{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T06:42:47Z","timestamp":1740120167629,"version":"3.37.3"},"reference-count":28,"publisher":"World Scientific Pub Co Pte Ltd","issue":"10","funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11672007"],"award-info":[{"award-number":["11672007"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11832002"],"award-info":[{"award-number":["11832002"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100004826","name":"Natural Science Foundation of Beijing Municipality","doi-asserted-by":"publisher","award":["3172003"],"award-info":[{"award-number":["3172003"]}],"id":[{"id":"10.13039\/501100004826","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Bifurcation Chaos"],"published-print":{"date-parts":[[2019,9]]},"abstract":"<jats:p> In this paper, we investigate systematically the vibration of a typical 2DOF nonlinear system with repeated linearized natural frequencies. By application of Descartes\u2019 rule of signs, we demonstrate that there are 14 types of roots describing different modal motions for varying nonlinear parameters. The method of multiple scales is used to obtain the amplitude-phase portraits by introducing the energy ratios and phase differences. The typical nonlinear in-unison and elliptic out-of-unison modal motions are located for the 14 types of roots and then validated by numerical simulations. It is found that the normal in-unison modal motions, elliptic out-of-unison modal motions are analogous to the polarization of classical optic theory. Further, some kinds of periodic and chaotic motions under out-of-unison and in-unison excitations are investigated numerically. The result of this study offers a detailed discussion of nonlinear modal motions and responses of 2DOF systems with cubic nonlinear terms. <\/jats:p>","DOI":"10.1142\/s0218127419501323","type":"journal-article","created":{"date-parts":[[2019,9,24]],"date-time":"2019-09-24T05:04:31Z","timestamp":1569301471000},"page":"1950132","source":"Crossref","is-referenced-by-count":0,"title":["On Nonlinear Motions of Two-Degree-of-Freedom Nonlinear Systems with Repeated Linearized Natural Frequencies"],"prefix":"10.1142","volume":"29","author":[{"given":"Hua-Zhen","family":"An","sequence":"first","affiliation":[{"name":"Institute of Advanced Structure Technology, Beijing Institute of Technology, Beijing 100081, P. R. 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