Gyrocompass - Wikipedia

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class="vector-toc-list"> <li id="toc-First_time-dependent_rotation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#First_time-dependent_rotation"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>First time-dependent rotation</span> </div> </a> <ul id="toc-First_time-dependent_rotation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Second_and_third_fixed_rotations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Second_and_third_fixed_rotations"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Second and third fixed rotations</span> </div> </a> <ul id="toc-Second_and_third_fixed_rotations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Constant_translation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Constant_translation"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>Constant translation</span> </div> </a> <ul id="toc-Constant_translation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Fourth_time-dependent_rotation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Fourth_time-dependent_rotation"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.4</span> <span>Fourth time-dependent rotation</span> </div> </a> <ul id="toc-Fourth_time-dependent_rotation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Last_time-dependent_rotation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Last_time-dependent_rotation"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.5</span> <span>Last time-dependent rotation</span> </div> </a> <ul id="toc-Last_time-dependent_rotation-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Dynamics_of_the_system" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Dynamics_of_the_system"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Dynamics of the system</span> </div> </a> <button aria-controls="toc-Dynamics_of_the_system-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 Dynamics of the system subsection</span> </button> <ul id="toc-Dynamics_of_the_system-sublist" class="vector-toc-list"> <li id="toc-Particular_case:_the_poles" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Particular_case:_the_poles"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Particular case: the poles</span> </div> </a> <ul id="toc-Particular_case:_the_poles-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-The_general_and_physically_relevant_case" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#The_general_and_physically_relevant_case"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>The general and physically relevant case</span> </div> </a> <ul id="toc-The_general_and_physically_relevant_case-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">6</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-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">8</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bibliography" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Bibliography"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Bibliography</span> </div> </a> <ul id="toc-Bibliography-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">10</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">Gyrocompass</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. Available in 37 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-37" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">37 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%A8%D9%88%D8%B5%D9%84%D8%A9_%D8%AC%D9%8A%D8%B1%D9%88%D8%B3%D9%83%D9%88%D8%A8%D9%8A%D8%A9" title="بوصلة جيروسكوبية – Arabic" lang="ar" hreflang="ar" data-title="بوصلة جيروسكوبية" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%96%D0%B8%D1%80%D0%BE%D0%BA%D0%BE%D0%BC%D0%BF%D0%B0%D1%81" title="Жирокомпас – Bulgarian" lang="bg" hreflang="bg" data-title="Жирокомпас" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Girocomp%C3%A0s" title="Girocompàs – Catalan" lang="ca" hreflang="ca" data-title="Girocompàs" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Gyrokompas" title="Gyrokompas – Czech" lang="cs" hreflang="cs" data-title="Gyrokompas" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Kreiselkompass" title="Kreiselkompass – German" lang="de" hreflang="de" data-title="Kreiselkompass" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%93%CF%85%CF%81%CE%BF%CF%83%CE%BA%CE%BF%CF%80%CE%B9%CE%BA%CE%AE_%CF%80%CF%85%CE%BE%CE%AF%CE%B4%CE%B1" title="Γυροσκοπική πυξίδα – Greek" lang="el" hreflang="el" data-title="Γυροσκοπική πυξίδα" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Girocomp%C3%A1s" title="Girocompás – Spanish" lang="es" hreflang="es" data-title="Girocompás" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Girokonpas" title="Girokonpas – Basque" lang="eu" hreflang="eu" data-title="Girokonpas" 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/%D9%82%D8%B7%D8%A8%E2%80%8C%D9%86%D9%85%D8%A7%DB%8C_%DA%AF%D8%B1%D8%AF%D8%B4%E2%80%8C%D8%B3%D9%86%D8%AC" title="قطب‌نمای گردش‌سنج – Persian" lang="fa" hreflang="fa" data-title="قطب‌نمای گردش‌سنج" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Gyrocompas" title="Gyrocompas – French" lang="fr" hreflang="fr" data-title="Gyrocompas" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Comp%C3%A1s_g%C3%ADreasc%C3%B3pach" title="Compás gíreascópach – Irish" lang="ga" hreflang="ga" data-title="Compás gíreascópach" data-language-autonym="Gaeilge" data-language-local-name="Irish" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%ED%9A%8C%EC%A0%84_%EB%82%98%EC%B9%A8%EB%B0%98" title="회전 나침반 – Korean" lang="ko" hreflang="ko" data-title="회전 나침반" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%B3%D5%AB%D6%80%D5%B8%D5%AF%D5%B8%D5%B2%D5%B4%D5%B6%D5%A1%D6%81%D5%B8%D6%82%D5%B5%D6%81" title="Գիրոկողմնացույց – Armenian" lang="hy" hreflang="hy" data-title="Գիրոկողմնացույց" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%98%E0%A5%82%E0%A4%B0%E0%A5%8D%E0%A4%A3%E0%A4%BE%E0%A4%95%E0%A5%8D%E0%A4%B7%E0%A4%B8%E0%A5%8D%E0%A4%A5%E0%A4%BE%E0%A4%AA%E0%A5%80_%E0%A4%A6%E0%A4%BF%E0%A4%95%E0%A5%8D%E0%A4%B8%E0%A5%82%E0%A4%9A%E0%A4%95" title="घूर्णाक्षस्थापी दिक्सूचक – Hindi" lang="hi" hreflang="hi" data-title="घूर्णाक्षस्थापी दिक्सूचक" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/%C5%BDirokompas" title="Žirokompas – Croatian" lang="hr" hreflang="hr" data-title="Žirokompas" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Girokompas" title="Girokompas – Indonesian" lang="id" hreflang="id" data-title="Girokompas" 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/Girobussola" title="Girobussola – Italian" lang="it" hreflang="it" data-title="Girobussola" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%A6%D7%A4%D7%9F_%D7%92%D7%99%D7%A8%D7%95%D7%A1%D7%A7%D7%95%D7%A4%D7%99" title="מצפן גירוסקופי – Hebrew" lang="he" hreflang="he" data-title="מצפן גירוסקופי" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%93%D0%B8%D1%80%D0%BE%D0%BA%D0%BE%D0%BC%D0%BF%D0%B0%D1%81" title="Гирокомпас – Kazakh" lang="kk" hreflang="kk" data-title="Гирокомпас" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%93%D0%B8%D1%80%D0%BE%D0%BA%D0%BE%D0%BC%D0%BF%D0%B0c" title="Гирокомпаc – Kyrgyz" lang="ky" hreflang="ky" data-title="Гирокомпаc" data-language-autonym="Кыргызча" data-language-local-name="Kyrgyz" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/%C5%BDirokompass" title="Žirokompass – Latvian" lang="lv" hreflang="lv" data-title="Žirokompass" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-lb mw-list-item"><a href="https://lb.wikipedia.org/wiki/Gyrokompass" title="Gyrokompass – Luxembourgish" lang="lb" hreflang="lb" data-title="Gyrokompass" data-language-autonym="Lëtzebuergesch" data-language-local-name="Luxembourgish" class="interlanguage-link-target"><span>Lëtzebuergesch</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Gyrokompas" title="Gyrokompas – Dutch" lang="nl" hreflang="nl" data-title="Gyrokompas" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%82%B8%E3%83%A3%E3%82%A4%E3%83%AD%E3%82%B3%E3%83%B3%E3%83%91%E3%82%B9" title="ジャイロコンパス – Japanese" lang="ja" hreflang="ja" data-title="ジャイロコンパス" 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-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Gyrokompass" title="Gyrokompass – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Gyrokompass" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Girokompas" title="Girokompas – Uzbek" lang="uz" hreflang="uz" data-title="Girokompas" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Kompas_%C5%BCyroskopowy" title="Kompas żyroskopowy – Polish" lang="pl" hreflang="pl" data-title="Kompas żyroskopowy" 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/B%C3%BAssola_girosc%C3%B3pica" title="Bússola giroscópica – Portuguese" lang="pt" hreflang="pt" data-title="Bússola giroscópica" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Girocompas" title="Girocompas – Romanian" lang="ro" hreflang="ro" data-title="Girocompas" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%93%D0%B8%D1%80%D0%BE%D0%BA%D0%BE%D0%BC%D0%BF%D0%B0%D1%81" title="Гирокомпас – Russian" lang="ru" hreflang="ru" data-title="Гирокомпас" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Gyrocompass" title="Gyrocompass – Simple English" lang="en-simple" hreflang="en-simple" data-title="Gyrocompass" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Hyrr%C3%A4kompassi" title="Hyrräkompassi – Finnish" lang="fi" hreflang="fi" data-title="Hyrräkompassi" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%9A%E0%AF%81%E0%AE%B4%E0%AE%BF%E0%AE%A4%E0%AE%BF%E0%AE%9A%E0%AF%88%E0%AE%95%E0%AE%BE%E0%AE%9F%E0%AF%8D%E0%AE%9F%E0%AE%BF" title="சுழிதிசைகாட்டி – Tamil" lang="ta" hreflang="ta" data-title="சுழிதிசைகாட்டி" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-tg mw-list-item"><a href="https://tg.wikipedia.org/wiki/%D2%9A%D1%83%D1%82%D0%B1%D0%BD%D0%B0%D0%BC%D0%BE%D0%B8_%D0%B3%D0%B0%D1%80%D0%B4%D0%B8%D1%88%D1%81%D0%B0%D0%BD%D2%B7" title="Қутбнамои гардишсанҷ – Tajik" lang="tg" hreflang="tg" data-title="Қутбнамои гардишсанҷ" data-language-autonym="Тоҷикӣ" data-language-local-name="Tajik" class="interlanguage-link-target"><span>Тоҷикӣ</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Cayro" title="Cayro – Turkish" lang="tr" hreflang="tr" data-title="Cayro" 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%93%D1%96%D1%80%D0%BE%D0%BA%D0%BE%D0%BC%D0%BF%D0%B0%D1%81" title="Гірокомпас – Ukrainian" lang="uk" hreflang="uk" data-title="Гірокомпас" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E9%99%80%E8%9E%BA%E7%BE%85%E7%B6%93" title="陀螺羅經 – Cantonese" lang="yue" 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id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Type of non-magnetic compass based on the rotation of the Earth</div> <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">This article is about devices used on ships. For the type of gyroscope used to determine aircraft heading, see <a href="/wiki/Heading_indicator" title="Heading indicator">Heading indicator</a>.</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Kreiselkompass_Schnitt_Ansch%C3%BCtz.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/eb/Kreiselkompass_Schnitt_Ansch%C3%BCtz.jpg/220px-Kreiselkompass_Schnitt_Ansch%C3%BCtz.jpg" decoding="async" width="220" height="256" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/eb/Kreiselkompass_Schnitt_Ansch%C3%BCtz.jpg/330px-Kreiselkompass_Schnitt_Ansch%C3%BCtz.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/eb/Kreiselkompass_Schnitt_Ansch%C3%BCtz.jpg/440px-Kreiselkompass_Schnitt_Ansch%C3%BCtz.jpg 2x" data-file-width="1062" data-file-height="1238" /></a><figcaption>Cutaway of an Anschütz gyrocompass</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Algonquin_gyro_compass2.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/66/Algonquin_gyro_compass2.jpg/220px-Algonquin_gyro_compass2.jpg" decoding="async" width="220" height="165" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/66/Algonquin_gyro_compass2.jpg/330px-Algonquin_gyro_compass2.