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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>A Brief Explanation of NVH Theory </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">Within this section, the theoretical background of NVH analysis is briefly described.</span>

</p>
<p>This page discusses: </p><ul><li><a href = "#tso-c-usr-sizing-nvh-bckgrnd__tso-c-usr-sizing-nvh-modalformulation" id = "toc_rg" title = "">Theoretical Background of Structural NVH Simulations </a></li><li><a href = "#tso-c-usr-sizing-nvh-bckgrnd__tso-c-usr-sizing-nvh-acousticformulation" id = "toc_rg" title = "">Theoretical Background of Structural-Acoustic NVH Simulation</a></li><li><a href = "#tso-c-usr-sizing-nvh-bckgrnd__tso-c-usr-sizing-nvh-applications" id = "toc_rg" title = "">Possible Applications of NVH Simulation in Engineering</a></li></ul>
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<div class = "section" id = "tso-c-usr-sizing-nvh-bckgrnd__tso-c-usr-sizing-nvh-modalformulation"><h2 class = "title sectiontitle">Theoretical Background of Structural NVH Simulations </h2>

<p>To explain the concept of structural NVH, an idealization of an automotive vehicle is explained below. 
The chassis and tires are considered and the vibrations from driving on a road are analyzed.</p>

<table class = "table" id = "tso-c-usr-sizing-nvh-bckgrnd__table_91EF4D6F6EDA4E8498E11B7903116357"><caption/><colgroup><col/></colgroup><tbody class = "tbody">
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<td class = "entry"><br/><img class = "image" id = "tso-c-usr-sizing-nvh-bckgrnd__image_1348CFAEA8314DD4898D1716EE34EDE2" src = "../TsoUserImages/free_body_diagram_car_nvh.png"/><br/></td>
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<p> The image above shows an automotive modeled as a simple system of masses, springs, and
dashpots. The dashpot is used to provide damping and has a damping coefficient C. The two
springs represent the stiffness of the tires and the suspension. The masses represent the
mass of the tires and the chassis. A free body force representation is made where the
displacements, velocity and accelerations of each mass are also considered. Then the
equations of motion for this system are written as follows: </p>
      
