Hydrofoil Motions in a Random Seaway 



equations were sufficiently accurate for the initial studies, but proved to be in- 

 adequate when more detailed information became available and a more accurate 

 simulation was required. Varying coefficients had to be introduced. All of the 

 derivatives are functions of immersion depth and second order derivatives had 

 to be introduced to account for some of the more nonlinear functions. This re- 

 sults in a set of complicated equations. In fact, each variable has to be written 

 in the form of a Taylor series and linearisation of even the second order terms 

 can lead to significant errors, particularly in the roll derivatives. 



These equations became very cumbersome, difficult to mechanize on the 

 computer and still had significant inaccuracies in the roll terms. Because of 

 the complex analog computer set-up required and the inaccuracies that were 

 still present in the nonlinear "Taylor Series" equations, the so-called "explicit 

 variable" method of simulation was developed, in order to simplify the compu- 

 tation and to achieve greater coherence in the derivation of the longitudinal and 

 lateral equations for the surface piercing hydrofoil. Each derivative is a func- 

 tion of immersion depth, which in turn is a function of heave, pitch and wave ef- 

 fects, all of which are derived from the longitudinal equations. In the explicit 

 variable simulation, this coherence can be achieved because all forces are de- 

 rived from two parameters; the lift-curve slope for a given foil element, and 

 the total angle of attack on that element due to all motions about the craft centre 

 of gravity. 



The development of these equations from Euler's basic equations of motion 

 is outlined below for the axis convention of Fig. (i). 



(fert^itr^) 



Assume a rigid body with OJC^ as a plane of symmetry. 



Figure (i) 



'Je 



l\ 



'-If 



I -J> 



'1 



Assume a rigid body with Oxz as a plane of symmetry. Euler's equations 



are: 



Linear Motions and Forces 



m(U + QW - RV) = X - mg sin 



615 



(1) 



