centrifugal force will cause strains, vibration and increased friction, and a tendency of the shafts to jump out of their bearings.
§ 111. Centrifugal Couples of a Rotating Body.—Besides the tendency (if any) of the combined centrifugal forces of the particles of a rotating body to shift the axis of rotation, they may also tend to turn it out of its original direction. The latter tendency is called a centrifugal couple, and vanishes for rotation about a principal axis.
It is essential to the steady motion of every rapidly rotating piece in a machine that its axis of rotation should not merely traverse its centre of gravity, but should be a permanent axis; for otherwise the centrifugal couples will increase friction, produce oscillation of the shaft and tend to make it leave its bearings.
The principles of this and the preceding section are those which regulate the adjustment of the weight and position of the counterpoises which are placed between the spokes of the driving-wheels of locomotive engines.
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| (From Balancing of Engines, by permission of Edward Arnold.) |
| Fig. 130. |
§ 112.* Method of computing the position and magnitudes of balance weights which must be added to a given system of arbitrarily chosen rotating masses in order to make the common axis of rotation a permanent axis.—The method here briefly explained is taken from a paper by W. E. Dalby, “The Balancing of Engines with special reference to Marine Work,” Trans. Inst. Nav. Arch. (1899). Let the weight (fig. 130), attached to a truly turned disk, be rotated by the shaft OX, and conceive that the shaft is held in a bearing at one point, O. The force required to constrain the weight to move in a circle, that is the deviating force, produces an equal and opposite reaction on the shaft, whose amount F is equal to the centrifugal force Wa2r/g ℔, where r is the radius of the mass centre of the weight, and a is its angular velocity in radians per second. Transferring this force to the point O, it is equivalent to, (1) a force at O equal and parallel to F, and, (2) a centrifugal couple of Fa foot-pounds. In order that OX may be a permanent axis it is necessary that there should be a sufficient number of weights attached to the shaft and so distributed that when each is referred to the point O
(1) ΣF = 0
(2) ΣFa = 0
The plane through O to which the shaft is perpendicular is called
the reference plane, because all the transferred forces act in that plane
at the point O. The plane through the radius of the weight containing
the axis OX is called the axial plane because it contains the forces
forming the couple due to the transference of F to the reference plane.
Substituting the values of F in (a) the two conditions become
| (1) (W1r1 + W2r2 + W3r3 + . . .) α2g = 0 |
| (2) (W1a1r1 + W2a2r2 + . . .) α2g = 0 |
In order that these conditions may obtain, the quantities in the
brackets must be zero, since the factor α2/g is not zero. Hence finally
the conditions which must be satisfied by the system of weights in
order that the axis of rotation may be a permanent axis is
| (1) (W1r1 + W2r2 + W3r3) = 0 |
| (2) (W1a1r1 + W2a2r2 + W3a3r3) = 0 |
It must be remembered that these are all directed quantities, and
that their respective sums are to be taken by drawing vector polygons.
In drawing these polygons the magnitude of the vector of
the type Wr is the product Wr, and the direction of the vector
is from the shaft outwards towards the weight W, parallel to the
radius r. For the vector representing a couple of the type War,
if the masses are all on the same side of the reference plane, the
direction of drawing is from the axis outwards; if the masses are
some on one side of the reference plane and some on the other side,
the direction of drawing is from the axis outwards towards the
weight for all masses on the one side, and from the mass inwards
towards the axis for all weights on the other side, drawing always
parallel to the direction defined by the radius r. The magnitude
of the vector is the product War. The conditions (c) may thus be
expressed: first, that the sum of the vectors Wr must form a closed
polygon, and, second, that the sum of the vectors War must form a
closed polygon. The general problem in practice is, given a system
of weights attached to a shaft, to find the respective weights and
positions of two balance weights or counterpoises which must be
added to the system in order to make the shaft a permanent axis,
the planes in which the balance weights are to revolve also being
given. To solve this the reference plane must be chosen so that it
coincides with the plane of revolution of one of the as yet unknown
balance weights. The balance weight in this plane has therefore
no couple corresponding to it. Hence by drawing a couple polygon
for the given weights the vector which is required to close the polygon
is at once found and from it the magnitude and position of the balance
weight which must be added to the system to balance the couples
follow at once. Then, transferring the product Wr corresponding
with this balance weight to the reference plane, proceed to draw
the force polygon. The vector required to close it will determine the
second balance weight, the work may be checked by taking the
reference plane to coincide with the plane of revolution of the second
balance weight and then re-determining them, or by taking a reference
plane anywhere and including the two balance weights trying
if condition (c) is satisfied.
