BARS 



125 



Take first the case of a perfectly free bar, of length I, say. 

 If we take the origin at the middle*, these conditions are, by 

 45(14), 



," = 0, ,'"=0 [c-i<] (4) 



The normal modes fall naturally into two classes ; in one of 

 these 77 is an even, in the other an odd function of x. For the 

 symmetrical vibrations we have 



77 = A cosh mx + C cos mx, (5) 



with the terminal conditions 



A cosh J ml G cos ^ml = 0, j 



A sinh ^ml + Csin |w = 0,J 

 whence tanh ^ml= tan ^ml (7) 



+ 1 



-1 



Fig. 44. 



The roots of this equation are easily found approximately by 

 graphical construction, viz. as the abscissae of the intersections 

 of the curves y = tan #, y = tanh x, the latter of which is 



* This improvement on the ordinary procedure is due to Sir A. G. Greenhill, 

 Mess, of Math. vol. xvi., p. 115 (1886). 



