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      <title>Degree-of-Freedom Set Definitions</title>
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      <div class="title_topic2" id="xps10_pagetitle">Degree-of-Freedom Set Definitions </div>
      <hr noshade="true">
      <p class="para_topic">Each degree-of-freedom is a member of one mutually exclusive set.  Set names have the following definitions. </p>
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               <th align="center" colspan="1" rowspan="1">
                  <p class="para_th"><b class="uiTerm">Set Name </b></p>
               </th>
               <th align="center" colspan="1" rowspan="1">
                  <p class="para_th"><b class="uiTerm">Definition </b></p>
               </th>
            </tr>
         </thead>
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               <td align="center" colspan="1" rowspan="1">
                  <p class="para_td">m </p>
               </td>
               <td align="left" colspan="1" rowspan="1">
                  <p class="para_td">degrees-of-freedom eliminated by multipoint constraints </p>
               </td>
            </tr>
            <tr>
               <td align="center" colspan="1" rowspan="1">
                  <p class="para_td">sb* </p>
               </td>
               <td align="left" colspan="1" rowspan="1">
                  <p class="para_td">degrees-of-freedom eliminated by single-point constraints that are included in boundary condition changes and by the AUTOSPC feature </p>
               </td>
            </tr>
            <tr>
               <td align="center" colspan="1" rowspan="1">
                  <p class="para_td">sg* </p>
               </td>
               <td align="left" colspan="1" rowspan="1">
                  <p class="para_td">degrees-of-freedom eliminated by single-point constraints that are specified on the PS field on GRID Bulk Data entries </p>
               </td>
            </tr>
            <tr>
               <td align="center" colspan="1" rowspan="1">
                  <p class="para_td">o </p>
               </td>
               <td align="left" colspan="1" rowspan="1">
                  <p class="para_td">degrees-of-freedom omitted by structural matrix partitioning </p>
               </td>
            </tr>
            <tr>
               <td align="center" colspan="1" rowspan="1">
                  <p class="para_td">q </p>
               </td>
               <td align="left" colspan="1" rowspan="1">
                  <p class="para_td">generalized degrees-of-freedom for dynamic reduction or component mode synthesis </p>
               </td>
            </tr>
            <tr>
               <td align="center" colspan="1" rowspan="1">
                  <p class="para_td">r </p>
               </td>
               <td align="left" colspan="1" rowspan="1">
                  <p class="para_td">reference degrees-of-freedom used to determine free body motion </p>
               </td>
            </tr>
            <tr>
               <td align="center" colspan="1" rowspan="1">
                  <p class="para_td">c </p>
               </td>
               <td align="left" colspan="1" rowspan="1">
                  <p class="para_td">degrees-of-freedom that are free during component mode synthesis or dynamic reduction </p>
               </td>
            </tr>
            <tr>
               <td align="center" colspan="1" rowspan="1">
                  <p class="para_td">b </p>
               </td>
               <td align="left" colspan="1" rowspan="1">
                  <p class="para_td">degrees-of-freedom fixed during component mode analysis or dynamic reduction </p>
               </td>
            </tr>
            <tr>
               <td align="center" colspan="1" rowspan="1">
                  <p class="para_td">e </p>
               </td>
               <td align="left" colspan="1" rowspan="1">
                  <p class="para_td">extra degrees-of-freedom introduced in dynamic analysis </p>
               </td>
            </tr>
            <tr>
               <td align="center" colspan="1" rowspan="1">
                  <p class="para_td">sa </p>
               </td>
               <td align="left" colspan="1" rowspan="1">
                  <p class="para_td">permanently constrained aerodynamic degrees-of-freedom </p>
               </td>
            </tr>
            <tr>
               <td align="center" colspan="1" rowspan="1">
                  <p class="para_td">k </p>
               </td>
               <td align="left" colspan="1" rowspan="1">
                  <p class="para_td">aerodynamic degrees-of-freedom </p>
               </td>
            </tr>
         </tbody>
      </table><br><p class="para_topic">* Strictly speaking, sb and sg are not exclusive with respect to one another.  Degrees-of-freedom may exist in both sets simultaneously.  Since these sets are not used explicitly in the solution sequences, this need not concern the user.  However, those who use these sets in their own DMAPs should avoid redundant specifications when using these sets for partitioning or merging operations.  That is, a degree-of-freedom should not be specified on both a PS field of a GRID entry (sg set) and on a selected SPC entry (sb set).  Redundant specifications will cause UFM 2120 in the VEC module and behavior listed in <em><a class="links" href="javascript:void(0)" onclick="top.popUpPage('QRG/linkInter_target_not_resolved.html#dmap:ref86112:-aux,-home','linkInter','menubar=yes,status=yes,resizable=yes,toolbar=yes,scrollbars=yes,location=no,directories=no');return(false);">NX Nastran DMAP Programmer&rsquo;s Guide</a></em> for the UPARTN module.  These sets are exclusive, however, from the other mutually exclusive sets. 
