Beek and Tuck 
where 
M = ship mass 
x ,0,z_.) = coordinates of centre of gravity 
Gi625G : 
k, = roll radius of gyration 
k, = pitch radius of gyration 
kigeks = yaw radius of gyration 
I4¢ = roll-yaw product of inertia (small). 
C;. is a matrix of restoring force coefficients, including all hydro- 
static effects and mooring forces, if any, but no hydrodynamic effects. 
The hydrostatic contributions to Cij are zero except for 
= A 
sii er WL, 
C35= C53 mMSPE sit Aw 
2 (2.4) 
= A 
C55 pek, Aw 
Cry, = Mg(z,, ~- Zo) 
where 
Avr, = waterplane area 
x = x coordinate of centre of flotation 
BS (centroid of A ) 
WL 
2 : ; oA 
kp radius of gyration of WL 
24 = z co-ordinate of meta-centre. 
Possible mooring force contributions to Cij are discussed in Section 
4. 
The remaining terms in the equations of motion are hydro- 
dynamic in nature, consisting of the hydrodynamic forces involving 
T;; and exciting forces of amplitude Fj. Ti; is a complex-valued 
transfer function equal to the hydrodynamic force in the ith mode 
due to a unit amplitude movement of the ship in the jth mode, and 
can be written (Tuck 1970) in the form 
Dc = 8 eae eae. (2.'5) 
