470 



KENNELLY AND AFFEL. 



(17). This term is a damped, or logarithmically decaying, periodic 

 velocity, effected not at the angular velocity co of impressed force, 

 but at the free angular velocity of the system co^ = Vcoo^ — A^ = 

 ojo sin7 radians per second, unless A is larger than coo, in which case the 

 second term of (17) is an ultraperiodic velocity, which may be repre- 

 sented by the projection of a uniformly damped uniform angular 

 velocity in a right hyperbola. -^^ For the particular case, cuo = A, 

 or the aperiodic case, the motion may be represented by the projec- 

 tion of uniformly damped uniform angular velocity in a parabola. In 

 every case, therefore, the second term of (17) corresponds to the pro- 

 jection of uniformly damped uniform angular velocity in a circle, 

 hyperbola or parabola, i. e., a conic section. In the cases of the tele- 

 phone-receiver diaphragms examined experimentally, A was much 

 less than coo; so that only the periodic interpretation of damped uni- 

 form angular velocity in a circle needs here to be considered. 



Damped Free-Vibration Vector Diagram. 



In Figure 30, let a particle at p, of mass m, be attracted cen- 

 tripetally toward the center O, with a force varying directly as the 

 displacement x, and be subjected to a frictional retarding force, oppo- 



Q 



Fig. 30. Fig. 31. 



Locus OF Damped Oscillatory Displacement, Velocity and 

 Acceleration Force. ' 



site to the instantaneous velocity; as well as to an inertia force opposite 

 to the acceleration. Then because there is no impressed force to give 

 energy to the particle, energy will continually be absorbed from it, and 



19 Bibliography No. 2 and No. 10. 



