Bronson — Transverse Vibrations of Helical Springs. 63 



(2) n = \ K /ll^ 



W V L M 



where M is the total mass of the spring. If T is now elimi- 

 nated between (1) and (2) the relation between L and n takes the 

 form 



(3) n = i t / mh + x JL 



The values of n calculated by this formula differed consider- 

 ably from the observed values, as might have been expected, 

 for in the first place the values used for T entirely ignored the 

 weight of the spring and were therefore always too small, and 

 in the second place the length of the spring was taken as the 

 distance between the two blocks A and B (fig. 1), which was 

 evidently somewhat greater than the true vibrating length. 

 In order to avoid these two difficulties, another constant was 

 added to equation (3). Two equations similar to (3), but with 

 this added constant, were then used to solve for the unknown 

 quantities x 1 and y. 



(4). y w 



9 

 M 



^ V L^ + y M 



Solving for y and x 1 gives 



(5) v ._ y.-y. + K.-g, 



(6) a .. = ^(K 1 +4 Wl V-N 1 ) 



where JT, = ^L, ; N 2 = ^L, ; K, = 4n? L, ; K, = 4*,' L 2 . 



The value of m is always taken as the slope of the straight 

 part of the length-tension curve ; L„ n^ L 2 and n^ are corre- 

 sponding values of L and n taken from the length-frequency 

 curve. The values of the constants for spring 1 are as follows : 



M= 1-2834 m— 7'00 



Lj= 12 n x = 20-37 



L 2 = 17 n 2 = 26-24 



Substituting these values in (5) and (6) gives 



y = — 0-1714 

 x l = —58-29 