<table class = "table" id = "tso-c-usr-sizing-nvh-bckgrnd__table_C2F18A3CBAD2450AA57FA1A1D7DD4AB0"><caption/><colgroup><col/></colgroup><tbody class = "tbody">
<tr class = "row">
<td class = "entry"><p><span class = "ph inlineequation"><math class = "- topic/foreign "><mrow class = "- topic/foreign "><msub class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">m</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">u</mi></mrow></msub><msub class = "- topic/foreign "><mrow class = "- topic/foreign "><mover class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">y</mi></mrow><mrow class = "- topic/foreign "><mo class = "- topic/foreign ">¨</mo></mrow></mover></mrow><mrow class = "- topic/foreign "><mn class = "- topic/foreign ">1</mn></mrow></msub><mo class = "- topic/foreign ">−</mo><msub class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">k</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">t</mi></mrow></msub><mo class = "- topic/foreign ">(</mo><msub class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">y</mi></mrow><mrow class = "- topic/foreign "><mn class = "- topic/foreign ">0</mn></mrow></msub><mo class = "- topic/foreign ">−</mo><msub class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">y</mi></mrow><mrow class = "- topic/foreign "><mn class = "- topic/foreign ">1</mn></mrow></msub><mo class = "- topic/foreign ">)</mo><mo class = "- topic/foreign ">+</mo><msub class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">k</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">s</mi></mrow></msub><mo class = "- topic/foreign ">(</mo><msub class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">y</mi></mrow><mrow class = "- topic/foreign "><mn class = "- topic/foreign ">1</mn></mrow></msub><mo class = "- topic/foreign ">−</mo><msub 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></msub><mo class = "- topic/foreign ">)</mo><mo class = "- topic/foreign ">+</mo><mi class = "- topic/foreign ">c</mi><mo class = "- topic/foreign ">(</mo><msub class = "- topic/foreign "><mrow class = "- topic/foreign "><mover class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">y</mi></mrow><mrow class = "- topic/foreign "><mo class = "- topic/foreign ">˙</mo></mrow></mover></mrow><mrow class = "- topic/foreign "><mn class = "- topic/foreign ">1</mn><mo class = "- topic/foreign ">−</mo></mrow></msub><msub class = "- topic/foreign "><mrow class = "- topic/foreign "><mover class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">y</mi></mrow><mrow class = "- topic/foreign "><mo class = "- topic/foreign ">˙</mo></mrow></mover></mrow><mrow class = "- topic/foreign "><mn class = "- topic/foreign ">2</mn></mrow></msub><mo class = "- topic/foreign ">)</mo><mo class = "- topic/foreign ">=</mo><mn class = "- topic/foreign ">0</mn><mo class = "- topic/foreign ">,</mo></mrow></math></span></p>
                      <p><span class = "ph inlineequation"><math class = "- topic/foreign "><mrow class = "- topic/foreign "><mtext class = "- topic/foreign "> </mtext><msub class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">m</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">s</mi></mrow></msub><msub class = "- topic/foreign "><mrow class = "- topic/foreign "><mover class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">y</mi></mrow><mrow class = "- topic/foreign "><mo class = "- topic/foreign ">¨</mo></mrow></mover></mrow><mrow class = "- topic/foreign "><mn class = "- topic/foreign ">2</mn></mrow></msub><mo class = "- topic/foreign ">−</mo><msub class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">k</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">s</mi></mrow></msub><mo class = "- topic/foreign ">(</mo><msub class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">y</mi></mrow><mrow class = "- topic/foreign "><mn class = "- topic/foreign ">1</mn></mrow></msub><mo class = "- topic/foreign ">−</mo><msub 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></msub><mo class = "- topic/foreign ">)</mo><mo class = "- topic/foreign ">−</mo><mi class = "- topic/foreign ">c</mi><mo class = "- topic/foreign ">(</mo><msub class = "- topic/foreign "><mrow class = "- topic/foreign "><mover class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">y</mi></mrow><mrow class = "- topic/foreign "><mo class = "- topic/foreign ">˙</mo></mrow></mover></mrow><mrow class = "- topic/foreign "><mn class = "- topic/foreign ">1</mn><mo class = "- topic/foreign ">−</mo></mrow></msub><msub class = "- topic/foreign "><mrow class = "- topic/foreign "><mover class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">y</mi></mrow><mrow class = "- topic/foreign "><mo class = "- topic/foreign ">˙</mo></mrow></mover></mrow><mrow class = "- topic/foreign "><mn class = "- topic/foreign ">2</mn></mrow></msub><mo class = "- topic/foreign ">)</mo><mo class = "- topic/foreign ">=</mo><mn class = "- topic/foreign ">0.</mn></mrow></math></span></p></td>
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<p> A matrix representation of the considered system of differential equations reads: </p>

