370 



THE CIVIL ENGINEER AND ARCHITECTS JOURNAL. 



[Dec. 



WORKING STEAM EXPANSIVELY. 



When a steam-engine is working at any given speed, the pressure on the 

 prank-pin is equal to the pressure on the piston resolved into the direction 

 of the length of the connecting-rod, minus the force of inertia of the re- 

 ciprocating parts when their velocity is increasing, or plus the via insita of 

 those parts when their velocity is decreasing : — it is required to ascertain the 

 amount of this + pressure. 



If the square of the velocity of any mass of matter increases in an ele- 

 mentary space n times as much as it would increase by falling through that 

 space, — then the force for that point or elementary space must be n times 

 the force of gravity, or n times the weight of the mass ; that is, putting 

 V = the velocity due to falling a given space, and V = the actual velocity, 



VrfV 

 as d V- • dv'', or as V(/V : vdv ''. n\\, or ■ — z— •= n ; 



van 



but v' ■= 2r/s, and differentiating vdv = gds, 



VrfV , -VdV 



, and for the destruction of motion 



gds 



Let : = the angle passed through by the crank; 

 r = the length of the crank j 

 s -= the space travelled by the piston ; 



ds 



V .= the velocity of the piston ■= — ; 



d i 



V = the velocity of the crank-pin in the arc z 



— gds 



rdz 

 ~di ' 



C = the length of the connecting-rod ; 



Q 



c = -.or the value of C in terms of the length of the crank; 



r 



n = the force of inertia or insita in terms of the weight ; 

 P = the pressure on the crank-pin caused by the insita or inertia, or 

 the value of n reduced to the mechanical conditions. 



-e- 



The space described by the piston is = ab; 



.•. s = r I ver ; + c — {(p — sia'z)- \ ; 



rdz / sin 2 cos z \ 



-r- X I sin I -h TTTi I • 



dt \ (e-'-sin=z)5/ 



ds 



differentiating r-. 



therefore, V 



dV = V cos 2 dz + V 



sin z + 

 V sin r cos z 



- J and differentiating 

 (c2-sin2i)5 



( jsing2r + cos 2z (c°-sin°;) >v 

 (e2- sm-z)i ) 



VdV 



The theorem = n may be put in a more convenient form, for 



yds 



V n \ V 



= dt, and - = dt; therefore, — = -3-;; and, by substituting. 



ds 



rdz 



ds 



rdz 



vdV 



we have 



grdz 



The motion of the piston-rod and appendages is vertical ; therefore, n 



must be resolved into the direction of the length of the connecting-rod. By 



the mechanical theorem sometimes called the triangle of forces, we have 



nc „ cvdV 



P; or ; = P; 



(c-— sia" z)i 



cv" cos z 



{c'-iia^z)i grdz 



isin=2i-f cos2r (e' 



consequently, 

 sin'r) 



32r (c=-8in=2)4 

 weight being considered unity. 



32 r(c2 _sin»z) = 



, the 



For the beam, let s be a fraction expressing the distance of the centre of 



gyration from the centre gudgeon when the length of the radius of the beam 



is I. Let n represent the force of inertia of the beam at the point g : then 



2vsd\ s»dV , 



g^rdz gfdz 



but of this force, a portion = (1 — s) n' will be sustained by the centre gud. 



geon ; the remainder, or j n', will be sustained by the top of the connecting- 



c 



rod, which, multiplied by -j , gives the pressure on the crank-pin 



(c*— 8in'z)i 



due to the inertia of the beam, which we will call P'; therefore 



P' = 7 3 r , or P' = s'P. on the supposition that the 



(i^—a\a'z)igrdz 



end of the beam describes a straight line instead of an arc, which supposi- 

 tion has been made by all writers on the theory of the crank. 



The connecting-rod has a compound motion — namely, vertical at the top 

 (neglecting the arc), and circular at the bottom : these two motions may be 

 resolved into vertical and horizontal. The sum of the inertia in the vertical 

 and horizontal directions, resolved in the direction of the length of the rod, 

 will give the value of P". Let the centre of inertia, in the vertical sense, 

 be supposed to be concentrated in an undetermined point p ; this point, 

 when the upper end is moving vertically with greater velocity than the lower 

 end, will be between the top and the centre of gravity ; and when the lower 

 end is moving vertically with greater velocity than the upper end, it will be 

 between the bottom and the centre of gravity — practically, it may be con- 

 sidered to be in the centre of gravity. 



The upper end will have passed the space s, and the lower end the vertical 

 space r ver z, the point p will have passed a vertical space «', and 



«' = «—/) (s—verz) = {l—p)s+pr\eTZ, 

 when p is a fraction expressing the distance of the aforesaid point from the 

 top, the length of the connecting-rod being unity ; inserting the value of », 

 and diiferentiating 



(I — p) sinz cos z 

 ds' •= rsiazdz + ~- . dz, 



(c^— sm=;)J 



and the vertical velocity of the pointy will be 



ds' 



rdz 

 dt 



(• 



(l—p) sin z cosz 

 (c2- sm'z)i 



) 



rdz 



Substituting v for -_—, differentiating and reducing by the " triangle of 



forces," we have 

 P„ = 



cvdV 



cti'cos; 



{c'—am''z)igrdz 32r {c'-sm' z)i 



i sin^ 2z + cos 2 j (c'— sin'j) 



""(1-^)-^ 32r(c=-sin".)^ 



which needs no further reduction, inasmuch as there is no vertical support 

 to the top end of the connecting-rod ; consequently, the whole of the in- 

 ertia or insita concentrated in the point p is sustained by the crank-pin. 



For the horizontal motion of the connecting-rod, the inertia is concen- 

 trated in the centre of gyration, and the space described horizontally by that 

 point will be jrsin z : differentiating and substituting, we have 

 V" = sJ'cosz; and ultimately we obtain 

 vdV" j'^ssinz _ „_ 



grdz 32 r 



which will need reducing, because (1 — s) "" will be supported by the end of 

 the beam laterally ; the remainder, j n", reduced into the direction of the 



sin z 

 length of the connecting-rod, by multipljing by ■ gives 



_ (_S g s'" -) _ p . therefore, for the connecting-rod we have P + P 

 32 re '' 



+ ci'il-p). 



}sin»2i 4- eos2z (c'-sin^z) 



(gpsinz)' 

 32 re 



32r(c=- sin=.-)4 ' "^" '''' 32 r (c=-sin»z)-^ 



Let W = the weight of the piston and rod and appendages ; M'' that of 

 the beam ; and W" that of the connecting-rod ;— then collecting the above 

 results, we have 



+ p=(w-H9"\r + w"j 



32r(e2-sin'r)4 

 I 8in'2 z + C0s2z (c' — sin'z) 

 32r(c>-Bin=-')'^ 



+ (w-l-s'W' + (l-;))W"); 



\V"(SVsmzy 

 32rc 



