{"id":599,"date":"2024-08-29T09:54:55","date_gmt":"2024-08-29T07:54:55","guid":{"rendered":"https:\/\/site.ellistat.com\/?p=599"},"modified":"2025-02-05T09:55:42","modified_gmt":"2025-02-05T08:55:42","slug":"megbizhatosag-weibull-torveny","status":"publish","type":"post","link":"https:\/\/ellistat.com\/hu\/fiabilite-loi-de-weibull\/","title":{"rendered":"Megb\u00edzhat\u00f3s\u00e1g, Weibull-t\u00f6rv\u00e9ny"},"content":{"rendered":"\n<p>La loi de Weibull est la plus souvent utilis\u00e9e pour mod\u00e9liser des donn\u00e9es de fiabilit\u00e9.&nbsp;<a href=\"https:\/\/ellistat.com\/solutions-data-analysis\/\">Le module Data Analysis d\u2019Ellistat&nbsp;<\/a>vous permet de r\u00e9aliser ces analyses en toute simplicit\u00e9.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">D\u00e9finitions : Fiabilit\u00e9, Loi de Weibull, D\u00e9faillance<\/h2>\n\n\n\n<p><strong>Fiabilit\u00e9<\/strong>:<br>La fiabilit\u00e9 est l\u2019aptitude d\u2019une entit\u00e9 \u00e0 accomplir une fonction requise, dans des conditions donn\u00e9es, pendant un intervalle de temps donn\u00e9.<\/p>\n\n\n\n<p><strong>D\u00e9faillance:<\/strong><br>La d\u00e9faillance est la fin de l\u2019aptitude d\u2019un dispositif \u00e0 accomplir une fonction que l\u2019on attendait. Elle peut \u00eatre partielle (alt\u00e9ration des performances) ou totale (fin de la fonction).<\/p>\n\n\n\n<p><strong>R(t)<\/strong>: Probabilit\u00e9 qu\u2019une entit\u00e9 E soit non d\u00e9faillante sur la dur\u00e9e [0 ; t ], en supposant qu\u2019elle n\u2019est pas d\u00e9faillante \u00e0 l\u2019instant t = 0<\/p>\n\n\n\n<p><strong>F(t)<\/strong>: F(t) est la fonction cumul\u00e9e des d\u00e9faillances \u00e8 F(t) = 1 \u2013 R(t)<\/p>\n\n\n\n<p><strong>Densit\u00e9 de probabilit\u00e9<\/strong>\u00a0: f(t) repr\u00e9sente le taux de d\u00e9faillance d\u2019un produit, c\u2019est-\u00e0-dire la probabilit\u00e9 qu\u2019un produit soit d\u00e9faillant sur l\u2019intervalle de temps [t,t+dt]\u200b<\/p>\n\n\n\n<p><strong>\u03bb(t) ou taux de d\u00e9faillance :\u00a0<\/strong>\u03bb(t) repr\u00e9sente le taux de d\u00e9faillance d\u2019un produit, c\u2019est-\u00e0-dire la probabilit\u00e9 qu\u2019un produit soit d\u00e9faillant sur l\u2019intervalle de temps [t,t+dt] sachant qu\u2019il n\u2019\u00e9tait pas d\u00e9faillant \u00e0 l\u2019instant t\u200b<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>\u03bb(t) diminue<\/strong>&nbsp;: le taux de d\u00e9faut diminue avec le temps, cela correspond en g\u00e9n\u00e9ral \u00e0 des d\u00e9fauts de jeunesse. Les produits avec des d\u00e9faillances intrins\u00e8ques s\u2019alt\u00e8rent rapidement alors que les autres produits durent beaucoup plus longtemps.<\/li>\n\n\n\n<li><strong>\u03bb(t) constant<\/strong>&nbsp;: le taux de d\u00e9faillance ne d\u00e9pend pas du temps. Il y a autant de risque qu\u2019un produit soit d\u00e9faillant \u00e0 l\u2019instant t quelle que soit sa dur\u00e9e de vie. Il s\u2019agit des d\u00e9faillances intrins\u00e8ques. C\u2019est le type de d\u00e9faillance que l\u2019on retrouve souvent sur des produits \u00e9lectroniques.