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class="vector-toc-list"> <li id="toc-Dimensions_and_terms" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Dimensions_and_terms"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Dimensions and terms</span> </div> </a> <ul id="toc-Dimensions_and_terms-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Relationships" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Relationships"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>Relationships</span> </div> </a> <ul id="toc-Relationships-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Gear_or_speed_ratio" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Gear_or_speed_ratio"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.3</span> <span>Gear or speed ratio</span> </div> </a> <ul id="toc-Gear_or_speed_ratio-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Torque_ratio_analysis_using_virtual_work" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Torque_ratio_analysis_using_virtual_work"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.4</span> <span>Torque ratio analysis using virtual work</span> </div> </a> <ul id="toc-Torque_ratio_analysis_using_virtual_work-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Mechanical_advantage" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Mechanical_advantage"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.5</span> <span>Mechanical advantage</span> </div> </a> <ul id="toc-Mechanical_advantage-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Hunting_and_non-hunting_gear_sets" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Hunting_and_non-hunting_gear_sets"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.6</span> <span>Hunting and non-hunting gear sets</span> </div> </a> <ul id="toc-Hunting_and_non-hunting_gear_sets-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Implementations" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Implementations"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Implementations</span> </div> </a> <button aria-controls="toc-Implementations-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Implementations subsection</span> </button> <ul id="toc-Implementations-sublist" class="vector-toc-list"> <li id="toc-Gear_trains_with_two_gears" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Gear_trains_with_two_gears"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Gear trains with two gears</span> </div> </a> <ul id="toc-Gear_trains_with_two_gears-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Idler_gears" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Idler_gears"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Idler gears</span> </div> </a> <ul id="toc-Idler_gears-sublist" class="vector-toc-list"> <li id="toc-Formula" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Formula"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2.1</span> <span>Formula</span> </div> </a> <ul id="toc-Formula-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Example" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Example"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2.2</span> <span>Example</span> </div> </a> <ul id="toc-Example-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Double_reduction_gear" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Double_reduction_gear"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Double reduction gear</span> </div> </a> <ul id="toc-Double_reduction_gear-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Belt_and_chain_drives" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Belt_and_chain_drives"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.4</span> <span>Belt and chain drives</span> </div> </a> <ul id="toc-Belt_and_chain_drives-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Automotive_applications" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Automotive_applications"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Automotive applications</span> </div> </a> <button aria-controls="toc-Automotive_applications-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Automotive applications subsection</span> </button> <ul id="toc-Automotive_applications-sublist" class="vector-toc-list"> <li id="toc-Example_2" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Example_2"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Example</span> </div> </a> <ul id="toc-Example_2-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Wide-ratio_vs._close-ratio_transmission" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Wide-ratio_vs._close-ratio_transmission"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Wide-ratio vs. close-ratio transmission</span> </div> </a> <ul id="toc-Wide-ratio_vs._close-ratio_transmission-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Gear train</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. 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mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%97%D1%83%D0%B1%D1%87%D0%B0%D1%81%D1%82%D0%B0%D1%8F_%D0%BF%D0%B5%D1%80%D0%B0%D0%B4%D0%B0%D1%87%D0%B0" title="Зубчастая перадача – Belarusian" lang="be" hreflang="be" data-title="Зубчастая перадача" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Reductor_de_velocitat" title="Reductor de velocitat – Catalan" lang="ca" hreflang="ca" data-title="Reductor de velocitat" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-de badge-Q70894304 mw-list-item" title=""><a href="https://de.wikipedia.org/wiki/Zahnradgetriebe" title="Zahnradgetriebe – German" lang="de" hreflang="de" data-title="Zahnradgetriebe" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Hammas%C3%BClekanne" title="Hammasülekanne – Estonian" lang="et" hreflang="et" data-title="Hammasülekanne" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Reductores_de_velocidad" title="Reductores de velocidad – Spanish" lang="es" hreflang="es" data-title="Reductores de velocidad" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Rapidmalpliigilo" title="Rapidmalpliigilo – Esperanto" lang="eo" hreflang="eo" data-title="Rapidmalpliigilo" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Engranaje_tren" title="Engranaje tren – Basque" lang="eu" hreflang="eu" data-title="Engranaje tren" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AF%D9%86%D8%A8%D8%A7%D9%84%D9%87_%DA%86%D8%B1%D8%AE%E2%80%8C%D8%AF%D9%86%D8%AF%D9%87" title="دنباله چرخ‌دنده – Persian" lang="fa" hreflang="fa" data-title="دنباله چرخ‌دنده" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EA%B8%B0%EC%96%B4_%ED%8A%B8%EB%A0%88%EC%9D%B8" title="기어 트레인 – Korean" lang="ko" hreflang="ko" data-title="기어 트레인" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Rangkaian_roda_gigi" title="Rangkaian roda gigi – Indonesian" lang="id" hreflang="id" data-title="Rangkaian roda gigi" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Riduttore_di_velocit%C3%A0" title="Riduttore di velocità – Italian" lang="it" hreflang="it" data-title="Riduttore di velocità" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%99%E1%83%91%E1%83%98%E1%83%9A%E1%83%90%E1%83%9C%E1%83%A3%E1%83%A0%E1%83%98_%E1%83%92%E1%83%90%E1%83%93%E1%83%90%E1%83%AA%E1%83%94%E1%83%9B%E1%83%90" title="კბილანური გადაცემა – Georgian" lang="ka" hreflang="ka" data-title="კბილანური გადაცემა" data-language-autonym="ქართული" data-language-local-name="Georgian" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%91%D2%B1%D1%80%D0%B0%D0%BC%D0%B0_%D1%82%D1%96%D1%81%D1%82%D1%96_%D0%B1%D0%B5%D1%80%D1%96%D0%BB%D1%96%D1%81" title="Бұрама тісті беріліс – Kazakh" lang="kk" hreflang="kk" data-title="Бұрама тісті беріліс" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%82%AE%E3%82%A2%E3%83%88%E3%83%AC%E3%83%BC%E3%83%B3" title="ギアトレーン – Japanese" lang="ja" hreflang="ja" data-title="ギアトレーン" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Przek%C5%82adnia_z%C4%99bata" title="Przekładnia zębata – Polish" lang="pl" hreflang="pl" data-title="Przekładnia zębata" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Redutor_de_velocidade" title="Redutor de velocidade – Portuguese" lang="pt" hreflang="pt" data-title="Redutor de velocidade" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%97%D1%83%D0%B1%D1%87%D0%B0%D1%82%D0%B0%D1%8F_%D0%BF%D0%B5%D1%80%D0%B5%D0%B4%D0%B0%D1%87%D0%B0" title="Зубчатая передача – Russian" lang="ru" hreflang="ru" data-title="Зубчатая передача" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Kugghjulsv%C3%A4xel" title="Kugghjulsväxel – Swedish" lang="sv" hreflang="sv" data-title="Kugghjulsväxel" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Di%C5%9Fli_tak%C4%B1m%C4%B1" title="Dişli takımı – Turkish" lang="tr" hreflang="tr" data-title="Dişli takımı" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%97%D1%83%D0%B1%D1%87%D0%B0%D1%81%D1%82%D0%B0_%D0%BF%D0%B5%D1%80%D0%B5%D0%B4%D0%B0%D1%87%D0%B0" title="Зубчаста передача – Ukrainian" lang="uk" hreflang="uk" data-title="Зубчаста передача" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E8%BD%AE%E7%B3%BB" title="轮系 – Chinese" lang="zh" hreflang="zh" data-title="轮系" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a 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searchaux" style="display:none">Mechanical transmission using multiple gears</div> <style data-mw-deduplicate="TemplateStyles:r1237032888/mw-parser-output/.tmulti">.mw-parser-output .tmulti .multiimageinner{display:flex;flex-direction:column}.mw-parser-output .tmulti .trow{display:flex;flex-direction:row;clear:left;flex-wrap:wrap;width:100%;box-sizing:border-box}.mw-parser-output .tmulti .tsingle{margin:1px;float:left}.mw-parser-output .tmulti .theader{clear:both;font-weight:bold;text-align:center;align-self:center;background-color:transparent;width:100%}.mw-parser-output .tmulti .thumbcaption{background-color:transparent}.mw-parser-output .tmulti .text-align-left{text-align:left}.mw-parser-output .tmulti .text-align-right{text-align:right}.mw-parser-output .tmulti .text-align-center{text-align:center}@media all and (max-width:720px){.mw-parser-output .tmulti .thumbinner{width:100%!important;box-sizing:border-box;max-width:none!important;align-items:center}.mw-parser-output .tmulti .trow{justify-content:center}.mw-parser-output .tmulti .tsingle{float:none!important;max-width:100%!important;box-sizing:border-box;text-align:center}.mw-parser-output .tmulti .tsingle .thumbcaption{text-align:left}.mw-parser-output .tmulti .trow>.thumbcaption{text-align:center}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .tmulti .multiimageinner img{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .tmulti .multiimageinner img{background-color:white}}</style><div class="thumb tmulti tright"><div class="thumbinner multiimageinner" style="width:392px;max-width:392px"><div class="trow"><div class="tsingle" style="width:232px;max-width:232px"><div class="thumbimage" style="height:254px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:Transmission_of_motion_by_compund_gear_train_(Army_Service_Corps_Training,_Mechanical_Transport,_1911).jpg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8f/Transmission_of_motion_by_compund_gear_train_%28Army_Service_Corps_Training%2C_Mechanical_Transport%2C_1911%29.jpg/230px-Transmission_of_motion_by_compund_gear_train_%28Army_Service_Corps_Training%2C_Mechanical_Transport%2C_1911%29.jpg" decoding="async" width="230" height="254" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8f/Transmission_of_motion_by_compund_gear_train_%28Army_Service_Corps_Training%2C_Mechanical_Transport%2C_1911%29.jpg/345px-Transmission_of_motion_by_compund_gear_train_%28Army_Service_Corps_Training%2C_Mechanical_Transport%2C_1911%29.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8f/Transmission_of_motion_by_compund_gear_train_%28Army_Service_Corps_Training%2C_Mechanical_Transport%2C_1911%29.jpg/460px-Transmission_of_motion_by_compund_gear_train_%28Army_Service_Corps_Training%2C_Mechanical_Transport%2C_1911%29.jpg 2x" data-file-width="541" data-file-height="597" /></a></span></div><div class="thumbcaption">Transmission of motion and force by gear wheels, compound train.