Prediction of Steering and Manoeuvring of Ships 



= phase angle by which the forward scotch-yoke leads the aft, 



CO = rate of rotation of the shaft connecting the scotch-yokes, 



= angular displacement of the model centre line relative to the tank 

 centre line, and 



t = time. 



Then 



77p = a sin (wt + 4>) 

 Vft - 3 sin ait 



d (■nF_^\ 4> / , , 



"r " dt \ — 2 — / " ^"^ "^"^ T """^ 1'^* T 



Also 



sin \p - — 2^ — - 'J sin — cos I oJt + -T ) • ^ •' 



Substituting for sin ^ and u^ in Eq. (6), 



1 a . / \ aw 



sin — cos [cot + 77= t;— cos -;r cos I Oit + 



cos d 2 ^^-^ \ ^ 2/ Up 2 \ 2 



(8) 



^d 



2 " U, 



cos 



Since i/' is small, we make the approximation cos ^ = l, giving 



cod , . 



tan| = -. (9) 



Thus the phase-angle is not a cyclic quantity, since it only depends on 

 frequency of oscillation and carriage speed, the distance d being fixed in the 

 model. 



The present HyA planar -motion mechanism has maximum amplitude a at 

 the scotch-yokes of 100 mm, and the distance d is 1050 mm, giving a maximum 

 possible value of 5.46 degrees for when the error in the approximation 

 cos i/' = 1 is less than 0.5 percent. This value is obtained only at zero speed 

 when the phase-angle between the scotch-yokes is 180 degrees. In a normal 

 operating condition, the maximum value of is reduced by the factor sin (0/2) 

 to approximately half this value when the error is 0.1 percent. While these 

 considerations give a good indication that the error involved is negligible, the 

 only rigorous way of checking this is to analyze the "contaminating" cyclic mo- 

 tions and corresponding impure forces by carrying through the exact calcula- 

 tion. This has been done in another connection and has shown that the error is 

 indeed negligible. 



327 