<\/li>\n\n\n\n<li><strong>\u03bb(t) augmente<\/strong>&nbsp;: le taux de d\u00e9faillance augmente avec le temps. Il y a de plus en plus de risque qu\u2019un produit soit d\u00e9faillant lorsque sa dur\u00e9e de vie augmente. Il s\u2019agit de la fin de vie du produit.<\/li>\n<\/ul>\n\n\n\n<p><strong>MTTF (Mean Time To Failure)<\/strong><\/p>\n\n\n\n<p>Le MTTF plut\u00f4t utilis\u00e9 en Fiabilit\u00e9 des produits est la moyenne des temps avant l\u2019apparition de la premi\u00e8re d\u00e9faillance.<\/p>\n\n\n\n<p><strong>MTBF (Mean Time Between Failure)<\/strong><\/p>\n\n\n\n<p>Le MTBF (Temps moyen de bon fonctionnement) est la moyenne des intervalles de temps entre deux apparitions de d\u00e9faillances.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Loi de Weibull<\/h2>\n\n\n\n<p>Pour mod\u00e9liser la loi de d\u00e9faillance d\u2019un produit, il faut \u00eatre capable de mod\u00e9liser de multiple types de d\u00e9faillance :<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"458\" src=\"https:\/\/ellistat.com\/wp-content\/uploads\/2024\/08\/Exemple-de-loi-1.webp\" alt=\"\" class=\"wp-image-1146\" srcset=\"https:\/\/ellistat.com\/wp-content\/uploads\/2024\/08\/Exemple-de-loi-1.webp 1024w, https:\/\/ellistat.com\/wp-content\/uploads\/2024\/08\/Exemple-de-loi-1-300x134.webp 300w, https:\/\/ellistat.com\/wp-content\/uploads\/2024\/08\/Exemple-de-loi-1-768x344.webp 768w, https:\/\/ellistat.com\/wp-content\/uploads\/2024\/08\/Exemple-de-loi-1-18x8.webp 18w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/figure>\n<\/div>\n\n\n<p>Il faut ainsi une grande flexibilit\u00e9 dans les formes, pour cela, la loi principalement utilis\u00e9e est la loi de Weibull, car elle permet une grande variabilit\u00e9 de forme.<\/p>\n\n\n\n<p>Gr\u00e2ce \u00e0 sa grande flexibilit\u00e9, la loi de Weibull permet de mod\u00e9liser le comportement de nombreux types de d\u00e9faillances telles que :<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>La r\u00e9sistance \u00e0 la rupture des composants ou l\u2019effort requis pour fatiguer des m\u00e9taux<\/li>\n\n\n\n<li>Le temps de d\u00e9faillance d\u2019un composant \u00e9lectronique<\/li>\n\n\n\n<li>Le temps de d\u00e9faillance pour des articles utilis\u00e9s en ext\u00e9rieur tels que les pneumatiques automobiles<\/li>\n\n\n\n<li>Les syst\u00e8mes qui faillissent lorsque le composant le plus faible dans le syst\u00e8me a une d\u00e9faillance<\/li>\n<\/ul>\n\n\n\n<p>La loi de Weibull permet \u00e9galement de mod\u00e9liser le comportement de diff\u00e9rentes situations de vie d\u2019un m\u00eame composant<br>La fonction de la loi de Weibull est la suivante :<\/p>\n\n\n\n<p class=\"has-text-align-center\"><span class=\"wp-katex-eq\" data-display=\"false\">R\\left( t \\right)=e-\\left( \\frac{t-\\delta}{\\phi} \\right)^{\\beta}<\/span><\/p>\n\n\n\n<p>Elle comporte 3 param\u00e8tres :<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><span class=\"wp-katex-eq\" data-display=\"false\">\\beta<\/span>: param\u00e8tre de forme\u00a0<\/li>\n\n\n\n<li>\u03b8\u00a0: param\u00e8tre d\u2019\u00e9chelle<\/li>\n\n\n\n<li><span class=\"wp-katex-eq\" data-display=\"false\">\\delta<\/span>: param\u00e8tre de retard<\/li>\n<\/ul>\n\n\n\n<h2 