{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,19]],"date-time":"2025-10-19T14:50:46Z","timestamp":1760885446140},"reference-count":22,"publisher":"World Scientific Pub Co Pte Lt","issue":"01","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Bifurcation Chaos"],"published-print":{"date-parts":[[2009,1]]},"abstract":"<jats:p> The nonlinear dynamic responses of a string-beam coupled system subjected to harmonic external and parametric excitations are studied in this work in the case of 1:2 internal resonance between the modes of the beam and string. First, the nonlinear governing equations of motion for the string-beam coupled system are established. Then, the Galerkin's method is used to simplify the nonlinear governing equations to a set of ordinary differential equations with four-degrees-of-freedom. Utilizing the method of multiple scales, the eight-dimensional averaged equation is obtained. The case of 1:2 internal resonance between the modes of the beam and string \u2014 principal parametric resonance-1\/2 subharmonic resonance for the beam and primary resonance for the string \u2014 is considered. Finally, nonlinear dynamic characteristics of the string-beam coupled system are studied through a numerical method based on the averaged equation. The phase portrait, Poincare map and power spectrum are plotted to demonstrate that the periodic and chaotic motions exist in the string-beam coupled system under certain conditions. <\/jats:p>","DOI":"10.1142\/s0218127409022841","type":"journal-article","created":{"date-parts":[[2009,3,20]],"date-time":"2009-03-20T06:36:08Z","timestamp":1237530968000},"page":"225-243","source":"Crossref","is-referenced-by-count":3,"title":["NONLINEAR RESPONSES OF A STRING-BEAM COUPLED SYSTEM WITH FOUR-DEGREES-OF-FREEDOM AND MULTIPLE PARAMETRIC EXCITATIONS"],"prefix":"10.1142","volume":"19","author":[{"given":"D. X.","family":"CAO","sequence":"first","affiliation":[{"name":"College of Mechanical Engineering, Beijing University of Technology, Beijing 100124, P. R. China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"W.","family":"ZHANG","sequence":"additional","affiliation":[{"name":"College of Mechanical Engineering, Beijing University of Technology, Beijing 100124, P. R. 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