III. CALCULATION ON TMNSLATIONAL MOTION 



In this section we undertake to calculate the shape of the 

 tether line, the tension in the line, and the offset of the buoy. 

 The calculation ejqjlicitly includes the drag forces on the buoy 

 and, in distributed form, on the anchor line. It ed-so includes 

 the buoyancy of the anchor line. It is assumed that the drag 

 forces will not change due to non-vertical slope of the cable, 

 but an extension can be maxie for this, if necessary. The cable 

 form can deviate substsmtially from the vertical in our basic 

 equations but we have chosen to use a small angle approximation 

 for cable slope in carrying out their solutions. Although extens- 

 ive tables of buoy cable parameters found in Taylor Model Basin 

 report 687 by Pode obviate the above calculation, we found it 

 siuipler to calculate than to interpolate with the tables. 



Figure 3 illustrates the geometry of the problem. The buoy 

 has a buoyancy force B. and a drag Dj^, which must be determined 

 independently. The resultant force of these two is F, and is 

 collinear with the tether line at the point of attachment. At a 

 depth X below the buoy the tether line has an offset distance of 



y. 



Figure k shows a short section of the line of length dx, in 

 the x direction. It is acted on by the four forces: T(x) pulling 

 generally upward, T+dT pulling downward, Ddx the drag on the line 

 segment, and Wdx the buoyant weight. The latter two have a 

 resultsuat which we call Fdx in a direction /6 as shown. We can 

 immediately write the eqioations for static equilibrivmi of this 

 line segment. 



Vertical: (T+dT) cos (e+dO) - T cos = -Wdx 



(1) 

 Horizontal: (T+dT) sin(e+dO) - T sin Q = Ddx 



On expanding the trigonometric quantities and dropping products 

 of two infinitesim8d.s we obtain: 



II cos - T sin © 1^ = _W 

 dx dx 



(2) 



^ sin + T cos © ~ = D 

 dx <i^ 



Two alternate forms may be obtained by first multiplying the upper 

 by cos © and the lower by sin © and adding, thus 



= D sin © - W cos © (3a) 



dx 



dT 

 dx 



or, on combining in the reverse manner we obtain 



^ = D cos © + S sin © . (3b) 



dx T ~ 



288 



