Harmoninen värähtelijä

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class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="Sivusto"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="Sisällysluettelo" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">Sisällysluettelo</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">siirrä sivupalkkiin</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">piilota</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">Johdanto</div> </a> </li> <li id="toc-Vaimenematon_harmoninen_värähtelijä" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Vaimenematon_harmoninen_värähtelijä"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Vaimenematon harmoninen värähtelijä</span> </div> </a> <ul id="toc-Vaimenematon_harmoninen_värähtelijä-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Pakotettu_harmoninen_värähtelijä" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Pakotettu_harmoninen_värähtelijä"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Pakotettu harmoninen värähtelijä</span> </div> </a> <ul id="toc-Pakotettu_harmoninen_värähtelijä-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Vaimennettu_harmoninen_värähtelijä" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Vaimennettu_harmoninen_värähtelijä"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Vaimennettu harmoninen värähtelijä</span> </div> </a> <button aria-controls="toc-Vaimennettu_harmoninen_värähtelijä-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>Vaihda alaosio Vaimennettu harmoninen värähtelijä</span> </button> <ul id="toc-Vaimennettu_harmoninen_värähtelijä-sublist" class="vector-toc-list"> <li id="toc-Ylivaimennus" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Ylivaimennus"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Ylivaimennus</span> </div> </a> <ul id="toc-Ylivaimennus-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Alivaimennus" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Alivaimennus"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Alivaimennus</span> </div> </a> <ul id="toc-Alivaimennus-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Kriittinen_vaimennus" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Kriittinen_vaimennus"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Kriittinen vaimennus</span> </div> </a> <ul id="toc-Kriittinen_vaimennus-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Vaimennettu_ja_pakotettu_harmoninen_värähtelijä" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Vaimennettu_ja_pakotettu_harmoninen_värähtelijä"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Vaimennettu ja pakotettu harmoninen värähtelijä</span> </div> </a> <ul id="toc-Vaimennettu_ja_pakotettu_harmoninen_värähtelijä-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Katso_myös" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Katso_myös"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Katso myös</span> </div> </a> <ul id="toc-Katso_myös-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Lähteet" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Lähteet"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Lähteet</span> </div> </a> <ul id="toc-Lähteet-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Aiheesta_muualla" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Aiheesta_muualla"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Aiheesta muualla</span> </div> </a> <ul id="toc-Aiheesta_muualla-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="Sisällysluettelo" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" title="Sisällysluettelo" > <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="Vaihda sisällysluettelo" > <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">Vaihda sisällysluettelo</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">Harmoninen värähtelijä</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="Mene artikkeliin toisella kielellä. Saatavilla 46 kielellä" > <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-46" 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">46 kieltä</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/%D9%87%D8%B2%D8%A7%D8%B2_%D8%AA%D9%88%D8%A7%D9%81%D9%82%D9%8A" title="هزاز توافقي — arabia" lang="ar" hreflang="ar" data-title="هزاز توافقي" data-language-autonym="العربية" data-language-local-name="arabia" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ast badge-Q17437798 badge-goodarticle mw-list-item" title="hyvä artikkeli"><a href="https://ast.wikipedia.org/wiki/Oscilador_harm%C3%B3nicu" title="Oscilador harmónicu — asturia" lang="ast" hreflang="ast" data-title="Oscilador harmónicu" data-language-autonym="Asturianu" data-language-local-name="asturia" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%A6%E0%A7%8B%E0%A6%B2_%E0%A6%97%E0%A6%A4%E0%A6%BF" title="দোল গতি — bengali" lang="bn" hreflang="bn" data-title="দোল গতি" data-language-autonym="বাংলা" data-language-local-name="bengali" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Harmonijski_oscilator" title="Harmonijski oscilator — bosnia" lang="bs" hreflang="bs" data-title="Harmonijski oscilator" data-language-autonym="Bosanski" data-language-local-name="bosnia" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A5%D0%B0%D1%80%D0%BC%D0%BE%D0%BD%D0%B8%D1%87%D0%B5%D0%BD_%D0%BE%D1%81%D1%86%D0%B8%D0%BB%D0%B0%D1%82%D0%BE%D1%80" title="Хармоничен осцилатор — bulgaria" lang="bg" hreflang="bg" data-title="Хармоничен осцилатор" data-language-autonym="Български" data-language-local-name="bulgaria" 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/Oscil%C2%B7lador_harm%C3%B2nic" title="Oscil·lador harmònic — katalaani" lang="ca" hreflang="ca" data-title="Oscil·lador harmònic" data-language-autonym="Català" data-language-local-name="katalaani" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%93%D0%B0%D1%80%D0%BC%D0%BE%D0%BD%D0%B8%D0%BB%D0%BB%D0%B5_%D0%BE%D1%81%D1%86%D0%B8%D0%BB%D0%BB%D1%8F%D1%82%D0%BE%D1%80" title="Гармонилле осциллятор — tšuvassi" lang="cv" hreflang="cv" data-title="Гармонилле осциллятор" data-language-autonym="Чӑвашла" data-language-local-name="tšuvassi" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Harmonick%C3%BD_oscil%C3%A1tor" title="Harmonický oscilátor — tšekki" lang="cs" hreflang="cs" data-title="Harmonický oscilátor" data-language-autonym="Čeština" data-language-local-name="tšekki" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Osgiliadur_harmonig" title="Osgiliadur harmonig — kymri" lang="cy" hreflang="cy" data-title="Osgiliadur harmonig" data-language-autonym="Cymraeg" data-language-local-name="kymri" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Harmonisk_oscillator" title="Harmonisk oscillator — tanska" lang="da" hreflang="da" data-title="Harmonisk oscillator" data-language-autonym="Dansk" data-language-local-name="tanska" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Harmonischer_Oszillator" title="Harmonischer Oszillator — saksa" lang="de" hreflang="de" data-title="Harmonischer Oszillator" data-language-autonym="Deutsch" data-language-local-name="saksa" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Harmooniline_v%C3%B5nkumine" title="Harmooniline võnkumine — viro" lang="et" hreflang="et" data-title="Harmooniline võnkumine" data-language-autonym="Eesti" data-language-local-name="viro" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Harmonic_oscillator" title="Harmonic oscillator — englanti" lang="en" hreflang="en" data-title="Harmonic oscillator" data-language-autonym="English" data-language-local-name="englanti" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-es badge-Q17437798 badge-goodarticle mw-list-item" title="hyvä artikkeli"><a href="https://es.wikipedia.org/wiki/Oscilador_arm%C3%B3nico" title="Oscilador armónico — espanja" lang="es" hreflang="es" data-title="Oscilador armónico" data-language-autonym="Español" data-language-local-name="espanja" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Simpla_vibra_movo" title="Simpla