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Leveraging this parametric modeling, the study successfully derived an explicit analytical formula reflecting the quantitative relationship between spanwise twist and veins by performing high-fidelity FSI simulations on only a minimal set of key parameter points. This facilitates a transition from costly physical simulations to efficient mathematical expressions. Subsequently, this formula was integrated into the quasi-steady blade element method alongside electromechanical constraints, such as motor torque saturation, to construct a fully coupled system optimization model. The optimized wing, identified via global search, achieved an 11% efficiency improvement and a lift-to-weight ratio exceeding 1.3, as validated by bench and guide-rail flight tests. This research transforms complex empirical design into deterministic analytical optimization, providing an efficient design tool for high-payload, high-efficiency FWMAVs.<\/jats:p>","DOI":"10.1088\/1748-3190\/ae9020","type":"journal-article","created":{"date-parts":[[2026,7,24]],"date-time":"2026-07-24T22:53:27Z","timestamp":1784933607000},"page":"046032","update-policy":"https:\/\/doi.org\/10.1088\/crossmark-policy","source":"Crossref","is-referenced-by-count":0,"title":["An explicit analytical model for passive twist and systematic optimization of membrane flapping wings"],"prefix":"10.1088","volume":"21","author":[{"ORCID":"https:\/\/orcid.org\/0009-0000-5749-6295","authenticated-orcid":true,"given":"Zhaotong","family":"Chen","sequence":"first","affiliation":[{"name":"Beihang University"},{"name":"Beihang University"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Zongxia","family":"Jiao","sequence":"additional","affiliation":[{"name":"Beihang University"},{"name":"Beihang University"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Wuyao","family":"Jiang","sequence":"additional","affiliation":[{"name":"Beihang University"},{"name":"Beihang University"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0044-5267","authenticated-orcid":true,"given":"Longfei","family":"Zhao","sequence":"additional","affiliation":[{"name":"Beihang University"},{"name":"Beihang University"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"266","published-online":{"date-parts":[[2026,8,10]]},"reference":[{"key":"bbae9020bib1","doi-asserted-by":"publisher","first-page":"603","DOI":"10.1126\/science.1231806","type":"journal-article","article-title":"Controlled flight of a biologically inspired, insect-scale robot","volume":"340","author":"Ma","year":"2013","journal-title":"Science"},{"key":"bbae9020bib2","doi-asserted-by":"publisher","DOI":"10.1088\/1748-3190\/aa5c9e","type":"journal-article","article-title":"Experimental optimization of wing shape for a hummingbird-like flapping wing micro air vehicle","volume":"12","author":"Nan","year":"2017","journal-title":"Bioinspir. 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