A Comparative Review on Operational Modal Analysis Methods
Year 2015,
Volume: 36 Issue: 3, 3302 - 3311, 13.05.2015
Farbod Ahmadıfar
,
Morteza Payab
Abstract
Abstract. In the present study we investigate development of operational modal analysis (OMA), different OMA techniques as well as relevant issues and experimental studies. Furthermore, we classify previously performed studies have been conducted so far and consider the most important one. So, first we review the fundamental concepts. Then, the history and application of the OMA method will be examined comprehensively to observe their role in the development of the technique. Three basic methods of FDD, SSI, and Next will be discussed and the research works associated to them will be analyzed.
References
- normalization of mode shape, load estimation, and analysis in the presence of harmonic forces
- were evaluated. After, we discussed about studies on OMA methods based on wavelet
- transform, using mode indexes in OMA, order and accuracy estimation, methods based on
- cepstrum, and so forth. Furthermore, we tested one building piece of seven RC shear wall with
- full index over vibration tale NEES. Three Sys ID methods of output-only were utilized for
- extraction of modal parameters, i.e. natural frequency, damping ratio, and modes shape taken
- from the building sample. These methods include: a) natural excitation technique integrated
- with eigensystem realization algorithm (NExT-ERA), b) data-driven stochastic subspace
- identification (SSI-Data), and c) enhanced frequency domain decomposition (EFDD). In the
- current research, an analysis of variability or uncertainty of these is identification methods of
- system were done in two stages.
- The first stage is when these methods are employed on measured response of the structure and
- the second stage is when these methods are applied on response of the structure simulated using
- a 3D non-linear finite element model which is calibrated and confirmed. the input factors we
- considered in the first stage are 1) amplitude of input excitation, A) a level of non-linearity of
- response, 2) spatial density, S)number of sensors, 3) the length of the structural response data
- used in the process of data identification (L) and 4) model order utilized in identification
- methods of parametric system (O).
- In the second stage of uncertainty analysis, in addition to four input factors in the first stage
- such as measurement noise, (N) is also included. Using ANOVA test for system identification
- results based on experimental measured data I the first stage, we observed that input factor A
- has the greatest impact on changeability of modal parameters which were identified through
- using these three methods (especially the natural frequency of the first mode). In the second
- stage of uncertainty analysis, ANOVA test was used on standard deviation and mean scores (for
- a set of 100 identification runs for random description of measurement noise)of modal
- parameters which were identified through finite element simulated data . We understood that
- variability and mean scores of identified modal parameters (especially natural frequencies)
- show the greatest sensivitiy towards input factor A for all methods which shows a good
- agreement with the results of the first stage. Input factors of S and N showed the least possible
- impact on mean scores of modal parameters which are identified through Next-ERA and EFDD.
- Although the level of measurement noise (N) greatly contribute (compared with input factors) to
- variability of standard deviations in identified modal parameters, but it does not work for mean
- scores of identified modal parameters. Also, meta-models are in good agreement with identified
- modal parameters in the second stage. According to relative amplitude of coefficients ß
- (regression) of meta-models, we found out that identified natural frequencies show the highest
- sensivitiy towards input factor A (as the variance analysis results indicated). Next, input factor
- L and linear interactions AL revealed the highest sensivitiy. Moreover, we observed that
- generally modal damping ratios and MAC values showed greater sensivitiy towards input
- factors S, N, and O against natural frequencies.
- Relative amplitudes of coefficients ß relative amplitude of coefficients demonstrate that
- specified input factors have more significant impact on variability of identified modal
- parameters of the first mode compared with the parameters of higher order modes. Thus, we
- conclude that level of accuracy/certainty in system identification results not only depends upon
- estimation error of used identification methods as well as measurement noise, but it also is a
- variable of test plans (e.g. excitation domain, sensors spatial density, response length of
- measurement data, and model order). Consequently, dynamic tests must be designed so that the
- most influential input factors place in optimum or appropriate levels in order to have more
- precise and meaningful system identification results.
- James, G. H., Carne, T.G., Lauffer, J.P. and Nard, A. R. 1992. Modal Testing Using Natural Excitation Proceeding of the 10th IMAC. San Diego. CA. USA.
- Cole, H. A. 1973. On-Line Failure Detection and Damping Measurements of Aerospace Structures by Random Decrement Signature. NASA CR-2205.
