{"id":"https://openalex.org/W4414330422","doi":"https://doi.org/10.3390/sym17091563","title":"An Innovated Vibration Equation for Longitudinal Plate by Using the Symmetric and Asymmetric Spectral Decomposition","display_name":"An Innovated Vibration Equation for Longitudinal Plate by Using the Symmetric and Asymmetric Spectral Decomposition","publication_year":2025,"publication_date":"2025-09-18","ids":{"openalex":"https://openalex.org/W4414330422","doi":"https://doi.org/10.3390/sym17091563"},"language":"en","primary_location":{"id":"doi:10.3390/sym17091563","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym17091563","pdf_url":"https://www.mdpi.com/2073-8994/17/9/1563/pdf?version=1758193229","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"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.mdpi.com/2073-8994/17/9/1563/pdf?version=1758193229","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5074165918","display_name":"Jun Yin","orcid":"https://orcid.org/0000-0002-1749-1120"},"institutions":[{"id":"https://openalex.org/I158842170","display_name":"Chongqing University","ror":"https://ror.org/023rhb549","country_code":"CN","type":"education","lineage":["https://openalex.org/I158842170"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Jun Yin","raw_affiliation_strings":["School of Electrical Engineering, Chongqing University, Chongqing 400044, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Electrical Engineering, Chongqing University, Chongqing 400044, China","institution_ids":["https://openalex.org/I158842170"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5101510070","display_name":"Chuanping Zhou","orcid":"https://orcid.org/0000-0002-5495-3654"},"institutions":[{"id":"https://openalex.org/I50760025","display_name":"Hangzhou Dianzi University","ror":"https://ror.org/0576gt767","country_code":"CN","type":"education","lineage":["https://openalex.org/I50760025"]}],"countries":["CN"],"is_corresponding":true,"raw_author_name":"Chuanping Zhou","raw_affiliation_strings":["School of Mechanical Engineering, Hangzhou Dianzi University Information Engineering College, Hangzhou 310018, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Mechanical Engineering, Hangzhou Dianzi University Information Engineering College, Hangzhou 310018, China","institution_ids":["https://openalex.org/I50760025"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5044950437","display_name":"Changyong Chu","orcid":"https://orcid.org/0000-0003-1492-6464"},"institutions":[{"id":"https://openalex.org/I50760025","display_name":"Hangzhou Dianzi University","ror":"https://ror.org/0576gt767","country_code":"CN","type":"education","lineage":["https://openalex.org/I50760025"]}],"countries":["CN"],"is_corresponding":true,"raw_author_name":"Changyong Chu","raw_affiliation_strings":["School of Mechanical Engineering, Hangzhou Dianzi University Information Engineering College, Hangzhou 310018, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Mechanical Engineering, Hangzhou Dianzi University Information Engineering College, Hangzhou 310018, China","institution_ids":["https://openalex.org/I50760025"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5101504374","display_name":"Huipeng Chen","orcid":"https://orcid.org/0000-0002-0157-4896"},"institutions":[{"id":"https://openalex.org/I50760025","display_name":"Hangzhou Dianzi University","ror":"https://ror.org/0576gt767","country_code":"CN","type":"education","lineage":["https://openalex.org/I50760025"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Huipeng Chen","raw_affiliation_strings":["School of Mechanical Engineering, Hangzhou Dianzi University Information Engineering College, Hangzhou 310018, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Mechanical Engineering, Hangzhou Dianzi University Information Engineering College, Hangzhou 310018, China","institution_ids":["https://openalex.org/I50760025"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5101951353","display_name":"Fan