394 Alex Goodman 
motion is produced by the Planar Motion Mechanism when the two struts move sinusoidally 
180 degrees out of phase with each other. 
The pure heaving motion shown in Fig. 3b is obtained when both struts move sinusoi- 
dally in phase with each other. This results in a motion whereby the model CG moves in a 
sinusoidal path while the pitch angle @ is invariant with time. 
The pure pitching motion shown in Fig. 3c is obtained by moving both struts out of 
phase with each other; the phase angle between struts, ¢,, is dependent upon frequency of 
oscillation, forward speed, and distance of each strut from the CG. This relationship can 
be expressed as 
and is derived as Eq. (A20) in Appendix A. The resulting motion is one in which the model 
CG moves in a sinusoidal path with the model axis always tangent to the path (angle of at- 
tack a = 0). 
The process for obtaining translatory acceleration derivatives from pure heaving tests 
is represented diagrammatically in Fig. 4. The diagrams across the top of the figure show 
the motions of the aft and forward struts with respect to each other. Corresponding posi- 
tions of a component resolver (which has been replaced by a sine-cosine potentiometer in 
the new system), provided with the electrical system to rectify the sinusoidal signals from 
the force balances, are also shown. At the left is a column of graphs showing the resulting 
motions and forces at the CG. The right-hand column contains the mathematical relationships 
represented by each graph. Descending from the top of Fig. 4, there is the vertical displace- 
ment z curve, the associated velocity 2 curve, the associated acceleration z curve, and the 
vertical force Zp curve. It may be noted that the Zp curve is displaced in point of time 
from the z curve by phase angle ¢. Thus Zr can be considered as being made up of two 
components, one in phase with the motion at the CC, Z;,, and the other in quadrature with 
the motion at the CG, Z,,,. The shaded area per cycle under each curve represents the mag- 
nitudes of Z;, and Z,,,, respectively. 
The process for obtaining rotary and angular acceleration derivatives from pure pitching 
tests.is represented diagrammatically in Fig. 5. The order followed is similar to that shown 
in Fig. 4. In this case, the pitch angle traces (0, 0, and 6) are of primary interest. The Zp 
curve is displaced in point of time from the 0 curve by phase angle ¢. The procedure for 
resolving the resultant force into in-phase and quadrature components is similar to that for 
the pure heaving case. The shaded area per cycle under each curve represents the magni- 
tudes of Z;, and Z,,,, respectively. 
In the pure heaving case the in-phase component of force is directly related to the linear 
acceleration and, therefore, can be used to compute explicitly the associated acceleration 
derivatives. Similarly, in the pure pitching case the in-phase component of force is directly 
related to the angular acceleration and the quadrature component is directly related to the 
angular acceleration and the quadrature component is directly related to the angular velocity. 
Thus both the angular acceleration and rotary derivatives can be computed explicitly. The 
relationships between the various rotary and acceleration derivatives and the respective 