- Ibrahim, S.R., Brincker, R. and Asmussen, J.C. 1996. Modal Parameter Identification from Responses of General Unknown Random Inputs. Proceeding of the 14th IMAC.
- Ibrahim, S.R., Asmussen, J.C. and Brincker, R. 1997. Vector Triggering Random Decrement Technique for Higher Identification Accuracy. Proceeding of the IMAC XV. Orlando. Florida.
- Ibrahim, S. R. 1977. Random Decrement Technique for Modal Identification of Structures. Journal of Spacecraft and Rockets. Vol. 14. No. 11.
- Vandiver, J. K. et al. 1982. A Mathematical Basis for the Random Decrement Vibration Signature Analysis Technique. ASME J. of Mechanical Design. Vol. 104.
- Vandiver, J. K. et al. 1982. A Mathematical Basis for the Random Decrement Vibration Signature Analysis Technique. ASME J. of Mechanical Design. Vol. 104.
- Brincker, R., Krenk, S. and Jensen, J.L. 1991b. Estimation of Correlation Functions by the Random Decrement Technique. Proceedings of the 9th IMAC.
- Brincker, R. and Andersen, P. 1999. ARMA Models in Modal Space. Proceeding of the 17th IMAC.
- Brincker, R. and Andersen, P. 2006. Understanding Stochastic Subspace Identification. Proceeding of the 24th IMAC.
- Brincker, R., Kirkegaard, P.H. and Rytter, A. 1991a. Identification of System Parameters by the Random Decrement Technique. Proceedings of the Florence Modal Analysis Conference. Florence.Italy.
- Brincker, R. 2007. Automated Frequency Domain Decomposition. Proceeding of the 25th IMAC.
- Brincker, R., Ventura, C. and Andersen, P. 2001b. Damping Estimation by Frequency Domain Decomposition Proceeding of the IMAC XIX. Kissimmee. USA.
- Brincker, R. and Asmussen, J.C. 1997. Random Decrement Based FRF Estimation. Proceeding of the IMAC XV. Orlando. Florida.
- Brincker, R., Rodrigues, J. and Andersen, P. 2004. Scaling the Mode Shapes of a Building Model by Mass Changes. Proceeding of the 22nd IMAC.
- Brincker, R. and Andersen, P. 2003. A Way of Getting Scaled Mode Shapes in Output Only Modal Testing. Proceeding of the 21st IMAC.
- Brincker, R., Andersen, P. and Mooller, N. 2000b. An Indicator For Separation of Structural and Harmonic Modes in Output-Only Modal Testing. Proceeding of the 18th IMAC.
- Brincker, R., Herlufsen, H. and Andersen, P. 2001a. Modal testing of mechanical structures to operational excitation forces. Proceeding of the IMAC-XIX. Vol. 1. Kissimmee. USA. 715
- Brincker, R., Zhang, L.M. and Anderson, P. 2000a. Modal Identification from Ambient Response using Frequency Domain Decomposition. Proceeding of the 18th IMAC. San Antonio. TX. USA
- Asmussen, J.C., Ibrahim, S, R. and Brincker, R. 1997. Application Of Vector Triggering Random Decrement. Proceeding. Of the IMAC XV. Orlando. Florida.
- Asmussen, J.C. and Brincker, R. 1998a. A New Approach for Predicting the Variance of Random Decrement Functions. Proceeding. Of the 16th IMAC.
- Asmussen, J.C., Ibrahi, S.R. and Brincker, R. 1998b. Random Decrement: Identification ofStructures Subjected to Ambient Excitation. Proceeding Of the 16th IMAC.
- Van Overchee, P. and De Moor, B. 1996. Subspace identification for linear systems – Theory, Implementation, Applications. Kluwer Academic Publishers. ISB N0-7923-9717- 7.
- Van Overchee, P. and De Moor, B. 1993. Subspace Algorithms for the Stochastic Identification Problem. Automatica. Vol. 29. No. 3.
- Jiang, D. and Ren, W-X. 2005. EMD-Based Stochastic Subspace Identification of Structures from Operational Vibration Measurements. Engineering Structures. Vol. 27. No. 12.
- Rodrigues, J., Brincker, R. and Andersen, P. 2004. Improvement of Frequency Domain Output-Only Modal Identification from the Application of the Random Decrement Technique. Proceeding of the 23rd IMAC.
- Rodrigues, J. and Brincker, R. 2005. Application of the Random Decrement Technique in Operational Modal Analysis. Proceeding of the 1st IOMAC.
