{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T02:13:15Z","timestamp":1760235195169,"version":"build-2065373602"},"reference-count":39,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2021,7,28]],"date-time":"2021-07-28T00:00:00Z","timestamp":1627430400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100012659","name":"Foundation for Innovative Research Groups of the National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["No. 51775270"],"award-info":[{"award-number":["No. 51775270"]}],"id":[{"id":"10.13039\/501100012659","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100008982","name":"Qatar National Research Fund","doi-asserted-by":"publisher","award":["NPRP11S-1220-170112"],"award-info":[{"award-number":["NPRP11S-1220-170112"]}],"id":[{"id":"10.13039\/100008982","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100012221","name":"Jinling Institute of Technology","doi-asserted-by":"publisher","award":["No. jit-fhxm-201914"],"award-info":[{"award-number":["No. jit-fhxm-201914"]}],"id":[{"id":"10.13039\/501100012221","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Dynamic load identification is an inverse problem concerned with finding the load applied on a structure when the dynamic characteristics and the response of the structure are known. In engineering applications, some of the structure parameters such as the mass or the stiffness may be unknown and\/or may change in time. In this paper, an online dynamic load identification algorithm based on an extended Kalman filter is proposed. The algorithm not only identifies the load by measuring the structural response but also identifies the unknown structure parameters and tracks their changes. We discuss the proposed algorithm for the cases when the unknown parameters are the stiffness or the mass coefficients. Furthermore, for a system with many degrees of freedom and to achieve online computations, we implement the model reduction theory. Thus, we reduce the number of degrees of freedom in the resulting symmetric system before applying the proposed extended Kalman filter algorithm. The algorithm is used to recover the dynamic loads in three numerical examples. It is also used to identify the dynamic load in a lab experiment for a structure with varying parameters. The simulations and the experimental results show that the proposed algorithm is effective and can simultaneously identify the parameters and any changes in them as well as the applied dynamic load.<\/jats:p>","DOI":"10.3390\/sym13081372","type":"journal-article","created":{"date-parts":[[2021,7,28]],"date-time":"2021-07-28T21:21:04Z","timestamp":1627507264000},"page":"1372","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":10,"title":["Online Dynamic Load Identification Based on Extended Kalman Filter for Structures with Varying Parameters"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8685-2201","authenticated-orcid":false,"given":"Hongqiu","family":"Li","sequence":"first","affiliation":[{"name":"Mechatronic Engineering College, Jinling Institute of Technology, Nanjing 211169, China"}]},{"given":"Jinhui","family":"Jiang","sequence":"additional","affiliation":[{"name":"State Key Laboratory of Mechanics and Control of Mechanical Structures, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0152-8478","authenticated-orcid":false,"given":"M Shadi","family":"Mohamed","sequence":"additional","affiliation":[{"name":"Institute for Infrastructure and Environment, Heriot-Watt University, Edinburgh EH14 4AS, UK"}]}],"member":"1968","published-online":{"date-parts":[[2021,7,28]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"61","DOI":"10.1061\/(ASCE)0733-9399(2004)130:1(61)","article-title":"Flexibility Based Approach for Damage Characterization: Benchmark Application","volume":"130","author":"Bernal","year":"2004","journal-title":"J. Eng. Mech."},{"key":"ref_2","first-page":"967","article-title":"A Time Domain Identification Method for Distributed Dynamic Loads in Spatial Structures","volume":"38","author":"Cheng","year":"2017","journal-title":"Appl. Math. Mech."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"787","DOI":"10.1002\/eqe.4290200806","article-title":"Estimation of structural Parameters in time domain: A substructure approach","volume":"20","author":"Koh","year":"1991","journal-title":"Earthq. Eng. Struct. Dyn."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"441","DOI":"10.1088\/0964-1726\/10\/3\/303","article-title":"Modal identification of output-only systems using frequency domain decomposition","volume":"10","author":"Brincker","year":"2001","journal-title":"Smart Mater. Struct."