Rotatory Inertia on the Vibrations of Bars. 37 



Also 



M = EA <g; w 



d 2 v ^ . d*v 2 d 4 v T 



dF +EA ' c ' f I7 = m ' c -^W + " iY - • • ( 5 ) 



W 



The first term in the right-hand member represents the 

 rotatory inertia, and is usually disregarded. 



2. Normal Functions. 

 In the last equation take Y = and put 



v = Xf p (x)(A p sm l: t-l-B p cosyt). ... (6) 



P 



Thus, on equating terms of type p, 



mp J p -BA^^=m^p d 



•7 ? 



or, putting 



and ?»k 2 /j 2 V (7) 



EA 



=x 4 ,} 



. g^ +X ^-X%=0 (8) 



The solution o£ this equation is 

 /p = J P cos 7^ 4 dp' sin 7^ -f d P " cosh y 2 |r + <V" sinh 7 2 f, (A) 

 where y! and y 2 are given by the roots of 



7 4 +xy-x 4 =o. 



J liCv/^+^-x*]" 



(9) 



/2 



3. Terminal Conditions. 



The end of the rod may be clamped, supported, or free; 

 and these conditions determine the ratio of any three of the 

 constants d in (A) to the fourth. 



d v 

 (i.) At a clamped end v and — are always zero. 



