{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,7]],"date-time":"2026-10-07T18:49:47Z","timestamp":1791398987664,"version":"4.3.3"},"reference-count":38,"publisher":"Public Library of Science (PLoS)","issue":"10","license":[{"start":{"date-parts":[[2026,10,7]],"date-time":"2026-10-07T00:00:00Z","timestamp":1791331200000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100004242","name":"Princess Nourah Bint Abdulrahman University","doi-asserted-by":"publisher","award":["PNURSP2026R764"],"award-info":[{"award-number":["PNURSP2026R764"]}],"id":[{"id":"10.13039\/501100004242","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["www.plosone.org"],"crossmark-restriction":false},"short-container-title":["PLoS One"],"abstract":"<jats:p>This study examines the vibrational dynamics of elastically constrained beam foundation systems by considering Timoshenko, shear, Rayleigh, and Euler\u2013Bernoulli beam models resting on viscoelastic Filonenko-Borodich and Het\u00e9nyi foundations. Emphasis is placed on understanding how damping, foundation stiffness, shear deformation, and rotary inertia govern dynamic transitions across undamped, underdamped, and overdamped regimes. Using separation of variables for the analytical solution and the Galerkin finite element method for the numerical solution, non\u2013classical boundary conditions are used for the formulation of the equations. Analytical and numerical solutions are employed to explore frequency characteristics, oscillation amplitudes, and stability behavior. The results reveal that increasing foundation stiffness enhances natural frequencies and contributes to effective vibration control, while increasing damping induces a gradual transition from sustained oscillations to complete vibration suppression. These damping-driven transitions highlight complex dynamic behavior relevant to nonlinear systems theory. Excellent agreement between analytical and numerical findings confirms the robustness of the proposed formulation. The study underscores the critical role of viscoelastic foundations and damping mechanisms in shaping nonlinear vibration responses, offering insights applicable to the design of advanced structural systems. Extensions to non\u2013uniform geometries, heterogeneous materials, and environmental effects are expected to further enrich the dynamical behavior and practical relevance of the model.<\/jats:p>","DOI":"10.1371\/journal.pone.0359573","type":"journal-article","created":{"date-parts":[[2026,10,7]],"date-time":"2026-10-07T17:53:08Z","timestamp":1791395588000},"page":"e0359573","update-policy":"https:\/\/doi.org\/10.1371\/journal.pone.corrections_policy","source":"Crossref","is-referenced-by-count":0,"title":["A computational dynamics and damping\u2013induced transitions in beam\u2013foundation systems with viscoelastic constraints"],"prefix":"10.1371","volume":"21","author":[{"given":"Muna","family":"Elsadig","sequence":"first","affiliation":[{"name":"Princess Nourah bint Abdulrahman University","place":["Riyadh, Saudi Arabia"],"department":["Department of Information Systems, College of Computer and Information Sciences"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Shamsa","family":"Kanwal","sequence":"additional","affiliation":[{"name":"COMSATS University Islamabad","place":["Tarlai Kalan, Islamabad, Pakistan"],"department":["Department of Mathematics"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9707-0608","authenticated-orcid":true,"given":"Rab","family":"Nawaz","sequence":"additional","affiliation":[{"name":"COMSATS University Islamabad","place":["Tarlai Kalan, Islamabad, Pakistan"],"department":["Department of Mathematics"]},{"name":"Gulf University for Science and Technology","place":["Hawally, Kuwait"],"department":["Center for Applied Mathematics and Bioinformatics (CAMB)"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Qazi Muhammad Zaigham","family":"Zia","sequence":"additional","affiliation":[{"name":"COMSATS University Islamabad","place":["Tarlai Kalan, Islamabad, Pakistan"],"department":["Department of Mathematics"]}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"340","published-online":{"date-parts":[[2026,10,7]]},"reference":[{"issue":"2","key":"pone.0359573.ref001","first-page":"55","article-title":"On