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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>Moment of Inertia</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">The moments of inertia can be applied as <code class = "ph codeph">DRESP</code> (topology, sizing, and
   bead optimization) and as <code class = "ph codeph">VARIABLE</code> (topology, sizing, shape, and bead
   optimization). The moments of inertia are defined using <code class = "ph codeph">INERTIA_XX</code>,
    <code class = "ph codeph">INERTIA_XY</code> (<code class = "ph codeph">INERTIA_YX</code>), <code class = "ph codeph">INERTIA_XZ</code>
    (<code class = "ph codeph">INERTIA_ZX</code>), <code class = "ph codeph">INERTIA_YY</code>, <code class = "ph codeph">INERTIA_YZ</code>
    (<code class = "ph codeph">INERTIA_ZY</code>), and <code class = "ph codeph">INERTIA_ZZ</code>, respectively.</span>

</p>
<p>This page discusses: </p><ul><li><a href = "#tso-c-usr-terms-inertiaMoment__tso-c-usr-terms-inertiaMoment-anaType" id = "toc_rg" title = "">Analysis-Independent Design Response</a></li><li><a href = "#tso-c-usr-terms-inertiaMoment__tso-c-usr-terms-inertiaMoment-expl" id = "toc_rg" title = "">Examples of Commands</a></li></ul>
</p></td></tr></table></td></tr></table>




