Analysis Used in Submerged Body Research 433 
The system that will be analyzed uses a slider crank mechanism having a large ratio of 
length of connecting rod to eccentricity. There is no appreciable error introduced, therefore, 
by assuming that the motion is sinusoidal for purposes of analysis. Referring to the notation 
and schematic diagram shown in Fig. 5, the motion of the forward and aft struts (taken with 
respect to the midposition of the forward strut) can be expressed as follows: 
Z, =| ay Sinuat (Al) 
Z, = a, sin (at - ,) (A2) 
where the crank arms @, and a, are assumed to be variables. The vertical displacement of 
the body CG is, therefore, 
= Pde ag ie 2 
per Naa ee ee (43) 
Expanding Eq. (A3) results in 
igo ta OY aes ame ae ae 
= ee os ets: t 
Zz, b x, + a, cos d,} sin wt as sin ?, cos w (A4) 
where the strut spacing with respect to the body CG, x, and x, are assumed also to be 
variables. The vertical velocity of the body CG can be obtained by differentiating Eq. (4) 
and expressed as 
: a,X, [/X, a, a, 
2 laa ae x, ee GES cos ES es sin wt]. (A5) 
For the case of pure pitching the struts are out of phase with each other, and a body 
pitch angle results and can be expressed in terms of the motion of each strut as 
g = ++ (A6) 
or expanding: 
eet a2 mes es t 
= 5 ( ae @ sin 2 ess cos wt| . (A7) 
The vertical velocity of the CG with respect to the inertial axes can be written as 
Zz = wcos 0 = a sin ¢ (A8) 
oO 
and for small angles, where cos @ = 1, sin 0 = 0, and u = U, can be simplified to 
