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It is further clear, and an established principle in mechanics, that if the pressure of the moving power be uniformly the same, the amount of power, actually expended, will be in direct proportion to the distance the pressure has moved.
2. Therefore, when the power has moved from a to c, the amount of power or momentum, actually applied, is represented by the square a c bd=R2; and if we denote the radius by 1, the unity will represent that value.
Let us now examine the effect of the power upon the crank at the intermediate points 1, 2, 3, 4, when moving from a towards b.
The direction of the pressure upon the crank at 1 will coincide with the line 11' parallel to a c; therefore, the leverage of that pressure, with relation to the centre, must be l'c=11′′
The different leverage with which the pressure will act upon the crank to produce an useful result, for its motion around the centre, can therefore be represented by an indefinitely great number of ordinates, of the quadrant, as represented in the diagram, and the intermediate abscissas, corresponding to these ordinates, and forming infinitely small portions of the line a c, will represent the different extents of spaces through which the motive power has been travelling when acting upon the different levers.
The products of the different ordinates in their corresponding abscissas,
will represent the different momenta, which have been produced upon the crank with relation to its motion around the centre, and the sum of these products will give us the whole amount of useful effect of the crank during one quarter revolution.
But the products of an ordinate and its abscissa is the area of the trapezoid formed by them, and the sum of these areas is equal to the area of the quadrant, therefore:
3. The total useful effect of the crank through one quadrant, is represented by the area of the quadrant, or
4. The power applied being R2=12, therefore the useful power lost is
Suppose the crank moving from g to b and passing the points h and i. We may represent the pressure upon the crank at these points by the lines hn and i l. Now the pressure il can be resolved into the two pressures i m
and ik, at right angles to each other, and forming sides of the parallelogram of forces i klm. The pressure i m being directed to the centre, will be useless for the motion of the crank. The pressure i k, in the direction of the tangent, is the only useful pressure, resulting for the motion of the crank.
The momentum of the crank at i is therefore=ik×ic=ik× R.
and so in all other points.
The triangles i k land i l m, are similar to the triangle ie 3', therefore li : i k :: ci c 3' :: R: c 3' :: R: 3" 3
nh: hp: che l' :: R: e l' : : R : 1" 1
From this follows, that, if we represent the actual pressure of the moving power by R, then the useful tangential pressures, resulting for the motion of the crank at the different points hi, will be to the actual pressure as the ordinates 1" 1, 3" 3 are to the radius R.
We can further represent the amount of power actually expended at each of the different points h, i, by multiplying the pressure R, by an infinitely small extent of the line c g.
The useful momenta of the tangential forces, resulting for the motion of the crank, will therefore be obtained by multiplying the different ordinates with the same differential extent of c g. But these differentials are the abscissas belonging to the ordinates, and their aggregate products are represented by the area of the quadrant, therefore;
The useful effect of the Crank is equal to the area of the quadrant, if the power actually employed be equal to the square of the Radius.
The distance the crank travels is equal to the circumference ab of the quadrant, for the same space of time in which the power applied moves from a to c.
5. Therefore, the travel of the piston is to the travel of the crank, as is the radius to the circumference of the quadrant; or as the stroke to half the circumference of the circle; or,
2:31416+≈ 1 : 1·5708 +.
Since the whole useful effect of the crank during one quarter revolution is equal to the area of the quadrant, we shall find the average force which carries the crank around by dividing the effect by the extent of the travel. Therefore
is the expression for the average force: or the average useful pressure of the crank is equal to one-half of the pressure applied.
All the conclusions we have arrived at with respect to the crank, will apply as well to a cylinder, the radius of which is equal to the length of the crank, or the diameter of which is equal to the stroke of the piston.
If we examine the action of the crank by the principle of virtual velocities, according to which, the effect produced, should be equal to the power applied, and we denote the pressure of the power by P, its velocity by v, the useful propelling pressure of the crank by p' and its velocity by V, then the following equations should be correct:
Pxv=p x v
and by substiuting for v its value R, and for V its value R we have PX R=p' × ↓ R «
If we denote the correct pressure of the crank by p, we have according to No. 6,
The law of virtual velocities gives therefore the propelling power of the crank 0.2146 + too great.