},{"key":"ref_5","unstructured":"Weixin, R., and Harik, I.E. (2001, January 5\u20138). Modal analysis of the Cumberland river bridge on I-24 highway in West Kentucky. Proceedings of the 19th International Modal Analysis Conference, Kissimmee, FL, USA."},{"key":"ref_6","first-page":"149","article-title":"Operational Modal Parameter Identification of High Arch Dam under Flood Discharge Excitation Based on Frequency Domain Decomposition Method","volume":"27","author":"Huokun","year":"2008","journal-title":"J. Vib. Shock"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"244","DOI":"10.1016\/j.ymssp.2011.07.008","article-title":"Structural parameters and dynamic loading identification from incomplete measurements: Approach and validation","volume":"28","author":"Xu","year":"2012","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_8","first-page":"288","article-title":"On Line Failure Detection and Damping Measurement of Aerospace Structure by Random Decrement Signature","volume":"68","author":"Cole","year":"1968","journal-title":"AIAA"},{"key":"ref_9","unstructured":"Ibrahim, S.R. (2001, January 5\u20138). Efficient Random Decrement Computation for identification of Ambient Responses. Proceedings of the 19th International Modal Analysis Conference, Kissimmee, FL, USA."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"855","DOI":"10.1006\/mssp.1999.1249","article-title":"Reference-based stochastic subspace identification for out-put-only modal analysis","volume":"13","author":"Peeters","year":"1999","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"620","DOI":"10.2514\/3.20031","article-title":"An eigensystem realization algorithm for modal parameter identification and model reduction","volume":"8","author":"Juang","year":"1985","journal-title":"J. Guid. Control Dyn."},{"key":"ref_12","first-page":"260","article-title":"The natural excitation technique(NEXT)for modal parameter extraction from operating structures","volume":"10","author":"James","year":"1995","journal-title":"Int. J. Anal. Exp. Modal Anal."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1006\/jsvi.1997.0977","article-title":"System identification from ambient vibration measurements on a Bridge","volume":"205","author":"Arrar","year":"1997","journal-title":"J. Sound Vib."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1757","DOI":"10.1061\/(ASCE)0733-9399(1984)110:12(1757)","article-title":"Structural Identification by Extended Kalman Filter","volume":"110","author":"Hoshiya","year":"1984","journal-title":"J. Eng. Mech."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"693","DOI":"10.1061\/(ASCE)0733-9399(2000)126:7(693)","article-title":"Time Domain Identification of Frames under Earthquake Loading","volume":"126","author":"Loh","year":"2000","journal-title":"J. Eng. Mech."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1233","DOI":"10.1061\/(ASCE)0733-9399(1998)124:11(1233)","article-title":"Adaptive H\u221e Filter: Its Application to Structural Identification","volume":"124","author":"Sato","year":"1998","journal-title":"J. Eng. Mech."},{"key":"ref_17","unstructured":"Yoshida, I. (2001, January 17\u201322). Damage detection using Monte Carlo filter based on non-Gaussian noise. Proceedings of the Structural Safety and Reliability, ICOSSAR\u201901, Newport Beach, CA, USA."},{"key":"ref_18","unstructured":"Ellis, N. (1973, January 10\u201312). A technique for evaluating the fluctuating aerodynamic forces on a flexible building. Proceedings of the International Symposium on Vibrating Problems in Industry, Keswick, UK."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"10","DOI":"10.4050\/JAHS.24.2.10","article-title":"Model verification of force determination for measuring vibration loads","volume":"19","author":"Bartlett","year":"1979","journal-title":"J. Am. Helicopter Soc."},{"key":"ref_20","first-page":"453","article-title":"Measurement of Impact Load by Using an Inverse Analysis Technique","volume":"34","author":"Hirotsugu","year":"1991","journal-title":"JSME Int. J. Ser. 1 Solid Mech. Strength Mater."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"3833","DOI":"10.1121\/1.4988515","article-title":"Load identification by coherence analysis of structural response","volume":"141","author":"Milana","year":"2017","journal-title":"Acoust. Soc. Am. J."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"124","DOI":"10.1016\/j.ijnonlinmec.2005.06.006","article-title":"Sequential non-linear least-square estimation for damage identification of structures","volume":"41","author":"Yang","year":"2006","journal-title":"Int. J. Non-Linear Mech."