{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,17]],"date-time":"2026-02-17T04:10:02Z","timestamp":1771301402797,"version":"3.50.1"},"reference-count":33,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2023,6,2]],"date-time":"2023-06-02T00:00:00Z","timestamp":1685664000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Zhejiang public welfare technology application research project","award":["LGF22E080021"],"award-info":[{"award-number":["LGF22E080021"]}]},{"name":"Zhejiang public welfare technology application research project","award":["202003N4169"],"award-info":[{"award-number":["202003N4169"]}]},{"name":"Zhejiang public welfare technology application research project","award":["11202138"],"award-info":[{"award-number":["11202138"]}]},{"name":"Zhejiang public welfare technology application research project","award":["52008215"],"award-info":[{"award-number":["52008215"]}]},{"name":"Zhejiang public welfare technology application research project","award":["LQ20E080013"],"award-info":[{"award-number":["LQ20E080013"]}]},{"name":"Zhejiang public welfare technology application research project","award":["2019B10076"],"award-info":[{"award-number":["2019B10076"]}]},{"name":"Ningbo natural science foundation project","award":["LGF22E080021"],"award-info":[{"award-number":["LGF22E080021"]}]},{"name":"Ningbo natural science foundation project","award":["202003N4169"],"award-info":[{"award-number":["202003N4169"]}]},{"name":"Ningbo natural science foundation project","award":["11202138"],"award-info":[{"award-number":["11202138"]}]},{"name":"Ningbo natural science foundation project","award":["52008215"],"award-info":[{"award-number":["52008215"]}]},{"name":"Ningbo natural science foundation project","award":["LQ20E080013"],"award-info":[{"award-number":["LQ20E080013"]}]},{"name":"Ningbo natural science foundation project","award":["2019B10076"],"award-info":[{"award-number":["2019B10076"]}]},{"name":"Natural Science Foundation of China","award":["LGF22E080021"],"award-info":[{"award-number":["LGF22E080021"]}]},{"name":"Natural Science Foundation of China","award":["202003N4169"],"award-info":[{"award-number":["202003N4169"]}]},{"name":"Natural Science Foundation of China","award":["11202138"],"award-info":[{"award-number":["11202138"]}]},{"name":"Natural Science Foundation of China","award":["52008215"],"award-info":[{"award-number":["52008215"]}]},{"name":"Natural Science Foundation of China","award":["LQ20E080013"],"award-info":[{"award-number":["LQ20E080013"]}]},{"name":"Natural Science Foundation of China","award":["2019B10076"],"award-info":[{"award-number":["2019B10076"]}]},{"name":"Natural Science Foundation of Zhejiang Province, China","award":["LGF22E080021"],"award-info":[{"award-number":["LGF22E080021"]}]},{"name":"Natural Science Foundation of Zhejiang Province, China","award":["202003N4169"],"award-info":[{"award-number":["202003N4169"]}]},{"name":"Natural Science Foundation of Zhejiang Province, China","award":["11202138"],"award-info":[{"award-number":["11202138"]}]},{"name":"Natural Science Foundation of Zhejiang Province, China","award":["52008215"],"award-info":[{"award-number":["52008215"]}]},{"name":"Natural Science Foundation of Zhejiang Province, China","award":["LQ20E080013"],"award-info":[{"award-number":["LQ20E080013"]}]},{"name":"Natural Science Foundation of Zhejiang Province, China","award":["2019B10076"],"award-info":[{"award-number":["2019B10076"]}]},{"name":"\u201cNingbo science and technology innovation 2025\u201d","award":["LGF22E080021"],"award-info":[{"award-number":["LGF22E080021"]}]},{"name":"\u201cNingbo science and technology innovation 2025\u201d","award":["202003N4169"],"award-info":[{"award-number":["202003N4169"]}]},{"name":"\u201cNingbo science and technology innovation 2025\u201d","award":["11202138"],"award-info":[{"award-number":["11202138"]}]},{"name":"\u201cNingbo science and technology innovation 2025\u201d","award":["52008215"],"award-info":[{"award-number":["52008215"]}]},{"name":"\u201cNingbo science and technology innovation 2025\u201d","award":["LQ20E080013"],"award-info":[{"award-number":["LQ20E080013"]}]},{"name":"\u201cNingbo science and technology innovation 2025\u201d","award":["2019B10076"],"award-info":[{"award-number":["2019B10076"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>The diagnosis of structural damage usually belongs to a mathematical inverse problem. This work presents a novel frequency-shift flexibility sensitivity algorithm for structural damage assessment using only the first-order vibration mode to achieve the goal of successfully identifying structural damage with fewer modal parameters. The core idea of the proposed method is to make the first-order vibration mode contribute the most to a structural flexibility matrix through the frequency-shift operation. A high-precision flexibility matrix can be obtained after the frequency-shift operation, which only needs the first mode of structural free vibration. Through this special advantage, structural damage coefficients can be accurately calculated by the frequency-shift flexibility sensitivity equation. Thus, a reliable identification result can be obtained according to the values of the calculated damage coefficients. In some engineering applications, another advantage of the proposed method is that it does not require a complete finite element modeling process, as long as a few lower-frequency vibration modes of the intact structure are measured. A truss structure and a beam structure are used as two numerical examples to demonstrate the proposed approach. The results show that the proposed method has higher calculation accuracy than the ordinary flexibility sensitivity method by using only the first-order vibration mode. The proposed method can overcome possible misdiagnosis of the ordinary flexibility sensitivity method. It also has been shown that the proposed method may have the potential to identify minor damage in a structure. Using the experimental data of a steel frame structure, the effectiveness and reliability of the proposed method have been further verified. The proposed method provides a simple way for structural damage identification with only a few vibration modal data.<\/jats:p>","DOI":"10.3390\/axioms12060551","type":"journal-article","created":{"date-parts":[[2023,6,5]],"date-time":"2023-06-05T02:57:47Z","timestamp":1685933867000},"page":"551","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Structural Damage Identification Using the First-Order Vibration-Mode-Based Frequency-Shift Flexibility Sensitivity Algorithm"],"prefix":"10.3390","volume":"12","author":[{"given":"Shanshan","family":"Cao","sequence":"first","affiliation":[{"name":"Experimental College, The Open University of China, Beijing 100039, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Qiuwei","family":"Yang","sequence":"additional","affiliation":[{"name":"School of Civil and Transportation Engineering, Ningbo University of Technology, Ningbo 315211, China"},{"name":"Engineering Research Center of Industrial Construction in Civil Engineering of Zhejiang, Ningbo University of Technology, Ningbo 315211, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3104-5434","authenticated-orcid":false,"given":"Xi","family":"Peng","sequence":"additional","affiliation":[{"name":"School of Civil and Transportation Engineering, Ningbo University of Technology, Ningbo 315211, China"},{"name":"Engineering Research Center of Industrial Construction in Civil Engineering of Zhejiang, Ningbo University of Technology, Ningbo 315211, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,6,2]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"110376","DOI":"10.1016\/j.ymssp.2023.110376","article-title":"A multi-fidelity surrogate model for structural health monitoring exploiting model order reduction and artificial neural networks","volume":"197","author":"Torzoni","year":"2023","journal-title":"Mech. 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