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Strain predictions at unmeasured locations of a substructure using sparse response-only vibration measurements

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Abstract

Structural health monitoring of complex structures is often limited by restricted accessibility to locations of interest within the structure and availability of operational loads. In this work, a novel output-only virtual sensing scheme is proposed. This scheme involves the implementation of the modal expansion in an augmented Kalman filter. The performance of the proposed scheme is compared with two existing methods. Method 1 relies on a finite element model updating, batch data processing, and modal expansion (MUME) procedure. Method 2 employs a recursive sequential estimation algorithm, which feeds a substructure model of the instrumented system into an augmented Kalman filter (AKF). The new scheme referred to as Method 3 (ME-AKF), implements strain estimates generated via Modal Expansion into an AKF as virtual measurements. To demonstrate the applicability of the aforementioned methods, a rollercoaster connection was instrumented with accelerometers, strain rosettes, and an optical sensor. A comparison of estimated dynamic strain response at unmeasured locations using three alternative schemes is presented. Although acceleration measurements are used indirectly for model updating, the response-only methods presented in this research use only measurements from strain rosettes for strain history predictions and require no prior knowledge of input forces. Predicted strains using all methods are shown to sufficiently predict the measured strain time histories from a control location and lie within a 95% confidence interval calculated based on modal expansion equations. In addition, the proposed ME-AKF method shows improvement in strain predictions at unmeasured locations without the necessity of batch data processing. The proposed scheme shows high potential for real-time dynamic estimation of the strain and stress state of complex structures at unmeasured locations.

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Data availability

Limited availability to strain, acceleration, and optical sensor data measured from the rollercoaster support bracket used during this research. Some data can be made available by a request to the corresponding author and upon approval from the amusement park management.

Code availability

Relevant code to this research is limited. Availability can be granted by a request to the corresponding author and upon approval from the amusement park management.

References

  1. ASTM (2018) ASTM F2291–18 standard practice for design of amusement. Rides Dev 1:65

    Google Scholar 

  2. ASTM (2019) F770–19 Standard practice for ownership , operation , maintenance , and inspection of amusement rides and devices. www.astm.org, West Conshohocken, PA

  3. AASHTO (2018) Manual for Bridge Evaluation (3rd Edition). Am Assoc State Highw Transp Off

  4. ABSG Consulting Inc (2015) Offshore wind energy inspection procedure assessment

  5. Kullaa J (2016) Virtual sensing of structural vibrations using dynamic substructuring. Mech Syst Signal Proc 79:203–224. https://doi.org/10.1016/j.ymssp.2016.02.045

    Article  Google Scholar 

  6. Vettori S, Lorenzo E Di, Cumbo R, Musella U, Tamarozzi T, Peeter B, Chatzi E (2019) Kalman-based virtual sensing for improvement of service responses replication in environmental tests. In: IMAC XXXVIII Conference. The Society of Experimental Mechanics, Orange County, California, Houston, Texas, p #8071

  7. Tarpø M, Nabuco B, Georgakis C, Brincker R (2019) The effect of operational modal analysis in strain estimation using the modal expansion. In: 8th IOMAC - International Operational Modal Analysis Conference, Proceedings. Copenhagen, pp 699–705

  8. Yu H, Mohammed MA, Mohammadi ME, Moaveni B, Barbosa AR, Stavridis A, Wood RL (2017) Structural identification of an 18-story RC building in Nepal using post-earthquake ambient vibration and lidar data. Front Built Environ 3:1–15. https://doi.org/10.3389/fbuil.2017.00011

    Article  Google Scholar 

  9. Iliopoulos A, Weijtjens W, Van Hemelrijck D, Devriendt C (2016) Full-Field strain prediction applied to an offshore wind turbine bt - model validation and uncertainty quantification, volume 3. In: Schoenherr T, Moaveni B, Papadimitriou C (eds) Atamturktur S. Springer, Cham, pp 349–357

    Google Scholar 

  10. Soman R, Ostachowicz W (2018) Kalman filter based load monitoring in beam like structures using fibre-optic strain sensors. Sensors (Basel) 19:103. https://doi.org/10.3390/s19010103

    Article  Google Scholar 

  11. Eftekhar Azam S, Chatzi E, Papadimitriou C (2015) A dual Kalman filter approach for state estimation via output-only acceleration measurements. Mech Syst Signal Proc 60:866–886. https://doi.org/10.1016/j.ymssp.2015.02.001

