Normalni mod – Wikipedija/Википедија

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class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">premjesti na bočnu traku</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">sakrij</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">Početak</div> </a> </li> <li id="toc-Spregnuti_oscilatori" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Spregnuti_oscilatori"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Spregnuti oscilatori</span> </div> </a> <ul id="toc-Spregnuti_oscilatori-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Stacionarni_talasi" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Stacionarni_talasi"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Stacionarni talasi</span> </div> </a> <ul id="toc-Stacionarni_talasi-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Elastična_čvrsta_tela" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Elastična_čvrsta_tela"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Elastična čvrsta tela</span> </div> </a> <ul id="toc-Elastična_čvrsta_tela-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Kvantna_mehanika" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Kvantna_mehanika"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Kvantna mehanika</span> </div> </a> <ul id="toc-Kvantna_mehanika-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Povezano" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Povezano"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Povezano</span> </div> </a> <ul id="toc-Povezano-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Literatura" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Literatura"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Literatura</span> </div> </a> <ul id="toc-Literatura-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Spoljašnje_veze" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Spoljašnje_veze"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Spoljašnje veze</span> </div> </a> <ul id="toc-Spoljašnje_veze-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="Sadržaj" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" lang="sh-Latn" dir="ltr"> <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="Prikaži/sakrij sadržaj" > <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">Prikaži/sakrij sadržaj</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" lang="sh-Latn" dir="ltr"><span class="mw-page-title-main">Normalni mod</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" lang="sh-Latn" dir="ltr"> <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="Prijedi na druge jezične varijante članka. Dostupno je na 22 jezika" > <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-22" 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">22 jezika</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Mode_normal" title="Mode normal — Katalonski" lang="ca" hreflang="ca" data-title="Mode normal" data-language-autonym="Català" data-language-local-name="Katalonski" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Vlastn%C3%AD_m%C3%B3d" title="Vlastní mód — Češki" lang="cs" hreflang="cs" data-title="Vlastní mód" data-language-autonym="Čeština" data-language-local-name="Češki" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%9D%D0%BE%D1%80%D0%BC%D0%B0%D0%BB%D0%BB%C4%95_%D1%81%D1%83%D0%BB%D0%BB%D0%B0%D0%BD%D1%83%D1%81%D0%B5%D0%BC" title="Нормаллĕ сулланусем — Čuvaški" lang="cv" hreflang="cv" data-title="Нормаллĕ сулланусем" data-language-autonym="Чӑвашла" data-language-local-name="Čuvaški" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Eigenmode" title="Eigenmode — Nemački" lang="de" hreflang="de" data-title="Eigenmode" data-language-autonym="Deutsch" data-language-local-name="Nemački" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Normal_mode" title="Normal mode — Engleski" lang="en" hreflang="en" data-title="Normal mode" data-language-autonym="English" data-language-local-name="Engleski" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Modo_normal" title="Modo normal — Španski" lang="es" hreflang="es" data-title="Modo normal" data-language-autonym="Español" data-language-local-name="Španski" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Normaalv%C3%B5nkumine" title="Normaalvõnkumine — Estonski" lang="et" hreflang="et" data-title="Normaalvõnkumine" data-language-autonym="Eesti" data-language-local-name="Estonski" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%85%D8%AF_%D9%86%D8%B1%D9%85%D8%A7%D9%84" title="مد نرمال — Persijski" lang="fa" hreflang="fa" data-title="مد نرمال" data-language-autonym="فارسی" data-language-local-name="Persijski" 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/Mode_normal" title="Mode normal — Francuski" lang="fr" hreflang="fr" data-title="Mode normal" data-language-autonym="Français" data-language-local-name="Francuski" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%90%D7%95%D7%A4%D7%A0%D7%99_%D7%AA%D7%A0%D7%95%D7%93%D7%94_%D7%A2%D7%A6%D7%9E%D7%99%D7%99%D7%9D" title="אופני תנודה עצמיים — Hebrejski" lang="he" hreflang="he" data-title="אופני תנודה עצמיים" data-language-autonym="עברית" data-language-local-name="Hebrejski" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%8D%D5%A5%D6%83%D5%A1%D5%AF%D5%A1%D5%B6_%D5%BF%D5%A1%D5%BF%D5%A1%D5%B6%D5%B8%D6%82%D5%B4%D5%B6%D5%A5%D6%80" title="Սեփական տատանումներ — Jermenski" lang="hy" hreflang="hy" data-title="Սեփական տատանումներ" data-language-autonym="Հայերեն" data-language-local-name="Jermenski" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%9B%BA%E6%9C%89%E6%8C%AF%E5%8B%95" title="固有振動 — Japanski" lang="ja" hreflang="ja" data-title="固有振動" data-language-autonym="日本語" data-language-local-name="Japanski" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%9C%D0%B5%D0%BD%D1%88%D1%96%D0%BA%D1%82%D1%96_%D0%B6%D0%B8%D1%96%D0%BB%D1%96%D0%BA" title="Меншікті жиілік — Kozački" lang="kk" hreflang="kk" data-title="Меншікті жиілік" data-language-autonym="Қазақша" data-language-local-name="Kozački" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Normalusis_svyravimas" title="Normalusis svyravimas — Litvanski" lang="lt" hreflang="lt" data-title="Normalusis svyravimas" data-language-autonym="Lietuvių" data-language-local-name="Litvanski" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Eigensvinging" title="Eigensvinging — Norveški njorsk" lang="nn" hreflang="nn" data-title="Eigensvinging" data-language-autonym="Norsk nynorsk" data-language-local-name="Norveški njorsk" 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/Drgania_swobodne" title="Drgania swobodne — Poljski" lang="pl" hreflang="pl" data-title="Drgania swobodne" data-language-autonym="Polski" data-language-local-name="Poljski" 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/Modo_normal" title="Modo