E = M^gy - 



PgBA^S^ 



K^-p) 



2„2 



S^0 



COS 2at + constant. 



(12) 



Thus, the variation of E with time is proportional to cos 2at, 

 a simple form made possible by the prior choice of the rate of 

 rotation of the active coionterweight as twice a, in anticipa- 

 tion of the identities in equations (9) and (10) • 



Equation (1) is then satisfied, and the power input is 

 minimized if the coefficient of cos 2at in equation (12) is 

 made to vanish, i.e., if the active counterweight is set to 

 satisfy 



M^gy 



pgpA^S' 



K^-p) 



2^2 



S^a 



This expression is simplified by introducing y^, the maximum 

 y required when a = and when S = Sm, the maximum available 

 amplitude, that is. 



(13) 



p3A S2 

 cm 



2M 



(14) 



and by introducing Oq, the resonance frequency when y = 0, 

 that is. 



PgBA, 



/(^-p) 



Equation (13) can then be written as 



^ = 



Si 



s , 



(-S). 



where Tq/T has replaced o/oq. y is positive or negative as 

 T is greater or less than Tq. 



In sximmary, for any given S and T, the tunnel operates at 

 resonance when y is given by equation (16). The constants y^j, 

 Sjh, and Tq are determined by equations (7), CH), (14), and (15) 

 and by the dimensions and masses of the components of the tunnel 

 (see Table). 



(15) 



(16) 



16 



