Abstract
Experimental modal analysis is one of the key technologies in structural dynamics analysis. However, in cases involving extremely high or low modal damping, it is difficult to accurately identify all the modal parameters. In particular, for systems with extremely low damping, there may not be sufficient data to allow curve fitting in the vicinity of the resonant peaks. To overcome this difficulty, we propose a linear fit method of modal parameters on a new mapping plane. This method uses a basic equation linearized from the nonlinear equation of the frequency response function (FRF) by erasing the residue, which is a modal parameter. Then, the basic equation becomes linear on a mapping plane related to the ratios of the real and imaginary parts of the FRF. The linearized basic equation can identify the modal parameters of a vibration system with extremely low damping. It was observed that the influence of the measurement noise degrades the identification accuracy of the linear fit method. Consequently, it was confirmed that the identification accuracy deteriorates when data with low coherence and far from the natural frequency are used. Thus, a weighted least squares method using the coherence and Gaussian kernel function was proposed for the linear fit method. Finally, the modal parameters obtained using the proposed method and the conventional least-squares complex frequency (LSCF) method, from the FRF including noise, were compared, which indicated that the proposed method can produce estimation results with an accuracy comparable to that pertaining to the LSCF method.
Similar content being viewed by others
Data availability
The raw/processed data required to reproduce these findings cannot be shared at this time as the data are a part of an ongoing study as well.
References
Ewins, D.J.: Modal Testing: Theory and Practice. Research Studies Press, LTD, Baldock (2001)
Iiyama, K., Kurita, S., Motosaka, M., Chiba, K., Hiramatsu, H., Mitsuji, K.: Model identification of a heavily damaged nine-story steel-reinforced concrete building by ambient vibration measurement. J. Jpn. Ass. Earthq. Eng. 15(3), 1884–6246 (2015). https://doi.org/10.5610/jaee.15.3_78
Hsu, T.Y., Shih, Y.C., Pham, Q.V.: Damage detection of a thin plate using modal curvature via macrostrain measurement. Earthq. Eng. Eng. Vib. 18(2), 409–424 (2019). https://doi.org/10.1007/s11803-019-0512-y
Magalhaes, F., Cunha, A., Caetano, E.: Vibration based structural health monitoring of an arch bridge: from automated OMA to damage detection. Mech. Syst. Signal Process. 28, 212–228 (2012). https://doi.org/10.1016/j.ymssp.2011.06.011
Navabian, N., Bozorgnasab, M., Taghipour, R., et al.: Damage identification in plate-like structure using mode shape derivatives. Arch. Appl. Mech. 86, 819–830 (2016). https://doi.org/10.1007/s00419-015-1064-x
Kompalka, A.S., Reese, S., Bruhns, O.T.: Experimental investigation of damage evolution by data-driven stochastic subspace identification and iterative finite element model updating. Arch. Appl. Mech. 77, 559–573 (2007). https://doi.org/10.1007/s00419-007-0114-4
Mottershead, J.E., Link, M., Friswell, M.I.: The sensitivity method in finite element model updating: a tutorial. Mech. Syst. Signal Process. 25(5), 2275–2296 (2011). https://doi.org/10.1016/j.ymssp.2010.10.012
Basaga, H.B., Turker, T., Bayraktar, A.: A model updating approach based on design points for unknown structural parameters. Appl. Math. Model. 35(12), 5872–5883 (2011)
Sehgal, S., Kumar, H.: Structural dynamic model updating techniques: a state of the art review. Arch. Comput. Methods Eng. 23(3), 515–533 (2016)