vibra movo — esperanto" lang="eo" hreflang="eo" data-title="Simpla vibra movo" data-language-autonym="Esperanto" data-language-local-name="esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Osziladore_harmoniko" title="Osziladore harmoniko — baski" lang="eu" hreflang="eu" data-title="Osziladore harmoniko" data-language-autonym="Euskara" data-language-local-name="baski" 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%86%D9%88%D8%B3%D8%A7%D9%86%DA%AF%D8%B1_%D9%87%D9%85%D8%A7%D9%87%D9%86%DA%AF" title="نوسانگر هماهنگ — persia" lang="fa" hreflang="fa" data-title="نوسانگر هماهنگ" data-language-autonym="فارسی" data-language-local-name="persia" 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/Oscillateur_harmonique" title="Oscillateur harmonique — ranska" lang="fr" hreflang="fr" data-title="Oscillateur harmonique" data-language-autonym="Français" data-language-local-name="ranska" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Oscilador_harm%C3%B3nico" title="Oscilador harmónico — galicia" lang="gl" hreflang="gl" data-title="Oscilador harmónico" data-language-autonym="Galego" data-language-local-name="galicia" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko badge-Q17437796 badge-featuredarticle mw-list-item" title="suositeltu artikkeli"><a href="https://ko.wikipedia.org/wiki/%EC%A1%B0%ED%99%94_%EC%A7%84%EB%8F%99%EC%9E%90" title="조화 진동자 — korea" lang="ko" hreflang="ko" data-title="조화 진동자" data-language-autonym="한국어" data-language-local-name="korea" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%B8%E0%A4%B0%E0%A4%B2_%E0%A4%86%E0%A4%B5%E0%A4%B0%E0%A5%8D%E0%A4%A4%E0%A5%80_%E0%A4%A6%E0%A5%8B%E0%A4%B2%E0%A4%95" title="सरल आवर्ती दोलक — hindi" lang="hi" hreflang="hi" data-title="सरल आवर्ती दोलक" 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/Harmonijsko_titranje" title="Harmonijsko titranje — kroatia" lang="hr" hreflang="hr" data-title="Harmonijsko titranje" data-language-autonym="Hrvatski" data-language-local-name="kroatia" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Moto_armonico" title="Moto armonico — italia" lang="it" hreflang="it" data-title="Moto armonico" data-language-autonym="Italiano" data-language-local-name="italia" 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%AA%D7%A0%D7%93_%D7%94%D7%A8%D7%9E%D7%95%D7%A0%D7%99" title="מתנד הרמוני — heprea" lang="he" hreflang="he" data-title="מתנד הרמוני" data-language-autonym="עברית" data-language-local-name="heprea" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%B0%E1%83%90%E1%83%A0%E1%83%9B%E1%83%9D%E1%83%9C%E1%83%98%E1%83%A3%E1%83%9A%E1%83%98_%E1%83%9D%E1%83%A1%E1%83%AA%E1%83%98%E1%83%9A%E1%83%90%E1%83%A2%E1%83%9D%E1%83%A0%E1%83%98" title="ჰარმონიული ოსცილატორი — georgia" lang="ka" hreflang="ka" data-title="ჰარმონიული ოსცილატორი" data-language-autonym="ქართული" data-language-local-name="georgia" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Harmoninis_osciliatorius" title="Harmoninis osciliatorius — liettua" lang="lt" hreflang="lt" data-title="Harmoninis osciliatorius" data-language-autonym="Lietuvių" data-language-local-name="liettua" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%A5%D0%B0%D1%80%D0%BC%D0%BE%D0%BD%D0%B8%D1%81%D0%BA%D0%B8_%D1%82%D1%80%D0%B5%D0%BF%D0%B5%D1%80%D0%BD%D0%B8%D0%BA" title="Хармониски треперник — makedonia" lang="mk" hreflang="mk" data-title="Хармониски треперник" data-language-autonym="Македонски" data-language-local-name="makedonia" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Harmonische_oscillator" title="Harmonische oscillator — hollanti" lang="nl" hreflang="nl" data-title="Harmonische oscillator" data-language-autonym="Nederlands" data-language-local-name="hollanti" 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/%E8%AA%BF%E5%92%8C%E6%8C%AF%E5%8B%95%E5%AD%90" title="調和振動子 — japani" lang="ja" hreflang="ja" data-title="調和振動子" data-language-autonym="日本語" data-language-local-name="japani" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Harmonisk_oscillator" title="Harmonisk oscillator — norjan bokmål" lang="nb" hreflang="nb" data-title="Harmonisk oscillator" data-language-autonym="Norsk bokmål" data-language-local-name="norjan bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Harmonisk_oscillator" title="Harmonisk oscillator — norjan nynorsk" lang="nn" hreflang="nn" data-title="Harmonisk oscillator" data-language-autonym="Norsk nynorsk" data-language-local-name="norjan nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Oscylator_harmoniczny" title="Oscylator harmoniczny — puola" lang="pl" hreflang="pl" data-title="Oscylator harmoniczny" data-language-autonym="Polski" data-language-local-name="puola" 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/Oscilador_harm%C3%B4nico" title="Oscilador harmônico — portugali" lang="pt" hreflang="pt" data-title="Oscilador harmônico" data-language-autonym="Português" data-language-local-name="portugali" 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/Oscilator_armonic" title="Oscilator armonic — romania" lang="ro" hreflang="ro" data-title="Oscilator armonic" data-language-autonym="Română" data-language-local-name="romania" 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%B0%D1%80%D0%BC%D0%BE%D0%BD%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%B8%D0%B9_%D0%BE%D1%81%D1%86%D0%B8%D0%BB%D0%BB%D1%8F%D1%82%D0%BE%D1%80" title="Гармонический осциллятор — venäjä" lang="ru" hreflang="ru" data-title="Гармонический осциллятор" data-language-autonym="Русский" data-language-local-name="venäjä" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-si mw-list-item"><a href="https://si.wikipedia.org/wiki/%E0%B6%85%E0%B6%B1%E0%B7%94%E0%B7%80%E0%B6%BB%E0%B7%8A%E0%B6%AD%E0%B7%93_%E0%B6%AF%E0%B7%9D%E0%B6%BD%E0%B6%9A%E0%B6%BA" title="අනුවර්තී දෝලකය — sinhala" lang="si" hreflang="si" data-title="අනුවර්තී දෝලකය" data-language-autonym="සිංහල" data-language-local-name="sinhala" class="interlanguage-link-target"><span>සිංහල</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Harmoni%C4%8Dni_oscilator" title="Harmonični oscilator — sloveeni" lang="sl" hreflang="sl" data-title="Harmonični oscilator" data-language-autonym="Slovenščina" data-language-local-name="sloveeni" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/Harmonijski_oscilator" title="Harmonijski oscilator — serbia" lang="sr" hreflang="sr" data-title="Harmonijski oscilator" data-language-autonym="Српски / srpski" data-language-local-name="serbia" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Harmonijski_oscilator" title="Harmonijski oscilator — serbokroaatti" lang="sh" hreflang="sh" data-title="Harmonijski oscilator" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="serbokroaatti" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Harmonisk_oscillator" title="Harmonisk oscillator — ruotsi" lang="sv" hreflang="sv" data-title="Harmonisk oscillator" data-language-autonym="Svenska" data-language-local-name="ruotsi" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tt mw-list-item"><a href="https://tt.wikipedia.org/wiki/%D0%93%D0%B0%D1%80%D0%BC%D0%BE%D0%BD%D0%B8%D0%BA_%D1%82%D0%B8%D1%80%D0%B1%D3%99%D0%BD%D2%AF%D0%BB%D3%99%D1%80" title="Гармоник тирбәнүләр — tataari" lang="tt" hreflang="tt" data-title="Гармоник тирбәнүләр" data-language-autonym="Татарча / tatarça" data-language-local-name="tataari" class="interlanguage-link-target"><span>Татарча / tatarça</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Dao_%C4%91%E1%BB%99ng_%C4%91i%E1%BB%81u_h%C3%B2a" title="Dao động điều hòa — vietnam" lang="vi" hreflang="vi" data-title="Dao động điều hòa" data-language-autonym="Tiếng Việt" data-language-local-name="vietnam" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Harmonik_osilat%C3%B6r" title="Harmonik osilatör — turkki" lang="tr" hreflang="tr" data-title="Harmonik osilatör" 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hankkeissa </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="wb-otherproject-link wb-otherproject-commons mw-list-item"><a href="https://commons.wikimedia.org/wiki/Category:Harmonic_oscillators" hreflang="en"><span>Wikimedia Commons</span></a></li><li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q190070" title="Linkki yhdistettyyn keskustietovaraston kohteeseen [g]" accesskey="g"><span>Wikidata-kohde</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="Sivutyökalut"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Ulkoasu"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Ulkoasu</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">siirrä sivupalkkiin</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">piilota</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">Wikipediasta</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="fi" dir="ltr"><div class="noprint article-about-box" style="background-color: var( --background-color-neutral-subtle, #f9f9f9 ); color: var( --color-emphasized, #000000 ); font-size: 95%; padding: 0.2em 0.2em 0.2em 2em; margin-bottom: 1em; border: 1px solid #b6b6b6;"><i>Tämä artikkeli käsittelee klassista harmonista värähtelijää. Harmoninen värähtelijä voi tarkoittaa myös <a href="/wiki/Kvanttimekaaninen_harmoninen_v%C3%A4r%C3%A4htelij%C3%A4" title="Kvanttimekaaninen harmoninen värähtelijä">kvanttimekaanista harmonista värähtelijää</a>.</i></div> <figure class="mw-halign-right" typeof="mw:File/Frame"><a href="/wiki/Tiedosto:HarmOsc1.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/2/2a/HarmOsc1.png" decoding="async" width="397" height="194" class="mw-file-element" data-file-width="397" data-file-height="194" /></a><figcaption>Jousi-massasysteemi värähtelee sinimuotoisesti.</figcaption></figure> <p><b>Harmoninen värähtelijä</b> on <a href="/wiki/Fysiikka" title="Fysiikka">fysiikassa</a> järjestelmä, jossa kappaleeseen vaikuttaa harmoninen <a href="/wiki/Voima_(fysiikka)" title="Voima (fysiikka)">voima</a>. Harmonisessa värähtelijässä voiman suuruus on <a href="/wiki/Suoraan_verrannollisuus" title="Suoraan verrannollisuus">suoraan verrannollinen</a> kappaleen etäisyyteen tasapainoasemasta: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {F}=-k{x},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>F</mi> </mrow> <mo>=</mo> <mo>−</mo> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {F}=-k{x},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ef2d27be8b2e2859e500fb8786fb0467e888f33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.835ex; height:2.509ex;" alt="{\displaystyle {F}=-k{x},}"></span></dd></dl> <p>missä </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {F}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>F</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {F}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/265b6cfb92e9e697cf7051b0e829d117748377cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle {F}}"></span> on kappaleeseen kohdistunut voima,</li> <li><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/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span> on vakio (esimerkiksi <a href="/wiki/Jousi" title="Jousi">jousille</a> <a href="/wiki/Jousivakio" title="Jousivakio">jousivakio</a>, joka ilmaisee jousen jäykkyyttä),</li> <li>ja <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}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5cb8cd0cfa94e69432c076ca30c3bd6facaabb93" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle {x}}"></span> poikkeama tasapainoasemasta.</li></ul> <p>Tällaista voimaa sanotaan <a href="/wiki/Harmoninen_voima" title="Harmoninen voima">harmoniseksi voimaksi</a>. Harmoninen voima suuntautuu aina kohti tasapainoasemaa, sillä voimasta aiheutuva kiihtyvyys on jatkuvasti kohti tasapainoasemaa. <a href="/wiki/Heiluri" title="Heiluri">Heiluri</a> ja <a href="/wiki/Jousi" title="Jousi">jousen</a> päässä värähtelevä punnus ovat hyviä esimerkkejä harmonisesta värähtelijästä. </p><p>Jos F on ainoa systeemiin vaikuttava voima, kutsutaan systeemiä silloin vaimentumattomaksi tai ideaaliseksi harmoniseksi värähtelijäksi. Tällaisella värähtelijällä on vakioamplitudi ja –taajuus, joka ei riipu amplitudista. Värähtely on tällöin sinimuotoista. </p><p>Jos systeemiin vaikuttaa nopeuteen verrannollinen voima (<a href="/wiki/Kitka" title="Kitka">kitkavoima</a>), kutsutaan värähtelijää silloin vaimennetuksi harmoniseksi värähtelijäksi. Systeemillä on tällöin mahdollisuus käyttäytyä eri tavoin riippuen kitkakertoimen arvosta. </p><p>Jos systeemiin vaikuttaa ulkoinen ajasta riippuva voima (ns. pakkovoima), kutsutaan värähtelijää silloin pakotetuksi harmoniseksi värähtelijäksi. Pakkovoima tuo systeemiin uutta energiaa, joka voi estää vaimennetun harmonisen värähtelijän amplitudin pienenemisen ajan kuluessa. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Vaimenematon_harmoninen_värähtelijä"><span id="Vaimenematon_harmoninen_v.C3.A4r.C3.A4htelij.C3.A4"></span>Vaimenematon harmoninen värähtelijä</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Harmoninen_v%C3%A4r%C3%A4htelij%C3%A4&veaction=edit&section=1" title="Muokkaa osiota Vaimenematon harmoninen värähtelijä" class="mw-editsection-visualeditor"><span>muokkaa</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Harmoninen_v%C3%A4r%C3%A4htelij%C3%A4&action=edit&section=1" title="Muokkaa osion lähdekoodia: Vaimenematon harmoninen värähtelijä"><span>muokkaa wikitekstiä</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Frame"><a href="/wiki/Tiedosto:Simple_harmonic_oscillator.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/9/9d/Simple_harmonic_oscillator.gif" decoding="async" width="116" height="359" class="mw-file-element" data-file-width="116" data-file-height="359" /></a><figcaption>Kitkaton jousi-massasysteemi on vaimenematon harmoninen värähtelijä.</figcaption></figure> <p>Vaimentumattomaan harmoniseen värähtelijään ei vaikuta kitka- eikä pakkovoimaa, jolloin systeemiin vaikuttava voima on muotoa: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F=-kx.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo>=</mo> <mo>−</mo> <mi>k</mi> <mi>x</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F=-kx.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3409466df110132704130356881eec63a7abaad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.835ex; height:2.343ex;" alt="{\displaystyle F=-kx.}"></span></dd></dl> <p>Newtonin 2. laki: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F=ma=-kx.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo>=</mo> <mi>m</mi> <mi>a</mi> <mo>=</mo> <mo>−</mo> <mi>k</mi> <mi>x</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F=ma=-kx.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0c9d66a9f0d0536b14b077b4c1d6f7b236d2faf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:16.204ex; height:2.343ex;" alt="{\displaystyle F=ma=-kx.}"></span></dd></dl> <p><a href="/wiki/Kiihtyvyys" title="Kiihtyvyys">Kiihtyvyys</a> <i>a</i> on paikan <i>x</i> toinen aika<a href="/wiki/Derivaatta" title="Derivaatta">derivaatta</a> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m{\frac {\mathrm {d} ^{2}x}{\mathrm {d} t^{2}}}=-kx.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>x</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mi>k</mi> <mi>x</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m{\frac {\mathrm {d} ^{2}x}{\mathrm {d} t^{2}}}=-kx.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4d33d93f3e1f57663a6cd85d9ea8f4007aaa7b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:14.647ex; height:6.009ex;" alt="{\displaystyle m{\frac {\mathrm {d} ^{2}x}{\mathrm {d} t^{2}}}=-kx.