{"id":"https://openalex.org/W4417316362","doi":"https://doi.org/10.1142/s0218127426500379","title":"Nonlinear Dynamics and Primary Resonance Characteristics of Spiral Bevel Gear Systems","display_name":"Nonlinear Dynamics and Primary Resonance Characteristics of Spiral Bevel Gear Systems","publication_year":2025,"publication_date":"2025-12-13","ids":{"openalex":"https://openalex.org/W4417316362","doi":"https://doi.org/10.1142/s0218127426500379"},"language":"en","primary_location":{"id":"doi:10.1142/s0218127426500379","is_oa":false,"landing_page_url":"https://doi.org/10.1142/s0218127426500379","pdf_url":null,"source":{"id":"https://openalex.org/S81011612","display_name":"International Journal of Bifurcation and Chaos","issn_l":"0218-1274","issn":["0218-1274","1793-6551"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319815","host_organization_name":"World Scientific","host_organization_lineage":["https://openalex.org/P4310319815"],"host_organization_lineage_names":["World Scientific"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"International Journal of Bifurcation and Chaos","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5051232072","display_name":"Shuai Mo","orcid":"https://orcid.org/0000-0003-3305-004X"},"institutions":[{"id":"https://openalex.org/I150807315","display_name":"Guangxi University","ror":"https://ror.org/02c9qn167","country_code":"CN","type":"education","lineage":["https://openalex.org/I150807315"]},{"id":"https://openalex.org/I47720641","display_name":"Huazhong University of Science and Technology","ror":"https://ror.org/00p991c53","country_code":"CN","type":"education","lineage":["https://openalex.org/I47720641"]}],"countries":["CN"],"is_corresponding":true,"raw_author_name":"Shuai Mo","raw_affiliation_strings":["State Key Laboratory of Featured Metal Materials and Life-cycle Safety for Composite Structures, School of Mechanical Engineering, Guangxi University, Nanning 530004, P. R. China","State Key Laboratory of Intelligent Manufacturing, Equipment and Technology, Huazhong University of Science, and Technology, Wuhan 430074, P. R. China","State Key Laboratory of Intelligent Manufacturing, Equipment and Technology, Huazhong University of Science and Technology, Wuhan 430074, P. R. China"],"raw_orcid":"https://orcid.org/0000-0003-3305-004X","affiliations":[{"raw_affiliation_string":"State Key Laboratory of Featured Metal Materials and Life-cycle Safety for Composite Structures, School of Mechanical Engineering, Guangxi University, Nanning 530004, P. R. China","institution_ids":["https://openalex.org/I150807315"]},{"raw_affiliation_string":"State Key Laboratory of Intelligent Manufacturing, Equipment and Technology, Huazhong University of Science, and Technology, Wuhan 430074, P. R. China","institution_ids":["https://openalex.org/I47720641"]},{"raw_affiliation_string":"State Key Laboratory of Intelligent Manufacturing, Equipment and Technology, Huazhong University of Science and Technology, Wuhan 430074, P. R. China","institution_ids":["https://openalex.org/I47720641"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5079504398","display_name":"Rundong Zhang","orcid":"https://orcid.org/0009-0000-7493-615X"},"institutions":[{"id":"https://openalex.org/I150807315","display_name":"Guangxi University","ror":"https://ror.org/02c9qn167","country_code":"CN","type":"education","lineage":["https://openalex.org/I150807315"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Rundong Zhang","raw_affiliation_strings":["State Key Laboratory of Featured Metal Materials and Life-cycle Safety for Composite Structures, School of Mechanical Engineering, Guangxi University, Nanning 530004, P. R. China"],"raw_orcid":"https://orcid.org/0009-0000-7493-615X","affiliations":[{"raw_affiliation_string":"State Key Laboratory of Featured Metal Materials and Life-cycle Safety for Composite Structures, School of Mechanical Engineering, Guangxi University, Nanning 530004, P. R. China","institution_ids":["https://openalex.org/I150807315"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5016419598","display_name":"Shikui Dong","orcid":"https://orcid.org/0009-0001-8186-9917"},"institutions":[{"id":"https://openalex.org/I150807315","display_name":"Guangxi University","ror":"https://ror.org/02c9qn167","country_code":"CN","type":"education","lineage":["https://openalex.org/I150807315"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Shunfang