Estimation of Rotational Degrees of Freedom by EMA and FEM Mode Shapes

Transcription

1 Estimation of Rotational Degrees of Freedom by EMA and FEM Mode Shapes A. Sestieri, W. D Ambrogio, R. Brincker, A. Skafte, A. Culla Dipartimento di Ingegneria Meccanica e Aerospaziale, Università di Roma La Sapienza Via Eudossiana 18, I Rome, Italy, antonio.culla@uniroma1.it Dipartimento di Ingegneria Industriale e dell Informazione e di Economia, Università dell Aquila Via G. Gronchi, 18 - I-67100, L Aquila (AQ), Italy Department of Engineering, Aarhus University Nordre Ringgade 1 - DK-8000 Aarhus C Denmark ABSTRACT In this paper a new technique is presented to estimate the rotational degrees of freeedom of a flexural structure, using only a limited number of sensors that measure the translational DoFs of the system. A set of flexural mode shapes in a limited number of nodes is obtained by modal testing, while a different set of approximated mode is calculated by a Finite Element Model (FEM) at all the nodes and degrees of freedom of the structure. The technique is based on the classical assumption that the response can be determined by a linear combination of the structure s mode shapes. The structure s mode shapes are approximated by using the local correspondence principle for mode shapes, i.e. by using an optimally selected set of finite element mode shapes as Ritz vectors for the true mode shapes. This allows to obtain the rotational response at unmeasured DoFs. The technique is validated by comparing predicted and experimental results. Keywords: Experimental Mode Shapes, FE Mode Shapes, Rotational DoFs, Expansion 1 INTRODUCTION Rotational DoFs are generally involved in substructuring applications aimed to predict the dynamic behavior of coupled subsystems characterized by flexural behavior. In fact, neglecting rotational DoFs, an important part of compatibility and equilibrium conditions at the joint are lost, and the obtained results become meaningless. Several techniques have been devised to estimate rotational FRFs from measurement of translational FRFs: they range from the use of T-blocks [1] to the use of finite differences [2,3]. Other techniques are aimed at directly obtaining rotational FRFs using particular transducers [4]. Some more literature about experimental devices to obtain rotational FRFs is listed in [5], where a number of expansion techniques are proposed as a valid alternative to either unsatisfactory or expensive rotational FRF measurements. Notwithstanding several years have passed, the situation is more or less the same, and the requirement for efficient expansion techniques capable of estimating FRFs or mode shapes at rotational DoFs is very strong. In this paper a new technique is introduced to estimate unmeasured degrees of freedom of a flexural structure,

2 with particular emphasis to rotational DoFs. The method is based on the Local Correspondence (LC) Principle, presented in [6, 7] to estimate a linear transformation between mode shapes identified by experimental modal analysis (EMA) and modes from a Finite Element model. The technique is validated by comparing experimental results with those determined by both this new technique and SEREP expansion [8]. This comparison seems to be especially appropriate because, under particular conditions, both LC and SEREP provide very similar results. 2 THEORY In modal analysis theory, the eigenvectors, or mode shapes, of an N-DoF system form a basis in the N-dimensional vector space, i.e. any N-dimensional vector can be expressed as a linear combination of the eigenvectors. This is also known as the expansion theorem. For instance, it is possible to consider as a basis the set of mode shapes deriving from an FE model, organized in the modal matrix [Ψ] FE,complete, where any column represents a different eigenvector {ψ} FE,r, r = 1,...,N. Therefore, any N-dimensional vector {ψ} can be expressed as: {ψ} = [Ψ] FE,complete {t} (1) where the N-dimensional vector {t}, containing the linear combination coefficients, is given by: {t} = [Ψ] 1 FE,complete {ψ} (2) because [Ψ] FE,complete is a square N N non-singular matrix (the eigenvectors form a basis in the N-dimensional space). If m < N eigenvectors are considered, Eq. (1) holds approximately and is rewritten: {ψ} = [Ψ] FE {ˆt } (3) where [Ψ] FE is now an incomplete (N m) modal matrix, and {ˆt} is given by: [ˆt ] = [Ψ] + FE {ψ} (4) being [Ψ] + FE = ([Ψ]T FE [Ψ] FE ) 1 [Ψ] T FE the pseudo-inverse of the N m matrix [Ψ] FE, with N > m. Eq. (3) can also be applied to a reduced set of n < N DoFs, e.g. those measured in a modal test. In this case, it is possible to consider an n-dimensional vector {ψ} X,r corresponding to an identified mode shape, and to write: {ψ} X,r = [Ψ]FE,ex {ˆt } (5) where the subscript, ex, indicates that the DoFs selected from the FE mode shapes correspond to those of the identified modes shape, so that [Ψ] FE,ex is an n m matrix. If n m, {ˆt} is given by: {ˆt } = [Ψ] + FE,ex {ψ} X,r (6) being [Ψ] + FE,ex = ([Ψ]T FE,ex [Ψ] FE,ex ) 1 [Ψ] T FE,ex. (Note that [Ψ]+ FE,ex = [Ψ] 1 FE,ex if n = m.)

