140 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1954 



eter defining a specific case is tM/tE , or the corresponding value of K 

 {2CLtMll\:)- For the two selected values of Cl , solutions were obtained 

 for the foUowhig values of K: 0.05, 0.1, 0.3, 1.0 and 10. 



The computer set-up for this problem is indicated in Fig. 9 of the 

 article by E. Lakatos cited above.^ Solutions were obtained to (27) for 

 / = 1 and the values of Cl and K cited above, for the given boundary 

 conditions. The solutions were obtained in the form of machine drawn 

 curves giving, u versus r and du/dr versus r, over the range < ii < 3. 

 In accordance with the character of the net accelerating force, all these 

 curves were smooth and monotonic. 



For the conditions covered by the solution, these results give the mo- 

 tion time for a given travel directly in the form of the relation between 

 r and u. It is convenient to express the values of r in terms of the cor- 

 responding values of t/tjn ■ The results for Cl = 5 are shown in Fig. 10 

 as curves of t/tu versus u for various values of tu/tE . 



The other result of interest is the ratio of the magnetic drag W to 

 the spring load energy F, or of {V — T)/V, where T is the kinetic 

 energy at the end of a given travel: the total travel in an actual case. 

 T equals 7n{dx/dt)y2, or 2m{CLA(Ro-du/dTf/tl . For / = 1, V equals 

 FqX, or FouA(Ro ■ It follows that T/V is given by: 



T ^ mA(Ro (2Cjy fdu 

 V ~ 2FoU \ Ie / \dT 



and therefore by: 



T _ K^ (du 

 V ~ 2u [dr 



(28) 



Corresponding values of u and du/dr were read from the computer 

 results and substituted in (28) to determine T/V and thus W/V. The 

 resulting values of W/V for the case Cl = 5 are shown in Fig. 11 plotted 

 against tm/tE for various values of u. 



DISCUSSION OF COMPUTER RESULTS 



The significance of the results shown in Figs. 10 and 11 can be more 

 readily grasped by reference to representative values of the parameters 

 involved. For relay electromagnets, representative values of (Ro and A 

 are 0.04 cm~^ and 1 cm^, respectively, for which ASio = 0.04 cm, or 16 

 mil-in. This distance is, for this case, the travel for which u = 1. For 

 this value oi A(Ro , an effective mass ni of 10 gm, and an operated load 

 Fo oi 2 X 10^ dynes (200 gm wt), Im = 2 millisec. If the load were 



