398 J. H. ROHRS, ESQ. ON THE OSCILLATIONS OF A SUSPENSION CHAIN. 



determine L t , M , L and Af t , but I think the former would be found in practice the shorter 

 method. When the disturbance arises from a travelling load, we should assume for 



dV 

 ds 



a series 2 A n sin + 2C„ cos , 



\ 2 a I a 



„ „ s 2 . Mi (a 2 ia 2 I its 1 2tts \ 1 



for M , — a series — < — + — - - cos — + - cos &c. > l, 



c («,(». 'V \ a 4 a j) 



and f(s) would, as heretofore, be expanded in a series 2. sines , and we should have 



2 a 



to examine the two branches of the chain at the same time, the calculation would be lengthy, 



and the results of no more interest than those we have already obtained. Without going any 



way into the calculation ; we can however see, that the types of vibration of the 2.4 n series 



d 2 

 would be but slightly modified, for since the introduction of — L x s scarcely changes the types 



of A 2 , A 3 , &c, when A 2 , A 3 , &c. are the leading terms from their values 



3 7T / 5 7T / . 



— v eg — V eg , when , 

 2a 6 2a s dt 2 



and for all practical purposes produces no effect on the higher types of A b , &c. ; a slight 

 variation in i„ occasioned by the disturbance being not symmetrical, would make but little 

 difference. Consequently we may state roughly, that whether the oscillations be symmetrical 

 or unsymmetrical about the vertical axis, the time of a vibration arising from the causes 



we have considered, and which are expressible convergently in a 2^4„ sin series, 



will be nearly = 



4a 



(2m + 1) \/cg 



where n is a positive integer. 



J. H. ROHRS. 



INDEX TO PARAGRAPHS. 



1. Subject of Paper generally described. 



2. Property of /* expanded in a series 2.4„ 



. 2n+lws 



sin . 



a 



3. Equation to disturbed chain. 



4. & 5. Short methods of obtaining the types of 



vibration for a chain tied at its ends. 



6. Approximation for small terms. 



7. Disturbance in a chain, when M =O. 



8. Calculation of the effect produced by transit 



of loads and gusts of wind on a suspension 

 bridge. 



9 



10. 



Calculation of the effect of troops marching in 

 time along the platform. 



A chain disturbed by a periodic force (coin- 

 ciding in period with a possible type of 

 vibration) which acts for a long time, and 

 then ceases, will very approximately have 

 but one type of vibration the same as that 

 of the disturbing force. 



Approximation to second order. 



12. Recapitulation of results obtained. 



Addenda. 



11 