Yang","orcid":"https://orcid.org/0000-0001-7965-7184"},"institutions":[{"id":"https://openalex.org/I158842170","display_name":"Chongqing University","ror":"https://ror.org/023rhb549","country_code":"CN","type":"education","lineage":["https://openalex.org/I158842170"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Fan Yang","raw_affiliation_strings":["School of Electrical Engineering, Chongqing University, Chongqing 400044, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Electrical Engineering, Chongqing University, Chongqing 400044, China","institution_ids":["https://openalex.org/I158842170"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5044950437","https://openalex.org/A5101510070"],"corresponding_institution_ids":["https://openalex.org/I50760025"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2165},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2165},"fwci":0.5378,"has_fulltext":true,"cited_by_count":1,"citation_normalized_percentile":{"value":0.63909729,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":94,"max":97},"biblio":{"volume":"17","issue":"9","first_page":"1563","last_page":"1563"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10221","display_name":"Composite Structure Analysis and Optimization","score":0.9995999932289124,"subfield":{"id":"https://openalex.org/subfields/2211","display_name":"Mechanics of Materials"},"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/T10221","display_name":"Composite Structure Analysis and Optimization","score":0.9995999932289124,"subfield":{"id":"https://openalex.org/subfields/2211","display_name":"Mechanics of Materials"},"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/T10822","display_name":"Acoustic Wave Phenomena Research","score":0.9950000047683716,"subfield":{"id":"https://openalex.org/subfields/2204","display_name":"Biomedical 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/T10842","display_name":"Railway Engineering and Dynamics","score":0.9950000047683716,"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/vibration","display_name":"Vibration","score":0.6614000201225281},{"id":"https://openalex.org/keywords/differential-equation","display_name":"Differential equation","score":0.5354999899864197},{"id":"https://openalex.org/keywords/plane","display_name":"Plane (geometry)","score":0.4424000084400177},{"id":"https://openalex.org/keywords/equations-of-motion","display_name":"Equations of motion","score":0.4307999908924103},{"id":"https://openalex.org/keywords/dispersion","display_name":"Dispersion (optics)","score":0.41940000653266907},{"id":"https://openalex.org/keywords/wave-equation","display_name":"Wave equation","score":0.41609999537467957},{"id":"https://openalex.org/keywords/operator","display_name":"Operator (biology)","score":0.41370001435279846},{"id":"https://openalex.org/keywords/a-priori-and-a-posteriori","display_name":"A priori and a posteriori","score":0.40540000796318054},{"id":"https://openalex.org/keywords/series","display_name":"Series (stratigraphy)","score":0.40149998664855957}],"concepts":[{"id":"https://openalex.org/C198394728","wikidata":"https://www.wikidata.org/wiki/Q3695508","display_name":"Vibration","level":2,"score":0.6614000201225281},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.5877000093460083},{"id":"https://openalex.org/C78045399","wikidata":"https://www.wikidata.org/wiki/Q11214","display_name":"Differential equation","level":2,"score":0.5354999899864197},{"id":"https://openalex.org/C17825722","wikidata":"https://www.wikidata.org/wiki/Q17285","display_name":"Plane (geometry)","level":2,"score":0.4424000084400177},{"id":"https://openalex.org/C14037181","wikidata":"https://www.wikidata.org/wiki/Q215007","display_name":"Equations of motion","level":2,"score":0.4307999908924103},{"id":"https://openalex.org/C177562468","wikidata":"https://www.wikidata.org/wiki/Q182893","display_name":"Dispersion (optics)","level":2,"score":0.41940000653266907},{"id":"https://openalex.org/C59696629","wikidata":"https://www.wikidata.org/wiki/Q193846","display_name":"Wave