Supercritical nonlinear transverse vibration of a hyperelastic beam under harmonic axial loading
Dynamics of a hyperelastic beam in a buckled state subjected to a harmonic axial load is investigated in this work. For the static case, the buckled configuration of the hyperelastic beam is first determined through an asymptotic method when the axial load is in excess of its critical load. Then, th...
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Veröffentlicht in: | Communications in nonlinear science & numerical simulation 2022-09, Vol.112, p.106536, Article 106536 |
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description | Dynamics of a hyperelastic beam in a buckled state subjected to a harmonic axial load is investigated in this work. For the static case, the buckled configuration of the hyperelastic beam is first determined through an asymptotic method when the axial load is in excess of its critical load. Then, the governing equation of vibration of the buckled beam is derived, which is a complex nonlinear partial differential equation with varying coefficients. The first natural frequency of the buckled beam is obtained by linear analysis and effects of material and geometrical parameters on it are numerically investigated. By applying the Galerkin method, the governing equation for nonlinear transverse vibration of the beam in the buckled state is transformed into a series of strongly nonlinear ordinary differential equations. Dynamic characteristics of the hyperelastic beam are investigated by the Runge–Kutta method. Bifurcation diagrams, time traces, phase-plane portraits, and Poincare sections are obtained for different values of the external excitation frequency and amplitude of variation of the axial load. Results show that different dynamic behaviors of the hyperelastic buckled beam such as periodic, multi-periodic, quasiperiodic, and chaotic motions can be found when the amplitude of the axial load is varied. Amplitude–frequency responses of the hyperelastic buckled beam are obtained by the Runge–Kutta method and harmonic balance method. Results from the Runge–Kutta method are in good agreement with those from the harmonic balance method. Lastly, effects of the external mean axial load, amplitude of variation of the axial load, geometrical and material parameters on amplitude–frequency responses of the buckled beam are numerically investigated.
•Nonlinear dynamics of a hyperelastic buckled beam is studied.•Natural frequencies of the buckled beam are determined.•Periodic, multi-periodic, quasiperiodic, and chaotic motions are found.•Amplitude–frequency responses of the buckled beam are presented.•Results from numerical and harmonic balance methods are in good agreement. |
doi_str_mv | 10.1016/j.cnsns.2022.106536 |
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•Nonlinear dynamics of a hyperelastic buckled beam is studied.•Natural frequencies of the buckled beam are determined.•Periodic, multi-periodic, quasiperiodic, and chaotic motions are found.•Amplitude–frequency responses of the buckled beam are presented.•Results from numerical and harmonic balance methods are in good agreement.</description><identifier>ISSN: 1007-5704</identifier><identifier>EISSN: 1878-7274</identifier><identifier>DOI: 10.1016/j.cnsns.2022.106536</identifier><language>eng</language><publisher>Amsterdam: Elsevier B.V</publisher><subject>Amplitudes ; Amplitude–frequency response ; Asymptotic methods ; Axial loads ; Dynamic characteristics ; Galerkin method ; Harmonic analysis ; Harmonic balance method ; Hyperelastic buckled beam ; Linear analysis ; Nonlinear differential equations ; Nonlinear equations ; Nonlinear systems ; Nonlinear transverse vibration ; Parameters ; Partial differential equations ; Resonant frequencies ; Runge-Kutta method ; Transverse oscillation ; Vibration ; Vibration analysis</subject><ispartof>Communications in nonlinear science & numerical simulation, 2022-09, Vol.112, p.106536, Article 106536</ispartof><rights>2022 Elsevier B.V.