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342 lines
14 KiB
342 lines
14 KiB
\chapter{Scaling Behavior}
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\section{Introduction}
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In the field of radioactive materials transportation cask design,
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testing of scale models is very important. In order to successfully
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utilize the results of scale model tests, the appropriate parameters
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and techniques for extrapolating the results to full-scale behavior
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must be identified. Because the accelerations of practical interest
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are sufficiently high, the effects of a one G (gravity) field are
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ignored in the following discussion.
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\section{Simplified Analysis for Linear Systems}
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For linear behavior, a simplified engineering approach to scaling
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relations can be applied. For radioactive materials transportation
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cask applications, the linear accelerations at various locations along
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the cask length tend to be the most important parameters. These
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accelerations are used primarily to assess inertial loads on bolted
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closures, cask contents and other components. Therefore, the
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discussion below centers on the scaling relations for acceleration
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values. A more complete description of scaling relations can be found
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in Table 7.1.
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Linear acceleration of the center-of-gravity is described
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by the equation relating force ($F$), mass ($M$),
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and acceleration ($a$):
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\begin{equation}
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F = M \cdot a.
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\end{equation}
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Angular acceleration about the center-of-gravity is similarly
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described by the equation relating torque ($T$), moment of
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inertia ($I$), and angular acceleration ($\alpha$):
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\begin{equation}
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T = I \cdot \alpha.
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\end{equation}
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For linear acceleration, the forces are proportional to the
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square of the linear dimensions of the body. The mass is proportional
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to the cube of the linear dimensions. Therefore, the acceleration of
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a full-scale object should be half that of a half-scale object.
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For angular acceleration, the torques are proportional to the
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cube of the body scale
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(T = force $\cdot$ length, where force $\propto$ length$^{2}$).
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The moment of inertia is
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proportional to the fifth power of the body scale
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(I = mass $\cdot$ length$^{2}$, where mass $\propto$ length$^{3}$).
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Therefore, the angular acceleration of a full-scale object
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should be one fourth that of a half-scale object. However, the linear
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acceleration of the end of an object is equal to the angular
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acceleration times a length dimension. Thus, the linear acceleration
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of the tip of a rotating full-scale object should again be half that
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of a half-scale object. Note, for simplicity of the argument and
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because we are primarily concerned with the tangential component of
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acceleration for shallow angle slapdown events, the inward component
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of acceleration (centripetal acceleration) due to the square of the
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angular velocity times the length has been neglected in this argument.
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Inclusion of centripetal acceleration does not change the stated
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conclusion.
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\section{Numerical Confirmation Using \SLAP}
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Nonlinear behavior of the structure at the impact locations is
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much more common than linear behavior. Nonlinear behavior arises from
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both geometric and material properties sources. The nonlinear
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behavior is easily incorporated into numeric solution procedures such
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as those found in the slapdown analysis program, \SLAP , discussed in
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this report. \SLAP\ will therefore be used to address the effects
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of nonlinear behavior on the scaling relations for slapdown impact
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events. Two sets of analyses will be performed. First, an analysis
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of linear behavior will be compared to the results of the simple
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engineering analysis described above. Then, an analysis of nonlinear
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behavior will determine the effects of nonlinearity on the scaling
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relations derived for linear behavior.
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\subsection{Linear Springs}
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A simple solid cylinder was selected to demonstrate the
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scaling relations. No system of units was assumed, thus any
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consistent units may be applied. In full-scale, the cylinder
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was given
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a mass of 80, a length of 120 units, and a radius of
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30 units resulting in a
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moment of inertia of 114,000.
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An identical spring at each end of the
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cylinder had a travel of 10 units with a spring rate of
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600,000 and an identical
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unloading modulus (that is the spring was linear on both loading and
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unloading with no energy dissipation). This cylinder
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was impacted onto a
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rigid target at an initial angle of 10$^\circ$ with an initial velocity
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of 527.45. This velocity was selected,
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based on the author's bias, because it is consistent, in units of
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in/sec, with the NRC 30-foot free drop
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hypothetical accident condition. A solid cylinder consistent
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with the half-
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scale of the cylinder described above has a mass of 10,
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a length of 60 units,
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and a radius of 15 units
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for a moment of inertia of 3563. In half-scale,
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the spring travel was 5 units with a spring rate of
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300,000 for both loading
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and unloading. The initial impact conditions were identical with
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those of the
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full-scale cylinder. The model description parameters are listed in
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Table 7.1 for easy comparison.
