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462 lines
16 KiB
462 lines
16 KiB
\chapter{Effect of Aspect Ratio}
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\section{Introduction}
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An important issue for radioactive materials transportation casks
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is the aspect ratio at which shallow angle slapdown impact events
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result in a higher velocity secondary impact than primary impact. Two
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primary parameters have been identified, slenderness and location of
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the center of gravity.
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\section{Slenderness}
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The slenderness issue was studied using solid and hollow cylinders
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(simple models which capture the essential geometry of most
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radioactive materials transportation casks). For the solid cylinder
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model, the initial geometry had a radius of 30 and a length of 120.
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The length was reduced in steps to 50 while the radius was increased
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to keep the overall model volume constant. The
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hollow cylinder model was treated similarly. The initial
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hollow cylinder model
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had a length of 120 with an outer radius of 30 and an inner
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radius of 21.21. These radii gave a cross-sectional area of the
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hollow cylinder equal to half the area enclosed by the outer radius.
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As for the solid cylinder, the length was reduced to 50 while the
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overall model volume and the relationship between cross-sectional
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areas were held constant. The details of the model geometries are
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shown in Tables 4.1 and 4.3.
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The slapdown analysis was performed for an
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initial angle of 15$^\circ$. Two linear springs were investigated, one of
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which was elastic and the other plastic. The elastic spring had a
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linear spring constant for both loading and unloading of 600,000.
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Thus no energy was absorbed in the spring. The plastic spring had a
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loading constant of 600,000 (same as the elastic spring) but had an
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unloading constant of 600,000,000,000 (6 orders of magnitude higher
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than the loading constant). The effect of this high unloading
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constant was to prevent energy being returned to the structure on
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spring unloading thus giving an almost perfectly plastic spring. The
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initial angle of 15$^\circ$ was sufficient to ensure that for all cases,
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the nose was rebounding from the target prior to the tail impact. No
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frictional effects were considered here.
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\begin{table}
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\begin{center}
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\caption{Effect of Aspect Ratio on Slapdown for Solid Cylinder Model
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with a Linear-Elastic Spring}
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\makeqnum
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\begin{tabular}{||r|r|r|r|r|r|r||}
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\hline
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&\multicolumn{1}{c|}{Outer}
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&\multicolumn{1}{c|}{Moment of}
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&\multicolumn{1}{c|}{Radius of}
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&\multicolumn{1}{c|}{Aspect}
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&\multicolumn{1}{c|}{Tail}
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&\multicolumn{1}{c||}{Tail}\\
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\multicolumn{1}{||c|}{Length}
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&\multicolumn{1}{c|}{Radius}
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&\multicolumn{1}{c|}{Inertia}
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&\multicolumn{1}{c|}{Gyration}
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&\multicolumn{1}{c|}{(L/r)}
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&\multicolumn{1}{c|}{Velocity}
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&\multicolumn{1}{c||}{Displ}\\
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\hline
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$120$ &$30.00$ &$??114,000$ &$?37.75$ &$3.18$ &$?-979$ &$6.028$\\
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$?80$ &$36.75$ &$???69,700$ &$?29.51$ &$2.71$ &$?-831$ &$5.689$\\
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$?60$ &$42.33$ &$???60,000$ &$?27.39$ &$2.19$ &$?-613$ &$4.745$\\
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$?55$ &$44.31$ &$???59,400$ &$?27.25$ &$2.02$ &$?-526$ &$4.243$\\
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$?50$ &$46.48$ &$???61,200$ &$?27.26$ &$1.83$ &$?-409$ &$3.468$\\
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\hline
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\end{tabular}
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\end{center}
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Analysis performed at the following initial conditions:
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\makeqnum
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\begin{tabular}{lll}
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Initial vertical velocity &= &$-527.5$\\
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Initial angle &= &$???15.0^\circ$\\
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Total mass &= &$???80.0$\\
