The solution to Equation (14) is cumbersome; however, it can be accomplished using 
standard numerical techniques. Introducing the state vector [x;, X4,X3,X4. Xe, X¢] 
where X) = Voc 
Xy = 20g 
or 
ACG 
CELA 
X¢ = 
Equation (14) can be rewritten, using matrix algebra, as 
> > 
Ax = g (15) 
so that 
Te eA eal ee 
x=A’g (16) 
where A“! is inverse of the inertial matrix A. Equation (16) is now in a form that lends 
itself to integration by using a numerical method such as the Runge-Kutta-Merson integration 
routine. 
EQUATIONS OF MOTION, SIMPLIFIED FOR CONSTANT SPEED 
Assuming that the perturbation velocities in the forward direction are small in comparison 
to the speed of the craft, the equations of motion may be further simplified by neglecting 
the perturbations and setting the forward velocity equal to a constant, i.e. 
og = CONSTANT 
If it is also assumed that the thrust and drag forces are small in comparison to the hydrody- 
namic forces and that they are acting through the center of gravity, the equations of motion 
may be written as 
