supported on two bearings to carry a disk of weight W at its centre,
and let the centre of gravity of the disk be at a distance e from the
axis of rotation, this small distance being due to imperfections of
material or faulty construction. Neglecting the mass of the shaft
itself, when the shaft rotates with an angular velocity a, the centrifugal
force Wa2e/g will act upon the shaft and cause its axis to deflect
from the axis of rotation a distance, y say. The elastic resistance
evoked by this deflexion is proportional to the deflexion, so that if
c is a constant depending upon the form, material and method of
support of the shaft, the following equality must hold if the shaft
is to rotate stably at the stated speed—
| W | (y + e) a2 = cy, |
| g |
from which y = Wa2e / (gc − Wa2).
This expression shows that as a increases y increases until when Wa2 = gc, y becomes infinitely large. The corresponding value of a, namely , is called the critical velocity of the shaft, and is the speed at which the shaft ceases to rotate stably and at which centrifugal whirling begins. The general problem is to find the value of a corresponding to all kinds of loadings on shafts supported in any manner. The question was investigated by Rankine in an article in the Engineer (April 9, 1869). Professor A. G. Greenhill treated the problem of the centrifugal whirling of an unloaded shaft with different supporting conditions in a paper “On the Strength of Shafting exposed both to torsion and to end thrust,” Proc. Inst. Mech. Eng. (1883). Professor S. Dunkerley (“On the Whirling and Vibration of Shafts,” Phil. Trans., 1894) investigated the question for the cases of loaded and unloaded shafts, and, owing to the complication arising from the application of the general theory to the cases of loaded shafts, devised empirical formulae for the critical speeds of shafts loaded with heavy pulleys, based generally upon the following assumption, which is stated for the case of a shaft carrying one pulley: If N1, N2 be the separate speeds of whirl of the shaft and pulley on the assumption that the effect of one is neglected when that of the other is under consideration, then the resulting speed of whirl due to both causes combined may be taken to be of the form N1N2 √(N21 + N12) where N means revolutions per minute. This form is extended to include the cases of several pulleys on the same shaft. The interesting and important part of the investigation is that a number of experiments were made on small shafts arranged in different ways and loaded in different ways, and the speed at which whirling actually occurred was compared with the speed calculated from formulae of the general type indicated above. The agreement between the observed and calculated values of the critical speeds was in most cases quite remarkable. In a paper by Dr C. Chree, “The Whirling and Transverse Vibrations of Rotating Shafts,” Proc. Phys. Soc. Lon., vol. 19 (1904); also Phil. Mag., vol. 7 (1904), the question is investigated from a new mathematical point of view, and expressions for the whirling of loaded shafts are obtained without the necessity of any assumption of the kind stated above. An elementary presentation of the problem from a practical point of view will be found in Steam Turbines, by Dr A. Stodola (London, 1905).
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| Fig. 131. |
§ 114. Revolving Pendulum. Governors.—In fig. 131 AO represents an upright axis or spindle; B a weight called a bob, suspended by rod OB from a horizontal axis at O, carried by the vertical axis. When the spindle is at rest the bob hangs close to it; when the spindle rotates, the bob, being made to revolve round it, diverges until the resultant of the centrifugal force and the weight of the bob is a force acting at O in the direction OB, and then it revolves steadily in a circle. This combination is called a revolving, centrifugal, or conical pendulum. Revolving pendulums are usually constructed with pairs of rods and bobs, as OB, Ob, hung at opposite sides of the spindle, that the centrifugal forces exerted at the point O may balance each other.
In finding the position in which the bob will revolve with a given angular velocity, a, for most practical cases connected with machinery the mass of the rod may be considered as insensible compared with that of the bob. Let the bob be a sphere, and from the centre of that sphere draw BH = y perpendicular to OA. Let OH = z; let W be the weight of the bob, F its centrifugal force. Then the condition of its steady revolution is W : F :: z : y; that is to say, y/z = F/W = yα2/g; consequently
Or, if n = α 2π = α/6.2832 be the number of turns or fractions of a turn in a second,
| z = | g | = | 0.8165 ft. | = | 9.79771 in. | |
| 4π2n2 | n2 | n2 |
z is called the altitude of the pendulum.
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| Fig. 132. |
If the rod of a revolving pendulum be jointed, as in fig. 132, not to a point in the vertical axis, but to the end of a projecting arm C, the position in which the bob will revolve will be the same as if the rod were jointed to the point O, where its prolongation cuts the vertical axis.
A revolving pendulum is an essential part of most of the contrivances called governors, for regulating the speed of prime movers, for further particulars of which see Steam Engine.
Division 3. Working of Machines of Varying Velocity.
§ 115. General Principles.—In order that the velocity of every piece of a machine may be uniform, it is necessary that the forces acting on each piece should be always exactly balanced. Also, in order that the forces acting on each piece of a machine may be always exactly balanced, it is necessary that the velocity of that piece should be uniform.
An excess of the effort exerted on any piece, above that which is necessary to balance the resistance, is accompanied with acceleration; a deficiency of the effort, with retardation.
When a machine is being started from a state of rest, and brought by degrees up to its proper speed, the effort must be in excess; when it is being retarded for the purpose of stopping it, the resistance must be in excess.
An excess of effort above resistance involves an excess of energy exerted above work performed; that excess of energy is employed in producing acceleration.
An excess of resistance above effort involves an excess of work performed above energy expended; that excess of work is performed by means of the retardation of the machinery.
When a machine undergoes alternate acceleration and retardation, so that at certain instants of time, occurring at the end of intervals called periods or cycles, it returns to its original speed, then in each of those periods or cycles the alternate excesses of energy and of work neutralize each other; and at the end of each cycle the principle of the equality of energy and work stated in § 87, with all its consequences, is verified exactly as in the case of machines of uniform speed.
At intermediate instants, however, other principles have also to be taken into account, which are deduced from the second law of motion, as applied to direct deviation, or acceleration and retardation.
§ 116. Energy of Acceleration and Work of Retardation for a Shifting Body.—Let w be the weight of a body which has a motion of translation in any path, and in the course of the interval of time Δt let its velocity be increased at a uniform rate of acceleration from v1 to v2. The rate of acceleration will be
and to produce this acceleration a uniform effort will be required, expressed by
(The product wv/g of the mass of a body by its velocity is called its momentum; so that the effort required is found by dividing the increase of momentum by the time in which it is produced.)
To find the energy which has to be exerted to produce the acceleration from v1 to v2, it is to be observed that the distance through which the effort P acts during the acceleration is
consequently, the energy of acceleration is
being proportional to the increase in the square of the velocity, and independent of the time.
In order to produce a retardation from the greater velocity v2 to the less velocity v1, it is necessary to apply to the body a resistance connected with the retardation and the time by an equation identical in every respect with equation (71), except by the substitution of a in every respect with equation (71), except by the substitution of a resistance for an effort; and in overcoming that resistance the body performs work to an amount determined by equation (72), putting Rds for Pas.
§ 117. Energy Stored and Restored by Deviations of Velocity.—Thus a body alternately accelerated and retarded, so as to be brought back to its original speed, performs work during its retardation exactly equal in amount to the energy exerted upon it during its acceleration; so that that energy may be considered as stored during the acceleration, and restored during the retardation, in a manner analogous to the operation of a reciprocating force (§ 108).
Let there be given the mean velocity V = 12 (v2 + v1) of a body whose weight is w, and let it be required to determine the fluctuation of velocity v2 − v1, and the extreme velocities v1, v2, which that body must have, in order alternately to store and restore an amount of energy E. By equation (72) we have
which, being divided by V = 12(v2 + v1), gives
