382 BELL SYSTEM TECHNICAL JOURNAL 



inertia about these are equal, .1 and B are equal. By virtue of the frictionless ^^ 

 bearings the external torques L and X about 1 and 3 are zero. 

 Introducing these relations we have 



A^-^+ (C - .Oco^co, = 0, (la) 



at 



A^ - (C - .Ocoico;, = M, (2a) 



dt 



C^^ = 0. (3a) 



dt 



From (3a) the velocity of spin w-i remains constant. The torque M about y 

 is then to be found from (la) and (2a). For very small displacements, 



C02 = (p. 



Putting this in (la) and integrating from zero to /, assuming tp to be zero 

 at / = 0, gives 



(2a) then becomes 



C - A 



COi = ~ 0)z(p. 



.... (c - aY o 



A(p -\- ~ 053^9 = M. 



This represents an angular inertia A and stiffness -. The system 



will therefore resonate at a frequency . If the frequencies in- 

 volved in the variation of ^ are small compared with this, the inertia torque 

 will be negligible, and the system will behave as a stiffness. If the displace- 

 ments about A associated with wi are very small the restoring torque M 

 will act substantially about the y a.xis. That is, the lattice will encounter a 

 stiffness to rotation. 



Since the large number of gyrostats in an element of the model are oriented 

 in all directions, an angular displacement of the lattice about y will gen- 

 erally not be about the B a.xis for each gyrostat. If it makes an angle a with 

 this a.xis, then only the component (p cos a of the angular displacement will 

 be transmitted to the rotor. The resulting torque will then be S cos a, where 



S = 



iC - Afo^l 

 A 



It will be directed about B and so will not be parallel to the applied dis- 

 placement. However, if a second gyrostat has the position which the first 



