{"id":"https://openalex.org/W4283166220","doi":"https://doi.org/10.3390/sym14061247","title":"Harmonic Balance Method to Analyze the Steady-State Response of a Controlled Mass-Damper-Spring Model","display_name":"Harmonic Balance Method to Analyze the Steady-State Response of a Controlled Mass-Damper-Spring Model","publication_year":2022,"publication_date":"2022-06-16","ids":{"openalex":"https://openalex.org/W4283166220","doi":"https://doi.org/10.3390/sym14061247"},"language":"en","primary_location":{"id":"doi:10.3390/sym14061247","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym14061247","pdf_url":"https://www.mdpi.com/2073-8994/14/6/1247/pdf?version=1655361391","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Symmetry","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.mdpi.com/2073-8994/14/6/1247/pdf?version=1655361391","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5003966268","display_name":"Ali Kandil","orcid":"https://orcid.org/0000-0002-7176-2489"},"institutions":[{"id":"https://openalex.org/I32619867","display_name":"Egypt-Japan University of Science and Technology","ror":"https://ror.org/02x66tk73","country_code":"EG","type":"education","lineage":["https://openalex.org/I32619867"]},{"id":"https://openalex.org/I63601056","display_name":"Menoufia University","ror":"https://ror.org/05sjrb944","country_code":"EG","type":"education","lineage":["https://openalex.org/I63601056"]}],"countries":["EG"],"is_corresponding":true,"raw_author_name":"Ali Kandil","raw_affiliation_strings":["Department of Mathematics, Institute of Basic and Applied Sciences, Egypt-Japan University of Science and Technology (E-JUST), New Borg El-Arab City 21934, Egypt","Department of Physics and Engineering Mathematics, Faculty of Electronic Engineering, Menoufia University, Menouf 32952, Egypt"],"raw_orcid":"https://orcid.org/0000-0002-7176-2489","affiliations":[{"raw_affiliation_string":"Department of Mathematics, Institute of Basic and Applied Sciences, Egypt-Japan University of Science and Technology (E-JUST), New Borg El-Arab City 21934, Egypt","institution_ids":["https://openalex.org/I32619867"]},{"raw_affiliation_string":"Department of Physics and Engineering Mathematics, Faculty of Electronic Engineering, Menoufia University, Menouf 32952, Egypt","institution_ids":["https://openalex.org/I63601056"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5033097731","display_name":"Y. S. Hamed","orcid":"https://orcid.org/0000-0002-0365-0282"},"institutions":[{"id":"https://openalex.org/I179331831","display_name":"Taif University","ror":"https://ror.org/014g1a453","country_code":"SA","type":"education","lineage":["https://openalex.org/I179331831"]}],"countries":["SA"],"is_corresponding":false,"raw_author_name":"Y. S. Hamed","raw_affiliation_strings":["Department of Mathematics and Statistics, College of Science, Taif University, P. O. Box 11099, Taif 21944, Saudi Arabia"],"raw_orcid":"https://orcid.org/0000-0002-0365-0282","affiliations":[{"raw_affiliation_string":"Department of Mathematics and Statistics, College of Science, Taif University, P. O. Box 11099, Taif 21944, Saudi Arabia","institution_ids":["https://openalex.org/I179331831"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5025074757","display_name":"Jan Awrejcewicz","orcid":"https://orcid.org/0000-0003-0387-921X"},"institutions":[{"id":"https://openalex.org/I188884621","display_name":"Lodz University of Technology","ror":"https://ror.org/00s8fpf52","country_code":"PL","type":"education","lineage":["https://openalex.org/I188884621"]}],"countries":["PL"],"is_corresponding":false,"raw_author_name":"Jan Awrejcewicz","raw_affiliation_strings":["Department of Automation, Biomechanics, and Mechatronics, Faculty of Mechanical Engineering, Lodz University of Technology, 90-924 Lodz, Poland"],"raw_orcid":"https://orcid.org/0000-0003-0387-921X","affiliations":[{"raw_affiliation_string":"Department of Automation, Biomechanics, and Mechatronics, Faculty of Mechanical Engineering, Lodz University of Technology, 90-924 Lodz, Poland","institution_ids":["https://openalex.org/I188884621"]}]}],"institutions":[],"countries_distinct_count":3,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5003966268"],"corresponding_institution_ids":["https://openalex.org/I32619867","https://openalex.org/I63601056"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2165},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2165},"fwci":1.1527,"has_fulltext":false,"cited_by_count":12,"citation_normalized_percentile":{"value":0.74697902,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":99},"biblio":{"volume":"14","issue":"6","first_page":"1247","last_page":"1247"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10690","display_name":"Vibration Control and Rheological Fluids","score":0.9997000098228455,"subfield":{"id":"https://openalex.org/subfields/2205","display_name":"Civil and Structural Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10690","display_name":"Vibration Control and Rheological Fluids","score":0.9997000098228455,"subfield":{"id":"https://openalex.org/subfields/2205","display_name":"Civil