normal — Portugalski" lang="pt" hreflang="pt" data-title="Modo normal" data-language-autonym="Português" data-language-local-name="Portugalski" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9D%D0%BE%D1%80%D0%BC%D0%B0%D0%BB%D1%8C%D0%BD%D1%8B%D0%B5_%D0%BA%D0%BE%D0%BB%D0%B5%D0%B1%D0%B0%D0%BD%D0%B8%D1%8F" title="Нормальные колебания — Ruski" lang="ru" hreflang="ru" data-title="Нормальные колебания" data-language-autonym="Русский" data-language-local-name="Ruski" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/Normalni_mod" title="Normalni mod — Srpski" lang="sr" hreflang="sr" data-title="Normalni mod" data-language-autonym="Српски / srpski" data-language-local-name="Srpski" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9D%D0%BE%D1%80%D0%BC%D0%B0%D0%BB%D1%8C%D0%BD%D1%96_%D0%BA%D0%BE%D0%BB%D0%B8%D0%B2%D0%B0%D0%BD%D0%BD%D1%8F" title="Нормальні коливання — Ukrajinski" lang="uk" hreflang="uk" data-title="Нормальні коливання" data-language-autonym="Українська" data-language-local-name="Ukrajinski" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Normal_moda" title="Normal moda — Uzbečki" lang="uz" hreflang="uz" data-title="Normal moda" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbečki" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E7%AE%80%E6%AD%A3%E6%A8%A1" title="简正模 — Kineski" lang="zh" hreflang="zh" data-title="简正模" data-language-autonym="中文" data-language-local-name="Kineski" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q900488#sitelinks-wikipedia" title="Uredi međujezične poveznice" class="wbc-editpage">Uredi veze</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Imenski prostori"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" lang="sh-Latn" dir="ltr"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Normalni_mod" title="Vidi stranicu sadržaja [c]" accesskey="c"><span>Stranica</span></a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/Razgovor:Normalni_mod" rel="discussion" title="Razgovarajte o sadržini ove stranice [t]" accesskey="t"><span>Razgovor</span></a></li> </ul> </div> </div> <div id="vector-variants-dropdown" class="vector-dropdown " lang="sh-Latn" dir="ltr"> <input type="checkbox" id="vector-variants-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-variants-dropdown" class="vector-dropdown-checkbox " aria-label="Promijeni jezičnu varijantu" > <label 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<div id="p-views" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-views" lang="sh-Latn" dir="ltr"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-view" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Normalni_mod"><span>Prikaži</span></a></li><li id="ca-ve-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Normalni_mod&veaction=edit" title="Uredite ovu stranicu [v]" accesskey="v"><span>Uredi</span></a></li><li id="ca-edit" class="collapsible vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Normalni_mod&action=edit" title="Uredite izvorni kod ove stranice [e]" accesskey="e"><span>Uredi kod</span></a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Normalni_mod&action=history" title="Pogledajte prethodne verzije ove stranice [h]" accesskey="h"><span>Historija</span></a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" 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typeof="mw:File/Thumb"><a href="/wiki/Datoteka:1D_normal_modes_(280_kB).gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/9/9b/1D_normal_modes_%28280_kB%29.gif" decoding="async" width="275" height="275" class="mw-file-element" data-file-width="275" data-file-height="275" /></a><figcaption>Različiti normalni modovi u 1D-rešetki.</figcaption></figure> <p><b>Normalni mod</b> jednog <a href="/wiki/Frekvencija" title="Frekvencija">oscilujućeg sistema</a> je način kretanja u kome se svi delovi sistema kreću <a href="/wiki/Sinusoida" title="Sinusoida">sinusoidno</a> na istoj frekvenciji. Frekvencije normalnih modova sistema su poznate kao prirodne ili rezonantne frekvencije. Jedan fizički objekat, kao što je zgrada, most ili molekul, poseduje skup normalnih modova (i korespondirajućih frekvencija) koje su zavisne od njegove strukture i sastava. </p><p>Normalni modovi mehaničkih sistema su jedno-frekventna rešenja jednačina kretanja. Najgeneralnije kretanje sistema je superpozicija njegovih normalnih modova. Modovi su nazvani <i>normalni</i> zato što oni mogu da se kreću nezavisno. Eksitacija jednog moda neće nikad prouzrokovati kretanje drugog moda. U mnogim sistemima to je ekvivalentno redukovanju kolekcije združenih oscilacija u skup razdvojenih efektivnih oscilatora. </p><p>Uobičajeno je da se koristi sistem sastavljen od tela i opruge u ilustrovanju elastičnih struktura. Kada je takav sistem pobuđen na jednoj od njegovih prirodnih frekvencija, sve mase u njegovom sastavu se kreću na istoj frekvenciji. Faze tih masa su iste, tako da one sve prolaze kroz ekvilibrijum i maksimalnu aplitudu istovremeno. Praktični značaj toga se može ilustrovati modelom zgrade. Ako zemljotres pobudi sistem blizo jedne od prirodnih frekvencija, pomeranje jednog sprata u odnosu na drugi - u zavisnosti of moda - može biti maksimalan. Očevidno, zgrade mogu da podnesu ovakva pomeranja do određene tačke. Modelovanje zgrada putem nalaženja njihovih normalnih modova je jedan lak način da se proveri bezbednost građevinskog dizajna. Koncept normalinih modova isto tako nalazi primenu u <a href="/wiki/Talas" class="mw-redirect" title="Talas">talasnoj teoriji</a>, <a href="/wiki/Optika" title="Optika">optici</a>, <a href="/wiki/Kvantna_mehanika" title="Kvantna mehanika">kvantnoj mehanici</a>, i <a href="/wiki/Molekulska_dinamika" title="Molekulska dinamika">molekularnoj dinamici</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Spregnuti_oscilatori">Spregnuti oscilatori</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Normalni_mod&veaction=edit&section=1" title="Uredi odjeljak Spregnuti oscilatori" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Normalni_mod&action=edit&section=1" title="Uredi kôd odjeljka Spregnuti oscilatori"><span>uredi kod</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Posmatrajmo dva ekvivalentana tela (koja nisu pod uticajem gravitacije), jednakih <a href="/wiki/Masa" title="Masa">masa</a> <i>M</i>, koja su povezana sa tri opruge, svaka od kojih ima <a href="/wiki/Opruga" title="Opruga">konstantu opruge</a> <i>K</i>. Tela su povezana na sledeći način: </p> <dl><dd><span typeof="mw:File"><a href="/wiki/Datoteka:Coupled_Harmonic_Oscillator.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3f/Coupled_Harmonic_Oscillator.svg/300px-Coupled_Harmonic_Oscillator.svg.png" decoding="async" width="300" height="78" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3f/Coupled_Harmonic_Oscillator.svg/450px-Coupled_Harmonic_Oscillator.