<table class = "table" id = "tso-c-usr-sizing-nvh-bckgrnd__table_0EB29E7980BD403BB87A3AD196D4F34D"><caption/><colgroup><col/></colgroup><tbody class = "tbody">
<tr class = "row">
<td class = "entry"><p><span class = "ph inlineequation"><math class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">M</mi><mover class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">y</mi></mrow><mrow class = "- topic/foreign "><mo class = "- topic/foreign ">¨</mo></mrow></mover><mo class = "- topic/foreign ">+</mo><mi class = "- topic/foreign ">C</mi><mover class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">y</mi></mrow><mrow class = "- topic/foreign "><mo class = "- topic/foreign ">˙</mo></mrow></mover><mo class = "- topic/foreign ">+</mo><mi class = "- topic/foreign ">K</mi><mi class = "- topic/foreign ">y</mi><mo class = "- topic/foreign ">=</mo><mi class = "- topic/foreign ">F</mi></mrow></math></span></p>
                      <p><span class = "ph inlineequation"><math class = "- topic/foreign "><mrow class = "- topic/foreign "><mo class = "- topic/foreign ">⇔</mo><mo class = "- topic/foreign ">[</mo><mrow class = "- topic/foreign "><mtable class = "- topic/foreign "><mtr class = "- topic/foreign "><mtd class = "- topic/foreign "><msub class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">m</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">u</mi></mrow></msub></mtd><mtd class = "- topic/foreign "><mn class = "- topic/foreign ">0</mn></mtd></mtr><mtr class = "- topic/foreign "><mtd class = "- topic/foreign "><mn class = "- topic/foreign ">0</mn></mtd><mtd class = "- topic/foreign "><msub class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">m</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">s</mi></mrow></msub></mtd></mtr></mtable></mrow><mo class = "- topic/foreign ">]</mo><mo class = "- topic/foreign ">[</mo><mrow class = "- topic/foreign "><mtable class = "- topic/foreign "><mtr class = "- topic/foreign "><mtd class = "- topic/foreign "><msub class = "- topic/foreign "><mrow class = "- topic/foreign "><mover class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">y</mi></mrow><mrow class = "- topic/foreign "><mo class = "- topic/foreign ">¨</mo></mrow></mover></mrow><mrow class = "- topic/foreign "><mn class = "- topic/foreign ">1</mn></mrow></msub></mtd></mtr><mtr class = "- topic/foreign "><mtd class = "- topic/foreign "><msub class = "- topic/foreign "><mrow class = "- topic/foreign "><mover class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">y</mi></mrow><mrow class = "- topic/foreign "><mo class = "- topic/foreign ">¨</mo></mrow></mover></mrow><mrow class = "- topic/foreign "><mn class = "- topic/foreign ">2</mn></mrow></msub></mtd></mtr></mtable></mrow><mo class = "- topic/foreign ">]</mo><mo class = "- topic/foreign ">+</mo><mo class = "- topic/foreign ">[</mo><mrow class = "- topic/foreign "><mtable class = "- topic/foreign "><mtr class = "- topic/foreign "><mtd class = "- topic/foreign "><mi class = "- topic/foreign ">c</mi></mtd><mtd class = "- topic/foreign "><mo class = "- topic/foreign ">−</mo><mi class = "- topic/foreign ">c</mi></mtd></mtr><mtr class = "- topic/foreign "><mtd class = "- topic/foreign "><mo class = "- topic/foreign ">−</mo><mi class = "- topic/foreign ">c</mi></mtd><mtd class = "- topic/foreign "><mi class = "- topic/foreign ">c</mi></mtd></mtr></mtable></mrow><mo class = "- topic/foreign ">]</mo><mo class = "- topic/foreign ">[</mo><mrow class = "- topic/foreign "><mtable class = "- topic/foreign "><mtr class = "- topic/foreign "><mtd class = "- topic/foreign "><msub class = "- topic/foreign "><mrow class = "- topic/foreign "><mover class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">y</mi></mrow><mrow class = "- topic/foreign "><mo class = "- topic/foreign ">˙</mo></mrow></mover></mrow><mrow class = "- topic/foreign "><mn class = "- topic/foreign ">1</mn></mrow></msub></mtd></mtr><mtr class = "- topic/foreign "><mtd class = "- topic/foreign "><msub class = "- topic/foreign "><mrow class = "- topic/foreign "><mover class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">y</mi></mrow><mrow class = "- topic/foreign "><mo class = "- topic/foreign ">˙</mo></mrow></mover></mrow><mrow class = "- topic/foreign "><mn class = "- topic/foreign ">2</mn></mrow></msub></mtd></mtr></mtable></mrow><mo class = "- topic/foreign ">]</mo><mo class = "- topic/foreign ">+</mo><mo class = "- topic/foreign ">[</mo><mrow class = "- topic/foreign "><mtable class = "- topic/foreign "><mtr class = "- topic/foreign "><mtd class = "- topic/foreign "><mo class = "- topic/foreign ">(</mo><msub class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">k</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">t</mi></mrow></msub><mo class = "- topic/foreign ">+</mo><msub class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">k</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">s</mi></mrow></msub><mo class = "- topic/foreign ">)</mo></mtd><mtd class = "- topic/foreign "><mo class = "- topic/foreign ">−</mo><msub class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">k</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">s</mi></mrow></msub></mtd></mtr><mtr class = "- topic/foreign "><mtd class = "- topic/foreign "><mo class = "- topic/foreign ">−</mo><msub class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">k</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">s</mi></mrow></msub></mtd><mtd class = "- topic/foreign "><msub class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">k</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">s</mi></mrow></msub></mtd></mtr></mtable></mrow><mo class = "- topic/foreign ">]</mo><mo class = "- topic/foreign ">[</mo><mrow class = "- topic/foreign "><mtable class = "- topic/foreign "><mtr class = "- topic/foreign "><mtd class = "- topic/foreign "><msub class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">y</mi></mrow><mrow class = "- topic/foreign "><mn class = "- topic/foreign ">1</mn></mrow></msub></mtd></mtr><mtr class = "- topic/foreign "><mtd class = "- topic/foreign "><msub 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></msub></mtd></mtr></mtable></mrow><mo class = "- topic/foreign ">]</mo><mo class = "- topic/foreign ">=</mo><mo class = "- topic/foreign ">[</mo><mrow class = "- topic/foreign "><mtable class = "- topic/foreign "><mtr class = "- topic/foreign "><mtd class = "- topic/foreign "><msub class = "- topic/foreign "><mrow class = "- topic/foreign "><mtable class = "- topic/foreign "><mtr class = "- topic/foreign "><mtd class = "- topic/foreign "><mi class = "- topic/foreign ">k</mi></mtd></mtr></mtable></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">t</mi></mrow></msub><msub class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">y</mi></mrow><mrow class = "- topic/foreign "><mn class = "- topic/foreign ">0</mn></mrow></msub></mtd></mtr><mtr class = "- topic/foreign "><mtd class = "- topic/foreign "><mn class = "- topic/foreign ">0</mn></mtd></mtr></mtable></mrow><mo class = "- topic/foreign ">]</mo></mrow></math></span></p></td>
</tr>
</tbody></table>
<p> The right side of these equations represents the enforced load, which is time-dependent.
In case of harmonic loads these differential equations can be transformed to a system of
linear equations. Solving this system for a range of load frequencies provides the
corresponding amplitudes of displacements. Such a procedure is called frequency response
analysis. In case that the right side of the above equation is not a harmonic one, it can be
transformed by means of the Fourier transform and then analyzed. </p>
</div>


