64 Bronson — Transverse Vibrations of Helical Springs. 



and putting these values in (4) gives the equation in its final 

 form 



(7) n = ±i/ L - 8 ' 328 • ^ QQ X 98Q 



" y L — 0-1714 1-2834 



The values of n calculated by this formula are given in Table 

 III and are plotted in fig. 4. 



The results obtained with springs 2 and 6 were so similar to 

 those obtained with 1 that it is unnecessary to give them here. 



In order to study the behavior of a spring beyond the point 

 where the length and tension were linearly related, it was 

 necessary to modify the method of making the observations. 

 The series with decreasing lengths was in all cases omitted, 

 and several hours were left between successive sets of observa- 

 tions, so that the spring might recover as far as possible its 

 original condition. Table IV and ^. 5 give the results for 

 spring 3 obtained in this way. 







Table IV. (Spring 



3.) 







T. 



L. 



L. 



Observed n. 



Calculated n. 



Differences. 







7-74 



8 



12-45 





13-05 



-0-60 



10 



8-62 



9 



16-89 





17-09 



— 0-20 



20 



9-51 



10 



19-71 





19-71 



—o-oo 



30 



10-41 



11 



21-62 





21*61 



+ 0-01 



40 



11-30 



12 



23*05 





23-06 



—o-oi 



50 



12-20 



13 



24-21 





24-22 



—o-oi 



60 



13-10 



14 



25-15 





25-16 



—o-oi 



70 



14-00 



15 



25-96 





25-95 



+ 0-01 



80 



14-91 



16 



26-62 





26-62 



—o-oo 



90 



15.84 



17 . 



27-16 





27-19 



— 0-03 



100 



1675 



18 



27-64 





27-69 



■ —0-05 



110 



17-69 



19 



28-07 





28-13 



-0-06 



120 



18*62 



20 



28-41 





28-52 



— 0-11 



130 



19-59 



21 



28-73 





28-87 



— 0-14 



140 



20*50 



22 



28-98 





29-18 



-0-20 



150 



21-45 



23 



39"19 





29-46 



— 0-27 







24 



29-30 





29-71 



— 0-41 



Data 



used in 



the calculations for spring 



3: 







M 



= 2-210 





m 



= 11-04 







L, 



= 10 





n 1 



= 19*71 







L 2 



= 16 





n i 



= 26-62 







y 



= —0-333 





x 1 



= —76-55 





In fig. 5 it is seen that the difference between the observed 

 and calculated values of n becomes greater the further the 

 length-tension curve departs from a straight line. This is just 

 what would be expected, if springs obey the same law as vibra- 



