\chapter{Analytical Prediction of Slapdown Magnitude}
\section{Introduction}

In this chapter, simple equations are developed that can economically
be used to estimate the ratio of the initial vertical velocity to the
secondary impact velocity during a slapdown impact orientation.  The
equations can also be used to determine the critical impact angle
$\theta_0$ producing the largest secondary impact velocity. The only
variables required are geometric and mass quantities; the
force-displacement behavior of the impact is not required. 


\section{Derivation of Slapdown Equation}

The body is approximated by the three degree-of-freedom system shown
in Figure~\ref{f:gi} on page~\pageref{f:gi}.  The force-displacement
behavior is approximated by springs located at the initial and
secondary impact points. The slapdown equation is developed assuming
that (1)~the springs are linear elastic, (2)~there are no horizontal
forces due to friction between the rigid surface and the body, and
(3)~at most one spring is in contact with the rigid surface at any
time.  Assumption~(2) reduces the body to a two degree-of-freedom
system, and assumption~(3) further reduces it to a sequence of one
degree-of-freedom systems. It will be shown that the linear elastic
behavior is a conservative assumption that maximizes the secondary
impact velocity. 

In the text that follows, the nose and tail of the body refer to
the initial and secondary impact points, respectively.  The sign
convention used is that displacements, velocities and accelerations are
positive downward; and angular positions, velocities and accelerations
are positive clockwise. 

\subsection{Nomenclature}

The nomenclature used in the derivation of the Slapdown equation is
defined below.
\begin{description}
\item[$y$,$\dot y$,$\ddot y$] Vertical displacement, velocity, and
acceleration.  Positive toward rigid surface,
\item[$\theta$,$\omega$,$\alpha$] Angular inclination, velocity, and
acceleration of the axis of the body with respect to the rigid
surface, positive counterclockwise if the nose is to the left of the
tail, 
\item[$m$,$I_{CG}$]
Body mass and moment of inertia about the center of
gravity,
\item[$l$]       Length from the center of gravity to the impact point
measured along the axis of the body,
\item[$k$]       Spring constant (force/length),
\item[$v_0$,$\theta_0$]     Initial vertical velocity and angular
position of the body, 
\item[$\omega_n$] Natural frequency of the differential equation, and 
\item[$t$]        Time.
\end{description}
For the geometric variables and the linear displacement quantities, the
subscripts $(\cdot)_n$ and $(\cdot)_t$ designate the nose and tail 
of the body, respectively; a superposed dot indicates the time
derivative of the dotted quantity.

