456 Undulation of an Unstiffcned Roadway in a Suspension Bridge, 

 stance y to the left or right of the point of impulse will be 



•(i-0 ««(i-£).' 



and therefore the momentum of an elementary section of the 

 girder of width dy, being integrated, will give 



momentum of whole girder = -njr\ 1 ( 1— uajty 



+ i( 1 - J^) % > « |-M«, the limits being 0, #, and 0, 6". 



If the train reaches the point of interruption with a velocity v, 

 it moves on beyond that point with the diminished velocity vcosO, 

 and it impinges on the roadway with a velocity v sin 6. The 

 momentum thus communicated to the girder will be Tv sin 6, T 

 being the mass of the train ; 



.-. f Mw=TV sin 6 } ,\u=^v sin 6. 



I will now find the depression which a blow communicating a 

 velocity u at the point of interruption will cause : I will take the 

 worst position for the interruption, viz. the middle point. The 

 girder unsupported by a chain will bend, as experiment shows, 

 through 1*6 foot under its own weight, together with the weight 

 of the maximum load, 400 tons. Let z be the additional depres- 

 sion which would be caused by a pressure P ; then 



P= 750 tt = 470* tons nearly, 



and the mass of the girder set in motion is equivalent to 

 f + 350 tons +g y 

 d 2 z 3^x470^ 



* dt 2 2 x 350 



-64z; 



Hence if h be the extent of the vibration caused by the blow, 



h= x ^u — ^v sin 6 feet. If v be 88 feet, or the velocity 60 miles 



an hour, h = 4 inches, if #=1°, or for a sudden change of 1 in 



57 in the gradient of the rail. The depression will be greater or 



less in proportion to the change in the gradient. 



The time in which the girder, on receiving the blow, passes 



tt" 1" 

 through its depression is — = — -. Each carriage is 25 feet 



long, and has two pairs of wheels. Hence the interval between 



