{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:32:36Z","timestamp":1760059956682,"version":"build-2065373602"},"reference-count":16,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2025,7,23]],"date-time":"2025-07-23T00:00:00Z","timestamp":1753228800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>To characterize a phenomenological model of a mechanical oscillator, it is important to know the properties of the envelope of the three main physical motion variables: deviation from equilibrium, velocity, and acceleration. Experimental data show that friction forces restrict the shape of these functions. A linear, exponential, or more abrupt decay can be observed depending on the different physical systems and conditions. This paper aimed to contribute to clarifying the role that some types of friction forces play in these shapes. Three types of friction\u2014constant sliding friction, pressure drag proportional to the square of velocity, and friction drag proportional to velocity\u2014were considered to characterize the line connecting the maxima and minima of displacement for a generic mechanical harmonic oscillator. The ordinary differential equation (ODE), describing the harmonic oscillator simultaneously containing the three types of dissipative forces (constant, viscous, and quadratic), was numerically solved to obtain energy dissipation, and the extrema of both displacement and velocity. The differential equation ruling the behavior of the amplitude, as a function of the friction force coefficients, was obtained from energy considerations. Solving this equation, we obtained analytical functions, parametrized by the force coefficients that describe the oscillator tail. A comparison between these functions and the predicted oscillator ODE extrema was made, and the results were in agreement for all the situations tested. Information from the velocity extrema and nulls was enough to obtain a second function that rules completely the ODE solution. The correlations obtained allow for the reverse operation: from the identified extremum data, it was possible to identify univocally the three friction coefficients fitting used in the model. Motion equations were solved, and some physical properties, namely energy conservation and work of friction forces, were revisited.<\/jats:p>","DOI":"10.3390\/axioms14080554","type":"journal-article","created":{"date-parts":[[2025,7,23]],"date-time":"2025-07-23T10:49:17Z","timestamp":1753267757000},"page":"554","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["The Characterization of the Mechanical Harmonic Oscillator Extremum Envelope Shape According to Different Friction Types"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3841-0983","authenticated-orcid":false,"given":"Jo\u00e3o C.","family":"Fernandes","sequence":"first","affiliation":[{"name":"Physics Department, Instituto Superior T\u00e9cnicon, Av. Rovisco Pais, P-1049-001 Lisboa, Portugal"}]}],"member":"1968","published-online":{"date-parts":[[2025,7,23]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"4","DOI":"10.1088\/1361-6404\/aa6c52","article-title":"Study of large-angle anharmonic oscillations of a physical pendulum using an acceleration sensor","volume":"38","author":"Fernandes","year":"2017","journal-title":"Eur. J. Phys."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"155","DOI":"10.1088\/0143-0807\/23\/2\/309","article-title":"Oscillations with three damping effects","volume":"23","author":"Wang","year":"2002","journal-title":"Eur. J. Phys."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"577","DOI":"10.1119\/1.5034345","article-title":"Combined viscous and dry friction damping of oscillatory motion","volume":"86","author":"Hinrichsen","year":"2018","journal-title":"Am. J. Phys."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"32","DOI":"10.1119\/1.4935358","article-title":"Forced oscillations with linear and nonlinear damping","volume":"84","author":"Li","year":"2016","journal-title":"Am. J. Phys."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"112","DOI":"10.1119\/1.14703","article-title":"The pendulum-rich physics from a simple system","volume":"54","author":"Nelson","year":"1986","journal-title":"Am. J. Phys."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"984","DOI":"10.1119\/1.14838","article-title":"Pendulum damping","volume":"54","author":"Squire","year":"1986","journal-title":"Am. J. Phys."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"3726","DOI":"10.1038\/s41598-017-03999-w","article-title":"Effect of sliding friction in harmonic oscillators","volume":"7","author":"Vitorino","year":"2017","journal-title":"Sci. Rep."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1243","DOI":"10.1088\/0143-0807\/34\/5\/1243","article-title":"Oscillations of a quadratically damped pendulum","volume":"34","author":"Mungan","year":"2013","journal-title":"Eur. J. Phys."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1208","DOI":"10.1016\/j.jsv.2004.04.034","article-title":"Identifying Coulomb and viscous damping from free-vibration acceleration decrements","volume":"282","author":"Liang","year":"2005","journal-title":"J. Sound Vib."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"119","DOI":"10.2298\/PIM0999119C","article-title":"Oscillator with strong quadratic damping force","volume":"85","author":"Cveticanin","year":"2009","journal-title":"Publ. L\u2019Institute Math."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"118216","DOI":"10.1016\/j.jsv.2023.118216","article-title":"Prediction of energy dissipation by analytical solution to combined viscous and Coulomb damping","volume":"573","author":"Karthik","year":"2024","journal-title":"J. Sound Vib."},{"key":"ref_12","unstructured":"Rogalsky, M.S., and Palmer, S. (2006). Advanced University Physics, Taylor & Francis."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"700","DOI":"10.1016\/j.jsv.2007.12.019","article-title":"A study of a nonlinear vibration isolator with a quasi-zero stiffness characteristic","volume":"315","author":"Kovacic","year":"2008","journal-title":"J. Sound Vib."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Kovacic, I., and Brennan, M.J. (2011). The Duffing Equation: Nonlinear Oscillators and Their Behavior, John Wiley & Sons.","DOI":"10.1002\/9780470977859"},{"key":"ref_15","unstructured":"(2025, July 18). LabVIEWTM is a Product from National Instruments. Available online: http:\/\/www.ni.com."},{"key":"ref_16","unstructured":"(2025, July 18). ORIGINTM is a Product from Origin Lab\u2014Data Analysis and Graphical Software. Available online: http:\/\/www.originlab.com."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/8\/554\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T18:14:25Z","timestamp":1760033665000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/8\/554"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,7,23]]},"references-count":16,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2025,8]]}},"alternative-id":["axioms14080554"],"URL":"https:\/\/doi.org\/10.3390\/axioms14080554","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2025,7,23]]}}}