4.4.1 To accurately predict the launch behavior
of a free-fall lifeboat using a scale model, the second moment of
mass of the model and the full-scale boat must be in the proper proportion.
The second moment of mass of the model (or the full-scale lifeboat)
can be measured by treating the lifeboat as physical pendulum. A physical
pendulum is a rigid body which is mounted so that it can swing freely
in a vertical plane about some axis. The physical pendulum is a generalization
of the simple pendulum which is a mass supported at the end of a weightless
cord (Resnick and Halliday, 1966). Other means are available to measure
the second moment of mass but these require a more difficult experimental
procedure and the resulting data are therefore more difficult to interpret.
Figure 4.1 Geometry for Measuring the Second Moment of Mass
4.4.2 To measure the second moment of mass by
treating the free-fall lifeboat as a physical pendulum, consider the
geometry presented in figure 4.1. The lifeboat is suspended from a
fixed point on the upper canopy. This point could be the recovery
hook in a full-scale boat or a U-bolt in a scale model. This point
becomes the point of rotation. When the lifeboat is suspended in this
manner, the CG is located at a distance d directly beneath the point
of rotation.
4.4.3 If the boat were to be pushed some amount
and then allowed to swing freely, it would oscillate about the point
of rotation as shown in figure 4.1. As the lifeboat oscillates, it
is in a state of harmonic motion that can be described by the differential
equation:
The first term on the left side of the equation is the angular
acceleration and θ is the angle formed between the lifeboat
in the free hanging position and the position at some other time when
it is oscillating. Because the equation of motion involves the term
sinθ, the lifeboat is not undergoing simple harmonic motion.
However, for small angles of displacement, sinθ is nearly equal
to θ (in radians). If it is assumed that the lifeboat undergoes
small rotations, the differential equation of motion can be more conveniently
expressed in the form:
This equation is valid, for practical purposes, as
long as the arc through which the lifeboat swings is less than about
20 degrees. From Equation 4.16, the period of the harmonic oscillation
is found to be:
The period-of harmonic motion is the time required
for one complete oscillation to occur. Equation 4.17 can be solved
for the second moment of mass which is found to be:
Using Equation 4.18, the second moment of mass can
be computed if the mass of the lifeboat, the period of oscillation,
and distance from the point of rotation to the CG are known.
4.4.4 To physically measure the second moment
of mass, then, the lifeboat should be suspended as shown in figure
4.1. It is then pushed some amount and released so that it oscillates
freely in the vertical plane. As the lifeboat oscillates, the point
of rotation should not move. Because small displacements are assumed
in the analysis discussed previously, the arc through which the boat
swings should not be greater than 20 degrees. The total time for the
lifeboat to complete at least five oscillations is measured. Because
of error inherently introduced into the measurement by starting and
stopping the stopwatch, the accuracy of the measurement increases
as the number of cycles during which the time is measured increases.
The error associated with starting and stopping the stopwatch is distributed
over more cycles so the error per cycle is smaller. After the time
required to complete a number of oscillations has been determined,
the second moment of mass of the lifeboat can be computed using the
following modified form of Equation 4.18:
In this equation, T' is the total time required for
n complete cycles of oscillation. The quantity I is the second moment
of mass of the model (or the full-scale lifeboat if it was used during
the measurement) about the CG.