Dong","raw_affiliation_strings":["State Key Laboratory of Featured Metal Materials and Life-cycle Safety for Composite Structures, School of Mechanical Engineering, Guangxi University, Nanning 530004, P. R. China"],"raw_orcid":"https://orcid.org/0009-0001-8186-9917","affiliations":[{"raw_affiliation_string":"State Key Laboratory of Featured Metal Materials and Life-cycle Safety for Composite Structures, School of Mechanical Engineering, Guangxi University, Nanning 530004, P. R. China","institution_ids":["https://openalex.org/I150807315"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5035297747","display_name":"Bowei Yao","orcid":"https://orcid.org/0009-0007-8266-9507"},"institutions":[{"id":"https://openalex.org/I4210090327","display_name":"Liuzhou Maternal and Child Health Hospital","ror":"https://ror.org/00fbwv278","country_code":"CN","type":"healthcare","lineage":["https://openalex.org/I4210090327"]},{"id":"https://openalex.org/I4210090903","display_name":"Liuzhou General Hospital","ror":"https://ror.org/00er4d216","country_code":"CN","type":"healthcare","lineage":["https://openalex.org/I4210090903"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Bowei Yao","raw_affiliation_strings":["Liuzhou Saike Technology Development Co., Ltd., Liuzhou 545000, P. R. China"],"raw_orcid":"https://orcid.org/0009-0007-8266-9507","affiliations":[{"raw_affiliation_string":"Liuzhou Saike Technology Development Co., Ltd., Liuzhou 545000, P. R. China","institution_ids":["https://openalex.org/I4210090327","https://openalex.org/I4210090903"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5113415495","display_name":"Sujiao Chen","orcid":"https://orcid.org/0009-0008-2404-3802"},"institutions":[{"id":"https://openalex.org/I4210090327","display_name":"Liuzhou Maternal and Child Health Hospital","ror":"https://ror.org/00fbwv278","country_code":"CN","type":"healthcare","lineage":["https://openalex.org/I4210090327"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Sujiao Chen","raw_affiliation_strings":["Liugong Liuzhou Driveline Co., Ltd., Liuzhou 545007, P. R. China"],"raw_orcid":"https://orcid.org/0009-0008-2404-3802","affiliations":[{"raw_affiliation_string":"Liugong Liuzhou Driveline Co., Ltd., Liuzhou 545007, P. R. China","institution_ids":["https://openalex.org/I4210090327"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5102507684","display_name":"Yurong Huang","orcid":"https://orcid.org/0009-0001-1037-1062"},"institutions":[{"id":"https://openalex.org/I4210090327","display_name":"Liuzhou Maternal and Child Health Hospital","ror":"https://ror.org/00fbwv278","country_code":"CN","type":"healthcare","lineage":["https://openalex.org/I4210090327"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Yurong Huang","raw_affiliation_strings":["Fangsheng Axle (Liuzhou) Co., Ltd., Liuzhou 545006, P. R. China","Fangsheng Axle (Liuzhou\u200a) Co., Ltd., Liuzhou 545006, P. R. China"],"raw_orcid":"https://orcid.org/0009-0001-1037-1062","affiliations":[{"raw_affiliation_string":"Fangsheng Axle (Liuzhou) Co., Ltd., Liuzhou 545006, P. R. China","institution_ids":["https://openalex.org/I4210090327"]},{"raw_affiliation_string":"Fangsheng Axle (Liuzhou\u200a) Co., Ltd., Liuzhou 545006, P. R. China","institution_ids":["https://openalex.org/I4210090327"]}]},{"author_position":"middle","author":{"id":null,"display_name":"Wenai Shi","orcid":"https://orcid.org/0009-0002-2751-3204"},"institutions":[{"id":"https://openalex.org/I4210090327","display_name":"Liuzhou Maternal and Child Health Hospital","ror":"https://ror.org/00fbwv278","country_code":"CN","type":"healthcare","lineage":["https://openalex.org/I4210090327"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Wenai Shi","raw_affiliation_strings":["Fangsheng Axle (Liuzhou) Co., Ltd., Liuzhou 545006, P. R. China","Fangsheng Axle (Liuzhou\u200a) Co., Ltd., Liuzhou 545006, P. R. China"],"raw_orcid":"https://orcid.org/0009-0002-2751-3204","affiliations":[{"raw_affiliation_string":"Fangsheng Axle (Liuzhou) Co., Ltd., Liuzhou 545006, P. R. China","institution_ids":["https://openalex.org/I4210090327"]},{"raw_affiliation_string":"Fangsheng Axle (Liuzhou\u200a) Co., Ltd., Liuzhou 545006, P. R. China","institution_ids":["https://openalex.org/I4210090327"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5113404707","display_name":"Nanjiang Peng","orcid":"https://orcid.org/0009-0005-3855-7642"},"institutions":[{"id":"https://openalex.org/I4210108786","display_name":"SAIC-GM-Wuling (China)","ror":"https://ror.org/01g08q774","country_code":"CN","type":"company","lineage":["https://openalex.org/I4210108786"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Nanjiang Peng","raw_affiliation_strings":["Liuzhou Wuling Automobile Industry Co., Ltd., Liuzhou 545007, P. R. China"],"raw_orcid":"https://orcid.org/0009-0005-3855-7642","affiliations":[{"raw_affiliation_string":"Liuzhou Wuling Automobile Industry Co., Ltd., Liuzhou 545007, P. R. China","institution_ids":["https://openalex.org/I4210108786"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5087656173","display_name":"Haruo HOUJOH","orcid":"https://orcid.org/0009-0009-5400-7340"},"institutions":[{"id":"https://openalex.org/I114531698","display_name":"Tokyo Institute of Technology","ror":"https://ror.org/0112mx960","country_code":"JP","type":"education","lineage":["https://openalex.org/I114531698"]}],"countries":["JP"],"is_corresponding":false,"raw_author_name":"Haruo Houjoh","raw_affiliation_strings":["Precision and Intelligence Laboratory, Tokyo Institute of Technology, Yokohama 226-8503, Japan"],"raw_orcid":"https://orcid.org/0009-0009-5400-7340","affiliations":[{"raw_affiliation_string":"Precision and Intelligence Laboratory, Tokyo Institute of Technology, Yokohama 226-8503, Japan","institution_ids":["https://openalex.org/I114531698"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5100692237","display_name":"Wei Zhang","orcid":"https://orcid.org/0000-0003-0210-5642"},"institutions":[{"id":"https://openalex.org/I150807315","display_name":"Guangxi University","ror":"https://ror.org/02c9qn167","country_code":"CN","type":"education","lineage":["https://openalex.org/I150807315"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Wei Zhang","raw_affiliation_strings":["State Key Laboratory of Featured Metal Materials and Life-cycle Safety for Composite Structures, School of Mechanical Engineering, Guangxi University, Nanning 530004, P. R. China"],"raw_orcid":"https://orcid.org/0000-0003-0210-5642","affiliations":[{"raw_affiliation_string":"State Key Laboratory of Featured Metal Materials and Life-cycle Safety for Composite Structures, School of Mechanical Engineering, Guangxi University, Nanning 530004, P. R. China","institution_ids":["https://openalex.org/I150807315"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":10,"corresponding_author_ids":["https://openalex.org/A5051232072"],"corresponding_institution_ids":["https://openalex.org/I150807315","https://openalex.org/I47720641"],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":false,"cited_by_count":0,"citation_normalized_percentile":{"value":0.38884837,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"36","issue":"03","first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11062","display_name":"Gear and Bearing Dynamics Analysis","score":0.980400025844574,"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"}},"topics":[{"id":"https://openalex.org/T11062","display_name":"Gear and Bearing Dynamics Analysis","score":0.980400025844574,"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"}},{"id":"https://openalex.org/T10690","display_name":"Vibration Control and Rheological Fluids","score":0.0032999999821186066,"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/T12435","display_name":"Bladed Disk Vibration Dynamics","score":0.002300000051036477,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/nonlinear-system","display_name":"Nonlinear system","score":0.6539999842643738},{"id":"https://openalex.org/keywords/vibration","display_name":"Vibration","score":0.6177999973297119},{"id":"https://openalex.org/keywords/lyapunov-exponent","display_name":"Lyapunov exponent","score":0.5968999862670898},{"id":"https://openalex.org/keywords/chaotic","display_name":"Chaotic","score":0.5616000294685364},{"id":"https://openalex.org/keywords/control-theory","display_name":"Control theory (sociology)","score":0.5458999872207642},{"id":"https://openalex.org/keywords/bifurcation","display_name":"Bifurcation","score":0.5171999931335449},{"id":"https://openalex.org/keywords/bevel-gear","display_name":"Bevel gear","score":0.5146999955177307},{"id":"https://openalex.org/keywords/spiral","display_name":"Spiral (railway)","score":0.4717999994754791},{"id":"https://openalex.org/keywords/spiral-bevel-gear","display_name":"Spiral bevel