442 hersey: vibrations of elastic systems 



prisms, so far as any practical application to the stiffness of 

 beams or galvanometer suspensions is concerned. 



Formulas for frequency of a coupled system. From (6), 



J & + & / (a) (10) 



' rn., 4- Wo 



n = 



n mi + m 2 



because m = m x + m 2 , and, for forces acting at the coupling, 

 S = Si + So (equation (11) of previous paper). Since / 

 depends on the generalized shape, it will be different for different 

 values of the ratio R, and in any case it is multivalued. 



In the particular case of two bodies having the same generalized 

 shape and a, Si and *S 2 of (10) can be replaced by their equivalents 

 in terms of n\ and n 2 by virtue of (6), with the result 



n = VRn\+(l-R)nl F (R, a) (1 1) 



The function F depends only on the generalized shape; not at 

 all on the absolute sizes, masses, or elastic moduli of the bodies. 

 Formulas for change in frequency of coupled system with temper- 

 ature. Differentiating (10) logarithmically gives, in the case of 

 two bodies of different generalized shapes but of the same 

 material, 



n=h[{K+l)a-Kp + y] 

 in which 



K = v C 1 +(l- v )C 2 + Cn 



C 12 = 2(l + ,)^(l+.)ilog(^- 2 



and in which d and C 2 are the appropriate values for the ordi- 

 nary shape factor C for stiffness, as given in the previous paper ; 

 viz., 



Y\ aa V-MMi/ 



with a like expression for C 2 . 



