curve so obtained is, if V, be constant, a curve of piston velocity, scale OP=V。.

If V, be not constant, it is a curve of velocity ratio of piston to crank, scale OP=unity (compare page 174).

We can now see what effect obliquity has, by comparing this curve with the one obtained, neglecting obliquity. To do this draw the dotted circle on OB'. We then see that so long as OP is to the left of OB, the curve lies outside the circle, so the velocity is greater than that obtained neglecting obliquity. Also the maximum velocity is greater than Vo, and occurs at a crank position before OB. When OP is along OB, VV, as it did before. When OP is to the right of OB, then the velocities are all less than they were when we neglected obliquity.

The mean velocity is of course not affected by obliquity, and is always

and is represented by the dotted circle.

We can now obtain, if necessary, a linear diagram of velocity, on the stroke as base, referring to piston position.

Fig. 129 shows the construction, and any further explanation can be got by referring to page 170, where the diagram is explained for the case neglecting obliquity.

Angular Velocity of Connecting Rod.-The velocity here mentioned is sometimes required, and the

process of finding it will enable us to introduce an important method, so we will briefly consider it.

A pin in the end of the rod turns in a bearing in the crosshead, or vice versa, and the angular velocity we wish to obtain is that of the pin relative to the bearing. Now CP is the rod, and Vp, V, the velocities of its ends. The crosshead bearing has also the velocity V and this causes our difficulty, because we have a moving bearing. Now the method we use depends on the following principle: We cannot alter the relative motion of two bodies by imparting to them as a whole a

velocity in any given direction. This simply means that supposing the two bodies mounted say, on a stand, then we cannot by moving the stand about affect the relative motion of the bodies.

Apply this principle then, by imparting to the connecting rod and crosshead as a whole a velocity Vp parallel to OC.

Then this leaves C with resultant velocity zero, so the bearing is reduced to rest, while P has now a velocity composed of Vp and V。. To find the resultant velocity of P resolve Vp and V, along and at right angles to CP. Then along CP

which of course it should be, since, C being at rest, P can only move at right angles to CP.

principle. We have then

This verifies our

Velocity of P is at right – V, sin a+ V2. sin &,
angles to CP

and graphically

=

So the third side PT of the triangle OPT represents the linear velocity of P relative to the crosshead.

We can at once deduce the angular velocity of the rod or pin in the bearing.

So the velocity ratio of connecting rod-crosshead pair to crank-frame pair is PT: CP.

One result which we can at once obtain is the speed of rubbing of the gudgeon pin in the bearing at the instant. For if r be the radius of the bearing in feet,

We may if necessary construct curves showing this angular velocity, as has already been done for piston velocity.

EXAMPLES.

(All to be solved by construction when possible.)

Find

1. The stroke of an engine is 4 ft. 3 ins., length of connecting rod 8 ft. The steam is to be cut off at .6 of the stroke. the angle the crank then makes with the line of stroke.

N

If the

crank position be found neglecting obliquity, and the cut-off take place at the position so found, what will be the actual point of cut-off? Ans. 86° 13′; .665.

412

2. Show that the greatest error in piston position due to obliquity is very approximately;s being the stroke, and n the ratio of connecting rod to crank. Apply this to the preceding example. Ans. 3 ins. 3. In (1) the travel of the slide valve is 10 inches, and the angle of advance 20°. Find the piston position when the valve is in the middle of its stroke, and also when it has moved 21 inches either way from the central position.

Ans. 19.8; .41; 43.9 ins. from beginning of stroke.

4. The stroke of a pumping donkey is 8 inches, and it runs at 85 revolutions per minute. Find the speed of the piston at each eighth of its stroke.

Ans. 117.7, 154.2, 172.3, etc.,

ft. per

min.

Draw the

5. Draw to scale a curve of velocity for an engine 3′ 6′′ stroke, 7' 6" connecting rod, running at 110 revolutions. crank circle on a scale of 1 in. to I ft., and state what scale this causes the curve to be—1st, as a velocity ratio curve; 2d, as a velocity curve. Find the maximum piston velocity, and the velocity at each quarter stroke. Also find the piston position when the actual velocity equals the mean.

Ans. Scales, Ist, 1 ins. to unity; 2d, 1 inch to 691.4 ft. per min. Maximum velocity, 1332; Ist quarter, 1141; 2d, 1320; 3d, 968 ft. per min. Piston positions, 4, 35 ins. from beginning of stroke.

6. Construct a curve of angular velocity of connecting rod by marking off the values of PT (page 176) along the corresponding cranks for the preceding. State on what scale this curve gives, Ist, angular velocity of rod; 2d, speed of rubbing of gudgeon, diameter, 10 ins. Give numerical value of the maximum in each

case.

Ans. Scales, I inch to 1.536 radians per second; I inch to .64 f.s. Maximum values, 2.69 radians per sec.; 1.12 f.s.

7. Show, in a pumping donkey, that the curve giving the speed of rubbing of the crank pin brasses, in the slot in the piston rod head, is a pair of circles.

8. If A, be the angular velocity of the crank and A that of the connecting rod at a given instant; show that the angular velocity with which the crank pin is then rotating in its bear

ing is A.-A; regard being had to the sign of A, taking the direction of A, as positive. Ans. Applying a velocity V, to the whole engine at right angles to OP reduces the centre of the bearing to rest. O moves with velocity Vo, so the crank pin rotates round P with angular velocity A.; and C, having a velocity compounded of V, and Vo, it will be found is rotating round P with angular velocity A (page 177). Since pin and bearing both rotate round the same centre, the relative velocity of rotation is evidently the difference of A, and A if they are in the same direction, and the sum if in opposite directions. The relative velocity is then A. - Ä.

This is one case of a very important principle, viz. that the relative angular velocity of two bodies as above is the difference of their angular velocities, irrespective of what their centres of motion may be. The general principle can be proved in the above

manner.

9. Find the velocity of rubbing of the crank pin of No. 5 in its bearing when on either dead centre.

Diameter 14 ins.

Ans. 8.6; 5.33 f.s.