gear","score":0.42500001192092896}],"concepts":[{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.6539999842643738},{"id":"https://openalex.org/C198394728","wikidata":"https://www.wikidata.org/wiki/Q3695508","display_name":"Vibration","level":2,"score":0.6177999973297119},{"id":"https://openalex.org/C191544260","wikidata":"https://www.wikidata.org/wiki/Q1238630","display_name":"Lyapunov exponent","level":3,"score":0.5968999862670898},{"id":"https://openalex.org/C2777052490","wikidata":"https://www.wikidata.org/wiki/Q5072826","display_name":"Chaotic","level":2,"score":0.5616000294685364},{"id":"https://openalex.org/C47446073","wikidata":"https://www.wikidata.org/wiki/Q5165890","display_name":"Control theory (sociology)","level":3,"score":0.5458999872207642},{"id":"https://openalex.org/C2781349735","wikidata":"https://www.wikidata.org/wiki/Q4904824","display_name":"Bifurcation","level":3,"score":0.5171999931335449},{"id":"https://openalex.org/C62926433","wikidata":"https://www.wikidata.org/wiki/Q2144215","display_name":"Bevel gear","level":2,"score":0.5146999955177307},{"id":"https://openalex.org/C174128100","wikidata":"https://www.wikidata.org/wiki/Q846907","display_name":"Spiral (railway)","level":2,"score":0.4717999994754791},{"id":"https://openalex.org/C136905703","wikidata":"https://www.wikidata.org/wiki/Q1641343","display_name":"Spiral bevel gear","level":2,"score":0.42500001192092896},{"id":"https://openalex.org/C761482","wikidata":"https://www.wikidata.org/wiki/Q118093","display_name":"Transmission (telecommunications)","level":2,"score":0.4171999990940094},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.39899998903274536},{"id":"https://openalex.org/C2776652708","wikidata":"https://www.wikidata.org/wiki/Q584028","display_name":"Backlash","level":2,"score":0.38679999113082886},{"id":"https://openalex.org/C144171764","wikidata":"https://www.wikidata.org/wiki/Q48103","display_name":"Torque","level":2,"score":0.3797000050544739},{"id":"https://openalex.org/C112972136","wikidata":"https://www.wikidata.org/wiki/Q7595718","display_name":"Stability (learning theory)","level":2,"score":0.37369999289512634},{"id":"https://openalex.org/C167020490","wikidata":"https://www.wikidata.org/wiki/Q3457902","display_name":"Nonlinear resonance","level":3,"score":0.3650999963283539},{"id":"https://openalex.org/C78736273","wikidata":"https://www.wikidata.org/wiki/Q946764","display_name":"Natural frequency","level":3,"score":0.3479999899864197},{"id":"https://openalex.org/C139210041","wikidata":"https://www.wikidata.org/wiki/Q2145840","display_name":"Resonance (particle physics)","level":2,"score":0.3474000096321106},{"id":"https://openalex.org/C47432892","wikidata":"https://www.wikidata.org/wiki/Q831390","display_name":"Wavelet","level":2,"score":0.3343000113964081},{"id":"https://openalex.org/C40678858","wikidata":"https://www.wikidata.org/wiki/Q6934874","display_name":"Multiple-scale analysis","level":3,"score":0.31529998779296875},{"id":"https://openalex.org/C132323500","wikidata":"https://www.wikidata.org/wiki/Q675387","display_name":"Duffing equation","level":3,"score":0.3100000023841858},{"id":"https://openalex.org/C113608303","wikidata":"https://www.wikidata.org/wiki/Q1056835","display_name":"Transmission system","level":3,"score":0.3034000098705292},{"id":"https://openalex.org/C57879066","wikidata":"https://www.wikidata.org/wiki/Q41217","display_name":"Mechanics","level":1,"score":0.2994000017642975},{"id":"https://openalex.org/C8590192","wikidata":"https://www.wikidata.org/wiki/Q1054694","display_name":"Frequency response","level":2,"score":0.29829999804496765},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.2953000068664551},{"id":"https://openalex.org/C78045399","wikidata":"https://www.wikidata.org/wiki/Q11214","display_name":"Differential equation","level":2,"score":0.2921999990940094},{"id":"https://openalex.org/C83581075","wikidata":"https://www.wikidata.org/wiki/Q1361503","display_name":"Excitation","level":2,"score":0.27900001406669617},{"id":"https://openalex.org/C24890656","wikidata":"https://www.wikidata.org/wiki/Q82811","display_name":"Acoustics","level":1,"score":0.2773999869823456},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.2743000090122223},{"id":"https://openalex.org/C157626765","wikidata":"https://www.wikidata.org/wiki/Q2088081","display_name":"Phase plane","level":3,"score":0.27250000834465027},{"id":"https://openalex.org/C44280652","wikidata":"https://www.wikidata.org/wiki/Q104837","display_name":"Phase (matter)","level":2,"score":0.27160000801086426},{"id":"https://openalex.org/C60640748","wikidata":"https://www.wikidata.org/wiki/Q2337858","display_name":"Lyapunov function","level":3,"score":0.2605000138282776},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.25119999051094055}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1142/s0218127426500379","is_oa":false,"landing_page_url":"https://doi.org/10.1142/s0218127426500379","pdf_url":null,"source":{"id":"https://openalex.org/S81011612","display_name":"International Journal of Bifurcation and Chaos","issn_l":"0218-1274","issn":["0218-1274","1793-6551"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319815","host_organization_name":"World Scientific","host_organization_lineage":["https://openalex.org/P4310319815"],"host_organization_lineage_names":["World Scientific"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"International Journal of Bifurcation and Chaos","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G5626511922","display_name":null,"funder_award_id":"52265004","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"}],"funders":[{"id":"https://openalex.org/F4320321001","display_name":"National Natural Science Foundation of China","ror":"https://ror.org/01h0zpd94"}],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":29,"referenced_works":["https://openalex.org/W2778194965","https://openalex.org/W2793811941","https://openalex.org/W2889258544","https://openalex.org/W2889354910","https://openalex.org/W2898378142","https://openalex.org/W2940755566","https://openalex.org/W2973641672","https://openalex.org/W3005857807","https://openalex.org/W3025206414","https://openalex.org/W3096579225","https://openalex.org/W3155509538","https://openalex.org/W4306152416","https://openalex.org/W4306889077","https://openalex.org/W4307141907","https://openalex.org/W4353033187","https://openalex.org/W4376631533","https://openalex.org/W4380270557","https://openalex.org/W4386906157","https://openalex.org/W4388115169","https://openalex.org/W4388383298","https://openalex.org/W4391320624","https://openalex.org/W4392121623","https://openalex.org/W4400520796","https://openalex.org/W4400599045","https://openalex.org/W4401981465","https://openalex.org/W4403754431","https://openalex.org/W4403987524","https://openalex.org/W4404056210","https://openalex.org/W4405873785"],"related_works":[],"abstract_inverted_index":{"This":[0],"paper":[1],"investigates":[2],"the":[3,46,77,80,88,100,117,125,135,140],"nonlinear":[4,21,50],"vibration":[5,40],"characteristics":[6],"of":[7,90],"spiral":[8,109],"bevel":[9,110],"gear":[10,30,111],"transmission":[11,36,122,149],"systems.":[12],"A":[13],"dynamic":[14,51],"model":[15],"was":[16,56,84],"established":[17],"by":[18],"incorporating":[19],"key":[20],"factors,":[22],"including":[23],"bearing":[24],"support":[25],"forces,":[26],"time-varying":[27],"mesh":[28,32,91,93],"stiffness,":[29,94],"backlash,":[31],"damping,":[33,92],"and":[34,72,95,151],"static":[35,121],"error.":[37],"The":[38,49,105],"system\u2019s":[39,101],"differential":[41],"equations":[42],"were":[43],"solved":[44],"using":[45],"Runge\u2013Kutta":[47],"method.":[48],"behavior":[52],"under":[53],"varying":[54],"parameters":[55,154],"analyzed":[57],"through":[58],"time-domain":[59],"waveforms,":[60],"frequency":[61,119],"spectra,":[62],"phase":[63],"portraits,":[64],"Poincar\u00e9":[65],"sections,":[66],"bifurcation":[67],"diagrams,":[68],"wavelet":[69],"time-frequency":[70],"plots,":[71],"Lyapunov":[73],"exponent":[74],"spectra.":[75],"Employing":[76],"multiscale":[78],"method,":[79],"primary":[81,102],"resonance":[82,103],"equation":[83],"derived":[85],"to":[86,130,144],"examine":[87],"influence":[89],"external":[96],"load":[97],"fluctuations":[98],"on":[99],"characteristics.":[104],"results":[106],"indicate":[107],"that":[108],"systems":[112],"exhibit":[113],"significant":[114],"nonlinearity.":[115],"As":[116],"excitation":[118],"or":[120],"error":[123],"increases,":[124],"system":[126,141],"transitions":[127],"from":[128,142],"periodic":[129,145],"chaotic":[131,143],"motion.":[132,146],"Conversely,":[133],"increasing":[134],"driving":[136],"torque":[137],"can":[138],"shift":[139],"To":[147],"ensure":[148],"stability":[150],"reliability,":[152],"relevant":[153],"must":[155],"be":[156],"carefully":[157],"optimized.":[158]},"counts_by_year":[],"updated_date":"2026-03-27T05:58:40.876381","created_date":"2025-12-15T00:00:00"}