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

<table class="table-group" id="tab1"><tr><td><table class="table"><tr><td class="thead-hr" colspan="6"><hr/></td></tr><tr class="thead"><td align="left">Method</td><td align="center"><span style="width: 17.299999px;"><svg height="11.925" id="M73" 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">
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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"><span style="width: 17.299999px;"><svg height="11.925" id="M74" 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
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"><span style="width: 17.299999px;"><svg height="12.0625" id="M75" 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
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
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"><span style="width: 17.299999px;"><svg height="11.925" id="M76" 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
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>
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<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
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
l-26 -65h-233z" id="x35"></path></g>
</svg></span></td></tr><tr><td class="thead-hr" colspan="6"><hr/></td></tr><tr><td align="left">HPM </td><td align="center">9.92014</td><td align="center">39.4911</td><td align="center">88.8321</td><td align="center">157.9168</td><td align="center">246.7421</td></tr><tr><td align="left">DTM [<a href="/journals/mpe/2013/721294/#B1">1</a>]</td><td align="center">9.92014</td><td align="center">39.4911</td><td align="center">88.8321</td><td align="center">—</td><td align="center">—</td></tr><tr><td align="left">DQEM [<a href="/journals/mpe/2013/721294/#B7">7</a>]</td><td align="center">9.92014</td><td align="center">39.4913</td><td align="center">89.4002</td><td align="center">—</td><td align="center">—</td></tr><tr><td align="left">Exact solution [<a href="/journals/mpe/2013/721294/#B1">1</a>]</td><td align="center">9.92014</td><td align="center">39.4911</td><td align="center">88.8321</td><td align="center">—</td><td align="center">—</td></tr><tr class="table-tr"><td colspan="6"><hr class="tbody-hr"/></td></tr></table></td></tr></table>

Normalized free vibration frequencies of simply supported beam resting on the Winkler foundation.

Mathematical Problems in Engineering

tab1

Table 1

Table 1: Free Vibration Analysis of an Euler Beam of Variable Width on the Winkler Foundation Using Homotopy Perturbation Method 