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup></div></div><div class="tsingle" style="width:156px;max-width:156px"><div class="thumbimage" style="height:254px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:Fotothek_df_tg_0000387_Bergwerk_%5E_Bergbau_%5E_Tretrad_%5E_Entw%C3%A4sserung_%5E_Taschen.jpg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/47/Fotothek_df_tg_0000387_Bergwerk_%5E_Bergbau_%5E_Tretrad_%5E_Entw%C3%A4sserung_%5E_Taschen.jpg/154px-Fotothek_df_tg_0000387_Bergwerk_%5E_Bergbau_%5E_Tretrad_%5E_Entw%C3%A4sserung_%5E_Taschen.jpg" decoding="async" width="154" height="255" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/47/Fotothek_df_tg_0000387_Bergwerk_%5E_Bergbau_%5E_Tretrad_%5E_Entw%C3%A4sserung_%5E_Taschen.jpg/231px-Fotothek_df_tg_0000387_Bergwerk_%5E_Bergbau_%5E_Tretrad_%5E_Entw%C3%A4sserung_%5E_Taschen.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/47/Fotothek_df_tg_0000387_Bergwerk_%5E_Bergbau_%5E_Tretrad_%5E_Entw%C3%A4sserung_%5E_Taschen.jpg/308px-Fotothek_df_tg_0000387_Bergwerk_%5E_Bergbau_%5E_Tretrad_%5E_Entw%C3%A4sserung_%5E_Taschen.jpg 2x" data-file-width="482" data-file-height="798" /></a></span></div><div class="thumbcaption">Illustration by <a href="/wiki/Georgius_Agricola" title="Georgius Agricola">Georgius Agricola</a> (1580) showing a toothed wheel that engages a slotted cylinder to form a gear train that transmits power from a human-powered treadmill to mining pump.</div></div></div></div></div> <p>A <b>gear train</b> or <b>gear set</b> is a <a href="/wiki/Machine_element" title="Machine element">machine element</a> of a <a href="/wiki/Mechanical_system" class="mw-redirect" title="Mechanical system">mechanical system</a> formed by mounting two or more <a href="/wiki/Gear" title="Gear">gears</a> on a frame such that the teeth of the gears engage. </p><p>Gear teeth are designed to ensure the <a href="/wiki/Pitch_circle_diameter_(gears)" class="mw-redirect" title="Pitch circle diameter (gears)">pitch circles</a> of engaging gears roll on each other without slipping, providing a smooth transmission of rotation from one gear to the next.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> Features of gears and gear trains include: </p> <ul><li>The <b>gear ratio</b> of the pitch circles of mating gears defines the speed ratio and the <a href="/wiki/Mechanical_advantage" title="Mechanical advantage">mechanical advantage</a> of the gear set.</li> <li>A <a href="/wiki/Epicyclic_gearing" title="Epicyclic gearing">planetary gear train</a> provides high gear reduction in a compact package.</li> <li>It is possible to design gear teeth for gears that are <a href="/wiki/Non-circular_gear" title="Non-circular gear">non-circular</a>, yet still transmit torque smoothly.</li> <li>The speed ratios of <a href="/wiki/Chain_drive" title="Chain drive">chain</a> and <a href="/wiki/Belt_(mechanical)" title="Belt (mechanical)">belt drives</a> are computed in the same way as gear ratios. See <a href="/wiki/Bicycle_gearing" title="Bicycle gearing">bicycle gearing</a>.</li></ul> <p>The transmission of rotation between contacting toothed wheels can be traced back to the <a href="/wiki/Antikythera_mechanism" title="Antikythera mechanism">Antikythera mechanism</a> of Greece and the <a href="/wiki/South-pointing_chariot" title="South-pointing chariot">south-pointing chariot</a> of China. Illustrations by the Renaissance scientist <a href="/wiki/Georgius_Agricola" title="Georgius Agricola">Georgius Agricola</a> show gear trains with cylindrical teeth. The implementation of the <a href="/wiki/Involute_gear" title="Involute gear">involute tooth</a> yielded a standard gear design that provides a constant speed ratio. </p> <div style="clear:both;" class=""></div> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Gear_ratio">Gear ratio</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gear_train&action=edit&section=1" title="Edit section: Gear ratio"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Pignons_10_15_vitesse.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Pignons_10_15_vitesse.svg/330px-Pignons_10_15_vitesse.svg.png" decoding="async" width="330" height="256" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Pignons_10_15_vitesse.svg/495px-Pignons_10_15_vitesse.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Pignons_10_15_vitesse.svg/660px-Pignons_10_15_vitesse.svg.png 2x" data-file-width="120" data-file-height="93" /></a><figcaption>Two meshed <a href="/wiki/Spur_gear" title="Spur gear">spur gears</a> showing <a href="/wiki/Tangent_circles" title="Tangent circles">tangent contact</a> between their <i>pitch circles</i>, each illustrated with broken blue lines; the gear on the left has 10 teeth and the gear on the right has 15 teeth.</figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="Dimensions_and_terms">Dimensions and terms</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gear_train&action=edit&section=2" title="Edit section: Dimensions and terms"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <i>pitch circle</i> of a given gear is determined by the <a href="/wiki/Tangent_circles" title="Tangent circles">tangent point</a> contact between two meshing gears; for example, two <a href="/wiki/Spur_gear" title="Spur gear">spur gears</a> mesh together when their pitch circles are tangent, as illustrated.<sup id="cite_ref-Shigley_3-0" class="reference"><a href="#cite_note-Shigley-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 529">: 529 </span></sup> </p><p>The <i>pitch diameter</i> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e85ff03cbe0c7341af6b982e47e9f90d235c66ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}"></span> is the diameter of a gear's pitch circle, measured through that gear's rotational centerline, and the <i>pitch radius</i> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span> is the radius of the pitch circle.<sup id="cite_ref-Shigley_3-1" class="reference"><a href="#cite_note-Shigley-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 529">: 529 </span></sup> The distance between the rotational centerlines of two meshing gears is equal to the sum of their respective pitch radii.<sup id="cite_ref-Shigley_3-2" class="reference"><a href="#cite_note-Shigley-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 533">: 533 </span></sup> </p><p>The <i>circular pitch</i> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> is the distance, measured along the pitch circle, between one tooth and the corresponding point on an adjacent tooth.<sup id="cite_ref-Shigley_3-3" class="reference"><a href="#cite_note-Shigley-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 529">: 529 </span></sup> </p><p>The number of teeth <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}"></span> per gear is an integer determined by the pitch circle and circular pitch. </p> <div class="mw-heading mw-heading3"><h3 id="Relationships">Relationships</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gear_train&action=edit&section=3" title="Edit section: Relationships"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Spur_gear_tooth_dims.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5b/Spur_gear_tooth_dims.svg/220px-Spur_gear_tooth_dims.svg.png" decoding="async" width="220" height="293" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5b/Spur_gear_tooth_dims.svg/330px-Spur_gear_tooth_dims.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5b/Spur_gear_tooth_dims.svg/440px-Spur_gear_tooth_dims.svg.png 2x" data-file-width="120" data-file-height="160" /></a><figcaption>Spur gear tooth dimensions and how they are measured: <ul> <li><i>t</i> = tooth thickness, along the pitch circle</li> <li><i>p</i> = circular pitch, along the pitch circle</li> <li><i>a</i> = addendum, radially</li> <li><i>b</i> = dedendum, radially</li> </ul> In this example, the gear has 20 teeth.</figcaption></figure> <p>The circular pitch <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> of a gear can be defined as the circumference of the pitch circle using its pitch radius <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span> divided by the number of teeth <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}"></span>:<sup id="cite_ref-Shigley_3-4" class="reference"><a href="#cite_note-Shigley-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 530">: 530 </span></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\equiv {\frac {2\pi r}{N}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>≡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> <mi>r</mi> </mrow> <mi>N</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p\equiv {\frac {2\pi r}{N}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9149b4b3a4bfb1034fd6d6964acf0293b3135b4e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; margin-left: -0.089ex; width:8.737ex; height:5.176ex;" alt="{\displaystyle p\equiv {\frac {2\pi r}{N}}}"></span></dd></dl> <p>The thickness <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}"></span> of each tooth, measured through the pitch circle, is equal to the gap between neighboring teeth (also measured through the pitch circle) to ensure the teeth on adjacent gears, cut to the same tooth profile, can mesh without interference. This means the circular pitch <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> is equal to twice the thickness of a tooth,<sup id="cite_ref-Shigley_3-5" class="reference"><a href="#cite_note-Shigley-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 535">: 535 </span></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p=2t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <mn>2</mn> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p=2t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d646ea04ea4a1a4071f3a99a0c41a2f2c5088bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:6.359ex; height:2.509ex;" alt="{\displaystyle p=2t}"></span></dd></dl> <p>In the United States, the <i>diametral pitch</i> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></span> is the number of teeth on a gear divided by the pitch diameter; for SI countries, the <i>module</i> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span> is the reciprocal of this value.<sup id="cite_ref-Shigley_3-6" class="reference"><a href="#cite_note-Shigley-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 529">: 529 </span></sup> For any gear, the relationship between the number of teeth, diametral pitch or module, and pitch diameter is given by: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d={\frac {N}{P}}=N\cdot m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>N</mi> <mi>P</mi> </mfrac> </mrow> <mo>=</mo> <mi>N</mi> <mo>⋅</mo> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d={\frac {N}{P}}=N\cdot m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1cfb1dcf04897467da22fa756672b3307e9a4fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:16.096ex; height:5.176ex;" alt="{\displaystyle d={\frac {N}{P}}=N\cdot m}"></span></dd></dl> <p>Since the pitch diameter is related to circular pitch as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d\equiv 2r={\frac {Np}{\pi }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo>≡</mo> <mn>2</mn> <mi>r</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>N</mi> <mi>p</mi> </mrow> <mi>π</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d\equiv 2r={\frac {Np}{\pi }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f7703cdb5dbf666af8c67b0d6101349e51f5817" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.693ex; height:5.343ex;" alt="{\displaystyle d\equiv 2r={\frac {Np}{\pi }}}"></span></dd></dl> <p>this means </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {N}{P}}=N\cdot m={\frac {Np}{\pi }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>N</mi> <mi>P</mi> </mfrac> </mrow> <mo>=</mo> <mi>N</mi> <mo>⋅</mo> <mi>m</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>N</mi> <mi>p</mi> </mrow> <mi>π</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {N}{P}}=N\cdot m={\frac {Np}{\pi }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/37bfc2919149585cee207e8eec4547ffc569b310" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:18.949ex; height:5.343ex;" alt="{\displaystyle {\frac {N}{P}}=N\cdot m={\frac {Np}{\pi }}}"></span></dd></dl> <p>Rearranging, we obtain a relationship between diametral pitch and circular pitch:<sup id="cite_ref-Shigley_3-7" class="reference"><a href="#cite_note-Shigley-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 530">: 530 </span></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P={\frac {1}{m}}={\frac {\pi }{p}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>m</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mi>p</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P={\frac {1}{m}}={\frac {\pi }{p}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9aab25d34879748074cc9f88386a920629eae74" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:12.987ex; height:5.676ex;" alt="{\displaystyle P={\frac {1}{m}}={\frac {\pi }{p}}}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Gear_or_speed_ratio">Gear or speed ratio</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gear_train&action=edit&section=4" title="Edit section: Gear or speed ratio"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Frame"><a href="/wiki/File:Gears_animation.