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3f/Coupled_Harmonic_Oscillator.svg/600px-Coupled_Harmonic_Oscillator.svg.png 2x" data-file-width="597" data-file-height="156" /></a></span></dd></dl> <p>gde su krajnje tačke fiksirane i ne mogu se pomeriti. Mi ćemo koristiti <i>x</i><sub>1</sub>(<i>t</i>) da označimo horizontano pomeranje leve mase, i <i>x</i><sub>2</sub>(<i>t</i>) da označimo pomeranje desne mase. Ako obeležimo drugi <a href="/wiki/Izvod" title="Izvod">izvod</a> od <i>x</i>(<i>t</i>) u odnosu na vreme sa <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 {\ddot {x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo>¨</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\ddot {x}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06e0e705ddda28c6cd06cdc6e18be9abf88bb395" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:2.176ex;" alt="{\displaystyle {\ddot {x}}}"></span>, jednačine kretanja su: </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{\ddot {x}}_{1}=-Kx_{1}+K(x_{2}-x_{1})\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo>¨</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mi>K</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mi>K</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M{\ddot {x}}_{1}=-Kx_{1}+K(x_{2}-x_{1})\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dde71729ed33b88886f6472ebe5ba2c66e727437" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-right: -0.387ex; width:28.894ex; height:2.843ex;" alt="{\displaystyle M{\ddot {x}}_{1}=-Kx_{1}+K(x_{2}-x_{1})\,\!}"></span></dd> <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{\ddot {x}}_{2}=-Kx_{2}+K(x_{1}-x_{2})\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo>¨</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mi>K</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mi>K</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M{\ddot {x}}_{2}=-Kx_{2}+K(x_{1}-x_{2})\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d05415f50e472dd09a26c3a36984450b42d120a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-right: -0.387ex; width:28.894ex; height:2.843ex;" alt="{\displaystyle M{\ddot {x}}_{2}=-Kx_{2}+K(x_{1}-x_{2})\,\!}"></span></dd></dl> <p>Pošto očekujemo oscilatorno kretanje, možemo koristiti: </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_{1}(t)=A_{1}e^{i\omega t}\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>ω</mi> <mi>t</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{1}(t)=A_{1}e^{i\omega t}\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f37a5a994e464edbedefc401c36cc1f4e8aa8e84" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-right: -0.387ex; width:14.815ex; height:3.176ex;" alt="{\displaystyle x_{1}(t)=A_{1}e^{i\omega t}\,\!}"></span></dd> <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_{2}(t)=A_{2}e^{i\omega t}\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>ω</mi> <mi>t</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{2}(t)=A_{2}e^{i\omega t}\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48bb6eb5bc286586184866437978d819381ffc7d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-right: -0.387ex; width:14.815ex; height:3.176ex;" alt="{\displaystyle x_{2}(t)=A_{2}e^{i\omega t}\,\!}"></span></dd></dl> <p>Zamena ovih izraza u jednačine kretanja daje: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\omega ^{2}MA_{1}e^{i\omega t}=-2KA_{1}e^{i\omega t}+KA_{2}e^{i\omega t}\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>M</mi> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>ω</mi> <mi>t</mi> </mrow> </msup> <mo>=</mo> <mo>−</mo> <mn>2</mn> <mi>K</mi> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>ω</mi> <mi>t</mi> </mrow> </msup> <mo>+</mo> <mi>K</mi> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>ω</mi> <mi>t</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\omega ^{2}MA_{1}e^{i\omega t}=-2KA_{1}e^{i\omega t}+KA_{2}e^{i\omega t}\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0058ffdaee511e532101f800c80b47f7ff7188b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.387ex; width:39.068ex; height:3.009ex;" alt="{\displaystyle -\omega ^{2}MA_{1}e^{i\omega t}=-2KA_{1}e^{i\omega t}+KA_{2}e^{i\omega t}\,\!}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\omega ^{2}MA_{2}e^{i\omega t}=KA_{1}e^{i\omega t}-2KA_{2}e^{i\omega t}\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>M</mi> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>ω</mi> <mi>t</mi> </mrow> </msup> <mo>=</mo> <mi>K</mi> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>ω</mi> <mi>t</mi> </mrow> </msup> <mo>−</mo> <mn>2</mn> <mi>K</mi> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>ω</mi> <mi>t</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\omega ^{2}MA_{2}e^{i\omega t}=KA_{1}e^{i\omega t}-2KA_{2}e^{i\omega t}\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/36e8e122b49d5243240c07858b9bb1f33b1dbbb2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.387ex; width:37.26ex; height:3.009ex;" alt="{\displaystyle -\omega ^{2}MA_{2}e^{i\omega t}=KA_{1}e^{i\omega t}-2KA_{2}e^{i\omega t}\,\!}"></span></dd></dl> <p>Exponcijalni faktor je zajednički u svim sabircima, tako da se jednačine mogu pojednostaviti: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\omega ^{2}M-2K)A_{1}+KA_{2}=0\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>M</mi> <mo>−</mo> <mn>2</mn> <mi>K</mi> <mo stretchy="false">)</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mi>K</mi> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\omega ^{2}M-2K)A_{1}+KA_{2}=0\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee23db3cfa50096f451be3e3174d97d9022d0576" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-right: -0.387ex; width:27.969ex; height:3.176ex;" alt="{\displaystyle (\omega ^{2}M-2K)A_{1}+KA_{2}=0\,\!}"></span></dd> <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 KA_{1}+(\omega ^{2}M-2K)A_{2}=0\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mo stretchy="false">(</mo> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>M</mi> <mo>−</mo> <mn>2</mn> <mi>K</mi> <mo stretchy="false">)</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle KA_{1}+(\omega ^{2}M-2K)A_{2}=0\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c0b24101894ab024d09f616e7978982f4ef654f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-right: -0.387ex; width:27.969ex; height:3.176ex;" alt="{\displaystyle KA_{1}+(\omega ^{2}M-2K)A_{2}=0\,\!