Selection and Updating of Parameters for the GARTEUR SM-AG19 Testbed C. Mares, J. E. Mottershead Department of Mechanical Engineering, The University of Liverpool, UK e-mail :
[email protected],
[email protected] M.I. Friswell Department of Mechanical Engineering, The University of Wales Swansea, UK e-mail :
[email protected]
Abstract A finite element model of the GARTEUR SM-AG19 test structure is updated. A physical explanation is given on how the parameters for updating are selected. The sensitivity method is applied using six parameters and nine measured natural frequencies. Excellent convergenge of the finite element model onto the measured data is achieved.
1.
Introduction
The Structures and Materials Action Group (SMAG19) of the Group for Aeronautical Research and Technology in EURope (GARTEUR) was initiated in 1995 with the purpose of comparing a number of current measurement and identification techniques applied to a common structure [1-3]. The testbed was designed and manufactured by ONERA and has now been accepted as a benchmark structure by the working group on Finite Element Model Updating of the COST Action F3 on Strucural Dynamics [4,5]. The testbed represents a typical aircraft design with fuselage, wings and tail. Realistic damping levels are achieved by the application of a viscoelastic tape bonded to the upper surface of the wings and covered by a thin aluminium constraining layer. The overall dimensions of the testbed are: 2000 mm wingspan and 1500 mm fuselage. The experimental data-base (supplied by the University of Manchester and DLR) consists of 24 frequency response functions for an excitation at the right wing tip ranging from 0 80 Hz, 2 24 frequency response functions for excitations at the right and left wing tips from 4 65 Hz, and 14 normal modes covering a frequency range from 6:38 151:32 Hz. A finite element model containing 157 elements (beams rigid elements and lumped masses) is constructed for updating which is applied to six parameters using nine measured natural frequencies by the
sensitivity method. The physical reasoning underlying the choice of the parameters is explained. Excellent convergence of the finite element predictions onto the measured data is achieved.
2.
Modeling aspects
The test structure [1] consists of aluminium beams of rectangular cross section. The modelling uncertainty, which we aim to reduce by model updating, is mainly 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 wing 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 sections remaining plane. The counter arguement, for beam-like structures, is that a fine mesh of plate ele-
wing offset wing
wing fuselage connection
fuselage offset
fuselage
Figure 2: Wing-fuselage modelling aspects. constraining layer ISD112
offset
Figure 1: GARTEUR test - finite element model.
offset
wing
Figure 3: Wing modelling aspects. ments 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 5 lumped masses. The complete model, shown in Figure 1, has 534 degrees-of-freedom. An equivalent beam-model is used to represent the joint between the fuselage and wings as shown in Figure 2. 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. Beam offsets, shown with lengths exaggerated in Figure 2 are used. The damping and contraining layers on the wings are each represented by beams with nodes offset to the neutral axis of the main wing-beam. Again the offsets are shown with exaggerated lengths in Figure 3 for clarity. Rigid elements are used to connect the winglets at the wing-tips as shown in Figure 4. Offsets, as in Figure 5, 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 fin/tail-plane joint represents the mass of additional parts 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 as can be seen in Figure 1.
winglet rigid element (RBAR) wing
Figure 4: Wing modelling aspects. tail plane
joint mass
11 00 00 11 offsets
tail fin
tail offset
1 0 0 1 0 1 0 1 0 1
Figure 5: Tail modelling aspects.
m
11 00 00 11
generic element
bending offset a b
Figure 6: Tail parameters.
3.
Model updating
The choice of updating parameters is an important aspect of the finite element model updating process [6,7]. The sensitive parameters chosen must also be justified by engineering understanding of the test structure [8-12]. The flexural and torsional rigidities (E Iy and GIt ) 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. The beam-offsets, (ab )w , used to stiffen the wings at the joint with the fuselage represent an unrealistic rigid joint and must be adjusted. They are sensitive for the same reason that the wing rigidities are sensitive. At the seventh mode in-plane wing bending occurs with the two wing-tips in antiphase. This puts the equivalent beam separating the wings and fuselage into torsion and it is sensitive for that mode. The torsional eigenvalue, (q2 )con , of the stiffness matrix (equivalently the torsional rigidity) of the equivalent beam is updated. Two updating parameters at the tail are used. These are the offset, (ab )t , at the tail-fin/fuselage joint and the first in-plane bending eigenvalue ,(q1 )t , of the substructure stiffness matrix for the three elements at the tail-plane/fin joint. Both are shown in Figure 6 and are sensitive parameters for the fifth mode. The sensitivities are given in Table 1. The updating is carried out using the first nine natural frequencies. Modes 10 14 are used only for checking the capacity of the updated model to predict natural frequencies outside the frequency range.
4.
