XV. On the Oscillations of a Suspension Chain. By J. H. Rohrs, M.A., late 



Fellow of Jesus College, Cambridge. 



[Read Dec. 8, 1851.] 



1. In the following paper I propose to determine the tension and other circumstances 

 of motion at any point of a disturbed chain ; principally with a view to exhibit the causes 

 of fracture by vibration in suspension bridges, and to suggest means for obviating as much 

 as possible the danger arising from such vibrations. I shall consider the curvature of the 

 chain to be but small, as it is in practice, as also the extensibility of the material to be zero. 

 The linear expansion of wrought iron bars being only f^^th for a strain of a ton to the square 

 inch, would introduce additional terms into the equation for the motion of inextensible chains 

 too minute to be important. To apply the results of my investigation to the case of a suspen- 

 sion bridge, I shall suppose the platform perfectly flexible, and the weight of the platform and 

 load uniformly distributed throughout the chain. I shall likewise suppose the links of the 

 chain to be so numerous, that it may be considered simply as a flexible material line ; of course 

 these conditions are never entirely fulfilled in suspension chains ; but the object I propose to 

 myself is not so much to calculate exactly the amount of strain, to which any one particular 

 bridge is liable, but how far the suspension principle (supposing it could be perfectly carried 

 out) would give rise to strains from vibration, and hence to infer what we might expect would 

 take place, during the oscillations of bridges constructed mainly or partially on that principle. 

 Before I enter upon the analysis of the question, I shall prove a property of f(s) when 

 expanded by Fourier's method in a series of cosines of the form 



^C-^t)}- 



The mode of proving this property, and the propriety of doing so, were suggested to me by 

 the paper of Professor Stokes on the " Critical Values of the Sums of Periodic series." I 

 may observe at once, that the effect of the majority of the practical causes of disturbance in 

 suspension chains, such as a moving load, variable pressure of wind sweeping rapidly across the 

 platform, and the tramping of columns of troops moving across the bridge, are all easily and 

 rapidly expressed by series of sines 



2..A, 



. /2n + 1 tts\ 



• Sin h-Tj' 



n being any positive integer; at all events by far the greater part of the effect of such 

 disturbing causes can be most easily exhibited in a series of this description, as we shall 

 hereafter see. 



2. Let then /(*) vanish when s = a, 



/'(«) . . . . «-0, 



/"(*) . . . . s = a, 



/'"(•) . . . . s = 

 Vot. IX. Part III. 49 



