Str^m-Tejsen and Chislett 



When an oscillator technique is used to generate the yaw velocity, the force- 

 response at a particular instant is basically the same as in the steady state 

 provided frequency of oscillation is sufficiently low. If the sinusoidal angular 

 velocity is of small amplitude, the forces are proportional to the motion and are 

 also sinusoidal. This is the case within the range of linear force -response A-A. 

 Tests made within this area provide the linear terms Y^^ and N^ as discussed 

 above. It is, however, desirable to explore the full range of yaw velocities that 

 a vessel can experience, and this will probably extend into the nonlinear area 

 indicated by B-B. If an oscillatory test is made in which a sinusoidal angular 

 velocity of this amplitude is impressed on the model, the force -response will 

 be of sinusoidal character within the region of lesser yaw velocities, A-A, but 

 progressively deviate from sinusoidal in the regions A-B resulting in a gauge 

 force of the character illustrated in the bottom of Fig. 18. 



A curve of the character shown at the top in Fig. 18 can be expressed with 

 good accuracy by a linear and a cubic term: 



Y(r) = Y,r + Y^^^r^ . 



Since the yaw velocity impressed on the model is sinusoidal, described by 



•■ = '"max sin wt , 



it follows that the cyclic forces experienced by the force gauges (Fig. 18, bot- 

 tom) are expressed by 



Y(r) = Y, r^3, sin a,t + Y„, r^^^ sin^ ct 



or 



Y(r) = a sin wt + b sin"' wt , 



where a and b are constants in time. Integration over one period with a change 

 of polarity after half a period then gives 



Y(r) - (Yr) = 4a + | b , 



i.e., 



integrated value 2 

 4 = ^ " 3^ 



= Y r + — Y r^ 



r max 3 r r r max 



The true cubic term Y^^^ corresponding to the steady-state condition is 

 thus 1-1/2 times greater than the cubic term obtained by fairing force -values 

 which have been integrated over one period of sinusoidal yaw-velocity. 



344 



