Vibration Analysis of Multilink Manipulators Based on Timoshenko Beam Theory

<table>(a) Modes (Hz) of an industrial manipulator (see Table <a href="../tab8/">8</a>). (FEM: ABAQUS 6.6, B23, B22) left column: <i>EB</i>-model; right column: <svg height="10.325" id="M332" style="vertical-align:-0.0pt" version="1.1" viewbox="0 0 11.225 10.325" width="11.225" 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">𝑇</tspan>
</text>
</g>
</g>
</svg>-model: <svg height="15.075" id="M333" style="vertical-align:-3.25793pt" version="1.1" viewbox="0 0 87.550003 15.075" width="87.550003" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,15.075)">
<g transform="translate(72,-59.94)">
<text transform="matrix(1,0,0,-1,-71.95,63.24)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝜃</tspan>
</text>
<text transform="matrix(1,0,0,-1,-65.4,60.12)">
<tspan style="font-size: 8.75px; " x="0" y="0">2</tspan>
</text>
<text transform="matrix(1,0,0,-1,-57.05,63.24)">
<tspan style="font-size: 12.50px; " x="0" y="0">=</tspan>
<tspan style="font-size: 12.50px; " x="12.027886" y="0">𝜃</tspan>
</text>
<text transform="matrix(1,0,0,-1,-38.46,60.12)">
<tspan style="font-size: 8.75px; " x="0" y="0">3</tspan>
</text>
<text transform="matrix(1,0,0,-1,-30.11,63.24)">
<tspan style="font-size: 12.50px; " x="0" y="0">=</tspan>
<tspan style="font-size: 12.50px; " x="12.027886" y="0">4</tspan>
<tspan style="font-size: 12.50px; " x="18.279387" y="0">5</tspan>
</text>
<text transform="matrix(1,0,0,-1,-5.57,68.56)">
<tspan style="font-size: 8.75px; " x="0" y="0">∘</tspan>
</text>
</g>
</g>
</svg>. (b) Modes (Hz) of an industrial manipulator (see Table <a href="../tab8/">8</a>). (FEM: ABAQUS 6.6, B23, B22) left column: <i>EB</i>-model; right column: <svg height="10.325" id="M334" style="vertical-align:-0.0pt" version="1.1" viewbox="0 0 11.225 10.325" width="11.225" 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">𝑇</tspan>
</text>
</g>
</g>
</svg>-model: <svg height="15.075" id="M335" style="vertical-align:-3.25793pt" version="1.1" viewbox="0 0 87.550003 15.075" width="87.550003" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,15.075)">
<g transform="translate(72,-59.94)">
<text transform="matrix(1,0,0,-1,-71.95,63.24)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝜃</tspan>
</text>
<text transform="matrix(1,0,0,-1,-65.4,60.12)">
<tspan style="font-size: 8.75px; " x="0" y="0">2</tspan>
</text>
<text transform="matrix(1,0,0,-1,-57.05,63.24)">
<tspan style="font-size: 12.50px; " x="0" y="0">=</tspan>
<tspan style="font-size: 12.50px; " x="12.027886" y="0">𝜃</tspan>
</text>
<text transform="matrix(1,0,0,-1,-38.46,60.12)">
<tspan style="font-size: 8.75px; " x="0" y="0">3</tspan>
</text>
<text transform="matrix(1,0,0,-1,-30.11,63.24)">
<tspan style="font-size: 12.50px; " x="0" y="0">=</tspan>
<tspan style="font-size: 12.50px; " x="12.027886" y="0">4</tspan>
<tspan style="font-size: 12.50px; " x="18.279387" y="0">5</tspan>
</text>
<text transform="matrix(1,0,0,-1,-5.57,68.56)">
<tspan style="font-size: 8.75px; " x="0" y="0">∘</tspan>
</text>
</g>
</g>
</svg>.</table>

Journal of Robotics

fig5

Figure 5

Figure 5: Vibration Analysis of Multilink Manipulators Based on Timoshenko Beam Theory 