Appendix B 



EVALUATION OF THE NORMALIZED AMPLITUDE OF THE 

 MAXIMUM DYNAMIC STRESS AS w' ^ nTr 



As noted in the main text in the Discussion of results, the normalized amplitude 

 of the maximum dynamic stress, | ^maxl' '^ particularly sensitive to variations of 'to'. 

 For particular values of to', given specific values of j3 andjLl, it is desirable to eval- 

 uate the peak in I^qxI fTiore precisely than by an interpolation of the computer 

 output. This essentially requires the derivation of dl^g^^/d co' from Equation 7, 

 equating this to zero, and solving for to' as a function of j8 and jJi. Inspection of 

 Equation 7 indicates the complexity of this derivation, which is unilluminating in 

 terms of a proposed design procedure. As an alternative. Equation 7 was evaluated 

 in the limit as cc' approaches nTT, where n = 1, 2, 3, . . . In certain cases, this 

 corresponds to the peak in the maximum dynamic stress I^^qxI* ^^® analysis was 

 carried out in order to determine the inaccuracies involved in interpolating the 

 computer output, and is repeated here for completeness. 



Given Equations 7, 8, 9, and 10 below, it is required to determine the value 

 of ^Qx I °^ ^' approaches n TT, where n = 1, 2, 3, . . . 



(^ax)^ = (to')^U'i)^[l +tan(p(tan>I' + sec*)] 



(u;)2 = 



cos (co' +(p) 



2 2 2 



2j3 sin (psin to' 



1 + 



9 2 2 



p sin a:' sin 2(p 



cos^ (to' +<p) _ 



il/2 



- 1 



(7) 

 (8) 



cp = arc tan 



to' 



^cps 



^ = arc tan 



y r{^\) tancp - cot2<p 



2 2 



(9) 

 (10) 



From Equation 9, when to' ~* nTT, 



to' nTT 

 tancp = — = 



(B-1) 



62 



