{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,30]],"date-time":"2026-04-30T18:20:08Z","timestamp":1777573208449,"version":"3.51.4"},"reference-count":44,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2022,12,24]],"date-time":"2022-12-24T00:00:00Z","timestamp":1671840000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100000266","name":"Engineering and Physical Sciences Research Council","doi-asserted-by":"publisher","award":["EP\/R004900\/1"],"award-info":[{"award-number":["EP\/R004900\/1"]}],"id":[{"id":"10.13039\/501100000266","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100000266","name":"Engineering and Physical Sciences Research Council","doi-asserted-by":"publisher","award":["EP\/R003645\/1"],"award-info":[{"award-number":["EP\/R003645\/1"]}],"id":[{"id":"10.13039\/501100000266","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100000266","name":"Engineering and Physical Sciences Research Council","doi-asserted-by":"publisher","award":["EP\/N010884\/1"],"award-info":[{"award-number":["EP\/N010884\/1"]}],"id":[{"id":"10.13039\/501100000266","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>Ultrasonic guided waves offer a convenient and practical approach to structural health monitoring and non-destructive evaluation. A key property of guided waves is the fully defined relationship between central frequency and propagation characteristics (phase velocity, group velocity and wavenumber)\u2014which is described using dispersion curves. For many guided wave-based strategies, accurate dispersion curve information is invaluable, such as group velocity for localisation. From experimental observations of dispersion curves, a system identification procedure can be used to determine the governing material properties. As well as returning an estimated value, it is useful to determine the distribution of these properties based on measured data. A method of simulating samples from these distributions is to use the iterative Markov-Chain Monte Carlo (MCMC) procedure, which allows for freedom in the shape of the posterior. In this work, a scanning-laser Doppler vibrometer is used to record the propagation of Lamb waves in a unidirectional-glass-fibre composite plate, and dispersion curve data for various propagation angles are extracted. Using these measured dispersion curve data, the MCMC sampling procedure is performed to provide a Bayesian approach to determining the dispersion curve information for an arbitrary plate. The distribution of the material properties at each angle is discussed, including the inferred confidence in the predicted parameters. The percentage errors of the estimated values for the parameters were 10\u201315 points larger when using the most likely estimates, as opposed to calculating from the posterior distributions, highlighting the advantages of using a probabilistic approach.<\/jats:p>","DOI":"10.3390\/s23010185","type":"journal-article","created":{"date-parts":[[2022,12,27]],"date-time":"2022-12-27T02:55:07Z","timestamp":1672109707000},"page":"185","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["A Bayesian Method for Material Identification of Composite Plates via Dispersion Curves"],"prefix":"10.3390","volume":"23","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3384-2346","authenticated-orcid":false,"given":"Marcus","family":"Haywood-Alexander","sequence":"first","affiliation":[{"name":"Dynamics Research Group, Department of Mechanical Engineering, The University of Sheffield, Mappin Street, Sheffield S1 3JD, UK"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5712-7323","authenticated-orcid":false,"given":"Nikolaos","family":"Dervilis","sequence":"additional","affiliation":[{"name":"Dynamics Research Group, Department of Mechanical Engineering, The University of Sheffield, Mappin Street, Sheffield S1 3JD, UK"}]},{"given":"Keith","family":"Worden","sequence":"additional","affiliation":[{"name":"Dynamics Research Group, Department of Mechanical Engineering, The University of Sheffield, Mappin Street, Sheffield S1 3JD, UK"}]},{"given":"Robin S.","family":"Mills","sequence":"additional","affiliation":[{"name":"Dynamics Research Group, Laboratory for Verification and Validation (LVV), Europa Avenue, Sheffield S9 1ZA, UK"}]},{"given":"Purim","family":"Ladpli","sequence":"additional","affiliation":[{"name":"Siemens Gamesa Renewable Energy, Assensjev 11, 9220 Aalborg, Denmark"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3433-3247","authenticated-orcid":false,"given":"Timothy J.","family":"Rogers","sequence":"additional","affiliation":[{"name":"Dynamics Research Group, Department of Mechanical Engineering, The University of Sheffield, Mappin Street, Sheffield S1 3JD, UK"}]}],"member":"1968","published-online":{"date-parts":[[2022,12,24]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"14","DOI":"10.4028\/www.scientific.net\/KEM.270-273.14","article-title":"Ultrasonic guided waves in structural health monitoring","volume":"270\u2013273","author":"Rose","year":"2004","journal-title":"Key Eng. Mater."