beam is represented by the values of a^ ^ , &,„, ap^ , and a^p. Apparently, 

 in general, the relative influence of these parameters can be ascertained 

 only by making extensive numerical calciilations . A major source of 

 difficulty lies in the possible variation of the a's over all real values. 



A simple form of similitude, however, is readily discovered. In 

 Section 2 formulas are given for forced vibration of a uniform beam when 

 F and G vibrate at the same circular frequency w but perhaps in different 

 phases (for example, F = A sin wt , G = B sin (ut + i>)). The mode of 

 vibration of the beam was shown there to depend upon two dimsnsiontess 

 parameters q£ and C whose definitions, as given in Equations [58a,b], may 

 conveniently be rewritten as follows: 



e *■ *• 



Here i denotes the beam length and q£ may have any positive value. The 



formulas obtained for the a's can be written thus: 



ij 



^l^^llil' ^2 = ^2ll' ^21 = ^2liF' a^p = a22fe[66a,b,c,d: 



where a , a , a , and a are complicated functions of qX. and C but are 

 11 12 21 22 



otherwise independent of the beam parameters. 



To obtain simple similitude, variation of qii. and C mxist be avoided. A 

 group of foil-beam systems that can vibrate at the sajne q£ and C will be 

 compared, and the comparison will include only vibrations of these systems 



* To infer Formulas [66a,b,c,d] from those given below Equations [l2a,b] in 

 Section 2, q-, and q2 being defined there by Equations [5a,b], note that 

 q]_£ and qp£. are functions of qi and C and, consequently, the same is 

 true of the functions s, c, S, and C. Elsewhere, replace q-, and qg by 

 (q i)/2. or (qpJl)/J., and replace q^ by multiplying EI by (qJ.) /ii • 



65 