\subsection{Rigid Body Motion Equations}

Using D'Alembert's principle, the vertical acceleration $\ddot y_n$ at the
nose and the angular acceleration $\alpha$ about the nose during
the time interval that only the nose is in contact with the rigid
surface can be determined by the summation of moments and forces. 
\begin{eqnarray}
\sum M_n\rightarrow 0  & = & 2 \ddot y_n l_n +\alpha(l_n-r)^2
     +\alpha(l_n+r)^2\label{e1} \\ 
\sum F_n\rightarrow -ky_n & = & \frac{m}{2}\left\{\left[(l_n-r)\alpha 
     +\ddot y_n \right] +\left[(l_n+r)\alpha+\ddot y_n\right]\right\}\label{e2}
\end{eqnarray}
\begin{tabbing}
where: \=ZZ\= \kill
where: \>$r  $\> is the radius of gyration ${}=\sqrt{\frac{I_{CG}}{m}}$,\\
       \>$l_n$\> is the distance from the center of mass to the nose,\\ 
       \>$y_n$\> is the vertical displacement of the nose,\\
       \>$k  $\> is the elastic spring constant, and\\
       \>     \> it is assumed that $\cos\theta\approx 1$.
\end{tabbing}
Equation~\eref{e1} can be used to give $\alpha$ in terms of $\ddot y_n$:
\begin{equation}
\alpha = \frac{-\ddot y_n l_n}{l_n^2 + r^2}\label{eq:alpha}
\end{equation}
Using this result, Equation~\eref{e2} can be written as the single
degree-of-freedom equation: 
\begin{equation}
\ddot y_n\left[\frac{m}{\beta_n^2+1}\right] + ky_n = 0\label{eq:diff}
\end{equation}
where $\beta_n$ is defined to be $l_n/r$.  Equation~\eref{eq:diff} has the
solution
\begin{eqnarray}
     y_n & = & A\cos\omega_n t + B\sin\omega_n t\label{eq:acbs}\\
\omega_n & = & \sqrt{\frac{k}{m}\left(\beta_n^2+1\right)}
\end{eqnarray}
At time $t=0$, $\dot y_n = v_0$ and $y_n = 0$.  Substituting these 
conditions into Equation~\eref{eq:acbs} results in the following equations
for the vertical displacement, velocity, and acceleration of the nose: 
\begin{eqnarray}
      y_n & = &  \frac{v_0}{\omega_n}\sin\omega_n t\\
 \dot y_n & = &  v_0\cos\omega_n t\label{eq:vl}\\
\ddot y_n & = & -v_0\omega_n\sin\omega_n t\label{eq:ayl}
\end{eqnarray}
The angular quantities are determined by substituting 
Equation~\eref{eq:ayl} into Equation~\eref{eq:alpha} and integrating.
\begin{eqnarray}
\alpha & = & \frac{\beta_n v_0\omega_n}{r(\beta_n^2+1)}\sin\omega_n t\\
\omega & = & \frac{-\beta_n v_0}{r(\beta_n^2+1)}[\cos\omega_n t - 1]
               \label{eq:omega}\\
\theta & = & \frac{-\beta_n v_0}{r(\beta_n^2+1)}
                \left[\frac{\sin\omega_n t}{\omega_n} - t\right] + \theta_0
\label{etheta}
\end{eqnarray}

These results are only valid during the time interval ($0\leq t\leq
\pi / \omega_n$) when the spring is in contact with the rigid surface.
For times $t>\pi/\omega_n$ and before the tail contacts the rigid 
surface, 
\begin{eqnarray}
\alpha & = & 0, \\
\omega & = & \frac{2\beta_n v_0}{r(\beta_n^2+1)},\\
\theta & = & \frac{\beta_n v_0}{r(\beta_n^2+1)}\left[\frac{-\pi}{\omega_n}
      + 2t\right] + \theta_0\label{e12}
\end{eqnarray}

\subsection{Velocity of Tail at Impact}

The vertical velocity at the tail of the body $v_t$ is related to
the vertical velocity and angular velocity at the nose by 
\begin{equation}
v_t = v_n + \omega(l_n + l_t)\cos\theta
\end{equation}
where $v_n = \dot y_n$. 
Substituting Equations~\eref{eq:vl} and~\eref{eq:omega} into the above
equation and assuming that $\cos\theta\approx 1$, the ratio of the velocity
at the tail to the initial vertical velocity is:
\begin{equation}
\frac{v_t}{v_0} = \cos\omega_nt - \frac{\beta_n(\beta_n+\beta_t)}
{(1+\beta_n^2)}\left[\cos\omega_n t-1\right]\label{eq:vrv0t}
\end{equation}
where $\beta_t$ is defined to be $l_t/r$. This ratio is maximized at
time $t = \pi/\omega_n$ and gives a maximum velocity ratio of 
\begin{equation}
\frac{v_t}{v_0} = \frac{2\beta_n(\beta_n+\beta_t)}{1+\beta_n^2} - 1
\label{eq:vrv0}
\end{equation}

This result indicates that if linear elastic response is assumed, the
magnitude of the slapdown velocity of the body can be estimated using
only geometric and mass quantities; the force-displacement behavior of
the body is not needed.  Furthermore, Equation~\eref{eq:vrv0} can be
used to show that 
\begin{equation}
{\rm If} \quad\beta_n\beta_t \leq1, \quad{\rm then} 
\quad\frac{v_t}{v_0} \leq1\label{eq:blbr}
\end{equation}

\subsection{Effect of Inelastic Force-Displacement Behavior}

The results in the previous section were derived assuming a linear
elastic force-displacement behavior.  In general, the force-displacement
behavior is inelastic and energy is dissipated by plastic deformation.
In this section it will be shown that when energy is dissipated by
plastic deformation, the slapdown velocity ratio is closer to 
unity than if there is no energy dissipation. 

