.
4. MODIFIED-STRUCTURE RESULTS
A description of physical modifications applied (i) at the tail and (ii) at the wing was given by Link and Friswell [1], and the modifications thus prescribed were implemented in both the initial and updated finite element models. Model updating was not carried out again using the new measurements, which were used only to assess the quality of the updated model described in the previous section. In the case of the tail modification, the natural frequencies are shown in Table 6, and the correlation of the initial and updated mode shapes with measurements can be found in Tables 7 and 8. The rms error in the first 10 natural frequencies is 0:93%, the greatest individual error being 1:92% at the third mode. Table 7 shows interchanges involving modes 4–6, and separately modes 7 and 8 in the initial model, all of which are corrected by updating. For the wing modification, Table 9 shows the natural frequencies, and Tables 10 and 11 show the correlation of results from the initial and updated models with experimental modes. A significant error of 5:85% remains in predicting the third natural frequency after
Table 6 Measured natural frequencies and finite element predictions (Hz)}tail modification Mode no. f1 f2 f3 f4 f5 f6 f7 f8 f9 f10
Exper model
FE model initial
FE model final
Error initial (%)
Error final (%)
6.54 13.94 32.36 35.09 35.52 38.10 48.68 50.17 56.46 58.16
6.60 14.18 31.81 35.06 36.47 36.66 50.83 51.12 59.21 59.64
6.54 13.87 31.74 35.27 35.47 37.98 48.19 49.92 56.39 57.09
0.95 1.71 1.71 0.10 2.67 3.80 4.41 1.90 4.87 2.54
0.04 0.48 1.92 0.49 0.14 0.33 1.01 0.49 0.12 1.83
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GARTEUR TEST STRUCTURE
Table 7 Initial MAC}tail modification Exp. mode no. FE mode no. MAC
1 1 100
2 2 99
3 3 99
4 5 87
5 6 94
6 4 85
7 8 94
8 7 66
9 9 76
10 10 88
7 7 92
8 8 98
9 9 86
10 10 93
Table 8 Final MAC}tail modification Exp. mode no. FE mode no. MAC
1 1 100
2 2 99
3 3 99
4 4 96
5 5 67
6 6 97
Table 9 Measured natural frequencies and finite element predictions (Hz)}wing modification Mode no.
Exper model
FE model initial
FE model final
Error initial (%)
Error final (%)
6.31 16.43 27.28 35.36 36.46 48.92 50.01 54.34 64.54 69.55
6.60 14.18 31.81 35.06 36.47 36.66 50.83 51.12 59.21 59.64
6.26 16.57 28.88 35.08 36.15 48.92 49.72 54.39 64.41 69.61
4.69 13.69 16.59 0.86 0.04 25.08 1.63 5.91 8.25 14.25
0.81 0.88 5.85 0.80 0.83 0.01 0.59 0.09 0.20 0.09
f1 f2 f3 f4 f5 f6 f7 f8 f9 f10
Table 10 Initial MAC}wing modification Exp. mode no. FE mode no. MAC
1 1 79
2 2 95
3 3 96
4 4 94
5 5 90
6 7 95
7 6 84
8 8 97
9 10 95
10 9 79
7 7 94
8 8 99
9 9 93
10 10 95
Table 11 Final MAC}wing modification Exp. mode no. FE mode no. MAC
1 1 79
2 2 94
3 3 96
4 4 94
5 5 94
6 6 89
updating, although the initial error was over 16%. Otherwise the rms error over the first 10 modes is 0:59%. Modes 6 and 7, and separately modes 9 and 10, of the initial model, which are interchanged with experimental modes, are unscrambled in the updated model.
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C. MARES ET AL.
5. CONCLUSIONS
A finite element model of the GARTEUR SM-AG19 test structure is created and updated. Parameterisation of the joints, where the principal modelling uncertainties are located, is achieved by using eigenvalue sensitivities and mode shapes to obtain a physical understanding of the complete structure. The most important joint is where the wings connect to the fuselage and a good parameterisation of this joint is essential if an updated model with physical meaning is to be obtained. In the present study, it is demonstrated that a good updated model is achieved with meaningful parameters in the range of the first 10 modes. The behaviour of the structure with physical modifications is accurately predicted by the updated model.
