ESTIMATION AND CONTKOL OF OPERATE TEME OE liELAYS 129 



<fx I'Rtc 



df 2m6iiAi 



((1 - (1 - v^)e-r - vl). 



Substituting {tc + /*;) z for t, this equation may be integrated with 

 respect to t to obtain the following expressions for the velocity x and 

 the travel x in the second stage: 



/ Rtcitc -\- Ie) r f \ fnn\ 



■"= 2mai.A. -^'fa'^^- (22) 



/ Rtcytc ~\~ iEj J- r \ /ooN 



The functions /i(yi , z) and/2(i^i , z) are the integrals with respect to z of 

 the bracketed term in the preceding equation. These functions can be 

 evaluated from the ciu'ves of Fig. 6, which shows them plotted against 

 2 for several values of Vi . 



For a specific case, the values of i^i and ti will have been determined 

 from the relations for the first stage. Let X2 be the travel at the end of the 

 second stage. Then the value of /2(i'i , z) for.-c = X2 is given by (23), and 

 22 , the corresponding value of z, can be read from the curve of Fig. 6 

 for the value of vi applying. Then the time for the second stage, ^2 , is 

 given by Z2{tc + ts) ■ For 2 = Z2 , (20) gives 1^2 , the value of v at the end of 

 the second stage. This in turn gives the corresponding value of the flux 

 t'2^1 , and, from (21), the pull F at the end of this stage. The velocity at 

 the end of this stage is given by (22), with/i(?;i , z) read from the curves 

 of Fig. 6 for z = Z2 . 



THIRD STAGE 



The total operate time is U -{- (2 -{- tz , where ^3 is the time for the third 

 stage. In the relay case, this is the stagger time between the operation 

 of the first and last contacts. A first, and frequently adequate approxi- 

 mation to ts , is given by assuming the velocity in the third stage to be 

 constant at the value attained at the end of the second stage. For a 

 more exact determination, particularly in determining the final velocity 

 for complete operation, it may be assumed that the flux in the third 

 stage is constant at the value v^ipi attained at the end of the second stage. 

 The mechanical output V + T, in the third stage is then given by 

 equation (11), withy taken as V2 , and Xi and X2 replaced, respectively, by 

 X2 and .rs , the latter denoting the travel at the end of the third stage 

 (zero for complete operation). Knowing the spring load V between X2 

 and .Ts , ^3 and the final velocity can be computed from the increase in 

 kinetic energy T on the assumption of uniform acceleration. 