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\begin{table}
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\begin{center}
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\caption{Comparison of Full- and Half-Scale Linear Spring Models}
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\makeqnum
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\begin{tabular}{||l|r|r|r||}
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\hline
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&\multicolumn{1}{|c}{Full-Scale}
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&\multicolumn{1}{|c}{Half-Scale}
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&\multicolumn{1}{|c||}{Ratio}\\
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& & &\multicolumn{1}{|c||}{(Full/Half)}\\
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Geometry and & & & \\
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Initial Conditions: & & & \\
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\quad Overall Length &$???120.0$ &$??60.0$ &$?2.0?$\\
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\quad Mass &$????80.0$ &$??10.0$ &$?8.0?$\\
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\quad Moment of Inertia &$114000.0$ &$3563.0$ &$32.0?$\\
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\quad Spring Constants: & & &\\
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\quad \quad Loading &$600000.$ &$300000.$ &$2.0?$\\
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\quad \quad Unloading &$600000.$ &$300000.$ &$2.0?$\\
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\quad Initial Velocity &$-527.5$ &$-527.5$ &$1.0?$\\
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\quad Initial Angle &$??10.0$ &$??10.0$ &$1.0?$\\
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\hline
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Results: & & & \\
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Maximum Accelerations & & & \\
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\quad Nose &$?85720.$ &$?171400.$ &$0.5?$\\
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&$-67780.$ &$-135600.$ &$0.5?$\\
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\quad Tail &$158400.$ &$316700.$ &$0.5?$\\
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&$-36610.$ &$-73210.$ &$0.5?$\\
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\quad C.-G. &$45300.$ &$90590.$ &$0.5?$\\
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&$????0.$ &$????0.$ &$0.5?$\\
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\quad Angular &$?1893.$ &$?7583.$ &$0.25$\\
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&$-1023.$ &$-4090.$ &$0.25$\\
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Maximum Velocities & & & \\
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\quad Nose &$?536.4$ &$?536.4$ &$1.0?$\\
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&$-527.5$ &$-527.5$ &$1.0?$\\
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\quad Tail &$???1.7$ &$???1.7$ &$1.0?$\\
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&$-983.2$ &$-983.2$ &$1.0?$\\
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\quad C.-G. &$??56.7$ &$??56.7$ &$1.0?$\\
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&$-527.5$ &$-527.5$ &$1.0?$\\
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\quad Angular &$??0.0$ &$??0.0$ &$0.5?$\\
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&$-12.7$ &$-25.3$ &$0.5?$\\
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Maximum Displacements & & & \\
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\quad Nose &$3.28$ &$1.64$ &$2.0?$\\
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\quad Tail &$6.04$ &$3.02$ &$2.0?$\\
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\hline
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\end{tabular}
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\end{center}
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\end{table}
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\SLAP\ analysis results show that the sequence of
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events occurs at twice the speed for the half-scale cylinder
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as for the
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full-scale cylinder. The displacements for the half-scale cylinder
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are half the full-scale cylinder's
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displacements.
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Linear velocities are the same for
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both half- and full-scale cylinders, and linear accelerations of the
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half-scale cylinder are double those of the full-scale cylinder.
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The angular
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velocities of the half-scale cylinder are double those of the full-scale
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cylinder, and the angular accelerations for
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the half-scale cylinder are four times those
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for the full-scale cylinder.
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Thus, if an object has a non-zero angular velocity
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at impact, the half-scale object must have
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double the initial angular velocity of the full-scale object,
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for proper scaling. These
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results are shown in Figures 7.1-7.3. The maximum values of
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displacement, velocity, and acceleration are shown in Table 7.1.
