﻿952 On the Analysis of Damped Vibrations. 



Subtracting (1) from (2), and 



R 1 -R = A (l + g) (6) 



Subtracting (4) from (2), and 



:E 1 -R 3 = A S(l-a 2 ). .... . (7) 



Subtracting (3) from (5), and 



R 4 -R2 = A S 2 (1-S 2 ). ..... (8) 



Dividing (8) by (7), and 



RT=R 3 - 6 (9) 



Substituting in (6), we have the first amplitude : 

 A e =(R 1 -B )/l-.+« 

 _ (Rj— R )(R 1 ~R 3 ) 



( Rl _R 2 _R 3 + R 4 )- • • • ^ 

 Dividing (7) by (6) and adding (2) and (3), we have 



2E=E 1 + R 2 - ( Bi _i£l£ + R4) , 



which gives the position of true zero. 



The solid friction term S follows from (1), and since c 2 

 is easily found statically, F the solid friction of the system 

 per unit mass follows by division. 



Where the system is dead-beat, the foregoing method does 

 not apply, and one way of solution is then by tuning of the 

 system by adding mass or increasing o 2 or both so that 

 sufficient equations are determined for elimination. 



The curve, of which the vibration in II. is a projection, is 

 an equiangular spiral with alternating origins distant 

 2F/c 2 apart, and it may be traced in either of two ways 

 according to circumstances. In the first place an arithmetic 

 spiral (see Phil. Mag., July 1922, p. 284) may be drawn 

 and the radii vectores shortened logarithmically, or an equi- 

 angular spiral may be drawn and portions taken out each sub- 

 tending ir, and such that the initial radius vector of one 

 portion is 2F/r 2 less than the final radius vector of the 

 preceding portion. Clearly the parts run smoothly together 

 on account of the equiangular property of the spiral. 

 Here it may be added that since the evolute of the arith- 

 metic spiral is a straight line 2F/c 2 long, the curve can 

 be drawn mechanically by coiling a fine thread round two 

 pins 2F/c 2 apart. The same curve is described by the hand 

 of the housewife in winding up a card of " mending." 



