Transmission ratio in technopolymer gears: calculation and choice of tooth count<\/title>\n\n <link rel=\"preconnect\" href=\"https:\/\/fonts.googleapis.com\">\n <link rel=\"preconnect\" href=\"https:\/\/fonts.gstatic.com\" crossorigin=\"\">\n <link href=\"https:\/\/fonts.googleapis.com\/css2?family=Inter:wght@400;500;600;700&display=swap\" rel=\"stylesheet\">\n\n <link rel=\"stylesheet\" href=\"https:\/\/cdn.jsdelivr.net\/npm\/katex@0.16.10\/dist\/katex.min.css\">\n <script defer=\"\" src=\"https:\/\/cdn.jsdelivr.net\/npm\/katex@0.16.10\/dist\/katex.min.js\"><\/script>\n <script defer=\"\" src=\"https:\/\/cdn.jsdelivr.net\/npm\/katex@0.16.10\/dist\/contrib\/auto-render.min.js\" onload=\"renderMathInElement(document.body, {\n delimiters: [\n { left: '$$', right: '$$', display: true },\n { left: '$', right: '$', display: false }\n ]\n });\"><\/script>\n\n <style>\n *, *::before, *::after { box-sizing: border-box; margin: 0; padding: 0; }\n\n :root {\n --green: #1a6b3c;\n --green-dark: #0f4a28;\n --green-mid: #2d8a55;\n --green-light: #e8f4ed;\n --green-xlight: #f2faf5;\n --amber: #b45309;\n --amber-light: #fef3c7;\n\n --bg: #ffffff;\n --bg-soft: #f8f9fa;\n --bg-muted: #f1f3f5;\n --border: #e5e7eb;\n --border-mid: #d1d5db;\n\n --text-primary: #111827;\n --text-secondary: #4b5563;\n --text-muted: #9ca3af;\n\n --radius-sm: 6px;\n --radius-md: 8px;\n --radius-lg: 12px;\n --shadow-sm: 0 1px 3px rgba(0,0,0,.06), 0 1px 2px rgba(0,0,0,.04);\n --shadow-md: 0 4px 12px rgba(0,0,0,.08), 0 2px 4px rgba(0,0,0,.04);\n }\n\n body { min-height: 100vh; 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}\n\n .badge {\n display: inline-block;\n font-size: 12px;\n font-weight: 600;\n padding: 2px 8px;\n border-radius: 20px;\n background: var(--green-light);\n color: var(--green-dark);\n }\n\n \/* ============================================================\n RESPONSIVE\n ============================================================ *\/\n @media (max-width: 900px) {\n .article-body { flex-direction: column; padding: 20px 20px 48px; }\n }\n \n \/* Formula blocks *\/\n .formula-block {\n background: var(--bg-soft);\n border: 1px solid var(--border);\n border-radius: var(--radius-md);\n padding: 16px 20px;\n margin: 16px 0;\n text-align: center;\n overflow-x: auto;\n }\n .katex { font-size: 1.1em; }\n .formula-block .katex { font-size: 1.25em; }\n .formula-block .katex-display { margin: 0; }\n\n \/* Glossary tooltips *\/\n .glossary-term {\n border-bottom: 1px dotted var(--green);\n color: var(--green);\n font-style: italic;\n cursor: help;\n }\n .glossary-tooltip {\n position: fixed;\n z-index: 9999;\n background: #fff;\n border: 1px solid var(--border-mid);\n border-radius: var(--radius-md);\n box-shadow: var(--shadow-md);\n padding: 10px 14px;\n max-width: 300px;\n font-family: 'Inter', sans-serif;\n font-size: 13px;\n }\n .glossary-tooltip__title {\n font-weight: 600;\n color: var(--text-primary);\n margin-bottom: 4px;\n }\n .glossary-tooltip__def {\n font-size: 12.5px;\n color: var(--text-secondary);\n line-height: 1.6;\n }\n .glossary-tooltip__tag {\n display: inline-block;\n margin-top: 6px;\n font-size: 9px;\n font-weight: 600;\n padding: 2px 7px;\n border-radius: 20px;\n background: var(--green-light);\n color: var(--green-dark);\n text-transform: uppercase;\n letter-spacing: 0.04em;\n }\n\n <\/style>\n<\/head>\n<body>\n\n<div id=\"tt-diametro-primitivo\" class=\"glossary-tooltip\" hidden=\"\">\n <div class=\"glossary-tooltip__title\">Pitch diameter<\/div>\n <div class=\"glossary-tooltip__def\">Diameter of the theoretical circle at which contact occurs between two meshing gears. Obtained from the product of the module and the number of teeth.<\/div>\n <span class=\"glossary-tooltip__tag\">Glossary<\/span>\n<\/div>\n<div id=\"tt-gioco-di-flanchi-backlash\" class=\"glossary-tooltip\" hidden=\"\">\n <div class=\"glossary-tooltip__title\">Backlash<\/div>\n <div class=\"glossary-tooltip__def\">Free clearance between the flanks of two mating teeth, measured at the pitch circle. It prevents jamming due to thermal expansion or assembly errors.<\/div>\n <span class=\"glossary-tooltip__tag\">Glossary<\/span>\n<\/div>\n<div id=\"tt-angolo-di-pressione\" class=\"glossary-tooltip\" hidden=\"\">\n <div class=\"glossary-tooltip__title\">Pressure angle<\/div>\n <div class=\"glossary-tooltip__def\">Angle between the direction of the contact force between mating teeth and the tangent to the pitch circle at the point of contact. The standard value is 20\u00b0.<\/div>\n <span class=\"glossary-tooltip__tag\">Glossary<\/span>\n<\/div>\n<div id=\"tt-sottotaglio\" class=\"glossary-tooltip\" hidden=\"\">\n <div class=\"glossary-tooltip__title\">Undercut<\/div>\n <div class=\"glossary-tooltip__def\">Unintentional removal of material at the tooth root during profile generation. It occurs with low tooth counts and weakens the root cross-section.