Homotopy Analysis Method for Nonlinear Dynamical System of an Electrostatically Actuated Microcantilever

<table>Relative errors versus <svg height="10.95" id="M170" style="vertical-align:-0.1254pt" version="1.1" viewbox="0 0 9.1374998 10.95" width="9.1374998" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,10.95)">
<g transform="translate(72,-63.24)">
<text transform="matrix(1,0,0,-1,-71.95,63.41)">
<tspan style="font-size: 12.50px; " x="0" y="0">ℎ</tspan>
</text>
</g>
</g>
</svg> between 20th-order HAM solutions and numerical ones. Parameter values are <svg height="14.6" id="M171" style="vertical-align:-3.13504pt" version="1.1" viewbox="0 0 39.900002 14.6" width="39.900002" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,14.6)">
<g transform="translate(72,-60.32)">
<text transform="matrix(1,0,0,-1,-71.95,63.5)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝑐</tspan>
</text>
<text transform="matrix(1,0,0,-1,-66.76,60.37)">
<tspan style="font-size: 8.75px; " x="0" y="0">2</tspan>
</text>
<text transform="matrix(1,0,0,-1,-58.41,63.5)">
<tspan style="font-size: 12.50px; " x="0" y="0">=</tspan>
<tspan style="font-size: 12.50px; " x="12.027886" y="0">2</tspan>
</text>
</g>
</g>
</svg>, <svg height="10.8625" id="M172" style="vertical-align:-0.13794pt" version="1.1" viewbox="0 0 37.974998 10.8625" width="37.974998" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,10.8625)">
<g transform="translate(72,-63.31)">
<text transform="matrix(1,0,0,-1,-71.95,63.5)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝜔</tspan>
<tspan style="font-size: 12.50px; " x="11.977874" y="0">=</tspan>
<tspan style="font-size: 12.50px; " x="24.018263" y="0">1</tspan>
</text>
</g>
</g>
</svg>, <svg height="16.6" id="M173" style="vertical-align:-4.74141pt" version="1.1" viewbox="0 0 42.287498 16.6" width="42.287498" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,16.6)">
<g transform="translate(72,-58.72)">
<text transform="matrix(1,0,0,-1,-71.95,63.5)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝑉</tspan>
</text>
<text transform="matrix(1,0,0,-1,-64.76,60.37)">
<tspan style="font-size: 8.75px; " x="0" y="0">𝑝</tspan>
</text>
<text transform="matrix(1,0,0,-1,-56.51,63.5)">
<tspan style="font-size: 12.50px; " x="0" y="0">=</tspan>
<tspan style="font-size: 12.50px; " x="12.027886" y="0">1</tspan>
</text>
</g>
</g>
</svg>, and <svg height="10.875" id="M174" style="vertical-align:-0.15048pt" version="1.1" viewbox="0 0 35.637501 10.875" width="35.637501" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,10.875)">
<g transform="translate(72,-63.3)">
<text transform="matrix(1,0,0,-1,-71.95,63.5)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝑑</tspan>
<tspan style="font-size: 12.50px; " x="10.114927" y="0">=</tspan>
<tspan style="font-size: 12.50px; " x="22.155315" y="0">2</tspan>
</text>
</g>
</g>
</svg>, respectively.</table>

Mathematical Problems in Engineering

fig3

Figure 3

Figure 3: Homotopy Analysis Method for Nonlinear Dynamical System of an Electrostatically Actuated Microcantilever 