Modal Masses Estimation in OMA by a Consecutive Mass Change Method.

Transcription

1 Modal Masses Estiation in OMA by a Consecutive Mass Change Method. F. Pelayo University of Oviedo, Departent of Construction and Manufacturing Engineering, Gijón, Spain M. López-Aenlle University of Oviedo, Departent of Construction and Manufacturing Engineering, Gijón, Spain P. Reynolds. University of Sheffield, Departent of Civil and Structural Engineering, Sheffield. UK. A. Fernández-Canteli University of Oviedo, Departent of Construction and Manufacturing Engineering, Gijón, Spain ABSTRACT: Mode shapes in operational odal analysis (OMA) cannot be ass noralized since input forces are unknown, so that, if ass noralized ode shapes are needed the odal asses or scaling factors ust be estiated. A ethod for obtaining the odal asses in OMA is to odify the dynaic behavior of the structure by changing the ass or the stiffness and then using the odal paraeters of both the original and odified structure. In recent years, ass odification has been used and several expressions for the known ass change ethod have been developed, where a well distributed array of asses is usually needed for obtaining good accuracy. In this work a new ass change approach based on perforing several individual ass changes is presented and validated by experiental tests. This ethod requires only a sall nuber of asses that are located at different points in each individual experient and then the results of the individual test are cobined to estiate the scaling factors. 1 INTRODUCTION Operational odal analysis (OMA) ake uses of the natural or operational dynaic loads applied to the structure. This is an iportant advantage in large structures (Magalhaes et al. 2008), where the use of artificial excitation devices ay be expensive or ipractical (Brincker et al. 2003). Another advantage is that the odal tests can be perfored with the structure in its noral operation, so that vehicles and echanical systes can be tested in their in service condition (Parloo et al. 2003). When easureents are perfored in the lab, artificial devices ust be used in a proper way in order to reproduce a ulti input rando loading (Fernández et al. 2009). A drawback in output only analysis is that the ode shapes can not be ass noralized since input loads are unknown, so unity noralization is generally used (Brincker et al. 2003). In those applications where the FRF atrix of the structure is required, the scaling factors (or odal asses) ust be estiated. A ethod to estiate the scaling factors is to odify the dynaic behaviour of the structure by changing the stiffness, the ass (Aenlle et al. 2005, Parloo et al. 2001, Parloo et al. 2003) or both (Khatibi et al. 2009) and then perfor operational odal testing and analysis on both the original and the odified structure. The scaling factors can then be estiated using the odal paraeters of both structures. A siple and easily controlled ethod of odifying the dynaic behaviour of a structure is to change its ass. In recent years, soe different approaches (Aenlle et al. 2005b, Bernal 2004) and strategies (Aenlle et al. 2009) have been proposed based on the ass change ethod, which is relatively easy to use in sall structures where a large nuber of sall asses can easily be attached to the structure. However it has iportant liitations in ediu and large structures where several asses of large agnitude are needed. Moreover, if several odes are going to be scaled siultaneously, ore asses should be used.

2 2 IOMAC'11 4 th International Operational Modal Analysis Conference In this work a consecutive ass change ethod is presented, which consists of perforing different individual ass changes using a sall nuber of asses, which are attached to the structure at different locations in each experient. Then, the results for each location are cobined to estiate the scaling factors. The ethod is validated by experiental tests carried out on a 15 tonne prestressed concrete slab. 2 THEORY 2.1 Direct Mass Change Method The ass noralized ode shape,, and the arbitrary scaled ode shape,, corresponding to the ode, are related by the expression: where is the scaling factor. The scaling factor is a real or coplex nuber in case of real or coplex odes, respectively. The Mass Change Method consists of relating the odal paraeters (obtained by OMA) of both the original and the odified structure (figure 1). The odification is carried out by attaching asses to points on the structure where the ode shapes of the unodified structure are known. The user selects the nuber, the agnitude and the locations of the asses (Aenlle et al. 2009). In order to facilitate the ass odification and the calculation of the scaling factors, luped asses are often used, so that the ass change atrix becoes, in general, diagonal. A general expression for estiating the scaling factor of the ode shape is given by: (1) (2) where and are the natural frequencies of the original and odified structure, respectively, the ass change atrix and the ode shape. In Eq. (2), the unodified and odified ode shapes can be used which eans that this equation presents several alternatives (Aenlle et al. 2005b). However if large odifications in the ode shapes are expected the expression proposed by Bernal (2004) is recoended (Fernández et al. 2010), i.e.: (3) where ters are obtained fro the projection atrix. The basis of this approach is to introduce the projection of the odified ode shapes in the original ones. 2.2 The consecutive Mass Change Method The ass odification in the traditional ass change ethod, denoted here as direct ass change ethod, has generally been applied in only one step, i.e., by attaching all asses siultaneously at several points on the structure. The key idea of the consecutive ass change ethod is to use several sets of individual ass changes to estiate the scaling factors. Thus, instead of attaching asses siultaneously, the consecutive ass change ethod consists of perforing a series of consecutive ass odifications attaching a reduced nuber of asses,, ( ) so that the su of all the consecutive ass changes is, theoretically, equal to the direct ass change. The ethod is shown scheatically in Fig. 1.