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/1/14/Gears_animation.gif" decoding="async" width="172" height="120" class="mw-file-element" data-file-width="172" data-file-height="120" /></a><figcaption>Two meshing gears transmit rotational motion; note difference in rotational speeds is equal to the reciprocal of the ratio between the number of teeth on the two gears</figcaption></figure> <p>For a pair of meshing gears, the <i>angular speed ratio</i>, also known as the <i>gear ratio</i>, can be computed from the ratio of the pitch radii or the ratio of the number of teeth on each gear. Define the angular speed ratio <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{AB}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>B</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{AB}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/baa22b9617fe3558b77fc9e242bb5e851f495c55" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.476ex; height:2.509ex;" alt="{\displaystyle R_{AB}}"></span> of two meshed gears <i>A</i> and <i>B</i> as the ratio of the magnitude of their respective angular velocities: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{AB}\equiv {\frac {|\omega _{A}|}{|\omega _{B}|}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>B</mi> </mrow> </msub> <mo>≡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{AB}\equiv {\frac {|\omega _{A}|}{|\omega _{B}|}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12baf1ba484b3c142e41145f2070958c69dccbba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:12.63ex; height:6.509ex;" alt="{\displaystyle R_{AB}\equiv {\frac {|\omega _{A}|}{|\omega _{B}|}}}"></span></dd></dl> <p>Here, subscripts are used to designate the gear, so gear <i>A</i> has a radius of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{A}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{A}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dbdfd7231b267d01d3319642a4ff3c1382981702" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.513ex; height:2.009ex;" alt="{\displaystyle r_{A}}"></span> and <a href="/wiki/Angular_velocity" title="Angular velocity">angular velocity</a> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{A}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega _{A}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6edaeab4afc26cfafacb8ad0b72cdcacbfe76f7c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.911ex; height:2.009ex;" alt="{\displaystyle \omega _{A}}"></span> with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N_{A}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N_{A}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fa9233ef6a6a2c0576dbf65490ddb8307fde494" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.331ex; height:2.509ex;" alt="{\displaystyle N_{A}}"></span> teeth, which meshes with gear <i>B</i> which has corresponding values for radius <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{B}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{B}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8c6c549d02366b12961786dfac0732e531da234" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.528ex; height:2.009ex;" alt="{\displaystyle r_{B}}"></span>, angular velocity <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{B}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega _{B}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a45dc41211d296979700bf741caa3335023fa597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.925ex; height:2.009ex;" alt="{\displaystyle \omega _{B}}"></span>, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N_{B}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N_{B}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc141c4bf37d6a37464b3eabe8a30b2e1c9dbe29" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.346ex; height:2.509ex;" alt="{\displaystyle N_{B}}"></span> teeth. </p><p>When these two gears are meshed and turn without slipping, the velocity <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}"></span> of the tangent point where the two pitch circles come in contact is the same on both gears, and is given by:<sup id="cite_ref-Shigley_3-8" class="reference"><a href="#cite_note-Shigley-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 533">: 533 </span></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v=|r_{A}\omega _{A}|=|r_{B}\omega _{B}|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v=|r_{A}\omega _{A}|=|r_{B}\omega _{B}|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1a25f17caa5a0dbc75f6538cff01881e12f5f44" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.79ex; height:2.843ex;" alt="{\displaystyle v=|r_{A}\omega _{A}|=|r_{B}\omega _{B}|}"></span></dd></dl> <p>Rearranging, the ratio of angular velocity magnitudes is the inverse of the ratio of pitch circle radii: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {|\omega _{A}|}{|\omega _{B}|}}={\frac {r_{B}}{r_{A}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {|\omega _{A}|}{|\omega _{B}|}}={\frac {r_{B}}{r_{A}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23f837e7ef5f47c4e9f9c9c7c3d9bc019bc63222" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:11.518ex; height:6.509ex;" alt="{\displaystyle {\frac {|\omega _{A}|}{|\omega _{B}|}}={\frac {r_{B}}{r_{A}}}}"></span></dd></dl> <p>Therefore, the angular speed ratio can be determined from the respective pitch radii:<sup id="cite_ref-Shigley_3-9" class="reference"><a href="#cite_note-Shigley-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 533, 552">: 533, 552 </span></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{AB}\equiv \left|{\frac {\omega _{A}}{\omega _{B}}}\right|={\frac {r_{B}}{r_{A}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>B</mi> </mrow> </msub> <mo>≡</mo> <mrow> <mo>|</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> </mfrac> </mrow> <mo>|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{AB}\equiv \left|{\frac {\omega _{A}}{\omega _{B}}}\right|={\frac {r_{B}}{r_{A}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/446a7b42b32de4f86616619b9dee96ab8f970738" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:19.093ex; height:5.509ex;" alt="{\displaystyle R_{AB}\equiv \left|{\frac {\omega _{A}}{\omega _{B}}}\right|={\frac {r_{B}}{r_{A}}}}"></span></dd></dl> <p>For example, if gear <i>A</i> has a pitch circle radius of 1 in (25 mm) and gear <i>B</i> has a pitch circle radius of 2 in (51 mm), the angular speed ratio <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{AB}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>B</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{AB}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/baa22b9617fe3558b77fc9e242bb5e851f495c55" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.476ex; height:2.509ex;" alt="{\displaystyle R_{AB}}"></span> is 2, which is sometimes written as 2:1. Gear <i>A</i> turns at twice the speed of gear <i>B</i>. For every complete revolution of gear <i>A</i> (360°), gear <i>B</i> makes half a revolution (180°). </p><p>In addition, consider that in order to mesh smoothly and turn without slipping, these two gears <i>A</i> and <i>B</i> must have compatible teeth. Given the same tooth and gap widths, they also must have the same circular pitch <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span>, which means </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{A}=p_{B}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{A}=p_{B}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dfaf9537dab5dec9cd3cd7023d558ea68a2ad889" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:8.471ex; height:2.009ex;" alt="{\displaystyle p_{A}=p_{B}}"></span> or, equivalently <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {2\pi r_{A}}{N_{A}}}={\frac {2\pi r_{B}}{N_{B}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> </mrow> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> </mrow> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {2\pi r_{A}}{N_{A}}}={\frac {2\pi r_{B}}{N_{B}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b59831384aab2b38560d08aee80d77748c86896" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:14.801ex; height:5.676ex;" alt="{\displaystyle {\frac {2\pi r_{A}}{N_{A}}}={\frac {2\pi r_{B}}{N_{B}}}}"></span></dd></dl> <p>This equation can be rearranged to show the ratio of the pitch circle radii of two meshing gears is equal to the ratio of their number of teeth: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {r_{B}}{r_{A}}}={\frac {N_{B}}{N_{A}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {r_{B}}{r_{A}}}={\frac {N_{B}}{N_{A}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b8ba611b8b81e8c2a9a9991832a509061c5cb14" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:10.645ex; height:5.676ex;" alt="{\displaystyle {\frac {r_{B}}{r_{A}}}={\frac {N_{B}}{N_{A}}}}"></span></dd></dl> <p>Since the angular speed ratio <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{AB}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>B</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{AB}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/baa22b9617fe3558b77fc9e242bb5e851f495c55" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.476ex; height:2.509ex;" alt="{\displaystyle R_{AB}}"></span> depends on the ratio of pitch circle radii, it is equivalently determined by the ratio of the number of teeth: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{AB}\equiv {\frac {|\omega _{A}|}{|\omega _{B}|}}={\frac {r_{B}}{r_{A}}}={\frac {N_{B}}{N_{A}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>B</mi> </mrow> </msub> <mo>≡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{AB}\equiv {\frac {|\omega _{A}|}{|\omega _{B}|}}={\frac {r_{B}}{r_{A}}}={\frac {N_{B}}{N_{A}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f15af5ca8dfc8f85ccf9a0f12710bc0785ca71a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:26.373ex; height:6.509ex;" alt="{\displaystyle R_{AB}\equiv {\frac {|\omega _{A}|}{|\omega _{B}|}}={\frac {r_{B}}{r_{A}}}={\frac {N_{B}}{N_{A}}}}"></span></dd></dl> <p>In other words, the [angular] speed ratio is <a href="/wiki/Inversely_proportional#Inverse_proportionality" class="mw-redirect" title="Inversely proportional">inversely proportional</a> to the radius of the pitch circle and the number of teeth of gear <i>A</i>, and directly proportional to the same values for gear <i>B</i>. </p> <div class="mw-heading mw-heading3"><h3 id="Torque_ratio_analysis_using_virtual_work">Torque ratio analysis using virtual work</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gear_train&action=edit&section=5" title="Edit section: Torque ratio analysis using virtual work"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The gear ratio also determines the transmitted torque. The <i>torque ratio</i> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathrm {TR} }_{AB}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> <mi mathvariant="normal">R</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>B</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathrm {TR} }_{AB}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe6033d7a155dc9554e07b9f23caae4fc4f8be76" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.101ex; height:2.509ex;" alt="{\displaystyle {\mathrm {TR} }_{AB}}"></span> of the gear train is defined as the ratio of its output torque to its input torque. Using the principle of <a href="/wiki/Virtual_work" title="Virtual work">virtual work</a>, the gear train's <a href="/wiki/Torque" title="Torque">torque</a> ratio is equal to the gear ratio, or speed ratio, of the gear train. Again, assume we have two gears <i>A</i> and <i>B</i>, with subscripts designating each gear and gear <i>A</i> serving as the input gear. