On the detection of artifacts in Harmonic Balance solutions of nonlinear oscillators
•Addition of an error criterion allows the detection of artificial solutions.•Classification of solutions in large relative error, low relative error stable and low relative error unstable ones.•Application to nonlinear oscillators showing varieties of multiple solutions. Harmonic Balance is a very...
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| Veröffentlicht in: | Applied Mathematical Modelling 2019-01, Vol.65, p.408-414 |
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| creator | von Wagner, U. Lentz, L. |
| description | •Addition of an error criterion allows the detection of artificial solutions.•Classification of solutions in large relative error, low relative error stable and low relative error unstable ones.•Application to nonlinear oscillators showing varieties of multiple solutions.
Harmonic Balance is a very popular semi-analytic method in nonlinear dynamics. It is easy to apply and is known to produce good results for numerous examples. Adding an error criterion taking into account the neglected terms allows an evaluation of the results. Looking on the therefore determined error for increasing ansatz orders, it can be evaluated whether a solution really exists or is an artifact. For the low-error solutions additionally a stability analysis is performed which allows the classification of the solutions in three types, namely in large error solutions, low error stable solutions and low error unstable solution. Examples considered in this paper are the classical Duffing oscillator and an extended Duffing oscillator with nonlinear damping and excitation. Compared to numerical integration, the proposed procedure offers a faster calculation of existing multiple solutions and their character. |
| doi_str_mv | 10.1016/j.apm.2018.08.013 |
| format | Article |
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Harmonic Balance is a very popular semi-analytic method in nonlinear dynamics. It is easy to apply and is known to produce good results for numerous examples. Adding an error criterion taking into account the neglected terms allows an evaluation of the results. Looking on the therefore determined error for increasing ansatz orders, it can be evaluated whether a solution really exists or is an artifact. For the low-error solutions additionally a stability analysis is performed which allows the classification of the solutions in three types, namely in large error solutions, low error stable solutions and low error unstable solution. Examples considered in this paper are the classical Duffing oscillator and an extended Duffing oscillator with nonlinear damping and excitation. Compared to numerical integration, the proposed procedure offers a faster calculation of existing multiple solutions and their character.</description><identifier>ISSN: 0307-904X</identifier><identifier>ISSN: 1088-8691</identifier><identifier>EISSN: 0307-904X</identifier><identifier>DOI: 10.1016/j.apm.2018.08.013</identifier><language>eng</language><publisher>New York: Elsevier Inc</publisher><subject>Damping ; Duffing oscillator ; Duffing oscillators ; Error analysis ; Harmonic Balance method ; Historic artifacts ; Nonlinear analysis ; Nonlinear dynamics ; Nonlinear systems ; Numerical integration ; Oscillators ; Stability analysis</subject><ispartof>Applied Mathematical Modelling, 2019-01, Vol.65, p.408-414</ispartof><rights>2018 Elsevier Inc.</rights><rights>Copyright Elsevier BV Jan 2019</rights><lds50>peer_reviewed</lds50><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c368t-d24ddf930ad5c230f305553ea1ef45b5bb9aba6daf202ff455c420c741cb423e3</citedby><cites>FETCH-LOGICAL-c368t-d24ddf930ad5c230f305553ea1ef45b5bb9aba6daf202ff455c420c741cb423e3</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktohtml>$$Uhttps://www.sciencedirect.com/science/article/pii/S0307904X18304025$$EHTML$$P50$$Gelsevier$$H</linktohtml><link.rule.ids>314,776,780,3537,27901,27902,65306</link.rule.ids></links><search><creatorcontrib>von Wagner, U.</creatorcontrib><creatorcontrib>Lentz, L.</creatorcontrib><title>On the detection of artifacts in Harmonic Balance solutions of nonlinear oscillators</title><title>Applied Mathematical Modelling</title><description>•Addition of an error criterion allows the detection of artificial solutions.•Classification of solutions in large relative error, low relative error stable and low relative error unstable ones.•Application to nonlinear oscillators showing varieties of multiple solutions.
