These solutions are based upon the assumption that the fluid forces are independent of the 

 resonant motion of the cylinder. In reality, there is a complex nonlinear dependence between the fluid 

 forces and the displacement, c.f.. Fig. 2.7. Also, the maximum displacement is dependent upon the 

 mass ratio and structural damping of the cylinder and its mountings. When the force coefficients are 

 assumed to be known, as from experiments, as shown in Figs. 2.3 to 2.6, these analytical models can be 

 used to predict the cylinder displacement and vice versa. A more complete development of these equa- 

 tions is given by Griffin (E2, E7). 



It is a relatively simple matter to extend the analysis to the case of a flexible cylinder such as a 

 cable with a normal mode shape t//,(z), following the "wake-oscillator" formulations described in Appen- 

 dix D. If a pure mode response is assumed, then the displacement of the flexible cylinder is 



Y,= Yiljiiz) sin ar (E.1.11) 



at each spanwise point z = z/L, and the maximum displacement is scaled by the factor 



Yeff.max = Y i;' / \4']iz)\MAX-Y/yi (E.1.12) 



where 



■ \ 



/; ^} iz) dz 



/, = ^ and y, = '"','""" (E.1.13) 



for a cylinder of length L. For the special case of a circular cylinder /, = \^ih.)\MAX = 1> ^nd other 

 values of /, and i//, for a variety of flexible cylindrical cross-sections are tabulated in Table El. The 

 parameter y, can be calculated directly from the entries in the table. 



The model just discussed also can be extended to the case of a flexible cylinder by means of a 

 normal mode approach. The cross flow displacement of the cylinder is assumed to take the form 



y{z,t) = Y, Y,(t)^\,M^ (E.1.14) 



which is a standard expansion of the normal modes. When equation (E.1.14) is substituted into equa- 

 tion (E.1.1), the result is a generalized form for the equation of motion given by 



165 