<\/div>\n <span class=\"glossary-tooltip__tag\">Glossary<\/span>\n<\/div>\n\n <div class=\"article-page\">\n <div class=\"article-body\">\n <div class=\"article-body__main\"><script type=\"application\/ld+json\">\n{\n \"@context\": \"https:\/\/schema.org\",\n \"@graph\": [\n {\n \"@type\": \"TechArticle\",\n \"@id\": \"#article\",\n \"headline\": \"Transmission ratio in technopolymer gears: calculation and choice of tooth count\",\n \"description\": \"The transmission ratio is the ratio between the rotational speed of the input gear and that of the output gear. 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It occurs with low tooth counts and weakens the root cross-section.\",\n \"inDefinedTermSet\": \"#glossario\"\n }\n ]\n }\n ]\n}\n<\/script>\n\n\n\n <h1 class=\"article__title\">Transmission ratio in technopolymer gears: calculation and choice of tooth count<\/h1>\n<h2 class=\"article__h2\" id=\"what-the-transmission-ratio-is\">\n <span class=\"article__h2-icon\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"16\" height=\"16\" viewBox=\"0 0 24 24\" fill=\"none\" stroke=\"currentColor\" stroke-width=\"2\" stroke-linecap=\"round\" stroke-linejoin=\"round\"><line x1=\"6\" y1=\"3\" x2=\"6\" y2=\"15\"><\/line><circle cx=\"18\" cy=\"6\" r=\"3\"><\/circle><circle cx=\"6\" cy=\"18\" r=\"3\"><\/circle><path d=\"M18 9a9 9 0 0 1-9 9\"><\/path><\/svg><\/span>\n What the transmission ratio is\n<\/h2>\n<p class=\"article__subtitle\">The transmission ratio is the ratio between the rotational speed of the input gear and that of the output gear. For two meshing wheels, it’s calculated by dividing the number of teeth of the driven wheel by that of the driving one:<\/p>\n<div class=\"formula-block\">$$i = \\frac{n_1}{n_2} = \\frac{Z_2}{Z_1} = \\frac{d_2}{d_1}$$<\/div>\n<p class=\"article__p\">Where $n_1$ is the speed of the driving wheel, $n_2$ that of the driven one, $Z_1$ and $Z_2$ the respective tooth counts, $d_1$ and $d_2$ the pitch diameters.<\/p>\n<p class=\"article__p\">If the ratio is greater than 1, you get a speed reduction and an increase in output torque. If it’s less than 1, speed increases and torque decreases. You can express it as a fraction, a decimal number or a colon-separated pair: a 20-tooth pinion with a 60-tooth driven wheel gives a 3:1 ratio, that is, the output speed is one third of the input.<\/p>\n<div class=\"infobox infobox--grey\">\n <div class=\"infobox__title\">\n <svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"14\" height=\"14\" viewBox=\"0 0 24 24\" fill=\"none\" stroke=\"currentColor\" stroke-width=\"2\" stroke-linecap=\"round\" stroke-linejoin=\"round\"><path d=\"M14.7 6.3a1 1 0 0 0 0 1.4l1.6 1.6a1 1 0 0 0 1.4 0l3.77-3.77a6 6 0 0 1-7.94 7.94l-6.91 6.91a2.12 2.12 0 0 1-3-3l6.91-6.91a6 6 0 0 1 7.94-7.94l-3.76 3.76z\"><\/path><\/svg>\n Technical note\n <\/div>\n <p class=\"infobox__text\">Some catalogs and software use the inverse convention, dividing the driving teeth by the driven ones. Always specify whether the ratio indicates reduction, multiplication or inverse kinematic ratio, to avoid interpretation errors.<\/p>\n<\/div>\n<h2 class=\"article__h2\" id=\"why-in-technopolymers-the-calculation-isnt-enough\">\n <span class=\"article__h2-icon\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"16\" height=\"16\" viewBox=\"0 0 24 24\" fill=\"none\" stroke=\"currentColor\" stroke-width=\"2\" stroke-linecap=\"round\" stroke-linejoin=\"round\"><circle cx=\"12\" cy=\"12\" r=\"10\"><\/circle><line x1=\"12\" y1=\"8\" x2=\"12\" y2=\"12\"><\/line><line x1=\"12\" y1=\"16\" x2=\"12.01\" y2=\"16\"><\/line><\/svg><\/span>\n Why in technopolymers the calculation isn’t enough\n<\/h2>\n<p class=\"article__p\">With metal gears, once you’ve calculated the geometric ratio, the system behaves predictably. With technopolymers you can’t stop there.<\/p>\n<p class=\"article__p\">The effective ratio can vary based on temperature, thermal expansion, moisture absorption, applied load and wear over time. The transmission ratio is also a ratio between pitch diameters: any dimensional change, even small, alters the real behavior of the system.<\/p>\n<p class=\"article__p\">On top of that, friction generates heat that polymers dissipate less quickly than metals. This affects durability and behavior under continuous load, especially at high peripheral speed:<\/p>\n<div class=\"formula-block\">$$v = \\frac{\\pi \\cdot d \\cdot n}{60}$$<\/div>\n<p class=\"article__p\">Where v is the peripheral speed, d the <dfn id=\"def-diametro-primitivo\"><span class=\"glossary-term\" data-tooltip=\"tt-diametro-primitivo\" title=\"Diameter of the theoretical circle at which contact occurs between two meshing gears. Obtained from the product of the module and the number of teeth.\">pitch diameter<\/span><\/dfn> and n the rotational speed in rpm.