Page 287 - Introduction to Naval Architecture

272 MANOEUVRING

Figure 10.11 Submarine in vertical plane

requires two sets of control surface, the hydroplanes, one forward and
one aft.
Consider a submarine turning in the vertical plane as in Figure 10.11.
Taking the combined effects of the two sets of hydroplanes as
represented by a term in 6 H the equations of motion for the vertical
plane are given by:

These are similar to the equations for the directional stability of a
surface ship. In this case Z and M are the vertical force and pitching
moment. Subscripts w, q and #H denote differentiation with respect to
the variable concerned. In these equations:
mqV'm a centrifugal force term

mgBQB is a statical stability term, BG being the distance between B
andG.

This stability term is constant for all speeds whereas the moments M
vary with the square of the velocity. The stability term can normally be
ignored at speeds greater than 10 knots. Ignoring this term for the time
being and eliminating w between the two equations leads to the
condition that for the submarine to have positive dynamic stability:

This is termed the high speed stability criterion. If this criterion is met and
the submarine is statically stable, it will be stable at all speeds. If the
criterion is not met then a statically stable submarine will develop a
diverging oscillation at forward speeds above some critical value.