at a chosen qX, and E,, 



To facilitate thinking, choose one system out of the chosen group as 

 a reference system and indicate the quantities that refer to it by adding 

 a prime. For any other system of the group write i = XI' , EI = eA (EI) . 



Thus within the group: 



k 

 £ c^ X ; EI c^eX 



with X and e denoting arbitrary numbers that equal unity for the reference 



1+ 

 system. The factor X is introduced here because, in a simple change of 



k 

 scale with all dimensions increased in ratio X , I <>: X . The factor 



thus represents the effect on EI of any change in the shape of the beam 

 and perhaps in E. Then, from Equations [66a, b], among the beajns of the 

 chosen group: 



^11 '^ k ' "12 " ^21 - ;7 ' '22 ' ;^ [6Ta,b,c] 



To make C the same for all of the vibrations considered, the factor 



KAG l^/il,!) in Equation [65] must be the same for all beams. Hence, it must 



be that KAG « eX^. Also, y = p, A in terms of the density p of the beam 

 e b D 



materisd. Thus 



P^A P^K'G' KAG 



y _ b b e e 



K'A'G' 

 b 



p, A ^^KGe 



p ,2 b e 



-^ = TcX • T = — 



P' KG 

 b e 



For the reference beam, t = 1. Thus among the vibrating beams it must be 

 that 



KAG = eX^ : y « TcX ; to = — — [68a,b,c] 



^ X/T 



66 