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Viktorov, I.A. (1967). Rayleigh and Lamb Waves: Physical Theory and Applications, Plenum Press.","DOI":"10.1007\/978-1-4899-5681-1"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"167","DOI":"10.1111\/j.1475-1305.2001.tb01254.x","article-title":"Rayleigh and Lamb Waves - Basic Principles","volume":"37","author":"Worden","year":"2001","journal-title":"Strain"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Rose, J.L. (2014). Ultrasonic Waves in Solid Media, Cambridge University Press.","DOI":"10.1017\/CBO9781107273610"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"11","DOI":"10.1016\/0963-8695(92)90003-Y","article-title":"Optimization of Lamb wave inspection techniques","volume":"25","author":"Alleyne","year":"1992","journal-title":"NDT E Int."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"287","DOI":"10.1016\/0041-624X(96)00024-8","article-title":"The use of Lamb waves for the long range inspection of large structures","volume":"34","author":"Cawley","year":"1996","journal-title":"Ultrasonics"},{"key":"ref_7","first-page":"404","article-title":"Lamb wave reflection for the quick nondestructive evaluation of large composite laminates","volume":"52","author":"Guo","year":"1994","journal-title":"Mater. Eval."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"117063","DOI":"10.1016\/j.jsv.2022.117063","article-title":"Informative Bayesian Tools for Damage Localisation by Decomposition of Lamb Wave Signals","volume":"535","author":"Dervilis","year":"2022","journal-title":"J. Sound Vib."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"25","DOI":"10.1016\/j.ultras.2013.06.009","article-title":"Acoustic source localization","volume":"54","author":"Kundu","year":"2014","journal-title":"Ultrasonics"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"50","DOI":"10.1121\/1.1913575","article-title":"Elastic waves in free anisotropic plates","volume":"54","author":"Solie","year":"1973","journal-title":"J. Acoust. Soc. Am."},{"key":"ref_11","unstructured":"Kundu, T. (2019). Mechanics of Elastic Waves and Ultrasonic Nondestructive Evaluation, CRC Press."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1917","DOI":"10.1121\/1.1689342","article-title":"Wave propagation and damping in linear viscoelastic laminates","volume":"115","author":"Shorter","year":"2004","journal-title":"J. Acoust. Soc. Am."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1966","DOI":"10.1016\/j.ijsolstr.2013.02.016","article-title":"On the effect of damping on dispersion curves in plates","volume":"50","author":"Manconi","year":"2013","journal-title":"Int. J. Solids Struct."},{"key":"ref_14","unstructured":"Fan, Z. (2010). Applications of Guided Wave Propagation on Waveguides with Irregular Cross-Section. [Ph.D. Thesis, Imperial College London]."},{"key":"ref_15","unstructured":"Yang, Z., and Wu, Z. Guided Waves Dispersion Analysis in Composite Pipe Using the SAFE Method. Proceedings of the European Workshop on Structural Health Monitoring."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"704","DOI":"10.1016\/j.compstruct.2016.06.063","article-title":"Guided wave propagation in an infinite functionally graded magneto-electro-elastic plate by the Chebyshev spectral element method","volume":"153","author":"Xiao","year":"2016","journal-title":"Compos. Struct."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"045022","DOI":"10.1088\/0964-1726\/19\/4\/045022","article-title":"Multi-site damage localization in anisotropic plate-like structures using an active guided wave structural health monitoring system","volume":"19","author":"Moll","year":"2010","journal-title":"Smart Mater. Struct."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1332","DOI":"10.1109\/58.949742","article-title":"Acoustic wave propagation in continuous functionally graded plates: An extension of the Legendre polynomial approach","volume":"48","author":"Lefebvre","year":"2001","journal-title":"IEEE Trans. Ultrason. Ferroelectr. Freq. Control"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"449","DOI":"10.1016\/j.compstruct.2015.02.029","article-title":"Evaluation of effective elastic properties of layered composite fiber-reinforced plastic plates by piezoelectrically induced guided waves and laser Doppler vibrometry","volume":"125","author":"Eremin","year":"2015","journal-title":"Compos. Struct."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"257","DOI":"10.1016\/j.jsv.2016.11.037","article-title":"Estimation of plate material properties by means of a complex wavenumber fit using Hankel\u2019s functions and the image source method","volume":"390","author":"Roozen","year":"2017","journal-title":"J. Sound Vib."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"53","DOI":"10.1016\/j.ultras.2017.10.005","article-title":"Guided ultrasonic waves for determining effective orthotropic material parameters of continuous-fiber reinforced thermoplastic plates","volume":"84","author":"Webersen","year":"2018","journal-title":"Ultrasonics"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"110969","DOI":"10.1016\/j.compstruct.2019.110969","article-title":"On the identification of the elastic properties of composites by ultrasonic guided waves and optimization algorithm","volume":"223","author":"Cui","year":"2019","journal-title":"Compos. Struct."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Rautela, M., Gopalakrishnan, S., Gopalakrishnan, K., and Deng, Y. (2020, January 8\u201310). Ultrasonic guided waves based identification of elastic properties using 1d-convolutional neural networks. Proceedings of the 2020 IEEE International Conference on Prognostics and Health Management (ICPHM), Detroit, MI, USA.","DOI":"10.1109\/ICPHM49022.2020.9187057"},{"key":"ref_24","unstructured":"Gopalakrishnan, K., Rautela, M., and Deng, Y. Deep learning based identification of elastic properties using ultrasonic guided waves. Proceedings of the European Workshop on Structural Health Monitoring."