3 Eq. (5) can be also rewritten as: X,r = [Ψ] FE,ex {ˆt } (7) where {ˆΨ} X,r represents an estimate of {ψ} X,r. (This can be seen as a smoothing technique for experimental mode shapes.) In [6] it is shown that the mode shapes in the incomplete modal matrix should be selected so as their frequencies be the closest to the frequency of the identified mode shape. This is called the Local Correspondence Principle. In [9] a similar issue was raised, concerning the correlation between an experimental modal vector and a subspace spanned by several FE modal vectors. Once the vector of linear combination coefficients {ˆt} has been determined, it can be multiplied by [Ψ] FE, i.e. the FE mode shape matrix at the full set of DoFs, to expand the experimental mode shape at the full set of DoFs, including rotational DoFs whenever they are included in the FE model, as for beam and shell elements: = [Ψ] FE {ˆt } (8) X-full,r 2.1 Smart selection of mode shapes based on the Local Correspondence Principle The accuracy of the estimates provided by Eqs. (7) and (8) depends on several factors such as: the accuracy of the FE model; how many (m) and which modes shapes are retained in the incomplete FE modal matrix; how many (n) and which DoFs are measured in the modal test. Actually, if n = m, i.e. if the number of retained modes is equal to the number of measured DoFs, Eq. (7) provides exactly the experimental mode shape {ψ} X,r. Furthermore, the accuracy of the expansion provided by Eq. (8) is difficult to assess since the involved DoFs are not measured. The problem of finding how many and which FE mode shapes have to be retained to obtain the best fit for any given experimental mode is addressed in [6,7]. The trick is to partition the measured DoFs in two sets, the a-set of active DoFs and the d-set of deleted DoFs, so that a given identified mode is written as: { } {ψ}xa,r {ψ} X,r = (9) {ψ} Xd,r The deleted DoFs are used as reference, to assess the quality of the mode shape expansion. The active DoFs are used in the computation of the expanded mode shape. Therefore Eq. (6) is applied to the a-set, provided that m a, where a is the number of DoFs in the a-set. It is obtained: {ˆt } = [Ψ] + FE,a {ψ} Xa,r (10) The value of {ˆt} provided by Eq. (10) is used to estimate the identified mode shape both at the a-set and at the d-set of DoFs: Xa,r = [Ψ] FE,a {ˆt } Xd,r = [Ψ] FE,d {ˆt } (11)

