Skip to main content
SpringerLink
Log in

Online damage detection via a synergy of proper orthogonal decomposition and recursive Bayesian filters

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

In this paper, an approach based on the synergistic use of proper orthogonal decomposition and Kalman filtering is proposed for the online health monitoring of damaged structures. The reduced-order model of a structure is obtained during an (offline) initial training stage of monitoring; afterward, effective estimations of a possible structural damage are provided online by tracking the evolution in time of stiffness parameters and projection bases handled in the model order reduction procedure. Such tracking is accomplished via two Kalman filters: a first (extended) one to deal with the time evolution of a joint state vector, gathering the reduced-order state and the stiffness terms degraded by damage; a second one to deal with the update of the reduced-order model in case of damage evolution. Both filters exploit the information conveyed by measurements of the structural response to the external excitations. Results are reported for a (pseudo-experimental) benchmark test on an eight-story shear building. Capability and performance of the proposed approach are assessed in terms of tracked variation of the stiffness terms of the reduced-order model, identified damage location and speed-up of the whole health monitoring procedure.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13

Similar content being viewed by others

References

  1. Glaser, S.D., Li, H., Wang, M.L., Ou, J., Lynch, J.: Sensor technology innovation for the advancement of structural health monitoring: a strategic program of US-China research for the next decade. Smart Struct. Syst. 3(2), 221–244 (2007). doi:10.12989/sss.2007.3.2.221

    Article  Google Scholar 

  2. Stallings, J.M., Tedesco, J.W., El-Mihilmy, M., McCauley, M.: Field performance of FRP bridge repairs. J. Bridge Eng. 5, 107–113 (2000)

    Article  Google Scholar 

  3. Aktan, A., Catbas, F., Grimmelsman, K., Tsikos, C.: Issues in infrastructure health monitoring for management. J. Eng. Mech. 126(7), 711–724 (2000). doi:10.1061/(ASCE)0733-9399. (2000) 126:7(711)

    Article  Google Scholar 

  4. Ko, J.M., Ni, Y.Q.: Technology developments in structural health monitoring of large-scale bridges. Eng. Struct. 27(12), 1715–1725 (2005). doi:10.1016/j.engstruct.2005.02.021

    Article  Google Scholar 

  5. Glaser, S.D., Tolman, A.: Sense of sensing: from data to informed decisions for the built environment. J. Infrastruct. Syst. ACSE 14, 4–14 (2008)

    Article  Google Scholar 

  6. Yeum, C.M., Dyke, S.J.: Vision-based automated crack detection for bridge inspection. Comput. Aided Civil Infrastruct. Eng. 30(10), 759–770 (2015). doi:10.1111/mice.12141

    Article  Google Scholar 

  7. Memarzadeh, M., Pozzi, M.: Integrated inspection scheduling and maintenance planning for infrastructure systems. Comput. Aided Civil Infrastruct. Eng. (2015). doi:10.1111/mice.12178

    Google Scholar 

  8. Cho, S., Spencer, B.F.: Sensor attitude correction of wireless sensor network for acceleration-based monitoring of civil structures. Comput. Aided Civil Infrastruct. Eng. 30(11), 859–871 (2015). doi:10.1111/mice.12147

    Article  Google Scholar 

  9. Mariani, S., Corigliano, A., Caimmi, F., Bruggi, M., Bendiscioli, P., De Fazio, M.: MEMS-based surface mounted health monitoring system for composite laminates. Microelectron. J. 44(7), 598–605 (2013). doi:10.1016/j.mejo.2013.03.003

    Article  Google Scholar 

  10. Mariani, S., Bruggi, M., Caimmi, F., Bendiscioli, P., De Fazio, M.: Sensor deployment over damage-containing plates: a topology optimization approach. J. Intell. Mater. Syst. Struct. 24, 1105–1122 (2013)

    Article  Google Scholar 

  11. Chan, T.H.T., Yu, L., Tam, H.Y., Ni, Y.Q., Liu, S.Y., Chung, W.H., Cheng, L.K.: Fiber Bragg grating sensors for structural health monitoring of Tsing Ma bridge: background and experimental observation. Eng. Struct. 28(5), 648–659 (2006). doi:10.1016/j.engstruct.2005.09.018

