a = a 



11 ~ Elq^Ji y£a)^ ' l2 21 EIq^«.2 u«.^w^ 



= - 1^ = _ 12 



2 ••122 l32 



b^^ = - yi^o) ; b^2 = ^21 " ~ F ^^ "" •' ^22 " " 3 ^^ " 



1 3 

 In b , yJ^ is the total mass of the beam; in b , — pX is its moment 

 11 22 3 



1 2 

 of inertia as a rigid body rotating about one end; in b , — 'V^ equals 



mass times distance from either end of the beam to the center of mass. 



The validity of Equations [l5a,b] for F and G in this approximation is 



easily verified from elementary mechanics. Thus, when q£ << 1, the beam 



behaves, as it should, approximately like a uniform rigid rod of length 



X. and mass V^. 



2 



As q^ ->■ 0, a , a , a , and a, , a^^ - a all become infinite, 

 11 12 22 11 22 12 

 o 

 whereas b , b , b , and b b - b"^ all go to zero. 

 11' 12' 22' 11 22 12 



In the following discussion of other cases, the notation is often 

 shortened by denoting the first fraction in each formula preceding the 



EI fraction by the symbol for the coefficient with a bar over it. For 



- sC - cS ,- -7- 1 + cC • *. „ n 



example, a, -, = - . Also, A = ; or if qJ!, >> 1, 



11 1 - cC 1 - cC 



and A = - 777777; also if IcI >> (l/C), 



11 " c - (l/C) '"'^ '^ c - (1/c; 



a,, = tan qil -1 and A = - 1. 



(b) First "built-in" frequency: When qX, = 0.59T7r = 1.875, calcula- 

 tion shows that cC = cos q£ cosh q£ = -1. This is the familiar frequency 



12 