<\/p>\n<div class=\"warnbox\">\n <div class=\"warnbox__title\">\n <svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"14\" height=\"14\" viewBox=\"0 0 24 24\" fill=\"none\" stroke=\"currentColor\" stroke-width=\"2\" stroke-linecap=\"round\" stroke-linejoin=\"round\"><path d=\"M10.29 3.86L1.82 18a2 2 0 0 0 1.71 3h16.94a2 2 0 0 0 1.71-3L13.71 3.86a2 2 0 0 0-3.42 0z\"><\/path><line x1=\"12\" y1=\"9\" x2=\"12\" y2=\"13\"><\/line><line x1=\"12\" y1=\"17\" x2=\"12.01\" y2=\"17\"><\/line><\/svg>\n Warning\n <\/div>\n <p class=\"warnbox__text\">In technopolymers, underestimating the dimensional change with temperature leads to insufficient clearances and a risk of seizing. Always calculate the nominal ratio, then check under the real operating conditions.<\/p>\n<\/div>\n<h2 class=\"article__h2\" id=\"torque-speed-and-power\">\n <span class=\"article__h2-icon\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"16\" height=\"16\" viewBox=\"0 0 24 24\" fill=\"none\" stroke=\"currentColor\" stroke-width=\"2\" stroke-linecap=\"round\" stroke-linejoin=\"round\"><polyline points=\"22 12 18 12 15 21 9 3 6 12 2 12\"><\/polyline><\/svg><\/span>\n Torque, speed and power\n<\/h2>\n<p class=\"article__p\">The relation between angular speed and torque is direct: at equal power, reducing speed increases the transmitted torque. Torque depends on the tangential force and the pitch radius:<\/p>\n<div class=\"formula-block\">$$M = F \\cdot r$$<\/div>\n<p class=\"article__p\">In technopolymers these values aren’t static. Elastic deformation under load, thermal variations and progressive wear make them fluctuate during operation. Keep this in mind when you size the system.<\/p>\n<h2 class=\"article__h2\" id=\"williss-rule-for-epicyclic-systems\">\n <span class=\"article__h2-icon\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"16\" height=\"16\" viewBox=\"0 0 24 24\" fill=\"none\" stroke=\"currentColor\" stroke-width=\"2\" stroke-linecap=\"round\" stroke-linejoin=\"round\"><path d=\"M20 7h-9\"><\/path><path d=\"M14 17H5\"><\/path><circle cx=\"17\" cy=\"17\" r=\"3\"><\/circle><circle cx=\"7\" cy=\"7\" r=\"3\"><\/circle><\/svg><\/span>\n Willis’s rule for epicyclic systems\n<\/h2>\n<p class=\"article__p\">In epicyclic (or planetary) reducers, the wheels don’t rotate around fixed axes: part of the system is carried by the planet carrier. In these cases, Willis’s rule lets you calculate the ratio in the various operating configurations:<\/p>\n<div class=\"formula-block\">$$\\frac{(\\omega_A – \\omega_C)}{(\\omega_B – \\omega_C)} = \\frac{- Z_B}{Z_A}$$<\/div>\n<p class=\"article__p\">Where \u03c9A and \u03c9B are the angular speeds of two wheels, \u03c9C the planet-carrier speed, ZA and ZB the respective tooth counts.<\/p>\n<p class=\"article__p\">In technopolymer systems, Willis’s rule describes the ideal behavior. The real system can deviate from it due to deformation, <dfn id=\"def-gioco-di-flanchi-backlash\"><span class=\"glossary-term\" data-tooltip=\"tt-gioco-di-flanchi-backlash\" title=\"Free clearance between the flanks of two mating teeth, measured at the pitch circle. It prevents jamming due to thermal expansion or assembly errors.\">backlash<\/span><\/dfn>, wear and thermal variations: this deviation is called ratio error, and it’s the difference between the theoretical position and the actual position of the driven gear.<\/p>\n<p class=\"article__p\">To limit it: size the tooth backlash as a function of material, temperature and moisture; evaluate reinforced materials such as PA6+GF for greater dimensional stability; check dimensions and clearances after environmental conditioning.<\/p>\n<h2 class=\"article__h2\" id=\"how-many-teeth-to-choose\">\n <span class=\"article__h2-icon\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"16\" height=\"16\" viewBox=\"0 0 24 24\" fill=\"none\" stroke=\"currentColor\" stroke-width=\"2\" stroke-linecap=\"round\" stroke-linejoin=\"round\"><line x1=\"4\" y1=\"9\" x2=\"20\" y2=\"9\"><\/line><line x1=\"4\" y1=\"15\" x2=\"20\" y2=\"15\"><\/line><line x1=\"10\" y1=\"3\" x2=\"8\" y2=\"21\"><\/line><line x1=\"16\" y1=\"3\" x2=\"14\" y2=\"21\"><\/line><\/svg><\/span>\n How many teeth to choose\n<\/h2>\n<p class=\"article__p\">The number of teeth affects strength, quietness, durability, precision and the risk of interference (the abnormal contact between teeth that leads to seizing or accelerated wear).<\/p>\n<p class=\"article__p\">With a 20\u00b0 <dfn id=\"def-angolo-di-pressione\"><span class=\"glossary-term\" data-tooltip=\"tt-angolo-di-pressione\" title=\"Angle between the direction of the contact force between mating teeth and the tangent to the pitch circle at the point of contact. The standard value is 20\u00b0.\">pressure angle<\/span><\/dfn>, the minimum number to avoid <dfn id=\"def-sottotaglio\"><span class=\"glossary-term\" data-tooltip=\"tt-sottotaglio\" title=\"Unintentional removal of material at the tooth root during profile generation. It occurs with low tooth counts and weakens the root cross-section.\">undercut<\/span><\/dfn> is about 17 teeth. You can go down to 14-17 teeth using shifted profiles, but it requires accurate case-by-case verification.