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathrm {TR} }_{AB}\equiv {\frac {T_{B}}{T_{A}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> <mi mathvariant="normal">R</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>B</mi> </mrow> </msub> <mo>≡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathrm {TR} }_{AB}\equiv {\frac {T_{B}}{T_{A}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/708018a1f40bb6fd03f28d326e1b8a4be60f7d41" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:12.873ex; height:5.509ex;" alt="{\displaystyle {\mathrm {TR} }_{AB}\equiv {\frac {T_{B}}{T_{A}}}}"></span></dd></dl> <p>For this analysis, consider a gear train that has one degree of freedom, which means the angular rotation of all the gears in the gear train are defined by the angle of the input gear. The input torque <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{A}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{A}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/416a1dd7d799d8a81370d796e350420774dc2369" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.822ex; height:2.509ex;" alt="{\displaystyle T_{A}}"></span> acting on the input gear <i>A</i> is transformed by the gear train into the output torque <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{B}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{B}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b558df0361c34c9e93a63042e0d01b44e0295cc0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.837ex; height:2.509ex;" alt="{\displaystyle T_{B}}"></span> exerted by the output gear <i>B</i>. </p><p>Let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{AB}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>B</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{AB}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/baa22b9617fe3558b77fc9e242bb5e851f495c55" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.476ex; height:2.509ex;" alt="{\displaystyle R_{AB}}"></span> be the speed ratio, then by definition </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{AB}\equiv {\frac {|\omega _{A}|}{|\omega _{B}|}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>B</mi> </mrow> </msub> <mo>≡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{AB}\equiv {\frac {|\omega _{A}|}{|\omega _{B}|}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12baf1ba484b3c142e41145f2070958c69dccbba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:12.63ex; height:6.509ex;" alt="{\displaystyle R_{AB}\equiv {\frac {|\omega _{A}|}{|\omega _{B}|}}}"></span></dd></dl> <p>Assuming the gears are rigid and there are no losses in the engagement of the gear teeth, then the principle of <a href="/wiki/Virtual_work" title="Virtual work">virtual work</a> can be used to analyze the static equilibrium of the gear train. Because there is a single degree of freedom, the angle <i>θ</i> of the input gear completely determines the angle of the output gear and serves as the generalized coordinate of the gear train. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {d\theta }{dt}}=\omega _{A}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>θ</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d\theta }{dt}}=\omega _{A}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/490c3865d0c3746d079c1873141744c3be320d5e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:9.152ex; height:5.509ex;" alt="{\displaystyle {\frac {d\theta }{dt}}=\omega _{A}}"></span></dd></dl> <p>The speed ratio <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{AB}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>B</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{AB}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/baa22b9617fe3558b77fc9e242bb5e851f495c55" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.476ex; height:2.509ex;" alt="{\displaystyle R_{AB}}"></span> of the gear train can be rearranged to give the magnitude of angular velocity of the output gear in terms of the input gear velocity. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\omega _{B}|={\frac {|\omega _{A}|}{R_{AB}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>B</mi> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\omega _{B}|={\frac {|\omega _{A}|}{R_{AB}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/03611cc595f3167b25c40dbab14902e8b8532290" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:12.63ex; height:6.176ex;" alt="{\displaystyle |\omega _{B}|={\frac {|\omega _{A}|}{R_{AB}}}}"></span></dd></dl> <p>Rewriting in terms of a common angular velocity, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{A}=\omega ,\quad \omega _{B}=\omega /R_{AB}\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>=</mo> <mi>ω</mi> <mo>,</mo> <mspace width="1em" /> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo>=</mo> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>B</mi> </mrow> </msub> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega _{A}=\omega ,\quad \omega _{B}=\omega /R_{AB}\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b1fcc4ed9a06c109fa0e8b917dc05bac8b66ed5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-right: -0.387ex; width:23.92ex; height:2.843ex;" alt="{\displaystyle \omega _{A}=\omega ,\quad \omega _{B}=\omega /R_{AB}\!}"></span></dd></dl> <p>The principle of virtual work states the input force on gear A and the output force on gear B using applied torques will sum to zero:<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{\theta }=T_{A}{\frac {\partial \omega _{A}}{\partial \omega }}-T_{B}{\frac {\partial \omega _{B}}{\partial \omega }}=T_{A}-T_{B}/R_{AB}=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>ω</mi> </mrow> </mfrac> </mrow> <mo>−</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>ω</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>B</mi> </mrow> </msub> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{\theta }=T_{A}{\frac {\partial \omega _{A}}{\partial \omega }}-T_{B}{\frac {\partial \omega _{B}}{\partial \omega }}=T_{A}-T_{B}/R_{AB}=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/811fd302b3bbcbf2031c7dbe8257557ac2d92f02" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:46.386ex; height:5.509ex;" alt="{\displaystyle F_{\theta }=T_{A}{\frac {\partial \omega _{A}}{\partial \omega }}-T_{B}{\frac {\partial \omega _{B}}{\partial \omega }}=T_{A}-T_{B}/R_{AB}=0.}"></span></dd></dl> <p>This can be rearranged to: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{AB}={\frac {T_{B}}{T_{A}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>B</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{AB}={\frac {T_{B}}{T_{A}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77cda01c9ebcd89cc6b015b7ab7fdd75f0bee0e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:11.248ex; height:5.509ex;" alt="{\displaystyle R_{AB}={\frac {T_{B}}{T_{A}}}}"></span></dd></dl> <p>Since <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{AB}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>B</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{AB}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/baa22b9617fe3558b77fc9e242bb5e851f495c55" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.476ex; height:2.509ex;" alt="{\displaystyle R_{AB}}"></span> is the gear ratio of the gear train, the input torque <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{A}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{A}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/416a1dd7d799d8a81370d796e350420774dc2369" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.822ex; height:2.509ex;" alt="{\displaystyle T_{A}}"></span> applied to the input gear <i>A</i> and the output torque <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{B}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{B}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b558df0361c34c9e93a63042e0d01b44e0295cc0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.837ex; height:2.509ex;" alt="{\displaystyle T_{B}}"></span> on the output gear <i>B</i> are related by the same gear or speed ratio. </p> <div class="mw-heading mw-heading3"><h3 id="Mechanical_advantage">Mechanical advantage</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gear_train&action=edit&section=6" title="Edit section: Mechanical advantage"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The torque ratio of a gear train is also known as its <i><a href="/wiki/Mechanical_advantage" title="Mechanical advantage">mechanical advantage</a></i>; as demonstrated, the gear ratio and speed ratio of a gear train also give its mechanical advantage. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {MA} \equiv {\frac {T_{B}}{T_{A}}}=R_{AB}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">M</mi> <mi mathvariant="normal">A</mi> </mrow> <mo>≡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>B</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {MA} \equiv {\frac {T_{B}}{T_{A}}}=R_{AB}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74bc16b8233237398b68542d49badc6960ba563b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:18.22ex; height:5.509ex;" alt="{\displaystyle \mathrm {MA} \equiv {\frac {T_{B}}{T_{A}}}=R_{AB}}"></span></dd></dl> <p>The mechanical advantage <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {MA} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">M</mi> <mi mathvariant="normal">A</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {MA} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f63624c8f8823dc0cc018e6a73d44e64c70a87a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.874ex; height:2.176ex;" alt="{\displaystyle \mathrm {MA} }"></span> of a pair of meshing gears for which the input gear <i>A</i> has <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N_{A}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N_{A}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fa9233ef6a6a2c0576dbf65490ddb8307fde494" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.331ex; height:2.509ex;" alt="{\displaystyle N_{A}}"></span> teeth and the output gear <i>B</i> has <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N_{B}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N_{B}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc141c4bf37d6a37464b3eabe8a30b2e1c9dbe29" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.346ex; height:2.509ex;" alt="{\displaystyle N_{B}}"></span> teeth is given by<sup id="cite_ref-Basic_5-0" class="reference"><a href="#cite_note-Basic-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 74–76">: 74–76 </span></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {MA} =R_{AB}=\left|{\frac {\omega _{A}}{\omega _{B}}}\right|={\frac {N_{B}}{N_{A}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">M</mi> <mi mathvariant="normal">A</mi> </mrow> <mo>=</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>B</mi> </mrow> </msub> <mo>=</mo> <mrow> <mo>|</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> </mfrac> </mrow> <mo>|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {MA} =R_{AB}=\left|{\frac {\omega _{A}}{\omega _{B}}}\right|={\frac {N_{B}}{N_{A}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1367969cac597c6d43d574d9d0ad26b0db14950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:26.883ex; height:5.676ex;" alt="{\displaystyle \mathrm {MA} =R_{AB}=\left|{\frac {\omega _{A}}{\omega _{B}}}\right|={\frac {N_{B}}{N_{A}}}}"></span></dd></dl> <p>This shows that if the output gear <i>B</i> has more teeth than the input gear <i>A</i>, then the gear train <i>amplifies</i> the input torque. In this case, the gear train is called a <i>speed reducer</i> and since the output gear must have more teeth than the input gear, the speed reducer amplifies the input torque.