and Structural Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10805","display_name":"Vehicle Dynamics and Control Systems","score":0.9984999895095825,"subfield":{"id":"https://openalex.org/subfields/2203","display_name":"Automotive Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11372","display_name":"Hydraulic and Pneumatic Systems","score":0.9944999814033508,"subfield":{"id":"https://openalex.org/subfields/2210","display_name":"Mechanical Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/control-theory","display_name":"Control theory (sociology)","score":0.8458561897277832},{"id":"https://openalex.org/keywords/floquet-theory","display_name":"Floquet theory","score":0.7972574234008789},{"id":"https://openalex.org/keywords/harmonic-balance","display_name":"Harmonic balance","score":0.7559603452682495},{"id":"https://openalex.org/keywords/linear-variable-differential-transformer","display_name":"Linear variable differential transformer","score":0.6728792190551758},{"id":"https://openalex.org/keywords/nonlinear-system","display_name":"Nonlinear system","score":0.6313497424125671},{"id":"https://openalex.org/keywords/damper","display_name":"Damper","score":0.5927973389625549},{"id":"https://openalex.org/keywords/state-variable","display_name":"State variable","score":0.5311164855957031},{"id":"https://openalex.org/keywords/actuator","display_name":"Actuator","score":0.42699626088142395},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.39127421379089355},{"id":"https://openalex.org/keywords/transformer","display_name":"Transformer","score":0.3610452711582184},{"id":"https://openalex.org/keywords/mechanics","display_name":"Mechanics","score":0.3472439646720886},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.34125804901123047},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.30254924297332764},{"id":"https://openalex.org/keywords/engineering","display_name":"Engineering","score":0.25539371371269226},{"id":"https://openalex.org/keywords/structural-engineering","display_name":"Structural engineering","score":0.1749051809310913},{"id":"https://openalex.org/keywords/voltage","display_name":"Voltage","score":0.13094934821128845}],"concepts":[{"id":"https://openalex.org/C47446073","wikidata":"https://www.wikidata.org/wiki/Q5165890","display_name":"Control theory (sociology)","level":3,"score":0.8458561897277832},{"id":"https://openalex.org/C55649039","wikidata":"https://www.wikidata.org/wiki/Q902618","display_name":"Floquet theory","level":3,"score":0.7972574234008789},{"id":"https://openalex.org/C121758426","wikidata":"https://www.wikidata.org/wiki/Q5659248","display_name":"Harmonic balance","level":3,"score":0.7559603452682495},{"id":"https://openalex.org/C125884559","wikidata":"https://www.wikidata.org/wiki/Q369791","display_name":"Linear variable differential transformer","level":5,"score":0.6728792190551758},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.6313497424125671},{"id":"https://openalex.org/C140096630","wikidata":"https://www.wikidata.org/wiki/Q3423628","display_name":"Damper","level":2,"score":0.5927973389625549},{"id":"https://openalex.org/C129537906","wikidata":"https://www.wikidata.org/wiki/Q7603913","display_name":"State variable","level":2,"score":0.5311164855957031},{"id":"https://openalex.org/C172707124","wikidata":"https://www.wikidata.org/wiki/Q423488","display_name":"Actuator","level":2,"score":0.42699626088142395},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.39127421379089355},{"id":"https://openalex.org/C66322947","wikidata":"https://www.wikidata.org/wiki/Q11658","display_name":"Transformer","level":3,"score":0.3610452711582184},{"id":"https://openalex.org/C57879066","wikidata":"https://www.wikidata.org/wiki/Q41217","display_name":"Mechanics","level":1,"score":0.3472439646720886},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.34125804901123047},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.30254924297332764},{"id":"https://openalex.org/C127413603","wikidata":"https://www.wikidata.org/wiki/Q11023","display_name":"Engineering","level":0,"score":0.25539371371269226},{"id":"https://openalex.org/C66938386","wikidata":"https://www.wikidata.org/wiki/Q633538","display_name":"Structural engineering","level":1,"score":0.1749051809310913},{"id":"https://openalex.org/C165801399","wikidata":"https://www.wikidata.org/wiki/Q25428","display_name":"Voltage","level":2,"score":0.13094934821128845},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0},{"id":"https://openalex.org/C97355855","wikidata":"https://www.wikidata.org/wiki/Q11473","display_name":"Thermodynamics","level":1,"score":0.0},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.0},{"id":"https://openalex.org/C2775924081","wikidata":"https://www.wikidata.org/wiki/Q55608371","display_name":"Control (management)","level":2,"score":0.0},{"id":"https://openalex.org/C117323899","wikidata":"https://www.wikidata.org/wiki/Q1907989","display_name":"Distribution