<\/p>\n<p class=\"article__p\">A pinion with few teeth increases the risk of interference, reduces the resistant section and accelerates wear. In technopolymers this aspect is more critical than in metals: the lower stiffness of the material can cause additional deformation under load. If you need a small pinion, consider increasing the module, using shifted profiles or switching to a stiffer material.<\/p>\n<div class=\"infobox infobox--grey\">\n <div class=\"infobox__title\">\n <svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"14\" height=\"14\" viewBox=\"0 0 24 24\" fill=\"none\" stroke=\"currentColor\" stroke-width=\"2\" stroke-linecap=\"round\" stroke-linejoin=\"round\"><path d=\"M14.7 6.3a1 1 0 0 0 0 1.4l1.6 1.6a1 1 0 0 0 1.4 0l3.77-3.77a6 6 0 0 1-7.94 7.94l-6.91 6.91a2.12 2.12 0 0 1-3-3l6.91-6.91a6 6 0 0 1 7.94-7.94l-3.76 3.76z\"><\/path><\/svg>\n Technical note\n <\/div>\n <p class=\"infobox__text\">The module and the face width are chosen together. A small module allows more teeth but reduces the robustness of the single tooth. A large module increases robustness but also the footprint and the inertia. A wide face distributes the load, but in technopolymer it can retain heat and generate non-uniform pressure along the tooth.<\/p>\n<\/div>\n<h2 class=\"article__h2\" id=\"which-material-for-which-application\">\n <span class=\"article__h2-icon\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"16\" height=\"16\" viewBox=\"0 0 24 24\" fill=\"none\" stroke=\"currentColor\" stroke-width=\"2\" stroke-linecap=\"round\" stroke-linejoin=\"round\"><polygon points=\"12 2 2 7 12 12 22 7 12 2\"><\/polygon><polyline points=\"2 17 12 22 22 17\"><\/polyline><polyline points=\"2 12 12 17 22 12\"><\/polyline><\/svg><\/span>\n Which material for which application\n<\/h2>\n<p class=\"article__p\">The choice of material changes the bending strength and the tribological behavior of the tooth:<\/p>\n<ul class=\"article__list\">\n <li><strong>PA6+GF<\/strong> (glass-fiber reinforced polyamide): highest mechanical strength, good for high loads.<\/li>\n <li><strong>POM-C<\/strong> (polyoxymethylene): preferable when dimensional stability is the priority and loads are moderate.<\/li>\n <li><strong>PK<\/strong> (polyketone): suited to chemically aggressive or high-humidity environments.<\/li>\n<\/ul>\n<p class=\"article__p\">All three offer light weight, corrosion resistance and the option to run without lubrication: characteristics that make them suited to sectors such as food, chemical and medical.<\/p>\n<div class=\"infobox infobox--grey\">\n <div class=\"infobox__title\">\n <svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"14\" height=\"14\" viewBox=\"0 0 24 24\" fill=\"none\" stroke=\"currentColor\" stroke-width=\"2\" stroke-linecap=\"round\" stroke-linejoin=\"round\"><line x1=\"9\" y1=\"18\" x2=\"15\" y2=\"18\"><\/line><line x1=\"10\" y1=\"22\" x2=\"14\" y2=\"22\"><\/line><path d=\"M15.09 14c.18-.98.65-1.74 1.41-2.5A4.65 4.65 0 0 0 18 8 6 6 0 0 0 6 8c0 1 .23 2.23 1.5 3.5A4.61 4.61 0 0 1 8.91 14\"><\/path><\/svg>\n From the field\n <\/div>\n <p class=\"infobox__text\">The optimal module for a pair of technopolymer gears isn’t derived from geometric calculation alone. Load, speed, lubrication, temperature and expected service life all come into play. In many cases experimental tests show critical pairings different from the estimated ones, even among materials with similar nominal characteristics. Use simulations to orient yourself, but always verify in the field.<\/p>\n<\/div>\n<div class=\"infobox\">\n <div class=\"infobox__title\">\n <svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"14\" height=\"14\" viewBox=\"0 0 24 24\" fill=\"none\" stroke=\"currentColor\" stroke-width=\"2\" stroke-linecap=\"round\" stroke-linejoin=\"round\"><path d=\"M4 4h16c1.1 0 2 .9 2 2v12c0 1.1-.9 2-2 2H4c-1.1 0-2-.9-2-2V6c0-1.1.9-2 2-2z\"><\/path><polyline points=\"22,6 12,13 2,6\"><\/polyline><\/svg>\n Ask us\n <\/div>\n <p class=\"infobox__text\">If you’re sizing a technopolymer transmission and aren’t sure of the tooth count or the right module for your case, <a href=\"https:\/\/shop.stagnoligears.com\/en\/contacts\/\" target=\"_blank\" rel=\"noopener\" class=\"cta-link\">write<\/a> to the technical office: we assess load, material and operating conditions together.<\/p>\n<\/div>\n\n <\/div>\n <\/div>\n <\/div>\n\n\n<script>\n (function () {\n var hideTimers = {};\n function 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})();\n<\/script>\n\n<\/body><\/html>\n","protected":false},"excerpt":{"rendered":"<p>Mathematical calculation of the reduction ratio and geometric constraints on minimum tooth count to prevent kinematic interference.