<div class = "body conbody">
<table class = "table" id = "tso-c-usr-terms-inertiaMoment__xx944094"><caption/><colgroup><col/><col/></colgroup><tbody class = "tbody">
<tr class = "row">
<td class = "entry"><p>INERTIA_XX</p></td>
<td class = "entry"><span class = "ph inlineequation"><math class = "- topic/foreign "><mrow class = "- topic/foreign "><msub class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">I</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">x</mi></mrow></msub><mo class = "- topic/foreign ">=</mo><mrow class = "- topic/foreign "><mo class = "- topic/foreign ">∫</mo><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">ρ</mi><mrow class = "- topic/foreign "><mo class = "- topic/foreign ">(</mo><msup class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">y</mi></mrow><mrow class = "- topic/foreign "><mn class = "- topic/foreign ">2</mn></mrow></msup><mo class = "- topic/foreign ">+</mo><msup class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">z</mi></mrow><mrow class = "- topic/foreign "><mn class = "- topic/foreign ">2</mn></mrow></msup><mo class = "- topic/foreign ">)</mo></mrow></mrow><mo class = "- topic/foreign ">d</mo><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">V</mi></mrow></mrow></mrow></math></span></td>
</tr>
<tr class = "row">
<td class = "entry"><p>INERTIA_YY</p></td>
<td class = "entry"><span class = "ph inlineequation"><math class = "- topic/foreign "><mrow class = "- topic/foreign "><msub class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">I</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">y</mi></mrow></msub><mo class = "- topic/foreign ">=</mo><mrow class = "- topic/foreign "><mo class = "- topic/foreign ">∫</mo><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">ρ</mi><mrow class = "- topic/foreign "><mo class = "- topic/foreign ">(</mo><msup class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">x</mi></mrow><mrow class = "- topic/foreign "><mn class = "- topic/foreign ">2</mn></mrow></msup><mo class = "- topic/foreign ">+</mo><msup class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">z</mi></mrow><mrow class = "- topic/foreign "><mn class = "- topic/foreign ">2</mn></mrow></msup><mo class = "- topic/foreign ">)</mo></mrow></mrow><mo class = "- topic/foreign ">d</mo><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">V</mi></mrow></mrow></mrow></math></span></td>
</tr>
<tr class = "row">
<td class = "entry"><p>INERTIA_ZZ</p></td>
<td class = "entry"><span class = "ph inlineequation"><math class = "- topic/foreign "><mrow class = "- topic/foreign "><msub class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">I</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">z</mi></mrow></msub><mo class = "- topic/foreign ">=</mo><mrow class = "- topic/foreign "><mo class = "- topic/foreign ">∫</mo><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">ρ</mi><mrow class = "- topic/foreign "><mo class = "- topic/foreign ">(</mo><msup class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">x</mi></mrow><mrow class = "- topic/foreign "><mn class = "- topic/foreign ">2</mn></mrow></msup><mo class = "- topic/foreign ">+</mo><msup class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">y</mi></mrow><mrow class = "- topic/foreign "><mn class = "- topic/foreign ">2</mn></mrow></msup><mo class = "- topic/foreign ">)</mo></mrow></mrow><mo class = "- topic/foreign ">d</mo><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">V</mi></mrow></mrow></mrow></math></span></td>
</tr>
<tr class = "row">
<td class = "entry"><p>INERTIA_XY</p></td>
<td class = "entry"><span class = "ph inlineequation"><math class = "- topic/foreign "><mrow class = "- topic/foreign "><msub class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">I</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">x</mi><mi class = "- topic/foreign ">y</mi></mrow></msub><mo class = "- topic/foreign ">=</mo><msub class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">I</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">y</mi><mi class = "- topic/foreign ">x</mi></mrow></msub><mo class = "- topic/foreign ">=</mo><mrow class = "- topic/foreign "><mo class = "- topic/foreign ">−</mo><mo class = "- topic/foreign ">∫</mo><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">ρ</mi><mi class = "- topic/foreign ">x</mi><mi class = "- topic/foreign ">y</mi></mrow><mo class = "- topic/foreign ">d</mo><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">V</mi></mrow></mrow></mrow></math></span></td>
</tr>
<tr class = "row">
<td class = "entry"><p>INERTIA_XZ</p></td>
<td class = "entry"><span class = "ph inlineequation"><math class = "- topic/foreign "><mrow class = "- topic/foreign "><msub class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">I</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">x</mi><mi class = "- topic/foreign ">z</mi></mrow></msub><mo class = "- topic/foreign ">=</mo><msub class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">I</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">z</mi><mi class = "- topic/foreign ">x</mi></mrow></msub><mo class = "- topic/foreign ">=</mo><mrow class = "- topic/foreign "><mo class = "- topic/foreign ">−</mo><mo class = "- topic/foreign ">∫</mo><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">ρ</mi><mi class = "- topic/foreign ">x</mi><mi class = "- topic/foreign ">z</mi></mrow><mo class = "- topic/foreign ">d</mo><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">V</mi></mrow></mrow></mrow></math></span></td>
</tr>
<tr class = "row">
<td class = "entry"><p>INERTIA_YZ</p></td>
<td class = "entry"><span class = "ph inlineequation"><math class = "- topic/foreign "><mrow class = "- topic/foreign "><msub class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">I</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">y</mi><mi class = "- topic/foreign ">z</mi></mrow></msub><mo class = "- topic/foreign ">=</mo><msub class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">I</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">y</mi><mi class = "- topic/foreign ">x</mi></mrow></msub><mo class = "- topic/foreign ">=</mo><mrow class = "- topic/foreign "><mo class = "- topic/foreign ">−</mo><mo class = "- topic/foreign ">∫</mo><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">ρ</mi><mi class = "- topic/foreign ">y</mi><mi class = "- topic/foreign ">z</mi></mrow><mo class = "- topic/foreign ">d</mo><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">V</mi></mrow></mrow></mrow></math></span></td>
</tr>
</tbody></table>

<div class = "section" id = "tso-c-usr-terms-inertiaMoment__tso-c-usr-terms-inertiaMoment-anaType"><h2 class = "title sectiontitle">Analysis-Independent Design Response</h2>

