1843.] 



THE CIVIL ENGINEER AND ARCHITECT'S JOURNAL. 



307 



THE RELATIVE EFFICIENCY OF LONG AND SHORT 



CONNECTING RODS, CONSIDERED IN THE EXPOSITION 



OF CRANK AND CONNECTING ROD MOTION. 



By H. F. Clifford. 



The subject of the following paper is one of great importance to the 

 practical mechanic; and as we have never yet seen any satisfactory 

 solution of this long disputed question, we have endeavoured to draw 

 such conclusions from the investigation of the theory of crank and 

 connecting rod motion as we feel convinced will set at rest all pre- 

 vious doubts concerning the comparative advantages of long and 

 short connecting rods. With a given force in the direction of the 

 piston, our object in the present investigation is to ascertain the best 

 means of obtaining the greatest amount of that force in the direction 

 of rotation, and thus render friction as small as possible; in other 

 words, -whether is a short or long connecting rod the more effectual 

 method for fulfilling the conditions of our proposed inquiry? 



We shall first show that if F be the original force exerted by the 

 piston rod of a steam engine along its own axis, that there is more of 

 that force transmitted in the direction of the long rod than the short 

 one. 



Let P Q, Fig. 1, be a cylinder with piston, and let the piston rod 

 exert a force F, at the point B, where the rods B A, B D, connect it 

 respectively to the cranks A C, D E. 



Fig. 1. 



Let A B make an angle (<jj) with the horizon. 



BD .. (0) 



„. , , . f „ | in the direction of A B =: F cos tp. 



The resolved part of F j _ B D = F cos l 



Now the magnitude of the force in either direction depends on the 

 cosines of the angles, which the respective rods A B, B D, make with 

 the horizon, and since cos tp and cos become a maximum when <p 

 and = o, it follows the smaller the angle the larger the numerical 

 value of the cosine, and since is considerably less than tp, the long 

 rod has evidently more of the resolved part of F in its direction than 

 the short one. 



Again — 



mi , , . ,„ (in the direction of A C = F sin *. 



The resolved part of F ', ,-. „ „ . T 



1 | . . D E = F sin 0. 



The magnitude of the forces in this case depends upon the sines 

 of the angles, and since the larger the angle the greater the sine, the 

 resolved part of F in A* C, is much greater than that in D E ; in other 

 words, the pressure into the centre, or friction in the axle, is more in 

 the short than in the long rod. 



Having proved then that there is more force in the direction of the 

 long rod, we now proceed to show that the resolved part of the force in 

 B A, and B D, in the direction of rotation, is greater in the case of 

 the long than the short rod; or, 



Proposition. To find the part of the force exerted by the piston 

 rod of a steam engine which is perpendicular to the crank, in any 

 given position of the fly. 



Let E A, Fig. 2, be the piston rod, the direction of which produced 

 is supposed to pass through the centre of the fly C; A B, the con- 

 necting rod ; C B, the crank : B T D, the tangent at B. 



Let C B = a, A B = b, A C B = 8, B D — x, A D = z. 



L A B D = tp, Z the tangent makes with connecting rod 

 AB. 



The resolved part of F in B A = F cos B A C. 



= F cos (A D B + A B D.) 



= F cos (<p -f- - — 0) 



* i 



= F sin (0 — tp) 

 —f suppose. 

 The resolved part of F in B D =/cos tp. 



= F cos <p sin (0 — tp) 



Fig. 2. 



Investigating equation A, it is evident that the smaller the angle <p the 

 greater the force in B D, and the longer the rod A B, the smaller the 

 angle it makes with the tangent, and thus we have more of the re- 

 solved part of the force in B A, in the direction of rotation in the 

 long rod, a fortiori, how much more have we of the original force F, 

 transmitted by the piston in the direction of rotation in the case of 

 the long than the short rod. 



There is, however, a small arc in the crank's orbit, in which the 

 short rod possesses an advantage, for let E B, F B, be respectively a 

 long and short connecting rod, A G H, the fly, G C H, perpendicular to 

 A E, and let the crank (in Fig. 3) be in such a position that B T, the 

 tangent, bisects the L between the rods; let D be the point correspond- 



* The solution of this equation is rather intricate, as we have to express 

 the value of F in B D, in terms of the known quantities 6, a, and b ; but, as 

 it is desirable to know the result, we give the working. 

 b- + .r- — ~ 



NOW COS tp 



2bx 



B D = B C tan = a tan 0. 

 lin (p sin tp sin <p. 



cos 0. 



1. 



lin ( 2" 



z = b 



sin tp 



= b sin. tp sec. 



cos 9 



By substitution in equation 1, we have 



b- + a- tan. 2 — b- sin. 2 tp sec. 2 



Cos. i 



2a4 tan. 



2 ab tan. cos. tp = b 2 + a 2 tan. 2 — b 2 sec. 2 (1 — cos. 2 tp.) 

 .'. b- cos. 17 tp sec. 2 — lab tan. cos. tp = (A 2 — a 2 ) tan. 2 0. 



Dividing by b- sec. 2 and putting - 



we have 



Cos. 2 tp — 1m sin. 2 cos. tp = (1 — 4 m~) sin. 2 

 Completing the quadratic "| 

 and extracting the root j" 



^cos. tp — m sin. 2 0= x sin. -J 1 — 4 m- sin. 



Now the positive sign must be taken in order that cos. <p may he always 

 possible. 



.'. Cos. tp = m sin. 2 4- sin. V 1 — 4 m 2 sin. 2 0, 



And sin.<£ = V cos. 2 0—4 m sin. 2 (m cos. 2 + cos. 0) V 1 — 1 m a sin. 2 0. 



Let X = the resolved part of f, in B E, 

 then from equation A, X = F cos. tp sin. (0 — tp.) 



= F sin. cos. 2 tp — F cos. sin. tp cos. tp. 



Putting for sin. tp and cos. tp their respective values, X = F sin. 3 1 1 + 4 m 



(m cos. 2 4- cos. a/ 1 — 4 m 2 sin. 2 0) l — F cos. sin. (2 m cos. + 



*/l — 4 m 2 sin. 2 0) V cos. 2 — 4 m sin. 3 0(mcos.2 + cos. ■/ 1 — 4 m 3 sin. 2 0, 

 which is the required equation. 



42a 