3 3 Original Structure DOF 3 DOF 2 Individual Mass Change DOF 1 Original Structure DOF 3 Consecutive Mass Change Method DOF 2 DOF = ~ Original Structure DOF 3 DOF 2 Direct Mass Change Method DOF 1 Figure 1 : Relation between Direct and Consecutive Mass Change ethods (exaple with 3 asses). The consecutive ass change ethod proposed in this work is derived fro the Bernal equation. To illustrate the ethod, a siple case of three consecutive ass changes is used to derived equations and then the generalization to asses is presented. Taking into account that the scaling factor of the original structure is constant, we can establish the following relations for the scaling factor of the ode shape: (4) where supercripts 1, 2 and 3, represent the three consecutive individual ass changes (see Fig 1), superscript 123 eans that the three asses are attached siultaneously to the structure (direct ass change ethod), is the square frequency ratio and subscripts 0 and 1 indicate the original or odified structure, respectively. If the relation between odified and original ode shapes, e.g. is introduced, the denoinators of each individual ass change can be expressed as: (5) Taking into account that the denoinator of Eq. (4) corresponding to the direct ass change ethod is given by: (6) If only the diagonal ters of the atrix shapes are expressed as: (Bernal 2004) are considered, the odified ode

4 4 IOMAC'11 4 th International Operational Modal Analysis Conference (7) Now, if Eq. (6) is substituted in Eq. (4) using the relations (7), it is obtained that: (8) Fro which the following relation between the direct and individual square frequency ratios is inferred: On the other hand, fro the sensitivity equations of ode shapes (Aenlle et al. 2005, Parloo et al. 2003), it is inferred that the following approxiation can be applied in case of sall changes: Finally, if Eqs. (9) and (10) are substituted in Eq. (4), the relation between the individual and the direct case is: (9) (10) (11) which generalized to ass changes leads to: (12) Fro Eqs. (11) and (12), it can inferred that both the direct and consecutive ass change ethods should provide the sae results. However, larger uncertainty is expected in the consecutive ass change ethod because ore experiental tests ust be carried out with less asses, so that the frequency shift obtained in each individual experient is saller (Aenlle et al. 2009). One advantage of the consecutive ass change ethod is that only a few asses are needed in each experiental test and that total agnitude is less than that used in direct ass change ethod. On the other hand, ore experiental tests are needed and the asses ust be oved in each experient to different DOF s.