      </p>
      <p class="para_topic">Each degree-of-freedom is also a member of one or more combined sets called &ldquo;supersets.&rdquo;  Supersets have the following definitions: </p>
      <p align="center"><img align="bottom" src="graphics/dof_sets-ufig01.gif"></p>
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               <th align="left" colspan="1" rowspan="1">
                  <p class="para_th"><b class="uiTerm">Set Name </b></p>
                  <p class="para_th">(+ indicates union of two sets) </p>
               </th>
               <th align="left" colspan="1" rowspan="1">
                  <p class="para_th"><b class="uiTerm">Meaning </b></p>
               </th>
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         </thead>
         <tbody>
            <tr>
               <td align="left" colspan="1" rowspan="1">
                  <p class="para_td">s = sb + sg </p>
               </td>
               <td align="left" colspan="1" rowspan="1">
                  <p class="para_td">all degrees-of-freedom eliminated by single point constraints </p>
               </td>
            </tr>
            <tr>
               <td align="left" colspan="1" rowspan="1">
                  <p class="para_td"><em>l</em> = b + c 
                  </p>
               </td>
               <td align="left" colspan="1" rowspan="1">
                  <p class="para_td">the structural degrees-of-freedom remaining after the reference degrees-of-freedom are removed (degrees-of-freedom left over) </p>
               </td>
            </tr>
            <tr>
               <td align="left" colspan="1" rowspan="1">
                  <p class="para_td">t = <em>l</em> + r 
                  </p>
               </td>
               <td align="left" colspan="1" rowspan="1">
                  <p class="para_td">the total set of physical boundary degrees-of-freedom for superelements </p>
               </td>
            </tr>
            <tr>
               <td align="left" colspan="1" rowspan="1">
                  <p class="para_td">a = t + q </p>
               </td>
               <td align="left" colspan="1" rowspan="1">
                  <p class="para_td">the set assembled in superelement analysis </p>
               </td>
            </tr>
            <tr>
               <td align="left" colspan="1" rowspan="1">
                  <p class="para_td">d = a + e </p>
               </td>
               <td align="left" colspan="1" rowspan="1">
                  <p class="para_td">the set used in dynamic analysis by the direct method </p>
               </td>
            </tr>
            <tr>
               <td align="left" colspan="1" rowspan="1">
                  <p class="para_td">f = a + o </p>
               </td>
               <td align="left" colspan="1" rowspan="1">
                  <p class="para_td">unconstrained (free) structural degrees-of-freedom </p>
               </td>
            </tr>
            <tr>
               <td align="left" colspan="1" rowspan="1">
                  <p class="para_td">fe = f + s </p>
               </td>
               <td align="left" colspan="1" rowspan="1">
                  <p class="para_td">free structural degrees-of-freedom plus extra degrees-of-freedom </p>
               </td>
            </tr>
            <tr>
               <td align="left" colspan="1" rowspan="1">
                  <p class="para_td">n = f + e </p>
               </td>
               <td align="left" colspan="1" rowspan="1">
                  <p class="para_td">all structural degrees-of-freedom not constrained by multipoint constraints </p>
               </td>
            </tr>
            <tr>
               <td align="left" colspan="1" rowspan="1">
                  <p class="para_td">ne = n + e </p>
               </td>
               <td align="left" colspan="1" rowspan="1">
                  <p class="para_td">all structural degrees-of-freedom not constrained by multipoint constraints plus extra degrees-of-freedom </p>
               </td>
            </tr>
            <tr>
               <td align="left" colspan="1" rowspan="1">
                  <p class="para_td">g = n + m </p>
               </td>
               <td align="left" colspan="1" rowspan="1">
                  <p class="para_td">all structural (grid) degrees-of-freedom including scalar degrees-of-freedom </p>
               </td>
            </tr>
            <tr>
               <td align="left" colspan="1" rowspan="1">
                  <p class="para_td">p = g + e </p>
               </td>
               <td align="left" colspan="1" rowspan="1">
                  <p class="para_td">all physical degrees-of-freedom </p>
               </td>
            </tr>
            <tr>
               <td align="left" colspan="1" rowspan="1">
                  <p class="para_td">ps = p + sa </p>
               </td>
               <td align="left" colspan="1" rowspan="1">
                  <p class="para_td">physical and constrained (SPCi) aerodynamic degrees-of-freedom </p>
               </td>
            </tr>