the differential equation for the flexural vibrations of prismatic rods","volume":"32","author":"SP Timoshenko","year":"1920","journal-title":"Glasnik Hrvatskog Prirodoslovnog Dru\u0161tva"},{"issue":"245","key":"pone.0359573.ref002","doi-asserted-by":"crossref","first-page":"744","DOI":"10.1080\/14786442108636264","article-title":"LXVI. On the correction for shear of the differential equation for transverse vibrations of prismatic bars","volume":"41","author":"SP Timoshenko","year":"1921","journal-title":"The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science"},{"issue":"253","key":"pone.0359573.ref003","doi-asserted-by":"crossref","first-page":"125","DOI":"10.1080\/14786442208633855","article-title":"X. On the transverse vibrations of bars of uniform cross-section","volume":"43","author":"SP Timoshenko","year":"1922","journal-title":"The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science"},{"issue":"4","key":"pone.0359573.ref004","doi-asserted-by":"crossref","first-page":"1154","DOI":"10.1115\/1.3422854","article-title":"Free vibration of a beam with one end spring-hinged and the other free","volume":"39","author":"KR Chun","year":"1972","journal-title":"J Appl Mech"},{"issue":"3","key":"pone.0359573.ref005","doi-asserted-by":"crossref","first-page":"813","DOI":"10.1115\/1.3423097","article-title":"Vibration frequencies for a uniform beam with one end spring-hinged and carrying a mass at the other free end","volume":"40","author":"TW Lee","year":"1973","journal-title":"J Appl Mech"},{"issue":"12","key":"pone.0359573.ref006","doi-asserted-by":"crossref","first-page":"3204","DOI":"10.1016\/j.camwa.2008.07.029","article-title":"An innovative eigenvalue problem solver for free vibration of Euler\u2013Bernoulli beam by using the Adomian decomposition method","volume":"56","author":"HY Lai","year":"2008","journal-title":"Comput Math Appl"},{"issue":"2","key":"pone.0359573.ref007","doi-asserted-by":"crossref","first-page":"171","DOI":"10.1002\/num.1690080207","article-title":"A fully Sinc\u2010Galerkin method for Euler\u2013Bernoulli beam models","volume":"8","author":"RC Smith","year":"1992","journal-title":"Numerical Methods Partial"},{"issue":"4","key":"pone.0359573.ref008","doi-asserted-by":"crossref","first-page":"541","DOI":"10.1016\/0022-460X(84)90508-X","article-title":"Vibrations of Timoshenko beams with elastically restrained ends","volume":"97","author":"BAH Abbas","year":"1984","journal-title":"Journal of Sound and Vibration"},{"issue":"1","key":"pone.0359573.ref009","doi-asserted-by":"crossref","first-page":"123","DOI":"10.1016\/S0022-460X(77)80118-1","article-title":"The second frequency spectrum of timoshenko beams","volume":"51","author":"BAH Abbas","year":"1977","journal-title":"Journal of Sound and Vibration"},{"issue":"3","key":"pone.0359573.ref010","doi-asserted-by":"crossref","first-page":"502","DOI":"10.1016\/0022-460X(89)90615-9","article-title":"Frequency analysis of clamped-clamped uniform beams with intermediate elastic support","volume":"133","author":"C Kameswara Rao","year":"1989","journal-title":"Journal of Sound and Vibration"},{"issue":"1","key":"pone.0359573.ref011","doi-asserted-by":"crossref","first-page":"83","DOI":"10.1016\/0045-7949(93)90281-H","article-title":"A general solution to vibrations of beams on variable Winkler elastic foundation","volume":"47","author":"D Zhou","year":"1993","journal-title":"Comput Struct"},{"issue":"2","key":"pone.0359573.ref012","first-page":"72","article-title":"Response of non-uniform beam resting on an elastic foundation to several moving masses","volume":"24","author":"ST Oni","year":"1996","journal-title":"Abacus J Math Assoc Nigeria"},{"key":"pone.0359573.ref013","author":"S Adhikari","year":"2000"},{"issue":"1","key":"pone.0359573.ref014","doi-asserted-by":"crossref","DOI":"10.1007\/s10665-021-10133-3","article-title":"An asymptotic investigation of the dynamics and dispersion of an elastic five-layered plate for anti-plane shear vibration","volume":"128","author":"R Nawaz","year":"2021","journal-title":"J Eng Math"},{"issue":"12","key":"pone.0359573.ref015","doi-asserted-by":"crossref","first-page":"125026","DOI":"10.1088\/1402-4896\/ac36a1","article-title":"Dispersion of elastic waves in an inhomogeneous multilayered plate over