Draw the cranks O1, 02, 03, etc., and with fixed centre O draw arcs

I'I, cutting Or in I.

2'II, cutting O2 in II.

23'XXIII, cutting O23 in XXIII.

Now draw a fair curve through A, I, II . . . O . . . XXIH, A.

The process can be followed in the figure; and examples should be drawn to as large a scale as possible.

The part of the curve near O is not very clear in the main figure, so it is shown enlarged in Fig. 117 (a). Here we see that there are, we may say, two separate enclosed curves, having as common tangents the cranks corresponding to zero distance of the piston from the centre of its stroke, ¿.e. piston at O, taking AA' as the stroke. To find these crank positions we simply reverse our original construction for piston position, i.e. from O', the centre of o, 12 the stroke, we strike an arc with radius O'O, i.e. connecting rod length, cutting the crank circle in two points O1 and O2 (Fig. 118), and then OO, and OO2 are the two required cranks. The

curve which lies to the left of these
cranks shows distances to the left
of the centre, and that lying to the
right distances to the right of the A
centre. For cranks very near 001
or OO, it is not advisable to trust
to the curve but to perform the
original construction.

2

Fig. 118.

TA'

We can now use this curve to give us the piston position corresponding to a given crank, or vice versa. The former we have already considered, for the latter we proceed thus:

Let the given piston position be at a distance x from its mid-position.

Take a radius r and with centre O describe a circle. This circle cuts the curve in four points, either of which joined to O gives a crank position fulfilling the required condition. To determine which is the particu

lar one we must know on which side of the centre the piston is, and in which direction it is travelling.

Thus in Fig. 119 if the piston be In OA travelling to A', Or is the crank

from A', 04

OA'

3

of

Obliquity

Connecting

Rod. We have seen that the piston position corresponding to a given crank position depends on the ratio

of connecting rod to crank arm.

Now to see in what way an alteration of connecting

rod length does affect the position, let us examine Fig. 120, where we have two examples drawn, differing only in length of connecting rod.

In each figure O is the crank centre, OP a given crank, CP the connecting rod, and M the piston position, PM then being an arc with centre C.

In each figure drop PN perpendicular to the line of stroke.

In each figure the distance CN is less than CM or CP, the connecting rod length. The amount of the difference in each case depends on the angle PCO, i.e. on the obliquity of the connecting rod; and it varies from zero at the beginning and end of the stroke to a maximum somewhere between.

Contrasting the two figures we see that MN is much less in (6) than in (a), due to the fact that the connecting rod is longer in (b). If we made the connecting rod still longer, the difference MN would be less still; and when the ratio of connecting rod to crank reaches say about 12:1, the points M, N would be practically identical.

The process of dropping a perpendicular from P is a simpler one than that of finding the arc PM; and, moreover, the result obtained can be expressed in a simpler form. Hence, for many purposes, we treat the point N as the piston position instead of M; our error in doing so is MN; and we say MN is the error due to obliquity, and N is the piston position neglecting obliquity.

We can very easily calculate the amount of the error due to obliquity. For with the usual lettering (Fig. 120)

MN=CM-CN,

=na-na cos PCN.

PCN is usually denoted by and PON by 0,

.. MN=na (1 − cos ),

The greatest value of is when the crank is upright, i.e. 0=90°; and then

If n=4, the greatest error is only s/16, while if n=12 it is only s/48.

There are two cases in which there would be no error due to obliquity.

Ist. With an infinite connecting rod, for then M and N would be identical. This is of course an impossi

bility.

2d. With connecting rod of length zero.

This latter can be in a way effected, and there is a certain practical type of direct acting engine, which we may say has a connecting road of zero length, and the movement of which is unaffected by obliquity.

The figure shows a portion of the mechanism of such an engine, used as a pumping donkey.

A is an outside view of the cylinder, with guides

B

N

Fig. 121.

B is the piston rod,

fastened to the top and bottom.

the end being either forged solid or screwed into a crosspiece, thus forming a T head, in which is cut a slot at right angles to the line of stroke. C is a block sliding in the slot, which (C) in a way takes the place of the

connecting rod and contains the crank pin brasses. O is the centre of the crank shaft, the crank circle being dotted. The shaft is only shown by its centre, and the remainder of the mechanism is, for clearness, omitted.

Now, plainly this mechanism works as an ordinary direct actor neglecting obliquity.

For taking the centre point of the slot as defining the motion of the piston (page 154), this point always is at N, the foot of the perpendicular from P, the centre of the crank pin, on to the line of stroke. This point is also the centre of the end of the piston rod, so that it is C in our former figures, thus C and N are identical; and this is why we say that in a way the connecting rod is of zero length.

Having thus found that results obtained neglecting obliquity are not only approximate for the ordinary engine, but are also the actual results for another kind of engine, let us see what these results come to.

To find the piston position, we simply drop PN perpendicular to AA'.

But now let us proceed, as on

page 160, to draw a curve.

Then (Fig. 122), we make OQ=ON, and so on.

Doing this, AN

Therefore Q lies on a circle having OA for diameter, and similarly for the other side OA'.

We see then that the determination, neglecting