OSCILLATIONS OF A SUSPENSION CHAIN. 395 



I 8<\ CPV 



ds* 

 s*\ d?V 2 dr ,„, „ % d 3 V' 



I «*\ d'V 2 d* ._, ,. x 



d« 3 



As before, we must approximate by first neglecting and then taking into account the 



terms multiplied by - and its powers; the process would be simply laborious and inelegant, 



and as it is rather general laws than minute results, that we require, the labor of the cal- 

 culation would be scarcely repaid. It may be observed that the most important term and 

 the only one perhaps worth considering at all is 



2 d?_ U ,*V 



c df ° ds> ' 

 12. On the whole, then, we may state the following as the most important of the results 

 we have arrived at. 



(1) That for a slight disturbance of a chain of but small deflection the increase of 

 tension is nearly uniform throughout, and consequently if we cause the increase of tension 

 to be always small or zero at the points of suspension, we may be sure that it will not be 

 very great at any other point. 



(2) That the mere transit of loads at an ordinary pace across bridges of wide span is 

 not likely to produce a sensible vibration. 



(3) That stamping or tramping of any kind is dangerous in the extreme, as if it be 

 periodic and coincide in period with any of the possible types of vibration we have already 

 found, it will give rise to a continually increasing disturbance. 



(4) That the rapid sweeping of gusts of wind along the platform is apt, under the condi- 

 tions we have already found, to give rise to violent vibrations. 



(5) That when the vibration arises from the action of a periodic force (coinciding in type 

 with a type of vibration of the equation of disturbance,) continued for some time and then 

 ceasing, the disturbance (approximately) will consist of terms containing but one type of 

 vibration, and the chain will be separated by nodal points where there will be no vertical 

 motion. 



The most important points of difference between the motion of a suspension bridge and 

 that of a simple chain are, first, that in a suspension bridge, the rigidity of the platform tends 

 to check the vibrations of the chains, but probably not much in bridges of wide span : and 

 secondly, that the links of the chain are finite in length, about thirty being the average number 

 from pier to pier ; this latter point of difference would not however greatly alter the results. 

 A very common and practically one of the most important causes of danger I have not 

 alluded to, I mean the improper manner in which sometimes the chains are bolted together, 

 and which causes an enormous twisting force to be brought to bear upon the bolts. This is 

 however a question entirely independent of the problem of a vibrating chain. If by means 

 of buffers, or some kind of spring attachment, the increase of tension at the ends of the 

 chain could be kept within moderate limits, it is clear that but little danger need be ap- 

 prehended from the chain being thrown into a state of vibration ; the very great amount of 

 Vol. IX. Part III. 51 



