436 Alex Goodman 
sine-cosine potentiometer). The pitch angle trajectory of the model is a direct function of 
the phase angle between struts as shown by Eq. (A21). Therefore, the zero-pitch-angle point 
(wt), will vary with strut-phase angle. This requires that the resolver be repositioned and 
synchronized with the zero-pitch-angle point for each condition of pure pitching. 
The relationship between the zero-pitch-angle point (wt), and the strut-phase angle can 
be determined from Eq. (A5). At the point (w£), in the cycle, 6 = 0, z, = zo, and z,isa 
maximum. Therefore for x,/x, = @,/a, = 1, Eq. (A5) reduces to 
z, = 0 = (1 + cos $¢,) cos wt + sin ¢, sin ot (A22) 
and 
tan wt = seals Bs (A23) 
sin $, 
or 
(wt) = tan“? | 
sin ¢, (A24) 
Differentiating with respect to ¢, yields 
owt) 1 
3d, ra 2 : (A25) 
APPENDIX B 
Derivation of Reduction Equations 
The Planar Motion Mechanism separates the motions of a body moving through a fluid 
into the hydrodynamically pure pitching and pure heaving motions as defined in Fig. 3. The 
differential equations of motion referred to a moving body axis system are used to establish 
a direct and explicit relationship between the various rotary and acceleration derivatives and 
the measured quadrature and in-phase components of the forces and moments. The linear 
force and moment equations describing the body motions with respect to the initial equilibrium 
conditions can be written as: 
Transverse Force: 
Y= Y.r+ (5 ety, + n) rU + (Y.-m)v 
2 gent 4 
+ € pr uv, v + (5 et zi pt Y.p. (Bl) 
