Prediction of Steering and Manoeuvring of Ships 



Calculation of Coefficients in the X Equation 



When the ship is sailing straight ahead with constant velocity Uj, the propel- 

 ler thrust, modified by the thrust-deduction effect, exactly equals the resistance 

 of the ship: 



X = T( 1 - t ) - R^. = . 



This equilibrium condition defines the initial propeller thrust and the cor- 

 responding propeller torque and revolutions. 



As soon as a manoeuvre is initiated, the equilibrium condition is disturbed 

 and the x force, which represents the difference between the propeller thrust 

 and the ship resistance, will then vary as a function of speed u and propeller 

 revolutions. Approximating the x force by a third-order polynomial, 



X(u) = 80+ ajAu + ajAu^ + a ^/ki^ , 



where Au = (u - Uj)U, the dimensionless hydrodynamic coefficients X*, X^^, 

 x^^j, and x^^^ can be obtained directly as the coefficients of the approximating 

 polynomial, as follows: 



X* = Bg - 0; X^ = aj; X^^ = a^; X^^^ = a^ . 



It has been found that the X force and corresponding coefficients are com- 

 puted most accurately on the basis of the results from the open-water, resist- 

 ance, and self-propulsion tests. Table 5 shows examples of such calculations. 

 The thrust deduction coefficient t = 0.136 and wake coefficient w = 0.160 used in 

 the computation but not indicated in the table are taken from the self -propulsion 

 test. These values are assumed to be constant for all speeds. The correspond- 

 ing propeller thrust values are computed from the open-water propeller curves 

 assuming constant wake, and taking the type of engine and engine setting to be 

 maintained during the manoeuvre into account. As shown in the example, the 

 propeller thrust can be calculated assuming constant propeller revolutions, or 

 assuming the propeller torque to vary proportionally to the revolutions to a 

 certain power. If torque is assumed to vary inversely proportionally to pro- 

 peller revolutions, the thrust corresponding to a turbine powerplant capable of 

 maintaining constant power output would be obtained. If torque is assumed to 

 be constant during the manoeuvre, the corresponding condition for a diesel en- 

 gine would be obtained. Figure 25 illustrates the relationship between propeller 

 revolutions, torque and speed for these various conditions. 



The variation of propeller revolutions with speed as derived from this 

 computation can conveniently be used to obtain the correct propeller revolu- 

 tions, which have to be used when tests are executed at reduced speeds to obtain 

 the Au - and revolution-dependent coefficients ("rudder angle and speed" tests). 



The three coefficients, x^, x^^, and X^^^, can also be obtained experimen- 

 tally by fairing x-force measurements made for zero rudder angle in a "rudder 

 angle and speed" test. Such experimentally derived values have agreed well with 

 the calculated coefficients. 



359 