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/66/Algonquin_gyro_compass2.jpg/440px-Algonquin_gyro_compass2.jpg 2x" data-file-width="3072" data-file-height="2304" /></a><figcaption>A gyrocompass repeater</figcaption></figure> <p>A <b>gyrocompass</b> is a type of non-magnetic <a href="/wiki/Compass" title="Compass">compass</a> which is based on a fast-spinning disc and the rotation of the <a href="/wiki/Earth" title="Earth">Earth</a> (or another planetary body if used elsewhere in the universe) to find geographical <a href="/wiki/Direction_(geometry)" title="Direction (geometry)">direction</a> automatically. A gyrocompass makes use of one of the seven fundamental ways to determine the heading of a vehicle.<sup id="cite_ref-JournalOfNavigation2016_1-0" class="reference"><a href="#cite_note-JournalOfNavigation2016-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> A <a href="/wiki/Gyroscope" title="Gyroscope">gyroscope</a> is an essential component of a gyrocompass, but they are different devices; a gyrocompass is built to use the effect of <a href="/wiki/Gyroscopic_precession" class="mw-redirect" title="Gyroscopic precession">gyroscopic precession</a>, which is a distinctive aspect of the general <a href="/wiki/Gyroscopic_effect" class="mw-redirect" title="Gyroscopic effect">gyroscopic effect</a>.<sup id="cite_ref-an_2-0" class="reference"><a href="#cite_note-an-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-l_3-0" class="reference"><a href="#cite_note-l-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> Gyrocompasses, such as the <a href="/wiki/Fibre_optic_gyrocompass" title="Fibre optic gyrocompass">fibre optic gyrocompass</a> are widely used to provide a heading for <a href="/wiki/Navigation" title="Navigation">navigation</a> on <a href="/wiki/Ship" title="Ship">ships</a>.<sup id="cite_ref-SafeNavWatch_4-0" class="reference"><a href="#cite_note-SafeNavWatch-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> This is because they have two significant advantages over <a href="/wiki/Magnetic_compass" class="mw-redirect" title="Magnetic compass">magnetic compasses</a>:<sup id="cite_ref-l_3-1" class="reference"><a href="#cite_note-l-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <ul><li>they find <a href="/wiki/True_north" title="True north">true north</a> as determined by the axis of the <a href="/wiki/Earth%27s_rotation" title="Earth's rotation">Earth's rotation</a>, which is different from, and navigationally more useful than, <a href="/wiki/Magnetic_North_Pole#Magnetic_north_and_magnetic_declination" class="mw-redirect" title="Magnetic North Pole"><i>magnetic</i> north</a>, and</li> <li>they have a greater degree of accuracy because they are unaffected by <a href="/wiki/Ferromagnetic" class="mw-redirect" title="Ferromagnetic">ferromagnetic</a> materials, such as in a ship's <a href="/wiki/Steel" title="Steel">steel</a> <a href="/wiki/Hull_(watercraft)" title="Hull (watercraft)">hull</a>, which distort the <a href="/wiki/Magnetic_field" title="Magnetic field">magnetic field</a>.<sup id="cite_ref-SafeNavWatch_4-1" class="reference"><a href="#cite_note-SafeNavWatch-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup></li></ul> <p>Aircraft commonly use gyroscopic instruments (but not a gyrocompass) for navigation and attitude monitoring; for details, see <a href="/wiki/Flight_instruments" title="Flight instruments">flight instruments</a> (specifically the <a href="/wiki/Heading_indicator" title="Heading indicator">heading indicator</a>) and <a href="/wiki/Gyroscopic_autopilot" title="Gyroscopic autopilot">gyroscopic autopilot</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gyrocompass&action=edit&section=1" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The first, not yet practical,<sup id="cite_ref-hee_5-0" class="reference"><a href="#cite_note-hee-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> form of gyrocompass was patented in 1885 by Marinus Gerardus van den Bos.<sup id="cite_ref-hee_5-1" class="reference"><a href="#cite_note-hee-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> A usable gyrocompass was invented in 1906 in Germany by <a href="/wiki/Hermann_Ansch%C3%BCtz-Kaempfe" title="Hermann Anschütz-Kaempfe">Hermann Anschütz-Kaempfe</a>, and after successful tests in 1908 became widely used in the German Imperial Navy.<sup id="cite_ref-an_2-1" class="reference"><a href="#cite_note-an-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-hee_5-2" class="reference"><a href="#cite_note-hee-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> Anschütz-Kaempfe founded the company <a href="/wiki/Raytheon_Ansch%C3%BCtz" class="mw-redirect" title="Raytheon Anschütz">Anschütz & Co.</a> in <a href="/wiki/Kiel" title="Kiel">Kiel</a>, to mass produce gyrocompasses; the company is today Raytheon Anschütz GmbH.<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> The gyrocompass was an important invention for nautical navigation because it allowed accurate determination of a vessel’s location at all times regardless of the vessel’s motion, the weather and the amount of steel used in the construction of the ship.<sup id="cite_ref-maritime.org_8-0" class="reference"><a href="#cite_note-maritime.org-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p><p>In the United States, <a href="/wiki/Elmer_Ambrose_Sperry" title="Elmer Ambrose Sperry">Elmer Ambrose Sperry</a> produced a workable gyrocompass system (1908: <span><a rel="nofollow" class="external text" href="https://patents.google.com/patent/US1242065">U.S. patent 1,242,065</a></span>), and founded the <a href="/wiki/Sperry_Corporation" title="Sperry Corporation">Sperry Gyroscope Company</a>. The unit was adopted by the U.S. Navy (1911<sup id="cite_ref-l_3-2" class="reference"><a href="#cite_note-l-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>), and played a major role in World War I. The Navy also began using Sperry's "Metal Mike": the first gyroscope-guided autopilot steering system. In the following decades, these and other Sperry devices were adopted by steamships such as the <a href="/wiki/RMS_Queen_Mary" title="RMS Queen Mary">RMS <i>Queen Mary</i></a>, airplanes, and the warships of World War II. After his death in 1930, the Navy named the <a href="/wiki/USS_Sperry" title="USS Sperry">USS <i>Sperry</i></a> after him. </p><p>Meanwhile, in 1913, C. Plath (a Hamburg, Germany-based manufacturer of navigational equipment including sextants and magnetic compasses) developed the first gyrocompass to be installed on a commercial vessel. C. Plath sold many gyrocompasses to the Weems’ School for Navigation in Annapolis, MD, and soon the founders of each organization formed an alliance and became Weems & Plath.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:1889_Gymnote_Gyroscope.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c1/1889_Gymnote_Gyroscope.jpg/220px-1889_Gymnote_Gyroscope.jpg" decoding="async" width="220" height="269" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c1/1889_Gymnote_Gyroscope.jpg/330px-1889_Gymnote_Gyroscope.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c1/1889_Gymnote_Gyroscope.jpg/440px-1889_Gymnote_Gyroscope.jpg 2x" data-file-width="721" data-file-height="882" /></a><figcaption>The 1889 Dumoulin-Krebs gyroscope</figcaption></figure> <p>Before the success of the gyrocompass, several attempts had been made in Europe to use a gyroscope instead. By 1880, <a href="/wiki/William_Thomson,_1st_Baron_Kelvin" class="mw-redirect" title="William Thomson, 1st Baron Kelvin">William Thomson</a> (Lord Kelvin) tried to propose a <a href="/wiki/Gyrostat" class="mw-redirect" title="Gyrostat">gyrostat</a> to the British Navy. In 1889, <a href="/wiki/Arthur_Krebs" class="mw-redirect" title="Arthur Krebs">Arthur Krebs</a> adapted an electric motor to the Dumoulin-Froment marine gyroscope, for the French Navy. That gave the <a href="/wiki/French_submarine_Gymnote_(Q1)" title="French submarine Gymnote (Q1)"><i>Gymnote</i></a> submarine the ability to keep a straight line while underwater for several hours, and it allowed her to <a rel="nofollow" class="external text" href="http://rbmn.free.fr/Gymnote_Blocus_1890.jpg">force a naval block</a> in 1890. </p><p>In 1923 <a href="/wiki/Max_Schuler" title="Max Schuler">Max Schuler</a> published his paper containing his observation that if a gyrocompass possessed <a href="/wiki/Schuler_tuning" title="Schuler tuning">Schuler tuning</a> such that it had an oscillation period of 84.4 minutes (which is the orbital period of a notional satellite orbiting around the Earth at sea level), then it could be rendered insensitive to lateral motion and maintain directional stability.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Operation">Operation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gyrocompass&action=edit&section=2" title="Edit section: Operation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A <a href="/wiki/Gyroscope" title="Gyroscope">gyroscope</a>, not to be confused with a gyrocompass, is a spinning wheel mounted on a set of <a href="/wiki/Gimbals" class="mw-redirect" title="Gimbals">gimbals</a> so that its axis is free to orient itself in any way.<sup id="cite_ref-l_3-3" class="reference"><a href="#cite_note-l-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> When it is spun up to speed with its axis pointing in some direction, due to the law of conservation of <a href="/wiki/Angular_momentum" title="Angular momentum">angular momentum</a>, such a wheel will normally maintain its original orientation to a fixed point in <a href="/wiki/Outer_space" title="Outer space">outer space</a> (not to a fixed point on Earth). Since the Earth rotates, it appears to a stationary observer on Earth that a gyroscope's axis is completing a full rotation once every 24 hours.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>note 1<span class="cite-bracket">]</span></a></sup> Such a rotating gyroscope is used for navigation in some cases, for example on aircraft, where it is known as <a href="/wiki/Heading_indicator" title="Heading indicator">heading indicator</a> or directional gyro, but cannot ordinarily be used for long-term marine navigation. The crucial additional ingredient needed to turn a gyroscope into a gyrocompass, so it would automatically position to true north,<sup id="cite_ref-an_2-2" class="reference"><a href="#cite_note-an-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-l_3-4" class="reference"><a href="#cite_note-l-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> is some mechanism that results in an <a href="/wiki/Torque" title="Torque">application of torque</a> whenever the compass's axis is not pointing north. </p><p>One method uses <a href="/wiki/Friction" title="Friction">friction</a> to apply the needed torque:<sup id="cite_ref-maritime.org_8-1" class="reference"><a href="#cite_note-maritime.org-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> the gyroscope in a gyrocompass is not completely free to reorient itself; if for instance a device connected to the axis is immersed in a viscous fluid, then that fluid will resist reorientation of the axis. This friction force caused by the fluid results in a <a href="/wiki/Torque" title="Torque">torque</a> acting on the axis, causing the axis to turn in a direction orthogonal to the torque (that is, to <a href="/wiki/Precess" class="mw-redirect" title="Precess">precess</a>) along a <a href="/wiki/Line_of_longitude" class="mw-redirect" title="Line of longitude">line of longitude</a>. Once the axis points toward the celestial pole, it will appear to be stationary and won't experience any more frictional forces. This is because true north (or true south) is the only direction for which the gyroscope can remain on the surface of the earth and not be required to change. This axis orientation is considered to be a point of minimum <a href="/wiki/Potential_energy" title="Potential energy">potential energy</a>. </p><p>Another, more practical, method is to use weights to force the axis of the compass to remain horizontal (perpendicular to the direction of the center of the Earth), but otherwise allow it to rotate freely within the horizontal plane.