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Spring - elastic Load &= &$??600.0\times10^3$\\
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Spring - elastic Unload &= &$??600.0\times10^3$\\
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\end{tabular}
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\end{table}
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\begin{table}
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\begin{center}
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\caption{Effect of Aspect Ratio on Slapdown for Solid Cylinder Model
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with a Plastic (Energy-Absorbing) Spring}
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\makeqnum
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\begin{tabular}{||r|r|r|r|r|r|r||}
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\hline
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&\multicolumn{1}{c|}{Outer}
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&\multicolumn{1}{c|}{Moment of}
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&\multicolumn{1}{c|}{Radius of}
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&\multicolumn{1}{c|}{Aspect}
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&\multicolumn{1}{c|}{Tail}
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&\multicolumn{1}{c||}{Tail}\\
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\multicolumn{1}{||c|}{Length}
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&\multicolumn{1}{c|}{Radius}
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&\multicolumn{1}{c|}{Inertia}
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&\multicolumn{1}{c|}{Gyration}
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&\multicolumn{1}{c|}{(L/r)}
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&\multicolumn{1}{c|}{Velocity}
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&\multicolumn{1}{c||}{Displ}\\
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\hline
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$120$ &$30.00$ &$??114,000$ &$?37.75$ &$3.18$ &$-754$ &$5.031$\\
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$?80$ &$36.75$ &$???69,700$ &$?29.51$ &$2.71$ &$-678$ &$4.804$\\
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$?60$ &$42.33$ &$???60,000$ &$?27.39$ &$2.19$ &$-567$ &$4.426$\\
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$?55$ &$44.31$ &$???59,400$ &$?27.25$ &$2.02$ &$-523$ &$4.261$\\
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$?50$ &$46.48$ &$???61,200$ &$?27.26$ &$1.83$ &$-465$ &$3.990$\\
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\hline
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\end{tabular}
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\end{center}
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Analysis performed at the following initial conditions:
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\begin{tabular}{lll}
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Initial vertical velocity &= &$-527.5$\\
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Initial angle &= &$???15.0^\circ$\\
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Total mass &= &$???80.0$\\
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Spring - plastic Load &= &$??600.0\times10^3$\\
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Spring - plastic Unload &= &$??600.0\times10^9$\\
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\end{tabular}
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\end{table}
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\begin{figure}
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\vspace{3.5 in}
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\caption{Effect of Aspect Ratio (L/r) on Slapdown Severity}
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\end{figure}
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The results of the slapdown analysis for the solid cylinders
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with elastic and plastic springs
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are presented in Tables 4.1 and 4.2
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respectively. For the hollow cylinders, the results are shown
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in Tables 4.3 and 4.4.
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The ratio of length to
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radius of gyration (L/r) was
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selected to describe the slenderness of an object
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subjected to shallow angle slapdown.
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The vertical velocity of the tail
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at the secondary impact and the maximum tail spring displacement
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were chosen to represent the severity of the secondary
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impact event. The maximum spring displacement is directly related to
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the energy required to stop the tail of the object while the vertical
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velocity at impact provides a clear representation of the slapdown
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event independent of the tail spring characteristics. Tail vertical
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velocity at impact, non-dimensionalized by the initial velocity, is
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plotted against aspect ratio for both the solid and hollow
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cylinders in Figure 4.1.
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In Figure 4.1, non-dimensional tail velocities
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less than one, indicate that slapdown did not occur (secondary impact
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was less severe than primary impact). Slapdown did not occur when the
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aspect ratio was less than 2 for both model geometries and both
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spring types. The plastic spring brings the tail
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velocity at secondary impact closer to the initial velocity for all
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aspect ratios. Thus, with the plastic spring, secondary impact
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velocities are lower for aspect ratios greater than 2 and higher
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(but still less than the initial velocity) for aspect ratios less
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than 2.
