Vibration of Bars treated simply. 



581 



given by M = /> + / , 2 (.r-av ? ), and is thus dependent on # 2 , the 

 place of application of the second force. But, as seen from 



Fio-. 2. 



D/ST/1HC£ /I LONG 



the equation, or the graph as shown in fig. ;], the rate of 

 increase of M is independent of ./•., and depends only on the 

 sum of the forces. 



Fio-. 



Y 









k 







* 







S' 







£ 







<a 







S 









^ 







„ -- 



* 





s' ^ — - 





<* 





s^-* " 





1 



^C-'' 



i\ 



M 



jc 2 oc 



0/ST/WC£ ALONG SAR 



Thus, for values of x exceeding x. 2 , we have 



Or, if the angle of the final part of the graph is $, with the 

 horizontal, then 



tau(b 2 =/i+f.2. 



Hence for any number of isolated forces applied alono the 

 bar between the origin and x, we have 



dM/d**/i.+A+/ 8 . • - = F, say 



(2) 



In other words, if the magnitudes of the forces are specified, 

 the space rate of .increase of M, the bending moment at anv 



