{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,5]],"date-time":"2025-11-05T06:39:56Z","timestamp":1762324796278,"version":"build-2065373602"},"reference-count":35,"publisher":"MDPI AG","issue":"17","license":[{"start":{"date-parts":[[2020,8,27]],"date-time":"2020-08-27T00:00:00Z","timestamp":1598486400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100004281","name":"Narodowe Centrum Nauki","doi-asserted-by":"publisher","award":["2018\/29\/B\/ST8\/01293"],"award-info":[{"award-number":["2018\/29\/B\/ST8\/01293"]}],"id":[{"id":"10.13039\/501100004281","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Materials"],"abstract":"<jats:p>The middle ear is one of the smallest biomechanical systems in the human body and is responsible for the hearing process. Hearing is modelled in different ways and by various methods. In this paper, three-degree-of-freedom models of the human middle ear with different viscoelastic properties are proposed. Model 1 uses the Maxwell type viscoelasticity, Model 2 is based on the Kelvin\u2013Voigt viscoelasticity, and Model 3 uses the Kelvin\u2013Voigt viscoelasticity with relaxation effect. The primary aim of the study is to compare the models and their dynamic responses to a voice excitation. The novelty of this study lies in using different models of viscoelasticity and relaxation effect that has been previously unstudied. First, mathematical models of the middle ear were built, then they were solved numerically by the Runge\u2013Kutta procedure and finally, numerical results were compared with those obtained from experiments carried out on the temporal bone with the Laser Doppler Vibrometer. The models exhibit differences in the natural frequency and amplitudes near the second resonance. All analysed models can be used for modelling the rapidly changing processes that occur in the ear and to control active middle ear implants.<\/jats:p>","DOI":"10.3390\/ma13173779","type":"journal-article","created":{"date-parts":[[2020,8,27]],"date-time":"2020-08-27T08:05:18Z","timestamp":1598515518000},"page":"3779","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Biomechanics of the Human Middle Ear with Viscoelasticity of the Maxwell and the Kelvin\u2013Voigt Type and Relaxation Effect"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2808-2007","authenticated-orcid":false,"given":"Rafal","family":"Rusinek","sequence":"first","affiliation":[{"name":"Department of Applied Mechanics, Mechanical Engineering Faculty, Lublin University of Technology, 20-618 Lublin, Poland"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2777-1254","authenticated-orcid":false,"given":"Marcin","family":"Szymanski","sequence":"additional","affiliation":[{"name":"Department of Otolaryngology Head and Neck Surgery, Medical University of Lublin, 20-059 Lublin, Poland"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Robert","family":"Zablotni","sequence":"additional","affiliation":[{"name":"Department of Applied Mechanics, Mechanical Engineering Faculty, Lublin University of Technology, 20-618 Lublin, Poland"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,8,27]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"168","DOI":"10.1121\/1.1908610","article-title":"Network Model of the Middle Ear","volume":"33","author":"Moller","year":"1961","journal-title":"J. Acoust. Soc. Am."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1514","DOI":"10.1121\/1.1918382","article-title":"Analysis of the Middle-Ear Function. Part I: Input Impedance","volume":"34","author":"Zwislocki","year":"1962","journal-title":"J. Acoust. Soc. Am."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"43","DOI":"10.1016\/S0378-5955(00)00177-5","article-title":"Acoustic responses of the human middle ear","volume":"150","author":"Voss","year":"2000","journal-title":"Hear. Res."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Gyo, K., and Wada, H. (2004). Modeling Flexibility in the Human Ossicular Chain: Comparision to Ossicular Fixation Data. Middle Ear Mechanics in Research and Otology, Word Scientific.","DOI":"10.1142\/5389"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"60","DOI":"10.1016\/j.heares.2005.01.002","article-title":"Experimental ossicular fixations and the middle ears response to sound: Evidence for a flexible ossicular chain","volume":"204","author":"Nakajima","year":"2005","journal-title":"Hear. Res."