428 APPLIED MECHANICS 
bob-weights. If w, and w, are the weights of the bob-weights, including, 
the weights of the reciprocating parts connected to them, then w,7;=R,, 
and w,r,=R,, the various centrifugal forces being represented by the 
products of weight and radius. 7“ 
A four-cylinder engine with four cranks is one that lends itself to — 
complete balancing of the reciprocating parts without the addition of 
balance weights. The quantities to be considered are: first, the ratios 
the weights of the four sets of reciprocating parts to that of one of th 
(3 quantities); second, the ratios of the distances between the cen 
lines of the cylinders to one of the distances (3 quantities); third, 
angles between the cranks (3 quantities); in all 9 quantities, and if 
5 of these be given, the other 4 can be found. . 
In this Article reciprocating masses only have been referred to. 
Revolving masses, including any cranks or eccentrics introduced to drive 
bob-weights, must be balanced separately by other revolving masses. . 
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Exercises XXVIb. 
1. The reciprocating parts of a single cylinder horizontal steam-engine we 
200 lbs., and the remaining parts of the engine weigh 6400 lbs. The stroke 
the piston is 16 inches, and the crank shaft makes 300 revolutions per minute. 
Assuming that the engine is not bolted down, but is free to oscillate, find the - 
amplitude of the oscillations, and the magnitude of the displacing force at the — 
end of each oscillation. Assume that the reciprocating parts have harmonic — 
motion. — 
2. After the engine of the preceding exercise is bolted down, suppose re 
is found that a force of 2000 lbs., applied to the crank shaft in the line of stroke, — 
displaces the engine frame 0:001 inch, and that the displacement up to five time: 
this amount is proportional to the displacing force. Assuming that all the yield ~ 
takes place between the frame and the foundations, what will now be the 
amplitude of the oscillations of the engine frame? ia 
3. A connecting-rod, 6 feet 2 inches long between centres, was found to — 
balance in a horizontal position on a knife edge placed at 244 inches from the 
large end centre. When a weight of 14 lbs. was placed at the small end centre, — 
it was found that the whole balanced in a horizontal position on a knife i 
placed at 26} inches from the large end centre. Find the weight of the rod 
the weights of the parts which should be credited as revolving and reciprocating 
respectively. 5 *: 
4. Find the balance weights at crank radius, in order to balance all the © 
revolving masses and two-thirds of the reciprocating masses of an inside sings nad 
locomotive, having given the following data. Cranks at right angles, left- 
crank leading. Distance centre to centre of cylinders, 2 feet, Distance between 
planes containing mass centres of balance weights, 5 fect. Mass of reciprocating — 
parts per cylinder, reduced to crank radius, 500 Ibs. Mass of revolving = 
per cylinder, reduced to crank radius, 700 lbs. [Inst.C.E.] — 
5. The reciprocating parts of an inside cylinder uncoupled locomotive weigh — 
550 lbs. per cylinder. The revolving parts are equivalent to 650 lbs. per 
cylinder at crank pin. The stroke of the pistons is 24 inches, and the distance — 
between the centre lines of the cylinders is 25 inches. Find the balance vee 
which must be placed in the driving wheels at 2 feet 6 inches radius, their 
planes of revolution being 5 feet apart, in order to balance the whole of the ~ 
revolving parts and two-thirds of the reciprocating parts. Oranks at righ 
angles, left-hand crank leading. Ag 
6. The following particulars relate to an outside cylinder uncoupled loco-— 
motive. Stroke of pistons, 26 inches. Length of connecting-rod, 78 inches _ 
Distance of centre of gravity of connecting-rod from centre of large end, 26 — 
inches. Weight of connecting-rod, 450 lbs, Weight of reciprocating parts | 
cylinder, 400 lbs. Equivalent weight of one crank arm and the portion of t 
crank pin within it, 130 lbs. at 13 inches radius. Weight of one overhanging — 
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