a Winkler elastic foundation with imperfect interfacial conditions","volume":"96","author":"M Asif","year":"2021","journal-title":"Phys Scr"},{"issue":"1","key":"pone.0359573.ref016","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.cma.2003.08.011","article-title":"Vibration analysis of edge-cracked beam on elastic foundation with axial loading using the differential quadrature method","volume":"194","author":"M-H Hsu","year":"2005","journal-title":"Computer Methods in Applied Mechanics and Engineering"},{"issue":"6","key":"pone.0359573.ref017","doi-asserted-by":"crossref","first-page":"1457","DOI":"10.3390\/buildings13061457","article-title":"Analyzing the Effect of Rotary Inertia and Elastic Constraints on a Beam Supported by a Wrinkle Elastic Foundation: A Numerical Investigation","volume":"13","author":"G Kanwal","year":"2023","journal-title":"Buildings"},{"issue":"6","key":"pone.0359573.ref018","doi-asserted-by":"crossref","first-page":"065017","DOI":"10.1088\/1402-4896\/acd482","article-title":"Effects of shear deformation and rotary inertia on elastically constrained beam resting on Pasternak foundation","volume":"98","author":"G Kanwal","year":"2023","journal-title":"Phys Scr"},{"issue":"2","key":"pone.0359573.ref019","doi-asserted-by":"crossref","first-page":"368504231172585","DOI":"10.1177\/00368504231172585","article-title":"Dispersion of elastic waves in the three-layered inhomogeneous sandwich plate embedded in the Winkler foundations","volume":"106","author":"M Asif","year":"2023","journal-title":"Sci Prog"},{"issue":"4","key":"pone.0359573.ref020","doi-asserted-by":"crossref","first-page":"467","DOI":"10.1007\/BF02916477","article-title":"Natural frequencies of Euler-Bernoulli beam with open cracks on elastic foundations","volume":"20","author":"Y Shin","year":"2006","journal-title":"J Mech Sci Technol"},{"issue":"1","key":"pone.0359573.ref021","doi-asserted-by":"crossref","first-page":"151","DOI":"10.1016\/0045-7949(94)00594-S","article-title":"Free vibrations of Timoshenko beams on two-parameter elastic foundation","volume":"57","author":"MA De Rosa","year":"1995","journal-title":"Comput Struct"},{"key":"pone.0359573.ref022","volume-title":"Beams on Elastic Foundation: Theory with Applications in the Fields of Civil and Mechanical Engineering","author":"M Het\u00e9nyi","year":"1973"},{"issue":"5","key":"pone.0359573.ref023","doi-asserted-by":"crossref","first-page":"663","DOI":"10.1002\/eqe.4290100504","article-title":"Vibration of beams on partial elastic foundations","volume":"10","author":"PF Doyle","year":"1982","journal-title":"Earthquake Eng Struct Dyn"},{"issue":"3","key":"pone.0359573.ref024","doi-asserted-by":"crossref","first-page":"773","DOI":"10.3390\/mca16030773","article-title":"Free vibration analysis of beams on variable Winkler elastic foundation by using the differential transform method","volume":"16","author":"A Kacar","year":"2011","journal-title":"Math Comput Appl"},{"key":"pone.0359573.ref025","doi-asserted-by":"crossref","first-page":"1057","DOI":"10.1016\/j.jsv.2007.08.014","article-title":"Timoshenko beam-column with generalized end conditions on elastic foundation: Dynamic-stiffness matrix and load vector","volume":"310","author":"LG Arboleda-Monsalve","year":"2008","journal-title":"Journal of Sound and Vibration"},{"issue":"5","key":"pone.0359573.ref026","doi-asserted-by":"crossref","first-page":"743","DOI":"10.1016\/j.compgeo.2008.12.006","article-title":"Linear and nonlinear vibration of non-uniform beams on two-parameter foundations using p-elements","volume":"36","author":"B Zhu","year":"2009","journal-title":"Computers and Geotechnics"},{"issue":"3","key":"pone.0359573.ref027","doi-asserted-by":"crossref","first-page":"606","DOI":"10.1016\/j.apm.2005.11.023","article-title":"Nonlinear analysis of thin rectangular plates on Winkler\u2013Pasternak elastic foundations by DSC\u2013HDQ methods","volume":"31","author":"\u00d6 Civalek","year":"2007","journal-title":"Applied Mathematical Modelling"},{"issue":"2","key":"pone.0359573.ref028","doi-asserted-by":"crossref","first-page":"149","DOI":"10.1016\/S0022-460X(77)80029-1","article-title":"Natural frequencies of Timoshenko beams on pasternak foundations","volume":"51","author":"TM Wang","year":"1977","journal-title":"Journal