},{"key":"ref_23","first-page":"1","article-title":"The Identification Method Research of Dynamic Random Load Distributing on Structure","volume":"31","author":"Zhang","year":"2012","journal-title":"J. Vibrat. Shock"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"615","DOI":"10.1016\/j.cma.2018.06.014","article-title":"Identifying the wavenumber for the inverse Helmholtz problem using an enriched finite element formulation","volume":"340","author":"Jiang","year":"2018","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1333","DOI":"10.1007\/s00419-020-01670-x","article-title":"Inverse algorithm for real-time road roughness estimation for autonomous vehicles","volume":"90","author":"Jiang","year":"2020","journal-title":"Arch. Appl. Mech."},{"key":"ref_26","unstructured":"Ory, H., Glaser, H., and Holzdeppe, D. (1985, January 1\u20133). Reconstruction of forcing functions based on measured structural responses. Proceedings of the 2nd International Symposium on Aeroelasticity and Structural Dynamics, Aachen, Germany."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"11","DOI":"10.1007\/BF02325828","article-title":"Two methods for determining impact-force history on elastic plates","volume":"35","author":"Wu","year":"1995","journal-title":"Exp. Mech."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"1092","DOI":"10.1299\/jmmp.2.1092","article-title":"Impact force estimation by quadratic spline approximation","volume":"2","author":"Gunawan","year":"2008","journal-title":"J. Solid Mech. Mater. Eng."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"0328012","DOI":"10.3788\/AOS201838.0328012","article-title":"An Algorithm of Dynamic Load Identification Based on FGB Sensor and Kalman Filter","volume":"38","author":"Xuegang","year":"2018","journal-title":"Acta Opt. Sin."},{"key":"ref_30","first-page":"35","article-title":"A New Approach to Linear Filtering and Prediction Problems","volume":"82","author":"Kalman","year":"1960","journal-title":"J. Fluids Eng."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"497","DOI":"10.1002\/stc.171","article-title":"An adaptive extended Kalman filter for structural damage identifications II: Unknown inputs","volume":"14","author":"Yang","year":"2007","journal-title":"Struct. Control Health Monit."},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Lenkutis, T., Cer\u0161kus, A., and \u0160e\u0161ok, N. (2021). Road Surface Profile Synthesis: Assessment of Suitability for Simulation. Symmetry, 13.","DOI":"10.3390\/sym13010068"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"229","DOI":"10.1016\/j.ymssp.2011.07.026","article-title":"Structural damage detection with limited input and output measurement signals","volume":"28","author":"Lei","year":"2012","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"1167","DOI":"10.1016\/j.cma.2014.08.010","article-title":"An online coupled state\/input\/parameter estimation approach for structural dynamics","volume":"283","author":"Naets","year":"2015","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Du, H., Kong, W., Wang, Y., Liu, W., Huang, M., Zhang, W., and Tang, M. (2020). Research on Rotorcraft Blade Tip Vortex Identification and Motion Characteristics in Hovering State. Symmetry, 12.","DOI":"10.3390\/sym12020196"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"29","DOI":"10.1016\/j.engstruct.2015.04.019","article-title":"An efficient algorithm for simultaneous identification of time-varying structural parameters and unknown excitations of a building structure","volume":"98","author":"Huang","year":"2015","journal-title":"Eng. Struct."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"120","DOI":"10.1016\/j.ymssp.2019.03.013","article-title":"A novel unscented Kalman filter for recursive state-input-system identification of nonlinear systems","volume":"127","author":"Lei","year":"2019","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_38","first-page":"221","article-title":"Identification of Structural Parameters and Unknown Excitations Basend on the Extended Kalman Filter","volume":"36","author":"Xiaoxiong","year":"2019","journal-title":"Eng. Mech."},{"key":"ref_39","unstructured":"Anhua, Y. (2019). Parameter Identification of Shear-Type Frame Structures Based on Kalman Filter, Fujian Agriculture and Forestry University."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/8\/1372\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T06:36:08Z","timestamp":1760164568000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/8\/1372"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,7,28]]},"references-count":39,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2021,8]]}},"alternative-id":["sym13081372"],"URL":"https:\/\/doi.org\/10.3390\/sym13081372","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2021,7,28]]}}}