# Page:Atreatiseonstea00bourgoog.djvu/108

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Parallel Motion.

CHAPTER IV.

PARALLEL MOTION.

The forms of parallel motion which have been used are so various, that we shall not attempt to enumerate the whole. In what follows, however, we shall treat of those varieties that have been most commonly adopted in land or marine engines; and in doing so, it will be our object both to explain their principles of action, and give practical rules for calculating the lengths of the various parts in every assignable case.

As this proportion holds true at any part of the stroke, it proves that the point E, which divides the link in this proportion, moves in the perpendicular line R S′. When, therefore, the lengths of the levers are given, it is easy to find the point E; but in practice it generally happens (as will be shown subsequently) that the point E is fixed: and we have to find H F. This will be done from this proportion, F E : E G :: G C : H F = $\tfrac{GC \cdot EG}{FE}$ … (A). This proportion is founded on the assumption that the versed sines of the beam and radius bar are inversely as the radii. This, though nearly true, is not exactly so. The versed sines of the smaller radii increase in a greater proportion than the inverse ratio of the radii. The consequence of this is, that the point E in the link G F will be moved too much toward H, and will deviate from the perpendicular in that direction at each end of the stroke. In small engines, and when H F is nearly equal to G C, this deviation will be so small as scarcely to be worthy of notice; but in large engines, and when there is a considerable difference between the lengths of the radius rod and beam, it will be of some consequence. From what has already been said, it will be observed that the deviation is all on one side of the perpendicular; so that the curve described by the point E will be of the shape shown in fig. 8. By making a small addition to the length of the radius bar, we can make the deviation take place equally on each side of the perpendicular. We can add so much to its length as to make the point E move towards H too slowly at half stroke; or, in other words, deviate towards C till the beam has got about quarter stroke from the horizontal position, when the point E will cross the perpendicular, and deviate to an equal amount towards H. The curve described by E will then be of the shape shown in fig. 9, where the deviations at m, n, o, p, are all equal, and only half the deviation at r and s in the last figure.[2] To facilitate the making of this correction in the length of the radius bar, we give the following table of the amounts to be added to its length for the various proportions that may obtain between it and the beam C G. It is to be observed too, that when the radius bar H F is longer than the beam G C, these amounts are to be subtracted from its length.

TABLE (A.)
This column gives F HC G when C G is the greater, and C GF H when F H is the greater. Correction to be added to or subtracted from the calculated length of the radius bar, in decimal parts of its calculated length.
1.0   0
.9 0.0034
.8 0.0075
.7 0.0163
.6 0.0270
.5 0.0452
.4 0.0817
1. The beam of an engine is usually made at least three times the length of the stroke; so that the sine of the angle which it makes with the horizontal line at the extreme of its stroke, is equal to ½ of the radius. The angle whose sine is ½ of the radius, is 19° 28′.
2. Let $G C = r G' G'' = s$, and $\mathrm{arc}\ G'' G G' = \phi$; then $G K = \mathrm{vers.}\ \phi$;
but by Euclid (III. 35), $GK(2 G C-G K) = (\tfrac{1}{2} G' G^2)^2$
$\therefore \mathrm{vers.}\ \phi (2r - \mathrm{vers.}\ \phi) = \tfrac{1}{4}s^2 \therefore 2r\ \mathrm{vers.}\ \phi = \tfrac{1}{4}s^2 + \mathrm{vers.}^2\ \phi$
$\therefore r=\tfrac{\tfrac{1}{4}s^2 + \mathrm{vers.}^2\ \phi}{2 \mathrm{vers.}\ \phi}= \tfrac{\tfrac{1}{4}s^2 + \tfrac{1}{2} \mathrm{vers.}^2\ \phi}{\mathrm{vers.}\ \phi}$; or, $r=\tfrac{\tfrac{1}{2} s^2}{\mathrm{vers.}\ \phi}+\tfrac{1}{2} \mathrm{vers.}\ \phi$
$\tfrac{1}{2}s^2$ being a constant quantity, this equation shows that the radii are inversely as the versed