    Article  Google Scholar 

  12. Helmi, K., Taylor, T., Zarafshan, A., Ansari, F.: Reference free method for real time monitoring of bridge deflections. Eng. Struct. 103, 116–124 (2015). doi:10.1016/j.engstruct.2015.09.002

    Article  Google Scholar 

  13. Hampshire, T.A., Adeli, H.: Monitoring the behavior of steel structures using distributed optical fiber sensors. J. Constr. Steel Res. 53(3), 267–281 (2000)

    Article  Google Scholar 

  14. Gentile, C., Cabboi, A.: Vibration-based structural health monitoring of stay cables by microwave remote sensing. Smart Struct. Syst. 16(2), 263–280 (2015). doi:10.12989/sss.2015.16.2.263

    Article  Google Scholar 

  15. Farrar, C.R., Darling, T.W., Migliori, A., Baker, W.E.: Microwave interferometers for non-contact vibration measurements on large structures. Mech. Syst. Signal Process. 13(2), 241–253 (1999). doi:10.1006/mssp.1998.1216

    Article  Google Scholar 

  16. Laefer, D.F., Truong-Hong, L., Carr, H., Singh, M.: Crack detection limits in unit based masonry with terrestrial laser scanning. NDTE Int. 62, 66–76 (2014). doi:10.1016/j.ndteint.2013.11.001

    Article  Google Scholar 

  17. Breuer, P., Chmielewski, T., Górski, P., Konopka, E.: Application of GPS technology to measurements of displacements of high-rise structures due to weak winds. J. Wind Eng. Ind. Aerodyn. 90(3), 223–230 (2002). doi:10.1016/S0167-6105(01)00221-5

    Article  Google Scholar 

  18. Górski, P.: Investigation of dynamic characteristics of tall industrial chimney based on GPS measurements using Random Decrement Method. Eng. Struct. 83, 30–49 (2015). doi:10.1016/j.engstruct.2014.11.006

    Article  Google Scholar 

  19. Park, S.W., Park, H.S., Kim, J.H., Adeli, H.: 3D displacement measurement model for health monitoring of structures using a motion capture system. Measurement 59, 352–362 (2015). doi:10.1016/j.measurement.2014.09.063

    Article  Google Scholar 

  20. Lee, J.J., Shinozuka, M.: Real-time displacement measurement of a flexible bridge using digital image processing techniques. Exp. Mech. 46(1), 105–114 (2006). doi:10.1007/s11340-006-6124-2

    Article  Google Scholar 

  21. Hwa Kim, B.: Extracting modal parameters of a cable on shaky motion pictures. Mech. Syst. Signal Process. 49(1–2), 3–12 (2014). doi:10.1016/j.ymssp.2014.02.002

    Article  Google Scholar 

  22. Qarib, H., Adeli, H.: Recent advances in health monitoring of civil structures. Sci. Iran. 21(6), 1733–1742 (2014)

    Google Scholar 

  23. Bursi, O.S., Kumar, A., Abbiati, G., Ceravolo, R.: Identification, model updating, and validation of a steel twin deck curved cable-stayed footbridge. Comput. Aided Civil Infrastruct. Eng. 29(9), 703–722 (2014). doi:10.1111/mice.12076

    Article  Google Scholar 

  24. Fuggini, C., Chatzi, E., Zangani, D.: Combining genetic algorithms with a meso-scale approach for system identification of a smart polymeric textile. Comput. Aided Civil Infrastruct. Eng. 28(3), 227–245 (2013). doi:10.1111/j.1467-8667.2012.00789.x

    Article  Google Scholar 

  25. Moaveni, B., Conte, J.P., Hemez, F.M.: Uncertainty and sensitivity analysis of damage identification results obtained using finite element model updating. Comput. Aided Civil Infrastruct. Eng. 24(5), 320–334 (2009). doi:10.1111/j.1467-8667.2008.00589.x

    Article  Google Scholar 

  26. Moaveni, B., Behmanesh, I.: Effects of changing ambient temperature on finite element model updating of the Dowling Hall Footbridge. Eng. Struct. 43, 58–68 (2012)

    Article  Google Scholar 

  27. Farrar, C.R., Doebling, S.W., Nix, D.A.: Vibration-based structural damage identification. Philos. Trans. R. Soc. Lond. A Math. Phys. Eng. Sci. 359(1778), 131–149 (2001)

