ESTIMATION AND CONTROL OF OPERATK TIME OF RELAYS 139 



On substitutiii<j; this in the i)i'0(>odinj2; oquation, there is obtained: 



a+„)(c„+»)g-™;;;^ 



To reduce this expression to dimensionless form, the following sub- 

 stitutions are made: 



/' is wi'itten for — - / Fo , where Fo is the operated load, so that / 

 ax I 



= 1 at the start of the release motion. 

 T is written for 2Cz, tjtE , expressing the time i as a multiple of IeI i2Ci) ■ 

 til is written for 2mA(Sio/Fo. Thus Im is the time for travel of the mass 



through the distance A(Ro for a constant accelerating force i^o . 

 K is written for 2Cz, ImI^e- 

 The preceding equation then ])ecomes: 



(l + w)(C. + ^0(/-/v^ 



H.^((C, + .r(/-K^^)) = 0. 



(27) 



This is the form of the release motion equation to which analogue 

 computer solutions were obtained. It is a third order differential equa- 

 tion giving the travel, expressed as a multiple u of ^(Ro , as a function 

 of the time expressed as a multiple r of tsji'lCi). The boundary condi- 

 tions correspond to zero initial values of travel, velocity, and accelera- 

 tion, so that for r = 0, u = du/d.T = d u/dr = 0. 



ANALOGUE COMPUTER SOLUTION 



The analogTie computer gives solutions to specific cases, and a specific 

 case of (27) is defined by the values of C l and tM/lE applying, and by 

 the form of the relation between /and u, defining the shape of the spring 

 load. To confine the cases considered to a manageable category, they 

 were limited to those for which ./' = 1 for all values of u: the case of a 

 constant spring load. 



The constant Cl , the ratio (olo + <iiL)/o'ln , is an inverse measure of the 

 relative magnitude of the leakage field. As noted in Section 2, its value 

 is usually in excess of 4. Solutions were obtained for two cases: Cl = 3 

 and Cl = o. For/ = 1, and a fixed value of d. , the remaining param- 



