Vibrational Study of Fluid-Filled Functionally Graded Cylindrical Shells Resting on Elastic Foundations

<table class="table-group" id="tab4"><tr><td><table class="table"><tr><td class="thead-hr" colspan="10"><hr/></td></tr><tr class="thead"><td align="left">Shell</td><td align="center"><span style="width: 14.8375px;"><svg height="10.325" id="M173" style="vertical-align:-0.0pt" version="1.1" viewbox="0 0 14.8375 10.325" width="14.8375" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,10.325)">
<g transform="translate(72,-63.74)">
<text transform="matrix(1,0,0,-1,-71.95,63.79)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝑁</tspan>
</text>
</g>
</g>
</svg></span></td><td align="center"><span style="width: 42.637501px;"><svg height="13.775" id="M174" style="vertical-align:-0.0pt" version="1.1" viewbox="0 0 42.637501 13.775" width="42.637501" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,13.775)">
<g transform="translate(72,-60.98)">
<text transform="matrix(1,0,0,-1,-71.95,61.03)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝑁</tspan>
</text>
<text transform="matrix(1,0,0,-1,-60.18,66.03)">
<tspan style="font-size: 8.75px; " x="0" y="0">𝑆</tspan>
<tspan style="font-size: 8.75px; " x="5.6887999" y="0">𝑆</tspan>
<tspan style="font-size: 8.75px; " x="11.3776" y="0">=</tspan>
<tspan style="font-size: 8.75px; " x="17.372721" y="0">0</tspan>
</text>
</g>
</g>
</svg></span></td><td align="center"><span style="width: 37.724998px;"><svg height="13.775" id="M175" style="vertical-align:-0.0pt" version="1.1" viewbox="0 0 37.724998 13.775" width="37.724998" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,13.775)">
<g transform="translate(72,-60.98)">
<text transform="matrix(1,0,0,-1,-71.95,61.03)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝑁</tspan>
</text>
<text transform="matrix(1,0,0,-1,-60.18,66.03)">
<tspan style="font-size: 8.75px; " x="0" y="0">𝑁</tspan>
<tspan style="font-size: 8.75px; " x="7.4479518" y="0">=</tspan>
<tspan style="font-size: 8.75px; " x="13.443072" y="0">0</tspan>
</text>
</g>
</g>
</svg></span></td><td align="center"><span style="width: 52.349998px;"><svg height="11.1" id="M176" style="vertical-align:-0.17555pt" version="1.1" viewbox="0 0 52.349998 11.1" width="52.349998" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,11.1)">
<g transform="translate(72,-63.12)">
<text transform="matrix(1,0,0,-1,-71.95,63.35)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝑁</tspan>
<tspan style="font-size: 12.50px; " x="14.103384" y="0">=</tspan>
<tspan style="font-size: 12.50px; " x="26.143772" y="0">0</tspan>
<tspan style="font-size: 12.50px; " x="32.395271" y="0">.</tspan>
<tspan style="font-size: 12.50px; " x="35.521023" y="0">5</tspan>
</text>
</g>
</g>
</svg></span></td><td align="center"><span style="width: 40.625px;"><svg height="10.6875" id="M177" style="vertical-align:-0.0pt" version="1.1" viewbox="0 0 40.625 10.6875" width="40.625" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,10.6875)">
<g transform="translate(72,-63.45)">
<text transform="matrix(1,0,0,-1,-71.95,63.5)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝑁</tspan>
<tspan style="font-size: 12.50px; " x="14.103384" y="0">=</tspan>
<tspan style="font-size: 12.50px; " x="26.143772" y="0">1</tspan>
</text>
</g>
</g>
</svg></span></td><td align="center"><span style="width: 40.625px;"><svg height="10.6875" id="M178" style="vertical-align:-0.0pt" version="1.1" viewbox="0 0 40.625 10.6875" width="40.625" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,10.6875)">
<g transform="translate(72,-63.45)">
<text transform="matrix(1,0,0,-1,-71.95,63.5)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝑁</tspan>
<tspan style="font-size: 12.50px; " x="14.103384" y="0">=</tspan>
<tspan style="font-size: 12.50px; " x="26.143772" y="0">2</tspan>
</text>
</g>
</g>
</svg></span></td><td align="center"><span style="width: 40.625px;"><svg height="11.1" id="M179" style="vertical-align:-0.17555pt" version="1.1" viewbox="0 0 40.625 11.1" width="40.625" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,11.1)">
<g transform="translate(72,-63.12)">
<text transform="matrix(1,0,0,-1,-71.95,63.35)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝑁</tspan>
<tspan style="font-size: 12.50px; " x="14.103384" y="0">=</tspan>