<p>For the moment of inertia, 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-inertiaMoment__xx944204"><caption/><colgroup><col/><col/><col/><col/><col/></colgroup><thead class = "thead">
<tr class = "row">
<th class = "entry" id = "tso-c-usr-terms-inertiaMoment__xx944204__entry__1"/>
<th class = "entry" id = "tso-c-usr-terms-inertiaMoment__xx944204__entry__2"><p>TOPO</p></th>
<th class = "entry" id = "tso-c-usr-terms-inertiaMoment__xx944204__entry__3"><p>SHAPE</p></th>
<th class = "entry" id = "tso-c-usr-terms-inertiaMoment__xx944204__entry__4"><p>BEAD</p></th>
<th class = "entry" id = "tso-c-usr-terms-inertiaMoment__xx944204__entry__5"><p>SIZING</p></th>
</tr>
</thead><tbody class = "tbody">
<tr class = "row">
<td class = "entry" headers = "tso-c-usr-terms-inertiaMoment__xx944204__entry__1"><p>OBJ_FUNC</p></td>
<td class = "entry" headers = "tso-c-usr-terms-inertiaMoment__xx944204__entry__2"><p>S</p></td>
<td class = "entry" headers = "tso-c-usr-terms-inertiaMoment__xx944204__entry__3"><p>S</p></td>
<td class = "entry" headers = "tso-c-usr-terms-inertiaMoment__xx944204__entry__4"><p>S</p></td>
<td class = "entry" headers = "tso-c-usr-terms-inertiaMoment__xx944204__entry__5"><p>S</p></td>
</tr>
<tr class = "row">
<td class = "entry" headers = "tso-c-usr-terms-inertiaMoment__xx944204__entry__1"><p>CONSTRAINT</p></td>
<td class = "entry" headers = "tso-c-usr-terms-inertiaMoment__xx944204__entry__2"><p>S</p></td>
<td class = "entry" headers = "tso-c-usr-terms-inertiaMoment__xx944204__entry__3"><p>S</p></td>
<td class = "entry" headers = "tso-c-usr-terms-inertiaMoment__xx944204__entry__4"><p>S</p></td>
<td class = "entry" headers = "tso-c-usr-terms-inertiaMoment__xx944204__entry__5"><p>S</p></td>
</tr>
</tbody></table>

<p>The moments of inertia can be applied as <code class = "ph codeph">DRESP</code> (topology, sizing, and bead
    optimization) and as <code class = "ph codeph">VARIABLE</code> (topology, sizing, shape, and bead optimization).
    The moments of inertia are defined using <code class = "ph codeph">INERTIA_XX</code>,
     <code class = "ph codeph">INERTIA_XY</code> (<code class = "ph codeph">INERTIA_YX</code>), <code class = "ph codeph">INERTIA_XZ</code>
     (<code class = "ph codeph">INERTIA_ZX</code>), <code class = "ph codeph">INERTIA_YY</code>, <code class = "ph codeph">INERTIA_YZ</code>
     (<code class = "ph codeph">INERTIA_ZY</code>), and <code class = "ph codeph">INERTIA_ZZ</code>, respectively.</p>
<p>Mathematically, the moments of inertia about the origin of the coordinate system are given by the
    above integrals, which can be calculated in a global or a local coordinate system. The local
    coordinate system is defined using <code class = "ph codeph">CS_REF</code>. The volume for which the moments of
    inertia are calculated is defined using <code class = "ph codeph">EL_GROUP</code>. </p>
<div class = "note"><span class = "run-in.note">Note:
			</span><span class = "notecontent"><p><ol class = "ol">
<li class = "li">Only the elements of the element group (<code class = "ph codeph">EL_GROUP</code>) listed
in the tables of supported element types will be applied in the calculation
of the moments of inertia. </li>
<li class = "li">The product of inertia with respect to any two orthogonal axes is
zero if either of the axes is an axis of symmetry.</li>
<li class = "li">The physical density defined in the finite element input file will be used in the calculation of
       the moments of inertia.</li>
</ol></p></span></div>

<p>The design response (<code class = "ph codeph">DRESP</code>) for the moment of inertia
about the line through the origin, parallel to the x-axis is defined
like</p>
<pre class = "codeblock">
<code class = "ph codeph">
DRESP
 ID_NAME  = ...
 DEF_TYPE = SYSTEM
 TYPE     = INERTIA_XX
 EL_GROUP = ...
 CS_REF   = ...
END_
</code>
</pre>

