{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,14]],"date-time":"2026-07-14T17:08:46Z","timestamp":1784048926115,"version":"3.55.0"},"reference-count":34,"publisher":"MDPI AG","issue":"23","license":[{"start":{"date-parts":[[2021,11,28]],"date-time":"2021-11-28T00:00:00Z","timestamp":1638057600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12002298"],"award-info":[{"award-number":["12002298"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100002858","name":"China Postdoctoral Science Foundation","doi-asserted-by":"publisher","award":["2020M681578"],"award-info":[{"award-number":["2020M681578"]}],"id":[{"id":"10.13039\/501100002858","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Shandong Provincial Natural Science Foundation, China","award":["ZR2020QA038"],"award-info":[{"award-number":["ZR2020QA038"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Materials"],"abstract":"<jats:p>A dynamic model of an L-shaped multi-beam joint structure is presented to investigate the nonlinear dynamic behavior of the system. Firstly, the nonlinear partial differential equations (PDEs) of motion for the beams, the governing equations of the tip mass, and their matching conditions and boundary conditions are obtained. The natural frequencies and the global mode shapes of the linearized model of the system are determined, and the orthogonality relations of the global mode shapes are established. Then, the global mode shapes and their orthogonality relations are used to derive a set of nonlinear ordinary differential equations (ODEs) that govern the motion of the L-shaped multi-beam jointed structure. The accuracy of the model is verified by the comparison of the natural frequencies solved by the frequency equation and the ANSYS. Based on the nonlinear ODEs obtained in this model, the dynamic responses are worked out to investigate the effect of the tip mass and the joint on the nonlinear dynamic characteristic of the system. The results show that the inertia of the tip mass and the nonlinear stiffness of the joints have a great influence on the nonlinear response of the system.<\/jats:p>","DOI":"10.3390\/ma14237279","type":"journal-article","created":{"date-parts":[[2021,12,1]],"date-time":"2021-12-01T05:02:36Z","timestamp":1638334956000},"page":"7279","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":10,"title":["Nonlinear Dynamic Modeling and Analysis of an L-Shaped Multi-Beam Jointed Structure with Tip Mass"],"prefix":"10.3390","volume":"14","author":[{"given":"Jin","family":"Wei","sequence":"first","affiliation":[{"name":"School of Electromechanical and Automotive Engineering, Yantai University, Yantai 264005, China"},{"name":"State Key Laboratory of Mechanics and Control of Mechanical Structures, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Tao","family":"Yu","sequence":"additional","affiliation":[{"name":"School of Electromechanical and Automotive Engineering, Yantai University, Yantai 264005, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Dongping","family":"Jin","sequence":"additional","affiliation":[{"name":"State Key Laboratory of Mechanics and Control of Mechanical Structures, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Mei","family":"Liu","sequence":"additional","affiliation":[{"name":"School of Astronautics, Harbin Institute of Technology, Harbin 150001, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Dengqing","family":"Cao","sequence":"additional","affiliation":[{"name":"School of Astronautics, Harbin Institute of Technology, Harbin 150001, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jinjie","family":"Wang","sequence":"additional","affiliation":[{"name":"School of Electromechanical and Automotive Engineering, Yantai University, Yantai 264005, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2021,11,28]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"104214","DOI":"10.1016\/j.euromechsol.2021.104214","article-title":"Vibration analysis and distributed piezoelectric energy harvester design for the L-shaped beam","volume":"87","author":"Cao","year":"2021","journal-title":"Eur. 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