Further work will be simplified, however, as in the theory of the 

 undamped beam, if combinations of these four functions are used such that 

 two of them vanish when x = 0. The latter two combinations of the 

 exponential factors are: 



1 / iq,x -iq^xx 1 • ^ ."~ • ~ 



— 7" (e ^1 - e ^1 ; = — smh iq-j^x = sin q-j^x; 



(e^p"" - e -'12^) /2 = sinh q^x [37] 



The general solution of Equation [30] may then be written thus: 



M = e (d^ sin q^ x + d cos q x + d sinh q x + d, cosh q x) [38] 



Here d ... d, are four adjustable constants, probably complex. 



Substitution for M from Equation [38] and from Equations [29e,f] for 

 F and G in the boundary equations. Equations [29a,b,c,d], then gives the 

 following four equations , after canceling out e ^ : 



dq+dq = A; d+d, = -B [39a,b] 



11 3 2 2 i* 



d sin q £ + d cos q £ ^ ^o sinh q^^ "*" '^k '^^^^ 12^ ~ '^ [39c] 



d^q^ cos q^£ - d^q^ sin q^i + d^q^ cosh q^ii. + d^q^ sinh q_^i = [39d] 



The treatment may now proceed formally Just as for the undamped beam. 

 Equations for the complex displacement v and the complex slope 6 (= dv/dx) 

 are as follows (derived from Equation [28a]): 



/ 2 . '^x - 9^M ,2 . '^v - 3^M 



y U - 10) c ; V = — - ; p (oj - itoc ) e = - 



9 x^ 9 x^ 



Substitute here for M from Equation [38] and then let x ^ 0. Write v and 



o 



35 



