GO 



THE CIVIL ENGINEER AND ARCHITECTS JOURNAL. 



[February, 



pressure to the maximum pressure attained during the stroke. With 

 respect (o tlie gradual opening of the port, tliat would not of itself 

 a(Tpct the ]iressure of the steam in the cylinder in any degree, since 

 the degree of opening is constantly proportional to the velocity of the 

 jiiston, liy which means the supjily of steam is always equal to the de- 

 niand. This is ccpially true while the steam-port is being closed in 

 the last half of the stroke. We may therefore be permitted to assume, 

 as above, that tlie steam presses on the piston with its full force during 

 the whole time tliat the steam-port is open. 



We may also, for the sake of simplicity, assume the motion of the 

 crank to be strictly uniform; for the variations of power are so ex- 

 ceedingly slight in comparison witli the energy of the moving mass, 

 on account of the arrangement of the two cranks, that no sensible va- 

 riations in the velocity of the engine can possibly result therefrom. 



I'nder these circumstances, when the piston has just arrived at the 

 middle of the back stroke, its motion may be regarded for an instant 

 as uniform, since it is changing from an accelerated to a retarded mc- 

 tion; the strain on the key «diich connects the piston to the piston- 

 rod is therefore equal to the effective pressure of the steam on the 

 piston 7>ii)iiis the friction of the latter against the surface of the cylin- 

 der. But as soon as the piston has passed tlie middle of its stroke, its 

 motion begins to be retarded, and since the retarding force lias to be 

 transmitted from the piston-rod to the piston through the medium of 

 the key which connects them, the latter has to bear the strain of this 

 force in addition to the pressure of the steam on the piston, which it 

 liadtobearin the middle of the stroke. This retarding force must 

 evidently increase from the middle to the end of the stroke with the 

 rate of retardation of the )iiston's motion. The strain at the end of 

 the stroke will therefore be equal to the effective pressure of the 

 steam on the surface of the piston, mimis its friction, plua the greatest 

 retarding force, since the retardation is then the most rapid. At the 

 commencement of the fore stroke, the pressure of the steam is equal 

 on both sides of the piston, and there is consequently no power to 

 move the piston but such a portion of the momentum of tlie engine as 

 is snfKcient to overcome the inertia and friction of the piston. The 

 former requires a force precisely equal to the retarding force at the 

 end of the stroke, so that the strain is suddenly diminished by the 

 cHective pressure of steam on the piston minus twice its friction; and 

 the remaining strain is gradually, though quickly, taken off by the 

 escape of the waste steam allowing tiie steam on the other side of the 

 piston to exert sutiicient force to accelerate its motion. From the 

 moment when this is the case all the pressure is borne bv the conical 

 end of the piston-rod, until it arrives at tlie corresponding" point of the 

 back stroke, from which the strain on the key increases gradually until 

 it attains its maximum at the end of the stroke, as we have already 

 explained. 



In locomotive engines, wdiere the steam is used at very high pres- 

 sures and the pistons are light, the strain due to the retardation of the 

 motion of the piston is very inconsiderable in comparison with that 

 due to the pressure of the steam, as we shall presently prove, when 

 wc calculate the intensity of the retarding force; so that comparatively 

 very little increase of strain takes place during the last half of tlie 

 stroke, this increase being due to the retardation of the piston alone, 

 while the increase during the first half amounts, as we have already 

 mentioned, to the whole effective jiressure of the steam; for by the 

 time the piston has passed through the first half of the stroke, the 

 pressure of the waste steam must be reduced very nearly to that of the 

 atmosphere. 



The shocks comjilained of appear therefore to be due, not to the 

 inertia of the piston requiring a considerable force to change its direc- 

 tion, but to the alternate action of the steam on the two sides of the 

 jiiston producing a strain on one side of the key which reaches its 

 maximum at some point of the back stroke, and is taken oft'cntirely 

 during tlieyorestr<ike. 