<sup id="cite_ref-an_2-3" class="reference"><a href="#cite_note-an-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-l_3-5" class="reference"><a href="#cite_note-l-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> In this case, gravity will apply a torque forcing the compass's axis toward true north. Because the weights will confine the compass's axis to be horizontal with respect to the Earth's surface, the axis can never align with the Earth's axis (except on the Equator) and must realign itself as the Earth rotates. But with respect to the Earth's surface, the compass will appear to be stationary and pointing along the Earth's surface toward the true North Pole. </p><p>Since the gyrocompass's north-seeking function depends on the rotation around the axis of the Earth that causes <a href="/wiki/Gyroscopic_precession#Torque-induced" class="mw-redirect" title="Gyroscopic precession">torque-induced gyroscopic precession</a>, it will not orient itself correctly to true north if it is moved very fast in an east to west direction, thus negating the Earth's rotation. However, aircraft commonly use <a href="/wiki/Heading_indicator" title="Heading indicator">heading indicators or directional gyros</a>, which are not gyrocompasses and do not align themselves to north via precession, but are periodically aligned manually to magnetic north.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Errors">Errors</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gyrocompass&action=edit&section=3" title="Edit section: Errors"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A gyrocompass is subject to certain errors. These include steaming error, where rapid changes in course, speed and <a href="/wiki/Latitude" title="Latitude">latitude</a> cause <a href="/wiki/Magnetic_deviation" title="Magnetic deviation">deviation</a> before the gyro can adjust itself.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> On most modern ships the <a href="/wiki/GPS" class="mw-redirect" title="GPS">GPS</a> or other navigational aids feed data to the gyrocompass allowing a small computer to apply a correction. Alternatively a design based on a <a href="/wiki/Inertial_navigation_system#Strapdown_systems" title="Inertial navigation system">strapdown architecture</a> (including a triad of <a href="/wiki/Fibre_optic_gyroscope" class="mw-redirect" title="Fibre optic gyroscope">fibre optic gyroscopes</a>, <a href="/wiki/Ring_laser_gyroscope" title="Ring laser gyroscope">ring laser gyroscopes</a> or <a href="/wiki/Hemispherical_resonator_gyroscope" title="Hemispherical resonator gyroscope">hemispherical resonator gyroscopes</a> and a triad of accelerometers) will eliminate these errors, as they do not depend upon mechanical parts to determinate rate of rotation.<sup id="cite_ref-House_15-0" class="reference"><a href="#cite_note-House-15"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Mathematical_model">Mathematical model</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gyrocompass&action=edit&section=4" title="Edit section: Mathematical model"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>We consider a gyrocompass as a gyroscope which is free to rotate about one of its symmetry axes, also the whole rotating gyroscope is free to rotate on the horizontal plane about the local vertical. Therefore there are two independent local rotations. In addition to these rotations we consider the rotation of the Earth about its north-south (NS) axis, and we model the planet as a perfect sphere. We neglect friction and also the rotation of the Earth about the Sun. </p><p>In this case a non-rotating observer located at the center of the Earth can be approximated as being an inertial frame. We establish cartesian coordinates <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 (X_{1},Y_{1},Z_{1})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (X_{1},Y_{1},Z_{1})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4537f3cf3d7145d9b4505071e1061602bb874685" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.902ex; height:2.843ex;" alt="{\displaystyle (X_{1},Y_{1},Z_{1})}"></span> for such an observer (whom we name as 1-O), and the barycenter of the gyroscope is located at a distance <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/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span> from the center of the Earth. </p> <div class="mw-heading mw-heading3"><h3 id="First_time-dependent_rotation">First time-dependent rotation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gyrocompass&action=edit&section=5" title="Edit section: First time-dependent rotation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Consider another (non-inertial) observer (the 2-O) located at the center of the Earth but rotating about the NS-axis by <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 .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ω</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5dc55a0f828ae7d92ad5426ce7348d84ee62a70" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.325ex; height:2.176ex;" alt="{\displaystyle \Omega .}"></span> We establish coordinates attached to this observer as <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{pmatrix}X_{2}\\Y_{2}\\Z_{2}\end{pmatrix}}={\begin{pmatrix}\cos \Omega t&\sin \Omega t&0\\-\sin \Omega t&\cos \Omega t&0\\0&0&1\end{pmatrix}}{\begin{pmatrix}X_{1}\\Y_{1}\\Z_{1}\end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mi mathvariant="normal">Ω</mi> <mi>t</mi> </mtd> <mtd> <mi>sin</mi> <mo>⁡</mo> <mi mathvariant="normal">Ω</mi> <mi>t</mi> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mi mathvariant="normal">Ω</mi> <mi>t</mi> </mtd> <mtd> <mi>cos</mi> <mo>⁡</mo> <mi mathvariant="normal">Ω</mi> <mi>t</mi> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}X_{2}\\Y_{2}\\Z_{2}\end{pmatrix}}={\begin{pmatrix}\cos \Omega t&\sin \Omega t&0\\-\sin \Omega t&\cos \Omega t&0\\0&0&1\end{pmatrix}}{\begin{pmatrix}X_{1}\\Y_{1}\\Z_{1}\end{pmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8539af2b347d64258d8bea7a77d74e8470e30a9c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:43.29ex; height:9.176ex;" alt="{\displaystyle {\begin{pmatrix}X_{2}\\Y_{2}\\Z_{2}\end{pmatrix}}={\begin{pmatrix}\cos \Omega t&\sin \Omega t&0\\-\sin \Omega t&\cos \Omega t&0\\0&0&1\end{pmatrix}}{\begin{pmatrix}X_{1}\\Y_{1}\\Z_{1}\end{pmatrix}}}"></span> so that the unit <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 {\hat {X}}_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {X}}_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f6f599b2d5b886f5e950d58506e9c1fc1f7d3cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.034ex; height:3.176ex;" alt="{\displaystyle {\hat {X}}_{1}}"></span> versor <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 (X_{1}=1,Y_{1}=0,Z_{1}=0)^{T}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>,</mo> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (X_{1}=1,Y_{1}=0,Z_{1}=0)^{T}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/885ea293a7e45b8f80030cf4ebaec5a099565c70" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.074ex; height:3.176ex;" alt="{\displaystyle (X_{1}=1,Y_{1}=0,Z_{1}=0)^{T}}"></span> is mapped to the point <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 (X_{2}=\cos \Omega t,Y_{2}=-\sin \Omega t,Z_{2}=0)^{T}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mi mathvariant="normal">Ω</mi> <mi>t</mi> <mo>,</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mi mathvariant="normal">Ω</mi> <mi>t</mi> <mo>,</mo> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (X_{2}=\cos \Omega t,Y_{2}=-\sin \Omega t,Z_{2}=0)^{T}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c6552c6d33e3360a442a184b6a10a5866b276f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.721ex; height:3.176ex;" alt="{\displaystyle (X_{2}=\cos \Omega t,Y_{2}=-\sin \Omega t,Z_{2}=0)^{T}}"></span>. For the 2-O neither the Earth nor the barycenter of the gyroscope is moving. The rotation of 2-O relative to 1-O is performed with 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 {\vec {\Omega }}=(0,0,\Omega )^{T}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mi mathvariant="normal">Ω</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {\Omega }}=(0,0,\Omega )^{T}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b944b8b11b62ef4daace7ccb0825bd22ed0fcd60" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.046ex; height:3.509ex;" alt="{\displaystyle {\vec {\Omega }}=(0,0,\Omega )^{T}}"></span>. We suppose that the <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 X_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ad47c14b8a092f182512e76c96638aea6e3bea1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.979ex; height:2.509ex;" alt="{\displaystyle X_{2}}"></span> axis denotes points with zero longitude (the prime, or Greenwich, meridian). </p> <div class="mw-heading mw-heading3"><h3 id="Second_and_third_fixed_rotations">Second and third fixed rotations</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gyrocompass&action=edit&section=6" title="Edit section: Second and third fixed rotations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>We now rotate about the <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="{\textstyle Z_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle Z_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0f1d6d55d775fd8fa39d1fb474f9e743849f0d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.642ex; height:2.509ex;" alt="{\textstyle Z_{2}}"></span> axis, so that the <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="{\textstyle X_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle X_{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1692e646a5e7a9ec868f12f4952e63466df587d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.979ex; height:2.509ex;" alt="{\textstyle X_{3}}"></span>-axis has the longitude of the barycenter. In this case we have <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{pmatrix}X_{3}\\Y_{3}\\Z_{3}\end{pmatrix}}={\begin{pmatrix}\cos \Phi &\sin \Phi &0\\-\sin \Phi &\cos \Phi &0\\0&0&1\end{pmatrix}}{\begin{pmatrix}X_{2}\\Y_{2}\\Z_{2}\end{pmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mi mathvariant="normal">Φ</mi> </mtd> <mtd> <mi>sin</mi> <mo>⁡</mo> <mi mathvariant="normal">Φ</mi> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mi mathvariant="normal">Φ</mi> </mtd> <mtd> <mi>cos</mi> <mo>⁡</mo> <mi mathvariant="normal">Φ</mi> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}X_{3}\\Y_{3}\\Z_{3}\end{pmatrix}}={\begin{pmatrix}\cos \Phi &\sin \Phi &0\\-\sin \Phi &\cos \Phi &0\\0&0&1\end{pmatrix}}{\begin{pmatrix}X_{2}\\Y_{2}\\Z_{2}\end{pmatrix}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/36a309dce368ebd54050338ff79732c249dc50af" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:42.257ex; height:9.176ex;" alt="{\displaystyle {\begin{pmatrix}X_{3}\\Y_{3}\\Z_{3}\end{pmatrix}}={\begin{pmatrix}\cos \Phi &\sin \Phi &0\\-\sin \Phi &\cos \Phi &0\\0&0&1\end{pmatrix}}{\begin{pmatrix}X_{2}\\Y_{2}\\Z_{2}\end{pmatrix}}.