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"114178","DOI":"10.1016\/j.compstruct.2021.114178","article-title":"Elastic constants identification of fibre-reinforced composites by using guided wave dispersion curves and genetic algorithm for improved simulations","volume":"272","author":"Kudela","year":"2021","journal-title":"Compos. Struct."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"112569","DOI":"10.1016\/j.compstruct.2020.112569","article-title":"Elastic constants identification of woven fabric reinforced composites by using guided wave dispersion curves and genetic algorithm","volume":"249","author":"Kudela","year":"2020","journal-title":"Compos. Struct."},{"key":"ref_27","unstructured":"Rylander, B.I. (2001). Computational Complexity and the Genetic Algorithm, University of Idaho."},{"key":"ref_28","unstructured":"Achenbach, J.D. (1973). Wave Propagation in Elastic Solids, Elsevier."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"1370","DOI":"10.1016\/j.compscitech.2006.09.023","article-title":"Group velocity and characteristic wave curves of Lamb waves in composites: Modeling and experiments","volume":"67","author":"Wang","year":"2007","journal-title":"Compos. Sci. Technol."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"27","DOI":"10.1016\/j.mechrescom.2016.03.007","article-title":"Investigation of guided waves propagation in orthotropic viscoelastic carbon\u2013epoxy plate by Legendre polynomial method","volume":"74","author":"Othmani","year":"2016","journal-title":"Mech. Res. Commun."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"377","DOI":"10.1016\/j.ast.2018.04.041","article-title":"Influences of anisotropic fiber-reinforced composite media properties on fundamental guided wave mode behavior: A Legendre polynomial approach","volume":"78","author":"Othmani","year":"2018","journal-title":"Aerosp. Sci. Technol."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"152","DOI":"10.1002\/zamm.19290090206","article-title":"Praktische Verfahren der Gleichungsaufl\u00f6sung","volume":"9","author":"Mises","year":"1929","journal-title":"ZAMM J. Appl. Math. Mech. Z. Angew. Math. Mech."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"1046","DOI":"10.1109\/TASSP.1986.1164944","article-title":"Simple, effective computation of principal eigenvectors and their eigenvalues and application to high-resolution estimation of frequencies","volume":"34","author":"Tufts","year":"1986","journal-title":"IEEE Trans. Acoust. Speech Signal Process."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"1159","DOI":"10.1121\/1.400530","article-title":"A two-dimensional Fourier transform method for the measurement of propagating multimode signals","volume":"89","author":"Alleyne","year":"1991","journal-title":"J. Acoust. Soc. Am."},{"key":"ref_35","unstructured":"Boresi, A.P., Schmidt, R.J., and Sidebottom, O.M. (1985). Advanced Mechanics of Materials, Wiley."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"258","DOI":"10.1016\/j.ultras.2010.10.004","article-title":"Simulation of ultrasonic Lamb wave generation, propagation and detection for a reconfigurable air coupled scanner","volume":"51","author":"Dobie","year":"2011","journal-title":"Ultrasonics"},{"key":"ref_37","doi-asserted-by":"crossref","unstructured":"Le Cam, L. (1990). Maximum likelihood: An introduction. International Statistical Review\/Revue Internationale de Statistique, Wiley.","DOI":"10.2307\/1403464"},{"key":"ref_38","unstructured":"Lehmann, E.L., and Casella, G. (2006). Theory of Point Estimation, Springer Science & Business Media."},{"key":"ref_39","doi-asserted-by":"crossref","unstructured":"Gamerman, D., and Lopes, H.F. (2006). Markov Chain Monte Carlo: Stochastic Simulation for Bayesian Inference, CRC Press.","DOI":"10.1201\/9781482296426"},{"key":"ref_40","doi-asserted-by":"crossref","unstructured":"Gilks, W.R., Richardson, S., and Spiegelhalter, D. (1995). Markov Chain Monte Carlo in Practice, CRC Press.","DOI":"10.1201\/b14835"},{"key":"ref_41","doi-asserted-by":"crossref","unstructured":"Barber, D. (2012). Bayesian Reasoning and Machine Learning, Cambridge University Press.","DOI":"10.1017\/CBO9780511804779"},{"key":"ref_42","doi-asserted-by":"crossref","unstructured":"Neal, R. (2011). Handbook of Markov Chain Monte Carlo, CRC Press. chapter MCMC Using Hamiltonian Dynamics.","DOI":"10.1201\/b10905-6"},{"key":"ref_43","first-page":"2","article-title":"Hamiltonian Monte Carlo for hierarchical models","volume":"79","author":"Betancourt","year":"2015","journal-title":"Curr. Trends Bayesian Methodol. Appl."},{"key":"ref_44","unstructured":"Stan Development Team (2021, November 21). Stan Modelling Language Users Guide and Reference Manual. Available online: https:\/\/mc-stan.org."}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/23\/1\/185\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:50:11Z","timestamp":1760147411000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/23\/1\/185"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,12,24]]},"references-count":44,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2023,1]]}},"alternative-id":["s23010185"],"URL":"https:\/\/doi.org\/10.3390\/s23010185","relation":{},"ISSN":["1424-8220"],"issn-type":[{"value":"1424-8220","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,12,24]]}}}