class=\"wp-block-heading\"><em>\u03b2<\/em>: param\u00e8tre de forme<\/h2>\n\n\n\n<p>Il permet d\u2019adapter la forme de la loi pour \u00eatre le plus proche du taux de d\u00e9faillance constat\u00e9 :<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"846\" height=\"812\" src=\"https:\/\/ellistat.com\/wp-content\/uploads\/2024\/08\/Wiebull-beta-2.webp\" alt=\"\" class=\"wp-image-1147\" srcset=\"https:\/\/ellistat.com\/wp-content\/uploads\/2024\/08\/Wiebull-beta-2.webp 846w, https:\/\/ellistat.com\/wp-content\/uploads\/2024\/08\/Wiebull-beta-2-300x288.webp 300w, https:\/\/ellistat.com\/wp-content\/uploads\/2024\/08\/Wiebull-beta-2-768x737.webp 768w, https:\/\/ellistat.com\/wp-content\/uploads\/2024\/08\/Wiebull-beta-2-13x12.webp 13w\" sizes=\"auto, (max-width: 846px) 100vw, 846px\" \/><\/figure>\n\n\n\n<p><em>\u03b2<\/em>=1 : Le taux de d\u00e9faillance est constant (<em>\u03bb<\/em>&nbsp;constant)<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"846\" height=\"812\" src=\"https:\/\/ellistat.com\/wp-content\/uploads\/2024\/08\/wiebull-beta-3.webp\" alt=\"\" class=\"wp-image-1148\" srcset=\"https:\/\/ellistat.com\/wp-content\/uploads\/2024\/08\/wiebull-beta-3.webp 846w, https:\/\/ellistat.com\/wp-content\/uploads\/2024\/08\/wiebull-beta-3-300x288.webp 300w, https:\/\/ellistat.com\/wp-content\/uploads\/2024\/08\/wiebull-beta-3-768x737.webp 768w, https:\/\/ellistat.com\/wp-content\/uploads\/2024\/08\/wiebull-beta-3-13x12.webp 13w\" sizes=\"auto, (max-width: 846px) 100vw, 846px\" \/><\/figure>\n\n\n\n<p><em>\u03b2<\/em>&gt;1 : Le taux de d\u00e9faillance augmente avec le temps (<em>\u03bb<\/em>&nbsp;augmente \u2013 fin de vie d\u2019un produit)<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">\u03b8: param\u00e8tre d&rsquo;\u00e9chelle<\/h2>\n\n\n\n<p>Le param\u00e8tre \u03b8 permet d\u2019ajuster l\u2019\u00e9chelle de la loi de distribution \u00e0 l\u2019\u00e9chelle du probl\u00e8me constat\u00e9, par exemple :<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"746\" src=\"https:\/\/ellistat.com\/wp-content\/uploads\/2024\/08\/Weibull-5.png\" alt=\"\" class=\"wp-image-1151\" srcset=\"https:\/\/ellistat.com\/wp-content\/uploads\/2024\/08\/Weibull-5.png 1024w, https:\/\/ellistat.com\/wp-content\/uploads\/2024\/08\/Weibull-5-300x219.png 300w, https:\/\/ellistat.com\/wp-content\/uploads\/2024\/08\/Weibull-5-768x560.png 768w, https:\/\/ellistat.com\/wp-content\/uploads\/2024\/08\/Weibull-5-16x12.png 16w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/figure>\n\n\n\n<p>La d\u00e9faillance apparait autour de t = 90.4<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"746\" src=\"https:\/\/ellistat.com\/wp-content\/uploads\/2024\/08\/Weibull-5-2.png\" alt=\"\" class=\"wp-image-1154\" srcset=\"https:\/\/ellistat.com\/wp-content\/uploads\/2024\/08\/Weibull-5-2.png 1024w, https:\/\/ellistat.com\/wp-content\/uploads\/2024\/08\/Weibull-5-2-300x219.png 300w, https:\/\/ellistat.com\/wp-content\/uploads\/2024\/08\/Weibull-5-2-768x560.png 768w, https:\/\/ellistat.com\/wp-content\/uploads\/2024\/08\/Weibull-5-2-16x12.png 16w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/figure>\n\n\n\n<p>La d\u00e9faillance apparait autour de