ACKNOWLEDGEMENTS
The research described in this article was supported by the Engineering and Physical Sciences Research Council, grant number GR/M08622.
REFERENCES 1. M. Link and M. I. Friswell 2001 Mechanical Systems and Signal Processing 17, 9–20. Generation of validated structural dynamic models}results of the GARTEUR benchmark study. 2. J. E. Mottershead and M. I. Friswell 1993 Journal of Sound Vibration 162, 347–375. Model updating in structural dynamics: a survey. 3. M. I. Friswell and J. E. Mottershead 1995 Finite Element Model Updating in Structural Dynamics. Dordrecht: Kluwer Academic Publishers. 4. J. E. Mottershead and M. I. Friswell (Guest Eds.) 1998 Mechanical Systems and Signal Processing 12, 1–224. Special issue on model updating. 5. J. E. Mottershead, C. Mares, M. I. Friswell and S. James 2000 Mechanical Systems and Signal Processing 12, 923–944. Selection and updating of parameters for an aluminium spaceframe model. 6. C. Mares, J. E. Mottershead and M. I. Friswell 2000 Proceedings of ISMA 25, Leuven, Belgium. Selection and updating of parameters for the GARTEUR SM-AG19 testbed, 635–640. 7. C. Mares, M. I. Friswell and J. E. Mottershead 2002 Mechanical Systems and Signal Processing 16, 169–183. Model updating using robust estimation. 8. J. E. Mottershead, M. I. Friswell, G. H. T. Ng and J. A. Brandon 1996 Mechanical Systems and Signal Processing 10, 171–182. Geometric parameters for finite element updating of joints and constraints. 9. G. M. L. Gladwell and H. Ahmadian 1996 Mechanical Systems and Signal Processing 9, 601–614. Generic element matrices for finite element model updating. 10. H. Ahmadian, G. M. L. Gladwell and F. Ismail 1997 Journal of Vibration and Acoustics 119, 37–45. Parameter selection strategies in finite element updating.
RESULTS OBTAINED BY MINIMISING NATURAL-FREQUENCY ERRORS AND USING PHYSICAL REASONING C. Mares and J. E. Mottershead Department of Engineering, Mechanical Engineering Division, University of Liverpool, Brownlow Hill, Liverpool L69 3GH, UK. E-mail: [email protected]
and M. I. Friswell Department of Mechanical Engineering, University of Wales Swansea, Swansea SA2 8PP, UK (Received 28 March 2002, accepted 1 October 2002) A finite element model of the GARTEUR SM-AG19 test structure is updated. The philosophy applied in selecting the updating parameters is based on physical understanding, scrutiny of the mode shapes and sensitivity calculations. The ‘corrected’ model is shown not only to accurately reproduce test data used in the updating process, but also to provide results in agreement with test data (not used for updating) from the modified structure. # 2003 Published by Elsevier Science Ltd.
1. FINITE ELEMENT MODEL
The test structure consists of aluminium beams of rectangular cross-section as described by Link and Friswell [1] in first article of this special issue. The modelling uncertainty, which we aim to reduce by updating, is undoubtedly concentrated at the joints and, to a lesser extent, at the constrained viscoelastic layer that runs over the length of the wings. In the physical structure, the joint between the fuselage and the wings is achieved by screwed connections through a small plate which is sandwiched between the two. In several modes this joint is at a vibration node, in which case its representation in the finite element model is not critical. In other modes however the joint is strained considerably and the form of the model in the region of the joint may then be important. This joint is the main one that influences the choice of beams or plates for the finite element model. The wings and fuselage are long slender structures which might suggest the use of beam elements. However, they are not connected at the neutral axes. Therefore, locally at the joint, there will almost certainly be deformations that cannot be properly represented by the assumption of plane section remaining plane. If plate elements were to be used then a fine mesh would be needed to achieve the same accuracy that can be readily obtained from a small number of Hermitian beams. We choose to construct a finite element model consisting of 126 beams, 26 rigid elements and five lumped masses. The complete model has 534 degrees-of-freedom. An equivalent beam-model is used to represent the joint between the fuselage and wings as shown in Fig. 1. The fuselage imparts an added stiffness to the wings at the joint and similarly the wings have a stiffening effect on the fuselage as they pass over it, though to a 0888–3270/03/+$35.00/0