}"></span></dd></dl> <p>Ili u <a href="/wiki/Matrica_(matematika)" title="Matrica (matematika)">matričnoj</a> representaciji: </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 {\begin{bmatrix}\omega ^{2}M-2K&K\\K&\omega ^{2}M-2K\end{bmatrix}}{\begin{pmatrix}A_{1}\\A_{2}\end{pmatrix}}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>M</mi> <mo>−</mo> <mn>2</mn> <mi>K</mi> </mtd> <mtd> <mi>K</mi> </mtd> </mtr> <mtr> <mtd> <mi>K</mi> </mtd> <mtd> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>M</mi> <mo>−</mo> <mn>2</mn> <mi>K</mi> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}\omega ^{2}M-2K&K\\K&\omega ^{2}M-2K\end{bmatrix}}{\begin{pmatrix}A_{1}\\A_{2}\end{pmatrix}}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae13e35fcfd8b15595e4d7bed930f51a82b5bdf9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:38.782ex; height:6.176ex;" alt="{\displaystyle {\begin{bmatrix}\omega ^{2}M-2K&K\\K&\omega ^{2}M-2K\end{bmatrix}}{\begin{pmatrix}A_{1}\\A_{2}\end{pmatrix}}=0}"></span></dd></dl> <p>Da bi ova jednačina imala netrivijalna rešenja, leva matrica mora biti <a href="/wiki/Invertibilna_matrica" title="Invertibilna matrica">singularna</a>, tako da je <a href="/wiki/Determinanta" title="Determinanta">determinanta</a> te matrice jednaka nuli: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\omega ^{2}M-2K)^{2}-K^{2}=0\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>M</mi> <mo>−</mo> <mn>2</mn> <mi>K</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>0</mn> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\omega ^{2}M-2K)^{2}-K^{2}=0\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0985063944dbb3e5ca5edabd18c2d4c4f23238a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-right: -0.387ex; width:24.511ex; height:3.176ex;" alt="{\displaystyle (\omega ^{2}M-2K)^{2}-K^{2}=0\,\!}"></span></dd></dl> <p>Rešavajući po <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>, dobijaju se dva rešenja: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{1}={\sqrt {\frac {K}{M}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <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 \omega _{1}={\sqrt {\frac {K}{M}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40770ecae37ca9722aa86ccbe6c42fef848cf0e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:11.847ex; height:6.176ex;" alt="{\displaystyle \omega _{1}={\sqrt {\frac {K}{M}}},}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{2}={\sqrt {\frac {3K}{M}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mn>3</mn> <mi>K</mi> </mrow> <mi>M</mi> </mfrac> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega _{2}={\sqrt {\frac {3K}{M}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df9bb4b040d7eac95588c563a9da2c4730e3d7ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:12.634ex; height:6.176ex;" alt="{\displaystyle \omega _{2}={\sqrt {\frac {3K}{M}}}.}"></span></dd></dl> <p>Ako zamenimo ω<sub>1</sub> u matricu i rešimo za (<i>A</i><sub>1</sub>, <i>A</i><sub>2</sub>), dobijamo (1, 1). Ako zamenimo ω<sub>2</sub>, dobijamo (1, −1). (Ovi vektori su <a href="/w/index.php?title=Svojstveni_vektori&action=edit&redlink=1" class="new" title="Svojstveni vektori (stranica ne postoji)">svojstveni vektori</a>, i frekvencije su <a href="/w/index.php?title=Svojstvene_vrednosti&action=edit&redlink=1" class="new" title="Svojstvene vrednosti (stranica ne postoji)">svojstvene vrednosti</a>.) </p><p>Prvi normalni mod je: </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 {\begin{pmatrix}x_{1}(t)\\x_{2}(t)\end{pmatrix}}=c_{1}{\begin{pmatrix}1\\1\end{pmatrix}}\cos {(\omega _{1}t+\phi _{1})}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>=</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>t</mi> <mo>+</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}x_{1}(t)\\x_{2}(t)\end{pmatrix}}=c_{1}{\begin{pmatrix}1\\1\end{pmatrix}}\cos {(\omega _{1}t+\phi _{1})}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/616ea459d6e61f410b3202a76dc6534083f19fa8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:34.014ex; height:6.176ex;" alt="{\displaystyle {\begin{pmatrix}x_{1}(t)\\x_{2}(t)\end{pmatrix}}=c_{1}{\begin{pmatrix}1\\1\end{pmatrix}}\cos {(\omega _{1}t+\phi _{1})}}"></span></dd></dl> <p>To odgovara situaciji gde se obe mase kreću u istom smeru. Zbog toga, frekvencija je ista kao da su mase povezane čvrstom cevi. </p><p>Drugi normalni mod je: </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 {\begin{pmatrix}x_{1}(t)\\x_{2}(t)\end{pmatrix}}=c_{2}{\begin{pmatrix}1\\-1\end{pmatrix}}\cos {(\omega _{2}t+\phi _{2})}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>=</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>1</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mi>t</mi> <mo>+</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}x_{1}(t)\\x_{2}(t)\end{pmatrix}}=c_{2}{\begin{pmatrix}1\\-1\end{pmatrix}}\cos {(\omega _{2}t+\phi _{2})}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/936c41b96cd4a067b858e98007093c94948d7302" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:35.823ex; height:6.176ex;" alt="{\displaystyle {\begin{pmatrix}x_{1}(t)\\x_{2}(t)\end{pmatrix}}=c_{2}{\begin{pmatrix}1\\-1\end{pmatrix}}\cos {(\omega _{2}t+\phi _{2})}}"></span></dd></dl> <p>To je slućaj gde se mase kreću u suprotnim smerovima, dok je centar mase stacionaran. Generalno rešenje je superpozicija <b>normalnih modova</b> gde su <i>c</i><sub>1</sub>, <i>c</i><sub>2</sub>, φ<sub>1</sub>, i φ<sub>2</sub>, određeni <a href="/w/index.php?title=Po%C4%8Detnim_uslovima&action=edit&redlink=1" class="new" title="Početnim uslovima (stranica ne postoji)">početnim uslovima</a> problema. </p><p>Demonstrirani proces se može generalisati koristeći formalizam <a href="/w/index.php?title=Lagran%C5%BEeve_mehanike&action=edit&redlink=1" class="new" title="Lagranževe mehanike (stranica ne postoji)">Lagranževe mehanike</a> ili <a href="/w/index.php?title=Hamiltonove_mehanike&action=edit&redlink=1" class="new" title="Hamiltonove mehanike (stranica ne postoji)">Hamiltonove mehanike</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Stacionarni_talasi">Stacionarni talasi</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Normalni_mod&veaction=edit&section=2" title="Uredi odjeljak Stacionarni talasi" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Normalni_mod&action=edit&section=2" title="Uredi kôd odjeljka Stacionarni talasi"><span>uredi kod</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/w/index.php?title=Stacionarni_talas&action=edit&redlink=1" class="new" title="Stacionarni talas (stranica ne postoji)">Stacionarni talas</a> je kontinualna forma normalnog moda. U stojećem talasu, svi prostorni elementi (i.e. (<i>x</i>, <i>y</i>, <i>z</i>) koordinate) osciluju na istoj <a href="/wiki/Frekvencija" title="Frekvencija">frekvenciji</a> i <a href="/w/index.php?title=Faze_talasa&action=edit&redlink=1" class="new" title="Faze talasa (stranica ne postoji)">fazi</a> (dostižući <a href="/w/index.php?title=Mehani%C4%8Dki_ekvilibrijum&action=edit&redlink=1" class="new" title="Mehanički ekvilibrijum (stranica ne postoji)">ekvilibrijsku</a> tačku zajedno), ali imaju različite amplitude. </p><p><span class="mw-default-size" typeof="mw:File"><a href="/wiki/Datoteka:Standing_wave_2.