}"></span> </p><p>With the next rotation (about the axis <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="{\textstyle Y_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle Y_{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c96a426798a869b6b088ace618a50eef6b251e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.405ex; height:2.509ex;" alt="{\textstyle Y_{3}}"></span> of an angle <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="{\textstyle \delta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>δ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \delta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec578ebbf0a029f13dca70687f072742277a87ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:2.343ex;" alt="{\textstyle \delta }"></span>, the co-latitude) we bring the <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="{\textstyle Z_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle Z_{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3d55c4d75eedb0ad65c5cfcfc25f3c1df1d491c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.642ex; height:2.509ex;" alt="{\textstyle Z_{3}}"></span> axis along the local zenith (<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="{\textstyle Z_{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle Z_{4}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/836dbdc4c88a0823a5356bccdf1ea5095932f8db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.642ex; height:2.509ex;" alt="{\textstyle Z_{4}}"></span>-axis) of the barycenter. This can be achieved by the following orthogonal matrix (with unit determinant) <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{pmatrix}X_{4}\\Y_{4}\\Z_{4}\end{pmatrix}}={\begin{pmatrix}\cos \delta &0&-\sin \delta \\0&1&0\\\sin \delta &0&\cos \delta \end{pmatrix}}{\begin{pmatrix}X_{3}\\Y_{3}\\Z_{3}\end{pmatrix}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mi>δ</mi> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mi>δ</mi> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mi>sin</mi> <mo>⁡</mo> <mi>δ</mi> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mi>cos</mi> <mo>⁡</mo> <mi>δ</mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}X_{4}\\Y_{4}\\Z_{4}\end{pmatrix}}={\begin{pmatrix}\cos \delta &0&-\sin \delta \\0&1&0\\\sin \delta &0&\cos \delta \end{pmatrix}}{\begin{pmatrix}X_{3}\\Y_{3}\\Z_{3}\end{pmatrix}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73ad3095a406ba0efad58301a58e69aadf7bf7e2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:40.998ex; height:9.176ex;" alt="{\displaystyle {\begin{pmatrix}X_{4}\\Y_{4}\\Z_{4}\end{pmatrix}}={\begin{pmatrix}\cos \delta &0&-\sin \delta \\0&1&0\\\sin \delta &0&\cos \delta \end{pmatrix}}{\begin{pmatrix}X_{3}\\Y_{3}\\Z_{3}\end{pmatrix}},}"></span> </p><p>so that the <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="{\textstyle {\hat {Z}}_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Z</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\hat {Z}}_{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/063d9d581e79a852e9fa3c4b1d503bbc8e7145fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.735ex; height:3.176ex;" alt="{\textstyle {\hat {Z}}_{3}}"></span> versor <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="{\textstyle (X_{3}=0,Y_{3}=0,Z_{3}=1)^{T}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>,</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>,</mo> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle (X_{3}=0,Y_{3}=0,Z_{3}=1)^{T}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73b2f63900b49d0d80451d3b7d08df6bd9e86ee3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.074ex; height:3.009ex;" alt="{\textstyle (X_{3}=0,Y_{3}=0,Z_{3}=1)^{T}}"></span> is mapped to the point <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="{\textstyle (X_{4}=-\sin \delta ,Y_{4}=0,Z_{4}=\cos \delta )^{T}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mi>δ</mi> <mo>,</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>,</mo> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mi>δ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle (X_{4}=-\sin \delta ,Y_{4}=0,Z_{4}=\cos \delta )^{T}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b666b37f72226916a5287c20753f9db660a03fa7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.43ex; height:3.009ex;" alt="{\textstyle (X_{4}=-\sin \delta ,Y_{4}=0,Z_{4}=\cos \delta )^{T}.}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Constant_translation">Constant translation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gyrocompass&action=edit&section=7" title="Edit section: Constant translation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>We now choose another coordinate basis whose origin is located at the barycenter of the gyroscope. This can be performed by the following translation along the zenith axis <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{pmatrix}X_{5}\\Y_{5}\\Z_{5}\end{pmatrix}}={\begin{pmatrix}X_{4}\\Y_{4}\\Z_{4}\end{pmatrix}}-{\begin{pmatrix}0\\0\\R\end{pmatrix}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mi>R</mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}X_{5}\\Y_{5}\\Z_{5}\end{pmatrix}}={\begin{pmatrix}X_{4}\\Y_{4}\\Z_{4}\end{pmatrix}}-{\begin{pmatrix}0\\0\\R\end{pmatrix}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b85b692066c5014e55e4ccae46cadb94944763b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:28.762ex; height:9.176ex;" alt="{\displaystyle {\begin{pmatrix}X_{5}\\Y_{5}\\Z_{5}\end{pmatrix}}={\begin{pmatrix}X_{4}\\Y_{4}\\Z_{4}\end{pmatrix}}-{\begin{pmatrix}0\\0\\R\end{pmatrix}},}"></span> </p><p>so that the origin of the new system, <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 (X_{5}=0,Y_{5}=0,Z_{5}=0)^{T}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>,</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>,</mo> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (X_{5}=0,Y_{5}=0,Z_{5}=0)^{T}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd61f6897d8a107c85bd56e3c6b2a157aeead642" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.074ex; height:3.176ex;" alt="{\displaystyle (X_{5}=0,Y_{5}=0,Z_{5}=0)^{T}}"></span> is located at the point <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 (X_{4}=0,Y_{4}=0,Z_{4}=R)^{T},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>,</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>,</mo> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>=</mo> <mi>R</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (X_{4}=0,Y_{4}=0,Z_{4}=R)^{T},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a10d5a4f6d1a0cea45c339919b18013059490262" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.323ex; height:3.176ex;" alt="{\displaystyle (X_{4}=0,Y_{4}=0,Z_{4}=R)^{T},}"></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 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/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span> is the radius of the Earth. Now the <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 X_{5}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{5}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f14cb64bd8c186831d9e77d4394741f9f0ee3917" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.979ex; height:2.509ex;" alt="{\displaystyle X_{5}}"></span>-axis points towards the south direction. </p> <div class="mw-heading mw-heading3"><h3 id="Fourth_time-dependent_rotation">Fourth time-dependent rotation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gyrocompass&action=edit&section=8" title="Edit section: Fourth time-dependent rotation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Now we rotate about the zenith <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 Z_{5}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z_{5}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e3539f4e08101ff96b1587c843950789b72db0c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.642ex; height:2.509ex;" alt="{\displaystyle Z_{5}}"></span>-axis so that the new coordinate system is attached to the structure of the gyroscope, so that for an observer at rest in this coordinate system, the gyrocompass is only rotating about its own axis of symmetry. In this case we find <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{pmatrix}X_{6}\\Y_{6}\\Z_{6}\end{pmatrix}}={\begin{pmatrix}\cos \alpha &\sin \alpha &0\\-\sin \alpha &\cos \alpha &0\\0&0&1\end{pmatrix}}{\begin{pmatrix}X_{5}\\Y_{5}\\Z_{5}\end{pmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mi>α</mi> </mtd> <mtd> <mi>sin</mi> <mo>⁡</mo> <mi>α</mi> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mi>α</mi> </mtd> <mtd> <mi>cos</mi> <mo>⁡</mo> <mi>α</mi> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}X_{6}\\Y_{6}\\Z_{6}\end{pmatrix}}={\begin{pmatrix}\cos \alpha &\sin \alpha &0\\-\sin \alpha &\cos \alpha &0\\0&0&1\end{pmatrix}}{\begin{pmatrix}X_{5}\\Y_{5}\\Z_{5}\end{pmatrix}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/18993f476f3733eb7c140114b6c7037165563656" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:41.876ex; height:9.176ex;" alt="{\displaystyle {\begin{pmatrix}X_{6}\\Y_{6}\\Z_{6}\end{pmatrix}}={\begin{pmatrix}\cos \alpha &\sin \alpha &0\\-\sin \alpha &\cos \alpha &0\\0&0&1\end{pmatrix}}{\begin{pmatrix}X_{5}\\Y_{5}\\Z_{5}\end{pmatrix}}.}"></span> </p><p>The axis of symmetry of the gyrocompass is now along the <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 X_{6}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{6}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/489d3e6151450285af53210948fbb0706eb99711" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.979ex; height:2.509ex;" alt="{\displaystyle X_{6}}"></span>-axis. </p> <div class="mw-heading mw-heading3"><h3 id="Last_time-dependent_rotation">Last time-dependent rotation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gyrocompass&action=edit&section=9" title="Edit section: Last time-dependent rotation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The last rotation is a rotation on the axis of symmetry of the gyroscope as in <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{pmatrix}X_{7}\\Y_{7}\\Z_{7}\end{pmatrix}}={\begin{pmatrix}1&0&0\\0&\cos \psi &\sin \psi \\0&-\sin \psi &\cos \psi \end{pmatrix}}{\begin{pmatrix}X_{6}\\Y_{6}\\Z_{6}\end{pmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mi>cos</mi> <mo>⁡</mo> <mi>ψ</mi> </mtd> <mtd> <mi>sin</mi> <mo>⁡</mo> <mi>ψ</mi> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mi>ψ</mi> </mtd> <mtd> <mi>cos</mi> <mo>⁡</mo> <mi>ψ</mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}X_{7}\\Y_{7}\\Z_{7}\end{pmatrix}}={\begin{pmatrix}1&0&0\\0&\cos \psi &\sin \psi \\0&-\sin \psi &\cos \psi \end{pmatrix}}{\begin{pmatrix}X_{6}\\Y_{6}\\Z_{6}\end{pmatrix}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb92af26d01246b2184118b93e365cc10a5c7716" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:41.927ex; height:9.509ex;" alt="{\displaystyle {\begin{pmatrix}X_{7}\\Y_{7}\\Z_{7}\end{pmatrix}}={\begin{pmatrix}1&0&0\\0&\cos \psi &\sin \psi \\0&-\sin \psi &\cos \psi \end{pmatrix}}{\begin{pmatrix}X_{6}\\Y_{6}\\Z_{6}\end{pmatrix}}.}"></span> </p> <div class="mw-heading mw-heading2"><h2 id="Dynamics_of_the_system">Dynamics of the system</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gyrocompass&action=edit&section=10" title="Edit section: Dynamics of the system"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Since the height of the gyroscope's barycenter does not change (and the origin of the coordinate system is located at this same point), its <a href="/wiki/Gravitational_energy" title="Gravitational energy">gravitational potential energy</a> is constant. Therefore its Lagrangian <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 {\mathcal {L}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9027196ecb178d598958555ea01c43157d83597c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.604ex; height:2.176ex;" alt="{\displaystyle {\mathcal {L}}}"></span> corresponds to its kinetic energy <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 K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"></span> only. We have <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {L}}=K={\frac {1}{2}}{\vec {\omega }}^{T}I{\vec {\omega }}+{\frac {1}{2}}M{\vec {v}}_{\rm {CM}}^{2},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mo>=</mo> <mi>K</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ω</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ω</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>M</mi> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">C</mi> <mi mathvariant="normal">M</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}=K={\frac {1}{2}}{\vec {\omega }}^{T}I{\vec {\omega }}+{\frac {1}{2}}M{\vec {v}}_{\rm {CM}}^{2},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8751e942966b850dbabc0731ddbc821dc327691" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:29.347ex; height:5.176ex;" alt="{\displaystyle {\mathcal {L}}=K={\frac {1}{2}}{\vec {\omega }}^{T}I{\vec {\omega }}+{\frac {1}{2}}M{\vec {v}}_{\rm {CM}}^{2},}"></span> where <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/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></span> is the mass of the gyroscope, and <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {v}}_{\rm {CM}}^{2}=\Omega ^{2}R^{2}\sin ^{2}\delta ={\rm {constant}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">C</mi> <mi mathvariant="normal">M</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <msup> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>δ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">t</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {v}}_{\rm {CM}}^{2}=\Omega ^{2}R^{2}\sin ^{2}\delta ={\rm {constant}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ecd3e44c89bd3d80cf370fd7a9a488faebd28b9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.249ex; height:3.343ex;" alt="{\displaystyle {\vec {v}}_{\rm {CM}}^{2}=\Omega ^{2}R^{2}\sin ^{2}\delta ={\rm {constant}}}"></span> is the squared inertial speed of the origin of the coordinates of the final coordinate system (i.e. the