- Hoen, C. 2006c. Subspace Identification of Modal Coordinate Time Series. Proceeding of the 24th IMAC.
- Bandat, J. and Piersiol, A. 1986. Random Data Analysis and Measurement Procedures. John ‘Wiley & Son, New York. USA.
- Ventura, C. E. and Tomas H. 1997. Structural Assessment by Modal Analysis in Western Canada. Proceeding of the IMAC XV. Orlando. Florida.
- Ventura, C. E. and Tomas H. 1997. Structural Assessment by Modal Analysis in Western Canada. Proceeding of the IMAC XV. Orlando. Florida.
- Zhang, L.M., Wang, T. And Tamura, Y. 2005a. Frequency-spatial Domain Decomposition Technique with Application to Operational Modal Analysis of Civil Engineering Structures. Proceeding of the 1st IOMAC. Copenhagen. Denmark.
- Zhang, L. and Tamura, Y. 2003. Damping Estimation of Engineering Structures With Ambient Response Measurements. Proceeding of the 21st IMAC
- Zhang, L.M., Wang, T. And Tamura, Y. 2005a. Frequency-spatial Domain Decomposition Technique with Application to Operational Modal Analysis of Civil Engineering Structures. Proceeding of the 1st IOMAC. Copenhagen. Denmark.
- Zhang, L. and Tamura, Y. 2003. Damping Estimation of Engineering Structures With Ambient Response Measurements. Proceeding of the 21st IMAC.
- Zhang, L., Brincker, R. and Andersen, P. 2001. Modal Indicators for Operational Modal Identification. Proceeding of the 18th IMAC.
- Zhang, Y., Zhang, Z., Xu, X. and Hua, H. 2005b. Modal Parameter Identification Using Response Data Only. Journal of Sound and Vibration, Vol. 282. No. 1-2
- Zhang, Y., Zhang, Z., Xu, X. and Hua, H. 2005b. Modal Parameter Identification Using Response Data Only. Journal of Sound and Vibration, Vol. 282. No. 1-2
- Zhang, L., Brincker, R. and Andersen, P. 2001. Modal Indicators for Operational Modal Identification. Proceeding of the 18th IMAC.
- Andersen, P., Brincker, R., Rune, G. and Mevel, L. 2007. Automated Modal Parameter Estimation for Operational Modal Analysis of Large Systems. Proceeding of the 2nd IOMAC.
- Andersen, P., Brincker, R. and Kirkegaard, P.H. 1996. Theory of Covariance Equivalent ARMAV Models of Civil Engineering Structures. Proceeding of the 14th IMAC.
- Andersen, P. and Brincker, R. 1999. Estimation of Modal Parameters and Their Uncertainties. Proceeding of the 17th IMAC.
- Pedersen, I.B., Hansen, S.M., Brincker, R. and Aenlle, M.L. 2007. Load Estimation by FrequencyDomain Decomposition. Proceedings of the 2nd IMAC.
- Aenlle, M.L., Brincker, R. and Canteli, A.F. 2005a. Some Methods to Determine Scaled Mode Shapes in Natural Input Modal Analysis. Proceeding of the 23rd IMAC.
- Aenlle, M.L., Brincker, R. and Canteli, A.F. 2005c. Load Estimation from Natural input Modal Analysis. Proceeding of the 23rd IMAC.
- Jacobsen, N.J., Andersen, P. and Brincker, R. 2006. Using Enhanced Frequency Domain Decomposition as a Robust Technique to Harmonic Excitation in Operational Modal Analysis. Proceeding of the ISMA2006.
- Akaike, H. 1974. Stochastic Theory of Minimal Realization. IEEE Trans. Automatic Control. AC-19. Andersen, P., Brincker, R. and Kirkegaard, P.H. 1995. On the Uncertainty of Identification of Civil Engineering Structures Using ARMA Models. Proceeding of the 13th IMAC.
- Ljung, L. 1987. System Identification, Theory for the User. Prentice Hall. Englewood Cliffs. Magalhaes, F., Brincker, R. and Cunha, A. 2007. Damping Estimation Using Free Decays and Ambient Vibration Tests. Proceeding of the 2nd IOMAC.
