Free Vibration Analysis of an Euler Beam of Variable Width on the Winkler Foundation Using Homotopy Perturbation Method

<div class="floats-full-page floats-full-page-other"><h2>Table 5</h2><div class="floats-full-page-caption">Normalized free vibration frequencies of cantilever beam resting on Winkler foundation with linearly varying flexural stiffness.</div><div class="floats-full-page-content"><table class="table-group" id="tab5"><tr><td><table class="table"><tr><td class="thead-hr" colspan="7"><hr /></td></tr><tr class="thead"><td align="left"><span style="width: 9.7749996px;"><svg height="7.9499998" id="M93" style="vertical-align:-0.1638pt" version="1.1" viewbox="0 0 9.7749996 7.9499998" width="9.7749996" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,7.675)"><path d="M545 106q-67 -118 -134 -118q-24 0 -40 37.5t-30 129.5h-2q-47 -72 -103 -119.5t-108 -47.5q-47 0 -76 45.5t-29 119.5q0 113 85 204t174 91q47 0 70 -33.5t43 -119.5h3q32 47 80 140l55 13l10 -9q-47 -80 -138 -201q17 -99 27.5 -136t22.5 -37q23 0 69 61zM333 204&#10;q-14 98 -31 149.5t-50 51.5q-49 0 -94 -70t-45 -164q0 -55 15.5 -86t40.5 -31q70 0 164 150z" id="x1D6FC"></path></g>
</svg></span></td><td align="center">0.00</td><td align="center">0.10</td><td align="center">0.20</td><td align="center">0.30</td><td align="center">0.40</td><td align="center">0.50</td></tr><tr><td class="thead-hr" colspan="7"><hr /></td></tr><tr><td align="left"><span style="width: 17.299999px;"><svg height="11.925" id="M94" style="vertical-align:-3.27605pt" version="1.1" viewbox="0 0 17.299999 11.925" width="17.299999" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,7.775)"><path d="M615 290q0 -148 -114 -245q-65 -57 -125 -57q-39 0 -62 27t-29 70q-31 -41 -72 -69t-74 -28q-49 0 -82.5 43t-33.5 124q0 97 55 174.5t148 124.5l15 -24q-59 -45 -94 -113t-35 -146q0 -58 19.5 -88.5t48.5 -30.5q33 0 62 34.5t45 101.5q16 73 20 135l80 17l5 -6&#10;q-11 -37 -23.5 -100t-12.5 -92q0 -47 16.5 -71.5t44.5 -24.5q62 0 95.5 67t33.5 160q0 107 -63 152l1 9q14 14 45 14q29 0 57.5 -44t28.5 -114z" id="x1D714"></path></g>
<g transform="matrix(.012,-0,0,-.012,10.912,11.863)"><path d="M384 0h-275v27q67 5 81.5 18.5t14.5 68.5v385q0 38 -7.5 47.5t-40.5 10.5l-48 2v24q85 15 178 52v-521q0 -55 14.5 -68.5t82.5 -18.5v-27z" id="x31"></path></g>
</svg></span></td><td align="center">3.65546</td><td align="center">3.77785</td><td align="center">3.91814</td><td align="center">4.08113</td><td align="center">4.27363</td><td align="center">4.50571</td></tr><tr><td align="left"><span style="width: 17.299999px;"><svg height="11.925" id="M95" style="vertical-align:-3.27605pt" version="1.1" viewbox="0 0 17.299999 11.925" width="17.299999" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,7.775)"><path d="M615 290q0 -148 -114 -245q-65 -57 -125 -57q-39 0 -62 27t-29 70q-31 -41 -72 -69t-74 -28q-49 0 -82.5 43t-33.5 124q0 97 55 174.5t148 124.5l15 -24q-59 -45 -94 -113t-35 -146q0 -58 19.5 -88.5t48.5 -30.5q33 0 62 34.5t45 101.5q16 73 20 135l80 17l5 -6&#10;q-11 -37 -23.5 -100t-12.5 -92q0 -47 16.5 -71.5t44.5 -24.5q62 0 95.5 67t33.5 160q0 107 -63 152l1 9q14 14 45 14q29 0 57.5 -44t28.5 -114z" id="x1D714"></path></g>
<g transform="matrix(.012,-0,0,-.012,10.912,11.863)"><path d="M412 140l28 -9q0 -2 -35 -131h-373v23q112 112 161 170q59 70 92 127t33 115q0 63 -31 98t-86 35q-75 0 -137 -93l-22 20l57 81q55 59 135 59q69 0 118.5 -46.5t49.5 -122.5q0 -62 -29.5 -114t-102.5 -130l-141 -149h186q42 0 58.5 10.5t38.5 56.5z" id="x32"></path></g>