t = 904<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><em>\u03b4<\/em>: param\u00e8tre de retard<\/h2>\n\n\n\n<p>Le param\u00e8tre \u03b4 permet de d\u00e9placer la loi de distribution d\u2019un param\u00e8tre \u03b4<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"846\" height=\"812\" src=\"https:\/\/ellistat.com\/wp-content\/uploads\/2024\/08\/Weibull-6.webp\" alt=\"\" class=\"wp-image-1152\" srcset=\"https:\/\/ellistat.com\/wp-content\/uploads\/2024\/08\/Weibull-6.webp 846w, https:\/\/ellistat.com\/wp-content\/uploads\/2024\/08\/Weibull-6-300x288.webp 300w, https:\/\/ellistat.com\/wp-content\/uploads\/2024\/08\/Weibull-6-768x737.webp 768w, https:\/\/ellistat.com\/wp-content\/uploads\/2024\/08\/Weibull-6-13x12.webp 13w\" sizes=\"auto, (max-width: 846px) 100vw, 846px\" \/><\/figure>\n\n\n\n<p><em>\u0392 = 3,6 \u2013 \u03b8 = 100, \u03b4 = 0<\/em><\/p>\n\n\n\n<p>La d\u00e9faillance apparait autour de t = 90.4<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"746\" src=\"https:\/\/ellistat.com\/wp-content\/uploads\/2024\/08\/Weibull-7.png\" alt=\"\" class=\"wp-image-1153\" srcset=\"https:\/\/ellistat.com\/wp-content\/uploads\/2024\/08\/Weibull-7.png 1024w, https:\/\/ellistat.com\/wp-content\/uploads\/2024\/08\/Weibull-7-300x219.png 300w, https:\/\/ellistat.com\/wp-content\/uploads\/2024\/08\/Weibull-7-768x560.png 768w, https:\/\/ellistat.com\/wp-content\/uploads\/2024\/08\/Weibull-7-16x12.png 16w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/figure>\n\n\n\n<p><em>\u0392 = 3,6 \u2013 \u03b8 = 100, \u03b4 = 100<\/em><\/p>\n\n\n\n<p>La d\u00e9faillance apparait autour de t = 190.4<\/p>\n\n\n\n<p><a href=\"https:\/\/fr.wikipedia.org\/wiki\/Loi_de_Weibull\">En savoir plus sur la loi de Weibull.<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"<p>La loi de Weibull est la plus souvent utilis\u00e9e pour mod\u00e9liser des donn\u00e9es de fiabilit\u00e9.&nbsp;Le module Data Analysis d\u2019Ellistat&nbsp;vous permet de r\u00e9aliser ces analyses en toute simplicit\u00e9. D\u00e9finitions : Fiabilit\u00e9, Loi de Weibull, D\u00e9faillance Fiabilit\u00e9:La fiabilit\u00e9 est l\u2019aptitude d\u2019une entit\u00e9 \u00e0 accomplir une fonction requise, dans des conditions donn\u00e9es, pendant un intervalle de temps donn\u00e9. [&hellip;]<\/p>\n","protected":false},"author":2,"featured_media":2938,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[7],"tags":[],"class_list":["post-599","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-data-analysis"],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.3 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Fiabilit\u00e9, loi de Weibull - Ellistat<\/title>\n<meta name=\"description\" content=\"Loi de Weibull : Pur mod\u00e9liser la loi de d\u00e9faillance d&#039;un produit, il faut \u00eatre capable de mod\u00e9liser de multiples types de d\u00e9faillance.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/ellistat.com\/hu\/megbizhatosag-weibull-torveny\/\" \/>\n<meta property=\"og:locale\" content=\"hu_HU\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Fiabilit\u00e9, loi de Weibull - 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