Hybrid Optimization Approach for the Design of Mechanisms Using a New Error Estimator

<table class="table-group" id="tab5"><tr><td><table class="table"><tr><td class="thead-hr" colspan="13"><hr/></td></tr><tr class="thead"><td align="left">Design variables</td><td align="center"><svg height="11.9625" id="M86" style="vertical-align:-3.39066pt" version="1.1" viewbox="0 0 15.9625 11.9625" width="15.9625" 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)"><use xlink:href="#x1D465"></use></g>
<g transform="matrix(.012,-0,0,-.012,9.563,11.762)"><use xlink:href="#x30"></use></g>
</svg> (mm)</td><td align="center"><svg height="12.175" id="M87" style="vertical-align:-3.56265pt" version="1.1" viewbox="0 0 16.299999 12.175" width="16.299999" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,7.675)"><use xlink:href="#x1D466"></use></g>
<g transform="matrix(.012,-0,0,-.012,9.9,11.762)"><use xlink:href="#x30"></use></g>
</svg> (mm)</td><td align="center"><svg height="16.450001" id="M88" style="vertical-align:-3.39066pt" version="1.1" viewbox="0 0 20.637501 16.450001" width="20.637501" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,12.162)"><use xlink:href="#x1D703"></use></g>
<g transform="matrix(.012,-0,0,-.012,8.525,16.25)"><use xlink:href="#x31"></use></g><g transform="matrix(.012,-0,0,-.012,14.236,16.25)"><use xlink:href="#x30"></use></g>
</svg> (rad)</td><td align="center"><svg height="16.3125" id="M89" style="vertical-align:-3.276pt" version="1.1" viewbox="0 0 14.925 16.3125" width="14.925" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,12.162)"><use xlink:href="#x1D703"></use></g>
<g transform="matrix(.012,-0,0,-.012,8.525,16.25)"><use xlink:href="#x34"></use></g>
</svg> (rad)</td><td align="center"><svg height="15.25" id="M90" style="vertical-align:-3.276pt" version="1.1" viewbox="0 0 16.1625 15.25" width="16.1625" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,11.112)"><use xlink:href="#x1D43F"></use></g>
<g transform="matrix(.012,-0,0,-.012,9.763,15.187)"><use xlink:href="#x31"></use></g>
</svg> (mm)</td><td align="center"><svg height="15.25" id="M91" style="vertical-align:-3.276pt" version="1.1" viewbox="0 0 16.1625 15.25" width="16.1625" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,11.112)"><use xlink:href="#x1D43F"></use></g>
<g transform="matrix(.012,-0,0,-.012,9.763,15.187)"><use xlink:href="#x32"></use></g>
</svg> (mm)</td><td align="center"><svg height="15.4" id="M92" style="vertical-align:-3.39066pt" version="1.1" viewbox="0 0 16.1625 15.4" width="16.1625" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,11.112)"><use xlink:href="#x1D43F"></use></g>
<g transform="matrix(.012,-0,0,-.012,9.763,15.187)"><use xlink:href="#x33"></use></g>
</svg> (mm)</td><td align="center"><svg height="15.25" id="M93" style="vertical-align:-3.276pt" version="1.1" viewbox="0 0 16.1625 15.25" width="16.1625" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,11.112)"><use xlink:href="#x1D43F"></use></g>
<g transform="matrix(.012,-0,0,-.012,9.763,15.187)"><use xlink:href="#x34"></use></g>
</svg> (mm)</td><td align="center"><svg height="15.4" id="M94" style="vertical-align:-3.39066pt" version="1.1" viewbox="0 0 16.1625 15.4" width="16.1625" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,11.112)"><use xlink:href="#x1D43F"></use></g>
<g transform="matrix(.012,-0,0,-.012,9.763,15.187)"><use xlink:href="#x35"></use></g>
</svg> (mm)</td><td align="center"><svg height="15.4" id="M95" style="vertical-align:-3.39066pt" version="1.1" viewbox="0 0 16.1625 15.4" width="16.1625" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,11.112)"><use xlink:href="#x1D43F"></use></g>
<g transform="matrix(.012,-0,0,-.012,9.763,15.187)"><path d="M137 343l67 33q37 17 63 17q79 0 129.5 -53t50.5 -131q0 -92 -58 -156.5t-147 -64.5t-147 68t-58 182q0 63 17 119t43 95.5t61.5 72t69 52t67.5 31.5q62 22 128 33l6 -32q-56 -11 -108 -35q-149 -71 -184 -231zM227 337q-47 0 -95 -27q-6 -23 -6 -70q0 -93 36 -155.5
t96 -62.5q53 0 78 45.5t25 105.5q0 68 -35 116t-99 48z" id="x36"></path></g>
</svg> (mm)</td><td align="center"><svg height="7.9499998" id="M96" 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
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> (rad)</td><td align="center"><svg height="11.9625" id="M97" style="vertical-align:-3.39066pt" version="1.1" viewbox="0 0 15.9625 11.9625" width="15.9625" 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)"><use xlink:href="#x1D465"></use></g>
<g transform="matrix(.012,-0,0,-.012,9.563,11.762)"><use xlink:href="#x35"></use></g>
</svg> (mm)</td></tr><tr><td class="thead-hr" colspan="13"><hr/></td></tr><tr><td align="left">EA solution</td><td align="center">810.92</td><td align="center">311.51</td><td align="center">2.745</td><td align="center">−1.23</td><td align="center">517.03</td><td align="center">992.82</td><td align="center">991.66</td><td align="center">1227.4</td><td align="center">1161.7</td><td align="center">722.41</td><td align="center">−0.286</td><td align="center">100</td></tr><tr><td align="left">Hybrid solution</td><td align="center">537.39</td><td align="center">393.31</td><td align="center">2.785</td><td align="center">−1.4</td><td align="center">416.52</td><td align="center">603.75</td><td align="center">554.85</td><td align="center">724.41</td><td align="center">611.08</td><td align="center">372.27</td><td align="center">0.2437</td><td align="center">100</td></tr><tr class="table-tr"><td colspan="13"><hr class="tbody-hr"/></td></tr></table></td></tr></table>

Mathematical Problems in Engineering

tab5

Table 5

Table 5: Hybrid Optimization Approach for the Design of Mechanisms Using a New Error Estimator 