

(E.l.la) 



where, in general Q and Q are assumed to be functions of both z and r, and the mass per unit length 

 m is a function of z. If this equation is multiplied through by i//,(z) and integrated over the length 

 Liz = z/L) of the cylinder, then the balance of forces for the iih mode is 



(Y,„ + 2 a)„, Csi Yu + (Oni Y^){J^'ijjfiz)miz)dz) - 

 ^^[X,'q^,(z)^z-X,'c,0,(z)^z] 

 when the normal modes of the structure satisfy the orthogonality condition 



(E.1.15) 



m i — J 



J /M(z)t//,(z) \jjj(z)dz = 

 In terms of the variables given in the first paragraph of this section, equation (E.1.15) reduces to 



1 . 



(E.1.16) 



2, ,2 



,(}^. + 2^„ y, + Yi) 



pD'oi 



Jo Ciilji{z)dz Jq C,iiliiiz)dz 



rm{zHfiz)dz Vm{zH}{z)dz 



(E.1.17) 



'0 ^' .^0 



where K, is the generalized displacement and the integrals on the right-hand side are generalized force 



coefficients. When "equivalent" force coefficients and the "equivalent" mass are introduced by 



1 , 



C,F = 



Crf — 



and 



the equation of motion reduces to 



/"£ = 



J^ Q.A/(z) dz 



Jo CR^i{z)dz 

 j^ilii{z)dz 



Jq m(z)iljHz)dz 

 S'4'Hz)dz 



Y+lCsY+Y^fji 



(Q 



Cre) 



J,\iz)dz 



(E. 1.1 8a) 



(E. 1.1 8b) 



(E. 1.1 8c) 



(E.l.lSd) 



jyiz)dz 

 The subscript / is dropped since a single mode response is understood in this equation. In the case of 



the flexible cylindrical structure, such as a cable, a work and energy balance as discussed by Griffin (E2, 



166 



