<?xml version='1.0' encoding='UTF-8' ?>
<!DOCTYPE html
  SYSTEM "about:legacy-compat">
<html xmlns:mml = "http://www.w3.org/1998/Math/MathML" lang = "en"><head><meta charset = "UTF-8"/><meta name = "copyright" content = "(C) Copyright 2020"/><meta name = "DC.rights.owner" content = "(C) Copyright 2020"/><meta name = "DC.type" content = "concept"/><meta name = "abstract" content = "This section describes the theory of eigenfrequency."/><meta name = "description" content = "This section describes the theory of eigenfrequency."/><meta name = "DC.format" content = "HTML5"/><meta name = "DC.identifier" content = "tso-c-usr-terms-eigfreqOvw"/><meta name = "DC.language" content = "en"/><link rel = "stylesheet" type = "text/css" href = "../DSDocUI_XML34.css"/><title>Overview of Eigenfrequency</title>
<script type = "text/javascript" src = "../DSDocUI_Highlight34.js">
  	/* */
  	</script><script type = "text/javascript" src = "../MathJax/MathJax.js?config=DS-default,local/local">
  		/* */
  		</script></head><body onLoad = "highlightSearchTerms();" id = "tso-c-usr-terms-eigfreqOvw">
<a name = "hj-top"> </a><table class = "table1" id = "table11"><tr><td><table class = "DocHeader"><tr><td class = "DocHeader1" colspan = "2"><h1>Overview of Eigenfrequency</h1></td></tr><tr><td class = "DocHeader4" colspan = "2"/></tr><tr><td class = "DocHeader3" colspan = "2"><table class = "DocThemeIntro" id = "table12"><tr><td class = "Intro1Only"><p class = "header"><p class = "abstract">
<span class = "shortdesc">This section describes the theory of eigenfrequency.</span>

</p>
<ul><li><a href = "#tso-c-usr-terms-eigfreqOvw__tso-c-usr-terms-eigfreqOvw-anaType" id = "toc_rg" title = "">Analysis Types: Modal Analysis</a></li></ul>
</p></td></tr></table></td></tr></table>




<div class = "body conbody">
<table class = "table" id = "tso-c-usr-terms-eigfreqOvw__xx933895"><caption/><colgroup><col/><col/></colgroup><thead class = "thead">
<tr class = "row">
<th class = "entry" id = "tso-c-usr-terms-eigfreqOvw__xx933895__entry__1"><p>Parameter Name</p></th>
<th class = "entry" id = "tso-c-usr-terms-eigfreqOvw__xx933895__entry__2"><p>Formula</p></th>
</tr>
</thead><tbody class = "tbody">
<tr class = "row">
<td class = "entry" headers = "tso-c-usr-terms-eigfreqOvw__xx933895__entry__1"><p>DYN_FREQ</p></td>
<td class = "entry" headers = "tso-c-usr-terms-eigfreqOvw__xx933895__entry__2"><span class = "ph inlineequation"><math class = "- topic/foreign "><mrow class = "- topic/foreign "><msub class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">f</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">j</mi></mrow></msub></mrow></math></span></td>
</tr>
<tr class = "row">
<td class = "entry" headers = "tso-c-usr-terms-eigfreqOvw__xx933895__entry__1"><p>DYN_FREQ_KREISSEL</p></td>
<td class = "entry" headers = "tso-c-usr-terms-eigfreqOvw__xx933895__entry__2"><span class = "ph inlineequation"><math class = "- topic/foreign "><mo class = "- topic/foreign ">-</mo><mfrac class = "- topic/foreign "><mrow class = "- topic/foreign "><mn class = "- topic/foreign ">1</mn></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">k</mi></mrow></mfrac><mo class = "- topic/foreign ">ln</mo><mfenced open = "(" close = ")" separators = "" class = "- topic/foreign "><munder class = "- topic/foreign "><mo class = "- topic/foreign ">∑</mo><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">j</mi></mrow></munder><msup class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">e</mi></mrow><mrow class = "- topic/foreign "><mo class = "- topic/foreign ">−</mo><mi class = "- topic/foreign ">k</mi><msub class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">f</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">j</mi></mrow></msub></mrow></msup></mfenced></math></span>
by default
<span class = "ph inlineequation"><math class = "- topic/foreign "><mi class = "- topic/foreign ">k</mi><mo class = "- topic/foreign ">=</mo><mfrac class = "- topic/foreign "><mrow class = "- topic/foreign "><mn class = "- topic/foreign ">30</mn></mrow><mrow class = "- topic/foreign "><msub class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">f</mi></mrow><mrow class = "- topic/foreign "><mtext class = "- topic/foreign ">min</mtext></mrow></msub></mrow></mfrac></math></span>
</td>
</tr>
</tbody></table>