<\/p>\n","protected":false},"parent":0,"template":"","meta":{"_acf_changed":false},"campo-applicazione":[5455],"categoria-ct":[5444],"class_list":["post-25593","centro-tecnico","type-centro-tecnico","status-publish","hentry","campo-applicazione-design","categoria-ct-cylindrical-technopolymer-gears"],"acf":[],"yoast_head":"\n<title>Transmission Ratio in Technopolymer Gears | Stagnoli Gears Shop<\/title>\n<meta name=\"description\" content=\"Learn how to calculate the transmission ratio in technopolymer gears and choose the correct tooth count while considering load, speed and material limits.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" 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{"id":25593,"date":"2026-06-30T10:43:17","date_gmt":"2026-06-30T08:43:17","guid":{"rendered":"https:\/\/shop.stagnoligears.com\/?post_type=centro-tecnico&p=25593"},"modified":"2026-07-10T13:57:37","modified_gmt":"2026-07-10T11:57:37","slug":"transmission-ratio-in-technopolymer-gears-calculation-and-choice-of-tooth-count","status":"publish","type":"centro-tecnico","link":"https:\/\/shop.stagnoligears.com\/en\/technical-hub\/transmission-ratio-in-technopolymer-gears-calculation-and-choice-of-tooth-count\/","title":{"rendered":"Transmission ratio in technopolymer gears: calculation and choice of tooth count"},"content":{"rendered":"\n\n \n \n \n Transmission ratio in technopolymer gears: calculation and choice of tooth count<\/title>\n\n <link rel=\"preconnect\" href=\"https:\/\/fonts.googleapis.com\">\n <link rel=\"preconnect\" href=\"https:\/\/fonts.gstatic.com\" crossorigin=\"\">\n <link href=\"https:\/\/fonts.googleapis.com\/css2?family=Inter:wght@400;500;600;700&display=swap\" rel=\"stylesheet\">\n\n <link rel=\"stylesheet\" href=\"https:\/\/cdn.jsdelivr.net\/npm\/katex@0.16.10\/dist\/katex.min.css\">\n <script defer=\"\" src=\"https:\/\/cdn.jsdelivr.net\/npm\/katex@0.16.10\/dist\/katex.min.js\"><\/script>\n <script defer=\"\" src=\"https:\/\/cdn.jsdelivr.net\/npm\/katex@0.16.10\/dist\/contrib\/auto-render.min.js\" onload=\"renderMathInElement(document.body, {\n delimiters: [\n { left: '$$', right: '$$', display: true },\n { left: '$', right: '$', display: false }\n ]\n });\"><\/script>\n\n <style>\n *, *::before, *::after { box-sizing: border-box; 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}\n\n .badge {\n display: inline-block;\n font-size: 12px;\n font-weight: 600;\n padding: 2px 8px;\n border-radius: 20px;\n background: var(--green-light);\n color: var(--green-dark);\n }\n\n \/* ============================================================\n RESPONSIVE\n ============================================================ *\/\n @media (max-width: 900px) {\n .article-body { flex-direction: column; padding: 20px 20px 48px; }\n }\n \n \/* Formula blocks *\/\n .formula-block {\n background: var(--bg-soft);\n border: 1px solid var(--border);\n border-radius: var(--radius-md);\n padding: 16px 20px;\n margin: 16px 0;\n text-align: center;\n overflow-x: auto;\n }\n .katex { font-size: 1.1em; }\n .formula-block .katex { font-size: 1.25em; }\n .formula-block .katex-display { margin: 0; }\n\n \/* Glossary tooltips *\/\n .glossary-term {\n border-bottom: 1px dotted var(--green);\n color: var(--green);\n font-style: italic;\n cursor: help;\n }\n .glossary-tooltip {\n position: fixed;\n z-index: 9999;\n background: #fff;\n border: 1px solid var(--border-mid);\n border-radius: var(--radius-md);\n box-shadow: var(--shadow-md);\n padding: 10px 14px;\n max-width: 300px;\n font-family: 'Inter', sans-serif;\n font-size: 13px;\n }\n .glossary-tooltip__title {\n font-weight: 600;\n color: var(--text-primary);\n margin-bottom: 4px;\n }\n .glossary-tooltip__def {\n font-size: 12.5px;\n color: var(--text-secondary);\n line-height: 1.6;\n }\n .glossary-tooltip__tag {\n display: inline-block;\n margin-top: 6px;\n font-size: 9px;\n font-weight: 600;\n padding: 2px 7px;\n border-radius: 20px;\n background: var(--green-light);\n color: var(--green-dark);\n text-transform: uppercase;\n letter-spacing: 0.04em;\n }\n\n <\/style>\n<\/head>\n<body>\n\n<div id=\"tt-diametro-primitivo\" class=\"glossary-tooltip\" hidden=\"\">\n <div class=\"glossary-tooltip__title\">Pitch diameter<\/div>\n <div class=\"glossary-tooltip__def\">Diameter of the theoretical circle at which contact occurs between two meshing gears. Obtained from the product of the module and the number of teeth.<\/div>\n <span class=\"glossary-tooltip__tag\">Glossary<\/span>\n<\/div>\n<div id=\"tt-gioco-di-flanchi-backlash\" class=\"glossary-tooltip\" hidden=\"\">\n <div class=\"glossary-tooltip__title\">Backlash<\/div>\n <div class=\"glossary-tooltip__def\">Free clearance between the flanks of two mating teeth, measured at the pitch circle. It prevents jamming due to thermal expansion or assembly errors.<\/div>\n <span class=\"glossary-tooltip__tag\">Glossary<\/span>\n<\/div>\n<div id=\"tt-angolo-di-pressione\" class=\"glossary-tooltip\" hidden=\"\">\n <div class=\"glossary-tooltip__title\">Pressure angle<\/div>\n <div class=\"glossary-tooltip__def\">Angle between the direction of the contact force between mating teeth and the tangent to the pitch circle at the point of contact. The standard value is 20\u00b0.