    Article  MATH  Google Scholar 

  28. Haritos, N., Owen, J.S.: The use of vibration data for damage detection in bridges: a comparison of system identification and pattern recognition approaches. Struct. Health Monit. 3(2), 141–163 (2004). doi:10.1177/1475921704042698

    Article  Google Scholar 

  29. Farrar, C.R., Worden, K.: Structural Health Monitoring: A Machine Learning Perspective. Wiley Publishing, London (2012)

    Book  Google Scholar 

  30. Amezquita-Sanchez, J.P., Adeli, H.: Signal processing techniques for vibration-based health monitoring of smart structures. Arch. Comput. Methods Eng. 23(1), 1–15 (2016). doi:10.1007/s11831-014-9135-7

    Article  MathSciNet  MATH  Google Scholar 

  31. Dervilis, N., Worden, K., Cross, E.: On robust regression analysis as a means of exploring environmental and operational conditions for SHM data. J. Sound Vib. 347, 279–296 (2015)

    Article  Google Scholar 

  32. Spiridonakos, M.D., Chatzi, E.N., Sudret, B.: Polynomial Chaos expansion models for the monitoring of structures under operational variability. ASCE ASME J. Risk Uncertain. Eng. Syst. Part A Civil Eng. 2(3), B4016003 (2016)

  33. Reynders, E., Wursten, G., De Roeck, G.: Output-only structural health monitoring in changing environmental conditions by means of nonlinear system identification. Struct. Health Monit. 13(1), 82–93 (2014)

    Article  Google Scholar 

  34. Yang, J., Lin, S.: Identification of parametric variations of structures based on least squares estimation and adaptive tracking technique. J. Eng. Mech. 131(3), 290–298 (2005). doi:10.1061/(ASCE)0733-9399. (2005) 131:3(290)

    Article  Google Scholar 

  35. Van Overschee, P., De Moor, B.: Subspace Identification for Linear Systems: Theory-Implementation-Applications. Springer, New York (1996)

    Book  MATH  Google Scholar 

  36. Van Overschee, P., De Moor, B.: N4SID: subspace algorithms for the identification of combined deterministic-stochastic systems. Automatica 30(1), 75–93 (1994). doi:10.1016/0005-1098(94)90230-5

    Article  MathSciNet  MATH  Google Scholar 

  37. Chin-Hsiung, L., Jian-Huang, W., Yi-Cheng, L., Pei-Yang, L., Shieh-Kung, H.: Structural damage diagnosis based on on-line recursive stochastic subspace identification. Smart Mater. Struct. 20(5), 055004 (2011)

    Article  Google Scholar 

  38. Chatzis, M., Chatzi, E., Smyth, A.W.: An experimental validation of time domain system identification methods with fusion of heterogeneous data. Earthq. Eng. Struct. Dyn. 44(4), 523–547 (2015). doi:10.1002/eqe.2528

    Article  Google Scholar 

  39. Moaveni, B., He, X., Conte, J., Restrepo, J., Panagiotou, M.: System identification study of a 7-story full-scale building slice tested on the UCSD-NEES shake table. J. Struct. Eng. 137(6), 705–717 (2010). doi:10.1061/(ASCE)ST.1943-541X.0000300

    Article  Google Scholar 

  40. Kalman, R.E.: A new approach to linear filtering and prediction problems. J. Basic Eng. 82(1), 35–45 (1960)

    Article  Google Scholar 

  41. Julier, S.J., Uhlmann, J.K.: A new extension of the Kalman filter to nonlinear systems. In: International Symposium on Aerospace/Defence, Sensing, Simulation and Controls, vol. 26, p. 32. Orlando (1997)

  42. Gordon, N.J., Salmond, D.J., Smith, A.F.M.: Novel approach to nonlinear/non-Gaussian Bayesian state estimation. IEE Proc. F RadarSignal Process. 140(2), 107–113 (1993)

    Article  Google Scholar 

  43. Chatzi, E.N., Smyth, A.W.: Particle filter scheme with mutation for the estimation of time-invariant parameters in structural health monitoring applications. Struct. Control Health Monit. 20(7), 1081–1095 (2013)

    Article  Google Scholar 

  44. Li, B.: Multiple-model Rao-Blackwellized particle CPHD filter for multitarget tracking. Nonlinear Dyn. 79(3), 2133–2143 (2014). doi:10.1007/s11071-014-1799-x

