Thrust Vector Control of an Upper-Stage Rocket with Multiple Propellant Slosh Modes

<table>Time responses of <svg height="11.3375" id="M193" style="vertical-align:-3.25793pt" version="1.1" viewbox="0 0 35.174999 11.3375" width="35.174999" xmlns="http://www.w3.org/2000/svg">
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<text transform="matrix(1,0,0,-1,-71.95,66.22)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝑣</tspan>
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<text transform="matrix(1,0,0,-1,-65.62,63.1)">
<tspan style="font-size: 8.75px; " x="0" y="0">𝑥</tspan>
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<text transform="matrix(1,0,0,-1,-60.31,66.22)">
<tspan style="font-size: 12.50px; " x="0" y="0">,</tspan>
<tspan style="font-size: 12.50px; " x="5.2012482" y="0">𝑣</tspan>
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<text transform="matrix(1,0,0,-1,-48.78,63.1)">
<tspan style="font-size: 8.75px; " x="0" y="0">𝑧</tspan>
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</svg> and <svg height="10.7375" id="M194" style="vertical-align:-0.13794pt" version="1.1" viewbox="0 0 8.3125 10.7375" width="8.3125" xmlns="http://www.w3.org/2000/svg">
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<g transform="translate(72,-63.41)">
<text transform="matrix(1,0,0,-1,-71.95,63.6)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝜃</tspan>
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</svg> (zero control case).</table>

Mathematical Problems in Engineering

fig5

Figure 5

Figure 5: Thrust Vector Control of an Upper-Stage Rocket with Multiple Propellant Slosh Modes 