transformer","level":4,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.3390/sym14061247","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym14061247","pdf_url":"https://www.mdpi.com/2073-8994/14/6/1247/pdf?version=1655361391","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Symmetry","raw_type":"journal-article"},{"id":"pmh:oai:doaj.org/article:342fcca9c50c425eb53537be96853117","is_oa":true,"landing_page_url":"https://doaj.org/article/342fcca9c50c425eb53537be96853117","pdf_url":null,"source":{"id":"https://openalex.org/S4306401280","display_name":"DOAJ (DOAJ: Directory of Open Access Journals)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by-sa","license_id":"https://openalex.org/licenses/cc-by-sa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Symmetry, Vol 14, Iss 6, p 1247 (2022)","raw_type":"article"},{"id":"pmh:oai:mdpi.com:/2073-8994/14/6/1247/","is_oa":true,"landing_page_url":"https://dx.doi.org/10.3390/sym14061247","pdf_url":null,"source":{"id":"https://openalex.org/S4306400947","display_name":"MDPI (MDPI AG)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I4210097602","host_organization_name":"Multidisciplinary Digital Publishing Institute (Switzerland)","host_organization_lineage":["https://openalex.org/I4210097602"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Symmetry; Volume 14; Issue 6; Pages: 1247","raw_type":"Text"}],"best_oa_location":{"id":"doi:10.3390/sym14061247","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym14061247","pdf_url":"https://www.mdpi.com/2073-8994/14/6/1247/pdf?version=1655361391","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Symmetry","raw_type":"journal-article"},"sustainable_development_goals":[{"score":0.550000011920929,"display_name":"Affordable and clean energy","id":"https://metadata.un.org/sdg/7"}],"awards":[],"funders":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4283166220.pdf","grobid_xml":"https://content.openalex.org/works/W4283166220.grobid-xml"},"referenced_works_count":37,"referenced_works":["https://openalex.org/W26331898","https://openalex.org/W1967047715","https://openalex.org/W1979032227","https://openalex.org/W1982828276","https://openalex.org/W1985382566","https://openalex.org/W1993962765","https://openalex.org/W1995258548","https://openalex.org/W1996219043","https://openalex.org/W2011193027","https://openalex.org/W2029164661","https://openalex.org/W2034793886","https://openalex.org/W2039340649","https://openalex.org/W2045464437","https://openalex.org/W2049437341","https://openalex.org/W2056778766","https://openalex.org/W2058312852","https://openalex.org/W2068020688","https://openalex.org/W2072026997","https://openalex.org/W2072479289","https://openalex.org/W2077166574","https://openalex.org/W2093844619","https://openalex.org/W2131790882","https://openalex.org/W2223789761","https://openalex.org/W2342509101","https://openalex.org/W2428781232","https://openalex.org/W2477244626","https://openalex.org/W2477365427","https://openalex.org/W2754716071","https://openalex.org/W2945704773","https://openalex.org/W2968887522","https://openalex.org/W3096417982","https://openalex.org/W3120767514","https://openalex.org/W3161625411","https://openalex.org/W3199084263","https://openalex.org/W3199577973","https://openalex.org/W3212173758","https://openalex.org/W4253424795"],"related_works":["https://openalex.org/W1488175704","https://openalex.org/W4286849939","https://openalex.org/W3217258686","https://openalex.org/W2990905476","https://openalex.org/W3158508387","https://openalex.org/W4383815106","https://openalex.org/W4386135338","https://openalex.org/W2131815370","https://openalex.org/W2800488099","https://openalex.org/W2108707250"],"abstract_inverted_index":{"This":[0],"research":[1],"is":[2],"concerned":[3],"with":[4,26],"extracting":[5],"the":[6,15,27,53,57,69,73,84,93,98],"approximate":[7],"solutions":[8,23],"of":[9,21,29,56,97],"a":[10,37,44,78],"controlled":[11,95],"mass-damper-spring":[12],"model":[13],"via":[14],"harmonic":[16,74],"balance":[17,75],"method.":[18,76],"The":[19],"stability":[20],"these":[22],"was":[24,81],"checked":[25],"aid":[28],"Floquet":[30],"theory.":[31],"A":[32],"nonlinear":[33],"saturation":[34],"controller":[35],"(NSC),":[36],"linear":[38,46],"variable":[39],"differential":[40],"transformer":[41],"(LVDT)":[42],"and":[43,61],"servo-controlled":[45],"actuator":[47],"(SCLA),":[48],"were":[49],"applied":[50],"to":[51,91],"suppress":[52],"undesired":[54],"oscillations":[55],"harmonically-excited":[58],"car.":[59],"2D":[60],"3D":[62],"graphical":[63],"plots":[64],"are":[65],"included":[66],"based":[67],"upon":[68],"equations":[70],"resulting":[71],"from":[72],"Moreover,":[77],"numerical":[79],"simulation":[80],"established":[82],"using":[83],"fourth":[85],"order":[86,90],"Rung\u2013Kutta":[87],"technique":[88],"in":[89],"confirm":[92],"overall":[94],"behavior":[96],"studied":[99],"model.":[100]},"counts_by_year":[{"year":2026,"cited_by_count":2},{"year":2025,"cited_by_count":4},{"year":2024,"cited_by_count":3},{"year":2023,"cited_by_count":1},{"year":2022,"cited_by_count":2}],"updated_date":"2026-05-06T08:25:59.206177","created_date":"2025-10-10T00:00:00"}