# 2003 Published by Elsevier Science Ltd.
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C. MARES ET AL.
wing offset
θ1 wing
wing fuselage connection
fuselage offset
θ2
fuselage
Figure 1. Wing-fuselage model.
Tail plane.
generic element θ θ
Tail fin.
3 4
bending offset θ5 Figure 2. Tail model.
lesser extent. Beam offsets, shown with lengths exaggerated in Fig. 1, are used. The damping and constraining layers on the wings are each represented by beams with nodes offset to the neutral axis of the main wing-beam. Rigid elements are used to connect the winglets at the wing-tips. Offsets, as in Fig. 2, are used at the connection between the fin and the tail-plane and between the fin and the fuselage. A point mass (and a point inertia) at the tail-plane/tail-fin joint represents the mass of two steel strips used in the construction of the joint. The physical accelerometer positions such as on the wing leading and trailing edges are located using offset nodes.
2. UPDATING PHILOSOPHY AND PARAMETERISATION
The choice of updating parameters is an important aspect of the finite element model updating process [2–4]. In this article sensitive parameters are only chosen for updating if they can be justified on the basis of engineering understanding of the test structure. The parameterisation of joints in an aluminium space frame [5] is an example of how this
41
GARTEUR TEST STRUCTURE
Table 1 Table of sensitivities for the updating parameters Mode f1 f2 f3 f4 f5 f6 f7 f8 f9 f10 f11 f12 f13 f14
y1
y2
y3
y4
y5
y6
y7
2.27e+02 7.87e+02 1.38e+03 2.02e+03 1.11e+03 1.59e+04 1.06e+04 2.17e+04 1.07e+03 2.37e+04 1.20e+03 1.24e+05 6.37e+05 5.65e+05
0.0 3.55e+00 2.98e+01 0.0 2.58e+02 0.0 1.10e+04 0.0 6.11e+02 1.02e+03 8.29e+00 3.11e+01 8.51e+01 9.61e+01
0.0 1.13e+02 9.81e+02 0.0 1.30e+03 0.0 1.24e+03 0.0 9.10e+00 1.14e+04 8.18e+03 2.73e+03 0.0 3.77e+02
0.0 0.0 0.0 0.0 9.88e01 0.0 9.46e+02 0.0 4.76e+04 4.94e+02 5.33e+02 6.66e+02 0.0 1.83e+02
0.0 1.69e+03 7.55e+03 2.85e01 1.01e+04 5.51e01 3.08e+02 3.81e+01 1.11e+02 6.71e+03 6.05e+04 1.42e+05 8.51e02 7.80e+04
1.42e+03 7.64e+03 7.54e+03 4.03e+02 1.22e+04 8.24e+04 6.14e+03 7.89e+01 2.15e+03 9.82e+04 1.26e+05 1.76e+05 0.0 6.36e+04
8.92e01 1.90e+01 2.28e+04 3.97e+02 1.71e+04 7.74e+02 6.31e+02 4.99e+02 1.91e+01 6.03e+01 6.90e+03 1.37e+04 1.32e+05 1.57e+05
Figure 3. Finite element mode 7:
philosophy can be applied to obtain an updated model with physical meaning. The parameters chosen in the exercise reported here have been used consistently in a series of studies [6,7] using the original DLR measurements [1]. In [7] the test structure is used as an example of model updating by a method which is robust to measurement and parameter uncertainties. Here, we discuss each of the chosen updating parameters in turn. The first-order sensitivities of the first 14 finite-element modes to changes in the parameters are shown in Table 1. The wing-offset, y1 ; used to represent local stiffening (in bending) of the wings by the fuselage is the single parameter most responsible for the deviation between the finite element predictions and test modes. The use of offset parameters in model updating is described by Mottershead et al. [8]. Table 1 shows parameter y1 to be sensitive to all the modes, and in numerous updating exercises it was found to converge consistently to the same value. The torsional rigidity, y2 ; of the wing-fuselage connection is the controlling parameter for the seventh mode, characterised by in-plane wing bending with the two wing-tips in anti-phase as shown in Fig. 3.