5 5 3 EXPERIMENTAL VERIFICATION 3.1 Structure and Equipent The structure tested was a siply supported in situ cast post tensioned slab strip of span Its total length was 11.2, including 0.2 overhangs over the knife edge supports. It had a width of 2, depth of and a weight of approxiately 15 tonnes. In the operational odal analysis 21 Honeywell QA750 acceleroeters were used. They were located at degrees of freedo (DOF s) 4 to 24, as indicated in Fig. 2, and were installed on levelled Perspex base plates. For digital data acquisition, a Data Physics DP730 spectru analyzer (Mobilyzer II) with 24 input channels and 4 output channels (24 bit resolution) was used. The spectru analyser was controlled by a laptop PC connected to it via an Ethernet connection. The sapling rate and acquisition tie used in each odal test were 128 Hz and 15 inutes, respectively Shaker Shaker Figure 2: Concrete slab strip diensions and location of easureent DOF s (1 27). Since the abient excitation level is too low for testing (the slab is in a very quiet lab), artificial excitation was required during the tests. Due to the fact that there are not coupled or repeated odes (Herlfunsen et al. 2005), two APS Dynaics Model 113 shakers, located at DOF s 15 and 19, were used as a source of excitation, even though fixed loads are not recoended for OMA. The slab was driven with uncorrelated rando signals fro both shakers with the sae excitation level for all the experients. 3.2 Experiental Results 1.35 For the concrete slab strip the first four odes are considered for the analysis. The first, second and fourth odes are bending of the slab strip, whereas the third ode is torsional. For this structure, the odal paraeters were estiated by classical (Ewins 1984) and operational odal analysis with Enhanced Frequency Doain Decoposition (EFDD) (Brincker et al. 2007) and the Stochastic Subspace Identification (SSI) (Peeters and Roeck 2001, Van Overschee and De Moor 1996) both techniques ipleented in the ARTeMIS Extractor software. In classical odal analysis (CMA) the odal paraeters were estiated fro the frequency response function (FRF) data calculated using the excitation and response tie histories with the sae configuration of shakers and acceleroeters, as that used in OMA analysis, that is, shakers located at DOF s 15 and 19 and acceleroeters located at DOF s 4 to 24, respectively. The ME scopeves software was used in a Multiple Input/Multiple Output (MIMO) analysis with two inputs and 21 outputs to obtain the odal paraeters fro FRF data. The FRFs of soe channels (MIMO analysis) and the EFDD singular value decoposition (OMA) are presented in Fig. 3.

6 6 IOMAC'11 4 th International Operational Modal Analysis Conference Figure 3 : FRFs (left) and Singular Value Decoposition (right) for the Slab. The natural frequencies are presented in Table 1, fro which it can be inferred that a good agreeent existed between the results provided by CMA and OMA. Tabla 1 : Natural frequencies in the slab strip. Mode 1 (bending) 2 (bending) 3 (torsion) 4 (bending) Hz Hz Hz Hz CMA FRFs OMA EFDD SSI The ass odification was carried out placing 25 kg steel ass blocks on the top surface of the slab strip (in groups of 100 kg) in each individual experient. To reduce the uncertainty of the results, a study of the ass change strategy was carried out previously to the odal test. The best strategies can be obtained constructing a location strategy table (Aenlle et al. 2009), using the ode shape atrix of the original structure. The ass change strategies used in the easureents are presented in Table 2. Strategies Strategy: 1 Mass attached Kg Table 2 : Mass strategies for the concrete slab strip. DOF s with ass Test X X Test X X Test X X Test X X Test X X Test X X Test X X Strategy: 2 Test X X Test X X Test X X The frequency shifts,, obtained fro the EFDD identification technique, for each individual ass location, are presented in Table 3.

7 7 Table 3 : Frequency shifts in the slab strip for each consecutive ass experient (EFDD). Mode (%) Masses attached to DOF s Table 3 confirs that asses attached to DOF s near to antinodes of the corresponding odes, provide the largest. Moreover larger frequency shifts are obtained for the torsional ode (ode nuber 3 in Table 3) due to the lower odal ass associated with this ode copared with the bending ode. The scaling factors obtained using Eq. (12) for both strategies (1 and 2) are presented in Table 4. To estiate the accuracy of the results, the scaling factors were also deterined fro FRFs with CMA and by the direct ass change ethod using 21 asses located at DOF s 4 to 24 (50 kg at each DOF). In CMA the ode shapes obtained fro FRFs data are ass noralized. The scaling factors were estiated using Eq. (1) so that the ode shapes were also noralized with the largest coponent of each ode shape equal to unity. Table 2 : Estiated scaling factors of the slab strip. Method Mode FRFs (CMA) Direct ass change ethod (21 Masses) Consecutive ass change ethod (Strategy 1) Consecutive ass change ethod (Strategy 2) If the results obtained by CMA fro the FRFs and the direct and consecutive ass change ethods are copared, it can be seen that they provide siilar results except for the ode, which has an error of approxiately 12%. A likely reason for this error is the not optial location of the shaker used in the CMA. For other odes, the error between the consecutive ass change ethod and the FRFs (CMA) is less than 3%. If the consecutive ass change ethod (Strategy 1) and the direct ass change ethod are copared, Table 5 shows that the results of both ethods provide approxiately the sae accuracy for all the odes considered in the analysis, with the largest discrepancy of 2.25% for the ode. As regards the consecutive ass change ethod with Strategy 2, the results are a little less accurate than Strategy 1 due to the less refined ass change strategy. The largest errors correspond with the ode (8% error) and the ode (7% error). CONCLUSIONS A new ethod for scaling ode shapes in operational odal analysis, which consists of perforing different individual ass changes using a sall nuber of asses, which are attached to the structure at different locations in each experient, has been proposed and validated. The accuracy of the ethod is siilar to that obtained with other ethods, such as the direct ass change ethod or classical odal analysis. This ethod has any potential benefits for application in ediu and large structures, because only a sall nuber of asses, located at different DOF s in each experient, are needed in the tests.

