﻿subjected to Forces in th> Direction of their Axes. 237 



whence 



_^ = 500-fi-^+12-0-^ o -'171 ~~. . (8) 

 y x poor pool- pcohiL 



§ 7. Equation (8) is of course for the fundamental. 

 When denling with harmonics v ve may in all cases take 



tanh <j>=l, 



and, when P = 0, we have for the i th tone 



al = pi=^(2i +1)tt. 

 Hence r 7 ,-, 



1\ =i(M+l>r+(-l)« S x(»y) 



*-Hx-(-iy}(x+i)-^ 



Equations (5) then give 



7T 2i + 1 



4EI( x *-l; ' 



r here 



Expanding as before 

 Whence 



yi ""l ^* j J po,/ 4+:t? % + l ^ ( % -M) 2 ^El- 

 § 8. The solution is restricted, by the use of equation (7), 



W l 

 to small values of -^^y-. 

 Jiil 



If the axial force P is compressive, vibration will become 

 impossible when 



P=4EI^ (9) 



and the bar will be unstable. 



In practice, when using equation (9) a factor of safety of 

 at least 10 is allowed, in which case 



EI* 4 " 



It can be found by trial that in this case equation (8) is a 

 very close approximation. Generally it may be said that 

 for the problems met with in constructional work equation (8) 

 may be used with fair accuracy ; but when dealing with long 

 and tine rods or wires, as in acoustical problems, the 



