466 KENNELLY AND AFFEL. 



It is well known that the forces in the simple rectilinear vibratory 

 motion of a particle about a position of rest may be regarded as the 

 projection, upon a line through this position, of the corresponding 

 rotating system in a plane. The vibratory motions and forces of a 

 particle subjected to (1) a simple vibratory resilient force —5.r dynes, 

 (2) a motional retarding force —rx dynes, and (3) an inertia force 

 — vix dynes, together with (4) a simple harmonic impressed force main- 

 taining the motion, may therefore be considered as the projection of a 

 vector system like that in Figure 25, on the reference line COD, when 

 that system rotates counter-clockwise about the center O, with angu- 

 lar velocity co radians per second. The instantaneous projection of 

 OB will then be the inertia force opposing acceleration of the particle, 

 for the instant considered, that of OA the instantaneous resilient force, 

 that of OC the instantaneous frictional retarding force, and that of OD 

 the impressed vibratory force, or vibromotive force (vmf). The instan- 

 taneous velocity will be the projection of OD, when divided by r. 

 The instantaneous displacement will be the projection of OA reversed, 

 when divided by s. The instantaneous acceleration will be the pro- 

 jection of OB reversed, divided by in. 



Consequently, for the case indicated in Figure 25, with reactive equi- 

 librium, i. e., equality between the opposing reactive forces s.r/co 

 and wz^i, the vibrational velocity .r, will be in phase with the im- 

 pressed vmf. OD; while the vibrational displacement x will be 90° 

 retarded behind the impressed vmf. 



At the impressed angular velocity less than coo, of reactive equili- 

 brium, the reactive force (1) of resilience Jsx/cj: dynes, will be greater 

 than the reactive force (3) of inertia —jmoix dynes. Such a case is 

 indicated in Figure 26; where OAi, the resilient force, exceeds OBi 

 the inertia force. Their difference is Ox\i\ the resultant reactive force. 

 The impressed force F = ODi must now equilibrate the resultant of 

 OAi^ and the motional retarding or frictional force OCi. It will then 

 be seen that, in the steady state, the velocity .i: leads the impressed 

 force by an angle au The displacement, in line with OBi, is now less 

 than 90° behind the impressed force. 



At any impressed angular velocity greater than that of reactive 

 equilibrium coo, the inertia force will overcome the resilient force. 

 This is the case represented in Figure 27; where the vector inertia force 

 OBo exceeds the vector resilient force OA2. Their difference 0B2\ 

 is the resultant reactive force, which, combined with the frictional 

 force OC2, gives the resultant force Oc/o to be overcome, or to be 

 equilibrated by the impressed vector force F = OD2, which now leads 