4 1 1 MAC a 0.5 MAC d a m 0 1 m* a m Figure 1: MAC performed on the active DoFs (left) and deleted DoFs (right) by increasing m from 1 to a. The accuracy of the estimate can be assessed by considering the Modal Assurance Criterion (MAC) [10] among estimated and identified eigenvectors at active and deleted DoFs: ( } ) H 2 ( } ) H 2 {ˆψ {ˆψ MAC a = ( } H {ˆψ Xa,r Xa,r {ψ} Xa,r Xa,r ) ( {ψ} H Xa,r {ψ} Xa,r ) MAC d = ( } H {ˆψ Xd,r Xd,r {ψ} Xd,r Xd,r ) ( {ψ} H Xd,r {ψ} Xd,r ) (12) where the superscript H stands for the conjugate transpose. The procedure leading to the computation of MAC a and MAC d can be repeated by varying the number m of retained modes, e.g. by increasing m from 1 to a. Whilst MAC a monotonically increases with m until it becomes equal to 1 for m = a, MAC d usually does not reaches its maximum for m = a (Fig. 1). Strategies to select how to increase m, that is which mode shape should be added to [Ψ] FE at each step, are currently under investigation. At the same time, strategies to efficiently partition the measured set of DoFs into active and deleted DoFs are being investigated as well. The value m at which MAC d is maximum represents the optimum. The corresponding {ˆt} is then used to expand experimental mode shapes at any desired DoF, eventually including rotational DoFs: = [Ψ] FE {ˆt } (13) X-full,r The procedure can be repeated to expand all the identified modes. At the end, an optimal set of expanded mode shapes is obtained. Using the expanded mode shapes, the FRF matrix at the full set of DoFs can be expressed as: T [H(jω)] = r X-full,r X-full,r m r (ω 2 r ω2 +jη r ω 2 r ) (14) where ω r, η r and m r are the natural frequency, loss factor (having assumed to use structural damping) and modal mass of the r-th mode shape. If identified mode shapes are scaled to unit modal mass, it is m r = 1 for any r. It is now appropriate to compare the results obtained using the proposed smart mode expansion technique based on the Local Correspondence Principle with results obtained using other DoF expansion techniques (e.g Guyan

5 expansion, dynamic expansion, SEREP). Among them, SEREP [8] is specifically considered here since it is also based on a set of FE mode shapes and, under particular conditions, it provides the same results as the mode expansion technique. 2.2 SEREP expansion SEREP is conceived as a DoF expansion procedure where the expanded modes are a linear combination of a selected set of FE modes. Using this procedure, an expanded mode shape is written as: where the transformation matrix [T] is given by: X-full,r = [T]{ψ} Xa,r (15) [T] = [Ψ] FE [Ψ] + FE,a (16) By substituting Eq. (16) into Eq. (15) one obtains: X-full,r = [Ψ] FE [Ψ]+ FE,a {ψ} Xa,r (17) By recalling Eq. (10) it is easy to realize that the expanded mode shape obtained using SEREP is very similar to the one provided by the mode expansion technique. Actually, they are exactly the same if the set of FE mode shapes that is used to obtain the expanded mode shape contains all the initially selected mode shapes, i.e. it is not reduced by optimization. Therefore, SEREP expansion can be seen as a suboptimal solution to the considered expansion problem. 3 APPLICATION The interest here is mainly addressed to estimate the rotational DoFs of an aluminum plate of size 300 mm 400 mm 3mm. The plate is initially modeled by finite elements to calculate its natural frequencies and mode shapes. Subsequently, experiments are performed to determine the frequency response functions used to identify natural frequencies and mode shapes. 3.1 FEM reference model The finite element model consists of 63 nodes and 48 shell elements. Each element is 50 mm 50 mm, has 4 nodes and 6 degrees of freedom per node. Fig. 2 shows the finite element mesh and the considered reference frame. The nodes considered for further comparison with experimental results are those shown within the red rectangle in Fig. 2. The results obtained by this analysis, to be used for expansion, are the first nine elastic modes in the frequency range Hz. The first six mode shapes are shown in Fig. 3. The mode shapes are calculated on the full set of 210 degrees of freedom, including rotational DoFs. 3.2 Experimental results The experiments are performed on the plate suspended by a soft rubber band to simulate free boundary conditions (Fig. 4a). The plate is excited using the hammer impact technique, and the drive point is shown in Fig. 4a. The

6 Figure 2: Finite element mesh of the plate.!! 1 th mode! 2 nd mode!!! 3 th mode! 4 th mode!!! 5 th mode! 6 th mode! Figure 3: The first six elastic modes of the FE model.

7 !!a) Figure 4: Experimental setup: a) the suspended plate and the drive point, b) accelerometers placed to calculate the rotational accelerations. b)! 10 3 H Figure 5: Comparison between measured ( ) and fitted ( ) FRF H 24z 11z. acceleration along z direction is measured at all the 35 nodes. To estimate the rotational accelerations at node 24 (see Fig. 4b), three accelerometers along x direction and three along y direction are placed at a relative distance of 20 mm. These measurements allow to calculate by central finite difference the derivative of the flexural acceleration [2, 3], which provide an estimate of rotational accelerations around x and y axes. Accelerations and force are used to determine FRFs (inertances), from which 8 natural frequencies, structural dampings, and mode shapes are identified by a curve fitting technique. Fig. 5 shows the comparison between measured and fitted FRF at DoF 24z.