In this derivation, the springs are assumed to have an infinite
unloading modulus which means that all of the internal energy in the
spring is dissipated.  Equations \eref{e1} through~\eref{etheta} are
still valid, except that the time interval is reduced to ($0\leq t\leq
\pi / 2\omega_n$).  If the tail impacts the rigid surface at time
$t=\pi/2\omega_n$, then Equation~\eref{eq:vrv0t} gives the following
expression for the maximum inelastic velocity ratio: 
\begin{equation}
\frac{v_t}{v_0} = \frac{\beta_n(\beta_n+\beta_t)}{1+\beta_n^2}
\label{eq:vrv0i}
\end{equation}
which can be written as:
\begin{equation}
\left(\frac{v_t}{v_0}\right)_\infty = \left(\frac{v_t}{v_0}\right)_E + 
  \left(\frac{1-\beta_n\beta_t}{1+\beta_n^2}\right)\label{eisum}
\end{equation}
where the subscripts $(\ )_\infty$ and $(\ )_E$ denote infinite
unloading modulus and elastic unloading modulus, respectively.  The
last term on the right-hand-side is greater than zero if
$\beta_n\beta_t\leq1$ and is less than zero if $\beta_n\beta_t\geq1$.
Therefore, an infinite unloading modulus results in a tip velocity
ratio that is closer to unity than the velocity ratio calculated
assuming linear elastic impact behavior.  These two cases bound the
possible unloading behavior of all bodies, therefore 
\begin{displaymath}
\begin{array}{cccc}
{\rm If\;} &
   \left\{\begin{array}{rcl}
       \beta_n\beta_t & \leq & 1\\
       {} & {} & {}\\
       \beta_n\beta_t & \geq & 1
   \end{array}\right\}, &
{\quad\rm then\quad} &
   \left\{\begin{array}{rclcl}
       \left[\frac{v_t}{v_0}\right]_E & \leq & 
                       \left[\frac{v_t}{v_0}\right]_I & \leq & 1\\
       {} & {} & {} & {} & {}\\
       \left[\frac{v_t}{v_0}\right]_E & \geq & 
                       \left[\frac{v_t}{v_0}\right]_I & \geq & 1
   \end{array}\right\}
\end{array}
\end{displaymath}
where the subscripts $(\ )_E$ and $(\ )_I$ refer to the elastic and
inelastic responses, respectively. The above results have been verified
using the \SLAP\ program. Note that although the velocity
ratio for the inelastic case increases when $\beta_n\beta_t \leq  1$, it
does not increase to greater than unity.  If the body is symmetric,
then $\beta_n = \beta_t$ and this result states that there will be no
slapdown if $L/r < 2$.

\section{Conclusions}

An equation has been derived which gives a conservative estimate of
the velocity ratio for slapdown of a body impacting a rigid surface.
Only easily obtainable geometric and mass quantities are required.
This equation can be used during the design process to help determine
a body geometry which will minimize the slapdown potential. 


Equation~\eref{e12} can be used to determine the minimum initial drop
angle $\theta_0$ such that the secondary impact occurs at the same
time or after the initial impact point has rebounded from the rigid
surface. This angle is important for three-dimensional finite element
analyses of the slapdown event.  The drop angle is given by
\begin{equation}
\theta_0 = \frac{\beta_n v_0\pi}{r(\beta_n^2+1)\omega_n} 
\end{equation}
This angle is measured counter-clockwise from the horizontal and
assumes that the body has the same radius at the initial and secondary
impact points.  If the body is not symmetric, the angle
$\tan^{-1}\left(\frac{R_n-R_t}{l_n+l_t}\right)$ must be subtracted
from the initial angle $\theta_0$ to account for the asymmetric
geometry. $R_n$ and $R_t$ are the radii at the nose and tail of the
body, respectively, and the radius is defined as the perpendicular
distance from the longitudinal axis of the body to the impact point. 