    Article  Google Scholar 

  45. Eftekhar Azam, S., Mariani, S.: Dual estimation of partially observed nonlinear structural systems: a particle filter approach. Mech. Res. Commun. 46, 54–61 (2012)

    Article  Google Scholar 

  46. Chatzi, E.N., Smyth, A.W., Masri, S.F.: Experimental application of on-line parametric identification for nonlinear hysteretic systems with model uncertainty. Struct. Saf. 32(5), 326–337 (2010)

    Article  Google Scholar 

  47. Eftekhar Azam, S.: Online Damage Detection in Structural Systems. Springer Briefs in Applied Sciences and Technology. Springer, Berlin (2014)

    Book  Google Scholar 

  48. Eftekhar Azam, S., Mariani, S.: Investigation of computational and accuracy issues in POD-based reduced order modeling of dynamic structural systems. Eng. Struct. 54, 150–167 (2013)

    Article  Google Scholar 

  49. Kerschen, G., Golinval, G.C.: Physical interpretation of the proper orthogonal modes using the singular value decomposition. J. Sound Vib. 249, 849–865 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  50. Corigliano, A., Dossi, M., Mariani, S.: Model order reduction and domain decomposition strategies for the solution of the dynamic elastic-plastic structural problem. Comput. Methods Appl. Mech. Eng. 290, 127–155 (2015). doi:10.1016/j.cma.2015.02.021

    Article  MathSciNet  Google Scholar 

  51. Kerschen, G., Golinval, J.-C., Vakakis, A.F., Bergman, L.A.: The method of proper orthogonal decomposition for dynamical characterization and order reduction of mechanical systems: an overview. Nonlinear Dyn. 41(1–3), 147–169 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  52. Lu, K., Yu, H., Chen, Y., Cao, Q., Hou, L.: A modified nonlinear POD method for order reduction based on transient time series. Nonlinear Dyn. 79(2), 1195–1206 (2014). doi:10.1007/s11071-014-1736-z

    Article  MATH  Google Scholar 

  53. Lu, K., Jin, Y., Chen, Y., Cao, Q., Zhang, Z.: Stability analysis of reduced rotor pedestal looseness fault model. Nonlinear Dyn. 82(4), 1611–1622 (2015). doi:10.1007/s11071-015-2264-1

    Article  MathSciNet  MATH  Google Scholar 

  54. Zhao, X., Shang, P.: Principal component analysis for non-stationary time series based on detrended cross-correlation analysis. Nonlinear Dyn. 84(2), 1033–1044 (2015). doi:10.1007/s11071-015-2547-6

    Article  MathSciNet  MATH  Google Scholar 

  55. Liang, Y.C., Lin, W.Z., Lee, H.P., Lim, S.P., Lee, K.H., Sun, H.: Proper orthogonal decomposition and its applications-part II: model reduction for mems dynamical analysis. J. Sound Vib. 256(3), 515–532 (2002). doi:10.1006/jsvi.2002.5007

    Article  Google Scholar 

  56. Ruotolo, R., Surace, C.: Using svd to detect damage in structures with different operational conditions. J. Sound Vib. 226(3), 425–439 (1999). doi:10.1006/jsvi.1999.2305

    Article  Google Scholar 

  57. Vanlanduit, S., Parloo, E., Cauberghe, B., Guillaume, P., Verboven, P.: A robust singular value decomposition for damage detection under changing operating conditions and structural uncertainties. J. Sound Vib. 284(3–5), 1033–1050 (2005). doi:10.1016/j.jsv.2004.07.016

    Article  Google Scholar 

  58. Galvanetto, U., Violaris, G.: Numerical investigation of a new damage detection method based on proper orthogonal decomposition. Mech. Syst. Signal Process. 21(3), 1346–1361 (2007). doi:10.1016/j.ymssp.2005.12.007

    Article  Google Scholar 

  59. Shane, C., Jha, R.: Proper orthogonal decomposition based algorithm for detecting damage location and severity in composite beams. Mech. Syst. Signal Process. 25(3), 1062–1072 (2011). doi:10.1016/j.ymssp.2010.08.015

    Article  Google Scholar 

  60. Mariani, S., Ghisi, A.: Unscented Kalman filtering for nonlinear structural dynamics. Nonlinear Dyn. 49(1–2), 131–150 (2007)

