3.3.1 The geometry of a free-fall lifeboat as
it rotates at the end of the launch ramp is shown in figure 3.6. After
the CG has moved past the end of the launch ramp, the lifeboat begins
to rotate. The primary factors affecting rotational behavior are the
velocity of the lifeboat, the mass and mass distribution of the lifeboat,
and the distance from the CG to the end of launch rail. The equations
of motion describing the behavior of the lifeboat during the rotation
phase are (Nelson, et. al., 1991):
where Φ is the friction angle which is equal
to tan-1μ. In addition to these three equations, an
equation for the compatibility of displacements is required. This
equation is:
Figure 3.6 Geometry of a Free-Fall Lifeboat As It Rotates Off the End of the
Launch Ramp
3.3.2 As implied in Equation 3.7, rotation is
caused by a couple formed by the reaction force between the ramp and
the lifeboat and the weight of the lifeboat. This couple imparts angular
momentum to the lifeboat. The momentum increases during the rotation
phase and then remains constant during the free-fall phase.
3.3.3 Presented in figure 3.7 is the angular momentum
of a lifeboat versus ramp length for three launch angles (Nelson,
1992). The lifeboat for which these data were computed is about 10
meters long and weighs about 125,000 N. The end of the launch rail
is at the stern and the CG is 3.85 meters forward of the stern.
Figure 3.7 Variation of Angular Momentum at End of Rotation Phase
3.3.4 The angular momentum imparted to the lifeboat
during rotation decreases as the distance L increases. This occurs
because the velocity of the lifeboat at the beginning of the rotation
phase increases as the distance to the end of the ramp increases.
As such, the time during which it rotates, the time during which the
couple acts, decreases as L increases. Because the time of rotation
is reduced, the time during which the forces act, and therefore the
angular momentum imparted to the lifeboat, is reduced. Likewise, the
duration of the rotation, and therefore the angular momentum, increases
as the distance to the end of the launch rail increases. The angular
momentum increases until the time at which the lifeboat is no longer
in contact with the launch ramp. After leaving the launch ramp, the
lifeboat continues to rotate at constant angular velocity until it
impacts the water.
3.3.5 The differences in the time during which
the lifeboat rotates are evident from the force data presented in
figure 3.8 (Nelson, et. al., 1992). The relative values of Land D
were changed by moving the CG forward and aft. The data, which were
computed for a typical free-fall lifeboat that was launched from a
ramp at an angle of 30 degrees, were normalized by dividing by the
boat weight. Until the rotation phase begins, the force between the
boat and the ramp is constant. This force, which is about 87 per cent
of the lifeboat weight in this case, decreases during the rotation
phase and becomes zero at the end of the rotation.
Figure 3.8 Typical Forces Acting on a Free-Fall Lifeboat During Ramp and Rotation
Phases for Three Locations of the CG
3.3.6 For the three locations of the CG presented,
the rotation phase ends at approximately the same time. The time at
which rotation begins, however, changes with the location of the CG.
When the CG is in the forward location, the boat begins to rotate
about 1.75 seconds after it is released whereas rotation begins about
1.95 seconds after release when the CG is moved aft. Although this
difference in time is small; the difference in the resulting rate
of rotation during free-fall is significant. Presented in figure 3.9
is the angular velocity from a time just before the boat begins to
rotate until shortly after the free-fall phase begins. As can be seen
from this figure, the rate of rotation increases during the rotation
phase. At any time during the rotation phase, the rate of rotation
is higher when the CG is located forward than it is when the CG is
located aft. The constant angular velocity at the right side of the
curves is the rate at which lifeboat is rotating as it is falling
through the air. For the lifeboat represented by these data, there
is about a 20 per cent change in the angular velocity during free-fall
as the CG is moved from an aft location to a forward location.
Figure 3.9 Angular Velocity During Rotation for Three Locations of the CG