</rights><rights>Copyright Elsevier Science Ltd. Sep 2022</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c331t-cd980560b88123d353914322cf1fe0a3c2ca066fffeb92926df5d7266c7d8dea3</citedby><cites>FETCH-LOGICAL-c331t-cd980560b88123d353914322cf1fe0a3c2ca066fffeb92926df5d7266c7d8dea3</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktohtml>$$Uhttps://dx.doi.org/10.1016/j.cnsns.2022.106536$$EHTML$$P50$$Gelsevier$$H</linktohtml><link.rule.ids>314,780,784,3550,27924,27925,45995</link.rule.ids></links><search><creatorcontrib>Wang, Yuanbin</creatorcontrib><creatorcontrib>Zhu, Weidong</creatorcontrib><title>Supercritical nonlinear transverse vibration of a hyperelastic beam under harmonic axial loading</title><title>Communications in nonlinear science & numerical simulation</title><description>Dynamics of a hyperelastic beam in a buckled state subjected to a harmonic axial load is investigated in this work. For the static case, the buckled configuration of the hyperelastic beam is first determined through an asymptotic method when the axial load is in excess of its critical load. Then, the governing equation of vibration of the buckled beam is derived, which is a complex nonlinear partial differential equation with varying coefficients. The first natural frequency of the buckled beam is obtained by linear analysis and effects of material and geometrical parameters on it are numerically investigated. By applying the Galerkin method, the governing equation for nonlinear transverse vibration of the beam in the buckled state is transformed into a series of strongly nonlinear ordinary differential equations. Dynamic characteristics of the hyperelastic beam are investigated by the Runge–Kutta method. Bifurcation diagrams, time traces, phase-plane portraits, and Poincare sections are obtained for different values of the external excitation frequency and amplitude of variation of the axial load. Results show that different dynamic behaviors of the hyperelastic buckled beam such as periodic, multi-periodic, quasiperiodic, and chaotic motions can be found when the amplitude of the axial load is varied. Amplitude–frequency responses of the hyperelastic buckled beam are obtained by the Runge–Kutta method and harmonic balance method. Results from the Runge–Kutta method are in good agreement with those from the harmonic balance method. Lastly, effects of the external mean axial load, amplitude of variation of the axial load, geometrical and material parameters on amplitude–frequency responses of the buckled beam are numerically investigated.
•Nonlinear dynamics of a hyperelastic buckled beam is studied.•Natural frequencies of the buckled beam are determined.•Periodic, multi-periodic, quasiperiodic, and chaotic motions are found.•Amplitude–frequency responses of the buckled beam are presented.•Results from numerical and harmonic balance methods are in good agreement.</description><subject>Amplitudes</subject><subject>Amplitude–frequency response</subject><subject>Asymptotic methods</subject><subject>Axial loads</subject><subject>Dynamic characteristics</subject><subject>Galerkin method</subject><subject>Harmonic analysis</subject><subject>Harmonic balance method</subject><subject>Hyperelastic buckled beam</subject><subject>Linear analysis</subject><subject>Nonlinear differential equations</subject><subject>Nonlinear equations</subject><subject>Nonlinear systems</subject><subject>Nonlinear transverse vibration</subject><subject>Parameters</subject><subject>Partial differential equations</subject><subject>Resonant frequencies</subject><subject>Runge-Kutta method</subject><subject>Transverse oscillation</subject><subject>Vibration</subject><subject>Vibration analysis</subject><issn>1007-5704</issn><issn>1878-7274</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2022</creationdate><recordtype>article</recordtype><recordid>eNp9kEtPwzAQhC0EEqXwC7hY4pziR-IkBw6o4iUhcQDOxrHX1FFqFzup6L_HpZw57Wo036x2ELqkZEEJFdf9Qvvk04IRxrIiKi6O0Iw2dVPUrC6P805IXVQ1KU_RWUo9yVRblTP08TptIOroRqfVgH3wg_OgIh6j8mkLMQHeui6q0QWPg8UKr3aZgEGljOAO1BpP3kDEKxXXwWdNfbscNQRlnP88RydWDQku_uYcvd_fvS0fi-eXh6fl7XOhOadjoU3bkEqQrmko44ZXvKUlZ0xbaoEorplWRAhrLXQta5kwtjI1E0LXpjGg-BxdHXI3MXxNkEbZhyn6fFIy0ZSMtJSX2cUPLh1DShGs3ES3VnEnKZH7KmUvf6uU-yrlocpM3RwoyA9sHUSZtAOvwbgIepQmuH_5H9ovf2k</recordid><startdate>202209</startdate><enddate>202209</enddate><creator>Wang, Yuanbin</creator><creator>Zhu, Weidong</creator><general>Elsevier B.V</general><general>Elsevier Science Ltd</general><scope>AAYXX</scope><scope>CITATION</scope></search><sort><creationdate>202209</creationdate><title>Supercritical nonlinear transverse vibration of a hyperelastic beam under harmonic axial loading</title><author>Wang, Yuanbin ; Zhu, Weidong</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c331t-cd980560b88123d353914322cf1fe0a3c2ca066fffeb92926df5d7266c7d8dea3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2022</creationdate><topic>Amplitudes</topic><topic>Amplitude–frequency response</topic><topic>Asymptotic methods</topic><topic>Axial loads</topic><topic>Dynamic characteristics</topic><topic>Galerkin method</topic><topic>Harmonic analysis</topic><topic>Harmonic balance method</topic><topic>Hyperelastic buckled beam</topic><topic>Linear analysis</topic><topic>Nonlinear differential equations</topic><topic>Nonlinear equations</topic><topic>Nonlinear systems</topic><topic>Nonlinear transverse vibration</topic><topic>Parameters</topic><topic>Partial differential equations</topic><topic>Resonant frequencies</topic><topic>Runge-Kutta method</topic><topic>Transverse oscillation</topic><topic>Vibration</topic><topic>Vibration analysis</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Wang, Yuanbin</creatorcontrib><creatorcontrib>Zhu, Weidong</creatorcontrib><collection>CrossRef</collection><jtitle>Communications in nonlinear science & numerical simulation</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Wang, Yuanbin</au><au>Zhu, Weidong</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Supercritical nonlinear transverse vibration of a hyperelastic beam under harmonic axial loading</atitle><jtitle>Communications in nonlinear science & numerical simulation</jtitle><date>2022-09</date><risdate>2022</risdate><volume>112</volume><spage>106536</spage><pages>106536-</pages><artnum>106536</artnum><issn>1007-5704</issn><eissn>1878-7274</eissn><abstract>Dynamics of a hyperelastic beam in a buckled state subjected to a harmonic axial load is investigated in this work. For the static case, the buckled configuration of the hyperelastic beam is first determined through an asymptotic method when the axial load is in excess of its critical load. Then, the governing equation of vibration of the buckled beam is derived, which is a complex nonlinear partial differential equation with varying coefficients. The first natural frequency of the buckled beam is obtained by linear analysis and effects of material and geometrical parameters on it are numerically investigated. By applying the Galerkin method, the governing equation for nonlinear transverse vibration of the beam in the buckled state is transformed into a series of strongly nonlinear ordinary differential equations. Dynamic characteristics of the hyperelastic beam are investigated by the Runge–Kutta method. Bifurcation diagrams, time traces, phase-plane portraits, and Poincare sections are obtained for different values of the external excitation frequency and amplitude of variation of the axial load. Results show that different dynamic behaviors of the hyperelastic buckled beam such as periodic, multi-periodic, quasiperiodic, and chaotic motions can be found when the amplitude of the axial load is varied. Amplitude–frequency responses of the hyperelastic buckled beam are obtained by the Runge–Kutta method and harmonic balance method. Results from the Runge–Kutta method are in good agreement with those from the harmonic balance method. Lastly, effects of the external mean axial load, amplitude of variation of the axial load, geometrical and material parameters on amplitude–frequency responses of the buckled beam are numerically investigated.
•Nonlinear dynamics of a hyperelastic buckled beam is studied.•Natural frequencies of the buckled beam are determined.•Periodic, multi-periodic, quasiperiodic, and chaotic motions are found.•Amplitude–frequency responses of the buckled beam are presented.•Results from numerical and harmonic balance methods are in good agreement.</abstract><cop>Amsterdam</cop><pub>Elsevier B.V</pub><doi>10.1016/j.cnsns.2022.106536</doi></addata></record> |
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subjects | Amplitudes Amplitude–frequency response Asymptotic methods Axial loads Dynamic characteristics Galerkin method Harmonic analysis Harmonic balance method Hyperelastic buckled beam Linear analysis Nonlinear differential equations Nonlinear equations Nonlinear systems Nonlinear transverse vibration Parameters Partial differential equations Resonant frequencies Runge-Kutta method Transverse oscillation Vibration Vibration analysis |
title | Supercritical nonlinear transverse vibration of a hyperelastic beam under harmonic axial loading |
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