    Article  Google Scholar 

  12. Papadimitriou C, Fritzen C-P, Kraemer P, Ntotsios E (2011) Fatigue predictions in entire body of metallic structures from a limited number of vibration sensors using Kalman filtering. Struct Control Heal Monit 18:554–573. https://doi.org/10.1002/stc

    Article  Google Scholar 

  13. Astroza R, Ebrahimian H, Conte JP (2017) Batch and Recursive Bayesian estimation methods for nonlinear structural system identification BT - risk and reliability analysis: theory and applicationsn Honor of ProfArmen Der Kiureghian. In: Gardoni P (ed) Armen Der Kiureghian. Springer, Cham, pp 341–364

    Google Scholar 

  14. Briers M, Doucet A, Maskell S (2004) Smoothing Algorithms for State-Space Models. IEEE Trans SIGNAL Process

  15. Baqersad J, Bharadwaj K (2018) Strain expansion-reduction approach. Mech Syst Signal Process 101:156–167. https://doi.org/10.1016/j.ymssp.2017.08.023

    Article  Google Scholar 

  16. Pingle P, Avitabile P (2011) Full field dynamic stress/strain from limited sets of measured data. Conf Proc Soc Exp Mech Ser 2:187–200. https://doi.org/10.1007/978-1-4419-9305-2_13

    Article  Google Scholar 

  17. Avitabile P, Pingle P (2012) Prediction of full field dynamic strain from limited sets of measured data. Shock Vib 19:765–785. https://doi.org/10.3233/SAV-2012-0686

    Article  Google Scholar 

  18. Tarpø M, Nabuco B, Georgakis C, Brincker R (2020) Expansion of experimental mode shape from operational modal analysis and virtual sensing for fatigue analysis using the modal expansion method. Int J Fatigue 130:105280. https://doi.org/10.1016/j.ijfatigue.2019.105280

    Article  Google Scholar 

  19. Juang JN, Pappa RS (1985) An eigensystem realization algorithm for modal parameter identification and model reduction. J Guid Control Dyn 8:620–627. https://doi.org/10.2514/3.20031

    Article  MATH  Google Scholar 

  20. James GH, Carne TG, Lauffer JP (1995) the natural excitation technique (NExT) for modal parameter extraction from operating structures. Modal Anal Int J Anal Exp Modal Anal 10:260–277

    Google Scholar 

  21. Moncayo H, Marulanda J, Thomson P (2010) Identification and monitoring of modal parameters in aircraft structures using the natural excitation technique (NExT) combined with the eigensystem realization algorithm (ERA). J Aerosp Eng 23:99–104. https://doi.org/10.1061/(ASCE)AS.1943-5525.0000011

    Article  Google Scholar 

  22. Yingnan G, Fangyi W (2013) Research on modal parameters identification of wing structures with NExT-ERA. Appl Mech Mater 249–250:1025–1029

    Google Scholar 

  23. Moaveni B, Barbosa AR, Conte JP, Hemez FM (2014) Uncertainty analysis of system identification results obtained for a seven-story building slice tested on the UCSD-NEES shake table. Struct Control Heal Monit 21:466–483. https://doi.org/10.1002/stc.1577

    Article  Google Scholar 

  24. Mieloszyk M, Opoka S, Ostachowicz W (2015) Frequency domain decomposition performed on the strain data obtained from the aluminium model of an offshore support structure. J Phys Conf Ser. https://doi.org/10.1088/1742-6596/628/1/012111

    Article  Google Scholar 

  25. Kranjc T, Slavić J, Boltežar M (2016) A comparison of strain and classic experimental modal analysis. JVC/Journal Vib Control 22:371–381. https://doi.org/10.1177/1077546314533137

    Article  Google Scholar 

  26. Peng Z, Dong K, Yin H (2019) A modal-based kalman filter approach and Osp method for structural response reconstruction. Shock Vib 2019:5475686. https://doi.org/10.1155/2019/5475686

    Article  Google Scholar 

  27. Palanisamy RP, Cho S, Kim H, Sim SH (2015) Experimental validation of Kalman filter-based strain estimation in structures subjected to non-zero mean input. Smart Struct Syst 15:489–503

    Article  Google Scholar 

  28. Erazo K, Sen D, Nagarajaiah S, Sun L (2019) Vibration-based structural health monitoring under changing environmental conditions using Kalman filtering. Mech Syst Signal Process 117:1–15. https://doi.org/10.1016/j.ymssp.2018.07.041