<div class = "section" id = "tso-c-usr-sizing-nvh-bckgrnd__tso-c-usr-sizing-nvh-acousticformulation"><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">Theoretical Background of Structural-Acoustic NVH Simulation</h2>

<p>The governing equations for free time-harmonic vibrations in a solid-acoustic region are given as:</p>      
<table class = "table" id = "tso-c-usr-sizing-nvh-bckgrnd__table_186FCBC39B984313A142DE8B53C2B7A1"><caption/><colgroup><col style = "width:33.33333333333333%"/><col style = "width:33.33333333333333%"/><col style = "width:33.33333333333333%"/></colgroup><tbody class = "tbody">
<tr class = "row">
<td class = "entry"><p>Solid:</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 ">σ</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">i</mi><mi class = "- topic/foreign ">j</mi><mo class = "- topic/foreign ">,</mo><mi class = "- topic/foreign ">j</mi></mrow></msub><mo class = "- topic/foreign ">+</mo><msub class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">f</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">i</mi></mrow></msub><mo class = "- topic/foreign ">+</mo><msub class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">ρ</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">s</mi></mrow></msub><msub class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">ü</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">i</mi></mrow></msub><mo class = "- topic/foreign ">=</mo><mn class = "- topic/foreign ">0</mn></mrow></math></span></td>
<td class = "entry"><span class = "ph inlineequation"><math class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">x</mi><mo class = "- topic/foreign ">∈</mo><msub class = "- topic/foreign "><mrow class = "- topic/foreign "><mi mathvariant = "normal" class = "- topic/foreign ">Ω</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">s</mi></mrow></msub></mrow></math></span></td>
</tr>
<tr class = "row">
<td class = "entry"><p>Solid coupling:</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 ">σ</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">i</mi><mi class = "- topic/foreign ">j</mi></mrow></msub><msubsup class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">n</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">j</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">s</mi></mrow></msubsup><mo class = "- topic/foreign ">=</mo><mi class = "- topic/foreign ">p</mi><msubsup class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">n</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">i</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">a</mi></mrow></msubsup></mrow></math></span></td>
<td class = "entry"><span class = "ph inlineequation"><math class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">x</mi><mo class = "- topic/foreign ">∈</mo><mi mathvariant = "normal" class = "- topic/foreign ">Γ</mi></mrow></math></span></td>
</tr>
<tr class = "row">
<td class = "entry"><p>Acoustic:</p></td>
<td class = "entry"><span class = "ph inlineequation"><math class = "- topic/foreign "><mrow class = "- topic/foreign "><mo class = "- topic/foreign ">▵</mo><mi class = "- topic/foreign ">p</mi><mo class = "- topic/foreign ">+</mo><mi class = "- topic/foreign ">q</mi><mfrac class = "- topic/foreign "><mrow class = "- topic/foreign "><mn class = "- topic/foreign ">1</mn></mrow><mrow class = "- topic/foreign "><msup class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">c</mi></mrow><mrow class = "- topic/foreign "><mn class = "- topic/foreign ">2</mn></mrow></msup></mrow></mfrac><mover class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">p</mi></mrow><mrow class = "- topic/foreign "><mo class = "- topic/foreign ">¨</mo></mrow></mover><mo class = "- topic/foreign ">=</mo><mn class = "- topic/foreign ">0</mn></mrow></math></span></td>
<td class = "entry"><span class = "ph inlineequation"><math class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">x</mi><mo class = "- topic/foreign ">∈</mo><msub class = "- topic/foreign "><mrow class = "- topic/foreign "><mi mathvariant = "normal" class = "- topic/foreign ">Ω</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">α</mi></mrow></msub></mrow></math></span></td>
</tr>
<tr class = "row">
<td class = "entry"><p>Acoustic coupling:</p></td>
<td class = "entry"><span class = "ph inlineequation"><math class = "- topic/foreign "><mrow class = "- topic/foreign "><mo class = "- topic/foreign ">▿</mo><mi class = "- topic/foreign ">p</mi><msubsup class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">n</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">i</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">a</mi></mrow></msubsup><mo class = "- topic/foreign ">=</mo><msub class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">ρ</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">a</mi></mrow></msub><msub class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">ü</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">i</mi></mrow></msub><msubsup class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">n</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">j</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">a</mi></mrow></msubsup></mrow></math></span></td>