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\begin{table}
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\begin{center}
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\caption{Effect of Aspect Ratio on Slapdown for Hollow Cylinder Model
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with a Linear-Elastic Spring}
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\makeqnum
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\begin{tabular}{||r|r|r|r|r|r|r|r||}
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\hline
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&\multicolumn{1}{c|}{Outer}
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&\multicolumn{1}{c|}{Inner}
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&\multicolumn{1}{c|}{Moment of}
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&\multicolumn{1}{c|}{Radius of}
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&\multicolumn{1}{c|}{Aspect}
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&\multicolumn{1}{c|}{Tail}
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&\multicolumn{1}{c||}{Tail}\\
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\multicolumn{1}{||c|}{Length}
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&\multicolumn{1}{c|}{Radius}
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&\multicolumn{1}{c|}{Radius}
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&\multicolumn{1}{c|}{Inertia}
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&\multicolumn{1}{c|}{Gyration}
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&\multicolumn{1}{c|}{(L/r)}
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&\multicolumn{1}{c|}{Velocity}
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&\multicolumn{1}{c||}{Displ}\\
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\hline
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$120$ &$30.00$ &$21.21$ &$??123,000$ &$?39.21$ &$3.06$ &$?-944$
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&$5.974$\\
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$?80$ &$36.75$ &$25.99$ &$???83,200$ &$?32.25$ &$2.48$ &$?-741$
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&$5.360$\\
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$?62$ &$41.74$ &$29.52$ &$???77,900$ &$?31.20$ &$1.99$ &$?-508$
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&$4.128$\\
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$?55$ &$44.31$ &$31.34$ &$???79,100$ &$?31.44$ &$1.75$ &$?-373$
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&$3.202$\\
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$?50$ &$46.48$ &$32.87$ &$???81,500$ &$?31.91$ &$1.57$ &$?-261$
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&$2.315$\\
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\hline
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\end{tabular}
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\end{center}
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Analysis performed at the following initial conditions:
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\begin{tabular}{lll}
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Initial vertical velocity &= &$-527.5$\\
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Initial angle &= &$???15.0^\circ$\\
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Total mass &= &$???80.0$\\
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Spring - elastic Load &= &$??600.0\times10^3$\\
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Spring - elastic Unload &= &$??600.0\times10^3$\\
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\end{tabular}
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\end{table}
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\begin{table}
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\begin{center}
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\caption{Effect of Aspect Ratio on Slapdown for Hollow Cylinder Model
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with a Plastic (Energy-Absorbing) Spring}
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\makeqnum
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\begin{tabular}{||r|r|r|r|r|r|r|r||}
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\hline
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&\multicolumn{1}{c|}{Outer}
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&\multicolumn{1}{c|}{Inner}
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&\multicolumn{1}{c|}{Moment of}
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&\multicolumn{1}{c|}{Radius of}
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&\multicolumn{1}{c|}{Aspect}
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&\multicolumn{1}{c|}{Tail}
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&\multicolumn{1}{c||}{Tail}\\
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\multicolumn{1}{||c|}{Length}
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&\multicolumn{1}{c|}{Radius}
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&\multicolumn{1}{c|}{Radius}
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&\multicolumn{1}{c|}{Inertia}
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&\multicolumn{1}{c|}{Gyration}
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&\multicolumn{1}{c|}{(L/r)}
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&\multicolumn{1}{c|}{Velocity}
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&\multicolumn{1}{c||}{Displ}\\
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\hline
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$120$ &$30.00$ &$21.21$ &$??123,000$ &$?39.21$ &$3.06$ &$-736$
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&$4.962$\\
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$?80$ &$36.75$ &$25.99$ &$???83,200$ &$?32.25$ &$2.48$ &$-633$
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&$4.659$\\
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$?62$ &$41.74$ &$29.52$ &$???77,900$ &$?31.20$ &$1.99$ &$-515$
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&$4.224$\\
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$?55$ &$44.31$ &$31.34$ &$???79,100$ &$?31.44$ &$1.75$ &$-448$
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&$3.896$\\
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$?50$ &$46.48$ &$32.87$ &$???81,500$ &$?31.91$ &$1.57$ &$-391$
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&$3.562$\\
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\hline
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\end{tabular}
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\end{center}
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Analysis performed at the following initial conditions:
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\makeqnum
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\begin{tabular}{lll}
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Initial vertical velocity &= &$-527.5$\\
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Initial angle &= &$???15.0^\circ$\\
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Total mass &= &$???80.0$\\
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Spring - plastic Load &= &$??600.0\times10^3$\\
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Spring - plastic Unload &= &$??600.0\times10^9$\\
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\end{tabular}
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\end{table}
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A secondary impact less severe than the primary impact has been
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shown, in Chapter 2,
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to be a general relationship for aspect ratios less than two. In
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Chapter 2, the aspect ratio was defined in terms of $\beta_{n} =
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l_{n}/r$ and $\beta_{t} = l_{t}/r$ where:
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\begin{tabbing}
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$r \;$ \= is the radius of gyration,\\
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$l_{n}$ \> is the distance from the center of mass to the nose, and\\
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$l_{t}$ \> is the distance from the center of mass to the tail.\\
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\end{tabbing}
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Here $l_{n}$ = $l_{t}$ = ${{L}\over{2}}$, so that
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\begin{equation}
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\beta_{n} = \beta_{t} = {{L}\over{2r}}.
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\end{equation}
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Thus
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\begin{equation}
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\beta_{n}\cdot\beta_{t} = {{L^{2}}\over{4r^{2}}}.
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\end{equation}
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As shown in Equation 2.3.19, the tail impact velocity is less than the
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nose impact velocity when
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\begin{equation}
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\beta_{n} \cdot \beta_{t}\leq 1
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\end{equation}
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or when
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\begin{equation}
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{{L}\over{r}} \leq 2.
