(4) I need not refer to such matters as the use of gyrostats for the correction of compasses on board ship, referred to in page [111].
Problems (2) and (3) are those to which I wish to refer. For a ship of 6,000 tons Mr. Schlick would use a large wheel of 10 to 20 tons, revolving about an axis E F (fig. 1) whose mean position is vertical. Its bearings are in a frame E C F D which can move about a thwart-ship axis C D with a precessional motion. Its centre of gravity is below this axis, so that like the ship itself the frame is in stable equilibrium. Let the ship have rolled through an angle R from its upright position, and suppose the axis E F to have precessed through the angle P from a vertical position. Let the angular velocity of rolling be called R˙, and the angular velocity of precession P˙; let the moment of momentum of the wheel be m. For any vibrating body like a ship it is easy to write out the equation of motion; into this equation we have merely to introduce the moment m P˙ diminishing R; into the equation for P we merely introduce the moment m R˙ increasing P. As usual we introduce frictional terms; in the first place F R˙ (F being a constant co-efficient) stilling the roll of the ship; in the second case f P˙ a fluid friction introduced by a pair of dash pots applied at the pins A and B to still the precessional vibrations of the frame. It will be found that the angular motion P is very much greater than the roll R. Indeed, so great is P that there are stops to prevent its exceeding a certain amount. Of course so long as a stop acts, preventing precession, the roll of the ship proceeds as if the gyrostat wheel were not rotating. Mr. Schlick drives his wheels by steam; he will probably in future do as Mr. Brennan does, drive them by electromotors, and keep them in air-tight cases in good vacuums, because the loss of energy by friction against an atmosphere is proportional to the density of the atmosphere. The solution of the equations to find the nature of the R and P motions is sometimes tedious, but requires no great amount of mathematical knowledge. In a case considered by me of
a 6,000 ton ship, the period of a roll was increased from 14 to 20 seconds by the use of the gyrostat, and the roll rapidly diminished in amount. There was accompanying this slow periodic motion, one of a two seconds' period, but if it did appear it was damped out with great rapidity. Of course it is assumed that, by the use of bilge keels and rolling chambers, and as low a metacentre as is allowable, we have already lengthened the time of vibration, and damped the roll R as much as possible, before applying the gyrostat. I take it that everybody knows the importance of lengthening the period of the natural roll of a ship, although he may not know the reason. The reason why modern ships of great tonnage are so steady is because their natural periodic times of rolling vibration are so much greater than the probable periods of any waves of the sea, for if a series of waves acts upon a ship tending to make it roll, if the periodic time of each wave is not very different from the natural periodic time of vibration of the ship, the rolling motion may become dangerously great.
If we try to apply Mr. Schlick's method to Mr. Brennan's car it is easy to show that there is instability of motion, whether there is or is not friction. If there is no friction, and we make the gyrostat frame unstable by keeping its centre of gravity above the axis C D, there will be vibrations, but the smallest amount of friction will cause these vibrations to get greater and greater. Even without friction there will be instability if m, the moment of momentum of the wheel, is less than a certain amount. We see, then, that no form of the Schlick method, or modification of it, can be applied to solve the Brennan problem.