<td class = "entry"><span class = "ph inlineequation"><math class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">x</mi><mo class = "- topic/foreign ">∈</mo><mi mathvariant = "normal" class = "- topic/foreign ">Γ</mi></mrow></math></span></td>
</tr>
</tbody></table>  
<p> Quantities that appear here are:</p>
<table class = "table" id = "tso-c-usr-sizing-nvh-bckgrnd__table_C74B24D8D1944FEF9EA544C1BD97131B"><caption/><colgroup><col style = "width:50%"/><col style = "width:50%"/></colgroup><tbody class = "tbody">
<tr class = "row">
<td class = "entry"><span class = "ph inlineequation"><math class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">s</mi></mrow></math></span></td>
<td class = "entry"><p>solid domain</p></td>
</tr>
<tr class = "row">
<td class = "entry"><span class = "ph inlineequation"><math class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">a</mi></mrow></math></span></td>
<td class = "entry"><p>acoustic domain</p></td>
</tr>
<tr class = "row">
<td class = "entry"><span class = "ph inlineequation"><math class = "- topic/foreign "><mrow class = "- topic/foreign "><mi mathvariant = "normal" class = "- topic/foreign ">Γ</mi></mrow></math></span></td>
<td class = "entry"><p>interface between two domains</p></td>
</tr>
<tr class = "row">
<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 ">u</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">i</mi></mrow></msub></mrow></math></span></td>
<td class = "entry"><p>displacements</p></td>
</tr>
<tr class = "row">
<td class = "entry"><span class = "ph inlineequation"><math class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">p</mi></mrow></math></span></td>
<td class = "entry"><p>pressure</p></td>
</tr>
<tr class = "row">
<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 ">σ</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">i</mi><mi class = "- topic/foreign ">j</mi></mrow></msub></mrow></math></span></td>
<td class = "entry"><p>stress tensor of the solid domain</p></td>
</tr>
<tr class = "row">
<td class = "entry"><span class = "ph inlineequation"><math class = "- topic/foreign "><mrow class = "- topic/foreign "><msubsup class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">n</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">j</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">s</mi></mrow></msubsup></mrow></math></span></td>
<td class = "entry"><p>outward pointing normal to the solid domain</p></td>
</tr>
<tr class = "row">
<td class = "entry"><span class = "ph inlineequation"><math class = "- topic/foreign "><mrow class = "- topic/foreign "><msubsup class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">n</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">i</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">a</mi></mrow></msubsup></mrow></math></span></td>
<td class = "entry"><p>outward pointing normal to the acoustic domain</p></td>
</tr>
<tr class = "row">
<td class = "entry"><span class = "ph inlineequation"><math class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">i</mi><mo class = "- topic/foreign ">,</mo><mi class = "- topic/foreign ">j</mi></mrow></math></span></td>
<td class = "entry"><p>=1,2,3</p></td>
</tr>
<tr class = "row">
<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 ">ρ</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">s</mi></mrow></msub></mrow></math></span></td>
<td class = "entry"><p>density of the solid</p></td>
</tr><tr class = "row">
<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 ">ρ</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">a</mi></mrow></msub></mrow></math></span></td>
<td class = "entry"><p>density of the acoustic media</p></td>
</tr>
<tr class = "row">
<td class = "entry"><span class = "ph inlineequation"><math class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">c</mi></mrow></math></span></td>
<td class = "entry"><p>speed of sound in the acoustic media</p></td></tr>
<tr class = "row">
<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 ">f</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">i</mi></mrow></msub></mrow></math></span></td>
<td class = "entry"><p>loading on the solid</p></td>
</tr>
<tr class = "row">
<td class = "entry"><span class = "ph inlineequation"><math class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">q</mi></mrow></math></span></td>
<td class = "entry"><p>loading on the acoustic media</p></td>
</tr>
</tbody></table>
      
