[ 1018 ] 



CV. On the Foiled Vibrations of Bridges. 

 By Professor S. P. Timoshenko *. 



IT is now generally agreed that imperfect balance of the 

 locomotive driving-wheels is the principal source of 

 impact effect in bridges of long span. The laws governing 

 this effect have not yet been definitely formulated, and much 

 more information is needed on the experimental side t- 



Some idea of the forced vibrations which are thus induced 

 may be obtained by considering the bridge as a beam of 

 constant cross-section with supported ends (fig. 1). The 

 deflexion of the vibrating beam may be represented as 

 follows :— 



7TiV 2 TTiV 



y — $x sin-y- + cf> 2 sin —j— 



, . 3tt.x 

 <p 3 sin — - - + 



(1) 



where b <p 2 , ..., etc. are functions of t only. Then if EI 

 denotes the flexural rigidity of the beam, and w its weight 

 per unit length, the expressions for the potential and kinetic 

 energies will be 



V=|EI 



ay 



JllSO^i 



1EI^ 

 4 I s 



1 



2 a K * at - , L < ■ 



l- • (*) 



9 ,=i 



Fig. 1. 



Pcos SZ* 



I 



I — 



We suppose that a single variable force P cos 27rt/r moves- 

 along the beam with a constant velocity v (fig. 1). The 

 corresponding differential equations may be written J in J the- 

 form 



^+^^ = Pcos--sm T -. . . (3> 

 Then taking <j> n =<^ B = at the instant £ = (), and writing 



to 

 * Communicated by Mr. E. V. Southwell. 

 t Cf. ' Engineering/ vol. cxii. p. 80 (1921). 



