On the dynamics and unstable region of a class of parametrically excited systems similar to a beam with an axially reciprocating mid-support
By means of Euler–Bernoulli beam theory and the Lagrange equation of the first kind, the dynamic equation of a beam with an axially harmonic reciprocating moving mid-support (beam-AHRMS) is established. Using two modal coordinates, the equation of the beam-AHRMS is reduced to a special form of Hill’...
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Veröffentlicht in: | Nonlinear dynamics 2025, Vol.113 (1), p.87-108 |
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description | By means of Euler–Bernoulli beam theory and the Lagrange equation of the first kind, the dynamic equation of a beam with an axially harmonic reciprocating moving mid-support (beam-AHRMS) is established. Using two modal coordinates, the equation of the beam-AHRMS is reduced to a special form of Hill’s equation, the particularity of which is that its excitation, i.e., time-dependent variable coefficients in front of the state variable of the equation, is a nonlinear function of the excitation in Mathieu’s equation. Using the perturbation method, the approximate solution for the transition curves between the stable and unstable regions of the system is obtained. The most different behaviour of the special form of Hill’s equation is that there exists a knot in the principal unstable region, where the bandwidth of the unstable region becomes zero for any frequency, and the principal parameter resonance is suppressed. A theorem for the occurrence and location of knots in the principal unstable region is proposed and proven. All the results of the analysis and the theorem are verified via a numerical method. |
doi_str_mv | 10.1007/s11071-024-10248-z |
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Using two modal coordinates, the equation of the beam-AHRMS is reduced to a special form of Hill’s equation, the particularity of which is that its excitation, i.e., time-dependent variable coefficients in front of the state variable of the equation, is a nonlinear function of the excitation in Mathieu’s equation. Using the perturbation method, the approximate solution for the transition curves between the stable and unstable regions of the system is obtained. The most different behaviour of the special form of Hill’s equation is that there exists a knot in the principal unstable region, where the bandwidth of the unstable region becomes zero for any frequency, and the principal parameter resonance is suppressed. A theorem for the occurrence and location of knots in the principal unstable region is proposed and proven. 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Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><cites>FETCH-LOGICAL-c200t-da6ba619e94bf7f50a82c2e43177975285a48e3b1e6beedb5df40e60929c0b6b3</cites><orcidid>0000-0003-3901-7208</orcidid></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://link.springer.com/content/pdf/10.1007/s11071-024-10248-z$$EPDF$$P50$$Gspringer$$H</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1007/s11071-024-10248-z$$EHTML$$P50$$Gspringer$$H</linktohtml><link.rule.ids>314,776,780,27903,27904,41467,42536,51297</link.rule.ids></links><search><creatorcontrib>Ding, Weigao</creatorcontrib><creatorcontrib>Xie, Jin</creatorcontrib><title>On the dynamics and unstable region of a class of parametrically excited systems similar to a beam with an axially reciprocating mid-support</title><title>Nonlinear dynamics</title><addtitle>Nonlinear Dyn</addtitle><description>By means of Euler–Bernoulli beam theory and the Lagrange equation of the first kind, the dynamic equation of a beam with an axially harmonic reciprocating moving mid-support (beam-AHRMS) is established. Using two modal coordinates, the equation of the beam-AHRMS is reduced to a special form of Hill’s equation, the particularity of which is that its excitation, i.e., time-dependent variable coefficients in front of the state variable of the equation, is a nonlinear function of the excitation in Mathieu’s equation. Using the perturbation method, the approximate solution for the transition curves between the stable and unstable regions of the system is obtained. The most different behaviour of the special form of Hill’s equation is that there exists a knot in the principal unstable region, where the bandwidth of the unstable region becomes zero for any frequency, and the principal parameter resonance is suppressed. A theorem for the occurrence and location of knots in the principal unstable region is proposed and proven. All the results of the analysis and the theorem are verified via a numerical method.