<\/div>\n <span class=\"glossary-tooltip__tag\">Glossary<\/span>\n<\/div>\n<div id=\"tt-sottotaglio\" class=\"glossary-tooltip\" hidden=\"\">\n <div class=\"glossary-tooltip__title\">Undercut<\/div>\n <div class=\"glossary-tooltip__def\">Unintentional removal of material at the tooth root during profile generation. It occurs with low tooth counts and weakens the root cross-section.<\/div>\n <span class=\"glossary-tooltip__tag\">Glossary<\/span>\n<\/div>\n\n <div class=\"article-page\">\n <div class=\"article-body\">\n <div class=\"article-body__main\"><script type=\"application\/ld+json\">\n{\n \"@context\": \"https:\/\/schema.org\",\n \"@graph\": [\n {\n \"@type\": \"TechArticle\",\n \"@id\": \"#article\",\n \"headline\": \"Transmission ratio in technopolymer gears: calculation and choice of tooth count\",\n \"description\": \"The transmission ratio is the ratio between the rotational speed of the input gear and that of the output gear. 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It occurs with low tooth counts and weakens the root cross-section.\",\n \"inDefinedTermSet\": \"#glossario\"\n }\n ]\n }\n ]\n}\n<\/script>\n\n\n\n <h1 class=\"article__title\">Transmission ratio in technopolymer gears: calculation and choice of tooth count<\/h1>\n<h2 class=\"article__h2\" id=\"what-the-transmission-ratio-is\">\n <span class=\"article__h2-icon\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"16\" height=\"16\" viewBox=\"0 0 24 24\" fill=\"none\" stroke=\"currentColor\" stroke-width=\"2\" stroke-linecap=\"round\" stroke-linejoin=\"round\"><line x1=\"6\" y1=\"3\" x2=\"6\" y2=\"15\"><\/line><circle cx=\"18\" cy=\"6\" r=\"3\"><\/circle><circle cx=\"6\" cy=\"18\" r=\"3\"><\/circle><path d=\"M18 9a9 9 0 0 1-9 9\"><\/path><\/svg><\/span>\n What the transmission ratio is\n<\/h2>\n<p class=\"article__subtitle\">The transmission ratio is the ratio between the rotational speed of the input gear and that of the output gear. For two meshing wheels, it’s calculated by dividing the number of teeth of the driven wheel by that of the driving one:<\/p>\n<div class=\"formula-block\">$$i = \\frac{n_1}{n_2} = \\frac{Z_2}{Z_1} = \\frac{d_2}{d_1}$$<\/div>\n<p class=\"article__p\">Where $n_1$ is the speed of the driving wheel, $n_2$ that of the driven one, $Z_1$ and $Z_2$ the respective tooth counts, $d_1$ and $d_2$ the pitch diameters.<\/p>\n<p class=\"article__p\">If the ratio is greater than 1, you get a speed reduction and an increase in output torque. If it’s less than 1, speed increases and torque decreases. You can express it as a fraction, a decimal number or a colon-separated pair: a 20-tooth pinion with a 60-tooth driven wheel gives a 3:1 ratio, that is, the output speed is one third of the input.<\/p>\n<div class=\"infobox infobox--grey\">\n <div class=\"infobox__title\">\n <svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"14\" height=\"14\" viewBox=\"0 0 24 24\" fill=\"none\" stroke=\"currentColor\" stroke-width=\"2\" stroke-linecap=\"round\" stroke-linejoin=\"round\"><path d=\"M14.7 6.3a1 1 0 0 0 0 1.4l1.6 1.6a1 1 0 0 0 1.4 0l3.77-3.77a6 6 0 0 1-7.94 7.94l-6.91 6.91a2.12 2.12 0 0 1-3-3l6.91-6.91a6 6 0 0 1 7.94-7.94l-3.76 3.76z\"><\/path><\/svg>\n Technical note\n <\/div>\n <p class=\"infobox__text\">Some catalogs and software use the inverse convention, dividing the driving teeth by the driven ones. Always specify whether the ratio indicates reduction, multiplication or inverse kinematic ratio, to avoid interpretation errors.<\/p>\n<\/div>\n<h2 class=\"article__h2\" id=\"why-in-technopolymers-the-calculation-isnt-enough\">\n <span class=\"article__h2-icon\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"16\" height=\"16\" viewBox=\"0 0 24 24\" fill=\"none\" stroke=\"currentColor\" stroke-width=\"2\" stroke-linecap=\"round\" stroke-linejoin=\"round\"><circle cx=\"12\" cy=\"12\" r=\"10\"><\/circle><line x1=\"12\" y1=\"8\" x2=\"12\" y2=\"12\"><\/line><line x1=\"12\" y1=\"16\" x2=\"12.01\" y2=\"16\"><\/line><\/svg><\/span>\n Why in technopolymers the calculation isn’t enough\n<\/h2>\n<p class=\"article__p\">With metal gears, once you’ve calculated the geometric ratio, the system behaves predictably. With technopolymers you can’t stop there.<\/p>\n<p class=\"article__p\">The effective ratio can vary based on temperature, thermal expansion, moisture absorption, applied load and wear over time. The transmission ratio is also a ratio between pitch diameters: any dimensional change, even small, alters the real behavior of the system.<\/p>\n<p class=\"article__p\">On top of that, friction generates heat that polymers dissipate less quickly than metals. This affects durability and behavior under continuous load, especially at high peripheral speed:<\/p>\n<div class=\"formula-block\">$$v = \\frac{\\pi \\cdot d \\cdot n}{60}$$<\/div>\n<p class=\"article__p\">Where v is the peripheral speed, d the <dfn id=\"def-diametro-primitivo\"><span class=\"glossary-term\" data-tooltip=\"tt-diametro-primitivo\" title=\"Diameter of the theoretical circle at which contact occurs between two meshing gears. Obtained from the product of the module and the number of teeth.