- Aenlle, M.L., Fernandez, P.F., Brincker, R. and Canteli, A.F. 2007. Scaling Factor Estimation Using An Optimized Mass Change Strategy Part 1: Theory. Proceeding of the 2nd IOMAC
- Aenlle, M.L., Brincker, R., Canteli, A.F. and Villa Garcfa, L.M. 2005b. Scaling Factor Estimation by the Mass Change Method. Proceeding of the 1st IOMAC.
Year 2015,
Volume: 36 Issue: 3, 3302 - 3311, 13.05.2015
Farbod Ahmadıfar
,
Morteza Payab
References
- normalization of mode shape, load estimation, and analysis in the presence of harmonic forces
- were evaluated. After, we discussed about studies on OMA methods based on wavelet
- transform, using mode indexes in OMA, order and accuracy estimation, methods based on
- cepstrum, and so forth. Furthermore, we tested one building piece of seven RC shear wall with
- full index over vibration tale NEES. Three Sys ID methods of output-only were utilized for
- extraction of modal parameters, i.e. natural frequency, damping ratio, and modes shape taken
- from the building sample. These methods include: a) natural excitation technique integrated
- with eigensystem realization algorithm (NExT-ERA), b) data-driven stochastic subspace
- identification (SSI-Data), and c) enhanced frequency domain decomposition (EFDD). In the
- current research, an analysis of variability or uncertainty of these is identification methods of
- system were done in two stages.
- The first stage is when these methods are employed on measured response of the structure and
- the second stage is when these methods are applied on response of the structure simulated using
- a 3D non-linear finite element model which is calibrated and confirmed. the input factors we
- considered in the first stage are 1) amplitude of input excitation, A) a level of non-linearity of
- response, 2) spatial density, S)number of sensors, 3) the length of the structural response data
- used in the process of data identification (L) and 4) model order utilized in identification
- methods of parametric system (O).
- In the second stage of uncertainty analysis, in addition to four input factors in the first stage
- such as measurement noise, (N) is also included. Using ANOVA test for system identification
- results based on experimental measured data I the first stage, we observed that input factor A
- has the greatest impact on changeability of modal parameters which were identified through
- using these three methods (especially the natural frequency of the first mode). In the second
- stage of uncertainty analysis, ANOVA test was used on standard deviation and mean scores (for
- a set of 100 identification runs for random description of measurement noise)of modal
- parameters which were identified through finite element simulated data . We understood that
- variability and mean scores of identified modal parameters (especially natural frequencies)
- show the greatest sensivitiy towards input factor A for all methods which shows a good
- agreement with the results of the first stage. Input factors of S and N showed the least possible
- impact on mean scores of modal parameters which are identified through Next-ERA and EFDD.
- Although the level of measurement noise (N) greatly contribute (compared with input factors) to
- variability of standard deviations in identified modal parameters, but it does not work for mean
- scores of identified modal parameters. Also, meta-models are in good agreement with identified
- modal parameters in the second stage. According to relative amplitude of coefficients ß
- (regression) of meta-models, we found out that identified natural frequencies show the highest
- sensivitiy towards input factor A (as the variance analysis results indicated). Next, input factor
- L and linear interactions AL revealed the highest sensivitiy. Moreover, we observed that
- generally modal damping ratios and MAC values showed greater sensivitiy towards input
- factors S, N, and O against natural frequencies.
- Relative amplitudes of coefficients ß relative amplitude of coefficients demonstrate that
- specified input factors have more significant impact on variability of identified modal
- parameters of the first mode compared with the parameters of higher order modes. Thus, we
- conclude that level of accuracy/certainty in system identification results not only depends upon
- estimation error of used identification methods as well as measurement noise, but it also is a
- variable of test plans (e.g. excitation domain, sensors spatial density, response length of
- measurement data, and model order). Consequently, dynamic tests must be designed so that the
- most influential input factors place in optimum or appropriate levels in order to have more
- precise and meaningful system identification results.
- James, G. H., Carne, T.G., Lauffer, J.P. and Nard, A. R. 1992. Modal Testing Using Natural Excitation Proceeding of the 10th IMAC. San Diego. CA. USA.
- Cole, H. A. 1973. On-Line Failure Detection and Damping Measurements of Aerospace Structures by Random Decrement Signature. NASA CR-2205.
- Ibrahim, S.R., Brincker, R. and Asmussen, J.C. 1996. Modal Parameter Identification from Responses of General Unknown Random Inputs. Proceeding of the 14th IMAC.
- Ibrahim, S.R., Asmussen, J.C. and Brincker, R. 1997. Vector Triggering Random Decrement Technique for Higher Identification Accuracy. Proceeding of the IMAC XV. Orlando. Florida.
