IOMAC'15 6 th International Operational Modal Analysis Conference

Transcription

1 IOMAC'15 6 th International Operational Modal Analysis Conference 2015 May12-14 Gijón - Spain COMPARISON OF DIFFERENT TECHNIQUES TO SCALE MODE SHAPES IN OPERATIONAL MODAL ANALYSIS. Luis Borja Peral Mtnez 1, F. Pelayo 1, M. García 1, Manuel L. Aenlle 1*, G. Ismael 1. 1 Department of Construction and Manufacturing Engineering, University of Oviedo, Gijón, Spain, * aenlle@uniovi.es ABSTRACT Operational modal analysis (OMA) is nowadays widely used to identify the modal parameters in civil, mechanical and aerospace structures. In OMA the forces are not measured, i.e., only the responses of the structure are used in the modal identification and therefore the mode shapes cannot be mass normalized. In the last years, the mass change method (MCM) has been successfully applied to determine the modal masses in small structures and lab-scaled models. However, the practical application in medium and large structures presents some difficulties because several masses of large magnitude have to be attached. In this paper, a new technique to determine the modal masses in OMA, which consists of combining the mass normalized mode shapes of a finite element model and the experimental mode shapes identified by operational modal analysis is applied to a steel lab-scaled model. The results are compared with those provided by the mass change method. It is proven that if a reasonable correlation exists between the experimental and the numerical model, the results provided by this method are of the same order of accuracy as those provided by the mass change method. Keywords: Operational Modal Analysis, Modal scaling, Mode Shapes, Numerical Model. 1 INTRODUCTION In operational modal analysis the mode shapes cannot be mass normalized because the input information is unavailable. One way to generate the information needed to properly scale the mode shapes consists of modifying the dynamic behavior of the structure with known perturbations. The mass change method (MCM) [1, 2, 3, 4] consists of performing operational modal analysis on the original structure and then repeat the modal testing on the same structure but modified with masses attached to some points where the mode shapes are known. The modal masses are then estimated using the modal parameters of both the unperturbed and the perturbed system and the known modifications. The uncertainty with the mass change method depends mainly on the frequency shift between the unperturbed and the perturbed systems [5]. This means that in order to minimize the

2 uncertainty in the modal masses, we need to perform a good modal-parameter identification and use rather large mass changes to allow for a reasonably frequency shift. Moreover, a number of masses equal to or greater than the number of peaks and valleys of the mode shape is recommended for each mode [6]. Thus, the practical application of the mass change method in medium and large structures can present some drawbacks since a cluster of large magnitude masses are needed to modify the dynamic behavior of the structure. In this paper, a new technique to determine the modal masses in OMA [7], which consists of combining the mass normalized mode shapes of a finite element (FE) model and the experimental mode shapes identified by operational modal analysis, is applied to a steel labscaled model. This technique is based on the assumption that the differences between the experimental and the finite element mass matrices are small. The results are compared with those provided by the mass change method. The results provided by this method are of the same order of accuracy as those provided by the mass change method if a reasonable correlation exists between the experimental and the numerical model. 2 THEORY If we consider an analytical or finite element model of a structure with no damping, the eigenvalue problem is given by: M FE φ FE ω 2 FE = K FE φ FE (1) where subscript FE indicates the finite element model and φ FE and ω FE are the FE mode shape and the FE natural frequency, respectively. If the experimental model is considered as a dynamic modification of the analytical one [8] and the modification is given by the mass M and stiffness K matrices, the experimental stiffness K X and mass M X matrices, where subscript x indicates experimental, are given by, M X = M FE + M (2) and K X = K FE + K (3) respectively. The eigenvalue equation of the experimental model is now expressed as: M X φ X ω X 2 = K X φ X (4) where φ x and ω X are the experimental mode shape and the experimental natural frequency, respectively. According to the structural modification theory (SDM), the unscaled mode shapes of the experimental model, ψ X, can be expressed as [8]: ψ X = φ FE R (5) Using the mass normalized FE mode shape matrix φ FE or by: ψ X = ψ FE B (6) Using the unscaled FE mode shape matrix ψ FE. The experimental un-scaled ψ x and the corresponding scaled or mass normalized φ x mode-shape matrices are related by the expression [3]: φ x = ψ x α x (7)