When a wheel is connected with the shaft of the crank, then the propelling power of the wheel is to the propelling power of the crank, as is the diameter of the crank, or the stroke of the piston, to the diameter of the wheel.
Let L = the stroke of the piston, and D= the diameter of the wheel, and P = the pressure of the piston, then is
Formula for calculating the propelling power of a Locomotive Engine. According to the principle of virtual velocities, the expression
gives the propelling power of the crank, and applying this to the action of the wheel, we have, according to No. 8,
as the value of the propelling power of the wheel.
Now let & be the diameter of the piston, the ratio of the circumference, then d2, will be the area of one of the pistons. Let p represent the ef fective pressure of the piston in pounds per square inch, all other dimensions being given in inches, then d2p will be the value of the total pressure of the two pistons.
By substituting this expression for P in the above formula, we have
as the value of the propelling power of the engine, expressed in pounds.
This formula corresponds exactly with that offered by De Pambour, in his work on Locomotive Engines, (see Am. edition, pp. 109, 110.) But we have proved that the principle of virtual velocities, gives the effect 0.2146 too great.
The correct expression for the propelling power of a wheel is according to No. 8,
which is the correct formula for the propelling power of a Locomotive En
gine, allowing only or 0-7854 +of the power, produced by De Pambour's formula.
Let P 50 lbs. per square inch, or 7,200 lbs. per square foot, L = 1·5, d=1, D = 5. By substituting these values in the correct formula,
d2 p L
3.14+ X 1
X 7,200 X 1.5 4 X 5
as the propelling power of the engine, leaving all other considerations out of view.
Or De Pambour's formula gives the power nearly one-fourth too great. De Pambour in his new publication on the Steam Engine, speaks of the great discrepancy between the theoretical and practical effect of Locomotive Engines, and he appears to acknowledge that the old formulas are insufficient. This distinguished Engineer presumes that the main cause of the error is to be found in the reduction of the pressure of the steam in the cylinders. There are, however, strong reasons to believe that this is not the case, if we take the evaporating power of the engine as the measure of its power, as we should do. One great loss of effect, is owing to
the use of the crank, as has been proved above, conclusively. We hope to be able to ascertain the amount of loss arising from other causes, particularly from the reaction of the waste steam, and may at some future period offer formulas which will very nearly agree with practice.
It remains yet to account for the loss of useful effect attending the use of the crank, or the change from a straight to a rotary motion, generally.
It appears that the law of virtual velocities is only applicable to analagous motions, with respect to actual and useful effect. Rotary motions and straight motions differ in their nature totally; they cannot be compared directly by that law.
The philosophical principle-what matter is in existence cannot be lost, nor destroyed, but can only be changed in form and space,-appears to be a grand universal law, rendering the existence of the Universe itself, permanent, and independent of time.
This principle, the truth of which will force itself upon the mind, by a concatenation of rigid deductions, must hold out in all cases, and therefore also in the case before us, where matter is brought into action to produce a mechanical effect. The laws of mechanics cannot be in contradiction to an universal law.
If therefore we cannot account for the loss of power in any way, we have strong reasons to doubt the accuracy of a demonstration which is to prove that loss of power.
On the other hand, no law has been established yet, to prove that the whole amount of power applied, can be made available for all purposes for which we want it, leaving of course friction out of view, and other circumstantial causes of loss.
In the case before us, let us suppose the tangential pressure ki, (see diagram,) is applied to the crank. If the impulse which the point i receives in consequence of that pressure, was allowed to be developed for any actual extent of space, in the line of the tangent, which is the direction of the impulse, we would be authorised to compare the quadrant to a succession of inclined planes, and we could prove no loss of power. But the impulse which is ready to act in the point i, is not allowed to develope itself in the direction of its natural tendency for any actual extent of space, without being checked. And since an inclined plane coincides with a straight line, and a straight line is the result of a point moving through an actual extent of space, in a straight course, it follows, conclusively, that we cannot compare the circumference of the quadrant with a succession of inclined planes.
We have made these remarks, in order to object to any demonstration which treats the circumference of the quadrant as a succession of inclined planes.
The impulse of the crank, with relation to the centre, can therefore only be considered for single mathematical points in the circumference, as h, i, without even allowing an initial extent for the display of the tangential forces.