Harmonic Balance is a very popular semi-analytic method in nonlinear dynamics. It is easy to apply and is known to produce good results for numerous examples. Adding an error criterion taking into account the neglected terms allows an evaluation of the results. Looking on the therefore determined error for increasing ansatz orders, it can be evaluated whether a solution really exists or is an artifact. For the low-error solutions additionally a stability analysis is performed which allows the classification of the solutions in three types, namely in large error solutions, low error stable solutions and low error unstable solution. Examples considered in this paper are the classical Duffing oscillator and an extended Duffing oscillator with nonlinear damping and excitation. Compared to numerical integration, the proposed procedure offers a faster calculation of existing multiple solutions and their character.</description><subject>Damping</subject><subject>Duffing oscillator</subject><subject>Duffing oscillators</subject><subject>Error analysis</subject><subject>Harmonic Balance method</subject><subject>Historic artifacts</subject><subject>Nonlinear analysis</subject><subject>Nonlinear dynamics</subject><subject>Nonlinear systems</subject><subject>Numerical integration</subject><subject>Oscillators</subject><subject>Stability analysis</subject><issn>0307-904X</issn><issn>1088-8691</issn><issn>0307-904X</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2019</creationdate><recordtype>article</recordtype><recordid>eNp9UE1LAzEQDaJg_fgB3gKed50km20XT1rUCoVeKngL2Xxglm1Sk1Tw35ulHjwJA_PBezPzHkI3BGoCpL0barnf1RTIooYShJ2gGTCYVx0076d_6nN0kdIAALx0M7TdeJw_DNYmG5Vd8DhYLGN2VqqcsPN4JeMueKfwoxylVwanMB4mZJqgPvjReSMjDkm5cZQ5xHSFzqwck7n-zZfo7flpu1xV683L6_JhXSnWLnKlaaO17RhIzRVlYBlwzpmRxNiG97zvO9nLVktLgdoy4qqhoOYNUX1DmWGX6Pa4dx_D58GkLIZwiL6cFJS0Dadz3nYFRY4oFUNK0Vixj24n47cgICbzxCCKeWIyT0AJwgrn_sgx5f0vZ6Io6kxRr10sPgkd3D_sHzq7eGc</recordid><startdate>201901</startdate><enddate>201901</enddate><creator>von Wagner, U.</creator><creator>Lentz, L.</creator><general>Elsevier Inc</general><general>Elsevier BV</general><scope>AAYXX</scope><scope>CITATION</scope><scope>7SC</scope><scope>8FD</scope><scope>JQ2</scope><scope>L7M</scope><scope>L~C</scope><scope>L~D</scope></search><sort><creationdate>201901</creationdate><title>On the detection of artifacts in Harmonic Balance solutions of nonlinear oscillators</title><author>von Wagner, U. ; Lentz, L.</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c368t-d24ddf930ad5c230f305553ea1ef45b5bb9aba6daf202ff455c420c741cb423e3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2019</creationdate><topic>Damping</topic><topic>Duffing oscillator</topic><topic>Duffing oscillators</topic><topic>Error analysis</topic><topic>Harmonic Balance method</topic><topic>Historic artifacts</topic><topic>Nonlinear analysis</topic><topic>Nonlinear dynamics</topic><topic>Nonlinear systems</topic><topic>Numerical integration</topic><topic>Oscillators</topic><topic>Stability analysis</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>von Wagner, U.</creatorcontrib><creatorcontrib>Lentz, L.</creatorcontrib><collection>CrossRef</collection><collection>Computer and Information Systems Abstracts</collection><collection>Technology Research Database</collection><collection>ProQuest Computer Science Collection</collection><collection>Advanced Technologies Database with Aerospace</collection><collection>Computer and Information Systems Abstracts Academic</collection><collection>Computer and Information Systems Abstracts Professional</collection><jtitle>Applied Mathematical Modelling</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>von Wagner, U.</au><au>Lentz, L.</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>On the detection of artifacts in Harmonic Balance solutions of nonlinear oscillators</atitle><jtitle>Applied Mathematical Modelling</jtitle><date>2019-01</date><risdate>2019</risdate><volume>65</volume><spage>408</spage><epage>414</epage><pages>408-414</pages><issn>0307-904X</issn><issn>1088-8691</issn><eissn>0307-904X</eissn><abstract>•Addition of an error criterion allows the detection of artificial solutions.•Classification of solutions in large relative error, low relative error stable and low relative error unstable ones.•Application to nonlinear oscillators showing varieties of multiple solutions.
Harmonic Balance is a very popular semi-analytic method in nonlinear dynamics. It is easy to apply and is known to produce good results for numerous examples. Adding an error criterion taking into account the neglected terms allows an evaluation of the results. Looking on the therefore determined error for increasing ansatz orders, it can be evaluated whether a solution really exists or is an artifact. For the low-error solutions additionally a stability analysis is performed which allows the classification of the solutions in three types, namely in large error solutions, low error stable solutions and low error unstable solution. Examples considered in this paper are the classical Duffing oscillator and an extended Duffing oscillator with nonlinear damping and excitation. Compared to numerical integration, the proposed procedure offers a faster calculation of existing multiple solutions and their character.</abstract><cop>New York</cop><pub>Elsevier Inc</pub><doi>10.1016/j.apm.2018.08.013</doi><tpages>7</tpages><oa>free_for_read</oa></addata></record> |
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| source | Elsevier ScienceDirect Journals; EZB-FREE-00999 freely available EZB journals; EBSCOhost Education Source; EBSCOhost Business Source Complete |
| subjects | Damping Duffing oscillator Duffing oscillators Error analysis Harmonic Balance method Historic artifacts Nonlinear analysis Nonlinear dynamics Nonlinear systems Numerical integration Oscillators Stability analysis |
| title | On the detection of artifacts in Harmonic Balance solutions of nonlinear oscillators |
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