{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,21]],"date-time":"2026-02-21T01:07:21Z","timestamp":1771636041976,"version":"3.50.1"},"reference-count":20,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2022,11,18]],"date-time":"2022-11-18T00:00:00Z","timestamp":1668729600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Key Laboratory of Science and Technology on Rotorcraft Aeromechanics","award":["61422202105"],"award-info":[{"award-number":["61422202105"]}]},{"name":"National Key Laboratory of Science and Technology on Rotorcraft Aeromechanics","award":["51775270"],"award-info":[{"award-number":["51775270"]}]},{"name":"Qing Lan Project and the National Natural Science Foundation of China","award":["61422202105"],"award-info":[{"award-number":["61422202105"]}]},{"name":"Qing Lan Project and the National Natural Science Foundation of China","award":["51775270"],"award-info":[{"award-number":["51775270"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The inverse problem and the direct problem are symmetrical to each other. As a mathematical method for inverse problems, dynamic load identification is applicable to the situation when the load acting on the structure is difficult to measure directly. In addition, in most practical fields, the exact value of the structural parameters cannot be obtained precisely, which makes the inverse problem beyond the capabilities of traditional dynamic load identification methods. Hence, in this work, we propose a dynamic load identification algorithm based on the extended Kalman filter (EKF) for a structure with unknown parameters. The algorithm is discussed under different conditions where the unknown parameters are either the stiffness or the mass of the structure. Such a case has not been considered in other literature yet. In order to verify the advantages of the proposed method, the recursive least square method was also used to compare the results. A 5-Dof symmetric system with unknown coefficients was selected for numerical simulation examples, and the influence of noise on the algorithm was also considered in the simulation. The results show that the proposed algorithm is effective for structures with unknown mass and stiffness coefficients. Compared with the recursive least square method, the method proposed in this paper has the higher accuracy and a wider application scope.<\/jats:p>","DOI":"10.3390\/sym14112449","type":"journal-article","created":{"date-parts":[[2022,11,18]],"date-time":"2022-11-18T03:57:44Z","timestamp":1668743864000},"page":"2449","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["Dynamic Load Identification for Structures with Unknown Parameters"],"prefix":"10.3390","volume":"14","author":[{"given":"Hongzhi","family":"Tang","sequence":"first","affiliation":[{"name":"State Key Laboratory of Mechanics and Control of Mechanical Structures, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"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"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"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"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Fang","family":"Zhang","sequence":"additional","affiliation":[{"name":"State Key Laboratory of Mechanics and Control of Mechanical Structures, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0680-6908","authenticated-orcid":false,"given":"Xu","family":"Wang","sequence":"additional","affiliation":[{"name":"College of Mechanical and Electronic Engineering, Nanjing Forestry University, Nanjing 210037, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,11,18]]},"reference":[{"key":"ref_1","first-page":"235","article-title":"Time series method for identifying load using impulse response function","volume":"8","author":"Shi","year":"1995","journal-title":"J. Vib. Eng."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"71","DOI":"10.1016\/j.jsv.2016.01.030","article-title":"Sparse regularization for force identification using dictionaries","volume":"368","author":"Qiao","year":"2016","journal-title":"J. Sound Vib."},{"key":"ref_3","first-page":"8261423","article-title":"Adaptive Boundary Control of Flexible Manipulators with Parameter Uncertainty Based on RBF Neural Network","volume":"2020","author":"He","year":"2020","journal-title":"Shock. Vib."},{"key":"ref_4","first-page":"967","article-title":"A Time Domain Identification Method for Distributed Dynamic Loads in Spatial Structures","volume":"38","author":"Wan","year":"2017","journal-title":"Appl. Math. Mech."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"107881","DOI":"10.1016\/j.ymssp.2021.107881","article-title":"A novel time-domain dynamic load identification numerical algorithm for continuous system","volume":"160","author":"Jiang","year":"2021","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"116465","DOI":"10.1016\/j.jsv.2021.116465","article-title":"One novel dynamic calibration method to identify two-dimensional distributed load","volume":"515","author":"Jiang","year":"2021","journal-title":"J. Sound Vib."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Jiang, J., Tang, H., Mohamed, M.S., Luo, S., and Chen, J. (2020). Augmented Tikhonov Regularization Method for Dynamic Load Identification. Appl. Sci., 10.","DOI":"10.3390\/app10186348"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"109757","DOI":"10.1016\/j.ymssp.2022.109757","article-title":"Dynamic load identification based on deep convolution neural network","volume":"185","author":"Yang","year":"2023","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"572","DOI":"10.1016\/j.jsv.2008.10.005","article-title":"Identification of linear time-varying mdof dynamic systems from forced excitation using Hilbert transform and EMD method","volume":"321","author":"Shi","year":"2009","journal-title":"J. Sound Vib."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"105007","DOI":"10.1088\/0964-1726\/21\/10\/105007","article-title":"A Hilbert transform method for parameter identification of time-varying structures with observer techniques","volume":"21","author":"Wang","year":"2012","journal-title":"Smart Mater. Struct."},{"key":"ref_11","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_12","unstructured":"Yoshida, I. (2001, January 17\u201322). Damage detection using Monte Carlo filter based on non-Gaussian noises. Proceedings of the 8th International Conference on Structural Safety and Reliability, Newport Beach, CA, USA."},{"key":"ref_13","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_14","doi-asserted-by":"crossref","first-page":"12","DOI":"10.1061\/(ASCE)0733-9399(2007)133:1(12)","article-title":"Least-Squares Estimation with Unknown Excitations for Damage Identification of Structures","volume":"133","author":"Yang","year":"2007","journal-title":"J. Eng. Mech."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"953","DOI":"10.1016\/S0022-460X(03)00797-1","article-title":"An inverse method for the estimation of input forces acting on non-linear structural systems","volume":"275","author":"Ma","year":"2004","journal-title":"J. Sound Vib."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"111","DOI":"10.1016\/j.automatica.2006.08.002","article-title":"Unbiased minimum-variance input and state estimation for linear discrete-time systems","volume":"43","author":"Gillijns","year":"2007","journal-title":"Automatica"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"866","DOI":"10.1016\/j.ymssp.2015.02.001","article-title":"A dual Kalman filter approach for state estimation via output-only acceleration measurements","volume":"60","author":"Azam","year":"2015","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_18","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_19","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_20","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."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/11\/2449\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:21:02Z","timestamp":1760145662000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/11\/2449"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,11,18]]},"references-count":20,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2022,11]]}},"alternative-id":["sym14112449"],"URL":"https:\/\/doi.org\/10.3390\/sym14112449","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,11,18]]}}}