center of mass). This constant term does not affect the dynamics of the gyroscope and it can be neglected. On the other hand, the tensor of inertia is given by <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I={\begin{pmatrix}I_{1}&0&0\\0&I_{2}&0\\0&0&I_{2}\end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I={\begin{pmatrix}I_{1}&0&0\\0&I_{2}&0\\0&0&I_{2}\end{pmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/050bbed2eaebac75848fbeab8099c5126dbe6faf" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:19.966ex; height:9.176ex;" alt="{\displaystyle I={\begin{pmatrix}I_{1}&0&0\\0&I_{2}&0\\0&0&I_{2}\end{pmatrix}}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\vec {\omega }}&={\begin{pmatrix}1&0&0\\0&\cos \psi &\sin \psi \\0&-\sin \psi &\cos \psi \end{pmatrix}}{\begin{pmatrix}{\dot {\psi }}\\0\\0\end{pmatrix}}+{\begin{pmatrix}1&0&0\\0&\cos \psi &\sin \psi \\0&-\sin \psi &\cos \psi \end{pmatrix}}{\begin{pmatrix}\cos \alpha &\sin \alpha &0\\-\sin \alpha &\cos \alpha &0\\0&0&1\end{pmatrix}}{\begin{pmatrix}0\\0\\{\dot {\alpha }}\end{pmatrix}}\\&\qquad +{\begin{pmatrix}1&0&0\\0&\cos \psi &\sin \psi \\0&-\sin \psi &\cos \psi \end{pmatrix}}{\begin{pmatrix}\cos \alpha &\sin \alpha &0\\-\sin \alpha &\cos \alpha &0\\0&0&1\end{pmatrix}}{\begin{pmatrix}\cos \delta &0&-\sin \delta \\0&1&0\\\sin \delta &0&\cos \delta \end{pmatrix}}{\begin{pmatrix}\cos \Phi &\sin \Phi &0\\-\sin \Phi &\cos \Phi &0\\0&0&1\end{pmatrix}}{\begin{pmatrix}\cos \Omega t&\sin \Omega t&0\\-\sin \Omega t&\cos \Omega t&0\\0&0&1\end{pmatrix}}{\begin{pmatrix}0\\0\\\Omega \end{pmatrix}}\\&={\begin{pmatrix}{\dot {\psi }}\\0\\0\\\end{pmatrix}}+{\begin{pmatrix}0\\{\dot {\alpha }}\sin \psi \\{\dot {\alpha }}\cos \psi \end{pmatrix}}+{\begin{pmatrix}-\Omega \sin \delta \cos \alpha \\\Omega (\sin \delta \sin \alpha \cos \psi +\cos \delta \sin \psi )\\\Omega (-\sin \delta \sin \alpha \sin \psi +\cos \delta \cos \psi )\end{pmatrix}}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ω</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mi>cos</mi> <mo>⁡</mo> <mi>ψ</mi> </mtd> <mtd> <mi>sin</mi> <mo>⁡</mo> <mi>ψ</mi> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mi>ψ</mi> </mtd> <mtd> <mi>cos</mi> <mo>⁡</mo> <mi>ψ</mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo>˙</mo> </mover> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mi>cos</mi> <mo>⁡</mo> <mi>ψ</mi> </mtd> <mtd> <mi>sin</mi> <mo>⁡</mo> <mi>ψ</mi> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mi>ψ</mi> </mtd> <mtd> <mi>cos</mi> <mo>⁡</mo> <mi>ψ</mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mi>α</mi> </mtd> <mtd> <mi>sin</mi> <mo>⁡</mo> <mi>α</mi> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mi>α</mi> </mtd> <mtd> <mi>cos</mi> <mo>⁡</mo> <mi>α</mi> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>α</mi> <mo>˙</mo> </mover> </mrow> </mrow> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mspace width="2em" /> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mi>cos</mi> <mo>⁡</mo> <mi>ψ</mi> </mtd> <mtd> <mi>sin</mi> <mo>⁡</mo> <mi>ψ</mi> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mi>ψ</mi> </mtd> <mtd> <mi>cos</mi> <mo>⁡</mo> <mi>ψ</mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mi>α</mi> </mtd> <mtd> <mi>sin</mi> <mo>⁡</mo> <mi>α</mi> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mi>α</mi> </mtd> <mtd> <mi>cos</mi> <mo>⁡</mo> <mi>α</mi> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mi>δ</mi> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mi>δ</mi> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mi>sin</mi> <mo>⁡</mo> <mi>δ</mi> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mi>cos</mi> <mo>⁡</mo> <mi>δ</mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mi mathvariant="normal">Φ</mi> </mtd> <mtd> <mi>sin</mi> <mo>⁡</mo> <mi mathvariant="normal">Φ</mi> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mi mathvariant="normal">Φ</mi> </mtd> <mtd> <mi>cos</mi> <mo>⁡</mo> <mi mathvariant="normal">Φ</mi> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mi mathvariant="normal">Ω</mi> <mi>t</mi> </mtd> <mtd> <mi>sin</mi> <mo>⁡</mo> <mi mathvariant="normal">Ω</mi> <mi>t</mi> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mi mathvariant="normal">Ω</mi> <mi>t</mi> </mtd> <mtd> <mi>cos</mi> <mo>⁡</mo> <mi mathvariant="normal">Ω</mi> <mi>t</mi> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mi mathvariant="normal">Ω</mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo>˙</mo> </mover> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>α</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mi>sin</mi> <mo>⁡</mo> <mi>ψ</mi> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>α</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mi>cos</mi> <mo>⁡</mo> <mi>ψ</mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>−</mo> <mi mathvariant="normal">Ω</mi> <mi>sin</mi> <mo>⁡</mo> <mi>δ</mi> <mi>cos</mi> <mo>⁡</mo> <mi>α</mi> </mtd> </mtr> <mtr> <mtd> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">(</mo> <mi>sin</mi> <mo>⁡</mo> <mi>δ</mi> <mi>sin</mi> <mo>⁡</mo> <mi>α</mi> <mi>cos</mi> <mo>⁡</mo> <mi>ψ</mi> <mo>+</mo> <mi>cos</mi> <mo>⁡</mo> <mi>δ</mi> <mi>sin</mi> <mo>⁡</mo> <mi>ψ</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">(</mo> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mi>δ</mi> <mi>sin</mi> <mo>⁡</mo> <mi>α</mi> <mi>sin</mi> <mo>⁡</mo> <mi>ψ</mi> <mo>+</mo> <mi>cos</mi> <mo>⁡</mo> <mi>δ</mi> <mi>cos</mi> <mo>⁡</mo> <mi>ψ</mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\vec {\omega }}&={\begin{pmatrix}1&0&0\\0&\cos \psi &\sin \psi \\0&-\sin \psi &\cos \psi \end{pmatrix}}{\begin{pmatrix}{\dot {\psi }}\\0\\0\end{pmatrix}}+{\begin{pmatrix}1&0&0\\0&\cos \psi &\sin \psi \\0&-\sin \psi &\cos \psi \end{pmatrix}}{\begin{pmatrix}\cos \alpha &\sin \alpha &0\\-\sin \alpha &\cos \alpha &0\\0&0&1\end{pmatrix}}{\begin{pmatrix}0\\0\\{\dot {\alpha }}\end{pmatrix}}\\&\qquad +{\begin{pmatrix}1&0&0\\0&\cos \psi &\sin \psi \\0&-\sin \psi &\cos \psi \end{pmatrix}}{\begin{pmatrix}\cos \alpha &\sin \alpha &0\\-\sin \alpha &\cos \alpha &0\\0&0&1\end{pmatrix}}{\begin{pmatrix}\cos \delta &0&-\sin \delta \\0&1&0\\\sin \delta &0&\cos \delta \end{pmatrix}}{\begin{pmatrix}\cos \Phi &\sin \Phi &0\\-\sin \Phi &\cos \Phi &0\\0&0&1\end{pmatrix}}{\begin{pmatrix}\cos \Omega t&\sin \Omega t&0\\-\sin \Omega t&\cos \Omega t&0\\0&0&1\end{pmatrix}}{\begin{pmatrix}0\\0\\\Omega \end{pmatrix}}\\&={\begin{pmatrix}{\dot {\psi }}\\0\\0\\\end{pmatrix}}+{\begin{pmatrix}0\\{\dot {\alpha }}\sin \psi \\{\dot {\alpha }}\cos \psi \end{pmatrix}}+{\begin{pmatrix}-\Omega \sin \delta \cos \alpha \\\Omega (\sin \delta \sin \alpha \cos \psi +\cos \delta \sin \psi )\\\Omega (-\sin \delta \sin \alpha \sin \psi +\cos \delta \cos \psi )\end{pmatrix}}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d2b1b0c0ca43b48de9a2200b8cf4f9623d243a74" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -14.005ex; width:130.48ex; height:29.176ex;" alt="{\displaystyle {\begin{aligned}{\vec {\omega }}&={\begin{pmatrix}1&0&0\\0&\cos \psi &\sin \psi \\0&-\sin \psi &\cos \psi \end{pmatrix}}{\begin{pmatrix}{\dot {\psi }}\\0\\0\end{pmatrix}}+{\begin{pmatrix}1&0&0\\0&\cos \psi &\sin \psi \\0&-\sin \psi &\cos \psi \end{pmatrix}}{\begin{pmatrix}\cos \alpha &\sin \alpha &0\\-\sin \alpha &\cos \alpha &0\\0&0&1\end{pmatrix}}{\begin{pmatrix}0\\0\\{\dot {\alpha }}\end{pmatrix}}\\&\qquad +{\begin{pmatrix}1&0&0\\0&\cos \psi &\sin \psi \\0&-\sin \psi &\cos \psi \end{pmatrix}}{\begin{pmatrix}\cos \alpha &\sin \alpha &0\\-\sin \alpha &\cos \alpha &0\\0&0&1\end{pmatrix}}{\begin{pmatrix}\cos \delta &0&-\sin \delta \\0&1&0\\\sin \delta &0&\cos \delta \end{pmatrix}}{\begin{pmatrix}\cos \Phi &\sin \Phi &0\\-\sin \Phi &\cos \Phi &0\\0&0&1\end{pmatrix}}{\begin{pmatrix}\cos \Omega t&\sin \Omega t&0\\-\sin \Omega t&\cos \Omega t&0\\0&0&1\end{pmatrix}}{\begin{pmatrix}0\\0\\\Omega \end{pmatrix}}\\&={\begin{pmatrix}{\dot {\psi }}\\0\\0\\\end{pmatrix}}+{\begin{pmatrix}0\\{\dot {\alpha }}\sin \psi \\{\dot {\alpha }}\cos \psi \end{pmatrix}}+{\begin{pmatrix}-\Omega \sin \delta \cos \alpha \\\Omega (\sin \delta \sin \alpha \cos \psi +\cos \delta \sin \psi )\\\Omega (-\sin \delta \sin \alpha \sin \psi +\cos \delta \cos \psi )\end{pmatrix}}\end{aligned}}}"></span> </p><p>Therefore we find <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\mathcal {L}}&={\frac {1}{2}}\left[I_{1}\omega _{1}^{2}+I_{2}\left(\omega _{2}^{2}+\omega _{3}^{2}\right)\right]\\&={\frac {1}{2}}I_{1}\left({\dot {\psi }}-\Omega \sin \delta \cos \alpha \right)^{2}+{\frac {1}{2}}I_{2}\left\{\left[{\dot {\alpha }}\sin \psi +\Omega (\sin \delta \sin \alpha \cos \psi +\cos \delta \sin \psi )\right]^{2}+\left[{\dot {\alpha }}\cos \psi +\Omega (-\sin \delta \sin \alpha \sin \psi +\cos \delta \cos \psi )\right]^{2}\right\}\\&={\frac {1}{2}}I_{1}\left({\dot {\psi }}-\Omega \sin \delta \cos \alpha \right)^{2}+{\frac {1}{2}}I_{2}\left\{{\dot {\alpha }}^{2}+\Omega ^{2}\left(\cos ^{2}\delta +\sin ^{2}\alpha \sin ^{2}\delta \right)+2{\dot {\alpha }}\Omega \cos \delta \right\}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow> <mo>[</mo> <mrow> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msubsup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <msubsup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> <mo>)</mo> </mrow> </mrow> <mo>]</mo> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo>−</mo> <mi mathvariant="normal">Ω</mi> <mi>sin</mi> <mo>⁡</mo> <mi>δ</mi> <mi>cos</mi> <mo>⁡</mo> <mi>α</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mrow> <mo>{</mo> <mrow> <msup> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>α</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mi>sin</mi> <mo>⁡</mo> <mi>ψ</mi> <mo>+</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">(</mo> <mi>sin</mi> <mo>⁡</mo> <mi>δ</mi> <mi>sin</mi> <mo>⁡</mo> <mi>α</mi> <mi>cos</mi> <mo>⁡</mo> <mi>ψ</mi> <mo>+</mo> <mi>cos</mi> <mo>⁡</mo> <mi>δ</mi> <mi>sin</mi> <mo>⁡</mo> <mi>ψ</mi> <mo stretchy="false">)</mo> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>α</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mi>cos</mi> <mo>⁡</mo> <mi>ψ</mi> <mo>+</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">(</mo> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mi>δ</mi> <mi>sin</mi> <mo>⁡</mo> <mi>α</mi> <mi>sin</mi> <mo>⁡</mo> <mi>ψ</mi> <mo>+</mo> <mi>cos</mi> <mo>⁡</mo> <mi>δ</mi> <mi>cos</mi> <mo>⁡</mo> <mi>ψ</mi> <mo stretchy="false">)</mo> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>}</mo> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo>−</mo> <mi mathvariant="normal">Ω</mi> <mi>sin</mi> <mo>⁡</mo> <mi>δ</mi> <mi>cos</mi> <mo>⁡</mo> <mi>α</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mrow> <mo>{</mo> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>α</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>δ</mi> <mo>+</mo> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>α</mi> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>δ</mi> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>α</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mi mathvariant="normal">Ω</mi> <mi>cos</mi> <mo>⁡</mo> <mi>δ</mi> </mrow> <mo>}</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\mathcal {L}}&={\frac {1}{2}}\left[I_{1}\omega _{1}^{2}+I_{2}\left(\omega _{2}^{2}+\omega _{3}^{2}\right)\right]\\&={\frac {1}{2}}I_{1}\left({\dot {\psi }}-\Omega \sin \delta \cos \alpha \right)^{2}+{\frac {1}{2}}I_{2}\left\{\left[{\dot {\alpha }}\sin \psi +\Omega (\sin \delta \sin \alpha \cos \psi +\cos \delta \sin \psi )\right]^{2}+\left[{\dot {\alpha }}\cos \psi +\Omega (-\sin \delta \sin \alpha \sin \psi +\cos \delta \cos \psi )\right]^{2}\right\}\\&={\frac {1}{2}}I_{1}\left({\dot {\psi }}-\Omega \sin \delta \cos \alpha \right)^{2}+{\frac {1}{2}}I_{2}\left\{{\dot {\alpha }}^{2}+\Omega ^{2}\left(\cos ^{2}\delta +\sin ^{2}\alpha \sin ^{2}\delta \right)+2{\dot {\alpha }}\Omega \cos \delta \right\}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99ea7fb4d7a5a7959dd44adbf50a88c24e3c5f8c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.205ex; margin-bottom: -0.3ex; width:130.508ex; height:16.176ex;" alt="{\displaystyle {\begin{aligned}{\mathcal {L}}&={\frac {1}{2}}\left[I_{1}\omega _{1}^{2}+I_{2}\left(\omega _{2}^{2}+\omega _{3}^{2}\right)\right]\\&={\frac {1}{2}}I_{1}\left({\dot {\psi }}-\Omega \sin \delta \cos \alpha \right)^{2}+{\frac {1}{2}}I_{2}\left\{\left[{\dot {\alpha }}\sin \psi +\Omega (\sin \delta \sin \alpha \cos \psi +\cos \delta \sin \psi )\right]^{2}+\left[{\dot {\alpha }}\cos \psi +\Omega (-\sin \delta \sin \alpha \sin \psi +\cos \delta \cos \psi )\right]^{2}\right\}\\&={\frac {1}{2}}I_{1}\left({\dot {\psi }}-\Omega \sin \delta \cos \alpha \right)^{2}+{\frac {1}{2}}I_{2}\left\{{\dot {\alpha }}^{2}+\Omega ^{2}\left(\cos ^{2}\delta +\sin ^{2}\alpha \sin ^{2}\delta \right)+2{\dot {\alpha }}\Omega \cos \delta \right\}\end{aligned}}}"></span> </p><p>The Lagrangian can be rewritten as <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {L}}={\mathcal {L}}_{1}+{\frac {1}{2}}I_{2}\Omega ^{2}\cos ^{2}\delta +{\frac {d}{dt}}(I_{2}\alpha \Omega \cos \delta ),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msup> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>δ</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mi>α</mi> <mi mathvariant="normal">Ω</mi> <mi>cos</mi> <mo>⁡</mo> <mi>δ</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}={\mathcal {L}}_{1}+{\frac {1}{2}}I_{2}\Omega ^{2}\cos ^{2}\delta +{\frac {d}{dt}}(I_{2}\alpha \Omega \cos \delta ),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/66eb0396af72a648152c36189b9a4b2495907c6f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:41.362ex; height:5.509ex;" alt="{\displaystyle {\mathcal {L}}={\mathcal {L}}_{1}+{\frac {1}{2}}I_{2}\Omega ^{2}\cos ^{2}\delta +{\frac {d}{dt}}(I_{2}\alpha \Omega \cos \delta ),}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {L}}_{1}={\frac {1}{2}}I_{1}\left({\dot {\psi }}-\Omega \sin \delta \cos \alpha \right)^{2}+{\frac {1}{2}}I_{2}\left({\dot {\alpha }}^{2}+\Omega ^{2}\sin ^{2}\alpha \sin ^{2}\delta \right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo>−</mo> <mi mathvariant="normal">Ω</mi> <mi>sin</mi> <mo>⁡</mo> <mi>δ</mi> <mi>cos</mi> <mo>⁡</mo> <mi>α</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>α</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>α</mi> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>δ</mi> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}_{1}={\frac {1}{2}}I_{1}\left({\dot {\psi }}-\Omega \sin \delta \cos \alpha \right)^{2}+{\frac {1}{2}}I_{2}\left({\dot {\alpha }}^{2}+\Omega ^{2}\sin ^{2}\alpha \sin ^{2}\delta \right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cbec529aee03f65dae8b32aac77987b605e616c7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:59.28ex; height:5.176ex;" alt="{\displaystyle {\mathcal {L}}_{1}={\frac {1}{2}}I_{1}\left({\dot {\psi }}-\Omega \sin \delta \cos \alpha \right)^{2}+{\frac {1}{2}}I_{2}\left({\dot {\alpha }}^{2}+\Omega ^{2}\sin ^{2}\alpha \sin ^{2}\delta \right)}"></span> is the part of the Lagrangian responsible for the dynamics of the system. Then, 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 \partial {\mathcal {L}}_{1}/\partial \psi =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi mathvariant="normal">∂</mi> <mi>ψ</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial {\mathcal {L}}_{1}/\partial \psi =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6e052148177e43bf533f5e7f88aef1536180969" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.231ex; height:2.843ex;" alt="{\displaystyle \partial {\mathcal {L}}_{1}/\partial \psi =0}"></span>, we find <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{x}\equiv {\frac {\partial {\mathcal {L}}_{1}}{\partial {\dot {\psi }}}}=I_{1}\left({\dot {\psi }}-\Omega \sin \delta \cos \alpha \right)=\mathrm {constant} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>≡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo>˙</mo> </mover> </mrow> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo>−</mo> <mi mathvariant="normal">Ω</mi> <mi>sin</mi> <mo>⁡</mo> <mi>δ</mi> <mi>cos</mi> <mo>⁡</mo> <mi>α</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">t</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{x}\equiv {\frac {\partial {\mathcal {L}}_{1}}{\partial {\dot {\psi }}}}=I_{1}\left({\dot {\psi }}-\Omega \sin \delta \cos \alpha \right)=\mathrm {constant} .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1804b54c6aee0dfbd2707e17d5eee88a1906a85" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:47.583ex; height:6.509ex;" alt="{\displaystyle L_{x}\equiv {\frac {\partial {\mathcal {L}}_{1}}{\partial {\dot {\psi }}}}=I_{1}\left({\dot {\psi }}-\Omega \sin \delta \cos \alpha \right)=\mathrm {constant} .}"></span> </p><p>Since the angular momentum <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 {\vec {L}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>L</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {L}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0c139fc28d6ca3873993892f44e7331e5ff18fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.843ex;" alt="{\displaystyle {\vec {L}}}"></span> of the gyrocompass is given by <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 {\vec {L}}=I{\vec {\omega }},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>L</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ω</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {L}}=I{\vec {\omega }},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47624ff3dd61f317c0d6e12e9e37d69544617f55" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.946ex; height:3.176ex;" alt="{\displaystyle {\vec {L}}=I{\vec {\omega }},}"></span> we see that the constant <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 L_{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ebc06e436594a115abefe1f78041e044bc4946f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.755ex; height:2.509ex;" alt="{\displaystyle L_{x}}"></span> is the component of the angular momentum about the axis of symmetry. Furthermore, we find the equation of motion for the variable <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 \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span> as <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {d}{dt}}\left({\frac {\partial {\mathcal {L}}_{1}}{\partial {\dot {\alpha }}}}\right)={\frac {\partial {\mathcal {L}}_{1}}{\partial \alpha }},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>α</mi> <mo>˙</mo> </mover> </mrow> </mrow> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>α</mi> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d}{dt}}\left({\frac {\partial {\mathcal {L}}_{1}}{\partial {\dot {\alpha }}}}\right)={\frac {\partial {\mathcal {L}}_{1}}{\partial \alpha }},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2468420b3f7d63e127ca723b88283d2d0d03d0ba" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:20.07ex; height:6.176ex;" alt="{\displaystyle {\frac {d}{dt}}\left({\frac {\partial {\mathcal {L}}_{1}}{\partial {\dot {\alpha }}}}\right)={\frac {\partial {\mathcal {L}}_{1}}{\partial \alpha }},}"></span> or <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}I_{2}{\ddot {\alpha }}&=I_{1}\Omega \left({\dot {\psi }}-\Omega \sin \delta \cos \alpha \right)\sin \delta \sin \alpha +{\frac {1}{2}}I_{2}\Omega ^{2}\sin ^{2}\delta \sin 2\alpha \\&=L_{x}\Omega \sin \delta \sin \alpha +{\frac {1}{2}}I_{2}\Omega ^{2}\sin ^{2}\delta \sin 2\alpha \end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>α</mi> <mo>¨</mo> </mover> </mrow> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi mathvariant="normal">Ω</mi> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo>−</mo> <mi mathvariant="normal">Ω</mi> <mi>sin</mi> <mo>⁡</mo> <mi>δ</mi> <mi>cos</mi> <mo>⁡</mo> <mi>α</mi> </mrow> <mo>)</mo> </mrow> <mi>sin</mi> <mo>⁡</mo> <mi>δ</mi> <mi>sin</mi> <mo>⁡</mo> <mi>α</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msup> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>δ</mi> <mi>sin</mi> <mo>⁡</mo> <mn>2</mn> <mi>α</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mi mathvariant="normal">Ω</mi> <mi>sin</mi> <mo>⁡</mo> <mi>δ</mi> <mi>sin</mi> <mo>⁡</mo> <mi>α</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msup> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>δ</mi> <mi>sin</mi> <mo>⁡</mo> <mn>2</mn> <mi>α</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}I_{2}{\ddot {\alpha }}&=I_{1}\Omega \left({\dot {\psi }}-\Omega \sin \delta \cos \alpha \right)\sin \delta \sin \alpha +{\frac {1}{2}}I_{2}\Omega ^{2}\sin ^{2}\delta \sin 2\alpha \\&=L_{x}\Omega \sin \delta \sin \alpha +{\frac {1}{2}}I_{2}\Omega ^{2}\sin ^{2}\delta \sin 2\alpha \end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a1461f4b8c8f504f62e4d1394d231e01dfea3de" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.485ex; margin-bottom: -0.187ex; width:61.956ex; height:10.509ex;" alt="{\displaystyle {\begin{aligned}I_{2}{\ddot {\alpha }}&=I_{1}\Omega \left({\dot {\psi }}-\Omega \sin \delta \cos \alpha \right)\sin \delta \sin \alpha +{\frac {1}{2}}I_{2}\Omega ^{2}\sin ^{2}\delta \sin 2\alpha \\&=L_{x}\Omega \sin \delta \sin \alpha +{\frac {1}{2}}I_{2}\Omega ^{2}\sin ^{2}\delta \sin 2\alpha \end{aligned}}}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Particular_case:_the_poles">Particular case: the poles</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gyrocompass&action=edit&section=11" title="Edit section: Particular case: the poles"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>At the poles we find <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 \sin \delta =0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mi>δ</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin \delta =0,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/612835461d89a24c551a6c83bc4a32a214349a93" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.199ex; height:2.676ex;" alt="{\displaystyle \sin \delta =0,}"></span> and the equations of motion become <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}L_{x}&=I_{1}{\dot {\psi }}=\mathrm {constant} \\I_{2}{\ddot {\alpha }}&=0\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">t</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>α</mi> <mo>¨</mo> </mover> </mrow> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>0</mn> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}L_{x}&=I_{1}{\dot {\psi }}=\mathrm {constant} \\I_{2}{\ddot {\alpha }}&=0\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90214ed23ca47b8dccc584e511b69dc2599e94bb" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:22.854ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}L_{x}&=I_{1}{\dot {\psi }}=\mathrm {constant} \\I_{2}{\ddot {\alpha }}&=0\end{aligned}}}"></span> </p><p>This simple solution implies that the gyroscope is uniformly rotating with constant angular velocity in both the vertical and symmetrical axis. </p> <div class="mw-heading mw-heading3"><h3 id="The_general_and_physically_relevant_case">The general and physically relevant case</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gyrocompass&action=edit&section=12" title="Edit section: The general and physically relevant case"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Let us suppose now that <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 \sin \delta \neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mi>δ</mi> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin \delta \neq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3860f3622d6df20c741b736139b40923e16e8137" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.552ex; height:2.843ex;" alt="{\displaystyle \sin \delta \neq 0}"></span> and that <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 \alpha \approx 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>≈</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha \approx 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6eedf022b3cf815dd4c4ba3e15a94d218ad42c1c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.749ex; height:2.176ex;" alt="{\displaystyle \alpha \approx 0}"></span>, that is the axis of the gyroscope is approximately along the north-south line, and let us find the parameter space (if it exists) for which the system admits stable small oscillations about this same line. If this situation occurs, the gyroscope will always be approximately aligned along the north-south line, giving direction. In this case we find <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}L_{x}&\approx I_{1}\left({\dot {\psi }}-\Omega \sin \delta \right)\\I_{2}{\ddot {\alpha }}&\approx \left(L_{x}\Omega \sin \delta +I_{2}\Omega ^{2}\sin ^{2}\delta \right)\alpha \end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>≈</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo>−</mo> <mi mathvariant="normal">Ω</mi> <mi>sin</mi> <mo>⁡</mo> <mi>δ</mi> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>α</mi> <mo>¨</mo> </mover> </mrow> </mrow> </mtd> <mtd> <mi></mi> <mo>≈</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mi mathvariant="normal">Ω</mi> <mi>sin</mi> <mo>⁡</mo> <mi>δ</mi> <mo>+</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msup> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>δ</mi> </mrow> <mo>)</mo> </mrow> <mi>α</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}L_{x}&\approx I_{1}\left({\dot {\psi }}-\Omega \sin \delta \right)\\I_{2}{\ddot {\alpha }}&\approx \left(L_{x}\Omega \sin \delta +I_{2}\Omega ^{2}\sin ^{2}\delta \right)\alpha \end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7500f5ac2e2a31f2fbd482c5e1976b340a4e8efe" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:33.914ex; height:8.176ex;" alt="{\displaystyle {\begin{aligned}L_{x}&\approx I_{1}\left({\dot {\psi }}-\Omega \sin \delta \right)\\I_{2}{\ddot {\alpha }}&\approx \left(L_{x}\Omega \sin \delta +I_{2}\Omega ^{2}\sin ^{2}\delta \right)\alpha \end{aligned}}}"></span> </p><p>Consider the case that <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{x}<0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo><</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{x}<0,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52d03fb4317bc2bb47ee8817941a13644fa614d0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.663ex; height:2.509ex;" alt="{\displaystyle L_{x}<0,}"></span> and, further, we allow for fast gyro-rotations, that is <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left|{\dot {\psi }}\right|\gg \Omega .