}"></span></dd></dl> <p>Jos määritellään <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 _{0}}^{2}=k/m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\omega _{0}}^{2}=k/m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7914107866e2ef07b7869c339ceb6f95775dc1a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.067ex; height:3.176ex;" alt="{\displaystyle {\omega _{0}}^{2}=k/m}"></span>, voidaan yhtälö kirjoittaa muotoon: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\mathrm {d} ^{2}x}{\mathrm {d} t^{2}}}+{\omega _{0}}^{2}x=0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>x</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>x</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\mathrm {d} ^{2}x}{\mathrm {d} t^{2}}}+{\omega _{0}}^{2}x=0,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e01e35c0280a4edbfb8814b9e00d041c3bc96f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:17.145ex; height:6.009ex;" alt="{\displaystyle {\frac {\mathrm {d} ^{2}x}{\mathrm {d} t^{2}}}+{\omega _{0}}^{2}x=0,}"></span></dd></dl> <p>jonka yleinen ratkaisu on </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=A\cos {(\omega _{0}t+\phi )}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mi>A</mi> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>t</mi> <mo>+</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=A\cos {(\omega _{0}t+\phi )}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7eca5f2fa80da442da4b45a5e288e95416e93d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.078ex; height:2.843ex;" alt="{\displaystyle x=A\cos {(\omega _{0}t+\phi )}.}"></span></dd></dl> <p>Amplitudi <i>A</i> ja vaihe <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 \phi \,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi \,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c69f1c4a95b2d750b30fa4cf5d5d068a573ac0d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.773ex; height:2.509ex;" alt="{\displaystyle \phi \,}"></span> määritetään alkuehdosta. </p><p>Yleinen ratkaisu voidaan esittää myös muodossa: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=A\sin {(\omega _{0}t+\phi )},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mi>A</mi> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>t</mi> <mo>+</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=A\sin {(\omega _{0}t+\phi )},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/706d3552541aa496e55aa5be5ed448b3543b1427" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.823ex; height:2.843ex;" alt="{\displaystyle x=A\sin {(\omega _{0}t+\phi )},}"></span></dd></dl> <p>missä <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 \phi \,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi \,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c69f1c4a95b2d750b30fa4cf5d5d068a573ac0d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.773ex; height:2.509ex;" alt="{\displaystyle \phi \,}"></span> on siirtynyt <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 \pi /2\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi /2\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5145b1ceeed3ff342b36ddf4c8629fa260edc3b6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.044ex; height:2.843ex;" alt="{\displaystyle \pi /2\,}"></span> verran. </p><p>Yleinen ratkaisu voidaan esittää myös muodossa </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=C_{1}\sin {\omega _{0}t}+C_{2}\cos {\omega _{0}t},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>t</mi> </mrow> <mo>+</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>t</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=C_{1}\sin {\omega _{0}t}+C_{2}\cos {\omega _{0}t},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b76b01ae60db490bc809cb14146be695bfd27e39" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:27.542ex; height:2.509ex;" alt="{\displaystyle x=C_{1}\sin {\omega _{0}t}+C_{2}\cos {\omega _{0}t},}"></span> </p><p>missä <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 C_{1}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{1}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3d4f3c02d474547c70098d2529c8636829776a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.103ex; height:2.509ex;" alt="{\displaystyle C_{1}\,}"></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 C_{2}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{2}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40f3e757e084c7f6e4a02cc971dea82db704a97b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.103ex; height:2.509ex;" alt="{\displaystyle C_{2}\,}"></span> ovat vakioita, jotka voidaan määrittää alkuehdosta. </p><p>Värähtelyn <a href="/wiki/Taajuus" title="Taajuus">taajuudeksi</a> saadaan: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f={\frac {\omega _{0}}{2\pi }}={\frac {1}{2\pi }}{\sqrt {\frac {k}{m}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mi>k</mi> <mi>m</mi> </mfrac> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f={\frac {\omega _{0}}{2\pi }}={\frac {1}{2\pi }}{\sqrt {\frac {k}{m}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79873e3e7632eec1c91a2d8ec32039eb5d5fa301" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:19.989ex; height:6.176ex;" alt="{\displaystyle f={\frac {\omega _{0}}{2\pi }}={\frac {1}{2\pi }}{\sqrt {\frac {k}{m}}}.}"></span></dd></dl> <p>Värähtelyn nopeudeksi <i>v</i> ja kiihtyvyydeksi <i>a</i> saadaan </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v={\frac {\mathrm {d} x}{\mathrm {d} t}}=-A\omega _{0}\sin(\omega _{0}t+\phi )=C_{1}\omega _{0}\cos {\omega _{0}t}-C_{2}\omega _{0}\sin {\omega _{0}t}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mi>A</mi> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>t</mi> <mo>+</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>t</mi> </mrow> <mo>−</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>t</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v={\frac {\mathrm {d} x}{\mathrm {d} t}}=-A\omega _{0}\sin(\omega _{0}t+\phi )=C_{1}\omega _{0}\cos {\omega _{0}t}-C_{2}\omega _{0}\sin {\omega _{0}t}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/757a9863a6687176506af8a2545deb06ed171ea6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:60.017ex; height:5.509ex;" alt="{\displaystyle v={\frac {\mathrm {d} x}{\mathrm {d} t}}=-A\omega _{0}\sin(\omega _{0}t+\phi )=C_{1}\omega _{0}\cos {\omega _{0}t}-C_{2}\omega _{0}\sin {\omega _{0}t}}"></span> ja</dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a={\frac {\mathrm {d} ^{2}x}{\mathrm {d} t^{2}}}=-\omega _{0}^{2}x=-A\omega _{0}^{2}\cos(\omega _{0}t+\phi )=-C_{1}\omega _{0}^{2}\sin {\omega _{0}t}-C_{2}\omega _{0}^{2}\cos {\omega _{0}t}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>x</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <msubsup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mi>x</mi> <mo>=</mo> <mo>−</mo> <mi>A</mi> <msubsup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>t</mi> <mo>+</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msubsup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>t</mi> </mrow> <mo>−</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msubsup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>t</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a={\frac {\mathrm {d} ^{2}x}{\mathrm {d} t^{2}}}=-\omega _{0}^{2}x=-A\omega _{0}^{2}\cos(\omega _{0}t+\phi )=-C_{1}\omega _{0}^{2}\sin {\omega _{0}t}-C_{2}\omega _{0}^{2}\cos {\omega _{0}t}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/426b3b8d94c234576b42e6b0f6a655b7b85ef4cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:72.62ex; height:6.009ex;" alt="{\displaystyle a={\frac {\mathrm {d} ^{2}x}{\mathrm {d} t^{2}}}=-\omega _{0}^{2}x=-A\omega _{0}^{2}\cos(\omega _{0}t+\phi )=-C_{1}\omega _{0}^{2}\sin {\omega _{0}t}-C_{2}\omega _{0}^{2}\cos {\omega _{0}t}.