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\begin{figure}
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\vspace{3.5 in}
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\caption{Nose, Center-of-Gravity, and Tail Displacements versus Time
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for Full- and Half-Scale Cylinders with Linear Springs}
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\end{figure}
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\begin{figure}
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\vspace{3.5 in}
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\caption{Nose, Center-of-Gravity, and Tail Velocities versus Time
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for Full- and Half-Scale Cylinders with Linear Springs}
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\end{figure}
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\begin{figure}
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\vspace{3.5 in}
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\caption{Nose, Center-of-Gravity, and Tail Accelerations versus Time
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for Full- and Half-Scale Cylinders with Linear Springs}
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\end{figure}
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\subsection{Nonlinear Springs}
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To investigate the nonlinear behavior, the cylinders described
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above were again used. The only differences were the springs. For
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the full-scale cylinder,
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a spring with an initial spring rate of 750,000 for
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the initial travel of 2 units
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followed by a tangent spring rate of 0 for the
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next 8 units
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was selected. Thus the force curve rose linearly to a value of
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1,500,000
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and then remained at that value for the remainder of the
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compression. An unloading modulus of 600,000 was again selected.
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Since unloading occurs linearly from the maximum load reached, the
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spring selected has the potential to absorb a considerable amount of
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energy. The half-scale spring has an initial spring rate of 375,000
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over the first unit
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of displacement. This force remains constant for the
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next 4 units of displacement. The unloading modulus is 300,000. The
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initial conditions for the impact event were identical with those
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described above for the linear analysis. The impact initial
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conditions and model description are shown in Table 7.2.
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\begin{table}
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\begin{center}
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\caption{Comparison of Full- and Half-Scale Nonlinear Spring Models}
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\makeqnum
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\begin{tabular}{||l|r|r|r||}
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\hline
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&\multicolumn{1}{|c}{Full-Scale}
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&\multicolumn{1}{|c}{Half-Scale}
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&\multicolumn{1}{|c||}{Ratio}\\
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& & &\multicolumn{1}{|c||}{(Full/Half)}\\
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Geometry and & & & \\
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Initial Conditions: & & & \\
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\quad Overall Length &$???120.0$ &$??60.0$ &$?2.0?$\\
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\quad Mass &$????80.0$ &$??10.0$ &$?8.0?$\\
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\quad Moment of Inertia &$114000.0$ &$3563.0$ &$32.0?$\\
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\quad Spring Constants: & & &\\
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\quad \quad Loading section 1 &$750000.?$ &$375000.?$ &$2.0?$\\
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\quad \quad Loading section 2 &$ 0.?$ &$ 0.?$ &$2.0?$\\
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\quad \quad Unloading &$600000.?$ &$300000.?$ &$2.0?$\\
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\quad Initial Velocity &$-527.5$ &$-527.5$ &$1.0?$\\
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\quad Initial Angle &$??10.0$ &$??10.0$ &$1.0?$\\
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\hline
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Results: & & & \\
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Maximum Accelerations & & & \\
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\quad Nose &$65630.?$ &$131300.?$ &$0.5?$\\
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&$-28760.?$ &$-57530.?$ &$0.5?$\\
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\quad Tail &$66260.?$ &$132500.?$ &$0.5?$\\
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&$-28130.?$ &$-56260.?$ &$0.5?$\\
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\quad C.-G. &$18750.?$ &$37500.?$ &$0.5?$\\
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&$0.?$ &$0.?$ &$0.5?$\\
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\quad Angular &$788.5$ &$3154.$ &$0.25$\\
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&$-781.5$ &$-3126.$ &$0.25$\\
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Maximum Velocities & & & \\
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\quad Nose &$409.5???$ &$409.5???$ &$1.0?$\\
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&$-527.5???$ &$-527.5???$ &$1.0?$\\
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\quad Tail &$7.5???$ &$7.5???$ &$1.0?$\\
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&$-928.8???$ &$-928.8???$ &$1.0?$\\
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\quad C.-G. &$6.6???$ &$6.6???$ &$1.0?$\\
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&$-527.5???$ &$-527.5???$ &$1.0?$\\
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\quad Angular &$0.0158$ &$???0.0316$ &$0.5?$\\
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&$-11.15??$ &$-22.3???$ &$0.5?$\\
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Maximum Displacements & & & \\
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\quad Nose &$3.16$ &$1.58$ &$2.0?$\\
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\quad Tail &$7.53$ &$3.77$ &$2.0?$\\
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\hline
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\end{tabular}
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\end{center}
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\end{table}
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The results for the nonlinear springs showed exactly the same
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behavior as described for the linear springs. Nonlinear behavior is
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shown in Figures 7.4-7.6 and the maximum values in Table 7.2.