</description><subject>Applications of Nonlinear Dynamics and Chaos Theory</subject><subject>Beam theory (structures)</subject><subject>Classical Mechanics</subject><subject>Control</subject><subject>Dependent variables</subject><subject>Dynamical Systems</subject><subject>Engineering</subject><subject>Euler-Bernoulli beams</subject><subject>Euler-Lagrange equation</subject><subject>Excitation</subject><subject>Knots</subject><subject>Methods</subject><subject>Numerical analysis</subject><subject>Numerical methods</subject><subject>Original Paper</subject><subject>Perturbation methods</subject><subject>Physics</subject><subject>Physics and Astronomy</subject><subject>Statistical Physics and Dynamical Systems</subject><subject>Theorems</subject><subject>Vibration</subject><issn>0924-090X</issn><issn>1573-269X</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2025</creationdate><recordtype>article</recordtype><recordid>eNp9kMtqHDEQRUWIIRPHP5CVIOtOSuqHWstg8gKDNwl4J0rq6rFMv6LSEI-_wR9t2RPILhuVoO65EkeI9wo-KgDziZUCoyrQTaXK0VcPr8ROtaaudGdvXosd2LICCzdvxFvmOwCoNfQ78Xi9yHxLcjguOMfAEpdBHhbO6CeSifZxXeQ6SpRhQubn64YJZ8opBpymo6T7EDMNko-caWbJcY4TJpnXAnnCWf6J-bb0SryPL0SiELe0Bsxx2cs5DhUftm1N-Z04G3Fiuvg7z8Wvr19-Xn6vrq6__bj8fFUFDZCrATuPnbJkGz-asQXsddDU1MoYa1rdt9j0VHtFnScafDuMDVBXFNgAvvP1ufhw6i2_-H0gzu5uPaSlPOlqpW1vegO2pPQpFdLKnGh0W4ozpqNT4J6tu5N1V4S7F-vuoUD1CeISXvaU_lX_h3oChhSIxw</recordid><startdate>2025</startdate><enddate>2025</enddate><creator>Ding, Weigao</creator><creator>Xie, Jin</creator><general>Springer Netherlands</general><general>Springer Nature B.V</general><scope>AAYXX</scope><scope>CITATION</scope><orcidid>https://orcid.org/0000-0003-3901-7208</orcidid></search><sort><creationdate>2025</creationdate><title>On the dynamics and unstable region of a class of parametrically excited systems similar to a beam with an axially reciprocating mid-support</title><author>Ding, Weigao ; Xie, Jin</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c200t-da6ba619e94bf7f50a82c2e43177975285a48e3b1e6beedb5df40e60929c0b6b3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2025</creationdate><topic>Applications of Nonlinear Dynamics and Chaos Theory</topic><topic>Beam theory (structures)</topic><topic>Classical Mechanics</topic><topic>Control</topic><topic>Dependent variables</topic><topic>Dynamical Systems</topic><topic>Engineering</topic><topic>Euler-Bernoulli beams</topic><topic>Euler-Lagrange equation</topic><topic>Excitation</topic><topic>Knots</topic><topic>Methods</topic><topic>Numerical analysis</topic><topic>Numerical methods</topic><topic>Original Paper</topic><topic>Perturbation methods</topic><topic>Physics</topic><topic>Physics and Astronomy</topic><topic>Statistical Physics and Dynamical Systems</topic><topic>Theorems</topic><topic>Vibration</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Ding, Weigao</creatorcontrib><creatorcontrib>Xie, Jin</creatorcontrib><collection>CrossRef</collection><jtitle>Nonlinear dynamics</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Ding, Weigao</au><au>Xie, Jin</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>On the dynamics and unstable region of a class of parametrically excited systems similar to a beam with an axially reciprocating mid-support</atitle><jtitle>Nonlinear dynamics</jtitle><stitle>Nonlinear Dyn</stitle><date>2025</date><risdate>2025</risdate><volume>113</volume><issue>1</issue><spage>87</spage><epage>108</epage><pages>87-108</pages><issn>0924-090X</issn><eissn>1573-269X</eissn><abstract>By means of Euler–Bernoulli beam theory and the Lagrange equation of the first kind, the dynamic equation of a beam with an axially harmonic reciprocating moving mid-support (beam-AHRMS) is established. Using two modal coordinates, the equation of the beam-AHRMS is reduced to a special form of Hill’s equation, the particularity of which is that its excitation, i.e., time-dependent variable coefficients in front of the state variable of the equation, is a nonlinear function of the excitation in Mathieu’s equation. Using the perturbation method, the approximate solution for the transition curves between the stable and unstable regions of the system is obtained. The most different behaviour of the special form of Hill’s equation is that there exists a knot in the principal unstable region, where the bandwidth of the unstable region becomes zero for any frequency, and the principal parameter resonance is suppressed. A theorem for the occurrence and location of knots in the principal unstable region is proposed and proven. All the results of the analysis and the theorem are verified via a numerical method.</abstract><cop>Dordrecht</cop><pub>Springer Netherlands</pub><doi>10.1007/s11071-024-10248-z</doi><tpages>22</tpages><orcidid>https://orcid.org/0000-0003-3901-7208</orcidid></addata></record> |
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subjects | Applications of Nonlinear Dynamics and Chaos Theory Beam theory (structures) Classical Mechanics Control Dependent variables Dynamical Systems Engineering Euler-Bernoulli beams Euler-Lagrange equation Excitation Knots Methods Numerical analysis Numerical methods Original Paper Perturbation methods Physics Physics and Astronomy Statistical Physics and Dynamical Systems Theorems Vibration |
title | On the dynamics and unstable region of a class of parametrically excited systems similar to a beam with an axially reciprocating mid-support |
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