<div class = "section" id = "tso-c-usr-terms-eigfreqOvw__tso-c-usr-terms-eigfreqOvw-anaType"><h2 class = "title sectiontitle">Analysis Types: Modal Analysis</h2>

<table class = "table" id = "tso-c-usr-terms-eigfreqOvw__xx934194"><caption/><colgroup><col/></colgroup><tbody class = "tbody">
<tr class = "row">
<td class = "entry"><span class = "ph inlineequation"><math class = "- topic/foreign "><mrow class = "- topic/foreign "><mrow class = "- topic/foreign "><mo class = "- topic/foreign ">(</mo><mn class = "- topic/foreign ">4</mn><msup class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">π</mi></mrow><mrow class = "- topic/foreign "><mn class = "- topic/foreign ">2</mn></mrow></msup><msup class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">f</mi></mrow><mrow class = "- topic/foreign "><mn class = "- topic/foreign ">2</mn></mrow></msup><mi class = "- topic/foreign ">M</mi><mo class = "- topic/foreign ">−</mo><mi class = "- topic/foreign ">K</mi><mo class = "- topic/foreign ">)</mo><mi class = "- topic/foreign ">φ</mi><mo class = "- topic/foreign ">=</mo><mn class = "- topic/foreign ">0</mn></mrow></mrow></math></span></td>
</tr>
</tbody></table>

<p> For eigenvalues, the following table shows the allowed combinations between the strategy and the
    items <code class = "ph codeph">OBJ_FUNC</code> and <code class = "ph codeph">CONSTRAINT</code> with C for
    controller and S for sensitivity-based optimization. </p>
<table class = "table" id = "tso-c-usr-terms-eigfreqOvw__xx933952"><caption/><colgroup><col/><col/><col/><col/><col/></colgroup><thead class = "thead">
<tr class = "row">
<th class = "entry" id = "tso-c-usr-terms-eigfreqOvw__xx933952__entry__1"/>
<th class = "entry" id = "tso-c-usr-terms-eigfreqOvw__xx933952__entry__2"><p>TOPO</p></th>
<th class = "entry" id = "tso-c-usr-terms-eigfreqOvw__xx933952__entry__3"><p>SHAPE</p></th>
<th class = "entry" id = "tso-c-usr-terms-eigfreqOvw__xx933952__entry__4"><p>BEAD</p></th>
<th class = "entry" id = "tso-c-usr-terms-eigfreqOvw__xx933952__entry__5"><p>SIZING</p></th>
</tr>
</thead><tbody class = "tbody">
<tr class = "row">
<td class = "entry" headers = "tso-c-usr-terms-eigfreqOvw__xx933952__entry__1"><p>OBJ_FUNC</p></td>
<td class = "entry" headers = "tso-c-usr-terms-eigfreqOvw__xx933952__entry__2"><p>S*</p></td>
<td class = "entry" headers = "tso-c-usr-terms-eigfreqOvw__xx933952__entry__3"><p>C*, S*</p></td>
<td class = "entry" headers = "tso-c-usr-terms-eigfreqOvw__xx933952__entry__4"><p>C, S*</p></td>
<td class = "entry" headers = "tso-c-usr-terms-eigfreqOvw__xx933952__entry__5"><p>S*</p></td>
</tr>
<tr class = "row">
<td class = "entry" headers = "tso-c-usr-terms-eigfreqOvw__xx933952__entry__1"><p>CONSTRAINT</p></td>
<td class = "entry" headers = "tso-c-usr-terms-eigfreqOvw__xx933952__entry__2"><p>S</p></td>
<td class = "entry" headers = "tso-c-usr-terms-eigfreqOvw__xx933952__entry__3"><p>S</p></td>
<td class = "entry" headers = "tso-c-usr-terms-eigfreqOvw__xx933952__entry__4"><p>S</p></td>
<td class = "entry" headers = "tso-c-usr-terms-eigfreqOvw__xx933952__entry__5"><p>S</p></td>
</tr>
</tbody></table>
<p> *The Kreisselmeier-Steinhauser formulation is only allowed in objective function. </p>
<p>Eigenvalues are the simplest dynamic responses in structural mechanics.
Typical optimization tasks for modal analysis would be to:</p>
<ol class = "ol">
<li class = "li">Maximize the first eigenfrequency (first natural mode)</li>
<li class = "li">Constrain an eigenfrequency to be higher or lower than a given value</li>
<li class = "li">Maximize or minimize an eigenfrequency at a certain mode</li>
<li class = "li">Bandgap optimization: Force modes away from a certain frequency</li>
</ol>