equation","level":2,"score":0.41609999537467957},{"id":"https://openalex.org/C17020691","wikidata":"https://www.wikidata.org/wiki/Q139677","display_name":"Operator (biology)","level":5,"score":0.41370001435279846},{"id":"https://openalex.org/C75553542","wikidata":"https://www.wikidata.org/wiki/Q178161","display_name":"A priori and a posteriori","level":2,"score":0.40540000796318054},{"id":"https://openalex.org/C143724316","wikidata":"https://www.wikidata.org/wiki/Q312468","display_name":"Series (stratigraphy)","level":2,"score":0.40149998664855957},{"id":"https://openalex.org/C73905626","wikidata":"https://www.wikidata.org/wiki/Q206925","display_name":"Power series","level":2,"score":0.38679999113082886},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.3833000063896179},{"id":"https://openalex.org/C93779851","wikidata":"https://www.wikidata.org/wiki/Q271977","display_name":"Partial differential equation","level":2,"score":0.38089999556541443},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.35839998722076416},{"id":"https://openalex.org/C70915906","wikidata":"https://www.wikidata.org/wiki/Q1058681","display_name":"Differential operator","level":2,"score":0.3287000060081482},{"id":"https://openalex.org/C168110828","wikidata":"https://www.wikidata.org/wiki/Q1331626","display_name":"Spectral density","level":2,"score":0.3156000077724457},{"id":"https://openalex.org/C130076159","wikidata":"https://www.wikidata.org/wiki/Q7291997","display_name":"Random vibration","level":3,"score":0.2921000123023987},{"id":"https://openalex.org/C24890656","wikidata":"https://www.wikidata.org/wiki/Q82811","display_name":"Acoustics","level":1,"score":0.289000004529953},{"id":"https://openalex.org/C93226319","wikidata":"https://www.wikidata.org/wiki/Q193137","display_name":"Differential (mechanical device)","level":2,"score":0.28850001096725464},{"id":"https://openalex.org/C156778621","wikidata":"https://www.wikidata.org/wiki/Q1365748","display_name":"Spectrum (functional analysis)","level":2,"score":0.287200003862381},{"id":"https://openalex.org/C163258240","wikidata":"https://www.wikidata.org/wiki/Q25342","display_name":"Power (physics)","level":2,"score":0.2856000065803528},{"id":"https://openalex.org/C202579712","wikidata":"https://www.wikidata.org/wiki/Q590051","display_name":"Dispersion relation","level":2,"score":0.28529998660087585},{"id":"https://openalex.org/C40308292","wikidata":"https://www.wikidata.org/wiki/Q122518","display_name":"Plane wave","level":2,"score":0.28290000557899475},{"id":"https://openalex.org/C53744967","wikidata":"https://www.wikidata.org/wiki/Q1062971","display_name":"Characteristic equation","level":3,"score":0.28220000863075256},{"id":"https://openalex.org/C74650414","wikidata":"https://www.wikidata.org/wiki/Q11397","display_name":"Classical mechanics","level":1,"score":0.27970001101493835},{"id":"https://openalex.org/C104114177","wikidata":"https://www.wikidata.org/wiki/Q79782","display_name":"Motion (physics)","level":2,"score":0.2696000039577484},{"id":"https://openalex.org/C23463724","wikidata":"https://www.wikidata.org/wiki/Q2308831","display_name":"Spectral method","level":2,"score":0.2678999900817871},{"id":"https://openalex.org/C761482","wikidata":"https://www.wikidata.org/wiki/Q118093","display_name":"Transmission (telecommunications)","level":2,"score":0.26019999384880066},{"id":"https://openalex.org/C185184677","wikidata":"https://www.wikidata.org/wiki/Q1366795","display_name":"Exact differential equation","level":4,"score":0.2565999925136566},{"id":"https://openalex.org/C2778258933","wikidata":"https://www.wikidata.org/wiki/Q16918986","display_name":"Decomposition method (queueing