<tspan style="font-size: 12.50px; " x="26.143772" y="0">5</tspan>
</text>
</g>
</g>
</svg></span></td><td align="center"><span style="width: 48.4375px;"><svg height="11.1" id="M180" style="vertical-align:-0.17555pt" version="1.1" viewbox="0 0 48.4375 11.1" width="48.4375" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,11.1)">
<g transform="translate(72,-63.12)">
<text transform="matrix(1,0,0,-1,-71.95,63.35)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝑁</tspan>
<tspan style="font-size: 12.50px; " x="14.103384" y="0">=</tspan>
<tspan style="font-size: 12.50px; " x="26.143772" y="0">1</tspan>
<tspan style="font-size: 12.50px; " x="32.395271" y="0">5</tspan>
</text>
</g>
</g>
</svg></span></td><td align="center"><span style="width: 48.4375px;"><svg height="10.9125" id="M181" style="vertical-align:-0.17555pt" version="1.1" viewbox="0 0 48.4375 10.9125" width="48.4375" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,10.9125)">
<g transform="translate(72,-63.27)">
<text transform="matrix(1,0,0,-1,-71.95,63.5)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝑁</tspan>
<tspan style="font-size: 12.50px; " x="14.103384" y="0">=</tspan>
<tspan style="font-size: 12.50px; " x="26.143772" y="0">3</tspan>
<tspan style="font-size: 12.50px; " x="32.395271" y="0">0</tspan>
</text>
</g>
</g>
</svg></span></td></tr><tr><td class="thead-hr" colspan="10"><hr/></td></tr><tr><td align="left" rowspan="5">Type I</td><td align="center">1</td><td align="center">107.24</td><td align="center">102.72</td><td align="center">105.66</td><td align="center">104.91</td><td align="center">104.16</td><td align="center">103.43</td><td align="center">102.98</td><td align="center">102.85</td></tr><tr><td align="center">2</td><td align="center">136.27</td><td align="center">130.53</td><td align="center">134.27</td><td align="center">133.31</td><td align="center">132.36</td><td align="center">131.44</td><td align="center">130.87</td><td align="center">130.71</td></tr><tr><td align="center">3</td><td align="center">144.71</td><td align="center">138.61</td><td align="center">142.58</td><td align="center">141.56</td><td align="center">140.55</td><td align="center">139.57</td><td align="center">138.97</td><td align="center">138.79</td></tr><tr><td align="center">4</td><td align="center">148.13</td><td align="center">141.88</td><td align="center">145.95</td><td align="center">144.91</td><td align="center">143.88</td><td align="center">142.87</td><td align="center">142.25</td><td align="center">142.07</td></tr><tr><td align="center">5</td><td align="center">150.00</td><td align="center">143.67</td><td align="center">147.80</td><td align="center">146.73</td><td align="center">145.69</td><td align="center">144.67</td><td align="center">144.05</td><td align="center">143.86</td></tr><tr><td align="center" colspan="10"><hr/></td></tr><tr><td align="left" rowspan="5"> Type II</td><td align="center">1</td><td align="center">107.24</td><td align="center">102.72</td><td align="center">104.16</td><td align="center">104.91</td><td align="center">105.67</td><td align="center">106.45</td><td align="center">106.94</td><td align="center">107.09</td></tr><tr><td align="center">2</td><td align="center">136.27</td><td align="center">130.53</td><td align="center">132.36</td><td align="center">133.31</td><td align="center">134.28</td><td align="center">135.26</td><td align="center">135.89</td><td align="center">136.08</td></tr><tr><td align="center">3</td><td align="center">144.71</td><td align="center">138.61</td><td align="center">140.55</td><td align="center">141.56</td><td align="center">142.58</td><td align="center">143.63</td><td align="center">144.30</td><td align="center">144.49</td></tr><tr><td align="center">4</td><td align="center">148.13</td><td align="center">141.88</td><td align="center">143.88</td><td align="center">144.91</td><td align="center">145.96</td><td align="center">147.03</td><td align="center">147.71</td><td align="center">147.91</td></tr><tr><td align="center">5</td><td align="center">150.00</td><td align="center">143.67</td><td align="center">145.69</td><td align="center">146.74</td><td align="center">147.80</td><td align="center">148.88</td><td align="center">149.58</td><td align="center">149.78</td></tr><tr class="table-tr"><td colspan="10"><hr class="tbody-hr"/></td></tr></table></td></tr></table>