8 8 IOMAC'11 4 th International Operational Modal Analysis Conference ACKNOWLEDGEMENTS The partial econoic support given by the Spanish Ministry of Education through the project BIA C02-01and the European Social Fund (grant BES ) is gratefully appreciated. Special thanks also go to Dr Donald Nyawako fro the University of Sheffield for his support in the experiental part of this work. REFERENCES Magalhaes F., Caetano E., and Cunha A Online identification of the odal paraeters of a long span archbridge. Mechanical Systes and Signal Processing. Brincker, R., Ventura, C. and Andersen, P Why output-only odal testing is a desirable tool for a wide range of practical applications. In Proceedings of the International Modal Analysis Conference(IMACXXI). paper 265. Parloo E., Guillaue P., Anthonis J., Heylen W., and Swevers N Modelling of sprayer boo dynaics by eans of axiu likelihood identification techniques,part1: A coparison of input output and output-only odal testing. Biosyste Engineering, 85(2): Fernández P., Aenlle M. L., Brincker R., and Canteli A.F Artificial excitation in operational odal analysis. In Proc. of the International Modal Analysis Conference (IOMAC) III, (0): Aenlle M. L., Brincker R., and Canteli A. F Soe ethods to deterine scaled ode shapes in natural input odal analysis. In Proc. Of the International Modal Analysis Conference (IMAC) XXIII. Parloo E., Verboven P., Cuillae P., and Overeire M. V Sensitivity-based ass noralization of ode shape estiates fro output-only data. Proc. Int. Conf. on Structural Syste Identification, pages Parloo E., Guillaue P., Anthonis J., Heylen W., and Swevers N Modelling of sprayer boo dynaics by eans of axiu likelihood identification techniques, part 2: Sensitivity-based ode shape noralisation. Biosyste Engineering, 85(2): Khatibi M.M., Ashory M.R., and Malekjafarian A Scaling of ode shapes using ass-stiffness change ethod. In Proc. of the International Modal Analysis Conference (IOMAC) III, pages Aenlle M. L., Brincker R., Canteli A. F., and Villa L. M. 2005b. Scaling factor estiation by the ass change ethod. Proc. Of the International Operational Modal Analysis Conference (IOMAC). Bernal D Modal scaling fro known ass perturbation. Journal of Engineering Mechanics, 130(9):1083. López-Aenlle M., Fernández P., Brincker R., and Fernández-Canteli A Scaling factor estiation using an optiized ass-change strategy. Mechanical Systes and Signal Processing. Fernández P., Reynolds P., and López-Aenlle M Experiental evaluation of ass change approaches for scaling factors estiation. Proc. of the IMAC XXVIII. Herlfunsen, H., Andersen, P., Gade, S. and Møller N Identification techniques for operational odal analysis - an overview and practical experiencies. In Proc. Of the International Operational Modal Analysis Conference (IOMAC). Ewins D. J. (1984) Modal Testing_ Theory and Practice. Research Studies Press LTD and John Wiley & Sons INC. Brincker R., Andersen P., and Jacobsen N. J Autoated frequency doain decoposition for operational odal analysis. In Proceedings of Th e25th International Modal Analysis Conference(IMAC). Peeters B. and De Roeck G Stochastic syste identification for operational odal analysis_ a review. Journal of Dynaic Systes Measureent and Control Transactions of the ASME, 4(123): Van Overschee P. and De Moor B Subspace Identification for Linear Systes: Theory, Ipleentation & Applications. Dordrecht, Netherlands, Kluwer Acadeic Publishers.