- Ibrahim, S. R. 1977. Random Decrement Technique for Modal Identification of Structures. Journal of Spacecraft and Rockets. Vol. 14. No. 11.
- Vandiver, J. K. et al. 1982. A Mathematical Basis for the Random Decrement Vibration Signature Analysis Technique. ASME J. of Mechanical Design. Vol. 104.
- Vandiver, J. K. et al. 1982. A Mathematical Basis for the Random Decrement Vibration Signature Analysis Technique. ASME J. of Mechanical Design. Vol. 104.
- Brincker, R., Krenk, S. and Jensen, J.L. 1991b. Estimation of Correlation Functions by the Random Decrement Technique. Proceedings of the 9th IMAC.
- Brincker, R. and Andersen, P. 1999. ARMA Models in Modal Space. Proceeding of the 17th IMAC.
- Brincker, R. and Andersen, P. 2006. Understanding Stochastic Subspace Identification. Proceeding of the 24th IMAC.
- Brincker, R., Kirkegaard, P.H. and Rytter, A. 1991a. Identification of System Parameters by the Random Decrement Technique. Proceedings of the Florence Modal Analysis Conference. Florence.Italy.
- Brincker, R. 2007. Automated Frequency Domain Decomposition. Proceeding of the 25th IMAC.
- Brincker, R., Ventura, C. and Andersen, P. 2001b. Damping Estimation by Frequency Domain Decomposition Proceeding of the IMAC XIX. Kissimmee. USA.
- Brincker, R. and Asmussen, J.C. 1997. Random Decrement Based FRF Estimation. Proceeding of the IMAC XV. Orlando. Florida.
- Brincker, R., Rodrigues, J. and Andersen, P. 2004. Scaling the Mode Shapes of a Building Model by Mass Changes. Proceeding of the 22nd IMAC.
- Brincker, R. and Andersen, P. 2003. A Way of Getting Scaled Mode Shapes in Output Only Modal Testing. Proceeding of the 21st IMAC.
- Brincker, R., Andersen, P. and Mooller, N. 2000b. An Indicator For Separation of Structural and Harmonic Modes in Output-Only Modal Testing. Proceeding of the 18th IMAC.
- Brincker, R., Herlufsen, H. and Andersen, P. 2001a. Modal testing of mechanical structures to operational excitation forces. Proceeding of the IMAC-XIX. Vol. 1. Kissimmee. USA. 715
- Brincker, R., Zhang, L.M. and Anderson, P. 2000a. Modal Identification from Ambient Response using Frequency Domain Decomposition. Proceeding of the 18th IMAC. San Antonio. TX. USA
- Asmussen, J.C., Ibrahim, S, R. and Brincker, R. 1997. Application Of Vector Triggering Random Decrement. Proceeding. Of the IMAC XV. Orlando. Florida.
- Asmussen, J.C. and Brincker, R. 1998a. A New Approach for Predicting the Variance of Random Decrement Functions. Proceeding. Of the 16th IMAC.
- Asmussen, J.C., Ibrahi, S.R. and Brincker, R. 1998b. Random Decrement: Identification ofStructures Subjected to Ambient Excitation. Proceeding Of the 16th IMAC.
- Van Overchee, P. and De Moor, B. 1996. Subspace identification for linear systems – Theory, Implementation, Applications. Kluwer Academic Publishers. ISB N0-7923-9717- 7.
- Van Overchee, P. and De Moor, B. 1993. Subspace Algorithms for the Stochastic Identification Problem. Automatica. Vol. 29. No. 3.
- Jiang, D. and Ren, W-X. 2005. EMD-Based Stochastic Subspace Identification of Structures from Operational Vibration Measurements. Engineering Structures. Vol. 27. No. 12.
- Rodrigues, J., Brincker, R. and Andersen, P. 2004. Improvement of Frequency Domain Output-Only Modal Identification from the Application of the Random Decrement Technique. Proceeding of the 23rd IMAC.
- Rodrigues, J. and Brincker, R. 2005. Application of the Random Decrement Technique in Operational Modal Analysis. Proceeding of the 1st IOMAC.
- Hoen, C. 2006c. Subspace Identification of Modal Coordinate Time Series. Proceeding of the 24th IMAC.
- Bandat, J. and Piersiol, A. 1986. Random Data Analysis and Measurement Procedures. John ‘Wiley & Son, New York. USA.