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/7/7d/Standing_wave_2.gif" decoding="async" width="750" height="250" class="mw-file-element" data-file-width="750" data-file-height="250" /></a></span> </p><p>Opšti oblik stojećeg talasa je: </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 \Psi (t)=f(x,y,z)(A\cos(\omega t)+B\sin(\omega t))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ψ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>ω</mi> <mi>t</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>B</mi> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>ω</mi> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Psi (t)=f(x,y,z)(A\cos(\omega t)+B\sin(\omega t))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1161bbc044521e9dc6197dcfa8079a2581a02753" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:39.372ex; height:2.843ex;" alt="{\displaystyle \Psi (t)=f(x,y,z)(A\cos(\omega t)+B\sin(\omega t))}"></span></dd></dl> <p>gde <i>ƒ</i>(<i>x</i>, <i>y</i>, <i>z</i>) predstavlja zavisnost amplitude od lokacije, i cos\sin su oscilacije u funkciji vremena. </p><p>Fizički, stojeći talasi se formiraju <a href="/wiki/Interferencija" title="Interferencija">interferencijom</a> (superpozicijom) talasa i njihovih reflekcija (mada se može kazati i suprotno; da je kretanje talasa <a href="/w/index.php?title=Superpozicija&action=edit&redlink=1" class="new" title="Superpozicija (stranica ne postoji)">superpozicija</a> stacionarnih talasa). Geometrijski oblik sredine određuje oblik interferencije, i stoga određuje <i>ƒ</i>(<i>x</i>, <i>y</i>, <i>z</i>) formu stojećeg talasa. Ova prostorna zavisnost se naziva <b>normalni mod</b>. </p><p>Obično, problemi sa kontinuiranom zavisnošću od (<i>x</i>, <i>y</i>, <i>z</i>) nemaju konačan broj normalnih modova, nego imaju beskonačno mnogo normalnih modova. Ako je problem ograničen (tj. ako je definisan na konačnom prostornom segmentu) onda postoji <a href="/wiki/Prebrojiv_skup" title="Prebrojiv skup">prebrojivo mnogo</a> (diskretno beskonačno) normalnih modova (koji su obično numerisani sa <i>n</i> = 1, 2, 3, ...). Ako problem nije ograničen, postoji kontinualan <a href="/wiki/Elektromagnetski_spektar" title="Elektromagnetski spektar">spektar</a> normalnih modova. </p><p>Dozvoljene frekvencije su zavisne od normalnih modova, kao i od fizičkih konstanti problema (<a href="/wiki/Gustina" title="Gustina">gustina</a>, <a href="/wiki/Napon" class="mw-redirect" title="Napon">napon</a>, <a href="/wiki/Pritisak" class="mw-redirect" title="Pritisak">pritisak</a>, itd. ) koji određuju <a href="/w/index.php?title=Fazna_brzina&action=edit&redlink=1" class="new" title="Fazna brzina (stranica ne postoji)">faznu brzinu</a> talasa. Skup svih mogućih normalnih frekvencija se zove <a href="/w/index.php?title=Frekvenciski_spektar&action=edit&redlink=1" class="new" title="Frekvenciski spektar (stranica ne postoji)">frekvenciski spektar</a>. Obično je svaka frekvencija modulisana amplitudom na kojoj je nastala, što stvara grafikon [spektralne snage] oscilacija. </p><p>U <a href="/wiki/Muzika" title="Muzika">muzičkom</a> smislu, normalni modovi vibracionog instrumenta (gudačke žice, vazdušna cev, bubnjevi, etc.) se zovu "<a href="/w/index.php?title=Harmonik&action=edit&redlink=1" class="new" title="Harmonik (stranica ne postoji)">harmonici</a>" ili "<a href="/w/index.php?title=Vi%C5%A1i_harmonici&action=edit&redlink=1" class="new" title="Viši harmonici (stranica ne postoji)">viši harmonici</a>". </p> <div class="mw-heading mw-heading2"><h2 id="Elastična_čvrsta_tela"><span id="Elasti.C4.8Dna_.C4.8Dvrsta_tela"></span>Elastična čvrsta tela</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Normalni_mod&veaction=edit&section=3" title="Uredi odjeljak Elastična čvrsta tela" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Normalni_mod&action=edit&section=3" title="Uredi kôd odjeljka Elastična čvrsta tela"><span>uredi kod</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Vidi: <a href="/w/index.php?title=Ajn%C5%A1tajnovo_%C4%8Dvrsto_telo&action=edit&redlink=1" class="new" title="Ajnštajnovo čvrsto telo (stranica ne postoji)">Ajnštajnovo čvrsto telo</a> i <a href="/w/index.php?title=Debajev_model&action=edit&redlink=1" class="new" title="Debajev model (stranica ne postoji)">Debajev model</a> </p><p>U svakom čvrstom telu na bilo kojoj temperaturi, primarne čestice (tj. atomi ili molekule) nisu stacionarni, nego vibriraju oko središnih pozicija. Karakteristika izolacionih materijala da ne sprovode toplotnu energiju je skoro isključivo zasnovan na ovim vibracijama. Mnoge fizičke osobine čvrstih tela (kao što je elastični modul) mogu se predvideti ako su poznate frekvencije na kojima čestice osciluju. Najjednostavnija pretpostavka (po Ajnštajnu) je da sve čestice osciluju oko svojih srednjih pozicija na nakoj prirodnoj frekvenciji <i>ν</i>. To je ekvivalentno pretpostavci da svi atomi vibriraju nezavisno na frekvenciji <i>ν</i>. Ajnštajn je isto tako pretpostavio da su dozvoljena energetska stanja tih oscilacija harmonici, ili integralni proizvod od <i>hν</i>. Spektar talasnih formi se može matematički opisati koristeći Furijeve redove sinusoidnih gustinskih fluktuacija (ili termalnih fotona). </p> <figure typeof="mw:File/Thumb"><a href="/wiki/Datoteka:Harmonic_partials_on_strings.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c5/Harmonic_partials_on_strings.svg/250px-Harmonic_partials_on_strings.svg.png" decoding="async" width="250" height="238" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c5/Harmonic_partials_on_strings.svg/375px-Harmonic_partials_on_strings.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c5/Harmonic_partials_on_strings.svg/500px-Harmonic_partials_on_strings.svg.png 2x" data-file-width="620" data-file-height="590" /></a><figcaption><a class="external text" href="https://en.wikipedia.org/wiki/Fundamental_frequency">Fundamentalna frekvencija</a> i prvih šest harmonika vibrirajuće niti. Matematika talasne propagacije u kristalnim čvrstim telima se sastoji od tretiranja harmonika kao da su idealni <a href="/wiki/Furijeovi_redovi" class="mw-redirect" title="Furijeovi redovi">Furijeovi redovi</a> sinusoidnih gustinskih fluktuacija (ili atomskih talasa).