TELEPHONE DIAPHRAGMS. 469 



This gives one relation between s and m, in terms of the observed 

 angular velocity of resonance. Moreover, it can easily be shown that 

 the damping factor 



r (iP- — ui^ , . 



A = -— = per second (20) 



2m 2cotana *^ ^ 



where co is an angular velocity on the velocity circle, at a point making 

 an angle a with the diameter. If we select the angular velocity coi 

 at the upper quadrantal point on the circle, for which the impedance 

 angle a = —45° in Figure 28, the above formula becomes 



2 2 



A = — per second (21) 



2coi 



Similarly, the angular velocity 002 at the lower quadrantal point for 

 which a = + 45° gives 



A = — ;r per second (22) 



2aj2 



Summing the last two equations, we obtain 



A = — - — per second (23) 



so that the damping factor is half the difference between the angular 

 velocities at the quadrantal points. Formulas (20) to (23), or any of 

 them, furnish one relation between r and m, in terms of angular veloci- 

 ties at observed points on the velocity circle. Some third relation is, 

 however, required in order to evaluate m, r, and s. 



Transient Motion. 



We have seen that the second term of (17) becomes involved at 

 any sudden change or discontinuity in the impressed force /, except 

 in the case when the impressed force / happens to have the resonant 

 angular velocity coq. In such a case there is no disturbance in phase 

 relations; but there will be an instantaneous change in the lengths of 

 the vectors OC and OD, Fig. 25, the vectors OA and OB remaining 

 in equilibrium. It seems that in all other cases, a change in the im- 

 pressed force / must be accompanied by a transient change in the 

 motion of the system, due to the introduction of the second term in 