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C. MARES ET AL.
Two generic-element eigenvalues, y3 and y4 ; for the substructure stiffness matrix of the three elements at the tail-plane/tail-fin joint (including the offsets and joint mass described in Section 1) are important for the prediction of many modes with activity at the tail. y3 represents the stiffness of the first in-plane bending eigenvector of the joint separately from other ‘modes’ of the substructure stiffness matrix. y4 is the torsional ‘mode’ of the joint, which is important to the ninth and tenth mode, as well as to the seventh. Generic elements are described by Gladwell and Ahmadian [9] and Ahmadian et al. [10]. The tail bending offset, y5 ; at the connection with the fuselage affects modes 5; 11; 12 and 14 particularly strongly. The flexural and torsional rigidities (y6 and y7 ) of the wings are of course very sensitive because the wings are the most active components in all of the lower modes. Their use as updating parameters seems reasonable mainly because of uncertainty in the thickness of the viscoelastic and constraining layers. y6 and y7 are penalised in updating to prevent them taking unjustifiably large values. Updating was carried out using the first natural frequencies obtained by UWS and shown in column 7 of Table 1 [1]. It proved necessary to include 10 modes, rather than nine as specified in the ‘rules’ of the benchmark study, for reasons that will be elaborated in the following section. The measured modes were used only for the purpose of mode pairing. This leads to a 10 7 over-determined system of updating equations.
3. MODEL UPDATING RESULTS
The initial and updated finite element natural frequencies are given in Table 2. The initial and updated mode-shape correlations (MAC) are provided in Tables 3 and 4. The errors in natural frequency predictions of the first eight modes, from the initial model, are all positive, the largest being over 6% at the seventh mode, probably due to overestimation of most of the joint stiffnesses. Modes 3; 4 and 5 are close modes with significant cross coupling in the finite element model. Modes 9 and 10 are another set of close modes which change places to produce small diagonal and large off-diagonal terms in the MAC array of Table 3.
Table 2 Measured natural frequencies and finite element predictions (Hz) Mode no. f1 f2 f3 f4 f5 f6 f7 f8 f9 f10 f11 f12 f13 f14
Exper Model
FE Model initial
FE Model final
Error initial (%)
Error final (%)
6.55 16.61 34.88 35.36 36.71 50.09 50.72 56.44 65.14 69.64 105.47 134.68 145.87 155.82
6.61 16.95 35.98 36.45 37.22 50.88 53.94 59.30 64.48 66.94 105.71 133.09 146.33 158.68
6.54 16.77 34.95 35.29 36.35 49.91 50.73 56.42 65.16 69.63 101.22 130.04 143.25 153.18
0.88 2.05 3.15 3.09 1.40 1.58 6.34 5.07 1.01 3.87 0.22 1.18 0.31 1.84
0.07 0.97 0.20 0.20 0.97 0.35 0.02 0.02 0.03 0.02 4.03 3.45 1.79 1.69
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GARTEUR TEST STRUCTURE
Table 3 Initial MAC Exp. mode no. FE mode no. MAC
1 1 98
2 2 96
3 3 63
4 4 70
5 5 81
6 6 97
7 7 95
8 8 99
9 10 97
10 9 78
11 11 98
12 12 82
13 13 93
14 14 92
8 8 99
9 9 93
10 10 97
11 11 97
12 12 86
13 13 93
14 14 91
Table 4 Final MAC Exp. mode no. FE mode no. MAC
1 1 98
2 2 96
3 3 69
4 4 70
5 5 88
6 6 97
7 7 95
Table 5 Updated parameters y1 0.342
y2 0.538
y3 0.876
y4 1.957
y5 0.814
y6 1.081
y7 0.935