</figcaption></figure> <p>Debaj je kasnije uvideo da je svaki oscilator stalno i intimatno spregnut sa susednim oscilatorima. S tim na umu, i zamenjujući Anjštajnove identične nespregnute oscilatore sa istim brojem spregnutih oscilatora, Debaj je povezao elastične vibracije jedno-dimenzionalnog čvrstih tela sa brojem specijalnih vibracionih modova rastegnute niti (vidite sliku). Ćist tone najnižeg nivoa ili frekvencije se naziva fundamentalni ton, a umnošci te frekvencije se zovu viši harmonici. On je dodelio jednom oscilatory frekvenciju fundamentalne vibracije celog čvrstog tela. Ostalim oscilatorima je dodelio harmoničke frekvencije koje su relativne u odnosu na fundamentalnu frekvenciju, tako da je najviša frekvencija ograničena kretanjem najmanje primarne celine. </p><p>Normalni modovi vibracija kristala su u upšenom smislu superpozicije mnogih harmonika, gde svaki ima odgovarajuću amplitudu i fazu. Više talasne dužine (niže frekvencije) fotona su ekvivalentne akustičnim vibracijama. Obe vrste talasa, longitudalni i tranverzalni, se mogu propagirati kroz čvrsta tela, dok se u principu, samo longitudalni talasi prenose u tečnostima. </p><p>U longitudalnom (ili akustičkom) modu, pomeranje čestica oko njihovih ekvilibriskih pozicija je koinsidentno sa propagacionim pravcem talasa. Mehanički longitudalni talasi se mogu smatrati <i>kompresivnim talasima</i>. U transverzalnim (ili optičkim) modovima, individualne čestice se kreću perpendikularno na pravac propagacije talasa. </p><p>Sa gledišta kvantne teorije, srednja energija normalnog vibracionog moda kristala sa karakterističnom frekvencijom <i>υ</i> je: </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 E(v)={\frac {1}{2}}hv+{\frac {hv}{e^{hv/kt}-1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>h</mi> <mi>v</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>h</mi> <mi>v</mi> </mrow> <mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>k</mi> <mi>t</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E(v)={\frac {1}{2}}hv+{\frac {hv}{e^{hv/kt}-1}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/982b375bc40da4518a579e72239423ccff9a0b2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:25.288ex; height:5.843ex;" alt="{\displaystyle E(v)={\frac {1}{2}}hv+{\frac {hv}{e^{hv/kt}-1}}}"></span> </p><p>Član (1/2)<i>hυ</i> predstavlja "energiju nulte tačke", ili energiju koju će jedan oscilator imati u apsolutnoj nuli. <i>E</i> (<i>ν</i> ) postaje klasična vrednost <i>kT</i> na visokim temperaturama. </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 E(v)=kT\left[1+{\frac {1}{12}}{\frac {hv^{2}}{kT}}+O\left({\frac {hv}{kT}}\right)^{4}+\ldots \right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>k</mi> <mi>T</mi> <mrow> <mo>[</mo> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>12</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>h</mi> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mi>k</mi> <mi>T</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mi>O</mi> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>h</mi> <mi>v</mi> </mrow> <mrow> <mi>k</mi> <mi>T</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <mo>…</mo> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E(v)=kT\left[1+{\frac {1}{12}}{\frac {hv^{2}}{kT}}+O\left({\frac {hv}{kT}}\right)^{4}+\ldots \right]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/498384413b7c28ef5011979f4faab6e0d54656e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:43.613ex; height:7.509ex;" alt="{\displaystyle E(v)=kT\left[1+{\frac {1}{12}}{\frac {hv^{2}}{kT}}+O\left({\frac {hv}{kT}}\right)^{4}+\ldots \right]}"></span> </p><p>Entropija po normalnom modu je: </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 {\begin{aligned}S\left(v\right)&=\int _{0}^{T}{\frac {d}{dT}}E\left(v\right){\frac {dT}{T}}\\&={\frac {E\left(v\right)}{T}}-k\log \left(1-e^{-{\frac {hv}{kT}}}\right)\\\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>S</mi> <mrow> <mo>(</mo> <mi>v</mi> <mo>)</mo> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>T</mi> </mrow> </mfrac> </mrow> <mi>E</mi> <mrow> <mo>(</mo> <mi>v</mi> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>T</mi> </mrow> <mi>T</mi> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>E</mi> <mrow> <mo>(</mo> <mi>v</mi> <mo>)</mo> </mrow> </mrow> <mi>T</mi> </mfrac> </mrow> <mo>−</mo> <mi>k</mi> <mi>log</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>h</mi> <mi>v</mi> </mrow> <mrow> <mi>k</mi> <mi>T</mi> </mrow> </mfrac> </mrow> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}S\left(v\right)&=\int _{0}^{T}{\frac {d}{dT}}E\left(v\right){\frac {dT}{T}}\\&={\frac {E\left(v\right)}{T}}-k\log \left(1-e^{-{\frac {hv}{kT}}}\right)\\\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e37121733ea5205401de1854b076a5041fed21eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.838ex; width:34.509ex; height:12.843ex;" alt="{\displaystyle {\begin{aligned}S\left(v\right)&=\int _{0}^{T}{\frac {d}{dT}}E\left(v\right){\frac {dT}{T}}\\&={\frac {E\left(v\right)}{T}}-k\log \left(1-e^{-{\frac {hv}{kT}}}\right)\\\end{aligned}}}"></span> </p><p>Slobodna energija je: </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 F(v)=E-TS=kT\log \left(1-e^{-{\frac {hv}{kT}}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>E</mi> <mo>−</mo> <mi>T</mi> <mi>S</mi> <mo>=</mo> <mi>k</mi> <mi>T</mi> <mi>log</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>h</mi> <mi>v</mi> </mrow> <mrow> <mi>k</mi> <mi>T</mi> </mrow> </mfrac> </mrow> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(v)=E-TS=kT\log \left(1-e^{-{\frac {hv}{kT}}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/128e891fdcdadb65f1f050e70bdc52bce284f9b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:37.322ex; height:6.176ex;" alt="{\displaystyle F(v)=E-TS=kT\log \left(1-e^{-{\frac {hv}{kT}}}\right)}"></span> </p><p>koja, za <i>kT</i> >> <i>hν</i>, keži ka: </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 F(v)=kT\log \left({\frac {hv}{kT}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>k</mi> <mi>T</mi> <mi>log</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>h</mi> <mi>v</mi> </mrow> <mrow> <mi>k</mi> <mi>T</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(v)=kT\log \left({\frac {hv}{kT}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b0ce5ca288b626f5056218f208cf8359360750b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:21.087ex; height:6.176ex;" alt="{\displaystyle F(v)=kT\log \left({\frac {hv}{kT}}\right)}"></span> </p><p>Da bi smo izračunali unutrašnju energiju i specificičnu toplotu, moramo da znamo broj normalnih vibracionih modova frekvencije između vrednosti <i>ν</i> i <i>ν</i> + <i>dν</i>. Ako dozvolimo tom broju da bude <i>f</i> (ν)dν, i budući da je totalni broj normalnih modova 3<i>N</i>, funkcija <i>f</i> (ν) je data sa: </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 \int f(v)dv=3N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∫</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mi>d</mi> <mi>v</mi> <mo>=</mo> <mn>3</mn> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int f(v)dv=3N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1af5f2b0fd7e70543e726901591e20edf0106f51" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:15.464ex; height:5.676ex;" alt="{\displaystyle \int f(v)dv=3N}"></span> </p><p>Rešavajući integral se preko svih frekvencija kristala, dobija se unutranšnja energija, <i>U</i>: </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 U=\int f(v)E(v)dv}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo>=</mo> <mo>∫</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mi>E</mi> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mi>d</mi> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U=\int f(v)E(v)dv}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95b3d40dd0bcf10424054231f7db86921faee704" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:18.733ex; height:5.676ex;" alt="{\displaystyle U=\int f(v)E(v)dv}"></span> </p> <div class="mw-heading mw-heading2"><h2 id="Kvantna_mehanika">Kvantna mehanika</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Normalni_mod&veaction=edit&section=4" title="Uredi odjeljak Kvantna mehanika" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Normalni_mod&action=edit&section=4" title="Uredi kôd odjeljka Kvantna mehanika"><span>uredi kod</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>U <a href="/wiki/Kvantna_mehanika" title="Kvantna mehanika">kvantnoj mehanici</a>, stanje <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 \ |\psi \rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ |\psi \rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/343b1c721361b6e33502b16dd841a5c9f76dc4ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.645ex; height:2.843ex;" alt="{\displaystyle \ |\psi \rangle }"></span> sistema se opisuje <a href="/wiki/Talasna_funkcija" class="mw-redirect" title="Talasna funkcija">talasnom funkcijom</a> <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 \ \psi (x,t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \psi (x,t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/154e28619c82c083f51cab1d1741b9fac39f510e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.106ex; height:2.843ex;" alt="{\displaystyle \ \psi (x,t)}"></span> koja je rešenje <a href="/wiki/%C5%A0redingerova_jedna%C4%8Dina" class="mw-redirect" title="Šredingerova jednačina">Šredingerove jednačine</a>. Kvadrat apsolutne vrednosti od <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 \ \psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \psi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/133059010c33954db6f4c65aa66e9960807f4e2e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.094ex; height:2.509ex;" alt="{\displaystyle \ \psi }"></span> ,i.e. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ P(x,t)=|\psi (x,t)|^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ P(x,t)=|\psi (x,t)|^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4662d70fb95fde0fd05cc318fda0a6cf78934254" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.311ex; height:3.343ex;" alt="{\displaystyle \ P(x,t)=|\psi (x,t)|^{2}}"></span></dd></dl> <p>je <a href="/wiki/Verovatno%C4%87a" class="mw-redirect" title="Verovatnoća">verovatnoća</a> nalaženja čestice u mestu <i>x</i> u <a href="/wiki/Vreme" title="Vreme">vremenu</a> <i>t</i>. Obično, ova jednačina opisuje neku vrstu <a href="/wiki/Elektri%C4%8Dni_potencijal" title="Električni potencijal">potencijala</a>, u kom slučaju talasna funkcija se razlaže u superpoziciju energetskih <a class="external text" href="https://en.wikipedia.org/wiki/Eigenstate">eigen stanja</a>, koja osciluju sa frekvencijom <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 =E_{n}/\hbar }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> <mo>=</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi class="MJX-variant">ℏ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega =E_{n}/\hbar }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb847bc137d57d07c046d7563044972b81e384d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.947ex; height:2.843ex;" alt="{\displaystyle \omega =E_{n}/\hbar }"></span>. Na taj način se može napisati: </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 |\psi (t)\rangle =\sum _{n}|n\rangle \left\langle n|\psi (t=0)\right\rangle e^{-iE_{n}t/\hbar }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>n</mi> <mo fence="false" stretchy="false">⟩</mo> <mrow> <mo>⟨</mo> <mrow> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>=</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> <mo>⟩</mo> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi class="MJX-variant">ℏ</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\psi (t)\rangle =\sum _{n}|n\rangle \left\langle n|\psi (t=0)\right\rangle e^{-iE_{n}t/\hbar }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/327dbc3f91a445b2e1f18e0ee743d55ec58fde7c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:36.227ex; height:5.509ex;" alt="{\displaystyle |\psi (t)\rangle =\sum _{n}|n\rangle \left\langle n|\psi (t=0)\right\rangle e^{-iE_{n}t/\hbar }}"></span></dd></dl> <p>Eigen stanja imaju fizička značenja, i nisu samo <a class="external text" href="https://en.wikipedia.org/wiki/Orthonormal_basis">orto normalne osnove</a>. Kad se energija sistema meri, talasna funkcija se svodi u jedno od svojih eigen stanja, i tako se talasna funkcija čestice opisuje čistim eigen stanjem koje odgovara merenoj <a href="/wiki/Energija" title="Energija">energiji</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Povezano">Povezano</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Normalni_mod&veaction=edit&section=5" title="Uredi odjeljak Povezano" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Normalni_mod&action=edit&section=5" title="Uredi kôd odjeljka Povezano"><span>uredi kod</span></a><span class="mw-editsection-bracket">]</span></span></div> <div style="-moz-column-count:3; column-count:3;"> <ul><li>Specifični tipovi: <ul><li><a href="/w/index.php?title=Longitudalni_mod&action=edit&redlink=1" class="new" title="Longitudalni mod (stranica ne postoji)">Longitudalni mod</a></li> <li><a href="/w/index.php?title=Transverzni_mod&action=edit&redlink=1" class="new" title="Transverzni mod (stranica ne postoji)">Transverzni mod</a></li></ul></li> <li>Fizičke aplikacije: <ul><li><a href="/wiki/Talas_(fizika)" class="mw-redirect" title="Talas (fizika)">Talasi</a></li> <li><a href="/wiki/Optika" title="Optika">Optika</a></li> <li><a href="/wiki/Harmonijski_oscilatori" class="mw-redirect" title="Harmonijski oscilatori">Harmonijski oscilatori</a></li> <li><a href="/w/index.php?title=Vibraciona_spektroskopija&action=edit&redlink=1" class="new" title="Vibraciona spektroskopija (stranica ne postoji)">Vibraciona spektroskopija</a></li> <li><a href="/wiki/Kvantna_teorija" class="mw-redirect" title="Kvantna teorija">Kvantna teorija</a> <ul><li><a href="/wiki/%C5%A0redingerova_jedna%C4%8Dina" class="mw-redirect" title="Šredingerova jednačina">Šredingerova