<p>It is recommended to use the Kreisselmeier-Steinhauser formulation
when maximizing the first eigenfrequencies (especially for multiple eigenfrequencies)
given by

<table class = "table"><caption/><colgroup><col/></colgroup><tbody class = "tbody">
<tr class = "row">
<td class = "entry"><span class = "ph inlineequation"><math class = "- topic/foreign "><mfenced open = "" close = "" separators = " " class = "- topic/foreign "><mrow class = "- topic/foreign "><mo class = "- topic/foreign ">-</mo><mfrac class = "- topic/foreign "><mrow class = "- topic/foreign "><mn class = "- topic/foreign ">1</mn></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">k</mi></mrow></mfrac><mo class = "- topic/foreign ">ln</mo><mfenced open = "(" close = ")" separators = "" class = "- topic/foreign "><munder class = "- topic/foreign "><mo class = "- topic/foreign ">∑</mo><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">j</mi></mrow></munder><msup class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">e</mi></mrow><mrow class = "- topic/foreign "><mo class = "- topic/foreign ">−</mo><mi class = "- topic/foreign ">k</mi><msub class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">f</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">j</mi></mrow></msub></mrow></msup></mfenced></mrow><mrow class = "- topic/foreign "><mtext class = "- topic/foreign ">, by default:</mtext></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">k</mi><mo class = "- topic/foreign ">=</mo><mfrac class = "- topic/foreign "><mrow class = "- topic/foreign "><mn class = "- topic/foreign ">30</mn></mrow><mrow class = "- topic/foreign "><msub class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">f</mi></mrow><mrow class = "- topic/foreign "><mtext class = "- topic/foreign ">min</mtext></mrow></msub></mrow></mfrac></mrow></mfenced></math></span></td>
</tr>
</tbody></table>
</p>

<p>The Kreisselmeier-Steinhauser formulation is defined by <code class = "ph codeph">DYN_FREQ_KREISSEL</code>
in the design response. For this design response, mode tracking is not
needed.</p>
<p>For the other optimization tasks, mode tracking is often necessary because the modes and the
                eigenfrequencies might switch during the optimization.</p>
</div>

</div>

</td></tr></table><script type = "text/javascript" src = "../DSDocUI_Bottom34.js">/* */</script></body>
</html>
