MOMENT OF ROTATION PARALLEL FORCES. [CHAP. V 



torsiou P x R. Laying off C'C^ = 5 &', = m n, we have the 

 torsion rectangle C t b' b C'. 



Now the combined moment of torsion M,; and bending M b ia 

 We make then C' C equal to f C' C t = 



m n and C C/2 equal to f C C' = M b , and draw C 2 b. Then 

 ;iny segment of any ordinate, ssff^ is|- ofjf/". Revolve now 

 C' C with C' as a centre, round to C' and join C' C 2 . Then 

 C/2 C' is equal to f \/&tf> -f-Mt an( i therefore with C 2 as centre 

 revolving C 2 C' to C 3 , we find the point C 8 , C C 3 being equal to 



f M b +$ \/Ml -f M^. In the same way we find any other 

 point as^/g, by laying d&.f'f\ equal tof'f , joining ^/j and /"o and 

 making ^ ,/s equal to ^ ^. The line C 8 f% b thus found is a 

 hyperbola, and the ordinates between it and b C give the com- 

 bined moments [for pole distance O K\ at any point. 



[NOTE. We suppose the axle to turn freely at A, and the working point 

 or resistance beyond B ; hence the moments left of the wheel are given by 

 the ordinates to a O.] 



5XH. APPLICATION TO CRANK AND AXLE. 



The above finds special and important application in the case 

 of the crank and axle. 



Thus in PI. 8, Fig. 29, let E D C B be the centre line of crank 

 and shaft. Lay off a P equal to the force P acting at A, choose 

 a pole o and draw o a o P and the parallels o a and a E. Join 

 E and d and draw o P t parallel to E d. Then P P t is the 

 downward force at E and P x a the upward reaction at D. The 

 ordinates to E d a to pole distance o P, give the bending mo- 

 ments for the shaft. Make a F equal to the lever arm R, then 

 i? G is the moment P R, and we unite this as above with the 

 bending moments and thus find the curves c' d' e' the ordinates 

 to which give the combined moments at every point of the 

 shaft [see 4th]. 



For the arm B C, make the angle a B C equal to D a /, 

 and then the horizontal ordinates to a B give the bending mo- 

 ments for the arm. Make C c equal to C c and we have the 

 torsion rectangle C CQ b B, and as in the previous case we unite 

 the two and thus find the curve b h F, the horizontal ordinates 

 to which from B C give the required combined moments, to 



* Der Constructeur, Reuleaux, p. 52, Art. 18. 



