Oscillations of Chains. 739 



(C) Oscillations of a uniform chain loaded at the fee end. 



The Bessel functions J (z), Ji(s) 5 and J 2 (z), and the 

 Neumann functions Y {z), Yi{z), and Y 2 (V) appear in the 

 expression for the times of vibration of a loaded chain*. If 

 the load attached to the lowest point of the chain be n times 

 the mass of the chain, the periods of oscillation in the different 

 modes can be found from the roots of the equation 



(2) 



YqCAxO cY ( g )-2Y 1 ( g ) _¥,(*) 

 Jo(A^) "rJ (f)-2J 1 (^) ~J S (*V ' 



where z = (4tt/t,)(>V<7)^ *■= [(n + l)/w]* s 



and T,= time of complete vibration in the 5th mode. 



A bicycle-chain about 150 cms. long was suspended by 

 one extremity and a load was attached to the other. Through 

 two openings in the lower end of the lowest link of the 

 chain, a steel rod was passed which supported a number of 

 perforated iron disks about J cm. thick. The radii of the 

 disks varied from 1 cm, to 4 cms. The load could by this 

 means be made any multiple or submultiple of the mass of 

 the chain. 



Load equal to or greater than the mass of the chain. 



Mode of 

 vibration. 



Value 

 of n. 



Length of 



chain. 



cms. 



Time of vibration 



sees. 



Calculated. 



Time of vibration 



sees. 



Observed. 



1 

 2 

 1 

 2 



1 

 1 



219-9 

 1714 



2-812 

 •730 



2-484 

 •645 



2-809 

 •725 



2-472 

 •638 



1 

 o 



2 



148-8 



2-370 

 •475 



2-361 

 •472 



1 



2 



5 



148-8 



2410 

 •326 



2-410 

 •325 



1 

 9 



w 



10 



118-8 



2-425 

 •237 



2-425 

 •236 



* Routh, ' Advanced Rigid Dynamics,' 1905, p. 405. 