<p> We assume that structural stresses are given by a linear stress-strain constitutive
modeling and the acoustic domain pressure has a linear relation to the volumetric strain. In
addition, assume that the loading on the solid and acoustic domains is harmonic and the
solution to be steady-state time-harmonic. Thus, the following two harmonic functions:</p>   
      
<table class = "table" id = "tso-c-usr-sizing-nvh-bckgrnd__table_E210F8A5B80E4FECB7412644332A8E26"><caption/><colgroup><col/></colgroup><tbody class = "tbody">
<tr class = "row">
<td class = "entry"><p><span class = "ph inlineequation"><math class = "- topic/foreign "><mrow class = "- topic/foreign "><msub class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">u</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">t</mi></mrow></msub><mo class = "- topic/foreign ">=</mo><msub class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">u</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">A</mi></mrow></msub><msup class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">e</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">i</mi><mi mathvariant = "normal" class = "- topic/foreign ">Ω</mi><mi class = "- topic/foreign ">t</mi></mrow></msup></mrow></math></span>
                     and <span class = "ph inlineequation"><math class = "- topic/foreign "><mrow class = "- topic/foreign "><msub class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">P</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">t</mi></mrow></msub><mo class = "- topic/foreign ">=</mo><msub class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">P</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">A</mi></mrow></msub><msup class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">e</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">i</mi><mi mathvariant = "normal" class = "- topic/foreign ">Ω</mi><mi class = "- topic/foreign ">t</mi></mrow></msup></mrow></math></span></p></td>
</tr>
</tbody></table>

<p>are a solution to the problem defined above. Quantities that appear here are,</p>  
<table class = "table" id = "tso-c-usr-sizing-nvh-bckgrnd__table_BBBCB2BFE9B94A26BB435C97E78971FD"><caption/><colgroup><col style = "width:50%"/><col style = "width:50%"/></colgroup><tbody class = "tbody">
<tr class = "row">
<td class = "entry"><span class = "ph inlineequation"><math class = "- topic/foreign "><mrow class = "- topic/foreign "><mi mathvariant = "normal" class = "- topic/foreign ">Ω</mi></mrow></math></span></td><td class = "entry"><p>excitation frequency</p></td>
</tr>
<tr class = "row">
<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 ">u</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">A</mi></mrow></msub></mrow></math></span></td>
<td class = "entry"><p>amplitude for the structural domain</p></td>
</tr>
<tr class = "row">
<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 ">p</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">A</mi></mrow></msub></mrow></math></span></td>
<td class = "entry"><p>amplitude for the acoustic domain</p></td>
</tr>
</tbody></table>