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>|</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo>|</mo> </mrow> <mo>≫</mo> <mi mathvariant="normal">Ω</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left|{\dot {\psi }}\right|\gg \Omega .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/436f82d2ec066177df3f28f6459ee0e11a626212" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:8.828ex; height:3.509ex;" alt="{\displaystyle \left|{\dot {\psi }}\right|\gg \Omega .}"></span> </p><p>Therefore, for fast spinning rotations, <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 L_{x}<0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo><</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{x}<0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad90a2921b89963757ba5b25a1510a4f80051b85" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.016ex; height:2.509ex;" alt="{\displaystyle L_{x}<0}"></span> implies <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 {\dot {\psi }}<0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo><</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {\psi }}<0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a31dab440ff4ef4bad4d589d82b98327529aac5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.503ex; height:3.009ex;" alt="{\displaystyle {\dot {\psi }}<0.}"></span> In this case, the equations of motion further simplify to <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}L_{x}&\approx -I_{1}\left|{\dot {\psi }}\right|\approx \mathrm {constant} \\I_{2}{\ddot {\alpha }}&\approx -I_{1}\left|{\dot {\psi }}\right|\Omega \sin \delta \alpha \end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>≈</mo> <mo>−</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow> <mo>|</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo>|</mo> </mrow> <mo>≈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">t</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>α</mi> <mo>¨</mo> </mover> </mrow> </mrow> </mtd> <mtd> <mi></mi> <mo>≈</mo> <mo>−</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow> <mo>|</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo>|</mo> </mrow> <mi mathvariant="normal">Ω</mi> <mi>sin</mi> <mo>⁡</mo> <mi>δ</mi> <mi>α</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}L_{x}&\approx -I_{1}\left|{\dot {\psi }}\right|\approx \mathrm {constant} \\I_{2}{\ddot {\alpha }}&\approx -I_{1}\left|{\dot {\psi }}\right|\Omega \sin \delta \alpha \end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72e2535a4ede5d783652824977696adfb9dae5ef" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:26.343ex; height:7.509ex;" alt="{\displaystyle {\begin{aligned}L_{x}&\approx -I_{1}\left|{\dot {\psi }}\right|\approx \mathrm {constant} \\I_{2}{\ddot {\alpha }}&\approx -I_{1}\left|{\dot {\psi }}\right|\Omega \sin \delta \alpha \end{aligned}}}"></span> </p><p>Therefore we find small oscillations about the north-south line, as <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 \alpha \approx A\sin({\tilde {\omega }}t+B)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>≈</mo> <mi>A</mi> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ω</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mi>t</mi> <mo>+</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha \approx A\sin({\tilde {\omega }}t+B)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdc5b62d3a8715cbff71b7c8bb44d998acc2c916" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.271ex; height:2.843ex;" alt="{\displaystyle \alpha \approx A\sin({\tilde {\omega }}t+B)}"></span>, where the angular velocity of this harmonic motion of the axis of symmetry of the gyrocompass about the north-south line is given by <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {\omega }}={\sqrt {\frac {I_{1}\sin \delta }{I_{2}}}}{\sqrt {\left|{\dot {\psi }}\right|\Omega }},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ω</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>sin</mi> <mo>⁡</mo> <mi>δ</mi> </mrow> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mfrac> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow> <mo>|</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo>|</mo> </mrow> <mi mathvariant="normal">Ω</mi> </msqrt> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tilde {\omega }}={\sqrt {\frac {I_{1}\sin \delta }{I_{2}}}}{\sqrt {\left|{\dot {\psi }}\right|\Omega }},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/501027420402c7fada8ef527a336106804046353" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:22.385ex; height:7.676ex;" alt="{\displaystyle {\tilde {\omega }}={\sqrt {\frac {I_{1}\sin \delta }{I_{2}}}}{\sqrt {\left|{\dot {\psi }}\right|\Omega }},}"></span> which corresponds to a period for the oscillations given by <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T={\frac {2\pi }{\sqrt {\left|{\dot {\psi }}\right|\Omega }}}{\sqrt {\frac {I_{2}}{I_{1}\sin \delta }}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> </mrow> <msqrt> <mrow> <mo>|</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo>|</mo> </mrow> <mi mathvariant="normal">Ω</mi> </msqrt> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mrow> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>sin</mi> <mo>⁡</mo> <mi>δ</mi> </mrow> </mfrac> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T={\frac {2\pi }{\sqrt {\left|{\dot {\psi }}\right|\Omega }}}{\sqrt {\frac {I_{2}}{I_{1}\sin \delta }}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec6606a70423ce64f40b385b903b153e4327825e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:23.412ex; height:9.009ex;" alt="{\displaystyle T={\frac {2\pi }{\sqrt {\left|{\dot {\psi }}\right|\Omega }}}{\sqrt {\frac {I_{2}}{I_{1}\sin \delta }}}.}"></span> </p><p>Therefore <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 {\tilde {\omega }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ω</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tilde {\omega }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e60cbeebb31255ddce04c7526d2631bfd49fb8bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:2.176ex;" alt="{\displaystyle {\tilde {\omega }}}"></span> is proportional to the geometric mean of the Earth and spinning angular velocities. In order to have small oscillations we have required <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 {\dot {\psi }}<0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo><</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {\psi }}<0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/118428a80f9d251b622b8509b65bec638f1eda3c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.857ex; height:3.009ex;" alt="{\displaystyle {\dot {\psi }}<0}"></span>, so that the North is located along the right-hand-rule direction of the spinning axis, that is along the negative direction of the <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 X_{7}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{7}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c7d9af33b651d172949cee5952f57dbeb662eaf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.979ex; height:2.509ex;" alt="{\displaystyle X_{7}}"></span>-axis, the axis of symmetry. As a side result, on measuring <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/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span> (and knowing <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 {\dot {\psi }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo>˙</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {\psi }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c6912355df049856322d8d4d631156fa10f3a37" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.596ex; height:3.009ex;" alt="{\displaystyle {\dot {\psi }}}"></span>), one can deduce the local co-latitude <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 \delta .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d32ef13c6395c29916331212d0948171a62ac216" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.695ex; height:2.343ex;" alt="{\displaystyle \delta .}"></span> </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=Gyrocompass&action=edit&section=13" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Acronyms_and_abbreviations_in_avionics" class="mw-redirect" title="Acronyms and abbreviations in avionics">Acronyms and abbreviations in avionics</a></li> <li><a href="/wiki/Heading_indicator" title="Heading indicator">Heading indicator</a>, also known as direction indicator, a lightweight gyroscope (not a gyrocompass) used on aircraft</li> <li><a href="/wiki/HRG_gyrocompass" title="HRG gyrocompass">HRG gyrocompass</a></li> <li><a href="/wiki/Fluxgate_compass" title="Fluxgate compass">Fluxgate compass</a></li> <li><a href="/wiki/Fibre_optic_gyrocompass" title="Fibre optic gyrocompass">Fibre optic gyrocompass</a></li> <li><a href="/wiki/Inertial_navigation_system" title="Inertial navigation system">Inertial navigation system</a>, a more complex system that also incorporates accelerometers</li> <li><a href="/wiki/Schuler_tuning" title="Schuler tuning">Schuler tuning</a></li> <li><a href="/wiki/Binnacle" title="Binnacle">Binnacle</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gyrocompass&action=edit&section=14" title="Edit section: Notes"><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-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text">Although the effect is not visible in the specific case when the gyroscope's axis is precisely parallel to the Earth's rotational axis.</span> </li> </ol></div></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=Gyrocompass&action=edit&section=15" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-JournalOfNavigation2016-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-JournalOfNavigation2016_1-0">^</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="CITEREFGade2016" class="citation journal cs1">Gade, Kenneth (2016). <a rel="nofollow" class="external text" href="http://www.navlab.net/Publications/The_Seven_Ways_to_Find_Heading.pdf">"The Seven Ways to Find Heading"</a> <span class="cs1-format">(PDF)</span>. <i>The Journal of Navigation</i>. <b>69</b> (5). Cambridge University Press: 955–970. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1017%2FS0373463316000096">10.1017/S0373463316000096</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:53587934">53587934</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Journal+of+Navigation&rft.atitle=The+Seven+Ways+to+Find+Heading&rft.volume=69&rft.issue=5&rft.pages=955-970&rft.date=2016&rft_id=info%3Adoi%2F10.1017%2FS0373463316000096&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A53587934%23id-name%3DS2CID&rft.aulast=Gade&rft.aufirst=Kenneth&rft_id=http%3A%2F%2Fwww.navlab.net%2FPublications%2FThe_Seven_Ways_to_Find_Heading.