The convergence of the seven updating parameters is complete after about three iterations and their final values are given in Table 5. Except for the torsional eigenvalue of the generic element at the tail-plane/tail-fin joint, all the joint stiffnesses are reduced (yi 51; i ¼ 1; . . . ; 5; i=4) as is the torsional rigidity of the wings. The tail joint is stiffened by two steel strips of rectangular cross-section placed in the corners of the joint as shown in the top right-hand detail of Fig. 2 [1]. The steel strips impart considerable torsional rigidity to the joint over and above that of the finite element model, which includes only the mass of the two reinforcing strips. Torsion in the tail joint is the one exception to the general observation that the joints are over stiffened in the initial model. Modes 9; 10 and 12 are seen from Table 2 to be sensitive to y4 which explains the negative initial errors in the fifth column of Table 2. The converged value of y4 was different depending upon whether nine or ten modes were included in the objective function. The reason for including 10 modes was that the error after convergence was significantly less than when using nine modes. The flexural rigidity y6 is increased slightly. The torsional rigidity of the beam between the wing and fuselage y2 is substantially reduced, but this is an equivalent parameter having a numerical value that cannot be attributed any physically meaning. It was assigned an initial value in the range where mode 7 is sensitive. The seventh mode fails to converge if y2 is not selected as an updating parameter. The greatest stiffness reduction occurs in the wing-offset over the fuselage y1 : This indicates that the stiffening effect of the fuselage is considerably over-estimated in the initial model. y1 is the dominant updating parameter which, in numerous updating tests (using different candidate parameters), consistently converges to a normalised value close to 0.34. If y1 is set to 0:34 without changing the other parameters, modes 9 and 10; which had been interchanged, are swapped back to be in the same modal order as in the test. The rms error in the first 10 natural frequencies predicted by the updated model is 0:46%, and as can be seen in Table 2 the predictions of individual frequencies do not exceed 1% in the same range. Outside the range of the first 10 natural frequencies the
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predictions are less accurate, the 11th mode being predicted with an error that just exceeds 4%. The loss of accuracy in predicting modes 11–14 can be attributed to the fact that the mode shapes become more complicated at higher frequencies so that new parameters describing different joint behaviour become sensitive. An example of this can be seen in the sensitivities of the modes to the bending offset at the tail-fuselage joint y5 ; which take large values for the eleventh and twelfth natural frequencies outside the range of data used in the objective function. In this case, however, y5 must be included to obtain convergence of the fifth mode. It is seen from Table 4 that after updating some improvement in mode correlation is also achieved. As discussed above, the interchange in mode order between modes 9 and 10 is corrected, although the cross-coupling between the close modes 3 and 5 remains largely unchanged, because torsion cannot be observed fully by a single row of accelerometers along the wings (all three modes are sensitive to y7 ). There is also the possibility that non-proportional viscoelastic damping of the wings in torsion couples modes 3–5. The best way to form a judgement on the ‘quality’ of the updated model is not by inspecting the out-of-range predictions but by considering its performance in determining the natural frequencies and modes of the structure following a physical modification, which is the subject of the following section.