<sup id="cite_ref-Basic_5-1" class="reference"><a href="#cite_note-Basic-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 76">: 76 </span></sup> When the input gear rotates faster than the output gear, then the gear train amplifies the input torque. Conversely, if the output gear has fewer teeth than the input gear, then the gear train <i>reduces</i> the input torque;<sup id="cite_ref-Basic_5-2" class="reference"><a href="#cite_note-Basic-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 68">: 68 </span></sup> in other words, when the input gear rotates slower than the output gear, the gear train reduces the input torque. </p> <div class="mw-heading mw-heading3"><h3 id="Hunting_and_non-hunting_gear_sets">Hunting and non-hunting gear sets</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gear_train&action=edit&section=7" title="Edit section: Hunting and non-hunting gear sets"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A <b>hunting gear set</b> is a set of gears where the gear teeth counts are relatively <a href="/wiki/Prime_number" title="Prime number">prime</a> on each gear in an interfacing pair. Since the number of teeth on each gear have no common <a href="/wiki/Factorization" title="Factorization">factors</a>, then any tooth on one of the gears will come into contact with every tooth on the other gear before encountering the same tooth again. This results in less wear and longer life of the mechanical parts. A <b>non-hunting gear set</b> is one where the teeth counts are insufficiently prime. In this case, some particular gear teeth will come into contact with particular opposing gear teeth more times than others, resulting in more wear on some teeth than others.<sup id="cite_ref-amtech20231205_6-0" class="reference"><a href="#cite_note-amtech20231205-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Implementations">Implementations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gear_train&action=edit&section=8" title="Edit section: Implementations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Gear_trains_with_two_gears">Gear trains with two gears</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gear_train&action=edit&section=9" title="Edit section: Gear trains with two gears"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Animated_two_spur_gears_1_2.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9b/Animated_two_spur_gears_1_2.gif/220px-Animated_two_spur_gears_1_2.gif" decoding="async" width="220" height="153" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9b/Animated_two_spur_gears_1_2.gif/330px-Animated_two_spur_gears_1_2.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9b/Animated_two_spur_gears_1_2.gif/440px-Animated_two_spur_gears_1_2.gif 2x" data-file-width="1340" data-file-height="932" /></a><figcaption>Two meshed spur gears, with a 2:1 ratio</figcaption></figure> <p>The simplest example of a gear train has two gears. The <i>input gear</i> (also known as the <i>drive gear</i> or <i>driver</i>) transmits power to the <i>output gear</i> (also known as the <i>driven gear</i>). The input gear will typically be connected to a power source, such as a motor or engine. In such an example, the output of torque and rotational speed from the output (driven) gear depend on the ratio of the dimensions of the two gears or the ratio of the tooth counts. </p> <div class="mw-heading mw-heading3"><h3 id="Idler_gears">Idler gears</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gear_train&action=edit&section=10" title="Edit section: Idler gears"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Animated_3_Gear_Row.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/bd/Animated_3_Gear_Row.gif/250px-Animated_3_Gear_Row.gif" decoding="async" width="250" height="131" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/bd/Animated_3_Gear_Row.gif/375px-Animated_3_Gear_Row.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/bd/Animated_3_Gear_Row.gif/500px-Animated_3_Gear_Row.gif 2x" data-file-width="1546" data-file-height="808" /></a><figcaption>Gear train with an <a href="/wiki/Idler_gear" class="mw-redirect" title="Idler gear">idler gear</a> in the middle which does not affect the overall gear ratio but reverses the direction of rotation of the gear on the right.</figcaption></figure> <p>In a sequence of gears chained together, the ratio depends only on the number of teeth on the first and last gear. The intermediate gears, regardless of their size, do not alter the overall gear ratio of the chain. However, the addition of each intermediate gear reverses the direction of rotation of the final gear. </p><p>An intermediate gear which does not drive a shaft to perform any work is called an <a href="https://en.wiktionary.org/wiki/Idler" class="extiw" title="wikt:Idler">idler</a> gear. Sometimes, a single idler gear is used to reverse the direction, in which case it may be referred to as a <i>reverse idler</i>. For instance, the typical automobile <a href="/wiki/Manual_transmission" title="Manual transmission">manual transmission</a> engages reverse gear by means of inserting a reverse idler between two gears. </p><p>Idler gears can also transmit rotation among distant shafts in situations where it would be impractical to simply make the distant gears larger to bring them together. Not only do larger gears occupy more space, the mass and rotational inertia (<a href="/wiki/Moment_of_inertia" title="Moment of inertia">moment of inertia</a>) of a gear is proportional to the <a href="/wiki/Square_(algebra)" title="Square (algebra)">square</a> of its radius. Instead of idler gears, a toothed belt or chain can be used to transmit <a href="/wiki/Torque" title="Torque">torque</a> over distance. </p> <div class="mw-heading mw-heading4"><h4 id="Formula">Formula</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gear_train&action=edit&section=11" title="Edit section: Formula"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>If a simple gear train has three gears, such that the input gear <i>A</i> meshes with an intermediate gear <i>I</i> which in turn meshes with the output gear <i>B</i>, then the pitch circle of the intermediate gear rolls without slipping on both the pitch circles of the input and output gears. This yields the two relations </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {|\omega _{A}|}{|\omega _{I}|}}={\frac {N_{I}}{N_{A}}},\quad {\frac {|\omega _{I}|}{|\omega _{B}|}}={\frac {N_{B}}{N_{I}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> </mrow> </msub> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> </mfrac> </mrow> <mo>,</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> </mrow> </msub> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {|\omega _{A}|}{|\omega _{I}|}}={\frac {N_{I}}{N_{A}}},\quad {\frac {|\omega _{I}|}{|\omega _{B}|}}={\frac {N_{B}}{N_{I}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa83fdde32a5c2a8a9ea8111295e5f6734fcb675" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:28.645ex; height:6.509ex;" alt="{\displaystyle {\frac {|\omega _{A}|}{|\omega _{I}|}}={\frac {N_{I}}{N_{A}}},\quad {\frac {|\omega _{I}|}{|\omega _{B}|}}={\frac {N_{B}}{N_{I}}}.}"></span></dd></dl> <p>The speed ratio of the overall gear train is obtained by multiplying these two equations for each pair (<i>A</i>/<i>I</i> and <i>I</i>/<i>B</i>) to obtain </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R={\frac {|\omega _{A}|}{|\omega _{B}|}}={\frac {N_{B}}{N_{A}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R={\frac {|\omega _{A}|}{|\omega _{B}|}}={\frac {N_{B}}{N_{A}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e4d017f49e85bcffb2d0d7bc22847fbb183f060" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:17.845ex; height:6.509ex;" alt="{\displaystyle R={\frac {|\omega _{A}|}{|\omega _{B}|}}={\frac {N_{B}}{N_{A}}}.}"></span></dd></dl> <p>This is because the number of idler gear teeth <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N_{I}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N_{I}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/197ffa8f715850a0bea5c95947a8ed3ecffc1538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.927ex; height:2.509ex;" alt="{\displaystyle N_{I}}"></span> cancels out when the gear ratios of the two subsets are multiplied: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{final}=R_{AI}\cdot R_{IB}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> <mi>i</mi> <mi>n</mi> <mi>a</mi> <mi>l</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>I</mi> </mrow> </msub> <mo>⋅</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> <mi>B</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{final}=R_{AI}\cdot R_{IB}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/984b7b630562bcdd33ede9ff606cb5b72d481c4b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.721ex; height:2.843ex;" alt="{\displaystyle R_{final}=R_{AI}\cdot R_{IB}}"></span> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle =\left({\frac {N_{I}}{N_{A}}}\right)\cdot \left({\frac {N_{B}}{N_{I}}}\right)=\left({\frac {N_{B}}{N_{A}}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> </mrow> </msub> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>⋅</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> </mrow> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =\left({\frac {N_{I}}{N_{A}}}\right)\cdot \left({\frac {N_{B}}{N_{I}}}\right)=\left({\frac {N_{B}}{N_{A}}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a040ad93077b64e648da61ded779083fbac1a387" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:30.025ex; height:6.176ex;" alt="{\displaystyle =\left({\frac {N_{I}}{N_{A}}}\right)\cdot \left({\frac {N_{B}}{N_{I}}}\right)=\left({\frac {N_{B}}{N_{A}}}\right)}"></span></dd></dl></dd></dl> <p>Notice that this gear ratio is exactly the same as for the case when the gears <i>A</i> and <i>B</i> engage directly. The intermediate gear provides spacing but does not affect the gear ratio. For this reason it is called an <i>idler</i> gear. The same gear ratio is obtained for a sequence of idler gears and hence an idler gear is used to provide the same direction to rotate the driver and driven gear. If the driver gear moves in the clockwise direction, then the driven gear also moves in the clockwise direction with the help of the idler gear. </p> <div class="mw-heading mw-heading4"><h4 id="Example">Example</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gear_train&action=edit&section=12" title="Edit section: Example"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Gears_large.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/42/Gears_large.jpg/330px-Gears_large.jpg" decoding="async" width="330" height="449" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/42/Gears_large.jpg/495px-Gears_large.