equation for a imiform beam vibrating with one end built in and the other 



end free. Calculation from the formulas following Equations [lUa,b] gives: 



^11 "^ -2.069; a^2 ^ 1-520; a^^ =- I.II6; A = 



The corresponding values of b ^ , b-io' s^^ ^ o ^^^ ^^-^ infinite. It 



does not follow, however, from Equations [l5a,b] that v and 6 are 



00 



necessarily zero. This would be the case if only one of these variables 



were present; for Equations [lUa,b] furnish finite values of v and 9 to 

 '^ 00 



match any finite values of F and G. The ratio Qq/Vq, however, is fixed 



and cannot be varied by changing F and G. For, since A = 0, 

 2 r. . ^ 



ai2 a-22 



^11 ^12 



Thus, the ratio of the right-hand members of Equations [li+a,b] is -O.T35 q 

 for all finite values of F and G; it follows that Oq/v^ = -0.735 q also. 



If, however, the ratio F/G = + 0.735 q, then F/G = - ai2^^11~ ~ ^22^^ ^12' 

 so that a, ,F + a G = a F + a G = and Equations [lUa,b] give v = 6^ = 0. 

 The beajn is then vibrating as if it were built-in at x = 0. If it 

 actually is bmlt-in, F/G or 0.735 q is the ratio of the reactions -F and 

 -G on the supporting structure. 



Further illiimination resiilts from considering what happens when cC is 



merely very close to -1. Then "b , ^^.2 ' ^^^ ^?? ^^^ finite but large so 



that, in general, large forces are required to produce arbitrarily 



assigned values of v and 6 . The same conclusion can be derived from 

 •^ 00 



* The equation cC = -1 is assumed to be exactly true, the value of ql 

 being within 0.001 of that stated. Similarly, the same is assumed in 

 other cases. 



13 