Torres, W., Almazan, J.L., Sandoval, C., Boroschek, R.: Operational modal analysis and FE model updating of the Metropolitan Cathedral of Santiago, Chile. Eng. Struct. 143, 169–188 (2017). https://doi.org/10.1007/s11831-015-9150-3
Sun, W., Liu, R., Fan, Y.: Analytical modeling and damping optimization for a thin plate partially covered with hard coating. Arch. Appl. Mech. 88, 897–912 (2018). https://doi.org/10.1007/s00419-018-1348-z
Ibrahim, S.R., Mikulcik, E.C.: A method for the direct identification of vibration parameters from the free response. Shock Vib. Bull. 147, 183–198 (1977)
Malekjafarian A., Brincker R., Ashory M.R., Khatibi M.M.: Modified Ibrahim time domain method for identification of closely spaced modes: experimental results. In: Topics on the Dynamics of Civil Structures, vol. 1, pp. 443–449. Springer, New York. (2012). https://doi.org/10.1007/978-1-4614-2413-0_44
Lin, C.S.: Parametric estimation of systems with modal interference. Arch. Appl. Mech. 87(11), 1845–1857 (2017). https://doi.org/10.1007/s00419-017-1292-3
Bagheri, M., Jafari, A.: Analytical and experimental modal analysis of nonuniformly ring-stiffened cylindrical shells. Arch. Appl. Mech. 75, 177–191 (2006). https://doi.org/10.1007/s00419-005-0429-y
Yang, Y., Zhao, G., Ma, D., Xu, X.: Mode calculation and testing of a car body in white. Shock Vib. 18(1–2), 289–298 (2011). https://doi.org/10.3233/SAV-2010-0604
Yam, L.H., Guan, D.H., Zhang, A.Q.: Three-dimensional mode shapes of a tire using experimental modal analysis. Exp. Mech. 40(3), 369–375 (2000). https://doi.org/10.1007/BF02326482
Palanivelu, S., Narasimha, R.K.V., Ramarathnam, K.K.: Determination of rolling tyre modal parameters using finite element techniques and operational modal analysis. Mech. Syst. Signal Process. 64–65, 385–402 (2015). https://doi.org/10.1016/j.ymssp.2015.04.006
Saito, A., Suzuki, H., Kuroishi, M., Nakai, H.: Efficient forced vibration reanalysis method for rotating electric machines. J. Sound Vib. 334, 388–403 (2015). https://doi.org/10.1016/j.jsv.2014.09.004
Behnam, M.R., Khatibi, M.M., Malekjafarian, A.: An accurate estimation of frequency response functions in output-only measurements. Arch. Appl. Mech. 88, 837–853 (2018). https://doi.org/10.1007/s00419-018-1345-2
Matsubara, M., Kawamura, S.: Parameter identification of a three-dimensional flexible ring-based model of a tire using experimental modal analysis. Int. J. Automot. Eng. 10(2), 133–138 (2019). https://doi.org/10.20485/jsaeijae.10.2_133
Dobson, B.J.: A straight-line technique for extracting modal properties from frequency response data. Mech. Syst. Signal Process. 1(1), 29–40 (1987). https://doi.org/10.1016/0888-3270(87)90081-1
Phillips, A.W., Allemang, R.J., Fladung, W.A.: The Complex Mode Indicator Function (CMIF) as a parameter estimation method. Proc. SPIE Int. Soc. Opt. Eng. 3243, 705–710 (1998)
Lembregts, F.F., Leuridan, J., Zang, L., Kanda, H.: Multiple input modal analysis of frequency response functions based on direct parameter identification. Proc. Int. Modal Anal. Conf. IMAC 4, 589–598 (1986)
Yoshimura, T., Nagamatsu, A.: Research on modal analysis (7th report, estimation of variance of the frequency response function). Trans. Jpn. Soc. Mech. Eng. Ser. C 54, 2514–2521 (1988). (in Japanese)
Yoshimura, T.: Research on modal analysis (9th report, proposition of multireference curve-fittingbased on the maxima likelihood method—part 1). Trans. Jpn. Soc. Mech. Eng. Ser. C 56(523), 527–536 (1990). (in Japanese)
Guillaume, P., Verboven, P., Vanlanduit, S.: Frequency-domain maximum likelihood identification of modal parameters with confidence intervals. In: Proceedings of the International Seminar on Modal Analysis, vol. 1, pp. 359–366. Katholieke Universiteit Leuven (1998)