jednačina</a></li> <li><a href="/wiki/Talasna_funkcija" class="mw-redirect" title="Talasna funkcija">Talasna funkcija</a></li> <li><a href="/w/index.php?title=Merenja_u_kvantnoj_mehanici&action=edit&redlink=1" class="new" title="Merenja u kvantnoj mehanici (stranica ne postoji)">Merenja u kvantnoj mehanici</a></li></ul></li> <li><a href="/w/index.php?title=Harmoni%C4%8Dne_serije_(muzika)&action=edit&redlink=1" class="new" title="Harmonične serije (muzika) (stranica ne postoji)">Harmonične serije (muzika)</a></li> <li><a href="/wiki/Seizmologija" title="Seizmologija">Seizmologija</a></li> <li><a href="/w/index.php?title=Protiv_zemljotresni_en%C5%BEinjering&action=edit&redlink=1" class="new" title="Protiv zemljotresni enžinjering (stranica ne postoji)">Protiv zemljotresni enžinjering</a></li> <li><a href="/w/index.php?title=Mod_oblika&action=edit&redlink=1" class="new" title="Mod oblika (stranica ne postoji)">Mod oblika</a></li> <li><a href="/w/index.php?title=Sobna_akustika&action=edit&redlink=1" class="new" title="Sobna akustika (stranica ne postoji)">Sobna akustika</a></li></ul></li> <li>Matematički alati: <ul><li><a href="/wiki/Linearna_algebra" title="Linearna algebra">Linearna algebra</a></li> <li><a href="/w/index.php?title=Eigen_vektori&action=edit&redlink=1" class="new" title="Eigen vektori (stranica ne postoji)">Eigen vektori</a></li> <li><a href="/wiki/Diferencijalne_jedna%C4%8Dine" class="mw-redirect" title="Diferencijalne jednačine">Diferencijalne jednačine</a></li> <li><a href="/wiki/Furijeov_red" title="Furijeov red">Furijeova analiza</a></li> <li><a href="/w/index.php?title=Sturm%E2%80%93Liouville_teorija&action=edit&redlink=1" class="new" title="Sturm–Liouville teorija (stranica ne postoji)">Sturm–Liouville teorija</a></li> <li><a href="/w/index.php?title=Problem_Grani%C4%8Dnih_Vrednoti&action=edit&redlink=1" class="new" title="Problem Graničnih Vrednoti (stranica ne postoji)">Problem Graničnih Vrednoti</a></li></ul></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="Literatura">Literatura</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Normalni_mod&veaction=edit&section=6" title="Uredi odjeljak Literatura" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Normalni_mod&action=edit&section=6" title="Uredi kôd odjeljka Literatura"><span>uredi kod</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span class="citation book">Blevins, Robert D. (1979). <a rel="nofollow" class="external text" href="https://archive.org/details/formulasfornatur0000blev"><i>Formulas for natural frequency and mode shape</i></a>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Formulas+for+natural+frequency+and+mode+shape&rft.aulast=Blevins&rft.aufirst=Robert+D.&rft.au=Blevins%2C%26%2332%3BRobert+D.&rft.date=1979&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fformulasfornatur0000blev&rfr_id=info:sid/en.wikipedia.org:Normalni_mod"><span style="display: none;"> </span></span></li> <li><span class="citation book">Tzou, H. S.; Bergman, L. A.. <i>Dynamics and Control of Distributed Systems</i>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Dynamics+and+Control+of+Distributed+Systems&rft.aulast=Tzou&rft.aufirst=H.+S.&rft.au=Tzou%2C%26%2332%3BH.+S.&rft.au=Bergman%2C%26%2332%3BL.+A.&rfr_id=info:sid/en.wikipedia.org:Normalni_mod"><span style="display: none;"> </span></span></li> <li><span class="citation book">Deuss, Arwen (2010–2011). <i>Physics of the Earth as a Planet Lecture Notes</i>. Cambridge University.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Physics+of+the+Earth+as+a+Planet+Lecture+Notes&rft.aulast=Deuss&rft.aufirst=Arwen&rft.au=Deuss%2C%26%2332%3BArwen&rft.date=2010%E2%80%932011&rft.pub=Cambridge+University&rfr_id=info:sid/en.wikipedia.org:Normalni_mod"><span style="display: none;"> </span></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="Spoljašnje_veze"><span id="Spolja.C5.A1nje_veze"></span>Spoljašnje veze</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Normalni_mod&veaction=edit&section=7" title="Uredi odjeljak Spoljašnje veze" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Normalni_mod&action=edit&section=7" title="Uredi kôd odjeljka Spoljašnje veze"><span>uredi kod</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="http://www.falstad.com/coupled/">Simulacija spregnutih oscilatora</a>.</li> <li>Simulacija normalnih modova <a rel="nofollow" class="external text" href="http://www.falstad.com/loadedstring/">kanapa</a>, <a rel="nofollow" class="external text" href="http://www.falstad.com/circosc/">bubnja</a>, and <a rel="nofollow" class="external text" href="http://www.falstad.com/barwaves/">šipke</a>.</li> <li><a rel="nofollow" class="external text" href="http://public.fotki.com/ROBERT1010/scitech/coffee_a_la_mode_hi.html">Fotograpija šolje kafe koja vibrira na frekvenciji normalnog moda</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20100113151258/http://public.fotki.com/ROBERT1010/scitech/coffee_a_la_mode_hi.html">Arhivirano</a> 2010-01-13 na <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine-u</a></li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="" lang="sh-Latn" dir="ltr">Izvor: <a dir="ltr" href="https://sh.wikipedia.org/w/index.php?title=Normalni_mod&oldid=42183771">https://sh.wikipedia.org/w/index.php?title=Normalni_mod&oldid=42183771</a></div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Posebno:Kategorije" title="Posebno:Kategorije">Kategorije</a>: <ul><li><a href="/wiki/Kategorija:Obi%C4%8Dne_diferencijalne_jedna%C4%8Dine" title="Kategorija:Obične diferencijalne jednačine">Obične diferencijalne jednačine</a></li><li><a href="/wiki/Kategorija:Klasi%C4%8Dna_mehanika" title="Kategorija:Klasična mehanika">Klasična mehanika</a></li><li><a href="/wiki/Kategorija:Kvantna_mehanika" title="Kategorija:Kvantna mehanika">Kvantna mehanika</a></li><li><a href="/wiki/Kategorija:Spektroskopija" title="Kategorija:Spektroskopija">Spektroskopija</a></li><li><a href="/wiki/Kategorija:Dekompozicija_singularne_vrednosti" title="Kategorija:Dekompozicija singularne vrednosti">Dekompozicija singularne vrednosti</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Sakrivena kategorija: <ul><li><a href="/wiki/Kategorija:Webarchive_template_wayback_links" title="Kategorija:Webarchive template wayback links">Webarchive template wayback links</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" lang="sh-Latn" dir="ltr"> <ul id="footer-info"> <li id="footer-info-lastmod"> Ova stranica je posljednji put izmijenjena 18. augusta 2024. u 10:47.</li> <li id="footer-info-copyright">Tekst je dostupan pod licencom <a rel="nofollow" class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-ShareAlike</a>; mogu se primijeniti i dodatni uslovi. 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Normalni mod – Wikipedija/Википедија