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\end{equation}
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\section{Center of Gravity Location}
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In most
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transportation systems, due to the impact limiters, closure
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systems, gamma shielding, and contents, the center of gravity rarely
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coincides with the midpoint between loading springs. Thus,
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variation in the location of the center of gravity
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are of concern for transportation system design. Two of the solid
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cylinder models, those with length 120 and length 55, were chosen to
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investigate this phenomenon. These two models provide a wide
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variation in aspect ratio (3.18 and 2.02, respectively). The
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center of gravity was shifted independently of all other parameters
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including the moment of inertia. A physical interpretation of this
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shift is difficult to provide. It can be regarded as resulting from
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the appropriate distribution of a variable density material within the
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confines of the solid cylinder model geometry. Alternately, it can be
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regarded as a structure with the given mass and moment of inertia
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placed at varying locations along a rigid massless bar. The elastic
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and plastic springs described in Section 4.2 were also used here.
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\begin{figure}
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\vspace{3.5 in}
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\caption{Effect of the Location of the center of gravity on Slapdown
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Severity}
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\end{figure}
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The results of shifting the center of gravity axially on the two
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solid cylinder models are shown in Figure 4.2 and in
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Tables 4.5 and 4.6. As can be seen
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in these tables, both the aspect ratio and the nose spring
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characteristics, influence
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the effects shifting the center of gravity.
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For the long cylinder, the tail velocity is
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maximized when the center of gravity is at the 30\% location (closer to
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the nose than the tail) for both the linear and the nonlinear springs.
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For the short cylinder, maximum tail velocity occurs when the
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center of gravity
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coincides with the cylinder center (50\% location). As
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expected, for an arbitrary center of gravity location, maximum energy
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absorption at the tail does not coincide with maximum tail
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velocity. The maximum tail energy occurs at a different
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center of gravity
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location for each case investigated. For the long cylinder
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with an elastic nose spring, maximum tail energy occurs at a
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center of gravity
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location of 60\% (slightly toward the tail). With a plastic
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nose spring, maximum tail energy occurs at the 90\% location. For the
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short cylinder, the maximum tail energy center of gravity location is
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70\% for an elastic nose spring and 80\% for a plastic nose spring.
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\begin{table}
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\begin{center}
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\caption{Effect of Shift of center of gravity on Slapdown for Solid
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Cylinder Model Length=120 with Elastic and Plastic Spring}
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\makeqnum
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\begin{tabular}{||c|r|r|c|r|r||}
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\hline
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\multicolumn{3}{||c|}{Elastic Spring} &\multicolumn{3}{c||}{Plastic
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Spring}\\
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\hline
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\multicolumn{1}{||c|}{C. G.}
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& &
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&\multicolumn{1}{c|}{C. G.}
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& &\\
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\multicolumn{1}{||c|}{Location}
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&\multicolumn{1}{c|}{Tail}
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&\multicolumn{1}{c|}{Tail}
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&\multicolumn{1}{c|}{Location}
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&\multicolumn{1}{c|}{Tail}
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&\multicolumn{1}{c||}{Tail}\\
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\multicolumn{1}{||c|}{\% Length}
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&\multicolumn{1}{c|}{Velocity}
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&\multicolumn{1}{c|}{Displ}
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&\multicolumn{1}{c|}{\% Length}