3 Where α x is a diagonal matrix containing the experimental scaling factors. The modal mass matrix m x is related to the matrix of scaling factors by: α x = m x 1/2 (8) 3 MASS NORMALIZATION USING A FE MODEL The experimental modal mass matrix m X is given by: m X = ψ X T M X ψ X (9) If Eq. (2) is substituted in Eq.(9), it results in: m X = ψ X T M FE ψ X + ψ X T ΔM ψ X (10) From Eq. (10) it is derived that if the matrix ΔM is small, i.e., a reasonably good estimation of the mass matrix can be achieved, we can take the approximation M X M FE, i.e.: m X ψ X T M FE ψ X (11) where the symbol ^ indicates approximation. If Eq. (5) is substituted in Eq. (11), it becomes: m X = R T R (12) Due to the fact that the number of experimental modes N X and the number of measured coordinates a (sometimes referred to as the master or active) is limited, only an approximation of matrix R can be obtained with Eq. (5), i.e.: ψ a X = φ afe R (13) Where the subindex a indicates measured DOF s. The matrix R can be found by a classical least square (LS) solution as: R = φ + a ψ FE ax (14) Which is a square matrix size N X N X In order to prevent overfitting, the number of mode shapes in the subspace (matrix φ FE ) must be smaller than a (active DOF s). On the other hand, if the number of mode shapes in the FE subspace is too small for instance if only a single mode shape is included then the fit is also bad. In this paper, the matrix R has been estimated using the local correspondence (LC) principle [9] which is formulated as follows: for any perturbation of the mass or stiffness matrix, any perturbed mode shape can be expressed approximately as a linear combination of a limited set of unperturbed mode shapes. The limited set of mode shapes only need to consist of the corresponding unperturbed mode shape and a limited number of unperturbed mode shapes around (in terms of frequency) the unperturbed mode. This defines the sequence of FE mode shapes to be included in the FE subspace. 4 THE MASS CHANGE METHOD The mass change method is an experimental technique used to determine the modal masses of dynamic systems. A closed form expression for the j-th modal mass was derived [3] from the structural dynamic modification theory and it is expressed as:

4 1 m xj = (ω 2 2 x0j ωxii) Bji ψ T x0j (ω xii 2 M ) ψx Ii (15) where ω 0 and ω I are the natural frequencies of the unperturbed and the perturbed structures, respectively, M is the mass change matrix and B ji is the element in the i-th row and j-th column of the matrix B, which is estimated by a classical least square (LS) solution as: B = ψ + a ψ X0 axi (16) Eq. (15) coincides with that proposed by Bernal [10]. Moreover, the expressions presented by Brincker and Anderssen [2]: 1 m xj = 2 2 (ω 0j ωij) ψ T 0j (ω 2 xij M ) ψ 0j (17) or Aenlle et al. [3]: 1 m xj = 2 2 (ω 0j ωij) ψ T 0j (ω 2 xij M ) ψ Ij (18) can also be derived from Eq. (15) after some simplifications [3]. 5 NUMERICAL AND EXPERIMENTAL RESULTS 5.1 DESCRIPTION OF THE STRUCTURE A scaled model of a footbridge constructed with steel beams and plates was used in the experiments. Square hollow sections 30 x 30 x 3 mm were used in beams and columns and a plate 1800 x 250 x 5 mm was used to model the deck. Moreover, a small plate 160 x 60 x 3 mm was also welded at the bottom of the columns (See Figures 1 and 2). The total mass of the structure was 37,6 Kg. Figure 1. Model of the footbridge structure. Figure 2. Structure on the testing frame. 5.2 NUMERICAL MODEL A finite element model (FEM) of the structure was assembled in ABAQUS (see Figure 3). The beams and the columns were modelled with 3 node quadratic beam elements whereas the deck was modelled with 8 node quadratic shell elements (see Figure 4). The material was considered linear-elastic with the following mechanical properties: E= MPa and a Poisson ration ν = 0.3. The natural frequencies corresponding to the first seven modes are presented in Table 1 whereas the mode shapes are shown in Table 2.

5 Figure 3. ABAQUS model. Figure 4. FE mesh. 5.3 TEST SET UP The operational model testing was carried out measuring the responses in 50 degrees of freedom (DOFs) and using 20 accelerometers (sensitivity: 100 mv/g). Three data sets were used to cover all the measurements DOFs. The locations of the accelerometers in each data set are shown in figures 5, 6 and 7, respectively. Blue accelerometers indicate references sensors. The responses were recorded during approximately 10 minutes using a NI cdaq-9188 acquisition system with NI9234 acceleration modules. A sampling frequency of 2132 Hz was used in all the experiments. The structure was artificially excited applying repetitive hits with an impact hammer (figure 8), being the excitation random in time and space. Figure 5. Data Set 1. Figure 6. Data Set 2. Figure 7. Data Set 3. Figure 8. Impact hammer.

