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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>Maximize the First Natural Mode (First Eigenvalue)</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 shows you how to maximize the first eigenvalue with the sensitivity-based
   bead optimization approach.</span>

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<p>It is not recommended that you define a maximization of an eigenvalue problem similar to the
   controller-based bead algorithm. You have more control over the modes in the sensitivity-based
   algorithm. </p>
<p>A problem you usually want to avoid when optimizing eigenmodes is mode-switching because it
   destabilizes the optimization algorithm. The typical problem is by maximizing the first eigenmode
   it might "overtake" the second mode - hence, the modes switch place (previous second mode becomes
   the first mode) and the sensitivity algorithm must suddenly take a new mode into
   consideration.</p>
<p>Mode tracking can of course be used (see <a class = "xref" href = "tso-c-usr-beadParamSens-beadModeTracking.htm#tso-c-usr-beadParamSens-beadModeTracking" title = "This section describes how to control mode tracking in sensitivity-based bead optimizations.">Optimization Parameters For Mode Tracking</a>), but the computationally cheapest way
to push the first eigenmode up is to use the Kreisselmeier-Steinhauser
formulation (Type = DYN_FREQ_KREISSEL):</p>
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                      <p><span class = "ph inlineequation"><math class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">φ</mi><mo class = "- topic/foreign ">=</mo><mo class = "- topic/foreign ">−</mo><mfrac class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">α</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">k</mi></mrow></mfrac><mi class = "- topic/foreign ">ln</mi><mo class = "- topic/foreign ">⁡</mo><mo class = "- topic/foreign ">(</mo><mrow class = "- topic/foreign "><munderover class = "- topic/foreign "><mrow class = "- topic/foreign "><mo class = "- topic/foreign ">∑</mo></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">j</mi><mo class = "- topic/foreign ">=</mo><mn class = "- topic/foreign ">1</mn></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">N</mi></mrow></munderover><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></mrow><mo class = "- topic/foreign ">)</mo></mrow></math></span></p></td>
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<p>In the formula <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> is the eigenfrequency,
e the base of the natural logarithm and <span class = "ph inlineequation"><math class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">α</mi></mrow></math></span> and k are constants.</p>
<p>The following definition will enforce the 5 first modes to keep
their sequence. This is usually sufficient to avoid mode switching among
the first couple of modes. </p>
<pre class = "codeblock">
<code class = "ph codeph">
DRESP
 ID_NAME  = dresp_eig_kreissel
 DEF_TYPE = SYSTEM
 TYPE     = DYN_FREQ_KREISSEL
 LC_SET   = Modal,All,1-5
END_

OBJ_FUNC
 ID_NAME = max_dresp_eig_kreissel
 DRESP   = dresp_eig_kreissel
 TARGET  = MAX
END_
</code>
</pre>
<p>The first mode is maximized (<span class = "ph">TARGET=MAX</span>) until it
comes near the higher modes in which case they are being considered as well.</p>
<p>See the figure about the iteration history in
   <a class = "xref" href = "tso-c-usr-beadOptTask-diffBeadOptAlgos.htm#tso-c-usr-beadOptTask-diffBeadOptAlgos" title = "The user might choose between two bead optimization algorithms in Tosca Structure.bead. The algorithms have different ways to find the solution and their differences will be discussed in this chapter.">Differences between Bead Optimization Algorithms</a> 
   at iteration 13-14, where modes do not because of this formulation even though they a close to each other.</p></div>

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