<p>The design response (<code class = "ph codeph">DRESP</code>) for the moment of inertia
about the line through the origin, parallel to the y-axis is defined
like</p>
<pre class = "codeblock">
<code class = "ph codeph">
DRESP
 ID_NAME  = ...
 DEF_TYPE = SYSTEM
 TYPE     = INERTIA_YY
 EL_GROUP = ...
 CS_REF   = ...
END_
</code>
</pre>

<p>The design response (<code class = "ph codeph">DRESP</code>) for the moment of inertia
about the line through the origin, parallel to the z-axis is defined
like</p>
<pre class = "codeblock">
<code class = "ph codeph">
DRESP
 ID_NAME  = ...
 DEF_TYPE = SYSTEM
 TYPE     = INERTIA_ZZ
 EL_GROUP = ...
 CS_REF   = ...
END_
</code>
</pre>

<p>The design response (<code class = "ph codeph">DRESP</code>) for the moment of inertia
describing the coupling between the rotation parallel to the x-axis
and the rotation parallel to the y-axis yields:</p>
<pre class = "codeblock">
<code class = "ph codeph">
DRESP
 ID_NAME  = ...
 DEF_TYPE = SYSTEM
 TYPE     = INERTIA_XY
 (Alternatively, TYPE =INERTIA_YX)
 EL_GROUP = ...
 CS_REF   = ...
END_
</code>
</pre>
<p>The design response (<code class = "ph codeph">DRESP</code>) for the moment of inertia
describing the coupling between the rotation parallel to the x-axis
and the rotation parallel to the z-axis yields:</p>
<pre class = "codeblock">
<code class = "ph codeph">
DRESP
 ID_NAME  = ...
 DEF_TYPE = SYSTEM
 TYPE     = INERTIA_XZ
 (Alternatively, TYPE = INERTIA_ZX)
 EL_GROUP = ...
 CS_REF   = ...
END_
</code>
</pre>
<p>The design response (<code class = "ph codeph">DRESP</code>) for the moment of inertia
describing the coupling between the rotation parallel to the y-axis
and the rotation parallel to the z-axis yields:</p>
<pre class = "codeblock">
<code class = "ph codeph">
DRESP
 ID_NAME              = ...
 DEF_TYPE             = SYSTEM
 TYPE                 = INERTIA_YZ
 (Alternatively, TYPE = INERTIA_YZ)
 EL_GROUP             = ...
 CS_REF               = ...
END_
</code>
</pre>
</div>


<div class = "section" id = "tso-c-usr-terms-inertiaMoment__tso-c-usr-terms-inertiaMoment-expl"><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">Examples of Commands</h2>

<p>For example, the design response (<code class = "ph codeph">DRESP</code>) for the moment of
inertia of the entire structure (<code class = "ph codeph">ALL_ELEMENTS</code>) about
the line through the origin of the global coordinate system, parallel
to the y-axis is defined like</p>
<pre class = "codeblock">
<code class = "ph codeph">
DRESP
 ID_NAME  = DRESP_INERTIA_YY_GLOBAL
 DEF_TYPE = SYSTEM
 TYPE     = INERTIA_YY
 EL_GROUP = ALL_ELEMENTS
END_
</code>
</pre>

<p>For example, the definition of the design response (<code class = "ph codeph">DRESP</code>) for the moment of
    inertia of the substructure called <code class = "ph codeph">EL_GROUP_2</code> is calculated about the line
    through the origin of the local coordinate system number 23, parallel to the y-axis is like the
    following:</p>
<pre class = "codeblock">
<code class = "ph codeph">
DRESP
 ID_NAME  = DRESP_INERTIA_YY_LOCAL
 DEF_TYPE = SYSTEM
 TYPE     = CENTER_GRAVITY_X
 EL_GROUP = EL_GROUP_2
 CS_REF   = CS_23
END_
</code>
</pre>
</div>

</div>

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