The two points which this system is designed to illustrate, and which it is desirable to emphasise, are thus as follows. Firstly, as the whole system rotates, the movement of the pendulum masses M and M1 from the lower to the higher levels, or from the regions of low to those of higher velocity, is productive of a transformation of the rotatory energy of the system into energy of position—a transformation of the same nature as in the case of the simple pendulum system. Neglecting the minor transformations (§§ [24,] [29]), this energy process is a reversible one, and consequently, the return of the masses from the higher to the lower positions will be accompanied by the complete return of the transformed energy in its original form of energy of rotation. Secondly, the maximum amount of energy which can work in this reversible process is always less than the total energy of the system. The latter, therefore, conforms to the general condition of stability (§ [25]).
But this arrangement of rotating pendulums may be extended so as to include other features. To eliminate or in a manner replace the influence of gravitation, and to preserve the energy of position of the system—relative to the earth's surface—at a constant value, the pendulum arms may be assumed to be duplicated or extended to the points K and R ([Fig. 10]) respectively, where pendulum masses equal to M and M1 are attached.
The arms MK and M1R are thus continuous. Each arm is assumed to be pivoted at its middle point about a horizontal axis through N, and as the lower masses M and M1 rise in the course of the rotatory movement about AB the upper masses K and R will fall by corresponding amounts. The total energy of position of the system—referred to the earth's surface—thus remains constant whatever may be the position of the masses in the system. The restraining influence on the movement of the masses, formerly exercised by gravitation, is now furnished by means of a central spring F. A collar CD, connected as shown to the pendulum arms, slides on the spindle AB and compresses this spring as the masses move towards the horizontal level HL. As the masses return towards A and B the spring is released.
Fig. 10
If energy be applied to the system, so that it is caused to rotate about the central axis AB, the pendulum masses will tend to move outwards from that axis. Their movement may be said to be carried out over the surface of an imaginary sphere with centre on AB at N. The motion of the masses, as the velocity of rotation increases, is from the region of lower peripheral velocity, in the vicinity of the axis AB, to the regions of higher velocity, in the neighbourhood of H and L. This outward movement from the central axis towards H and L is representative of a transformation of energy of an exactly similar nature to that described above in the simple case. Part of the original energy of rotation of the system is now stored in the pendulum masses in virtue of their new position of displacement. But in this case, the movement is made, not against gravity, but against the central spring F. The energy, then, which in the former case might be said to be stored against gravitation (acting as an invisible spring) is in this case stored in the form of energy of strain or cohesion (§ [15]) in the central spring, which thus as it were takes the place of gravitation in the system. As in the previous case also, the operation is a reversible energy process. If the pendulum masses move in the opposite direction from the regions of higher velocity to those of lower velocity, the energy stored in the spring will be returned to the system in its original form of energy of motion. A vibratory motion of the pendulums to and from the central axis would thus be productive of an alternate storage and return of energy. It is obvious also, that due to the action of centrifugal force, the pendulum masses would tend to move radially outwards on the arms as they move towards the regions of highest velocity. Let this radial movement be carried out against the action of four radial springs S1, S2, S3, S4, as shown ([Fig. 11]). In virtue of the radial movement of the masses, these springs will be compressed and energy stored in them in the form of energy of strain or cohesion (§ [15]). The radial movement implies also that the masses will be elevated from the surface of the imaginary sphere over which they are assumed to move. The elevation from this surface will be greatest in the regions of high velocity in the neighbourhood of H and L, and least at A and B. As the masses move, therefore, from H and L towards the axis AB, they will also move inwards on the pendulum arms, relieving the springs, so that the energy stored in them is free to be returned to the system in its original form of energy of rotation. Every movement of the masses from the central axis outwards against the springs is thus made at the expense of the original energy of motion, and every movement inwards provokes a corresponding return of that energy to the system. Every movement also against the springs forms part of a reversible operation. The sum total of the energy which works in these reversible operations is always less than the complete energy of the rotatory system, and the latter is always stable (§ [25]), with respect to its energy properties. Let it now be assumed that the complete system as described is possessed of a precise and limited amount of energy of rotation, and that with the pendulum masses in an intermediate position as shown ([Fig. 11]) it is rotating with uniform angular velocity. The condition of the rotatory system might now be described as that of equilibrium. A definite amount of its original rotatory energy is now stored in the central spring and also in the radial springs. If now, without alteration in the intrinsic rotatory energy of the system, the pendulum masses were to execute a vibratory or pendulum motion about the position of equilibrium so that they move alternately to and from the central axis, then as they move inwards towards that axis the energy stored in the springs would be returned to the system in the original form of energy of rotation. This inward motion would, accordingly, produce acceleration. But, in the outward movement from the position of equilibrium, retardation would ensue on account of energy of motion being withdrawn from the system and stored in the springs.
Fig. 11