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\begin{figure}
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\vspace{3.5 in}
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\caption{Nose, Center-of-Gravity, and Tail Displacements versus Time
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for Full- and Half-Scale Cylinders with Nonlinear Springs}
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\end{figure}
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\begin{figure}
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\vspace{3.5 in}
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\caption{Nose, Center-of-Gravity, and Tail Velocities versus Time
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for Full- and Half-Scale Cylinders with Nonlinear Springs}
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\end{figure}
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\begin{figure}
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\vspace{3.5 in}
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\caption{Nose, Center-of-Gravity, and Tail Accelerations versus Time
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for Full- and Half-Scale Cylinders with Nonlinear Springs}
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\end{figure}
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\section{Conclusions}
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In this section, we have shown that the results of impact testing
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scale models can easily be related to the behavior of full-scale
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objects. Model displacements can be related to full-scale object
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displacements by multiplying by the model scale. Model linear
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accelerations can be related to full-scale object linear accelerations
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by dividing by the model scale. Model angular accelerations can be
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related to those of the full-scale object by dividing by the square of
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the model scale. Linear velocities are identical between model and
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full-scale objects regardless of scale. Angular velocities scale in
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the manner of linear accelerations. The scaling relations for
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velocities indicate that the initial conditions for a model impact
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test should be identical to those expected for the full-scale event
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except for initial angular velocity. Initial angular velocity for the
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model should be inversely proportional to the scale of the model used.
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These scaling relations are summarized in Table 7.2.
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\begin{table}
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\begin{center}
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\caption{Summary of Relationships for Scale Model Testing}
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\makeqnum
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\begin{tabular}{||l|c||}
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\hline
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\multicolumn{1}{|c}{Parameter}
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&\multicolumn{1}{|c||}{Scaling Relationships}\\
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Geometry and & \\
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Initial Conditions: &\\
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\quad Overall Length & $l_{sm} = l_{fs} \times (Scale)^{1}$\\
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\quad Mass & $M_{sm} = M_{fs} \times (Scale)^{3}$\\
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\quad Moment of Inertia & $I_{sm} = I_{fs} \times (Scale)^{5}$\\
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\quad Spring Constants & $K_{sm} = K_{fs} \times (Scale)^{1}$\\
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\quad Initial Velocity & $V_{sm} = V_{fs} \times (Scale)^{0}$\\
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\quad Initial Angle & $\theta _{sm} = \theta _{fs} \times
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(Scale)^{0}$\\
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\hline
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Results: & \\
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\quad Linear Accelerations & $a_{sm} = a_{fs} \times (Scale)^{-1}$\\
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\quad Angular Accelerations & $\alpha _{sm} = \alpha _{fs} \times
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(Scale)^{-2}$\\
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\quad Linear Velocities & $V_{sm} = V_{fs} \times (Scale)^{0}$\\
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\quad Angular Velocities & $\omega _{sm} = \omega _{fs} \times
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(Scale)^{-1}$\\
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\quad Linear Displacements & $\Delta _{sm} = \Delta _{fs} \times
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(Scale)^{1}$\\
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\quad Angular Displacements & $\theta _{sm} = \theta _{f} \times
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(Scale)^{0}$\\
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\hline
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\end{tabular}
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\end{center}
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\end{table}
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