When a weight is reciprocated, the equal and opposite force required for its acceleration at any instant appears as an unbalanced force on the frame of the machine to which the weight belongs. In the particular case, where the motion is of the kind known as “simple harmonic” the disturbing force on the frame due to the reciprocation of the weight is equal to the component of the centrifugal force in the line of stroke due to a weight equal to the reciprocated weight supposed concentrated at the crank pin. Using this principle the method of finding the balance weights to be added to a given system of reciprocating weights in order to produce a system of forces on the frame continuously in equilibrium is exactly the same as that just explained for a system of revolving weights, because for the purpose of finding the balance weights each reciprocating weight may be supposed attached to the crank pin which operates it, thus forming an equivalent revolving system. The balance weights found as part of the equivalent revolving system when reciprocated by their respective crank pins form the balance weights for the given reciprocating system. These conditions may be exactly realized by a system of weights reciprocated by slotted bars, the crank shaft driving the slotted bars rotating uniformly. In practice reciprocation is usually effected through a connecting rod, as in the case of steam engines. In balancing the mechanism of a steam engine it is often sufficiently accurate to consider the motion of the pistons as simple harmonic, and the effect on the framework of the acceleration of the connecting rod may be approximately allowed for by distributing the weight of the rod between the crank pin and the piston inversely as the centre of gravity of the rod divides the distance between the centre of the cross head pin and the centre of the crank pin. The moving parts of the engine are then divided into two complete and independent systems, namely, one system of revolving weights consisting of crank pins, crank arms, &c., attached to and revolving with the crank shaft, and a second system of reciprocating weights consisting of the pistons, cross-heads, &c., supposed to be moving each in its line of stroke with simple harmonic motion. The balance weights are to be separately calculated for each system, the one set being added to the crank shaft as revolving weights, and the second set being included with the reciprocating weights and operated by a properly placed crank on the crank shaft. Balance weights added in this way to a set of reciprocating weights are sometimes called bob-weights. In the case of locomotives the balance weights required to balance the pistons are added as revolving weights to the crank shaft system, and in fact are generally combined with the weights required to balance the revolving system so as to form one weight, the counterpoise referred to in the preceding section, which is seen between the spokes of the wheels of a locomotive. Although this method balances the pistons in the horizontal plane, and thus allows the pull of the engine on the train to be exerted without the variation due to the reciprocation of the pistons, yet the force balanced horizontally is introduced vertically and appears as a variation of pressure on the rail. In practice about two-thirds of the reciprocating weight is balanced in order to keep this variation of rail pressure within safe limits. The assumption that the pistons of an engine move with simple harmonic motion is increasingly erroneous as the ratio of the length of the crank r, to the length of the connecting rod l increases. A more accurate though still approximate expression for the force on the frame due to the acceleration of the piston whose weight is W is given by
| W | ω2r { cos θ + | r | cos 2θ } |
| g | l |
The conditions regulating the balancing of a system of weights reciprocating under the action of accelerating forces given by the above expression are investigated in a paper by Otto Schlick, “On Balancing of Steam Engines,” Trans, Inst. Nav. Arch. (1900), and in a paper by W. E. Dalby, “On the Balancing of the Reciprocating Parts of Engines, including the Effect of the Connecting Rod” (ibid., 1901). A still more accurate expression than the above is obtained by expansion in a Fourier series, regarding which and its bearing on balancing engines see a paper by J. H. Macalpine, “A Solution of the Vibration Problem” (ibid., 1901). The whole subject is dealt with in a treatise, The Balancing of Engines, by W. E. Dalby (London, 1906). Most of the original papers on this subject of engine balancing are to be found in the Transactions of the Institution of Naval Architects.
§ 113.* Centrifugal Whirling of Shafts.—When a system of revolving masses is balanced so that the conditions of the preceding section are fulfilled, the centre of gravity of the system lies on the axis of revolution. If there is the slightest displacement of the centre of gravity of the system from the axis of revolution a force acts on the shaft tending to deflect it, and varies as the deflexion and as the square of the speed. If the shaft is therefore to revolve stably, this force must be balanced at any instant by the elastic resistance of the shaft to deflexion. To take a simple case, suppose a shaft,