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 from the initial model are all positive, the largest being over 9 % at the seventh mode, probably due to considerable over-estimation of the joint stiffnesses. Modes 3; 4 and 5 are close modes which produce significant cross coupling with their finite element counterparts. Modes nine and ten 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. The convergence of the six updating parameters is shown in Figure 7 to be complete after about three iterations and their final values are given in Table 5. All the joint stiffnesses are reduced (< 1) as is the torsional rigidity of the wings. The flexural rigidity (EIy )w is increased slightly. The greatest stiffness reduction occurs in the wing-offset over the fuselage (ab )w . This indicates that the stiffening effect of the fuselage is small. The torsional eigenvalue of the beam between the wing and fuselage is also significantly reduced, but this is an equivalent parameter which we accepted at the time of mesh-building would not be physically meaningful. The first nine frequencies (in the updating frequency range) are all corrected to within an error of 1:05%. Small errors of the order of 3% persist outside the frequency range. It may be observed from the final MAC values in Table 4 that the ’swapped’ modes 9 and 10 have corrected themselves. This was done without deliberate pairing - the first 9 measured and the first 9 finite element modes only were used in the updating exercise. Fairly significant off-diagonal terms remain for modes 3 and 4. This might be improved by using mode-shape sensitivities whereas only eigenvalue sensitivities were applied in the exercise. However, since the modes are so very close it is quite possible that different combinations of these two might be obtained should the modal test which produced this data ever be repeated.
5.
Conclusions
The results of an updating exercise carried out on the GARTEUR SM-AG19 testbed are presented. The structure is modelled using beam elements and the first 9 natural frequencies. A set of 6 parameters
Mode
(E Iy )w
(GIt )w
(ab )w
(q2 )con
(ab )t
(q1 )t
f1
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 1.20e+03 1.24e+05 6.37e+05 5.65e+05
8.92e-01 1.90e+01 2.28e+04 3.97e+04 1.71e+04 7.74e+02 6.31e+02 4.99e+02 1.91e+01 6.03e+01 8.29e+00 3.11e+01 8.51e+01 9.61e+01
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 6.90e+03 1.37e+04 1.32e+05 1.57e+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 8.18e+03 2.73e+03 0.0 3.77e+02
0.0 1.69e+03 7.55e+03 2.85e-01 1.01e+04 5.51e-01 3.08e+02 3.81e+01 1.11e+02 6.71e+03 6.05e+04 1.42e+05 8.51e-02 7.80e+04
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 1.26e+05 1.76e+05 0.0 6.36e+04
f2 f3 f4 f5 f6 f7 f8 f9 f10 f11 f12 f13 f14
Table 1: Table of sensitivities for the updating parameters: wing bending stiffness; (GIt )w wing torsional stiffness; (ab )w - the wing bending offset; (q2 )con torsional eigenvalue of the connection beam (ab )t - tail bending offset. (q1 )t - the first eigenvalue of the tail generic element; (EIy )w
Mode No f1 f2 f3 f4 f5 f6 f7 f8 f9 f10 f11 f12 f13 f14
Exper Model 6.38 16.10 33.10 33.53 35.65 48.38 49.43 55.08 63.04 66.50 102.90 130.54 141.38 151.32
FeModel Initial 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
FeModel Final 6.32 16.27 33.36 33.54 35.37 47.96 49.65 55.37 63.29 64.39 102.48 129.41 137.66 148.00
Error Initial (%) 3.53 5.25 8.70 8.72 4.40 5.17 9.12 7.66 2.29 0.66 2.73 1.95 3.50 4.86
Error Final (%) -1.00 1.05 0.79 0.02 -0.80 -0.87 0.45 0.54 0.40 -3.17 -0.41 -0.86 -2.63 -2.19
Table 2: Measured natural fequencies and finite element predictions (Hz).
1 2 3 4 5 6 7 8 9 10 11 12 13 14
1 1.00 0.00 0.00 0.01 0.00 0.27 0.00 0.01 0.00 0.00 0.00 0.00 0.40 0.00
2 0.00 0.99 0.01 0.00 0.00 0.00 0.00 0.00 0.00 0.47 0.18 0.05 0.00 0.18
3 0.00 0.00 0.70 0.21 0.37 0.02 0.02 0.01 0.00 0.01 0.01 0.00 0.00 0.01
4 0.01 0.00 0.13 0.86 0.05 0.06 0.00 0.05 0.00 0.00 0.00 0.00 0.01 0.00
5 0.00 0.01 0.00 0.00 0.81 0.00 0.19 0.00 0.00 0.04 0.16 0.20 0.00 0.00
6 0.21 0.00 0.00 0.05 0.00 0.99 0.00 0.01 0.00 0.00 0.00 0.00 0.12 0.00
7 0.00 0.00 0.01 0.00 0.15 0.01 0.95 0.00 0.06 0.00 0.09 0.20 0.00 0.00
8 0.00 0.00 0.00 0.01 0.00 0.01 0.00 0.99 0.00 0.00 0.00 0.00 0.00 0.00
9 0.00 0.49 0.01 0.00 0.04 0.00 0.03 0.00 0.11 0.80 0.25 0.09 0.00 0.41
10 0.00 0.00 0.00 0.00 0.00 0.00 0.15 0.00 0.98 0.06 0.01 0.00 0.00 0.00
11 0.00 0.20 0.08 0.00 0.09 0.00 0.10 0.00 0.00 0.37 1.00 0.55 0.00 0.15
12 0.00 0.06 0.07 0.00 0.11 0.00 0.20 0.00 0.00 0.12 0.62 0.99 0.00 0.00
13 0.29 0.00 0.00 0.00 0.00 0.09 0.00 0.00 0.00 0.01 0.02 0.01 0.94 0.01
14 0.01 0.17 0.03 0.00 0.01 0.00 0.01 0.00 0.00 0.47 0.21 0.01 0.02 0.92
9 0.00 0.47 0.02 0.00 0.05 0.00 0.01 0.00 0.97 0.03 0.27 0.04 0.00 0.45
10 0.00 0.00 0.00 0.00 0.00 0.00 0.07 0.00 0.12 1.00 0.01 0.00 0.00 0.00
11 0.00 0.22 0.02 0.00 0.16 0.00 0.08 0.00 0.26 0.02 0.99 0.42 0.00 0.16
12 0.00 0.07 0.01 0.00 0.19 0.00 0.18 0.00 0.13 0.00 0.66 0.96 0.00 0.00
13 0.29 0.00 0.00 0.00 0.00 0.07 0.00 0.00 0.01 0.00 0.02 0.01 0.95 0.01
14 0.01 0.18 0.01 0.00 0.02 0.00 0.01 0.00 0.37 0.00 0.23 0.00 0.02 0.95
Table 3: Initial MAC.