    Article  MATH  Google Scholar 

  61. Hughes, T.J.R.: The Finite Element Method. Linear Static and Dynamic Finite Element Analysis. Dover, New York (2000)

    MATH  Google Scholar 

  62. Corigliano, A., Mariani, S.: Parameter identification in explicit structural dynamics: performance of the extended Kalman filter. Comput. Methods Appl. Mech. Eng. 193, 3807–3830 (2004)

    Article  MATH  Google Scholar 

  63. Sirovich, L.: Turbulence and the dynamics of coherent structures. I-coherent structures. II-symmetries and transformations. III-dynamics and scaling. Q. Appl. Math. 45(1), 573–590 (1987)

    Article  MATH  Google Scholar 

  64. Liang, Y.C., Lee, H.P., Lim, S.P., Lin, W.Z., Lee, K.H., Wu, C.G.: Proper orthogonal decomposition and its applications-part I: theory. J. Sound Vib. 252(3), 527–544 (2002). doi:10.1006/jsvi.2001.4041

    Article  MATH  Google Scholar 

  65. Butcher, E.A., Al-Shudeifat, M.A.: An efficient mode-based alternative to principal orthogonal modes in the order reduction of structural dynamic systems with grounded nonlinearities. Mech. Syst. Signal Process. 25(5), 1527–1549 (2011)

    Article  Google Scholar 

  66. Al-Shudeifat, M.A., Butcher, E.A.: Order reduction of forced nonlinear systems using updated LELSM modes with new Ritz vectors. Nonlinear Dyn. 62(4), 821–840 (2010)

    Article  MATH  Google Scholar 

  67. Kappagantu, R., Feeny, B.: An “optimal” modal reduction of a system with frictional excitation. J. Sound Vib. 224(5), 863–877 (1999)

    Article  Google Scholar 

  68. Al-Shudeifat, M.A., Butcher, E.A.: On the dynamics of a beam with switching crack and damaged boundaries. J. Vib. Control 19(1), 1077546311428640 (2013)

    Article  Google Scholar 

  69. Feeny, B., Kappagantu, R.: On the physical interpretation of proper orthogonal modes in vibrations. J. Sound Vib. 211(4), 607–616 (1998)

    Article  Google Scholar 

  70. Han, C.S., Feeny, B.: Enhanced proper orthogonal decomposition for the modal analysis of homogeneous structures. J. Vib. Control 8(1), 19–40 (2002)

    MATH  Google Scholar 

  71. Feeny, B.: On proper orthogonal co-ordinates as indicators of modal activity. J. Sound Vib. 255(5), 805–817 (2002)

    Article  Google Scholar 

  72. Bryson, A., Johansen, D.: Linear filtering for time-varying systems using measurements containing colored noise. IEEE Trans. Autom. Control 10(1), 4–10 (1965). doi:10.1109/TAC.1965.1098063

    Article  MathSciNet  Google Scholar 

  73. Geist, M., Pietquin, O.: Kalman filtering and colored noises: the (autoregressive) moving-average case. In: IEEE Workshop on Machine Learning Algorithms, Systems and Applications (MLASA 2011), Honolulu, United States. pp. 1–4 (2011)

  74. Grewal, M.S., Andrews, A.P.: Kalman Filtering: Theory and Practice Using MATLAB, 4th edn. Wiley Publishing, London (2011)

    MATH  Google Scholar 

  75. Wan, E.A., Nelson, A.T.: Dual Extended Kalman Filter Methods. In: Haykin, S. (ed.) Kalman Filtering and Neural Networks. Wiley Publishing, London (2001)

    Google Scholar 

  76. Capellari, G., Eftekhar Azam, S., Mariani, S.: Damage detection in flexible plates through reduced-order modeling and hybrid particle-Kalman filtering. Sensors 16(1), 2 (2016). doi:10.3390/s16010002

  77. Roffel, A.J., Narasimhan, S.: Extended Kalman filter for modal identification of structures equipped with a pendulum tuned mass damper. J. Sound Vib. 333(23), 6038–6056 (2014). doi:10.1016/j.jsv.2014.06.030

  78. Reif, K., Gunther, S., Yaz, E., Unbehauen, R.: Stochastic stability of the discrete-time extended Kalman filter. IEEE Trans. Autom. Control 44(4), 714–728 (1999). doi:10.1109/9.754809