Variation of natural frequencies (Hz) against circumferential wave number <svg height="7.1374998" id="M167" style="vertical-align:-0.10033pt" version="1.1" viewbox="0 0 7.8874998 7.1374998" width="7.8874998" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,7.1375)">
<g transform="translate(72,-66.29)">
<text transform="matrix(1,0,0,-1,-71.95,66.44)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝑛</tspan>
</text>
</g>
</g>
</svg> of an empty FG cylindrical shell on elastic foundation. (<svg height="10.9125" id="M168" style="vertical-align:-0.17555pt" version="1.1" viewbox="0 0 38.799999 10.9125" width="38.799999" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,10.9125)">
<g transform="translate(72,-63.27)">
<text transform="matrix(1,0,0,-1,-71.95,63.5)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝐺</tspan>
<tspan style="font-size: 12.50px; " x="12.640533" y="0">=</tspan>
<tspan style="font-size: 12.50px; " x="24.680922" y="0">0</tspan>
</text>
</g>
</g>
</svg>, <svg height="14.225" id="M169" style="vertical-align:-0.3135pt" version="1.1" viewbox="0 0 89.625 14.225" width="89.625" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,14.225)">
<g transform="translate(72,-60.62)">
<text transform="matrix(1,0,0,-1,-71.95,60.99)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝐾</tspan>
<tspan style="font-size: 12.50px; " x="12.990617" y="0">=</tspan>
<tspan style="font-size: 12.50px; " x="25.031006" y="0">1</tspan>
<tspan style="font-size: 12.50px; " x="31.282507" y="0">.</tspan>
<tspan style="font-size: 12.50px; " x="34.408257" y="0">5</tspan>
<tspan style="font-size: 12.50px; " x="43.435421" y="0">×</tspan>
<tspan style="font-size: 12.50px; " x="54.213009" y="0">1</tspan>
<tspan style="font-size: 12.50px; " x="60.464508" y="0">0</tspan>
</text>
<text transform="matrix(1,0,0,-1,-5.22,66.16)">
<tspan style="font-size: 8.75px; " x="0" y="0">7</tspan>
</text>
</g>
</g>
</svg>) Types I (<svg height="10.8125" id="M170" style="vertical-align:-0.10033pt" version="1.1" viewbox="0 0 38.424999 10.8125" width="38.424999" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,10.8125)">
<g transform="translate(72,-63.35)">
<text transform="matrix(1,0,0,-1,-71.95,63.5)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝑚</tspan>
<tspan style="font-size: 12.50px; " x="12.340461" y="0">=</tspan>
<tspan style="font-size: 12.50px; " x="24.38085" y="0">1</tspan>
</text>
</g>
</g>
</svg>, <svg height="11.0125" id="M171" style="vertical-align:-0.17555pt" version="1.1" viewbox="0 0 79.599998 11.0125" width="79.599998" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,11.0125)">
<g transform="translate(72,-63.19)">
<text transform="matrix(1,0,0,-1,-71.95,63.41)">
<tspan style="font-size: 12.50px; " x="0" y="0">ℎ</tspan>
<tspan style="font-size: 12.50px; " x="7.214231" y="0">/</tspan>
<tspan style="font-size: 12.50px; " x="10.690065" y="0">𝑅</tspan>
<tspan style="font-size: 12.50px; " x="23.405617" y="0">=</tspan>
<tspan style="font-size: 12.50px; " x="35.446007" y="0">0</tspan>
<tspan style="font-size: 12.50px; " x="41.697506" y="0">.</tspan>
<tspan style="font-size: 12.50px; " x="44.823254" y="0">0</tspan>
<tspan style="font-size: 12.50px; " x="51.074757" y="0">0</tspan>
<tspan style="font-size: 12.50px; " x="57.326256" y="0">2</tspan>
</text>
</g>
</g>
</svg>, <svg height="10.9125" id="M172" style="vertical-align:-0.17555pt" version="1.1" viewbox="0 0 62.112499 10.9125" width="62.112499" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,10.9125)">
<g transform="translate(72,-63.27)">
<text transform="matrix(1,0,0,-1,-71.95,63.5)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝐿</tspan>
<tspan style="font-size: 12.50px; " x="8.8521242" y="0">/</tspan>
<tspan style="font-size: 12.50px; " x="12.327958" y="0">𝑅</tspan>
<tspan style="font-size: 12.50px; " x="25.043509" y="0">=</tspan>
<tspan style="font-size: 12.50px; " x="37.083897" y="0">2</tspan>
<tspan style="font-size: 12.50px; " x="43.3354" y="0">0</tspan>
</text>
</g>
</g>
</svg>).

International Scholarly Research Notices

tab4

Table 4

Table 4: Vibrational Study of Fluid-Filled Functionally Graded Cylindrical Shells Resting on Elastic Foundations 