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&\multicolumn{1}{c|}{Velocity}
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&\multicolumn{1}{c||}{Displ}\\
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$10$ &$?-406$ &$1.557$ &$10$ &$-470$ &$1.791$\\
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$20$ &$?-967$ &$4.085$ &$20$ &$-743$ &$3.165$\\
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$30$ &$-1132$ &$5.357$ &$30$ &$-827$ &$4.080$\\
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$40$ &$-1093$ &$5.865$ &$40$ &$-809$ &$4.654$\\
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$50$ &$?-979$ &$6.028$ &$50$ &$-754$ &$5.031$\\
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$60$ &$?-847$ &$6.065$ &$60$ &$-689$ &$5.263$\\
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$70$ &$?-721$ &$6.049$ &$70$ &$-626$ &$5.466$\\
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$80$ &$?-608$ &$5.945$ &$80$ &$-570$ &$5.607$\\
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$90$ &$?-501$ &$5.613$ &$90$ &$-520$ &$5.722$\\
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\hline
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\end{tabular}
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\end{center}
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Analysis performed at the following initial conditions:
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\makeqnum
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\begin{tabular}{llllll}
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Initial vertical velocity &= &$-527.5$
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&Initial vertical velocity &= &$-527.5$\\
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Initial angle &= &$???15.0^\circ$ &Initial angle &= &$???15.0^\circ$\\
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Total mass &= &$???80.0$ &Total mass &= &$???80.0$\\
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Spring - elastic Load &= &$??600.0\times10^3$
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&Spring - plastic Load &= &$??600.0\times10^3$\\
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Spring - elastic Unload &= &$??600.0\times10^9$
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&Spring - plastic Unload &= &$??600.0\times10^9$\\
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\end{tabular}
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\end{table}
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\begin{table}
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\begin{center}
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\caption{Effect of Shift of center of gravity on Slapdown for Solid
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Cylinder Model Length=55 with Elastic and Plastic Spring}
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\makeqnum
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\begin{tabular}{||c|r|r|c|r|r||}
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\hline
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\multicolumn{3}{||c|}{Elastic Spring} &\multicolumn{3}{c||}{Plastic
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Spring}\\
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\hline
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\multicolumn{1}{||c|}{C. G.}
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& &
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&\multicolumn{1}{c|}{C. G.}
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& &\\
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\multicolumn{1}{||c|}{Location}
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&\multicolumn{1}{c|}{Tail}
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&\multicolumn{1}{c|}{Tail}
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&\multicolumn{1}{c|}{Location}
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&\multicolumn{1}{c|}{Tail}
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&\multicolumn{1}{c||}{Tail}\\
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\multicolumn{1}{||c|}{\% Length}
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&\multicolumn{1}{c|}{Velocity}
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&\multicolumn{1}{c|}{Displ}
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&\multicolumn{1}{c|}{\% Length}
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&\multicolumn{1}{c|}{Velocity}
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&\multicolumn{1}{c||}{Displ}\\
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$10$ &No Hit &$0.???$ &$10$ &$-198$ &$1.110$\\
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$20$ &$-193$ &$1.146$ &$20$ &$-358$ &$2.185$\\
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$30$ &$-401$ &$2.658$ &$30$ &$-460$ &$3.075$\\
|
|
$40$ &$-500$ &$3.655$ &$40$ &$-510$ &$3.755$\\
|
|
$50$ &$-526$ &$4.243$ &$50$ &$-523$ &$4.261$\\
|
|
$60$ &$-508$ &$4.522$ &$60$ &$-516$ &$4.636$\\
|
|
$70$ &$-467$ &$4.562$ &$70$ &$-496$ &$4.902$\\
|
|
$80$ &$-416$ &$4.398$ &$80$ &$-471$ &$5.044$\\
|
|
$90$ &$-361$ &$4.048$ &$90$ &$-444$ &$5.031$\\
|
|
\hline
|
|
\end{tabular}
|
|
\end{center}
|
|
|
|
Analysis performed at the following initial conditions:
|
|
|
|
\makeqnum
|
|
\begin{tabular}{llllll}
|
|
Initial vertical velocity &= &$-527.5$
|
|
&Initial vertical velocity &= &$-527.5$\\
|
|
Initial angle &= &$???15.0^\circ$ &Initial angle &= &$???15.0^\circ$\\
|
|
Total mass &= &$???80.0$ &Total mass &= &$???80.0$\\
|
|
Spring - elastic Load &= &$??600.0\times10^3$
|
|
&Spring - plastic Load &= &$??600.0\times10^3$\\
|
|
Spring - elastic Unload &= &$??600.0\times10^9$
|
|
&Spring - plastic Unload &= &$??600.0\times10^9$\\
|
|
\end{tabular}
|
|
\end{table}
|
|
|
|
\section{Conclusions}
|
|
The ratio of length to radius of gyration is the most appropriate
|
|
parameter to describe the body geometry for shallow angle
|
|
slapdown. The conclusion reached in Chapter 2 for linear behavior,
|
|
that for a length to radius of gyration ratio less than two, no
|
|
slapdown (increase in tail velocity at impact) can occur, has been
|
|
numerically confirmed for nonlinear spring behavior as well.
|
|
It is difficult to draw general conclusions concerning location of the
|
|
center of gravity. For
|
|
off-center location of the center of gravity, a simple relationship
|
|
between tail velocity at impact and required energy absorption no
|
|
longer exists. The maximum tail velocity occurs at a different
|
|
center of gravity location than does the maximum tail spring energy
|
|
absorption. Therefore, severity of the secondary (tail) impact is
|
|
difficult to define. Thus, off-center location of the
|
|
center of gravity can have an important effect on slapdown but each
|
|
case must be investigated individually.
|
|
|