</svg></span></td><td align="center">22.0572</td><td align="center">22.2779</td><td align="center">22.5271</td><td align="center">22.8129</td><td align="center">23.1475</td><td align="center">23.5506</td></tr><tr><td align="left"><span style="width: 17.299999px;"><svg height="12.0625" id="M96" style="vertical-align:-3.3907pt" version="1.1" viewbox="0 0 17.299999 12.0625" width="17.299999" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,7.775)"><path d="M615 290q0 -148 -114 -245q-65 -57 -125 -57q-39 0 -62 27t-29 70q-31 -41 -72 -69t-74 -28q-49 0 -82.5 43t-33.5 124q0 97 55 174.5t148 124.5l15 -24q-59 -45 -94 -113t-35 -146q0 -58 19.5 -88.5t48.5 -30.5q33 0 62 34.5t45 101.5q16 73 20 135l80 17l5 -6&#10;q-11 -37 -23.5 -100t-12.5 -92q0 -47 16.5 -71.5t44.5 -24.5q62 0 95.5 67t33.5 160q0 107 -63 152l1 9q14 14 45 14q29 0 57.5 -44t28.5 -114z" id="x1D714"></path></g>
<g transform="matrix(.012,-0,0,-.012,10.912,11.863)"><path d="M285 378v-2q65 -13 102 -54.5t37 -97.5q0 -57 -30.5 -104.5t-74 -75t-85.5 -42t-72 -14.5q-31 0 -59.5 11t-40.5 23q-19 18 -16 36q1 16 23 33q13 10 24 0q58 -51 124 -51q55 0 88 40t33 112q0 64 -39 96.5t-88 32.5q-29 0 -64 -11l-6 29q77 25 118 57.5t41 84.5&#10;q0 45 -26.5 69.5t-68.5 24.5q-67 0 -120 -79l-20 20l43 63q51 56 127 56h1q66 0 107 -37t41 -95q0 -42 -31 -71q-22 -23 -68 -54z" id="x33"></path></g>
</svg></span></td><td align="center">61.7053</td><td align="center">61.9182</td><td align="center">62.1616</td><td align="center">62.4458</td><td align="center">62.7867</td><td align="center">63.2104</td></tr><tr><td align="left"><span style="width: 17.299999px;"><svg height="11.925" id="M97" style="vertical-align:-3.27605pt" version="1.1" viewbox="0 0 17.299999 11.925" width="17.299999" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,7.775)"><path d="M615 290q0 -148 -114 -245q-65 -57 -125 -57q-39 0 -62 27t-29 70q-31 -41 -72 -69t-74 -28q-49 0 -82.5 43t-33.5 124q0 97 55 174.5t148 124.5l15 -24q-59 -45 -94 -113t-35 -146q0 -58 19.5 -88.5t48.5 -30.5q33 0 62 34.5t45 101.5q16 73 20 135l80 17l5 -6&#10;q-11 -37 -23.5 -100t-12.5 -92q0 -47 16.5 -71.5t44.5 -24.5q62 0 95.5 67t33.5 160q0 107 -63 152l1 9q14 14 45 14q29 0 57.5 -44t28.5 -114z" id="x1D714"></path></g>
<g transform="matrix(.012,-0,0,-.012,10.912,11.863)"><path d="M456 178h-96v-72q0 -51 12.5 -62.5t72.5 -16.5v-27h-256v27q65 5 78 17t13 62v72h-260v28q182 271 300 426h40v-407h96v-47zM280 225v295h-2q-107 -148 -196 -295h198z" id="x34"></path></g>
</svg></span></td><td align="center">120.9061</td><td align="center">121.1194</td><td align="center">121.3645</td><td align="center">121.6528</td><td align="center">122.0024</td><td align="center">122.4434</td></tr><tr><td align="left"><span style="width: 17.299999px;"><svg height="12.0625" id="M98" style="vertical-align:-3.3907pt" version="1.1" viewbox="0 0 17.299999 12.0625" width="17.299999" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,7.775)"><path d="M615 290q0 -148 -114 -245q-65 -57 -125 -57q-39 0 -62 27t-29 70q-31 -41 -72 -69t-74 -28q-49 0 -82.5 43t-33.5 124q0 97 55 174.5t148 124.5l15 -24q-59 -45 -94 -113t-35 -146q0 -58 19.5 -88.5t48.5 -30.5q33 0 62 34.5t45 101.5q16 73 20 135l80 17l5 -6&#10;q-11 -37 -23.5 -100t-12.5 -92q0 -47 16.5 -71.5t44.5 -24.5q62 0 95.5 67t33.5 160q0 107 -63 152l1 9q14 14 45 14q29 0 57.5 -44t28.5 -114z" id="x1D714"></path></g>
<g transform="matrix(.012,-0,0,-.012,10.912,11.863)"><path d="M153 550l-26 -186q79 31 111 31q90 0 141.5 -51t51.5 -119q0 -93 -89 -166q-85 -69 -173 -71q-32 0 -61.5 11.5t-41.5 23.5q-18 17 -17 34q2 16 22 33q14 9 26 -1q61 -50 124 -50q60 0 93 43.5t33 104.5q0 69 -41.5 110t-121.5 41q-53 0 -102 -20l38 305h286l6 -8&#10;l-26 -65h-233z" id="x35"></path></g>
</svg></span></td><td align="center">199.8620</td><td align="center">200.0755</td><td align="center">200.3213</td><td align="center">200.6117</td><td align="center">200.9661</td><td align="center">201.4172</td></tr><tr class="table-tr"><td colspan="7"><hr class="tbody-hr" /></td></tr></table></td></tr></table></div></div>

Table 5 | Free Vibration Analysis of an Euler Beam of Variable Width on the Winkler Foundation Using Homotopy Perturbation Method

title

Table 5: Free Vibration Analysis of an Euler Beam of Variable Width on the Winkler Foundation Using Homotopy Perturbation Method

description

Table 5

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