468 kennelly and affel. 



Solution in Terms of Velocity. 



The solution of the above differential equation, with reference to .r, 

 is 



x= / j^^^^-M+jH-^-^^ ^ / (17) 



, ./ 5A sec 



where Xi is an initial vector velocity, and j\ = r/{2m) a numeric, 

 the damping constant per second. 



The first term on the right hand of this equation indicates the condi- 

 tions for the steady state of motion, while the second term is the 

 transient term; i. e., comes into action only during changes of motion 

 in the system, and when/ is either varied or withdrawn, the coefficients 

 m, r, and s, remaining constant. We may, therefore, at present con- 

 sider only constant impressed vmf. and ignore the second or transient 

 term. The formula (17) then reduces to 



X = ^^- = ■'- — Z (18) 



2 z sec 



where z is the mechanical impedance at the impressed and sustained 

 angular velocity co. It is evident from Figure 28, that as w varies, the 

 vector impedance OZ travels over the straight line y'X.y^. It is well 

 known that the reciprocal of a variable having a straight-line locus is 

 a variable with a circular locus, so that as co varies from zero to in- 

 finity, the vector locus of x will be a circle OXP, Figure 29, with its 

 diameter OX on the axis of reals, and equal in magnitude to F/r cms. 

 per second. That is, the linear velocity of a simple vibratory sys- 

 tem, having a retarding force directly proportional to x, follows a cir- 

 cular locus in regard to magnitude and phase, coming into phase with 

 the impressed force at the resonant angular velocity coo of reactive 

 equilibrium. 



Certain relations between the fundamental constants m, r, and s 

 of the vibrator may be determined from the observed distribution of 

 angular velocities around the velocity circle. Thus, the angular 

 velocity of resonance coo is found at the point where the diameter 

 intersects the circle. Since, at resonance, the two reactive forces 



— = Tncoo, we have 



COo 



radians . 



<^o = Vs/m ~^^^ (19) 