- Ventura, C. E. and Tomas H. 1997. Structural Assessment by Modal Analysis in Western Canada. Proceeding of the IMAC XV. Orlando. Florida.
- Ventura, C. E. and Tomas H. 1997. Structural Assessment by Modal Analysis in Western Canada. Proceeding of the IMAC XV. Orlando. Florida.
- Zhang, L.M., Wang, T. And Tamura, Y. 2005a. Frequency-spatial Domain Decomposition Technique with Application to Operational Modal Analysis of Civil Engineering Structures. Proceeding of the 1st IOMAC. Copenhagen. Denmark.
- Zhang, L. and Tamura, Y. 2003. Damping Estimation of Engineering Structures With Ambient Response Measurements. Proceeding of the 21st IMAC
- Zhang, L.M., Wang, T. And Tamura, Y. 2005a. Frequency-spatial Domain Decomposition Technique with Application to Operational Modal Analysis of Civil Engineering Structures. Proceeding of the 1st IOMAC. Copenhagen. Denmark.
- Zhang, L. and Tamura, Y. 2003. Damping Estimation of Engineering Structures With Ambient Response Measurements. Proceeding of the 21st IMAC.
- Zhang, L., Brincker, R. and Andersen, P. 2001. Modal Indicators for Operational Modal Identification. Proceeding of the 18th IMAC.
- Zhang, Y., Zhang, Z., Xu, X. and Hua, H. 2005b. Modal Parameter Identification Using Response Data Only. Journal of Sound and Vibration, Vol. 282. No. 1-2
- Zhang, Y., Zhang, Z., Xu, X. and Hua, H. 2005b. Modal Parameter Identification Using Response Data Only. Journal of Sound and Vibration, Vol. 282. No. 1-2
- Zhang, L., Brincker, R. and Andersen, P. 2001. Modal Indicators for Operational Modal Identification. Proceeding of the 18th IMAC.
- Andersen, P., Brincker, R., Rune, G. and Mevel, L. 2007. Automated Modal Parameter Estimation for Operational Modal Analysis of Large Systems. Proceeding of the 2nd IOMAC.
- Andersen, P., Brincker, R. and Kirkegaard, P.H. 1996. Theory of Covariance Equivalent ARMAV Models of Civil Engineering Structures. Proceeding of the 14th IMAC.
- Andersen, P. and Brincker, R. 1999. Estimation of Modal Parameters and Their Uncertainties. Proceeding of the 17th IMAC.
- Pedersen, I.B., Hansen, S.M., Brincker, R. and Aenlle, M.L. 2007. Load Estimation by FrequencyDomain Decomposition. Proceedings of the 2nd IMAC.
- Aenlle, M.L., Brincker, R. and Canteli, A.F. 2005a. Some Methods to Determine Scaled Mode Shapes in Natural Input Modal Analysis. Proceeding of the 23rd IMAC.
- Aenlle, M.L., Brincker, R. and Canteli, A.F. 2005c. Load Estimation from Natural input Modal Analysis. Proceeding of the 23rd IMAC.
- Jacobsen, N.J., Andersen, P. and Brincker, R. 2006. Using Enhanced Frequency Domain Decomposition as a Robust Technique to Harmonic Excitation in Operational Modal Analysis. Proceeding of the ISMA2006.
- Akaike, H. 1974. Stochastic Theory of Minimal Realization. IEEE Trans. Automatic Control. AC-19. Andersen, P., Brincker, R. and Kirkegaard, P.H. 1995. On the Uncertainty of Identification of Civil Engineering Structures Using ARMA Models. Proceeding of the 13th IMAC.
- Ljung, L. 1987. System Identification, Theory for the User. Prentice Hall. Englewood Cliffs. Magalhaes, F., Brincker, R. and Cunha, A. 2007. Damping Estimation Using Free Decays and Ambient Vibration Tests. Proceeding of the 2nd IOMAC.
- Aenlle, M.L., Fernandez, P.F., Brincker, R. and Canteli, A.F. 2007. Scaling Factor Estimation Using An Optimized Mass Change Strategy Part 1: Theory. Proceeding of the 2nd IOMAC
- Aenlle, M.L., Brincker, R., Canteli, A.F. and Villa Garcfa, L.M. 2005b. Scaling Factor Estimation by the Mass Change Method. Proceeding of the 1st IOMAC.