EXPLANATION OF THE ACTION OF THE RECIPROCATING PARTS OF A HORIZONTAL STEAM ENGINE
Let us take a horizontal engine of 2 feet stroke, making 200 revolutions per minute, so having a piston travel or average velocity of 800 feet per minute, which was my engine in the Paris Exposition of 1867.
We will suppose the piston to be driven through the crank, by which its motion is controlled, the power being got from some other motor, and that the cylinder heads have been removed so that the piston meets no resistance. We will also disregard the effect of the angular vibration of the connecting-rod, and assume the motion of the piston to be the same at each end of the cylinder.
On each stroke the crank does two things: First, it increases the motion of the piston from a state of rest to a velocity equal to the uniform velocity of the crank-pin in its circular path: and, second, it brings the piston to rest again, ready to have the same operation repeated in the reverse direction during the return stroke.
At the mid-stroke the crank is at right angles with the line of centers, and the velocity of the piston is 800 × ¹⁄₂π = 1256.64 feet per minute, or 20.944 feet per second, and no pressure is being exerted on the piston either to accelerate or retard its motion.
The pressure of the crank during a stroke, first to impart motion to the piston and second to arrest this motion, is represented by two opposite and equal triangles. Let the line AB, in the above figure, be the center line of a cylinder and its length represent the length of the stroke. Let the line AC, normal to the line AB, represent the force required to start the piston from a state of rest. Then the triangle AOC will represent the accelerating force that must be exerted on the piston at every point in the half stroke to bring up its velocity, until at O this equals that of the crank-pin in its circle of revolution, and the accelerating force, diminishing uniformly, has ceased. The opposite equal triangle BOD shows the resistance of the crank required to bring the piston to rest again.
How do we know this?
I will answer this question by the graphical method, the only one I know, and which I think will be understood by readers generally.
First, we observe that the distance the piston must move from the commencement to any point in the first half of its stroke, in order that it shall keep up with the crank, is the versed sine of the angle which the crank then forms with the line of centers. So the table of versed sines tells us where the piston is when the crank is at any point in its revolution, from 0 to 90°.
For example, let the quadrant AB in the following figure represent the path of the crank, and the line AO that of the piston. Let OF be the position reached by the crank. AOF is the angle formed by the crank with line of centers, and supposed to be 60°. FE normal to AO is the sine of this angle, and AE the versed sine. The latter is the distance traveled by the piston from the point A, and is .5, the length of the crank being 1.
Secondly, we ascertain how far the piston must advance for every degree or minute or second of the revolution of the crank in its quadrant by merely subtracting from its versed sine that of the preceding one. Thus the versed sine of 60° being .5, and that of 59° being .4849619251, the difference .0150380749 is the motion of the piston, or its mean velocity while the crank is traversing the 60th degree of its revolution.
Thirdly, we want to know the rate at which the motion of the piston is accelerated during any interval.
This acceleration is found by subtracting from the motion during each interval that during the preceding one. For example, the motion of the piston during the 60th degree being, as already seen, .0150380749, and that during the 59th degree being .0148811893, the difference between them, .0001568856, is the acceleration or amount of motion added during the 60th degree.
By this simple process we find the acceleration of the piston during the first degree of the revolution of the crank to be .0003046096, and that during the 90th degree to be .0000053161. But this latter is the amount by which the acceleration was reduced during the preceding degree. Therefore at the end of this degree the acceleration has ceased entirely.
Now, by erecting on the center line AC, at the end of each degree, ordinates which are extensions of the sine of the angle, and the lengths of which represent the acceleration during that degree we find that these all terminate on the diagonal line CO. Thus, when the crank has reached the 60th degree, and the piston has advanced half the distance to the mid-stroke or to E, Fig. 32, the acceleration during the 60th degree has been .0001523049, or one half of that during the first degree.
But how do we know the amount of the accelerating force exerted by the crank at the beginning of the stroke? This question is answered as follows:
We find that for the first three degrees the accelerating force is, for the purpose of our computations, constant, the diminution not appearing until we have passed the sixth place of decimals.
Let us now suppose the crank 1 foot in length to make 1 revolution per minute, so moving through 6° of arc in 1 second. At this uniform rate of acceleration the piston would be moved in 1 second the versed sine of 1° .0001523048 × 6² = .0054829728 of a foot.