of Sound and Vibration"},{"issue":"6","key":"pone.0359573.ref029","doi-asserted-by":"crossref","first-page":"891","DOI":"10.1007\/s40032-018-0485-z","article-title":"Dynamic Response of a Damped Euler\u2013Bernoulli Beam Having Elastically Restrained Boundary Supports","volume":"100","author":"K Mahapatra","year":"2018","journal-title":"J Inst Eng India Ser C"},{"key":"pone.0359573.ref030","doi-asserted-by":"crossref","first-page":"106707","DOI":"10.1016\/j.ijmecsci.2021.106707","article-title":"Free and forced vibration of the double beam with arbitrary end conditions connected with a viscoelastic layer and discrete points","volume":"209","author":"X Zhao","year":"2021","journal-title":"Int J Mech Sci"},{"issue":"2","key":"pone.0359573.ref031","doi-asserted-by":"crossref","first-page":"460","DOI":"10.3390\/ma14020460","article-title":"Stress-Based FEM in the Problem of Bending of Euler-Bernoulli and Timoshenko Beams Resting on Elastic Foundation","volume":"14","author":"Z Wi\u0119ckowski","year":"2021","journal-title":"Materials (Basel)"},{"issue":"2","key":"pone.0359573.ref032","doi-asserted-by":"crossref","first-page":"969","DOI":"10.1177\/14613484241298998","article-title":"Analyzing the chaotic and stability behavior of a duffing oscillator excited by a sinusoidal external force","volume":"44","author":"MK Abohamer","year":"2025","journal-title":"Journal of Low Frequency Noise, Vibration and Active Control"},{"issue":"5","key":"pone.0359573.ref033","doi-asserted-by":"crossref","first-page":"2155","DOI":"10.1007\/s42417-022-00693-8","article-title":"Analytical Approximate Solutions of a Magnetic Spherical Pendulum: Stability Analysis","volume":"11","author":"GM Moatimid","year":"2022","journal-title":"J Vib Eng Technol"},{"issue":"7","key":"pone.0359573.ref034","doi-asserted-by":"crossref","first-page":"2550067","DOI":"10.1142\/S0219455425500671","article-title":"Vibrational Analysis of Axially Loaded Rayleigh Beams on Variable Winkler Foundations with Implications for Structural Design","volume":"25","author":"A Alruwaili","year":"2025","journal-title":"International Journal of Structural Stability and Dynamics"},{"issue":"3","key":"pone.0359573.ref035","first-page":"910","article-title":"Damping-based attitude control of a gyrostatic satellite in resistive media","volume":"12","author":"AH Elneklawy","year":"2026","journal-title":"Journal of Applied and Computational Mechanics"},{"key":"pone.0359573.ref036","doi-asserted-by":"crossref","first-page":"690","DOI":"10.1016\/j.actaastro.2026.07.061","article-title":"Parametric resonance and chaos in gyroscopic systems: Analysis of electromagnetically perturbed spacecraft dynamics","volume":"249","author":"AH Elneklawy","year":"2026","journal-title":"Acta Astronautica"},{"key":"pone.0359573.ref037","doi-asserted-by":"crossref","first-page":"93","DOI":"10.1007\/s42417-025-02256-z","article-title":"Energy decay in a gyrostatically influenced rigid body","volume":"14","author":"TS Amer","year":"2026","journal-title":"Journal of Vibration Engineering & Technologies"},{"issue":"5","key":"pone.0359573.ref038","doi-asserted-by":"crossref","DOI":"10.1007\/s42417-025-01894-7","article-title":"Damped Vibrational Response of Beams on Multi-Parametric Foundations: A Computational and Analytical Approach","volume":"13","author":"G Kanwal","year":"2025","journal-title":"J Vib Eng Technol"}],"container-title":["PLOS One"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/dx.plos.org\/10.1371\/journal.pone.0359573","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,10,7]],"date-time":"2026-10-07T17:53:16Z","timestamp":1791395596000},"score":1,"resource":{"primary":{"URL":"https:\/\/dx.plos.org\/10.1371\/journal.pone.0359573"}},"subtitle":[],"editor":[{"given":"Mohammadreza","family":"Vafaei","sequence":"first","affiliation":[{"name":"Universiti Teknologi Malaysia","place":["MALAYSIA"]}],"role":[{"vocabulary":"crossref","role":"editor"}]}],"short-title":[],"issued":{"date-parts":[[2026,10,7]]},"references-count":38,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2026,10,7]]}},"URL":"https:\/\/doi.org\/10.1371\/journal.pone.0359573","relation":{},"ISSN":["1932-6203"],"issn-type":[{"value":"1932-6203","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,10,7]]}}}