{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,24]],"date-time":"2025-12-24T12:36:13Z","timestamp":1766579773043,"version":"build-2065373602"},"reference-count":43,"publisher":"MDPI AG","issue":"22","license":[{"start":{"date-parts":[[2023,11,16]],"date-time":"2023-11-16T00:00:00Z","timestamp":1700092800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Chinese National Natural Science Fund","award":["11604033"],"award-info":[{"award-number":["11604033"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>Nonlinear ultrasonic non-destructive testing (NDT) is a widely used method for detecting micro-damages in various materials and structures due to its high sensitivity and directional capability. However, the extraction and modulation of extremely weak nonlinear ultrasonic signals is quite a challenge in practical applications. Therefore, this paper focuses on the second harmonic modulation signal method in nonlinear ultrasonic NDT and proposes the design of the phononic crystal filter (PC filter) to achieve this filtering function. Through finite element simulations, it is demonstrated that the filtering frequency of the filter is influenced by the structural configuration, material wave speed, and geometric characteristics. Then, the design method for cubic PC filters is established. Furthermore, a time-domain finite element method is introduced to verify the filtering ability of the filter and further validate the rationality of this design approach.<\/jats:p>","DOI":"10.3390\/s23229227","type":"journal-article","created":{"date-parts":[[2023,11,16]],"date-time":"2023-11-16T08:19:43Z","timestamp":1700122783000},"page":"9227","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Second Harmonic Modulation for Ultrasonic Signals Based on the Design of the Phononic Crystal Filter"],"prefix":"10.3390","volume":"23","author":[{"given":"Yue","family":"Zhu","sequence":"first","affiliation":[{"name":"School of Aeronautics, Northwestern Polytechnical University, Xi\u2019an 710072, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Youxuan","family":"Zhao","sequence":"additional","affiliation":[{"name":"College of Aerospace Engineering, Chongqing University, Chongqing 400044, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5645-734X","authenticated-orcid":false,"given":"Peng","family":"Cao","sequence":"additional","affiliation":[{"name":"Faculty of Architecture, Civil and Transportation Engineering, Beijing University of Technology, Beijing 100124, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,11,16]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"4184","DOI":"10.1103\/PhysRevLett.84.4184","article-title":"Composite Medium with Simultaneously Negative Permeability and Permittivity","volume":"84","author":"Smith","year":"2000","journal-title":"Phys. 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