},{"key":"ref_6","unstructured":"Feng, B., and Gan, R.Z. (2002, January 23\u201326). A lumped-parameter mechanical model of human ear for sound transmission. Proceedings of the Second Joint Embs-Bmes Conference 2002, Houston, TX, USA."},{"key":"ref_7","unstructured":"Sopakayang, R. (2010). Viscoelastic Models for Ligaments and Tendons. [Ph.D. Thesis, Virginia Polytechnic Institute and State University]."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"652","DOI":"10.1002\/jor.1100150504","article-title":"Ligament creep cannot be predicted from stress relaxation at low stress: A biomechanical study of the rabbit medial collateral ligament","volume":"15","author":"Thornton","year":"1997","journal-title":"J. Orthop. Res."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"908","DOI":"10.1114\/1.1408926","article-title":"Nonlinear Ligament Viscoelasticity","volume":"29","author":"Provenzano","year":"2001","journal-title":"Ann. Biomed. Eng."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"306","DOI":"10.1023\/B:ABME.0000012751.31686.70","article-title":"Nonlinear Viscoelasticity in Rabbit Medial Collateral Ligament","volume":"32","author":"Hingorani","year":"2004","journal-title":"Ann. Biomed. Eng."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"2034","DOI":"10.1016\/j.jbiomech.2005.06.025","article-title":"A novel microstructural approach in tendon viscoelastic modelling at the fibrillar level","volume":"39","author":"Ciarletta","year":"2006","journal-title":"J. Biomech."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1756","DOI":"10.1121\/1.4895696","article-title":"Nonlinear stiffness characteristics of the annular ligament","volume":"136","author":"Lauxmann","year":"2014","journal-title":"J. Acoust. Soc. Am."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"18","DOI":"10.1016\/j.cam.2012.07.010","article-title":"Nonlinear modelling of the middle ear as an elastic multibody system\u2014Applying model order reduction to acousto-structural coupled systems","volume":"246","author":"Ihrle","year":"2013","journal-title":"J. Comput. Appl. Math."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"4580467","DOI":"10.1155\/2020\/4580467","article-title":"Sound Transmission in the First Nonlinear Model of Middle Ear with an Active Implant","volume":"2020","author":"Rusinek","year":"2020","journal-title":"Math. Probl. Eng."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"20","DOI":"10.1016\/j.ijmecsci.2018.09.040","article-title":"Middle ear vibration with stiff and flexible shape memory prosthesis","volume":"150","author":"Rusinek","year":"2019","journal-title":"Int. J. Mech. Sci."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"163","DOI":"10.1016\/j.ijmecsci.2016.10.004","article-title":"Dynamics of the middle ear ossicles with an SMA prosthesis","volume":"127","author":"Rusinek","year":"2017","journal-title":"Int. J. Mech. Sci."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"387","DOI":"10.1007\/s10237-007-0094-x","article-title":"Mechanical properties of anterior malleolar ligament from experimental measurement and material modeling analysis","volume":"7","author":"Cheng","year":"2008","journal-title":"Biomech. Model. Mechanobiol."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"330","DOI":"10.1016\/j.medengphy.2010.10.022","article-title":"Mechanical properties of stapedial annular ligament","volume":"33","author":"Gan","year":"2011","journal-title":"Med Eng. Phys."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"358","DOI":"10.1016\/j.medengphy.2007.04.005","article-title":"Experimental measurement and modeling analysis on mechanical properties of tensor tympani tendon","volume":"30","author":"Cheng","year":"2008","journal-title":"Med Eng. Phys."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"913","DOI":"10.1115\/1.2800837","article-title":"Mechanical properties of stapedial tendon in human middle ear","volume":"129","author":"Cheng","year":"2007","journal-title":"J. Biomech. Eng."