    Article  Google Scholar 

  29. Simon D (2006) Optimal state estimation: kalman, h infinity, and nonlinear approaches. Wiley-Interscience

    Book  Google Scholar 

  30. Azam ES, Chatzi E, Papadimitriou C, Smyth A (2017) Experimental validation of the kalman-type filters for online and real-time state and input estimation. J Vib Control 23:2494–2519. https://doi.org/10.1177/1077546315617672

    Article  Google Scholar 

  31. Dertimanis V, Chatzi E, Eftekhar Azam S (2016) Papadimitriou C (2016) Output-only fatigue prediction of uncertain steel structures. Eur Work Struct Heal Monit EWSHM 2:5–8

    Google Scholar 

  32. Lagerblad U, Wentzel H, Kulachenko A (2018) Fatigue damage prediction based on strain field estimates using a smoothed kalman filter and sparse measurements. In: Proceedings Of ISMA2018 and USD2018. pp 2805–2818

  33. Tatsis KE, Dertimanis VK, Papadimitriou C, Lourens E, Chatzi EN (2021) A general substructure-based framework for input-state estimation using limited output measurements. Mech Syst Signal Process 150:107223. https://doi.org/10.1016/j.ymssp.2020.107223

    Article  Google Scholar 

  34. Noppe N, Tatsis K, Chatzi EN, Devriendt C, Weijtjens W (2018) Fatigue stress estimation of offshore wind turbine using a kalman filter in combination with accelerometers. In: PROCEEDINGS OF ISMA2018 AND USD2018. pp 4847–4856

  35. Ansys® (2017) ANSYS Documentation. DeignXplorer User’s Guide

  36. Puerto Tchemodanova S, Tatsis K, Dertimanis V, Chatzi E (2019) Remaining fatigue life prediction of a roller coaster subjected to multiaxial nonproportional loading using limited measured strains locations (Under review). Struct Congr 4:1–11

    Google Scholar 

  37. Caicedo JM (2011) Practical guidelines for the natural excitation technique (NExT) and the eigensystem realization algorithm (ERA) for modal identification using ambient vibration. Exp Tech 35:52–58. https://doi.org/10.1111/j.1747-1567.2010.00643.x

    Article  Google Scholar 

  38. Allemang RJ, Brown DL (1998) A unified matrix polynomial approach to modal identification. J Sound Vib 211:301–322. https://doi.org/10.1006/jsvi.1997.1321

    Article  MATH  Google Scholar 

  39. Saltelli A, Tarantola S, Campolongo F, Ratto M (2002) Sensitivity analysis in practice. John Wiley and Sons Ltd

    Book  Google Scholar 

  40. Cundy AL (2003) Use of response surface metamodels in damage identification of dynamic structures. Virginia Polytech Inst State Univ Master Sci Thesis

  41. Qintao G, Lingmi Z (2004) Finite element model updating based on response surface methodology. Proc 22nd IMAC Dearborn,:306–309

  42. Zong Z, Lin X, Niu J (2015) Finite element model validation of bridge based on structural health monitoring—Part I: Response surface-based finite element model updating. J Traffic Transp Eng (English Ed 2:258–278 . https://doi.org/https://doi.org/10.1016/j.jtte.2015.06.001

  43. Kianifar MR, Campean F (2020) Performance evaluation of metamodelling methods for engineering problems: towards a practitioner guide. Struct Multidiscip Optim 61:159–186. https://doi.org/10.1007/s00158-019-02352-1

    Article  Google Scholar 

  44. Skafte A, Kristoffersen J, Vestermark J, Tygesen UT, Brincker R (2017) Experimental study of strain prediction on wave induced structures using modal decomposition and quasi static Ritz vectors. Eng Struct 136:261–276. https://doi.org/10.1016/j.engstruct.2017.01.014

    Article  Google Scholar 

  45. Oppenheim AV, Schafer RW, Buck JR (1999) Discrete-time signal processing. Prentice Hall

    Google Scholar 

  46. Chandrupatla TR, Belegundu AD (2015) Introduction to finite elements in engineering, 4th edn. Prentice Hall

    MATH  Google Scholar 

  47. Lourens E, Reynders E, De RG, Degrande G, Lombaert G (2012) An augmented Kalman filter for force identification in structural dynamics. Mech Syst Signal Process 27:446–460. https://doi.org/10.1016/j.ymssp.2011.09.025