\">pitch diameter<\/span><\/dfn> and n the rotational speed in rpm.<\/p>\n<div class=\"warnbox\">\n <div class=\"warnbox__title\">\n <svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"14\" height=\"14\" viewBox=\"0 0 24 24\" fill=\"none\" stroke=\"currentColor\" stroke-width=\"2\" stroke-linecap=\"round\" stroke-linejoin=\"round\"><path d=\"M10.29 3.86L1.82 18a2 2 0 0 0 1.71 3h16.94a2 2 0 0 0 1.71-3L13.71 3.86a2 2 0 0 0-3.42 0z\"><\/path><line x1=\"12\" y1=\"9\" x2=\"12\" y2=\"13\"><\/line><line x1=\"12\" y1=\"17\" x2=\"12.01\" y2=\"17\"><\/line><\/svg>\n Warning\n <\/div>\n <p class=\"warnbox__text\">In technopolymers, underestimating the dimensional change with temperature leads to insufficient clearances and a risk of seizing. Always calculate the nominal ratio, then check under the real operating conditions.<\/p>\n<\/div>\n<h2 class=\"article__h2\" id=\"torque-speed-and-power\">\n <span class=\"article__h2-icon\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"16\" height=\"16\" viewBox=\"0 0 24 24\" fill=\"none\" stroke=\"currentColor\" stroke-width=\"2\" stroke-linecap=\"round\" stroke-linejoin=\"round\"><polyline points=\"22 12 18 12 15 21 9 3 6 12 2 12\"><\/polyline><\/svg><\/span>\n Torque, speed and power\n<\/h2>\n<p class=\"article__p\">The relation between angular speed and torque is direct: at equal power, reducing speed increases the transmitted torque. Torque depends on the tangential force and the pitch radius:<\/p>\n<div class=\"formula-block\">$$M = F \\cdot r$$<\/div>\n<p class=\"article__p\">In technopolymers these values aren’t static. Elastic deformation under load, thermal variations and progressive wear make them fluctuate during operation. Keep this in mind when you size the system.<\/p>\n<h2 class=\"article__h2\" id=\"williss-rule-for-epicyclic-systems\">\n <span class=\"article__h2-icon\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"16\" height=\"16\" viewBox=\"0 0 24 24\" fill=\"none\" stroke=\"currentColor\" stroke-width=\"2\" stroke-linecap=\"round\" stroke-linejoin=\"round\"><path d=\"M20 7h-9\"><\/path><path d=\"M14 17H5\"><\/path><circle cx=\"17\" cy=\"17\" r=\"3\"><\/circle><circle cx=\"7\" cy=\"7\" r=\"3\"><\/circle><\/svg><\/span>\n Willis’s rule for epicyclic systems\n<\/h2>\n<p class=\"article__p\">In epicyclic (or planetary) reducers, the wheels don’t rotate around fixed axes: part of the system is carried by the planet carrier. In these cases, Willis’s rule lets you calculate the ratio in the various operating configurations:<\/p>\n<div class=\"formula-block\">$$\\frac{(\\omega_A – \\omega_C)}{(\\omega_B – \\omega_C)} = \\frac{- Z_B}{Z_A}$$<\/div>\n<p class=\"article__p\">Where \u03c9A and \u03c9B are the angular speeds of two wheels, \u03c9C the planet-carrier speed, ZA and ZB the respective tooth counts.<\/p>\n<p class=\"article__p\">In technopolymer systems, Willis’s rule describes the ideal behavior. The real system can deviate from it due to deformation, <dfn id=\"def-gioco-di-flanchi-backlash\"><span class=\"glossary-term\" data-tooltip=\"tt-gioco-di-flanchi-backlash\" title=\"Free clearance between the flanks of two mating teeth, measured at the pitch circle. It prevents jamming due to thermal expansion or assembly errors.\">backlash<\/span><\/dfn>, wear and thermal variations: this deviation is called ratio error, and it’s the difference between the theoretical position and the actual position of the driven gear.<\/p>\n<p class=\"article__p\">To limit it: size the tooth backlash as a function of material, temperature and moisture; evaluate reinforced materials such as PA6+GF for greater dimensional stability; check dimensions and clearances after environmental conditioning.<\/p>\n<h2 class=\"article__h2\" id=\"how-many-teeth-to-choose\">\n <span class=\"article__h2-icon\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"16\" height=\"16\" viewBox=\"0 0 24 24\" fill=\"none\" stroke=\"currentColor\" stroke-width=\"2\" stroke-linecap=\"round\" stroke-linejoin=\"round\"><line x1=\"4\" y1=\"9\" x2=\"20\" y2=\"9\"><\/line><line x1=\"4\" y1=\"15\" x2=\"20\" y2=\"15\"><\/line><line x1=\"10\" y1=\"3\" x2=\"8\" y2=\"21\"><\/line><line x1=\"16\" y1=\"3\" x2=\"14\" y2=\"21\"><\/line><\/svg><\/span>\n How many teeth to choose\n<\/h2>\n<p class=\"article__p\">The number of teeth affects strength, quietness, durability, precision and the risk of interference (the abnormal contact between teeth that leads to seizing or accelerated wear).<\/p>\n<p class=\"article__p\">With a 20\u00b0 <dfn id=\"def-angolo-di-pressione\"><span class=\"glossary-term\" data-tooltip=\"tt-angolo-di-pressione\" title=\"Angle between the direction of the contact force between mating teeth and the tangent to the pitch circle at the point of contact. The standard value is 20\u00b0.