<p>Discretization of the above equations in context of finite element method leads to,</p> 
<table class = "table" id = "tso-c-usr-sizing-nvh-bckgrnd__table_90905795805646F78B3AE144EEB699EC"><caption/><colgroup><col/></colgroup><tbody class = "tbody">
<tr class = "row">
<td class = "entry"><p><span class = "ph inlineequation"><math class = "- topic/foreign "><mrow class = "- topic/foreign "><mo class = "- topic/foreign ">−</mo><msup class = "- topic/foreign "><mrow class = "- topic/foreign "><mi mathvariant = "normal" class = "- topic/foreign ">Ω</mi></mrow><mrow class = "- topic/foreign "><mn class = "- topic/foreign ">2</mn></mrow></msup><mo class = "- topic/foreign ">[</mo><mrow class = "- topic/foreign "><mtable class = "- topic/foreign "><mtr class = "- topic/foreign "><mtd class = "- topic/foreign "><msub class = "- topic/foreign "><mrow class = "- topic/foreign "><mtable class = "- topic/foreign "><mtr class = "- topic/foreign "><mtd class = "- topic/foreign "><mi class = "- topic/foreign ">M</mi></mtd></mtr></mtable></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">s</mi></mrow></msub></mtd><mtd class = "- topic/foreign "><mn class = "- topic/foreign ">0</mn></mtd></mtr><mtr class = "- topic/foreign "><mtd class = "- topic/foreign "><msup class = "- topic/foreign "><mrow class = "- topic/foreign "><mtable class = "- topic/foreign "><mtr class = "- topic/foreign "><mtd class = "- topic/foreign "><mi class = "- topic/foreign ">A</mi></mtd></mtr></mtable></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">T</mi></mrow></msup></mtd><mtd class = "- topic/foreign "><msub class = "- topic/foreign "><mrow class = "- topic/foreign "><mtable class = "- topic/foreign "><mtr class = "- topic/foreign "><mtd class = "- topic/foreign "><mi class = "- topic/foreign ">M</mi></mtd></mtr></mtable></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">a</mi></mrow></msub></mtd></mtr></mtable></mrow><mo class = "- topic/foreign ">]</mo><mo class = "- topic/foreign ">[</mo><mrow class = "- topic/foreign "><mtable class = "- topic/foreign "><mtr class = "- topic/foreign "><mtd class = "- topic/foreign "><msub class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">u</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">A</mi></mrow></msub></mtd></mtr><mtr class = "- topic/foreign "><mtd class = "- topic/foreign "><msub class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">P</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">a</mi></mrow></msub></mtd></mtr></mtable></mrow><mo class = "- topic/foreign ">]</mo><mo class = "- topic/foreign ">+</mo><mi class = "- topic/foreign ">i</mi><mi mathvariant = "normal" class = "- topic/foreign ">Ω</mi><mo class = "- topic/foreign ">[</mo><mrow class = "- topic/foreign "><mtable class = "- topic/foreign "><mtr class = "- topic/foreign "><mtd class = "- topic/foreign "><msub class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">C</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">s</mi></mrow></msub></mtd><mtd class = "- topic/foreign "><mn class = "- topic/foreign ">0</mn></mtd></mtr><mtr class = "- topic/foreign "><mtd class = "- topic/foreign "><mn class = "- topic/foreign ">0</mn></mtd><mtd class = "- topic/foreign "><msub class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">C</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">a</mi></mrow></msub></mtd></mtr></mtable></mrow><mo class = "- topic/foreign ">]</mo><mo class = "- topic/foreign ">[</mo><mrow class = "- topic/foreign "><mtable class = "- topic/foreign "><mtr class = "- topic/foreign "><mtd class = "- topic/foreign "><msub class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">u</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">A</mi></mrow></msub></mtd></mtr><mtr class = "- topic/foreign "><mtd class = "- topic/foreign "><msub class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">P</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">a</mi></mrow></msub></mtd></mtr></mtable></mrow><mo class = "- topic/foreign ">]</mo><mo class = "- topic/foreign ">+</mo><mo class = "- topic/foreign ">[</mo><mrow class = "- topic/foreign "><mtable class = "- topic/foreign "><mtr class = "- topic/foreign "><mtd class = "- topic/foreign "><msub class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">K</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">s</mi></mrow></msub></mtd><mtd class = "- topic/foreign "><mo class = "- topic/foreign ">−</mo><mi class = "- topic/foreign ">A</mi></mtd></mtr><mtr class = "- topic/foreign "><mtd class = "- topic/foreign "><mn class = "- topic/foreign ">0</mn></mtd><mtd class = "- topic/foreign "><msub class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">K</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">a</mi></mrow></msub></mtd></mtr></mtable></mrow><mo class = "- topic/foreign ">]</mo><mo class = "- topic/foreign ">[</mo><mrow class = "- topic/foreign "><mtable class = "- topic/foreign "><mtr class = "- topic/foreign "><mtd class = "- topic/foreign "><msub class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">u</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">A</mi></mrow></msub></mtd></mtr><mtr class = "- topic/foreign "><mtd class = "- topic/foreign "><msub class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">P</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">a</mi></mrow></msub></mtd></mtr></mtable></mrow><mo class = "- topic/foreign ">]</mo><mo class = "- topic/foreign ">=</mo><mo class = "- topic/foreign ">[</mo><mrow class = "- topic/foreign "><mtable class = "- topic/foreign "><mtr class = "- topic/foreign "><mtd class = "- topic/foreign "><msub class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">P</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">s</mi></mrow></msub></mtd></mtr><mtr class = "- topic/foreign "><mtd class = "- topic/foreign "><msub class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">P</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">a</mi></mrow></msub></mtd></mtr></mtable></mrow><mo class = "- topic/foreign ">]</mo></mrow></math></span></p></td>
</tr>
</tbody></table>
<p>with the following quantities:</p>
<table class = "table" id = "tso-c-usr-sizing-nvh-bckgrnd__table_9FF41D15D2414209A9F9CE4543498CB8"><caption/><colgroup><col style = "width:50%"/><col style = "width:50%"/></colgroup><tbody class = "tbody">
<tr class = "row">
<td class = "entry"><span class = "ph inlineequation"><math class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">M</mi></mrow></math></span></td>
<td class = "entry"><p>mass matrix</p></td>
</tr>
<tr class = "row">
<td class = "entry"><span class = "ph inlineequation"><math class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">C</mi></mrow></math></span></td>
<td class = "entry"><p>damping matrix</p></td>
</tr>
<tr class = "row">
<td class = "entry"><span class = "ph inlineequation"><math class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">K</mi></mrow></math></span></td>
<td class = "entry"><p>stiffness matrix</p></td>
</tr>
<tr class = "row">
<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 ">P</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">s</mi></mrow></msub></mrow></math></span></td>
<td class = "entry"><p>applied harmonic structural force</p></td>
</tr>
<tr class = "row">
<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 ">P</mi></mrow><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">a</mi></mrow></msub></mrow></math></span></td>
<td class = "entry"><p>applied harmonic acoustic pressure</p></td>
</tr>
<tr class = "row">
<td class = "entry"><span class = "ph inlineequation"><math class = "- topic/foreign "><mrow class = "- topic/foreign "><mi class = "- topic/foreign ">A</mi></mrow></math></span></td>
<td class = "entry"><p>coupling matrix that models the interface between acoustic domain and structural components</p></td>
</tr>
</tbody></table>