A falling body uniformly accelerated by a force equal to its own weight moves in 1 second 16.083 feet. Therefore this uniform stress on the crank is .0054829728 16.083 = .000341, which is the well-established coefficient of centrifugal force—the centrifugal force of one pound making one revolution per minute in a circle of one foot radius.
So we find that the height AC of this triangle represents the centrifugal force of the reciprocating parts which, in any case, we can ascertain by the formula
WRr²C,
W being the weight of the body;
R being the length of the crank;
r being the number of revolutions per minute, and
C being the coefficient .000341.
This accounts for the fact that the reciprocating parts are perfectly balanced by an equal weight revolving opposite the crank.
In my treatise on the Richards Indicator and the Development and Application of Force in the Steam-engine, I have given a full exposition of this action here briefly outlined, and to that the reader is referred.
I have only to add that this computation is for horizontal engines. In vertical engines the effect of gravity must be considered, adding on the upward stroke and deducting on the downward stroke. Also the counterbalance in the crank-disk of vertical engines must be limited to the horizontal fling of the crank end of the connecting-rod, and all balancing must be as nearly as possible in the same plane.
In this respect double-crank engines have this advantage, that one half of the counterweight can be put on each side of the center line.
It is evident that the heavier the reciprocating parts and the more rapid the speed the greater the security for smooth and silent running. However loose the brasses and however sudden the impact of the steam on the piston, and however early or late the admission, there can be no sound or jar, if the inertia of the reciprocating parts is sufficient to equal the force of the entering steam, and if this is in excess it can do no harm. It is also evident that under these conditions at any point in the stroke the change of pressure to the opposite side of the crank-pin is made insensibly.
Some two or three weeks after this exhibition I received a note from President Barnard asking me to call upon him. On my responding to this invitation, he said to me that he had listened to my exposition of this action before the Polytechnic Club of the Institute, but he did not understand it; he had witnessed the experiments with my shop engine, but while he could not question the action in silencing all knock on the centers, still he did not understand it, and not until he investigated the problem in his own way by the method of the calculus did it become plain to him, and he could not see how I had ever been able to arrive at the exposition of the action without employing that method. This explains why the subject had not been considered in the report of the judges. President Barnard afterward kindly gave me a copy of his demonstration, to insert in my book on the Richards Indicator.
It seems appropriate to insert here the following letter received long after from a very prominent engineer of that day.
“Long Branch, N. J., Aug. 7th, 1872.
“Mr. Chas. T. Porter:
“My dear Sir: Since I had the pleasure of reading the paper which you read before the Polytechnic Club last winter, I have regarded your demonstration as not less original than subversive. It is, for the first time I believe, apprehended and asserted, not merely that the vis inertia of the reciprocating masses is not primarily an adverse element in the economy of the crank-engine, but that a certain amount of weight in the piston and its connections, and in high-speed engines a very considerable amount, is an absolute theoretical necessity.
“As this will be deemed rank heresy by folks who have been making skeleton pistons of wrought iron, it is well perhaps that you are entrenched at the outset behind the experimentum crucis of loose brasses. “Very truly yours,
“Joseph Nason.”
The following [figures] represent an elegant invention of Mr. Edwin F. Williams, which exhibits graphically the acceleration and retardation of the reciprocating parts of an engine.
In these views, A is the cross-head in its mid-position; B is the lath by which the paper drum of an indicator is actuated through the cord n. The lower end of this lath is fixed in its position on the cross-head by the stud j, on which it turns freely. y is the end of a vibrating arm, which permits the point of suspension of the lath B to fall below the position shown, as required in the motion of the cross-head on account of the lower end of the lath being so fixed. d is a cylindrical box, partly open, which is secured on the side of the cross-head, in a position parallel with motion, by the arm P. The end of this arm is on the stud j, inside the lath B. It is prevented from turning on this stud by the set-screw K, and its fixed position is further assured by the stud r.
In the box d is the cylindrical weight h, running freely on rollers, not shown, and bored to receive a spring e, of known strength. This spring is secured in two heads, one of which is screwed into the box and the other into the weight. The force required to move the weight h is thus applied to it through the spring.