\">pressure angle<\/span><\/dfn>, the minimum number to avoid <dfn id=\"def-sottotaglio\"><span class=\"glossary-term\" data-tooltip=\"tt-sottotaglio\" title=\"Unintentional removal of material at the tooth root during profile generation. It occurs with low tooth counts and weakens the root cross-section.\">undercut<\/span><\/dfn> is about 17 teeth. You can go down to 14-17 teeth using shifted profiles, but it requires accurate case-by-case verification.<\/p>\n<p class=\"article__p\">A pinion with few teeth increases the risk of interference, reduces the resistant section and accelerates wear. In technopolymers this aspect is more critical than in metals: the lower stiffness of the material can cause additional deformation under load. If you need a small pinion, consider increasing the module, using shifted profiles or switching to a stiffer material.<\/p>\n<div class=\"infobox infobox--grey\">\n <div class=\"infobox__title\">\n <svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"14\" height=\"14\" viewBox=\"0 0 24 24\" fill=\"none\" stroke=\"currentColor\" stroke-width=\"2\" stroke-linecap=\"round\" stroke-linejoin=\"round\"><path d=\"M14.7 6.3a1 1 0 0 0 0 1.4l1.6 1.6a1 1 0 0 0 1.4 0l3.77-3.77a6 6 0 0 1-7.94 7.94l-6.91 6.91a2.12 2.12 0 0 1-3-3l6.91-6.91a6 6 0 0 1 7.94-7.94l-3.76 3.76z\"><\/path><\/svg>\n Technical note\n <\/div>\n <p class=\"infobox__text\">The module and the face width are chosen together. A small module allows more teeth but reduces the robustness of the single tooth. A large module increases robustness but also the footprint and the inertia. A wide face distributes the load, but in technopolymer it can retain heat and generate non-uniform pressure along the tooth.<\/p>\n<\/div>\n<h2 class=\"article__h2\" id=\"which-material-for-which-application\">\n <span class=\"article__h2-icon\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"16\" height=\"16\" viewBox=\"0 0 24 24\" fill=\"none\" stroke=\"currentColor\" stroke-width=\"2\" stroke-linecap=\"round\" stroke-linejoin=\"round\"><polygon points=\"12 2 2 7 12 12 22 7 12 2\"><\/polygon><polyline points=\"2 17 12 22 22 17\"><\/polyline><polyline points=\"2 12 12 17 22 12\"><\/polyline><\/svg><\/span>\n Which material for which application\n<\/h2>\n<p class=\"article__p\">The choice of material changes the bending strength and the tribological behavior of the tooth:<\/p>\n<ul class=\"article__list\">\n <li><strong>PA6+GF<\/strong> (glass-fiber reinforced polyamide): highest mechanical strength, good for high loads.<\/li>\n <li><strong>POM-C<\/strong> (polyoxymethylene): preferable when dimensional stability is the priority and loads are moderate.<\/li>\n <li><strong>PK<\/strong> (polyketone): suited to chemically aggressive or high-humidity environments.<\/li>\n<\/ul>\n<p class=\"article__p\">All three offer light weight, corrosion resistance and the option to run without lubrication: characteristics that make them suited to sectors such as food, chemical and medical.<\/p>\n<div class=\"infobox infobox--grey\">\n <div class=\"infobox__title\">\n <svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"14\" height=\"14\" viewBox=\"0 0 24 24\" fill=\"none\" stroke=\"currentColor\" stroke-width=\"2\" stroke-linecap=\"round\" stroke-linejoin=\"round\"><line x1=\"9\" y1=\"18\" x2=\"15\" y2=\"18\"><\/line><line x1=\"10\" y1=\"22\" x2=\"14\" y2=\"22\"><\/line><path d=\"M15.09 14c.18-.98.65-1.74 1.41-2.5A4.65 4.65 0 0 0 18 8 6 6 0 0 0 6 8c0 1 .23 2.23 1.5 3.5A4.61 4.61 0 0 1 8.91 14\"><\/path><\/svg>\n From the field\n <\/div>\n <p class=\"infobox__text\">The optimal module for a pair of technopolymer gears isn’t derived from geometric calculation alone. Load, speed, lubrication, temperature and expected service life all come into play. In many cases experimental tests show critical pairings different from the estimated ones, even among materials with similar nominal characteristics. Use simulations to orient yourself, but always verify in the field.<\/p>\n<\/div>\n<div class=\"infobox\">\n <div class=\"infobox__title\">\n <svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"14\" height=\"14\" viewBox=\"0 0 24 24\" fill=\"none\" stroke=\"currentColor\" stroke-width=\"2\" stroke-linecap=\"round\" stroke-linejoin=\"round\"><path d=\"M4 4h16c1.1 0 2 .9 2 2v12c0 1.1-.9 2-2 2H4c-1.1 0-2-.9-2-2V6c0-1.1.9-2 2-2z\"><\/path><polyline points=\"22,6 12,13 2,6\"><\/polyline><\/svg>\n Ask us\n <\/div>\n <p class=\"infobox__text\">If you’re sizing a technopolymer transmission and aren’t sure of the tooth count or the right module for your case, <a href=\"https:\/\/shop.stagnoligears.com\/en\/contacts\/\" target=\"_blank\" rel=\"noopener\" class=\"cta-link\">write<\/a> to the technical office: we assess load, material and operating conditions together.<\/p>\n<\/div>\n\n <\/div>\n <\/div>\n <\/div>\n\n\n<script>\n (function () {\n var hideTimers = {};\n function 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