6 The modal parameters corresponding to the first seven modes were estimated by the Frequency Domain Decomposition Technique (FDD) [11] using the ARTEMIS software. The singular value decomposition of the responses is shown in Figure 9. The natural frequencies are presented in Table 1 and the mode shapes in Table 2. From Table 1 is inferred that the discrepancies between the experimental and the numerical natural frequencies are less than 10% for all the modes considered in the investigation, the maximum error of 9.01 % corresponding to the second mode. With respect to the mode shapes, the MAC between the experimental and the numerical mode shapes is shown in Table 3. A good correlation has been achieved in modes 1, 2, 3, 4 and 6 but modes 5 and 7 show a relatively low correlation. Figure 9. Unperturbed singular value decomposition. Table 1. Natural frequencies of the structure. MODE OMA Error FEM Unperturbed Perturbed FE-OMA [Hz] FDD [Hz] FDD [Hz] [%]

7 Table 2. Mode shapes of the structure. Experimental Analysis Mode 1 Finite Element Model Mode 2 Mode 3 Mode 4

8 Mode 5 Mode 6 Mode 7

9 Table 3. MAC between experimental and FEM mode shapes. MODE In order to apply the MCM, the dynamic behavior of the structure was modified attaching masses (table 4) in 26 points coincident with the location of the sensors (see Figure 2). The total mass change was Kg, which represents a 7.11 % of the total mass of the structure. The singular value decomposition of the responses measured on the perturbed structure is shown in Figure 10 and the estimated natural frequencies are presented in Table 1. Table 4. Examples of the masses attached to the structure. Weight (g) Number of masses attached Figure 3. Perturbed singular value decomposition.

10 5.4 MODAL MASSES The modal masses corresponding to mode shapes normalized with the largest component of each mode shape equal to the unity, obtained from the FE model and those estimated with the mass change method (Eqs. (17) and (18)) as well as the obtained combining the experimental and the numerical mode shapes (Eq. 12), are presented in Table 5. Table 5. Modal masses. MODE OMA FE m X = R T R Mass change method model Eq. (12) Eq. (17) Eq. (18) The errors obtained with the different techniques are also shown in Table 6. It can be observed that the discrepancies between the modal masses estimated with MCM estimated with Eq. (17) and those estimated with Eq. (12) are less than 10%, except for the 5 th mode where the error is 13.78%. This error in the 5 th mode is explained by the low MAC value achieved in the 5 th mode (Table 3). The results presented in Tables 5 and 6 demonstrate that the modal masses estimated with Eq. (12) are of the same order of accuracy as those provided by the mass change method if the finite element model is an accurate representation of the real structure. Table 6. Errors in the Modal masses. MODE Error Eq.(18) - Eq.(12) [%] Error Eq.(18) FEM [%] Error Eq.(12) FEM [%]

11 6 CONCLUSIONS A methodology to scale mode shapes in operational modal analysis, combining the experimental mode shapes identified by operational modal analysis and the mass normalized mode shapes obtained with a finite element model has been validated by experimental tests carried out on a scaled model of a footbridge. The equations needed to apply the methodology are very simple and easy to use. The technique can be applied to any kind of structure but it can be used advantageously in medium and large structures where the mass change method has important limitation because several masses of large magnitude are needed to modify the dynamic behavior of the structure. It has been demonstrated that this technique provides the same level of precision as the mass change method if the FE model is an accurate representation of the real structure. ACKNOWLEDGMENTS The economic support given by the Spanish Ministry of Science and Innovation through the project BIA C02-01 is gratefully appreciated. REFERENCES [1] Parloo, E., Verboven, P., Guillaume, P., Van Overmeire M., (2002). Sensitivity- Based Operational Mode Shape Normalization. Mechanical Systems and Signal Processing, [2] Brincker, R., Andersen, P., (2003). A Way of Getting Scaled Mode Shapes in Output Only Modal Analysis, In: Proceedings of the International Modal Analysis Conference (IMAC) XXI, Orlando, USA, paper 141. [3] López Aenlle, M., Brincker, R., Pelayo, F., Fernández-Canteli, A. (2012) On exact and approximated formulations for scaling mode shapes in operational modal analysis by mass and stiffness change. Journal of Sound and Vibration, 331, [4] Bernal, D. (2004). Modal Scaling from Known Mass Perturbations. Journal of Engineering Mechanics, 130, [5] Pelayo, F., Reynolds P. and Aenlle, M.L. (2010). Experimental evaluation of mass change approaches for scaling factors estimation. In: Proc. of the IMAC XXVIII. [6] López-Aenlle, M., Pelayo, F., Brincker, R. and Fernández-Canteli, A. (2010). Scaling Factor Estimation Using An Optimized Mass Change Strategy, Mechanical Systems and Signal Processing, 24, [7] Aenlle, M. L., Brincker, R. (2013) Modal Scaling in Operational Modal Analysis using a Finite Element Model, International Journal of Mechanical Sciences, 76, [8] Sestieri, A. (2000) Structural Dynamic Modification. Academy Proceedings in Engineering Sciences, (Sadhana), 3, [9] Brincker, R., Skafte, A., Aenlle, M.L., Sestieri, A., D Ambrogio, W. and Cantili., A.F. (2014) A local correspondence principle for mode shapes in structural dynamics. Mechanical Systems and Signal Processing. 45, [10] Bernal, D. (2011) A Receptance Based Formulation for Modal Scaling using Mass Perturbations. Mechanical Systems and Signal Processing, 25, [11] Brincker, R., Zhang, L.-M and Anderson, P. (2000) Modal identification from ambient response using frequency domain decomposition. In: Proc. of the 18th IMAC.