            <tr>
               <td align="left" colspan="1" height="47" rowspan="1">
                  <p class="para_td">pa = ps + k </p>
               </td>
               <td align="left" colspan="1" height="47" rowspan="1">
                  <p class="para_td">physical set for aerodynamics </p>
               </td>
            </tr>
            <tr>
               <td align="left" colspan="1" rowspan="1">
                  <p class="para_td">(fr = o) + <em>l</em></p>
               </td>
               <td align="left" colspan="1" rowspan="1">
                  <p class="para_td">statically independent set minus the statically determinate supports (fr = f − q − r) </p>
               </td>
            </tr>
            <tr>
               <td align="left" colspan="1" rowspan="1">
                  <p class="para_td">v = o + c + r </p>
               </td>
               <td align="left" colspan="1" rowspan="1">
                  <p class="para_td">the set free to vibrate in dynamic reduction and component mode synthesis </p>
               </td>
            </tr>
         </tbody>
      </table><br><p class="para_topic">The a-set and o-set are created in the following ways: </p>
      <ol type="1" start="1">
         <li>
            <p class="para_item">If only OMITi entries are present, then the o-set consists of degrees-of-freedom listed explicitly on OMITi entries.  The remaining f-set degrees-of-freedom are placed in the b-set, which is a subset of the a-set. </p>
         </li>
         <li>
            <p class="para_item">If ASETi or QSETi entries are present, then the a-set consists of all degrees-of-freedom listed on ASETi entries and any entries listing its subsets, such as QSETi, SUPORTi, CSETi, and BSETi entries.  Any OMITi entries are redundant.  The remaining f-set degrees-of-freedom are placed in the o-set. </p>
         </li>
         <li>
            <p class="para_item">If there are no ASETi, QSETi, or OMITi entries present but there are SUPORTi, BSETi, or CSETi entries present, then the entire f-set is placed in the a-set and the o-set is not created. </p>
         </li>
         <li>
            <p class="para_item">There must be at least one explicit ASETi, QSETi, or OMITi entry for the o-set to exist, even if the ASETi, QSETi, or OMITi entry is redundant. </p>
         </li>
      </ol>
      <p class="para_topic">In dynamic analysis, additional vector sets are obtained by a modal transformation derived from real eigenvalue analysis of the a-set. These sets are as follows: </p>
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                  <p class="para_td">&xi;<em><sub class="subscript">o</sub></em></p>
               </td>
               <td align="center" colspan="1" rowspan="1">
                  <p class="para_td">= </p>
               </td>
               <td align="left" colspan="1" rowspan="1">
                  <p class="para_td">rigid body (zero frequency) modal degrees-of-freedom </p>
               </td>
            </tr>
            <tr>
               <td align="right" colspan="1" rowspan="1">
                  <p class="para_td">&xi;<em><sub class="subscript">f</sub></em></p>
               </td>
               <td align="center" colspan="1" rowspan="1">
                  <p class="para_td">= </p>
               </td>
               <td align="left" colspan="1" rowspan="1">
                  <p class="para_td">finite frequency modal degrees-of-freedom </p>
               </td>
            </tr>
            <tr>
               <td align="right" colspan="1" rowspan="1">
                  <p class="para_td">&xi;<em><sub class="subscript">i</sub></em></p>
               </td>
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                  <p class="para_td">= </p>
               </td>
               <td align="left" colspan="1" rowspan="1">
                  <p class="para_td">&xi;<em><sub class="subscript">o</sub></em> + &xi;<em><sub class="subscript">f</sub></em>, the set of all modal degrees-of-freedom 
                  </p>
               </td>
            </tr>
         </tbody>
      </table><br><p class="para_topic">One vector set is defined that combines physical and modal degrees-of-freedom: </p>
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               <td align="right" colspan="1" rowspan="1">
                  <p class="para_td"><em>u<sub class="subscript">h</sub></em></p>
               </td>
               <td align="center" colspan="1" rowspan="1">
                  <p class="para_td">= </p>
               </td>
               <td align="left" colspan="1" rowspan="1">
                  <p class="para_td">&xi;<em><sub class="subscript">i</sub></em> + <em>u<sub class="subscript">e</sub></em>, the set of all modal degrees-of-freedom 
                  </p>
               </td>
            </tr>
         </tbody>
      </table><br><p class="para_topic">The membership of each degree-of-freedom can be printed by use of the Bulk Data entries PARAM,USETPRT and PARAM,USETSEL. </p>
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