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/4/42/Gears_large.jpg 2x" data-file-width="500" data-file-height="680" /></a><figcaption>2 gears and an <a href="/wiki/Idler-wheel#Idler_gear" title="Idler-wheel">idler gear</a> on a piece of farm equipment, with a ratio of 42/13 = 3.23:1</figcaption></figure> <p>In the photo, assume the smallest gear (Gear <i>A</i>, in the lower right corner) is connected to the motor, which makes it the drive gear or input gear. The somewhat larger gear in the middle (Gear <i>I</i>) is called an <a href="/wiki/Idler-wheel" title="Idler-wheel">idler</a> gear. It is not connected directly to either the motor or the output shaft and only transmits power between the input and output gears. There is a third gear (Gear <i>B</i>) partially shown in the upper-right corner of the photo. Assuming that gear is connected to the machine's output shaft, it is the output or driven gear. </p><p>Considering only gears <i>A</i> and <i>I</i>, the gear ratio between the idler and the input gear can be calculated as if the idler gear was the output gear. The input gear <i>A</i> in this two-gear subset has 13 teeth (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N_{A}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N_{A}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fa9233ef6a6a2c0576dbf65490ddb8307fde494" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.331ex; height:2.509ex;" alt="{\displaystyle N_{A}}"></span>) and the idler gear <i>I</i> has 21 teeth (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N_{I}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N_{I}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/197ffa8f715850a0bea5c95947a8ed3ecffc1538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.927ex; height:2.509ex;" alt="{\displaystyle N_{I}}"></span>). Therefore, the gear ratio for this subset <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{AI}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>I</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{AI}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af7a94a936eae96c86f047b8264aba27ee53ad7b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.057ex; height:2.509ex;" alt="{\displaystyle R_{AI}}"></span> is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{AI}={\frac {N_{I}}{N_{A}}}={\frac {21}{13}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>I</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> </mrow> </msub> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>21</mn> <mn>13</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{AI}={\frac {N_{I}}{N_{A}}}={\frac {21}{13}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8f5e0227325909b05bce57e44feddac82874752" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:17.582ex; height:5.676ex;" alt="{\displaystyle R_{AI}={\frac {N_{I}}{N_{A}}}={\frac {21}{13}}}"></span></dd></dl> <p>This is approximately 1.62 or 1.62:1. At this ratio, it means the drive gear (<i>A</i>) must make 1.62 revolutions to turn the output gear (<i>I</i>) once. It also means that for every one <a href="/wiki/Revolution_(geometry)" class="mw-redirect" title="Revolution (geometry)">revolution</a> of the driver (<i>A</i>), the output gear (<i>I</i>) has made <style data-mw-deduplicate="TemplateStyles:r1154941027">.mw-parser-output .frac{white-space:nowrap}.mw-parser-output .frac .num,.mw-parser-output .frac .den{font-size:80%;line-height:0;vertical-align:super}.mw-parser-output .frac .den{vertical-align:sub}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style><span class="frac"><span class="num">13</span>⁄<span class="den">21</span></span> = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"><span class="frac"><span class="num">1</span>⁄<span class="den">1.62</span></span>, or 0.62, revolutions. The larger gear (<i>I</i>) turns slower. </p><p>The third gear in the picture (<i>B</i>) has <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N_{B}=42}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo>=</mo> <mn>42</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N_{B}=42}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4148e546843ddc0802f38c7512fa683efd190846" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.769ex; height:2.509ex;" alt="{\displaystyle N_{B}=42}"></span> teeth. Now consider the gear ratio for the subset consisting of gears <i>I</i> and <i>B</i>, with the idler gear <i>I</i> serving as the input and third gear <i>B</i> serving as the output. The gear ratio between the idler (<i>I</i>) and third gear (<i>B</i>) <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{IB}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> <mi>B</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{IB}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/048d4635d578523c7d320f233eee612df898ca43" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.072ex; height:2.509ex;" alt="{\displaystyle R_{IB}}"></span> is thus </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{IB}={\frac {N_{B}}{N_{I}}}={\frac {42}{21}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> <mi>B</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>42</mn> <mn>21</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{IB}={\frac {N_{B}}{N_{I}}}={\frac {42}{21}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/261033821ce73cd0dc55c875dc2040dd3e45302c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:17.612ex; height:5.509ex;" alt="{\displaystyle R_{IB}={\frac {N_{B}}{N_{I}}}={\frac {42}{21}}}"></span></dd></dl> <p>or 2:1. </p><p>The final gear ratio of the compound system is 1.62×2≈3.23. For every 3.23 revolutions of the smallest gear <i>A</i>, the largest gear <i>B</i> turns one revolution, or for every one revolution of the smallest gear <i>A</i>, the largest gear <i>B</i> turns 0.31 (1/3.23) revolution, a total <a href="/wiki/Reduction_drive" title="Reduction drive">reduction</a> of about 1:3.23 (Gear Reduction Ratio (GRR) = 1/Gear Ratio (GR)). </p><p>Since the idler gear <i>I</i> contacts directly both the smaller gear <i>A</i> and the larger gear <i>B</i>, it can be removed from the calculation, also giving a ratio of 42/13≈3.23. The idler gear serves to make both the drive gear and the driven gear rotate in the same direction, but confers no mechanical advantage. </p> <div class="mw-heading mw-heading3"><h3 id="Double_reduction_gear">Double reduction gear</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gear_train&action=edit&section=13" title="Edit section: Double reduction gear"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:AnimatedGears.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cd/AnimatedGears.gif/250px-AnimatedGears.gif" decoding="async" width="250" height="128" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cd/AnimatedGears.gif/375px-AnimatedGears.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cd/AnimatedGears.gif/500px-AnimatedGears.gif 2x" data-file-width="903" data-file-height="461" /></a><figcaption>Double reduction gears</figcaption></figure> <p>A double reduction gear set comprises two pairs of gears, each individually single reductions, in series. In the diagram, the red and blue gears give the first stage of reduction and the orange and green gears give the second stage of reduction. The total reduction is the <a href="/wiki/Product_(mathematics)" title="Product (mathematics)">product</a> of the first stage of reduction and the second stage of reduction. </p><p>It is essential to have two coupled gears, of different sizes, on the intermediate <a href="/wiki/Layshaft" title="Layshaft">layshaft</a>. If a single intermediate gear was used, the overall ratio would be simply that between the first and final gears, the intermediate gear would only act as an <a href="/wiki/Idler_gear" class="mw-redirect" title="Idler gear">idler gear</a>: it would reverse the direction of rotation, but not change the ratio. </p> <div class="mw-heading mw-heading3"><h3 id="Belt_and_chain_drives">Belt and chain drives</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gear_train&action=edit&section=14" title="Edit section: Belt and chain drives"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Bicycle_belt_drive_1.JPG" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/56/Bicycle_belt_drive_1.JPG/220px-Bicycle_belt_drive_1.JPG" decoding="async" width="220" height="165" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/56/Bicycle_belt_drive_1.JPG/330px-Bicycle_belt_drive_1.JPG 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/56/Bicycle_belt_drive_1.JPG/440px-Bicycle_belt_drive_1.JPG 2x" data-file-width="2560" data-file-height="1920" /></a><figcaption>Bicycle with toothed belt drive to transmit torque from crank to rear sprocket</figcaption></figure> <p>Special gears called sprockets can be coupled together with chains, as on <a href="/wiki/Bicycle" title="Bicycle">bicycles</a> and some <a href="/wiki/Motorcycle" title="Motorcycle">motorcycles</a>. Alternatively, belts can have teeth in them also and be coupled to gear-like pulleys. Again, exact accounting of teeth and revolutions can be applied with these machines. </p><p>For example, a belt with teeth, called the <a href="/wiki/Timing_belt_(camshaft)" title="Timing belt (camshaft)">timing belt</a>, is used in some internal combustion engines to synchronize the movement of the <a href="/wiki/Camshaft" title="Camshaft">camshaft</a> with that of the <a href="/wiki/Crankshaft" title="Crankshaft">crankshaft</a>, so that the <a href="/wiki/Poppet_valve" title="Poppet valve">valves</a> open and close at the top of each cylinder at exactly the right time relative to the movement of each <a href="/wiki/Piston" title="Piston">piston</a>. A chain, called a <a href="/wiki/Ignition_timing" title="Ignition timing">timing</a> chain, is used on some automobiles for this purpose, while in others, the camshaft and crankshaft are coupled directly together through meshed gears. Regardless of which form of drive is employed, the crankshaft-to-camshaft gear ratio is always 2:1 on <a href="/wiki/Four-stroke_engine" title="Four-stroke engine">four-stroke engines</a>, which means that for every two revolutions of the crankshaft the camshaft will rotate once. </p> <div class="mw-heading mw-heading2"><h2 id="Automotive_applications">Automotive applications</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gear_train&action=edit&section=15" title="Edit section: Automotive applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:NonSynchronousGearBoxSF.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/37/NonSynchronousGearBoxSF.jpg/260px-NonSynchronousGearBoxSF.jpg" decoding="async" width="260" height="243" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/37/NonSynchronousGearBoxSF.jpg/390px-NonSynchronousGearBoxSF.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/37/NonSynchronousGearBoxSF.jpg/520px-NonSynchronousGearBoxSF.jpg 2x" data-file-width="748" data-file-height="700" /></a><figcaption>Cutaway illustration of gears of an automotive transmission</figcaption></figure> <p>Automobile <a href="/wiki/Powertrain" title="Powertrain">powertrains</a> generally have two or more major areas where gear sets are used. </p><p>For <a href="/wiki/Internal_combustion_engine" title="Internal combustion engine">internal combustion engine</a> (ICE) vehicles, gearing is typically employed in the <a href="/wiki/Transmission_(mechanics)" class="mw-redirect" title="Transmission (mechanics)">transmission</a>, which contains a number of different sets of gears that can be changed to allow a wide range of vehicle speeds while operating the ICE within a narrower range of speeds, optimizing efficiency, power, and <a href="/wiki/Torque" title="Torque">torque</a>. Because <a href="/wiki/Electric_vehicle" title="Electric vehicle">electric vehicles</a> instead use one or more electric traction motor(s) which generally have a broader range of operating speeds, they are typically equipped with a single-ratio <a href="/wiki/Reduction_gear" class="mw-redirect" title="Reduction gear">reduction gear</a> set instead. </p><p>The second common gear set in almost all motor vehicles is the <a href="/wiki/Differential_(mechanical_device)" title="Differential (mechanical device)">differential</a>, which contains the <a href="/wiki/Final_drive" class="mw-redirect" title="Final drive">final drive</a> to and often provides additional speed reduction at the wheels. Moreover, the differential contains gearing that splits torque equally<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (December 2023)">citation needed</span></a></i>]</sup> between the two wheels while permitting them to have different speeds when traveling in a curved path. </p><p>The transmission and final drive might be separate and connected by a <a href="/wiki/Driveshaft" class="mw-redirect" title="Driveshaft">driveshaft</a>, or they might be combined into one unit called a <a href="/wiki/Transaxle" title="Transaxle">transaxle</a>. The gear ratios in transmission and final drive are important because different gear ratios will change the characteristics of a vehicle's performance. </p> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:FordtaunusV4front.