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"372","DOI":"10.12659\/MSM.882123","article-title":"Vibrations in the Human Middle Ear","volume":"17","author":"Rusinek","year":"2011","journal-title":"Med Sci. Monit."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"120013","DOI":"10.1063\/1.5019128","article-title":"Vibration of middle ear with shape memory prosthesis\u2014Experimental and numerical study","volume":"1922","author":"Rusinek","year":"2018","journal-title":"AIP Conf. Proc."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Fratzl, P. (2008). Collagen: Structure and Mechanics, Springer.","DOI":"10.1007\/978-0-387-73906-9"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1237","DOI":"10.1016\/S0094-114X(98)00068-8","article-title":"Nonlinear oscillations in machines and mechanisms theory","volume":"34","author":"Pust","year":"1999","journal-title":"Mech. Mach. Theory"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1016\/0021-9290(68)90032-8","article-title":"A rheological model for uncalcied parallel-bred collagenous tissue","volume":"1","author":"Viidik","year":"1967","journal-title":"J. Biomech."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"24","DOI":"10.1016\/j.heares.2007.03.006","article-title":"Fixation and detachment of superior and anterior malleolar ligaments in human middle ear: Experiment and modeling","volume":"230","author":"Dai","year":"2007","journal-title":"Hear. Res."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"178","DOI":"10.1159\/000013838","article-title":"Dynamics of Middle Ear Prostheses\u2014Simulations and Measurements","volume":"4","author":"Eiber","year":"1999","journal-title":"Audiol. Neurootol."},{"key":"ref_28","unstructured":"Gan, R.Z., and Sun, Q. (2002, January 23\u201326). Finite element modeling of human ear with external ear canal and middle ear cavity. Proceedings of the Second Joint Embs-Bmes Conference 2002, Houston, TX, USA."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"109","DOI":"10.1007\/s10237-002-0014-z","article-title":"Computer-integrated finite element modeling of human middle ear","volume":"1","author":"Sun","year":"2002","journal-title":"Biomech. Model. Mechanobiol."},{"key":"ref_30","first-page":"103","article-title":"A full-size physical model of the human middle ear","volume":"86","author":"Taschke","year":"2000","journal-title":"Acustica"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"103","DOI":"10.1016\/j.heares.2004.05.010","article-title":"Mechanisms of hearing loss resulting from middle-ear fluid","volume":"195","author":"Ravicz","year":"2004","journal-title":"Hear. Res."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"2541","DOI":"10.1007\/s10439-012-0596-2","article-title":"A Nonlinear Constitutive Model for Stress Relaxation in Ligaments and Tendons","volume":"40","author":"Davis","year":"2012","journal-title":"Ann. Biomed. Eng."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"653","DOI":"10.1007\/s10237-013-0525-9","article-title":"A three-dimensional constitutive model for the stress relaxation of articular ligaments","volume":"13","author":"Davis","year":"2014","journal-title":"Biomech. Model. Mechanobiol."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"1","DOI":"10.3233\/BIR-2010-0559","article-title":"Stress relaxation and recovery in tendon and ligament: Experiment and modeling","volume":"47","author":"Duenwald","year":"2010","journal-title":"Biorheology"},{"key":"ref_35","first-page":"593","article-title":"Movements of the ossicular chain during changes in static pressure","volume":"58","author":"Szymanski","year":"2004","journal-title":"Otolaryngol. Polska. Pol. Otolaryngol."}],"container-title":["Materials"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1996-1944\/13\/17\/3779\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T10:07:27Z","timestamp":1760177247000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1996-1944\/13\/17\/3779"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,8,27]]},"references-count":35,"journal-issue":{"issue":"17","published-online":{"date-parts":[[2020,9]]}},"alternative-id":["ma13173779"],"URL":"https:\/\/doi.org\/10.3390\/ma13173779","relation":{},"ISSN":["1996-1944"],"issn-type":[{"type":"electronic","value":"1996-1944"}],"subject":[],"published":{"date-parts":[[2020,8,27]]}}}