The operation of this instrument is as follows: The cross-head being at its mid-stroke, as represented, has acquired its full velocity. At this point no force is being exerted, either to impart or to arrest its motion. The same is the case with the free weight h. No pressure is here being exerted, either to compress or to elongate the spring e.
Joseph Nason
| Fig. 1 | Fig. 5 SCALE 40 265 REVS. PER MIN. 11¹⁄₄″ × 16″ PORTER-ALLEN. |
| Fig. 4 SCALE 40 265 REVS. PER MIN. 4.416 „ „ SEC. | |
| Fig. 2 | Fig. 3 |
Apparatus for Graphically Showing the Acceleration and Retardation of the Reciprocating Parts of an Engine.
Let the motion be in the direction from the crank. The crank now begins insensibly, by pulling through the spring e, to arrest the motion of the weight h. This pull will increase in intensity to the end of the stroke, when the weight is brought to rest, and the spring will become correspondingly elongated. Then, by a continuance of the same pull, the crank puts the cross-head and this free weight in motion in the reverse direction. This pull gradually relaxes, until at the mid-stroke it has ceased. The weight h has acquired its full velocity again; all stress is off the spring, and the spring and weight are back in the positions in the box d from which they started. This action is repeated during the opposite half of the revolution, but in the reverse direction, the pull being changed to a push, and the spring being compressed instead of elongated. Thus at every point the position of this free weight shows the amount of the accelerating or retarding force that is being exerted upon it at that point, elongating or compressing the spring.
This varying accelerating or retarding force is recorded as follows: A paper b, [Fig. 2], is stretched on the surface ff. This surface is the arc of a circle described about the center j, and is secured on the lath B, so that as this lath vibrates by the motion of the cross-head the different points in the length of the paper pass successively under the pencil. This is set in the end of the long arm a of the right-angled lever-arms 4 to 1 seen in [Fig. 2], which is actuated by the rod e passing centrally through the spring and secured in the head c. This pencil has thus imparted to it a transverse motion four times as great as the longitudinal motion of the weight h in the box d. The pencil is kept lifted from the paper (as permitted by the elasticity of the arm a) by the cord m. By letting the pencil down and turning the engine by hand, the neutral line x, [Fig. 2], is drawn. Then when the engine is running, on letting the pencil come in contact with the paper, the diagonal lines are drawn as shown on [Fig. 2].
Edwin F. Williams
If the rotation of the shaft were uniform and there were no lost motion in the shaft or connecting-rod, this diagonal line would repeat itself precisely, and would be a straight line modified by the angular vibration of the connecting-rod. On the other hand, these lost motions and the variations in the rotative speed must be exactly recorded, the latter being exhibited with a degree of accuracy not attainable by computation and plotting, and their correctness would be self-demonstrated. For this purpose this instrument must be found highly valuable, if it is really desired to have these variations revealed rather than concealed. [Fig. 5] represents the inertia diagram drawn by this instrument applied to a Porter-Allen engine running in the Boston Post Office at the speed of 265 revolutions per minute. [Fig. 4] shows the same diagram with the transverse motion of the pencil enlarged to correspond with the scale of the indicator, so exhibiting the force actually exerted on the crank-pin at every point, which is represented by the shaded area, and from which the rotative effect on the crank can be computed. The steam pressure absorbed at the commencement of the stroke by the inertia of these parts is represented by the blank area above the atmospheric line xx. This is not all imparted to the crank at the end on account of the compression.
I have myself had no experience in the use of this instrument, but I do not see why it might not be so made that the diagonal line or lines in [Fig. 4] would be drawn at once. The variations of motion would thus be shown much more accurately than they can be by the enlargement of these small indications. This would require the spring e to bear the same relation to the inertia of the weight h that the spring of the indicator bears to the steam pressure on its piston area. The steam diagram and the inertia diagram would then be drawn to the same scale. A separate instrument would be required for each scale. It would seem desirable that this instrument, which is not expensive, should be brought before the public in this practical shape.
The 16″×30″ engine exhibited at this fair of the American Institute was sold from the exhibition to the Arlington Mills, at Lawrence, Mass. For a reason that will appear later, I have always regarded this sale as the most important one that I ever made.