Dynamios of Submerged Towed Cylinders 



point of cessation of criss-crossing oscillation. Here it might be 

 argued that f_= 0,7, c^ = 0o3 may be too severe for low towing 

 speeds; comparison with theoretical values calculated with f g = 1 - C2 = 

 0,8 {shown in parentheses in Table 1) yields better agreement, as 

 anticipated. 



VI, DYNAMICS OF TOWED RIGID CYLINDERS 



6, 1 The Equations of Motion 



We consider exactly the same configuration as in Fig, Z, but 

 impose the restriction that the body be rigid. In this case the system 

 is reduced to one of two degrees of freedom. The generalized co- 

 ordinates may be taken to be the lateral displacement of the center 

 of mass, yc, and the angle that the body m.akes with the x-axis, <f>. 

 Accordingly, the displacement at any point is given by 



y = Yc + x4>» (14) 



For the sake of simplicity, we assume that the center of mass coin- 

 cides with the geometric center of the body, X| and Xg being small. 

 For convenience we now measure x from the center of mass, so 

 that the body extends from x = - L/2 to x = L/2, 



Instead of deriving the equations of force and moment balance 

 Independently of the previous work, we shall proceed as follows. 

 We shall integrate Eq, (10) or (10a) fornnally to obtain an equation 

 of force balance, and similarly integrate the product of the forces 

 in (10) or (10a) by x, to obtain the equation of moment balance. The 

 boundary conditions are incorporated through the integral of the 

 first term of these equations; alternatively, the shear forces at the 

 ends may be viewed as forces replacing the effect of nose and tail on 

 the main part of the body. 



Thus using Eqs. (10) to (14) in the manner described above, 

 we obtain the following two equations: 



[ M(L + x,f , + x^fp + m(L + X| + x^] y, 



+ [-|c^MUL/D +f, - f^] y^ +-^ M\j\^^ 



--|[m(x, - x^) +M(x,f, - X2f2)]L'<f+MUL[2 - ^•'^^] 5> 



+ Mu'[|c,^+f, -i2-\j cjcf>=0, (15) 



and 



1003 