Auweraer, H.V., Guillaume, P., Verboven, P., Vanlanduit, S.: Application of a fast-stabilizing frequency domain parameter estimation method. J. Dyn. Sys. Meas. Control 123(3), 651–658 (2001)
Peeters, B., Auweraer, H.V., Guillaume, P., Leuridan, J.: The PolyMAX frequency-domain method: a new standard for modal parameter estimation? Shock Vib. 113(3, 4), 395–409 (2004)
Peeters, B., Auweraer, H.V.: PloyMAX: a revolution in operational modal analysis. In: 1st International Operational Modal Analysis Conference, pp. 26–27 (2005)
Guillaume, P., Verboven, P., Vanlanduit, S., Auweraer, H.V., Peeters, B.: A ploy-reference implementation of the least-squares complex frequency-domain estimator. Proc. IMAC 21, 395–409 (2003)
Sitarz, P., Powalka, B.: Modal parameters estimation using ant colony optimization algorithm. Mech. Syst. Signal Process. 76–77, 531–554 (2016)
Sitarz, P., Powalka, B.: Dual ant colony operational modal analysis parameter estimation method. Mech. Syst. Signal Process. 98, 231–267 (2018)
Kawamura, S., Kato, Y., Harada, M., Minamoto, H.: Estimation of dynamic properties of a lightly damped element. In: Proceedings of Dynamics and Design Conference, vol. 322 (2013) (in Japanese)
Kitahara, A., Yoshimura, T.: Modal identification of cylindrical shell using circumference reduction method. Trans. JSME (2015). https://doi.org/10.1299/transjsme.14-00461. (in Japanese)
Kawamura, S., Kita, M., Matsubara, M., Ise, T.: Study of the effect of specimen size and frequency on the structural damping property of beam. Mech. Eng. J. (2016). https://doi.org/10.1299/mej.16-00446
Matsubara, M., Kawamura, S., Ise, T.: Application of modal properties identification to multi-degrees-of-freedom system using simultaneous of the real and imaginary parts of frequency response function. Trans. JSME (2018). https://doi.org/10.1299/transjsme.17-00540. (in Japanese)
Matsubara, M., Tajiri, D., Takehara, S., Kawamura, S.: Linear fit method for modal parameter estimation using the real and imaginary parts of frequency response function (Identification accuracy improvement based on weighted least square method). Trans. JSME (2019). https://doi.org/10.1299/transjsme.18-00433. (in Japanese)
Yin, H.P.: A new theoretical basis for the bandwidth method and optimal power ratios for the damping estimation. Mech. Syst. Signal Process. 22(6), 1869–1881 (2008). https://doi.org/10.1016/j.ymssp.2008.01.011
Srikanth, N., Gupta, M.: Damping characterization of magnesium based composites using an innovative circle-fit approach. Compos. Sci. Technol. 63(4), 559–568 (2003). https://doi.org/10.1016/S0266-3538(02)00231-2
Elsalama, A.A., Gohary, M.A., El-Gamal, H.A.: Modal analysis on tire with respect to different parameters. Alex. Eng. J. 56(3), 345–357 (2017). https://doi.org/10.1016/j.aej.2016.09.022
Zhang, G., Zang, C., Friswell, M.I.: Identification of weak nonlinearities in MDOF systems based on reconstructed constant response tests. Arch. Appl. Mech. 89, 2053–2074 (2019). https://doi.org/10.1007/s00419-019-01559-4
Wang, S., Sato, H., Ohori, M.: New approaches to the modal analysis for machine tool structure. J. Eng. Ind. 106(1), 40–47 (1984)
Funding
None.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no competing interests.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Matsubara, M., Saito, A. & Kawamura, S. Estimation of modal parameters by using the ratios of imaginary to real parts of frequency response functions. Arch Appl Mech 91, 1179–1191 (2021). https://doi.org/10.1007/s00419-020-01817-w
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00419-020-01817-w