174 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1952 



will be to minimize rebound motion at the front, since this is usually 

 near the point actuating the relay contacts. 



The basic problem is then to find the response of the armature subject 

 to aperiodic but well defined impulses, which are functions of the 

 positions and velocities of the system. 



III. ASSUMPTIONS 



In order to facilitate the solution of this problem, the following- 

 modifying assumptions are made: 



(1) As mentioned in the previous section, analysis is restricted to 

 planar motion. 



(2) The armature is assumed to be a rigid body. 



(3) Stops are assumed to be very stiff, massless springs capable of 

 energy absorption during impact with the armature. The associated 

 coefficient of restitution is assumed constant. Core and stop vibration 

 are neglected. 



(4) The tensioning forces Fi , F2 , F3 are assumed to be constant forces. 

 (This is fairly closely true for moderate rebound amplitudes of practical 

 relay structures.) 



(5) All displacements are small relative to the dimensions of the 

 system and in particular the angular displacement 6 is sufficiently small 

 so that 



cos 6 = 1 



smd = 6 



IV. DERIVATION OF EQUATIONS OF MOTION 



The derivation of the equations of motion resolves itself into the 

 solution of two different types of intervals: 



(1) Free Interval: This is the period during which the armature is 

 not in contact with any of its stops and only the tensioning forces are 

 acting. 



(2) Impact Interval: During such intervals the armature is in contact 

 with at least one of the stops. The stiffness of the latter is assumed so 

 high that the tensioning forces during this interval may be neglected. 



The three-degree-of-freedom case will be considered first and the 

 others subsequently deduced from it by allowing some of the constants 

 to approach zero. 



