502 



- 18 - 



Thus the curve of 3 against t will be a half sine c^ime as shown in Figure u, the motion 

 ceasing when 3 Decomes ztro at time t = vlzcx. with no further recommencement since °(t) = 

 thereafter. 



The geometrical criteria of the preceding section are now easily aoplied since we note from 

 (78) that the slope S decreases steadily from Aj V^ at t = to - A^ V" at 2 a t = tt. we thus hsve 

 the following cases: 



(a) A^ V^ 4 f. 



In this case criterion A.li(3) applies and the motion is of Type II throughout 

 so that the S. curve is identical with tne S curve. 



(b) A^ v^ > Y . 



In this case criterion A.ll(u) applies and the motion wilt then commence with 

 Type I, the S, curve then being the line OB of slope Y as illustrated. This type 

 ceases at the point B corresponding to time t given Oy: 



Y t^ = -J A^ a w^^ Sin 2 a t 



(79) 



ana the remaind;'r of the motion is of Type II since the slope of S curve dees not 

 again increase tc become > Y. The 3, is thus composed of the line OB and the curve 

 BC. 



If wfc now define w as the borderline value of W corresponding to A V^ = Y, then from (75), 

 (7!i) and (U6): 



tl^2 \]o_-_\'^ 



(80) 



1 f we further put: 



^1 = - 'l 



(81) 



then from (79) and (80) : 



20, 



W^2 Sin 2(Sj 



(82) 



For the rtrtximum values S and S' we then have from (70), (76) and Figure 4: 

 m m \ » , V # a 



i A. W ' 

 2 1m 



S* = S for W. < W. 



m m m 1 



^'m = 7 '' ^^ + S„,- 7 '', ''n,^ Sin' a t, for W > W, 

 milm2im 1 m 1 



(83) 



"m 2 1 m 

 Then using (79) the ratio S'^/S is given in terms of W^/**, by the following equations: 



-_!! = 1 for J? < 1 (84) 



m 1 



i (l + Ccs 20^*6^ Sin 2(9j) 



id. 



W^^ Sin 26^ 



W 



O^0j<7I-/2 



(85) 



LSingS as a parameter, we then obtain the values given in Table 2 and plotting in Figure 2. 



A.I3 