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGyrocompass" class="Z3988"></span></span> </li> <li id="cite_note-an-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-an_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-an_2-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-an_2-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-an_2-3"><sup><i><b>d</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFElliott-Laboratories2003" class="citation book cs1">Elliott-Laboratories (2003). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=VJ3WCpegQxwC"><i>The Anschutz Gyro-Compass and Gyroscope Engineering</i></a>. Watchmaker. pp. 7–24. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-929148-12-7" title="Special:BookSources/978-1-929148-12-7"><bdi>978-1-929148-12-7</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20170304174125/https://books.google.com/books?id=VJ3WCpegQxwC">Archived</a> from the original on 2017-03-04.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Anschutz+Gyro-Compass+and+Gyroscope+Engineering&rft.pages=7-24&rft.pub=Watchmaker&rft.date=2003&rft.isbn=978-1-929148-12-7&rft.au=Elliott-Laboratories&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DVJ3WCpegQxwC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGyrocompass" class="Z3988"></span></span> </li> <li id="cite_note-l-3"><span class="mw-cite-backlink">^ <a href="#cite_ref-l_3-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-l_3-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-l_3-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-l_3-3"><sup><i><b>d</b></i></sup></a> <a href="#cite_ref-l_3-4"><sup><i><b>e</b></i></sup></a> <a href="#cite_ref-l_3-5"><sup><i><b>f</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTime_Inc.1943" class="citation magazine cs1">Time Inc. (Mar 15, 1943). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=YlEEAAAAMBAJ&pg=PA82">"The gyroscope pilots ships & planes"</a>. <i>Life</i>. pp. 80–83. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20170227154326/https://books.google.com/books?id=YlEEAAAAMBAJ&pg=PA82">Archived</a> from the original on 2017-02-27.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Life&rft.atitle=The+gyroscope+pilots+ships+%26+planes&rft.pages=80-83&rft.date=1943-03-15&rft.au=Time+Inc.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DYlEEAAAAMBAJ%26pg%3DPA82&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGyrocompass" class="Z3988"></span></span> </li> <li id="cite_note-SafeNavWatch-4"><span class="mw-cite-backlink">^ <a href="#cite_ref-SafeNavWatch_4-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-SafeNavWatch_4-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation book cs1"><i>Safe Nav Watch</i>. Edinburgh: <a href="/wiki/Witherby_Publishing_Group" title="Witherby Publishing Group">Witherby Publishing Group</a>. 2023. pp. 26–27. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9781914993466" title="Special:BookSources/9781914993466"><bdi>9781914993466</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Safe+Nav+Watch&rft.place=Edinburgh&rft.pages=26-27&rft.pub=Witherby+Publishing+Group&rft.date=2023&rft.isbn=9781914993466&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGyrocompass" class="Z3988"></span></span> </li> <li id="cite_note-hee-5"><span class="mw-cite-backlink">^ <a href="#cite_ref-hee_5-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-hee_5-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-hee_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="CITEREFGalison1987" class="citation book cs1">Galison, Peter (1987). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=DN-9m2jSo8YC&pg=PA37"><i>How experiments end</i></a>. University of Chicago Press. pp. 34–37. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-226-27915-2" title="Special:BookSources/978-0-226-27915-2"><bdi>978-0-226-27915-2</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20120302022112/http://books.google.com/books?id=DN-9m2jSo8YC&pg=PA37">Archived</a> from the original on 2012-03-02.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=How+experiments+end&rft.pages=34-37&rft.pub=University+of+Chicago+Press&rft.date=1987&rft.isbn=978-0-226-27915-2&rft.aulast=Galison&rft.aufirst=Peter&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DDN-9m2jSo8YC%26pg%3DPA37&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGyrocompass" class="Z3988"></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</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="http://downloads.german-pavilion.com/downloads/pdf/exhibitor_24199.pdf">"Archived copy"</a> <span class="cs1-format">(PDF)</span>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20150629014757/http://downloads.german-pavilion.com/downloads/pdf/exhibitor_24199.pdf">Archived</a> <span class="cs1-format">(PDF)</span> from the original on 2015-06-29<span class="reference-accessdate">. Retrieved <span class="nowrap">2012-02-19</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Archived+copy&rft_id=http%3A%2F%2Fdownloads.german-pavilion.com%2Fdownloads%2Fpdf%2Fexhibitor_24199.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGyrocompass" class="Z3988"></span><span class="cs1-maint citation-comment"><code class="cs1-code">{{<a href="/wiki/Template:Cite_web" title="Template:Cite web">cite web</a>}}</code>: CS1 maint: archived copy as title (<a href="/wiki/Category:CS1_maint:_archived_copy_as_title" title="Category:CS1 maint: archived copy as title">link</a>)</span> Standard 22 Anschütz Gyro Compass [sic] System: Gyro Compass [sic] Technology [sic] for over than [sic] 100 years</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"><a rel="nofollow" class="external text" href="https://www.ihk-schleswig-holstein.de/news/ihk_kiel12955/Hermann-Anschuetz-Kaempfe/3405026">Chambers of Commerce and Industry in Schleswig-Holstein</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20170222061044/https://www.ihk-schleswig-holstein.de/news/ihk_kiel12955/Hermann-Anschuetz-Kaempfe/3405026">Archived</a> 2017-02-22 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a> Retrieved on February 22, 2017.</span> </li> <li id="cite_note-maritime.org-8"><span class="mw-cite-backlink">^ <a href="#cite_ref-maritime.org_8-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-maritime.org_8-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://www.maritime.org/fleetsub/elect/chap17.htm">Gyrocompass, Auxiliary Gyrocompass, and Dead Reckoning Analyzing Indicator and Tracer Systems</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20130601030840/http://www.maritime.org/fleetsub/elect/chap17.htm">Archived</a> 2013-06-01 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>, San Francisco Maritime National Park Association.</span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://www.weems-plath.com/weems-and-plath-story.php">The Invention of Precision Navigational Instruments for Air and Sea Navigation</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20110718024746/http://www.weems-plath.com/weems-and-plath-story.php">Archived</a> 2011-07-18 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>, Weems & Plath.</span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCollinson2003" class="citation cs2">Collinson, R. P. G. (2003), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=rbKkojYNjecC&pg=RA1-PA293"><i>Introduction to avionics systems</i></a>, Springer, p. 293, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4020-7278-9" title="Special:BookSources/978-1-4020-7278-9"><bdi>978-1-4020-7278-9</bdi></a>, <a rel="nofollow" class="external text" href="https://web.archive.org/web/20140707074016/http://books.google.com/books?id=rbKkojYNjecC&pg=RA1-PA293&lpg=RA1-PA293">archived</a> from the original on 2014-07-07</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+avionics+systems&rft.pages=293&rft.pub=Springer&rft.date=2003&rft.isbn=978-1-4020-7278-9&rft.aulast=Collinson&rft.aufirst=R.+P.+G.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DrbKkojYNjecC%26pg%3DRA1-PA293&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGyrocompass" class="Z3988"></span></span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text">NASA <a rel="nofollow" class="external text" href="https://asrs.arc.nasa.gov/publications/callback/cb_304.htm">NASA Callback: Heading for Trouble</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20110716165805/http://asrs.arc.nasa.gov/publications/callback/cb_304.htm">Archived</a> 2011-07-16 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>, NASA Callback Safety Bulletin website, December 2005, No. 305. Retrieved August 29, 2010.</span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text">Bowditch, Nathaniel. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=M54M8qjxLQMC&pg=PA93">American Practical Navigator</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20170307022825/https://books.google.com/books?id=M54M8qjxLQMC&pg=PA93">Archived</a> 2017-03-07 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>, Paradise Cay Publications, 2002, pp.93-94, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-939837-54-0" title="Special:BookSources/978-0-939837-54-0">978-0-939837-54-0</a>.</span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://www.navis.gr/navaids/gyro.htm">Gyrocompass: Steaming Error</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20081222083129/http://www.navis.gr/navaids/gyro.htm">Archived</a> 2008-12-22 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>, Navis. Accessed 15 December 2008.</span> </li> <li id="cite_note-House-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-House_15-0">^</a></b></span> <span class="reference-text">Seamanship Techniques:Shipboard and Marine Operations, D. J. House, Butterworth-Heinemann, 2004, p. 341</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Bibliography">Bibliography</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gyrocompass&action=edit&section=16" title="Edit section: Bibliography"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span><a rel="nofollow" class="external text" href="https://patents.google.com/patent/US1279471">U.S. patent 1,279,471</a></span> : "Gyroscopic compass" by <a href="/wiki/Elmer_Ambrose_Sperry" title="Elmer Ambrose Sperry">E. A. Sperry</a>, filed June, 1911; issued September, 1918</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTrainer2008" class="citation journal cs1">Trainer, Matthew (2008). "Albert Einstein's expert opinions on the Sperry vs. Anschütz gyrocompass patent dispute". <i>World Patent Information</i>. <b>30</b> (4): 320–325. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2008WPatI..30..320T">2008WPatI..30..320T</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.wpi.2008.05.003">10.1016/j.wpi.2008.05.003</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=World+Patent+Information&rft.atitle=Albert+Einstein%27s+expert+opinions+on+the+Sperry+vs.+Ansch%C3%BCtz+gyrocompass+patent+dispute&rft.volume=30&rft.issue=4&rft.pages=320-325&rft.date=2008&rft_id=info%3Adoi%2F10.1016%2Fj.wpi.2008.05.003&rft_id=info%3Abibcode%2F2008WPatI..30..320T&rft.aulast=Trainer&rft.aufirst=Matthew&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGyrocompass" class="Z3988"></span></li></ul> <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=Gyrocompass&action=edit&section=17" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="https://feynmanlectures.caltech.edu/TIPS_04.html#Ch4-S5">Feynman's Tips on Physics - The gyrocompass</a></li> <li><a rel="nofollow" class="external text" href="https://fi.edu/en/news/case-files-elmer-sperry-gyroscopic-compass">Case Files: Elmer A. Sperry</a> at the <a href="/wiki/Franklin_Institute" title="Franklin Institute">Franklin Institute</a> contains records concerning his 1914 Franklin Award for the gyroscopic compass</li> <li><a rel="nofollow" class="external text" href="https://www.britannica.com/technology/gyrocompass">Britannica - Gyrocompass </a></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist 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Gyrocompass - Wikipedia