Equations [li+a,b] by noting that the result of solving [lUa] for F and 

 substituting for F in [iHb], or solving for G and substituting for G, is: 



e^ = V + (a, -.app-a-ip) = v - (a a -a } 



^11 ° ^11 ^12 ^12 



Since a^ -, app - a is almost zero when cC is close to -1, moderate-sized 



F and G must leave 6q/"v close to a /a and also to 322/8-2^2 • "^^ produce 



considerably different values of 6q/Vq requires large-sized F and G. 



(c) First sliding-end frequency: When qX,, = O.T53it = 2.365, 

 calculation shows that sC = - cS or (multiplied by cC) tan ql = - tanh qi^ 

 and 



a^^ = -1.558 ; 1^2 = 0.767 ; ^22 = 

 Also, b -, = 0. If F = 0, then Equation [l^+b] gives 6=0. The beam is 

 then vibrating as if free to slide transversely (with F = 0) at x = 0; 

 however, it is constrained against rotation. Also, in this case, v^ = a 2^, 

 and -G or -v /a-,p represents the reaction upon the constraining 

 structure. Different vibratory motions occur if F is not zero. 



(d) As q2. passes throu^ tt or 3.1^2, a passes throup^ zero and 



changes sign (because sin q£ does), and b does likewise. Thus, at 



qJl = TT, V = a, ,F and 6 = a„„G. Calculation gives: 

 ^ ' o 11 o dd 



^11 " ' ^'^-^^ ' ^22 " "'^■'■'^ ' ■^ ^ ~ °'^^^ 



(e) First pin-end frequency: At q «, = 1.250it = 3.927, tan qi = 

 tanh q I and 



^1 = ^22 = ° ' \2^ ' °-^^^ ' ^22 = '^'^^^ ' ^ " " °-^^^ 



Ik 