4. MODIFIED-STRUCTURE RESULTS
A description of physical modifications applied (i) at the tail and (ii) at the wing was given by Link and Friswell [1], and the modifications thus prescribed were implemented in both the initial and updated finite element models. Model updating was not carried out again using the new measurements, which were used only to assess the quality of the updated model described in the previous section. In the case of the tail modification, the natural frequencies are shown in Table 6, and the correlation of the initial and updated mode shapes with measurements can be found in Tables 7 and 8. The rms error in the first 10 natural frequencies is 0:93%, the greatest individual error being 1:92% at the third mode. Table 7 shows interchanges involving modes 4–6, and separately modes 7 and 8 in the initial model, all of which are corrected by updating. For the wing modification, Table 9 shows the natural frequencies, and Tables 10 and 11 show the correlation of results from the initial and updated models with experimental modes. A significant error of 5:85% remains in predicting the third natural frequency after
Table 6 Measured natural frequencies and finite element predictions (Hz)}tail modification Mode no. f1 f2 f3 f4 f5 f6 f7 f8 f9 f10
Exper model
FE model initial
FE model final
Error initial (%)
Error final (%)
6.54 13.94 32.36 35.09 35.52 38.10 48.68 50.17 56.46 58.16
6.60 14.18 31.81 35.06 36.47 36.66 50.83 51.12 59.21 59.64
6.54 13.87 31.74 35.27 35.47 37.98 48.19 49.92 56.39 57.09
0.95 1.71 1.71 0.10 2.67 3.80 4.41 1.90 4.87 2.54
0.04 0.48 1.92 0.49 0.14 0.33 1.01 0.49 0.12 1.83
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GARTEUR TEST STRUCTURE
Table 7 Initial MAC}tail modification Exp. mode no. FE mode no. MAC
1 1 100
2 2 99
3 3 99
4 5 87
5 6 94
6 4 85
7 8 94
8 7 66
9 9 76
10 10 88
7 7 92
8 8 98
9 9 86
10 10 93
Table 8 Final MAC}tail modification Exp. mode no. FE mode no. MAC
1 1 100
2 2 99
3 3 99
4 4 96
5 5 67
6 6 97
Table 9 Measured natural frequencies and finite element predictions (Hz)}wing modification Mode no.
Exper model
FE model initial
FE model final
Error initial (%)
Error final (%)
6.31 16.43 27.28 35.36 36.46 48.92 50.01 54.34 64.54 69.55
6.60 14.18 31.81 35.06 36.47 36.66 50.83 51.12 59.21 59.64
6.26 16.57 28.88 35.08 36.15 48.92 49.72 54.39 64.41 69.61
4.69 13.69 16.59 0.86 0.04 25.08 1.63 5.91 8.25 14.25
0.81 0.88 5.85 0.80 0.83 0.01 0.59 0.09 0.20 0.09
f1 f2 f3 f4 f5 f6 f7 f8 f9 f10
Table 10 Initial MAC}wing modification Exp. mode no. FE mode no. MAC
1 1 79
2 2 95
3 3 96
4 4 94
5 5 90
6 7 95
7 6 84
8 8 97
9 10 95
10 9 79
7 7 94
8 8 99
9 9 93
10 10 95
Table 11 Final MAC}wing modification Exp. mode no. FE mode no. MAC
1 1 79
2 2 94
3 3 96
4 4 94
5 5 94
6 6 89
updating, although the initial error was over 16%. Otherwise the rms error over the first 10 modes is 0:59%. Modes 6 and 7, and separately modes 9 and 10, of the initial model, which are interchanged with experimental modes, are unscrambled in the updated model.
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5. CONCLUSIONS
A finite element model of the GARTEUR SM-AG19 test structure is created and updated. Parameterisation of the joints, where the principal modelling uncertainties are located, is achieved by using eigenvalue sensitivities and mode shapes to obtain a physical understanding of the complete structure. The most important joint is where the wings connect to the fuselage and a good parameterisation of this joint is essential if an updated model with physical meaning is to be obtained. In the present study, it is demonstrated that a good updated model is achieved with meaningful parameters in the range of the first 10 modes. The behaviour of the structure with physical modifications is accurately predicted by the updated model.
ACKNOWLEDGEMENTS
The research described in this article was supported by the Engineering and Physical Sciences Research Council, grant number GR/M08622.
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