Davis and Oates 

 m(V + RU - PW) = Y + mg cos sin $ (2) 



m(W + PV - QU) = Z + mg cos cos O (3) 



Angular Motions and Moments 



AP - ER + QR(C- B) - EPQ = L (4) 



BQ + RPCA-C) + P2-R2 = M (5) 



EP + CR + PQ(B - A) + EQR = N (6) 



Velocities Along Space Axes 



Xg = U cos cos W + V( sin $ sin cos 'I' - cos <1> sin ^f) 



+ W(cos O sin sin ^ - sin O cos W) (7) 



Yg = U cos sin W + V( sin $ sin sin ^ + cos $ cos f) 



+ W(cos sin sin ^ + sin cos W) (8) 



Zg = -U sin + V sin O cos + W cos cos (9) 



Relations between Angular Velocities 



P = <i) - w sin (10) 



Q = cos + ¥ cos sin $ (11) 



R = ¥ cos cos <[) - © sin $ (12) 



= Q cos - R sin <D (13) 



<i> = P + Q sin $ tan + R cos tan (14) 



W = (Q sin cl) + R cos O) sec 0. (15) 



All of the dynamic relationships that are necessary to investigate the mo- 

 tions of a body in response to impressed forces and moments are given in the 

 above equations. These equations are general and are accurate for motions of 

 any magnitude. The hydrofoil motions, however, are relatively limited, in which 

 case small angle approximations can be made for this craft without any signifi- 

 cant loss of accuracy, thus the equation can be simplified. This simplification 

 can be accomplished by writing all of the equations in terms of their deviations 

 from a fixed or reference condition with the exception of the craft forward speed 

 (u) and heading angle {^), which are subject to large changes. Small approxi- 

 mations cannot be applied to them. The parameters in the deviation equations 

 will be denoted by lower case letters. Reference values will be denoted by the 

 suffix zero. Thus (u,V,w) (p,Q,r) (©.O,*!*) are redefined as 



616 



