«=1 



On the Forced Vibrations of Bridges. 1019 



we obtain 



I 2 ~>oL tf r> 2tt£i . nirvt l . nVait — t^ . ... 



</>«= -5— r-.-^rl -t cos sin — j — sin ,:, -«m, (4) 



n^ir'a- tt'/J • r /. r 



and the expression (1) may be written as follows : — 



( / s iittv 2rr\ . ' niiro 2tt\ 



mrx 1 J sin -= H ■ U sm{ — = \t 



a— r-<\ / t / W t/ 



^ rc*-(/3 + rca) 2 "t n 4 -03-n«) s 

 / . n 2 ir 2 at . n 2 ir 2 at \ ) 



where a = — ; 8 = . 



air ttclt 



If the period t of the force is the same as the period 

 t x f = — J of the principal mode of vibration of the beam, 



resonance will occur, and the amplitude of the forced vibra- 

 tion will increase with t. Under these conditions, we have 



0=1, 



and at the instant when the periodic force ceases to act upon 

 the bridge we have 



t = ljv, 



so that a= — (in general, a small quantity). 



Zi t 



Then the first term (for n=.l) in the series on the right 

 of (5), which is the most important part of #, may be reduced 

 to the form 



2P/ 3 . <kx 1 . 2irt 

 ~^t sm T • ~ sm > 



and the maximum value of the deflexion is given by the 

 formula 



2P/ 3 



^ max -"a^El- ■*■•■••' ( 6 ) 



Since, as we have seen, a. is usually a small fraction, we may 

 conclude from (6) that the forced vibration produced by want 

 of balance in locomotives mav be of practical importance. 



The same method can be applied in other cases, where 

 more complicated expressions are required for the forces 

 which produce the vibration, and also in cases where variable 

 horizontal forces act on the beam. 



Zagreb, Yougoslavia. 

 Dec. 11th, 1921. 