<p>This system of equations is solved for each excitation frequency providing the corresponding amplitudes
of displacements and pressure. This procedure again corresponds to the frequency response
analysis but now with the possibility to compute the acoustic pressure.</p>  
</div>


<div class = "section" id = "tso-c-usr-sizing-nvh-bckgrnd__tso-c-usr-sizing-nvh-applications"><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">Possible Applications of NVH Simulation in Engineering</h2>

<p>Most applications of NVH are found in the automotive or aerospace fields. NVH analysis can
be carried out for both performance and comfort reasons. For example, modal analyses on cars
are carried out in order to prevent resonance of the engine and chassis frequencies. An
acoustic analysis, for example, could be performed to ensure cabin comfort. Apart from
engineering reasons, NVH analyses might be carried out when designing sound systems, or to
predict noise levels around industrial sites or airports etc. </p> The applications of NVH
are: <ul class = "ul" id = "tso-c-usr-sizing-nvh-bckgrnd__ul_4F07445B03C242BB82FA5B5E8F17961E">
<li class = "li">Engine noise vibration testing. </li>
<li class = "li">Acoustic performance testing. </li>
<li class = "li">Sound power testing. </li>
<li class = "li">Environmental noise measurements and noise field mapping. </li>
<li class = "li">Structural dynamics and vibration testing. </li>
<li class = "li">Occupational health and safety. </li>
</ul>
</div>


</div>

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