15 - 



499 



T d S + 



c 



T- d S. 



T S + T, S, 

 CO 11 



-2 ^0 *1 "' - (^0 - ^) Sx (") 



Alsc from (12) and (26) the kinetic energy at any time is: 

 f . i 2 



(63) 



while fron (17) the work done by the applied load is 



P(t) d w 



The energy equation is thus 



s n 



\ph 



A w2 + _L 

 2 /. 



-2^0*l^'-(^0-^l)=l 



"{t) d w 



(6t) 



This equation can also be obtained from the equations (21) and (27) by multiplying by W 

 and S./*, respectively, adding and integrating, and using equation (62). 



A.io Introduction of assumption that (T - T )/T is small. 



Now in either Type I or Type II motion we have T' = T in (27) and multiplying this equation 

 by S and integrating we obtain: 



4phS/ = A3 



f^ 



(T-T^)dS^ 



and since S > 0, T ^ T while initially S = S = we therefore obtain 



4 -^ PK S^' i A3 (T^ - T^) S^ 



(65) 



by use of (29). But the energy equation (64) can Be written with regrouped terms In the form: 

 i2 ^ 1 



5 PhAjW2.^lT^A^w2 (1-S) = A^ 



P{t) d w 



S = -i 



s 



,- _£iil 



^*3 t^o - ^' ^ 



• T. 



(65a) 



and we have shown that S^^ > S^ > throughout the motion so that S=S +s^S 50 and therefore, 

 using (65), we must h^ve: 



<: S ^ 'I2 — Li 



T. 



(65b) 



It therefore follows that If we new assumei 



T - T 



-2 1 small 



^0 



<66) 



then, neglecting terms jf this order in (65a), we obtain: 



P(t) d w 



u2 , 1 



f ph A^ W^ . ^ T^ A^ W' = ^ 



(67) 



for 