1 2 3 4 5 6 7 8 9 10 11 12 13 14
1 1.00 0.00 0.00 0.01 0.00 0.27 0.00 0.00 0.00 0.00 0.00 0.00 0.35 0.00
2 0.00 1.00 0.00 0.00 0.00 0.00 0.01 0.00 0.42 0.01 0.19 0.02 0.00 0.17
3 0.00 0.00 0.79 0.21 0.10 0.02 0.03 0.01 0.01 0.00 0.01 0.00 0.00 0.01
4 0.01 0.00 0.14 0.86 0.01 0.06 0.00 0.04 0.00 0.00 0.00 0.00 0.01 0.00
5 0.00 0.00 0.08 0.00 0.99 0.00 0.24 0.00 0.05 0.00 0.17 0.20 0.00 0.00
6 0.20 0.00 0.00 0.05 0.00 0.99 0.00 0.00 0.00 0.00 0.00 0.00 0.14 0.00
7 0.00 0.00 0.00 0.00 0.19 0.01 0.97 0.00 0.03 0.06 0.12 0.19 0.00 0.00
8 0.00 0.00 0.00 0.01 0.00 0.01 0.00 0.99 0.00 0.00 0.00 0.00 0.00 0.00
Table 4: Final MAC.
(E Iy )w
(GIt )w
(ab )w
(q2 )con
(ab )t
(q1 )t
1.020
0.822
0.094
0.489
0.736
0.984
Table 5: Updated parameters.
1.4 wing EIy wing GIt wing bending offset q2 connection beam tail bending offset q1 tail
1.2
5.
1
0.8
6. 0.6
7.
0.4
0.2
8. 0
0
1
2
3
4
5 Iterations
6
7
8
9
10
Figure 7: Parameters variation. 9. which can be related to the dynamical behaviour of the tested modes are updated, reducing the initial maximum error of 9% to 1%. 10.
Acknowledgements The research reported in this article is supported by EPSRC grant GR/M08622. Dr. Friswell gratefully acknowledges the support of the EPSRC through the award of an Advanced Fellowship.
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Engineering, Proc. of the Second International Conference, Swansea, UK, pp. 48-62, (1999). S. Keye, Prediction of modal and frequency response data from a validated finite element model, Identification in Engineering, Proc. of the Second International Conference, Swansea, UK, pp. 122-134, (1999). J. E. Mottershead, M. I. Friswell, Model updating in structural dynamics: a survey, J. of Sound and Vibration, 162(2), pp.347-375, (1993). M. I. Friswell, J. E. Mottershead, Finite Element Model Updating in Structural Dynamics, Kluwer Academic Press, Dordrecht, (1995). J. E. Mottershead, M. I. Friswell, G. H. T. Ng, J. A. Brandon, Geometric parameters for finite element updating of joints and constraints, Mechanical Systems and Signal Processing, Vol. 10(2), pp. 171-182, 1996. H. Ahmadian, J. E. Mottershead, M. I. Friswell, Parametrisation and identification of of a rubber-seal, Proc. of the 15th International Modal Analysis Conference, IMAC XV, Orlando, USA, (1997). J. E. Mottershead, C. Mares, M. I. Friswell, S. James, Selection and updating of parameters for an aluminium space-frame model, Mechanical Systems and Signal Processing, in press. G. M. Gladwell, H. Ahmadian, Generic element matrices for finite element model updating, Mechanical Systems and Signal Processing, Vol. 9, pp. 601-614, (1996). H. Ahmadian, G. M. L. Gladwell, F. Ismail, Parameter selction strategies in finite element updating, Trans. ASME, Vibration and Accoustics, Vol. 119(1), pp. 37-45, (1997).
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