    Article  Google Scholar 

  48. Maes K, Lourens E, Van Nimmen K, Reynders E, De Roeck G, Lombaert G (2015) Design of sensor networks for instantaneous inversion of modally reduced order models in structural dynamics. Mech Syst Signal Process 52–53:628–644. https://doi.org/10.1016/j.ymssp.2014.07.018

    Article  Google Scholar 

  49. Butler JL, Sherman CH (2016) Transducers and Arrays for Underwater Sound

  50. FARO Technologies Inc (2016) FARO Focus 3D X 330 HDR The Imaging Laser Scanner for Extended Ranges

  51. FARO Technologies Inc (2010) FaroArm® Platinum

  52. Horváth P (2018) Efficiency and accuracy investigation of the Craig-Bampton method through continuum vibration tests. AIP Conf Proc. https://doi.org/10.1063/1.5019121

    Article  Google Scholar 

  53. De KD, Rixen DJ, Voormeeren SN (2008) General framework for dynamic substructuring: history, review and classification of techniques. AIAA J 46:1169–1181. https://doi.org/10.2514/1.33274

    Article  Google Scholar 

  54. Bampton M, Craig R (1968) Coupling of substructures for dynamic analyses. AIAA J 6:1313–1319

    Article  Google Scholar 

  55. Papadimitriou C, Papadioti DC (2013) Component mode synthesis techniques for finite element model updating. Comput Struct 126:15–28. https://doi.org/10.1016/j.compstruc.2012.10.018

    Article  Google Scholar 

  56. Zhu D, Dong X, Wang Y (2016) Substructure stiffness and mass updating through minimization of modal dynamic residuals. J Eng Mech 142:04016013. https://doi.org/10.1061/(ASCE)EM.1943-7889.0001063

    Article  Google Scholar 

  57. Tatsis K, Dertimanis V, Abdallah I, Chatzi E (2017) A substructure approach for fatigue assessment on wind turbine support structures using output-only measurements. Proc Eng 199:1044–1049. https://doi.org/10.1016/j.proeng.2017.09.285

    Article  Google Scholar 

  58. Chang M, Pakzad SN (2013) Modified natural excitation technique for stochastic modal identification. J Struct Eng (United States) 139:1753–1762. https://doi.org/10.1061/(ASCE)ST.1943-541X.0000559

    Article  Google Scholar 

  59. He X, Moaveni B, Conte JP, Elgamal A, Masri SF (2009) System identification of alfred zampa memorial bridge using dynamic field test data. J Struct Eng 135:54–66. https://doi.org/10.1061/(ASCE)0733-9445(2009)135:1(54)

    Article  Google Scholar 

  60. Yam LH, Leung TP, Li DB, Xue KZ (1996) Theoretical and experimental study of modal strain analysis. J Sound Vib 191:251–260. https://doi.org/10.1006/jsvi.1996.0119

    Article  Google Scholar 

  61. Rhudy MB, Gu Y (2013) Online stochastic convergence analysis of the Kalman filter. Int J Stoch Anal. https://doi.org/10.1155/2013/240295

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

Data required to conduct the research presented in this paper would not have been possible without the collaboration of Mike Neuzil who provided the research team access and resources to accomplish data acquisition of the rollercoaster used in this research and also accommodating the use of LiDAR for geometric mapping. In addition, the authors would like to thank Dr. Jesse D. Sipple from BDI for helpful discussions and technical advice on setup of a wireless data acquisition system using various sensor types.

Funding

The writers are grateful for the funding of this research by NSF Division of Industrial Innovation and Partnership (IIP) Grant No. IIP-1640693. PFI: AIR-TT: Real-Time Fatigue Life Prediction for Decision-making and Asset Management. Any opinions, findings, and conclusions or recommendations expressed in this research are those of the authors and do not necessarily reflect the views of the National Science Foundation.

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Authors and Affiliations

  1. Department of Civil and Environmental Engineering, Tufts University, Medford, MA, 02155, USA

    Sofia Puerto Tchemodanova, Masoud Sanayei & Babak Moaveni

  2. Department of Civil, Environmental and Geomatic Engineering, ETH Zurich, Zürich, Switzerland

    Konstantinos Tatsis & Eleni Chatzi

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  1. Sofia Puerto Tchemodanova
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  2. Masoud Sanayei
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Correspondence to Masoud Sanayei.

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Tchemodanova, S.P., Sanayei, M., Moaveni, B. et al. Strain predictions at unmeasured locations of a substructure using sparse response-only vibration measurements. J Civil Struct Health Monit 11, 1113–1136 (2021). https://doi.org/10.1007/s13349-021-00476-x

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