The variation of ui is inferred from Equation [65b]. For any two beams 



made of the same material and with- cross sections of the same shape, so 



2 1+ 



that KG and t are the same, A ^ tX and I °: eX . 



The validity of the fundamental Equations [U6a,b], [UTa,b], eind [U8a,b] 

 for all beams must now be secured. Tentatively, the rule that all terms 

 in a given equation must vary in the same ratio from one vibrating system 

 to another will be followed. 



Equations [U8a,b] read, for the reference system (denoted here by 

 primes) and for any other chosen system, in view of Equations [67a,b,c]: 



^o' = ^11^' -^ 42^' ' 'o = 4l^' -^ ^22^' 



V =^ F + ^G ; e =^F^^G 



Here it can be assumed that the time factor in F is the same for all beams 

 (perhaps "sin wt") . Furthermore, since the amplitude of any vibration may 

 be arbitrarily varied, it does no harm to assume that the amplitudes are 

 such that (a' /eX) F is the same for all beams and hence equal to a.I,F'- 



Then F = eXF' . Also, according to the general rule being followed, the 



2 2 



second term (a' /eX ) G must then equal a' G' , so that G = eX G'. It 

 12 12 



2 

 follows that the phase of G is the same for all beams (perhaps G =: eX 



sin (u)t + (J)), 1}) constant). 



Thus V = v' for the vibrations considered, but substitution for F and 

 o 



G gives 6 = Q' /X. Summarizing, among the chosen vibrating beams: 



F « eX ; G ''^ eX ; v^ <^ 1 ; 6 <^ 1/X [68d,e,f,g] 



6T 



