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<a name = "hj-top"> </a><table class = "table1" id = "table11"><tr><td><table class = "DocHeader"><tr><td class = "DocHeader1" colspan = "2"><h1> Maximization of the First Eigenfrequencies </h1></td></tr><tr><td class = "DocHeader4" colspan = "2"/></tr><tr><td class = "DocHeader3"><table class = "DocHeaderIntro" id = "table12"><tr><td class = "Intro1Only"><p class = "header"><p class = "abstract">
<span class = "shortdesc">The following shows how several lowest eigenfrequencies are
increased.</span>

</p>
<p>This page discusses: </p><ul><li><a href = "#tso-c-usr-sizing-TasksModAna-maxFirstFreq__tso-c-usr-sizing-TasksModAna-maxFirstFreq-optProb" id = "toc_rg" title = "">Formulation of the Optimization Problem</a></li><li><a href = "#tso-c-usr-sizing-TasksModAna-maxFirstFreq__tso-c-usr-sizing-TasksModAna-maxFirstFreq-ResConv" id = "toc_rg" title = "">Result and Convergence</a></li></ul>
</p></td></tr></table></td><td class = "DocHeader2"><table class = "DocTopicsSeeAlso" id = "table13"><tr><td class = "TopicsTitle">See Also</td></tr><tr><td><a title = "This section describes the theory of eigenfrequency." href = "tso-c-usr-terms-eigfreqOvw.htm#tso-c-usr-terms-eigfreqOvw">Overview of Eigenfrequency</a></td></tr></table></td></tr></table>




<div class = "body conbody">
<p>It is important to consider more than the first natural eigenfrequency
as illustrated in the above figure (b) when increasing the natural
frequencies using optimization. At least, the next two first natural frequencies
should be considered in the optimization.</p>
<p>All natural eigenfrequencies requested in the FE model are applied
in the optimization if the <code class = "ph codeph">ALL</code> option is applied in the
<code class = "ph codeph">LC_SET</code> parameter.</p>
<p>During the optimization, the various natural frequencies are automatically
weighted by their distance from the lowest natural frequency, that is, when
the other natural frequencies approach the first natural frequency during
the optimization, the more they will be weighted. Generally, the first
natural frequency is always maximized. </p>
<p>The design response is defined using the Kreisselmeier-Steinhauser
 formulation.</p>
<p>Any number of natural frequencies in the design response can be specified
using the <code class = "ph codeph">DRESP</code> command.</p>
<div class = "section" id = "tso-c-usr-sizing-TasksModAna-maxFirstFreq__tso-c-usr-sizing-TasksModAna-maxFirstFreq-optProb"><h2 class = "title sectiontitle">Formulation of the Optimization Problem</h2>

<p>The optimization task is to maximize the lowest modal eigenfrequencies
with a volume constraint of 100%.  </p>
<table class = "table"><caption/><colgroup><col/></colgroup><tbody class = "tbody">
<tr class = "row">
<td class = "entry"><p>Model:</p><br/><img class = "image" src = "../TsoUserImages/size_typtasks_MinVol_modaltors_Obj2.png" width = "300"/><br/></td>
</tr>
<tr class = "row">
<td class = "entry"><p>f=20.4Hz:</p><br/><img class = "image" src = "../TsoUserImages/size_typtasks_MinVol_modal_Obj1.png" width = "300"/><br/></td>
</tr>
<tr class = "row">
<td class = "entry"><p>f=22.1Hz:</p><br/><img class = "image" src = "../TsoUserImages/size_typtasks_MinVol_modal_Obj2a.png" width = "300"/><br/></td>
</tr>
<tr class = "row">
<td class = "entry"><p>f=25.8Hz:</p><br/><img class = "image" src = "../TsoUserImages/size_typtasks_MinVol_modal_Obj4.png" width = "300"/><br/></td>
</tr>
<tr class = "row">
<td class = "entry"><p>f=26.4Hz:</p><br/><img class = "image" src = "../TsoUserImages/size_typtasks_MinVol_modal_Obj3.png" width = "300"/><br/></td>
</tr>
<tr class = "row">
<td class = "entry"><p>f=30.6Hz:</p><br/><img class = "image" src = "../TsoUserImages/size_typtasks_MinVol_modal_Obj5.png" width = "300"/><br/></td>
</tr>
<tr class = "row">
<td class = "entry"><p>f=35.0Hz:</p><br/><img class = "image" src = "../TsoUserImages/size_typtasks_MinVol_modal_Obj6.png" width = "300"/><br/></td>
</tr>

</tbody></table>

<p>In the above figure, you can see the model and the first six modal eigenfrequencies.</p>
</div>

<div class = "section" id = "tso-c-usr-sizing-TasksModAna-maxFirstFreq__tso-c-usr-sizing-TasksModAna-maxFirstFreq-ResConv"><p><map name = "FPMap1"><area href = "#hj-top" title = "Back to Top" shape = "rect" coords = "416, 0, 435, 10"/></map><span class = "itemsprite"/></p><h2 class = "title sectiontitle">Result and Convergence</h2>

<p>The following figure shows the objective function (maximize eigenfrequencies):</p>
<table class = "table"><caption/><colgroup><col/></colgroup><tbody class = "tbody">
<tr class = "row">
<td class = "entry"><br/><img class = "image" src = "../TsoUserImages/aa0c76f4.jpg" width = "450"/><br/></td>
</tr>
</tbody></table>
<p>The constraint (Mass) is shown in the following figure:</p>
<table class = "table"><caption/><colgroup><col/></colgroup><tbody class = "tbody">
<tr class = "row">
<td class = "entry"><br/><img class = "image" src = "../TsoUserImages/aa0c7702.jpg" width = "450"/><br/></td>
</tr>
</tbody></table>

<p> The gradient of three eigenfrequencies over 15 iterations and the
mass constraint are illustrated in the above figure. The constraint
is fulfilled.</p>
<p>In this example, all calculated natural frequencies are considered
for the objective function using the Kreisselmeier-Steinhauser formulation,
and the definition is as follows:</p>
<pre class = "codeblock">
<code class = "ph codeph">
DRESP
 ID_NAME  = all_lowest_eigenfrequencies
 DEF_TYPE = SYSTEM
 TYPE     = DYN_FREQ_KREISSEL
 LC_SET   = MODAL, ALL, ALL
END_

OBJ_FUNC
 ID_NAME = maximize_eigenfrequencies
 DRESP   = all_lowest_eigenfrequencies
 TARGET  = MAX
END_
</code>
</pre>
<p>If one has requested 10 eigenfrequencies in the finite element input model
but only the first 5 eigenfrequencies are to be used in the optimization
definition, then the design response for 5 eigenfrequencies is defined
as follows:</p>
<pre class = "codeblock">
<code class = "ph codeph">
DRESP
 ID_NAME  = all_lowest_eigenfrequencies_1_5
 DEF_TYPE = SYSTEM
 TYPE     = DYN_FREQ_KREISSEL
 LC_SET   = MODAL, ALL, 1-5
END_

OBJ_FUNC
 ID_NAME = maximize_eigenfrequencies_1_5
 DRESP   = all_lowest_eigenfrequencies_1_5
 TARGET  = MAX
END_
</code></pre>
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