Norrbin 



(1 - Y'')v = 1-(Y." - x^'H + (Yj; - l)ui + L-'/2 g-'/' • 1 Y^'^^u'i 



+ L ' Y" uv + L*^^g''^' . i Y" u^ + L-' • i Y" | v|v 

 uv ° Z uuv Z Iviv 



+ l»4y;;,^ lili-HY,-;,,^ Wi + y;;, v|i| 



(k', + N?H = L-' (N; - x^'); + L-" (N:, - x^)ui + L'^'^'g'"^ • InU,, u^i 

 + l'n'>v + l'/^-'^' . i N:^^u^ + L^ ^ N;;,^ |v|v 



4^.';.r lili^L-'. N;;jv|i+L-' . n;;, v|i| 



+ L''-I^"c.c8l-l-Se+L-\gT" 



(8.5) 



Eq. (8.5) is to be combined with Eqs. (7.3), (7.4) and (7.6). In 

 case of twin-screw ships (7.6) is to be properly modified and terms 

 corresponding to Sp • Tp and Sg • Tg are to be introduced in (8.5), 



Some Elementary Concepts 



So far as small motions are considered forward speed and 

 r.p.m. remain almost constant and the rudder force and moment 

 may be regarded as functions of nominal helm 6. The yaw- rate/helm 

 relation is given by the transfer function 



1 + T,s 



Yvg= K • 5 (8.6) 



'♦'^ 1 + (T, + T2)s + T, TgS^ 



and the open loop heading response by Y, = (l/s) • Yj^g, which may 

 be used with Yg from Eq. (7. 5) to study the closed-loop system 

 with transfer function F = Y, /(I + Y, Yg) . 



The static gain and the three time constants in (8.6) are built 

 up from the coefficient of Eq. (8.5). T3 is always positive. The 

 two constants T, and Tg are given by the roots of the characteristic 

 equation. If s = - (l/T ), the root to the right on the real axis, 

 turns positive the ship is inherently unstable. The analytical criterion 

 for dynamic stability suggests the dynanaic stability lever 



866 