The mode in which this intermittent strain on the key may cause it 

 t» work loose is evidently by the alternate compression and relaxation 

 of its'substance ; but the effect of this might be prevented by securing 

 the key with a screw in the same maimer as the key at the crank end 

 of the connecting rod. 



We have said that the chief part of the strain on the key of the 

 piston is due to the pressure of the steam; this investigation would 

 however be inconi|)lete without a calculation of the strain due to the 

 inertia of the piston in consequence of the great variations in its velo- 

 city. This strain is evidently equal to the force wdiich would be re- 

 quired to produce a certain acceleration or retardation in the motion 

 of the piston, and an accelerating or retarding force is proportional to 

 the rate of acceleration or retardation which it produces. 



Let V = the mean velocity of the piston in feet per minute, I = the 

 length of its stroke, !>= its velocity at any given instant, A = its dis- 

 tance from the end of the stroke, a =; the angle contained between 



the crank and the direction of the stroke, and .r = the rate of retarda- 

 tion per minute, at the given instant. 



Supposing, for the sake of simplicity, that the motion of the crank 

 is strictly uniform (which is very nearly true in reality), and that the 

 connecting rod is infinitely long in comparison with the crank, the cir- 



ir V 

 cuinferential velocity of the crank pin will be — — ■' and we shall have 



TT V 



■sill, f, 



whence we obtain by differentiation 

 a V ^= — r- 



COS ad a. 



The actual distance to be passed through by the crank pin before it 

 arrives at the dead centre is - o, which divided by its velocity--^ 

 gives for the time required to travel that distance 



'- ,rV' 



whence 



da= — — d I. 



Substituting this expression in the value of d r, and dividing by d I, 

 we obtain 



d V 

 dt' 



t' V2 cos a TT^ V2 (;— 2 A). 



2 1 IP 



And since this quantity expresses the retardation of the piston per 

 minute, we have also 



TT- V= (/— 2 A) 

 ^ = 2F 



The retardation per minute produced by the force of gravity is 

 about 1 15,884 feet, which if we call G, we shall have 



a: _ IT- V (/— 2 A) 



G~ 2317687^' 



If then we call n> the weight of the piston, and W the pressure due to 

 the retardation x, we shall evidently have 



W _ .r _ ttz V» (/— 2 A) 



w ~ G " 23I7ti8 i-' ' 



or the strain on the key of the piston is equal to the weight of the 



piston multiplied by the quantity — "i^^yj. . ,; • 

 As an example let V zi; 500, / = 1'5, and A : 



1 



48' 



. Supposing the 



driving wheels of the engine to be 5 feet in diameter, the speed under 

 these circumstances would be 2lV7r) miles per hour. The strain W on 

 the key of the piston is required, when the latter has arrived at a 

 quarter of an inch from the eiu) of the stroke, which is the lead usually 

 given to the slide in locomotives. By the preceding equation we 

 have 



W 



3-1415' X 500-' X 



) 



23I7G8 X l-5= 



= 6-9 w. 



On inspecting the general equation given above, it will be evident 

 that, all other circumstances remaining the same, the value of W varies 

 as the square of the velocitv of the piston, and that it increases as the 

 piston apjiroaches the end of the stroke, the strain at the very end 

 being equal to the weight of the piston multiplied by the quantity 



„, , ^ . , , . Under the circumstances assumed in the above example 

 2dl/bb t:- 



we should therefore have at the end of the stroke 



W = 7 w. 



It likewise appears that, with various lengths of stroke but the same 

 velocity, the strain is inversely as the lengtli of the stroke, when the 

 piston is at proportionate distances from the end. 



The same calculation applies, of course, as well to the first as to the 

 second half of the stroke of the piston, and is improperly omitted in 

 the cousideration of the unequal action of the steam on the crank and 

 the effect of fly-wheels ; but in these calculations it is not only the 

 weight of the piston, but that of all the alternatin^parts of the engine, 

 that must be taken into account. In the same manner the strain on 