8 MAC LC MAC SEREP Figure 6: MAC between the identified mode shapes and the expanded mode shapes: LC vs SEREP. 3.3 Numerical Results The smart mode expansion technique based on the Local Correspondence Principle (LC) is carried out by taking into account a set of 34 active DoFs over 35 measured DoFs every time: the deleted DoF is roved over the whole set. The results of LC are compared with the results of SEREP technique and with the measured FRFs. In Fig. 6 the MAC between the identified mode shapes and the expanded mode shapes using the LC technique (left) and SEREP (right) is shown. It can be noticed that the LC technique provides better results, especially for modes 3 and 4, and also for mode 7. This demonstrates the effectiveness of the smart mode expansion technique compared to SEREP. In Fig. 7 the experimental FRF H 24z 11z is compared with the corresponding FRF obtained from the FE model. In Fig. 8 the same experimental FRF is compared with the one given by from expression (14) using expanded mode shapes obtained using both LC and SEREP. No particular difference can be noticed in the synthesized FRFs. Both match the experimental FRF fairly well, and definitely better than the FRF obtained from the FE model. In Fig. 9 the experimentally derived FRF H 24ϑx 11z is compared with the corresponding FRF obtained from the FE model. It should be noticed that the experimentally derived rotational FRF shows a certain amount of scatter probably due to the procedure used to estimate such rotational FRFs (finite differences). In Fig. 10 the same experimental FRF is compared with the one given by expression (14) using expanded mode shapes obtained using both LC and SEREP. Now, some difference between LC synthesized and SEREP synthesized FRF is noticed, especially at one antiresonance. Both match the experimental FRF better than the FRF obtained from the FE model, especially in the frequency range up to 200 Hz. In Fig. 11 the experimentally derived FRF H 24ϑy 11z is compared with the corresponding FRF obtained from the FE model. Again, some amount of scatter is noticed. In Fig. 12 the same experimental FRF is compared with the one obtained from expression (14) using expanded mode shapes obtained using both LC and SEREP. Now, some difference between LC synthesized and SEREP synthesized FRF is noticed, especially at one antiresonance. Both match the experimental FRF better than the FRF obtained from the FE model, except at the resonance mimediately above 100 Hz.

9 10 3 H z 10 8 Figure 7: H 24z 11z : measured ( ) vs. FEM ( ) H z Figure 8: H 24z 11z : measured ( ) vs. LC ( ) and SEREP ( ). 4 CONCLUSIONS This work is an application of the Local Correspondence (LC) Principle developed to estimate unmeasured DoFs in a structure using only a limited amount of sensors. Specifically, here the technique is used to determine rotational mode shapes and related FRFs. The method is based on the classical assumption that the response can be determined by a linear combination of the structure s mode shapes. The FE mode shapes are then combined using a linear transformation to match the experimentally obtained mode shapes, thus creating a new set of modes having a large amount of DoFs.

10 H Rx 10 8 Figure 9: H 24ϑx 11z: measured ( ) vs. FEM ( ) H Rx 10 8 Figure 10: H 24ϑx 11z: measured ( ) vs. LC ( ) and SEREP ( ). The method is applied on a simple rectangular plate. A set of flexural mode shapes in a limited number of nodes is obtained by modal testing, while a different set of modes is calculated by a Finite Element Model (FEM) at all the nodes and DOFs of the structure. The results provide by the LC technique are compared with experimental results and with results of SEREP expansion. Although both LC and SEREP provide similar results, it is still difficult to asses how well they can estimate the rotational DoFs. In fact, the rotational DoFs experimentally presented here are not directly measured but rather reconstructed by a finite different scheme of translational DoFs along close measurement points. On the other side, the FE Model used here is not sufficiently accurate to provide reliable values of rotation for comparison and an updated FE model could be used. In any case, a challenging test would be to use the rotational FRFs in substruturing applications.

11 H Ry Figure 11: H 24ϑy 11z: measured ( ) vs. FEM ( ) H Ry 10 8 Figure 12: H 24ϑy 11z: measured ( ) vs. LC ( ) and SEREP ( ). ACKNOWLEDGEMENTS This research is supported by grants from University of Rome La Sapienza and University of L Aquila.

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