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a5/FordtaunusV4front.jpg/260px-FordtaunusV4front.jpg" decoding="async" width="260" height="198" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a5/FordtaunusV4front.jpg/390px-FordtaunusV4front.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/a/a5/FordtaunusV4front.jpg 2x" data-file-width="420" data-file-height="320" /></a><figcaption><a href="/wiki/Poppet_valve" title="Poppet valve">Valve</a> timing gears on a <a href="/wiki/Ford_Taunus_V4_engine" title="Ford Taunus V4 engine">Ford Taunus V4 engine</a> — the small gear is on the <a href="/wiki/Crankshaft" title="Crankshaft">crankshaft</a>, the larger gear is on the <a href="/wiki/Camshaft" title="Camshaft">camshaft</a>. The crankshaft gear has 34 teeth, the camshaft gear has 68 teeth and runs at half the crankshaft RPM.<br />(The small gear in the lower left is on the <a href="/wiki/Balance_shaft" title="Balance shaft">balance shaft</a>.)</figcaption></figure> <p>As noted, the ICE itself is often equipped with a gear train to synchronize valve operation with crankshaft speed. Typically, the camshafts are driven by gearing, chain, or toothed belt. </p> <div class="mw-heading mw-heading3"><h3 id="Example_2">Example</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gear_train&action=edit&section=16" title="Edit section: Example"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <table border="1" class="wikitable" style="font-size:100%;text-align:center;"> <caption>2004 <a href="/wiki/Chevrolet_Corvette_C5_Z06" class="mw-redirect" title="Chevrolet Corvette C5 Z06">Chevrolet Corvette C5 Z06</a>, six-speed <a href="/wiki/Manual_transmission" title="Manual transmission">manual transmission</a> </caption> <tbody><tr> <th>Gear </th> <td>1</td> <td>2</td> <td>3</td> <td>4</td> <td>5</td> <td>6</td> <td>R </td></tr> <tr> <th>Ratio </th> <td>2.97:1</td> <td>2.07:1</td> <td>1.43:1</td> <td>1:00:1</td> <td>0.84:1</td> <td>0.56:1</td> <td>-3.38:1 </td></tr></tbody></table> <p>In 1st gear, the engine makes 2.97 revolutions for every revolution of the transmission's output. In 4th gear, the gear ratio of 1:1 means that the engine and the transmission's output rotate at the same speed, referred to as the "direct drive" ratio. 5th and 6th gears are known as <a href="/wiki/Overdrive_(mechanics)" title="Overdrive (mechanics)">overdrive</a> gears, in which the output of the transmission is revolving faster than the engine's output. </p><p>The Corvette above is equipped with a differential that has a final drive ratio (or axle ratio) of 3.42:1, meaning that for every 3.42 revolutions of the transmission's output, the <a href="/wiki/Wheel" title="Wheel">wheels</a> make one revolution. The differential ratio multiplies with the transmission ratio, so in 1st gear, the engine makes 10.16 (= 2.97 × 3.42) revolutions for every revolution of the wheels. </p><p>The car's <a href="/wiki/Tire" title="Tire">tires</a> can almost be thought of as a third type of gearing. This car is equipped with 295/35-18 tires, which have a circumference of 82.1 inches. This means that for every complete revolution of the wheel, the car travels 82.1 inches (209 cm). If the Corvette had larger tires, it would travel farther with each revolution of the wheel, which would be like a higher gear. If the car had smaller tires, it would be like a lower gear. </p><p>With the gear ratios of the transmission and differential and the size of the tires, it becomes possible to calculate the speed of the car for a particular gear at a particular engine <a href="/wiki/Revolutions_per_minute" title="Revolutions per minute">RPM</a>. </p><p>For example, it is possible to determine the distance the car will travel for one revolution of the engine by dividing the circumference of the tire by the combined gear ratio of the transmission and differential. </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d={\frac {c_{t}}{gr_{t}\times gr_{d}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mrow> <mi>g</mi> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>×</mo> <mi>g</mi> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d={\frac {c_{t}}{gr_{t}\times gr_{d}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/963a43e33c81e01937be691c570d5d5c88fd80bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:14.238ex; height:5.176ex;" alt="{\displaystyle d={\frac {c_{t}}{gr_{t}\times gr_{d}}}}"></span> </p><p>It is also possible to determine a car's speed from the engine speed by multiplying the circumference of the tire by the engine speed and dividing by the combined gear ratio. </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{c}={\frac {c_{t}\times v_{e}}{gr_{t}\times gr_{d}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>×</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> </mrow> <mrow> <mi>g</mi> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>×</mo> <mi>g</mi> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{c}={\frac {c_{t}\times v_{e}}{gr_{t}\times gr_{d}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4faff0896643225f7acaea8b445920839f1b3750" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:15.094ex; height:5.343ex;" alt="{\displaystyle v_{c}={\frac {c_{t}\times v_{e}}{gr_{t}\times gr_{d}}}}"></span> </p><p>Note that the answer is in inches per minute, which can be converted to <a href="/wiki/Miles_per_hour" title="Miles per hour">mph</a> by dividing by 1056.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p> <table border="1" class="wikitable"> <tbody><tr> <th>Gear</th> <th>Distance per engine revolution</th> <th>Speed per 1000 RPM </th></tr> <tr> <td>1st gear</td> <td>8 in (200 mm)</td> <td>7.6 mph (12.2 km/h) </td></tr> <tr> <td>2nd gear</td> <td>11.5 in (290 mm)</td> <td>10.9 mph (17.5 km/h) </td></tr> <tr> <td>3rd gear</td> <td>16.6 in (420 mm)</td> <td>15.7 mph (25.3 km/h) </td></tr> <tr> <td>4th gear</td> <td>23.7 in (600 mm)</td> <td>22.5 mph (36.2 km/h) </td></tr> <tr> <td>5th gear</td> <td>28.3 in (720 mm)</td> <td>26.8 mph (43.1 km/h) </td></tr> <tr> <td>6th gear</td> <td>42.4 in (1,080 mm)</td> <td>40.1 mph (64.5 km/h) </td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Wide-ratio_vs._close-ratio_transmission">Wide-ratio vs. close-ratio transmission</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gear_train&action=edit&section=17" title="Edit section: Wide-ratio vs. close-ratio transmission"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-Original_research plainlinks metadata ambox ambox-content ambox-Original_research" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/b/b4/Ambox_important.svg/40px-Ambox_important.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/b/b4/Ambox_important.svg/60px-Ambox_important.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/b/b4/Ambox_important.svg/80px-Ambox_important.svg.png 2x" data-file-width="40" data-file-height="40" /></span></span></div></td><td class="mbox-text"><div class="mbox-text-span">This section <b>possibly contains <a href="/wiki/Wikipedia:No_original_research" title="Wikipedia:No original research">original research</a></b>.<span class="hide-when-compact"> Please <a class="external text" href="https://en.wikipedia.org/w/index.php?title=Gear_train&action=edit">improve it</a> by <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">verifying</a> the claims made and adding <a href="/wiki/Wikipedia:Citing_sources#Inline_citations" title="Wikipedia:Citing sources">inline citations</a>. Statements consisting only of original research should be removed.</span> <span class="date-container"><i>(<span class="date">April 2009</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1251242444"><table class="box-More_citations_needed plainlinks metadata ambox ambox-content ambox-Refimprove" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><a href="/wiki/File:Question_book-new.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">This article <b>needs additional citations for <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">verification</a></b>.<span class="hide-when-compact"> Please help <a href="/wiki/Special:EditPage/Gear_train" title="Special:EditPage/Gear train">improve this article</a> by <a href="/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">adding citations to reliable sources</a>. Unsourced material may be challenged and removed.<br /><small><span class="plainlinks"><i>Find sources:</i> <a rel="nofollow" class="external text" href="https://www.google.com/search?as_eq=wikipedia&q=%22Gear+train%22">"Gear train"</a> – <a rel="nofollow" class="external text" href="https://www.google.com/search?tbm=nws&q=%22Gear+train%22+-wikipedia&tbs=ar:1">news</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?&q=%22Gear+train%22&tbs=bkt:s&tbm=bks">newspapers</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?tbs=bks:1&q=%22Gear+train%22+-wikipedia">books</a> <b>·</b> <a rel="nofollow" class="external text" href="https://scholar.google.com/scholar?q=%22Gear+train%22">scholar</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.jstor.org/action/doBasicSearch?Query=%22Gear+train%22&acc=on&wc=on">JSTOR</a></span></small></span> <span class="date-container"><i>(<span class="date">April 2011</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Close-ratio_transmission" title="Close-ratio transmission">Close-ratio transmission</a></div> <p>A close-ratio transmission is a transmission in which there is a relatively little difference between the gear ratios of the gears. For example, a transmission with an engine shaft to drive shaft ratio of 4:1 in first gear and 2:1 in second gear would be considered wide-ratio when compared to another transmission with a ratio of 4:1 in first and 3:1 in second. This is because the close-ratio transmission has less of a progression between gears. For the wide-ratio transmission, the first gear ratio is 4:1 or 4, and in second gear it is 2:1 or 2, so the progression is equal to 4/2 = 2 (or 200%). For the close-ratio transmission, first gear has a 4:1 ratio or 4, and second gear has a ratio of 3:1 or 3, so the progression between gears is 4/3, or 133%. Since 133% is less than 200%, the transmission with the smaller progression between gears is considered close-ratio. However, the difference between a close-ratio and wide-ratio transmission is subjective and relative.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p><p>Close-ratio transmissions are generally offered in <a href="/wiki/Sports_car" title="Sports car">sports cars</a>, <a href="/wiki/Sport_bike" class="mw-redirect" title="Sport bike">sport bikes</a>, and especially in race vehicles, where the engine is tuned for maximum power in a narrow range of operating speeds, and the driver or rider can be expected to shift often to keep the engine in its <a href="/wiki/Power_band" title="Power band">power band</a>. </p><p>Factory 4-speed or 5-speed transmission ratios generally have a greater difference between gear ratios and tend to be effective for ordinary driving and moderate performance use. Wider gaps between ratios allow a higher 1st gear ratio for better manners in traffic, but cause engine speed to decrease more when shifting. Narrowing the gaps will increase acceleration at speed, and potentially improve top speed under certain conditions, but acceleration from a stopped position and operation in daily driving will suffer. </p><p><i>Range</i> is the torque multiplication difference between 1st and 4th gears; wider-ratio gear-sets have more, typically between 2.8 and 3.2. This is the single most important determinant of low-speed acceleration from stopped. </p><p><i>Progression</i> is the reduction or decay in the percentage drop in engine speed in the next gear, for example after shifting from 1st to 2nd gear. Most transmissions have some degree of progression in that the RPM drop on the 1-2 shift is larger than the RPM drop on the 2-3 shift, which is in turn larger than the RPM drop on the 3-4 shift. The progression may not be linear (continuously reduced) or done in proportionate stages for various reasons, including a special need for a gear to reach a specific speed or RPM for passing, racing and so on, or simply economic necessity that the parts were available. </p><p>Range and progression are not mutually exclusive, but each limits the number of options for the other. A wide range, which gives a strong torque multiplication in 1st gear for excellent manners in low-speed traffic, especially with a smaller motor, heavy vehicle, or numerically low axle ratio such as 2.50, means the progression percentages must be high. The amount of engine speed, and therefore power, lost on each up-shift is greater than would be the case in a transmission with less range, but less power in 1st gear. A numerically low 1st gear, such as 2:1, reduces available torque in 1st gear, but allows more choices of progression. </p><p>There is no optimal choice of transmission gear ratios or a final drive ratio for best performance at all speeds, as gear ratios are compromises, and not necessarily better than the original ratios for certain purposes. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gear_train&action=edit&section=18" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col" style="column-width: 22em;"> <ul><li><a href="/wiki/Machine_(mechanical)" class="mw-redirect" title="Machine (mechanical)">Machine (mechanical)</a></li> <li><a href="/wiki/Mechanism_(engineering)" title="Mechanism (engineering)">Mechanism (engineering)</a></li> <li><a href="/wiki/Powertrain" title="Powertrain">Powertrain</a></li> <li><a href="/wiki/Wheel_train_(horology)" class="mw-redirect" title="Wheel train (horology)">Wheel train (horology)</a></li> <li><a href="/wiki/Outline_of_machines" title="Outline of machines">Outline of machines</a></li> <li><a href="/wiki/Epicyclic_gearing" title="Epicyclic gearing">Epicyclic gearing</a> - related to <a href="/wiki/Turboprop" title="Turboprop">turboprop</a> reduction gear boxes</li> <li><a href="/wiki/Continuously_variable_transmission" title="Continuously variable transmission">Continuously variable transmission</a> (CVT)</li> <li><a href="/wiki/Virtual_work" title="Virtual work">Virtual work</a></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gear_train&action=edit&section=19" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text">Army Service Corps Training on Mechanical Transport, (1911), Fig. 112</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFUickerG._R._PennockJ._E._Shigley2003" class="citation book cs1">Uicker, J. J.; G. R. Pennock; J. E. Shigley (2003). <i>Theory of Machines and Mechanisms</i>. New York: Oxford University Press.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Theory+of+Machines+and+Mechanisms&rft.place=New+York&rft.pub=Oxford+University+Press&rft.date=2003&rft.aulast=Uicker&rft.aufirst=J.+J.&rft.au=G.+R.+Pennock&rft.au=J.+E.+Shigley&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGear+train" class="Z3988"></span></span> </li> <li id="cite_note-Shigley-3"><span class="mw-cite-backlink">^ <a href="#cite_ref-Shigley_3-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Shigley_3-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Shigley_3-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-Shigley_3-3"><sup><i><b>d</b></i></sup></a> <a href="#cite_ref-Shigley_3-4"><sup><i><b>e</b></i></sup></a> <a href="#cite_ref-Shigley_3-5"><sup><i><b>f</b></i></sup></a> <a href="#cite_ref-Shigley_3-6"><sup><i><b>g</b></i></sup></a> <a href="#cite_ref-Shigley_3-7"><sup><i><b>h</b></i></sup></a> <a href="#cite_ref-Shigley_3-8"><sup><i><b>i</b></i></sup></a> <a href="#cite_ref-Shigley_3-9"><sup><i><b>j</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFShigleyMischke1989" class="citation book cs1">Shigley, Joseph Edward; Mischke, Charles R. (1989). <a rel="nofollow" class="external text" href="https://archive.org/details/mechanicalengine0000shig_h9k1/page/526/mode/2up">"13: Gearing—General"</a>. <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/mechanicalengine0000shig_h9k1"><i>Mechanical Engineering Design</i></a></span> (Fifth ed.). New York, New York: McGraw-Hill Publishing Company. pp. 527–584. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-07-056899-5" title="Special:BookSources/0-07-056899-5"><bdi>0-07-056899-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=13%3A+Gearing%E2%80%94General&rft.btitle=Mechanical+Engineering+Design&rft.place=New+York%2C+New+York&rft.pages=527-584&rft.edition=Fifth&rft.pub=McGraw-Hill+Publishing+Company&rft.date=1989&rft.isbn=0-07-056899-5&rft.aulast=Shigley&rft.aufirst=Joseph+Edward&rft.au=Mischke%2C+Charles+R.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fmechanicalengine0000shig_h9k1%2Fpage%2F526%2Fmode%2F2up&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGear+train" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPaul1979" class="citation book cs1">Paul, Burton (1979). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/kinematicsdynami0000paul"><i>Kinematics and Dynamics of Planar Machinery</i></a></span>. Prentice Hall.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Kinematics+and+Dynamics+of+Planar+Machinery&rft.pub=Prentice+Hall&rft.date=1979&rft.aulast=Paul&rft.aufirst=Burton&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fkinematicsdynami0000paul&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGear+train" class="Z3988"></span></span> </li> <li id="cite_note-Basic-5"><span class="mw-cite-backlink">^ <a href="#cite_ref-Basic_5-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Basic_5-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Basic_5-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStandards_and_Curriculum_Division,_Bureau_of_Naval_Personnel1946" class="citation book cs1">Standards and Curriculum Division, Bureau of Naval Personnel (1946). <a rel="nofollow" class="external text" href="https://archive.org/details/BasicMachines10624/page/n73/mode/2up">"6: Gears, a topic with teeth in it"</a>. <a rel="nofollow" class="external text" href="https://archive.org/details/BasicMachines10624/"><i>Basic Machines</i></a>. Washington, D.C.: Government Printing Office. pp. 65–79.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=6%3A+Gears%2C+a+topic+with+teeth+in+it&rft.btitle=Basic+Machines&rft.place=Washington%2C+D.C.&rft.pages=65-79&rft.pub=Government+Printing+Office&rft.date=1946&rft.au=Standards+and+Curriculum+Division%2C+Bureau+of+Naval+Personnel&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2FBasicMachines10624%2Fpage%2Fn73%2Fmode%2F2up&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGear+train" class="Z3988"></span></span> </li> <li id="cite_note-amtech20231205-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-amtech20231205_6-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.amtechinternational.com/ring-pinion-gear-manufacturing/">"Why choose ring and pinion gears"</a>. <i>amtechinternational.com</i>. 5 December 2023.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=amtechinternational.com&rft.atitle=Why+choose+ring+and+pinion+gears&rft.date=2023-12-05&rft_id=https%3A%2F%2Fwww.amtechinternational.com%2Fring-pinion-gear-manufacturing%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGear+train" class="Z3988"></span></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.google.com/search?q=convert+in%2Fmin+to+mph&pws=0&gl=us">"Google: convert in/min to mph"</a><span class="reference-accessdate">. Retrieved <span class="nowrap">2018-11-24</span></span>. <q>Formula: divide the speed value by 1056</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Google%3A+convert+in%2Fmin+to+mph&rft_id=https%3A%2F%2Fwww.google.com%2Fsearch%3Fq%3Dconvert%2Bin%252Fmin%2Bto%2Bmph%26pws%3D0%26gl%3Dus&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGear+train" class="Z3988"></span></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCangialosi2001" class="citation web cs1">Cangialosi, Paul (2001). <a rel="nofollow" class="external text" href="http://www.5speeds.com/ratios.html">"TechZone Article: Wide and Close Gear Ratios"</a>. <i>5speeds.com</i>. Medatronics. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20120830130429/http://www.5speeds.com/ratios.html">Archived</a> from the original on 30 August 2012<span class="reference-accessdate">. Retrieved <span class="nowrap">28 October</span> 2012</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=5speeds.com&rft.atitle=TechZone+Article%3A+Wide+and+Close+Gear+Ratios&rft.date=2001&rft.aulast=Cangialosi&rft.aufirst=Paul&rft_id=http%3A%2F%2Fwww.5speeds.com%2Fratios.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGear+train" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gear_train&action=edit&section=20" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid 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template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Gears" title="Template talk:Gears"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Gears" title="Special:EditPage/Template:Gears"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Gears" style="font-size:114%;margin:0 4em"><a href="/wiki/Gear" title="Gear">Gears</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Systems</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Spur_gear" title="Spur gear">Spur gear systems</a></li> <li><a href="/wiki/Worm_drive" title="Worm drive">Worm drive</a></li> <li><a href="/wiki/Rack_and_pinion" title="Rack and pinion">Rack and pinion</a></li> <li><a href="/wiki/Epicyclic_gearing" title="Epicyclic gearing">Epicyclic (planetary) gearing</a></li> <li><a href="/wiki/Sun_and_planet_gear" title="Sun and planet gear">Sun and planet gear</a></li> <li><a href="/wiki/Strain_wave_gearing" title="Strain wave gearing">Harmonic drive</a></li> <li><a href="/wiki/Cycloidal_drive" title="Cycloidal drive">Cycloidal drive</a></li> <li><a href="/wiki/Non-circular_gear" title="Non-circular gear">Non-circular gear</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Shapes</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Spur_gear" title="Spur gear">Spur</a></li> <li><a href="/wiki/Bevel_gear" title="Bevel gear">Bevel</a></li> <li><a href="/wiki/Crown_gear" title="Crown gear">Crown</a></li> <li><a href="/wiki/Spiral_bevel_gear" title="Spiral bevel gear">Spiral bevel</a></li> <li><a href="/wiki/Gear#Helical" title="Gear">Helical</a></li> <li><a href="/wiki/Herringbone_gear" title="Herringbone gear">Herringbone</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Geartooth profiles</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Involute_gear" title="Involute gear">Involute</a></li> <li><a href="/wiki/Cycloid_gear" title="Cycloid gear">Cycloid</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Mechanics</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Propulsion_transmission" class="mw-redirect" title="Propulsion transmission">Transmission</a></li> <li><a href="/wiki/Differential_(mechanical_device)" title="Differential (mechanical device)">Differential</a></li> <li><a href="/wiki/Gear_coupling" class="mw-redirect" title="Gear coupling">Coupling</a></li> <li><a class="mw-selflink selflink">Train</a></li> <li><a href="/wiki/Bicycle_gearing" title="Bicycle gearing">Bicycle gearing</a></li> <li><a href="/wiki/Continuously_variable_transmission" title="Continuously variable transmission">Continuously variable transmission</a></li> <li><a href="/wiki/Offset_(gears)" class="mw-redirect" title="Offset (gears)">Offset</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Examples</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">Bicycles</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Cogset" title="Cogset">Cogset</a></li> <li><a href="/wiki/Derailleur" title="Derailleur">Derailleur</a></li> <li><a href="/wiki/Hub_gear" title="Hub gear">Hub gear</a></li> <li><a href="/wiki/Shaft-driven_bicycle" title="Shaft-driven bicycle">Shaft-driven bicycle</a></li> <li><a href="/wiki/Sprocket" title="Sprocket">Sprocket</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Horology" class="mw-redirect" title="Horology">Horology</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Wheel_train" title="Wheel train">Wheel train</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">See also</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Ball_screw" title="Ball screw">Ball screw</a></li> <li><a href="/wiki/Leadscrew" title="Leadscrew">Leadscrew</a></li> <li><a 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