formvUas hold as k k 



11 !2 



k^ -* ^11 ' k^ " ^"22 



a^ 



->■ D = a., a - a. _a_. ; D k^k^ ->■ D 



k^k^ -o "11^22 "12^21 ' " "1^2 



The limiting forms of most equations become evident after suitable mani- 

 pulations to make these formulas applicable. Equations [53c, d] being 

 divided through by k k . , > 



Equations [i+6a,b] and [i+8a,b] require no change. Equations [U7a,b] 

 divided by k^ or k give v = v and 6 = G. In Equations [^9a,b], [50a,b], 

 [51a,b], [53a,b,c,d], and [5^a,b], the final effect is simply to replace 

 D by D , S-, by a-i-i , and S2 by a22 > and to erase k, and k everywhere, the 

 term k,k2 in [53c] being replaced by 1. 



The equations do not seem, however, to be simpler in this case or 

 modified in a mathematically interesting way. 



6.5.3 Low-Frequency Approximation 



In the theory of a uniform beam under forcing, as given in Section 2, 

 formulais were found for a^ ^ , a p, b.^-. , and app in terms of E, I, the 

 length £ of the beam., and two dimensionless parameters q£ and 5 defined 

 thus : 



/y \1/1+ 1/2 EI 1 , .2 r.o .1 



q£ = (^) 0. £; 5 = ^^ J (qO [58a,b] 



Here \i is the effective mass of the beam per unit length. 



* y includes the effect of virtual mass when the beam is immersed in a 

 fluid. 



61 