theory)","level":2,"score":0.2563999891281128},{"id":"https://openalex.org/C57879066","wikidata":"https://www.wikidata.org/wiki/Q41217","display_name":"Mechanics","level":1,"score":0.2529999911785126}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.3390/sym17091563","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym17091563","pdf_url":"https://www.mdpi.com/2073-8994/17/9/1563/pdf?version=1758193229","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"}],"best_oa_location":{"id":"doi:10.3390/sym17091563","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym17091563","pdf_url":"https://www.mdpi.com/2073-8994/17/9/1563/pdf?version=1758193229","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":[],"awards":[{"id":"https://openalex.org/G932761477","display_name":null,"funder_award_id":"2021C03013","funder_id":"https://openalex.org/F8142712028","funder_display_name":"Key Research and Development Program of Zhejiang Province"}],"funders":[{"id":"https://openalex.org/F8142712028","display_name":"Key Research and Development Program of Zhejiang Province","ror":null}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4414330422.pdf","grobid_xml":"https://content.openalex.org/works/W4414330422.grobid-xml"},"referenced_works_count":22,"referenced_works":["https://openalex.org/W1971844758","https://openalex.org/W1974687249","https://openalex.org/W1979475746","https://openalex.org/W1998372789","https://openalex.org/W2006468147","https://openalex.org/W2016694788","https://openalex.org/W2057615590","https://openalex.org/W2058816900","https://openalex.org/W2063647445","https://openalex.org/W2088632212","https://openalex.org/W2156497553","https://openalex.org/W2261621682","https://openalex.org/W2266285536","https://openalex.org/W2330175941","https://openalex.org/W2545942550","https://openalex.org/W2742230224","https://openalex.org/W2897917526","https://openalex.org/W2980840341","https://openalex.org/W3014658328","https://openalex.org/W3126475386","https://openalex.org/W3137448318","https://openalex.org/W3187695498"],"related_works":[],"abstract_inverted_index":{"Thick":[0],"wall":[1],"structures":[2],"involving":[3],"longitudinal":[4,39,112,123,182],"wave":[5,129,134],"are":[6,114],"typically":[7],"utilized":[8],"in":[9,68,180],"aerospace":[10],"engineering,":[11,14],"nuclear":[12],"power":[13],"precision":[15,161],"transmission":[16],"device":[17],"design,":[18],"and":[19,90,98,131,162,170],"pressure":[20],"vessels":[21],"design.":[22],"Consequently,":[23],"developing":[24],"sophisticated":[25],"dynamic":[26,152,157],"models":[27],"for":[28,171],"thick":[29],"plates":[30,172],"is":[31,42,48,58,83,163,177],"of":[32,43,77,95,102,122,126,150,185],"paramount":[33],"importance.":[34],"However,":[35],"the":[36,44,65,69,74,88,100,138,148,154,181],"commonly":[37],"used":[38],"vibration":[40,113,124,183],"equation":[41,79,121,130,184],"second":[45],"order,":[46],"which":[47,62],"regarded":[49],"as":[50],"a":[51,59,127,132,142,166,178],"plane":[52],"stress":[53],"problem.":[54],"Its":[55],"dispersion":[56,67],"curve":[57],"straight":[60],"line,":[61],"cannot":[63],"describe":[64],"actual":[66],"plate.":[70],"In":[71],"this":[72],"paper,":[73],"spectral":[75,92],"analysis":[76],"Navier":[78],"describing":[80,111],"three-dimensional":[81],"elasto-dynamics":[82],"carried":[84],"out":[85],"by":[86],"using":[87],"symmetric":[89],"asymmetric":[91],"decomposition":[93],"theory":[94],"differential":[96,104],"operators":[97],"introducing":[99],"concept":[101],"virtual":[103],"operators.":[105],"The":[106,119],"infinite":[107],"product":[108],"operator":[109],"series":[110],"truncated":[115],"into":[116],"fourth":[117],"order.":[118],"governing":[120],"consists":[125],"fourth-order":[128],"second-order":[133],"equation.":[135],"Owing":[136],"to":[137],"fact":[139],"that":[140],"no":[141],"priori":[143],"assumptions":[144],"were":[145],"introduced":[146],"during":[147],"derivation":[149],"its":[151],"equations,":[153],"proposed":[155],"plate":[156],"model":[158],"boasts":[159],"higher":[160],"applicable":[164],"across":[165],"broader":[167],"frequency":[168],"spectrum":[169],"with":[173],"greater":[174],"thicknesses.":[175],"This":[176],"breakthrough":[179],"plates.":[186]},"counts_by_year":[{"year":2026,"cited_by_count":1}],"updated_date":"2026-06-15T08:34:33.830935","created_date":"2025-10-10T00:00:00"}