}"></span></dd></dl> <p>Värähtelijän <a href="/wiki/Kineettinen_energia" class="mw-redirect" title="Kineettinen energia">kineettinen energia</a> on </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K={\frac {1}{2}}m\left({\frac {\mathrm {d} x}{\mathrm {d} t}}\right)^{2}={\frac {1}{2}}kA^{2}\sin ^{2}(\omega _{0}t+\phi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>m</mi> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </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> <mi>k</mi> <msup> <mi>A</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> <mo stretchy="false">(</mo> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>t</mi> <mo>+</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K={\frac {1}{2}}m\left({\frac {\mathrm {d} x}{\mathrm {d} t}}\right)^{2}={\frac {1}{2}}kA^{2}\sin ^{2}(\omega _{0}t+\phi )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89e6052882ddc3562a2acf3330c358ef9636b97a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:39.915ex; height:6.509ex;" alt="{\displaystyle K={\frac {1}{2}}m\left({\frac {\mathrm {d} x}{\mathrm {d} t}}\right)^{2}={\frac {1}{2}}kA^{2}\sin ^{2}(\omega _{0}t+\phi )}"></span></dd></dl> <p>ja <a href="/wiki/Potentiaalienergia" title="Potentiaalienergia">potentiaalienergia</a> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U={\frac {1}{2}}kx^{2}={\frac {1}{2}}kA^{2}\cos ^{2}(\omega _{0}t+\phi ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>k</mi> <msup> <mi>x</mi> <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> <mi>k</mi> <msup> <mi>A</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> <mo stretchy="false">(</mo> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>t</mi> <mo>+</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U={\frac {1}{2}}kx^{2}={\frac {1}{2}}kA^{2}\cos ^{2}(\omega _{0}t+\phi ).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9e7a599b2ba1d8e4a3653c5acca491c509b2dea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:34.154ex; height:5.176ex;" alt="{\displaystyle U={\frac {1}{2}}kx^{2}={\frac {1}{2}}kA^{2}\cos ^{2}(\omega _{0}t+\phi ).}"></span></dd></dl> <p>Värähtelijän potentiaalienergia on siis suoraan verrannollinen tasapainopisteestä mitatun etäisyyden neliöön. </p><p>Värähtelijän potentiaali- ja kineettinen energia muuttuvat jatkuvasti toisikseen, mutta niiden summa on vakio: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E={\frac {1}{2}}kA^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>k</mi> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E={\frac {1}{2}}kA^{2}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/719b23efd300bf9a5dcb0414e531e8f764c294c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:11.528ex; height:5.176ex;" alt="{\displaystyle E={\frac {1}{2}}kA^{2}.}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Pakotettu_harmoninen_värähtelijä"><span id="Pakotettu_harmoninen_v.C3.A4r.C3.A4htelij.C3.A4"></span>Pakotettu harmoninen värähtelijä</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Harmoninen_v%C3%A4r%C3%A4htelij%C3%A4&veaction=edit&section=2" title="Muokkaa osiota Pakotettu harmoninen värähtelijä" class="mw-editsection-visualeditor"><span>muokkaa</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Harmoninen_v%C3%A4r%C3%A4htelij%C3%A4&action=edit&section=2" title="Muokkaa osion lähdekoodia: Pakotettu harmoninen värähtelijä"><span>muokkaa wikitekstiä</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Pakkovoima <i>F<sub>d</sub></i> on voima joka tuo systeemiin energiaa. Matemaattisesti yksinkertaisin tapaus on, kun pakkovoima värähtelee sinimuotoisesti. Kun kitkavoimaa eli vaimennusta ei oteta huomioon ja <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 \neq \omega _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> <mo>≠</mo> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega \neq \omega _{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/680014773edc43276271b9d2fa60588cd5b1e6c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.044ex; height:2.676ex;" alt="{\displaystyle \omega \neq \omega _{0}}"></span>, on systeemin liikeyhtälö muotoa </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\mathrm {d} ^{2}x}{\mathrm {d} t^{2}}}+{\omega _{0}}^{2}x=F_{0}\cos(\omega t),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>x</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>x</mi> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>ω</mi> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\mathrm {d} ^{2}x}{\mathrm {d} t^{2}}}+{\omega _{0}}^{2}x=F_{0}\cos(\omega t),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc706010f7d7476c1d7c464acbe86cebd89ff6d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:26.124ex; height:6.009ex;" alt="{\displaystyle {\frac {\mathrm {d} ^{2}x}{\mathrm {d} t^{2}}}+{\omega _{0}}^{2}x=F_{0}\cos(\omega t),}"></span></dd></dl> <p>missä <i>F<sub>0</sub></i> on pakkovoiman amplitudi ja <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>ω</mi> </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/48eff443f9de7a985bb94ca3bde20813ea737be8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }"></span> on pakkovoiman värähtelyn taajuus. Yhtälön yleinen ratkaisu voidaan esittää muodossa </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=C\cos {(\omega _{0}t-\delta )}+{\frac {F_{0}}{m({\omega _{0}}^{2}-{\omega }^{2})}}\cos {\omega t},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mi>C</mi> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>t</mi> <mo>−</mo> <mi>δ</mi> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow> <mi>m</mi> <mo stretchy="false">(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi>ω</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>ω</mi> <mi>t</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=C\cos {(\omega _{0}t-\delta )}+{\frac {F_{0}}{m({\omega _{0}}^{2}-{\omega }^{2})}}\cos {\omega t},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79f388531d8454a9583e9e06d6b45728ff70e4e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:42.356ex; height:6.176ex;" alt="{\displaystyle x=C\cos {(\omega _{0}t-\delta )}+{\frac {F_{0}}{m({\omega _{0}}^{2}-{\omega }^{2})}}\cos {\omega t},}"></span></dd></dl> <p>kun siis <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 \neq \omega _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> <mo>≠</mo> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega \neq \omega _{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/680014773edc43276271b9d2fa60588cd5b1e6c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.044ex; height:2.676ex;" alt="{\displaystyle \omega \neq \omega _{0}}"></span>. Jos tarkastellaan tapausta, jossa <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 =\omega _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> <mo>=</mo> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega =\omega _{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c93049cbc3eaea8d372ce40362944f1a8a67941" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.044ex; height:2.009ex;" alt="{\displaystyle \omega =\omega _{0}}"></span>, ylimmän kaavan yksittäisratkaisuksi saadaan </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x={\frac {F_{0}}{2m{\omega _{0}}}}t\sin {\omega _{0}t},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow> <mn>2</mn> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </mrow> </mfrac> </mrow> <mi>t</mi> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>t</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x={\frac {F_{0}}{2m{\omega _{0}}}}t\sin {\omega _{0}t},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ba274355dcc2f28a9bbd9a447a6e94886b36797" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:19.423ex; height:5.676ex;" alt="{\displaystyle x={\frac {F_{0}}{2m{\omega _{0}}}}t\sin {\omega _{0}t},}"></span></dd></dl> <p>josta huomataan, että värähtely kasvaa ajan <i>t</i> kuluessa. Tämä on matemaattinen selitys <a href="/wiki/Resonanssi" title="Resonanssi">resonanssi</a>-ilmiölle. Jos <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>ω</mi> </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/48eff443f9de7a985bb94ca3bde20813ea737be8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }"></span> on hyvin lähellä arvoa <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 _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega _{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a713d16c489051d4f515e12b1f86061c6be799b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.5ex; height:2.009ex;" alt="{\displaystyle \omega _{0}}"></span>, mutta ei aivan sama, saadaan ratkaisuksi </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x={\frac {F_{0}}{m({\omega _{0}}^{2}-{\omega }^{2})}}\sin {{\frac {\omega _{0}+\omega }{2}}t}\sin {{\frac {\omega _{0}-\omega }{2}}t}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow> <mi>m</mi> <mo stretchy="false">(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi>ω</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mi>ω</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mi>t</mi> </mrow> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>−</mo> <mi>ω</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mi>t</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x={\frac {F_{0}}{m({\omega _{0}}^{2}-{\omega }^{2})}}\sin {{\frac {\omega _{0}+\omega }{2}}t}\sin {{\frac {\omega _{0}-\omega }{2}}t}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd9242cfff3a8cbb3318a3da2ea910e72c64183b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:42.839ex; height:6.176ex;" alt="{\displaystyle x={\frac {F_{0}}{m({\omega _{0}}^{2}-{\omega }^{2})}}\sin {{\frac {\omega _{0}+\omega }{2}}t}\sin {{\frac {\omega _{0}-\omega }{2}}t}.}"></span></dd></dl> <p>Kun <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 _{0}-\omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>−</mo> <mi>ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega _{0}-\omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f60fa4f5689b4a9f6f19dcec19cc506ef8f5b6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.786ex; height:2.343ex;" alt="{\displaystyle \omega _{0}-\omega }"></span> on hyvin pieni eli pakkovoiman taajuus eroaa vain vähän värähtelijän ominaistaajuudesta, on jälkimmäisen sinifunktion jakso hyvin suuri. Tämä ilmenee huojumisena. Tätä muusikot käyttävät hyväksi <a href="/wiki/Viritys_(musiikki)" title="Viritys (musiikki)">virittäessään</a> soittimiaan. </p> <div class="mw-heading mw-heading2"><h2 id="Vaimennettu_harmoninen_värähtelijä"><span id="Vaimennettu_harmoninen_v.C3.A4r.C3.A4htelij.C3.A4"></span>Vaimennettu harmoninen värähtelijä</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Harmoninen_v%C3%A4r%C3%A4htelij%C3%A4&veaction=edit&section=3" title="Muokkaa osiota Vaimennettu harmoninen värähtelijä" class="mw-editsection-visualeditor"><span>muokkaa</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Harmoninen_v%C3%A4r%C3%A4htelij%C3%A4&action=edit&section=3" title="Muokkaa osion lähdekoodia: Vaimennettu harmoninen värähtelijä"><span>muokkaa wikitekstiä</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Frame"><a href="/wiki/Tiedosto:Damped_spring.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/2/2b/Damped_spring.gif" decoding="async" width="110" height="359" class="mw-file-element" data-file-width="110" data-file-height="359" /></a><figcaption>Vaimennettu jousi-massasysteemi.</figcaption></figure> <figure typeof="mw:File/Thumb"><a href="/wiki/Tiedosto:Damping.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e8/Damping.svg/300px-Damping.svg.png" decoding="async" width="300" height="209" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e8/Damping.svg/450px-Damping.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e8/Damping.svg/600px-Damping.svg.png 2x" data-file-width="512" data-file-height="356" /></a><figcaption>Systeemin käyttäytyminen riippuu vaimennuskertoimesta <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 \zeta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ζ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \zeta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5c3916703cae7938143d38865f78f27faadd4ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.095ex; height:2.509ex;" alt="{\displaystyle \zeta }"></span>.</figcaption></figure> <p>Käytännössä värähtelevään systeemiin vaikuttaa aina liikettä vastustavia kitkavoimia, joiden vaikutuksesta värähtely vaimenee ajan funktiona. Värähtelevän jousen asema noudattaa toisen kertaluvun lineaarista yhtälöä </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m{\frac {\mathrm {d} ^{2}x}{\mathrm {d} t^{2}}}+c{\frac {\mathrm {d} x}{\mathrm {d} t}}+kx=0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>x</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mi>k</mi> <mi>x</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m{\frac {\mathrm {d} ^{2}x}{\mathrm {d} t^{2}}}+c{\frac {\mathrm {d} x}{\mathrm {d} t}}+kx=0,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7aeea6e35e24f9b59f600434b127a9a0d868b84f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:24.148ex; height:6.009ex;" alt="{\displaystyle m{\frac {\mathrm {d} ^{2}x}{\mathrm {d} t^{2}}}+c{\frac {\mathrm {d} x}{\mathrm {d} t}}+kx=0,}"></span></dd></dl> <p>missä <i>c</i> on vaimennuskerroin. Yhtälöllä on kolme eri ratkaisua, riippuen vaimennuskertoimen <i>c</i> arvosta. Merkitään <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 a=c/2m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=c/2m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e1fef9a855f3f19e20851668776c82b628486b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.7ex; height:2.843ex;" alt="{\displaystyle a=c/2m}"></span> ja <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 b=1/2m{\sqrt {c^{2}-4mk}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>=</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>4</mn> <mi>m</mi> <mi>k</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b=1/2m{\sqrt {c^{2}-4mk}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/580f1cb5cf5c09c9f5932544cd8b89bd53af8ca1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.263ex; height:3.509ex;" alt="{\displaystyle b=1/2m{\sqrt {c^{2}-4mk}}}"></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Ylivaimennus">Ylivaimennus</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Harmoninen_v%C3%A4r%C3%A4htelij%C3%A4&veaction=edit&section=4" title="Muokkaa osiota Ylivaimennus" class="mw-editsection-visualeditor"><span>muokkaa</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Harmoninen_v%C3%A4r%C3%A4htelij%C3%A4&action=edit&section=4" title="Muokkaa osion lähdekoodia: Ylivaimennus"><span>muokkaa wikitekstiä</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Jos vaimennuskerroin on niin suuri, että <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 c^{2}>4mk}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>></mo> <mn>4</mn> <mi>m</mi> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c^{2}>4mk}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/be449b049c68481dcd876deda43bb07075294700" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.574ex; height:2.676ex;" alt="{\displaystyle c^{2}>4mk}"></span>, <a href="/wiki/Differentiaaliyht%C3%A4l%C3%B6" title="Differentiaaliyhtälö">differentiaaliyhtälön</a> ratkaisu on </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x(t)=c_{1}e^{-(a-b)t}+c_{2}e^{-(a+b)t},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>−</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mi>t</mi> </mrow> </msup> <mo>+</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mi>t</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x(t)=c_{1}e^{-(a-b)t}+c_{2}e^{-(a+b)t},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3cac843c81b44cb8896bcbb4a148139d6294adf5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.328ex; height:3.343ex;" alt="{\displaystyle x(t)=c_{1}e^{-(a-b)t}+c_{2}e^{-(a+b)t},}"></span></dd></dl> <p>josta huomataan, että mitään heilahtelua ei tapahdu, sillä molemmat eksponentit ovat negatiivisia, koska <i>a, b > 0</i> ja <i>b < a</i>. Tällöin molemmat termit lähestyvät nollaa, kun <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\to \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t\to \infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a34d7a61899d577d950881b4a44888d43f3fa93" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.777ex; height:2.009ex;" alt="{\displaystyle t\to \infty }"></span>. Heilahtelun rata voi ylittää tasapainoaseman <i>x = 0</i> korkeintaan kerran. </p> <div class="mw-heading mw-heading3"><h3 id="Alivaimennus">Alivaimennus</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Harmoninen_v%C3%A4r%C3%A4htelij%C3%A4&veaction=edit&section=5" title="Muokkaa osiota Alivaimennus" class="mw-editsection-visualeditor"><span>muokkaa</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Harmoninen_v%C3%A4r%C3%A4htelij%C3%A4&action=edit&section=5" title="Muokkaa osion lähdekoodia: Alivaimennus"><span>muokkaa wikitekstiä</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Jos vaimennuskerroin on niin pieni, että <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 c^{2}<4mk}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo><</mo> <mn>4</mn> <mi>m</mi> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c^{2}<4mk}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c6fd71e46a152f7057a334006fecaa8692a7a57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.574ex; height:2.676ex;" alt="{\displaystyle c^{2}<4mk}"></span>, differentiaaliyhtälön ratkaisu on </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x(t)=e^{-at}(A\cos \omega t+B\sin \omega t),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>a</mi> <mi>t</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>A</mi> <mi>cos</mi> <mo>⁡</mo> <mi>ω</mi> <mi>t</mi> <mo>+</mo> <mi>B</mi> <mi>sin</mi> <mo>⁡</mo> <mi>ω</mi> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x(t)=e^{-at}(A\cos \omega t+B\sin \omega t),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf72d21d8671d91670dcf2beea4d75e3c440e6e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.024ex; height:3.009ex;" alt="{\displaystyle x(t)=e^{-at}(A\cos \omega t+B\sin \omega t),}"></span></dd></dl> <p>jolloin syntyy vaimeneva värähdysliike, joka lähenee koko ajan tasapainoasemaa <i>x = 0</i>. </p> <div class="mw-heading mw-heading3"><h3 id="Kriittinen_vaimennus">Kriittinen vaimennus</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Harmoninen_v%C3%A4r%C3%A4htelij%C3%A4&veaction=edit&section=6" title="Muokkaa osiota Kriittinen vaimennus" class="mw-editsection-visualeditor"><span>muokkaa</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Harmoninen_v%C3%A4r%C3%A4htelij%C3%A4&action=edit&section=6" title="Muokkaa osion lähdekoodia: Kriittinen vaimennus"><span>muokkaa wikitekstiä</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Jos vaimennuskerroin on <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 c^{2}=4mk}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>4</mn> <mi>m</mi> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c^{2}=4mk}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83c0d212589b0cfbc96f8a3e4f8a0538d921c023" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.574ex; height:2.676ex;" alt="{\displaystyle c^{2}=4mk}"></span>, differentiaaliyhtälön ratkaisu on </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x(t)=(c_{1}+c_{2}t)\,e^{-at}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>a</mi> <mi>t</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x(t)=(c_{1}+c_{2}t)\,e^{-at}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae8fe1077f3b0d4263401908f2ac395afee85f4a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.78ex; height:3.009ex;" alt="{\displaystyle x(t)=(c_{1}+c_{2}t)\,e^{-at}.}"></span></dd></dl> <p>Tämän värähtelyn muoto on hyvin samanlainen kuin ylivaimennetunkin. Mitään heilahtelua ei synny ja rata voi ylittää tasapainoaseman <i>x = 0</i> tasan kerran ja <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\to 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\to 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/198f68f84d9d1c77e6d312f0c99bd0397e5ee49b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.106ex; height:2.176ex;" alt="{\displaystyle x\to 0}"></span>, kun <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\to \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t\to \infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a34d7a61899d577d950881b4a44888d43f3fa93" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.777ex; height:2.009ex;" alt="{\displaystyle t\to \infty }"></span>. </p> <div class="mw-heading mw-heading2"><h2 id="Vaimennettu_ja_pakotettu_harmoninen_värähtelijä"><span id="Vaimennettu_ja_pakotettu_harmoninen_v.C3.A4r.C3.A4htelij.C3.A4"></span>Vaimennettu ja pakotettu harmoninen värähtelijä</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Harmoninen_v%C3%A4r%C3%A4htelij%C3%A4&veaction=edit&section=7" title="Muokkaa osiota Vaimennettu ja pakotettu harmoninen värähtelijä" class="mw-editsection-visualeditor"><span>muokkaa</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Harmoninen_v%C3%A4r%C3%A4htelij%C3%A4&action=edit&section=7" title="Muokkaa osion lähdekoodia: Vaimennettu ja pakotettu harmoninen värähtelijä"><span>muokkaa wikitekstiä</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/Tiedosto:Mass-Spring-Damper.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Mass-Spring-Damper.png/300px-Mass-Spring-Damper.png" decoding="async" width="300" height="176" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Mass-Spring-Damper.png/450px-Mass-Spring-Damper.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Mass-Spring-Damper.png/600px-Mass-Spring-Damper.png 2x" data-file-width="763" data-file-height="448" /></a><figcaption>Massa m on kytketty jouseen ja vaimennukseen, jonka vaimennuskerroin on B ja F on ulkoinen pakkovoima.</figcaption></figure> <p>Jos halutaan estää vaimennetun värähtelijän amplitudin pieneneminen ajan kuluessa on systeemiin tuotava energiaa ulkoisella pakkovoimalla <i>F<sub>d</sub></i>. Kuten aikaisemmin kerrottiin, matemaattisesti yksinkertaisin tapaus on kun pakkovoima värähtelee sinimuotoisesti. Vaimennetun ja pakotetun värähtelijän liikeyhtälö on </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m{\frac {\mathrm {d} ^{2}x}{\mathrm {d} t^{2}}}+c{\frac {\mathrm {d} x}{\mathrm {d} t}}+kx=F_{0}\cos(\omega t),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>x</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mi>k</mi> <mi>x</mi> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>ω</mi> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m{\frac {\mathrm {d} ^{2}x}{\mathrm {d} t^{2}}}+c{\frac {\mathrm {d} x}{\mathrm {d} t}}+kx=F_{0}\cos(\omega t),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a267aed5989cee38dd6b36e0a1d5da8c7eabd2a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:33.127ex; height:6.009ex;" alt="{\displaystyle m{\frac {\mathrm {d} ^{2}x}{\mathrm {d} t^{2}}}+c{\frac {\mathrm {d} x}{\mathrm {d} t}}+kx=F_{0}\cos(\omega t),}"></span></dd></dl> <p>jonka ratkaisu muodostuu vaimennetun värähtelijän ja pakotetun värähtelijän liikeyhtälöiden ratkaisujen summasta. Kuten aikaisemmin osoitettiin, vaimennetun värähtelijän liikeyhtälön ratkaisu riippuu alkuehdoista. Epähomogeenisen liikeyhtälön yksittäisratkaisu taas ei riipu alkuehdoista, jolloin ratkaisuksi saadaan </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x(t)={\frac {F_{0}}{Z_{m}\omega }}\sin(\omega t-\phi ),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mi>ω</mi> </mrow> </mfrac> </mrow> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>ω</mi> <mi>t</mi> <mo>−</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x(t)={\frac {F_{0}}{Z_{m}\omega }}\sin(\omega t-\phi ),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d983e1da50de20a0327e94b42847937a5f65739e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:24.832ex; height:5.843ex;" alt="{\displaystyle x(t)={\frac {F_{0}}{Z_{m}\omega }}\sin(\omega t-\phi ),}"></span></dd></dl> <p>missä </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z_{m}={\sqrt {r^{2}+\left(\omega m-{\frac {k}{\omega }}\right)^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mrow> <mi>ω</mi> <mi>m</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>k</mi> <mi>ω</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z_{m}={\sqrt {r^{2}+\left(\omega m-{\frac {k}{\omega }}\right)^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8ce054143e7598e9c9a5367ebe70d6a079a6540" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:26.712ex; height:7.676ex;" alt="{\displaystyle Z_{m}={\sqrt {r^{2}+\left(\omega m-{\frac {k}{\omega }}\right)^{2}}}}"></span></dd></dl> <p>ja </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi =\arctan \left({\frac {\omega m-{\frac {k}{\omega }}}{r}}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> <mo>=</mo> <mi>arctan</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ω</mi> <mi>m</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>k</mi> <mi>ω</mi> </mfrac> </mrow> </mrow> <mi>r</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi =\arctan \left({\frac {\omega m-{\frac {k}{\omega }}}{r}}\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5c7f0083550ad82ea09297a6192cd349d3e3b92" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:24.299ex; height:7.676ex;" alt="{\displaystyle \phi =\arctan \left({\frac {\omega m-{\frac {k}{\omega }}}{r}}\right).}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Katso_myös"><span id="Katso_my.C3.B6s"></span>Katso myös</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Harmoninen_v%C3%A4r%C3%A4htelij%C3%A4&veaction=edit&section=8" title="Muokkaa osiota Katso myös" class="mw-editsection-visualeditor"><span>muokkaa</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Harmoninen_v%C3%A4r%C3%A4htelij%C3%A4&action=edit&section=8" title="Muokkaa osion lähdekoodia: Katso myös"><span>muokkaa wikitekstiä</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Kvanttimekaaninen_harmoninen_v%C3%A4r%C3%A4htelij%C3%A4" title="Kvanttimekaaninen harmoninen värähtelijä">Kvanttimekaaninen harmoninen värähtelijä</a></li> <li><a href="/wiki/Oskillaattori" title="Oskillaattori">Oskillaattori</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Lähteet"><span id="L.C3.A4hteet"></span>Lähteet</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Harmoninen_v%C3%A4r%C3%A4htelij%C3%A4&veaction=edit&section=9" title="Muokkaa osiota Lähteet" class="mw-editsection-visualeditor"><span>muokkaa</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Harmoninen_v%C3%A4r%C3%A4htelij%C3%A4&action=edit&section=9" title="Muokkaa osion lähdekoodia: Lähteet"><span>muokkaa wikitekstiä</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span class="kirjaviite" title="Kirjaviite">M. L. Boas: <i>Mathematical Methods in the Physical Sciences</i>, s. 297.  United States:  John Wiley & Sons, 1983.  <a href="/wiki/Toiminnot:Kirjal%C3%A4hteet/0-471-04409-1" title="Toiminnot:Kirjalähteet/0-471-04409-1">ISBN 0-471-04409-1</a>  <span style="font-size: 0.95em;">(englanniksi)</span></span></li> <li><span class="kirjaviite" title="Kirjaviite">G. R. Fowles & G. L. Cassiday: <i>Analytical Mechanics sixth edition</i>, s. 69.  United States:  Brooks/Cole Pub Co, 1998.  <a href="/wiki/Toiminnot:Kirjal%C3%A4hteet/0-03-022317-2" title="Toiminnot:Kirjalähteet/0-03-022317-2">ISBN 0-03-022317-2</a>  <span style="font-size: 0.95em;">(englanniksi)</span></span></li> <li><span class="kirjaviite" title="Kirjaviite">H. D. Young & R. A. Freedman & T. R. Sandin & A. L. Ford: <i>Sears and Zemansky's University Physics With Modern Physics</i>, s. 392.  United States:  Addison Wesley Publishing Company, 2000.  <a href="/wiki/Toiminnot:Kirjal%C3%A4hteet/0-201-60336-5" title="Toiminnot:Kirjalähteet/0-201-60336-5">ISBN 0-201-60336-5</a>  <span style="font-size: 0.95em;">(englanniksi)</span></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="Aiheesta_muualla">Aiheesta muualla</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Harmoninen_v%C3%A4r%C3%A4htelij%C3%A4&veaction=edit&section=10" title="Muokkaa osiota Aiheesta muualla" class="mw-editsection-visualeditor"><span>muokkaa</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Harmoninen_v%C3%A4r%C3%A4htelij%C3%A4&action=edit&section=10" title="Muokkaa osion lähdekoodia: Aiheesta muualla"><span>muokkaa wikitekstiä</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r22431496">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:#f9f9f9;display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><div class="side-box side-box-right plainlinks sistersitebox"><style data-mw-deduplicate="TemplateStyles:r22718453">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><a href="/wiki/Tiedosto:Commons-logo.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png" decoding="async" width="30" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/45px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/59px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></a></span></div> <div class="side-box-text plainlist"><a href="/wiki/Wikimedia_Commons" title="Wikimedia Commons">Wikimedia Commonsissa</a> on kuvia tai muita tiedostoja aiheesta <b><a href="https://commons.wikimedia.org/wiki/Category:Harmonic_oscillators" class="extiw" title="commons:Category:Harmonic oscillators">Harmoninen värähtelijä</a></b>.</div></div> </div> <ul><li><a rel="nofollow" class="external text" href="http://per.physics.helsinki.fi/kirjasto/ont/ah/ls_123_166.pdf">Harmoninen värähdysliike (pdf)</a> (<a rel="nofollow" class="external text" href="https://web.archive.org/web/20070621164922/http://per.physics.helsinki.fi/kirjasto/ont/ah/ls_123_166.pdf">Arkistoitu</a> – Internet Archive)</li> <li><a rel="nofollow" class="external text" href="http://hypertextbook.com/chaos/41.shtml">Artikkeli harmonisesta värähtelijästä Hypertextbookissa</a></li> <li><a rel="nofollow" class="external text" href="http://qbx6.ltu.edu/s_schneider/physlets/main/osc_damped_driven.shtml">Animaatio vaimennetusta ja pakotetusta harmonisesta värähtelijästä</a> (<a rel="nofollow" class="external text" href="https://web.archive.org/web/20090130041617/http://qbx6.ltu.edu/s_schneider/physlets/main/osc_damped_driven.shtml">Arkistoitu</a> – Internet Archive)</li></ul></div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; 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id="ca-history-sticky-header" tabindex="-1" data-event-name="history-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-history mw-ui-icon-wikimedia-wikimedia-history"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only mw-watchlink" id="ca-watchstar-sticky-header" tabindex="-1" data-event-name="watch-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-star mw-ui-icon-wikimedia-wikimedia-star"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-ve-edit-sticky-header" tabindex="-1" data-event-name="ve-edit-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-edit mw-ui-icon-wikimedia-wikimedia-edit"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-edit-sticky-header" tabindex="-1" data-event-name="wikitext-edit-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-wikiText mw-ui-icon-wikimedia-wikimedia-wikiText"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-viewsource-sticky-header" tabindex="-1" data-event-name="ve-edit-protected-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-editLock mw-ui-icon-wikimedia-wikimedia-editLock"></span> <span></span> </a> </div> <div class="vector-sticky-header-buttons"> <button class="cdx-button cdx-button--weight-quiet mw-interlanguage-selector" id="p-lang-btn-sticky-header" tabindex="-1" data-event-name="ui.dropdown-p-lang-btn-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-language mw-ui-icon-wikimedia-wikimedia-language"></span> <span>46 kieltä</span> </button> <a href="#" class="cdx-button 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Harmoninen värähtelijä – Wikipedia