364 Mr ROHRS, on THE MOTION OF BEAMS 



a 



Also between « = 0, and * = — , we have for the equation to the curve 



d'y ^ » _ Q _ 2^ 



da a6b^\ s) W 26V V2 8/ ' 



, ya Wa^ 5 ■ Qa" 



whence — = -— + — -: =6+6. 



2 6' 384, 4:8b^ 



If we had not assumed the pressure Q constant, we should have had to determine the 

 motion of the girder 



»re 32"= - 6 TT + — 2 sin sin (l), 



df ds* a a a 



Now y ^y + y", and y" = («), where (s) is the statical form of the girder at rest, 

 and 8= vt; 



.-. m'-^ +m'-^(l>(vt) = m'g-Q.. (2). 



The statical value of y is given by the equation 



" 3a \12 24a 24 / 



Hence between 1 and 2, eliminating Q if it were possible, we might solve the problem. 



An easier example where the pressure varies, is the motion of a beam suddenly loaded in 

 the middle and allowed to sink. 



Deflection of a girder suddenly loaded in the middle. 



Let M' be the mass of the load imposed, 



M the mass of the girder, 



Q the pressure at time t exerted by the load on the girder. Then supposing the 

 load collected at one point we shall have 



»»-—=- 6^—-+ — 2 —-sin -, 



dt^ da* a Z a 



where it must be observed that y is the part of the ordinate due only to the weight in the 

 middle, so that if y be the value of the ordinate when no weight is imposed on the girder, 

 y + y' is the complete ordinate. 