    Article  MathSciNet  MATH  Google Scholar 

  79. Sharma, G., Agarwala, A., Bhattacharya, B.: A fast parallel Gauss Jordan algorithm for matrix inversion using CUDA. Comput. Struct. 128, 31–37 (2013)

    Article  Google Scholar 

  80. De Callafon, R.A., Moaveni, B., Conte, J.P., He, X., Udd, E.: General realization algorithm for modal identification of linear dynamic systems. J. Eng. Mech. 134(9), 712–722 (2008)

    Article  Google Scholar 

  81. Krajcinovic, D.: Damage mechanics. Mech. Mater. 8(2–3), 117–197 (1989)

    Article  Google Scholar 

  82. Corigliano, A., Dossi, M., Mariani, S.: Domain decomposition and model order reduction methods applied to the simulation of multiphysics problems in MEMS. Comput. Struct. 122, 113–127 (2013)

    Article  Google Scholar 

  83. Brand, M.: Fast low-rank modifications of the thin singular value decomposition. Linear Algebra Appl. 415, 20–30 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  84. Bittanti, S., Savaresi, S.M.: On the parameterization and design of an extended Kalman filter frequency tracker. IEEE Trans. Autom. Control 45(9), 1718–1724 (2000)

    Article  MATH  Google Scholar 

  85. Kontoroupi, K., Smyth, A.W.: Online noise identification for joint state and parameter estimation of nonlinear systems. ASCE ASME J. Risk Uncertain. Eng. Syst. 2(3), B4015006 (2016). doi:10.1061/AJRUA6.0000839

    Google Scholar 

  86. Yuen, K.-V., Liang, P.F., Kuok, S.C.: Online estimation of noise parameters for Kalman filter. Struct. Eng. Mech. 47(3), 361–381 (2013)

    Article  Google Scholar 

  87. Yuen, K.-V., Kuok, S.-C.: Online updating and uncertainty quantification using nonstationary output-only measurement. Mech. Syst. Signal Process. 66–67, 62–77 (2016). doi:10.1016/j.ymssp.2015.05.019

    Article  Google Scholar 

  88. Lim, J.: Particle filtering for nonlinear dynamic state systems with unknown noise statistics. Nonlinear Dyn. 78(2), 1369–1388 (2014). doi:10.1007/s11071-014-1523-x

    Article  Google Scholar 

  89. Yang, Y., Gao, W.: An optimal adaptive Kalman filter. J. Geod. 80(4), 177–183 (2006)

    Article  MATH  Google Scholar 

  90. Boutayeb, M., Rafaralahy, H., Darouach, M.: Convergence analysis of the extended Kalman filter used as an observer for nonlinear deterministic discrete-time systems. IEEE Trans. Autom. Control 42(4), 581–586 (1997). doi:10.1109/9.566674

    Article  MathSciNet  MATH  Google Scholar 

  91. Zang, C., Imregun, M.: Structural damage detection using artificial neural networks and measured FRF data reduced via principal component projection. J. Sound Vib. 242(5), 813–827 (2001)

    Article  Google Scholar 

  92. Sahin, M., Shenoi, R.: Quantification and localisation of damage in beam-like structures by using artificial neural networks with experimental validation. Eng. Struct. 25(14), 1785–1802 (2003)

    Article  Google Scholar 

Download references

Acknowledgements

Financial support by Fondazione Cariplo through project Safer Helmets is gratefully acknowledged.

Author information

Authors and Affiliations

  1. Department of Mechanical Engineering, University of Thessaly, Vólos, Greece

    S. Eftekhar Azam

  2. Department of Civil and Environmental Engineering, Politecnico di Milano, Milan, Italy

    S. Mariani

  3. Structural Engineering Department, Building and Housing Research Center (BHRC), Tehran, Iran

    N. K. A. Attari

Authors
  1. S. Eftekhar Azam
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. S. Mariani
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. N. K. A. Attari
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S. Mariani.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Eftekhar Azam, S., Mariani, S. & Attari, N.K.A. Online damage detection via a synergy of proper orthogonal decomposition and recursive Bayesian filters. Nonlinear Dyn 89, 1489–1511 (2017). https://doi.org/10.1007/s11071-017-3530-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-017-3530-1

Keywords

Search

Navigation