456 F. W. Boggs and N. Tokita 
ae : 1/3 ; ; : : 
Retaining only the terms in (a, 0 R) /3 and those independent of this quantity, in other 
words neglecting the negative powers of @R, we will obtain the set of equations 
A [s' =auee Y,(C¢' + ¥¢)] 
+B {cayary? y' =e Yya| -éayy" + C(aR) vy] = 0 
(3.8) 
A [-ia¢ - U.P Y,,(Co' + y¢)] 
. TT . 1/3 f] 
tp 18) {- ay YP Woe [-ia, y" + C(a,aR) We sr vy |} = 0. 
We make the substitution 
e, = U,P Y 52 
z (3.9) 
en = ULP Yi. 
and rearrange terms, and we obtain 
A [(1 = ©)" = e,y¢| 
1/3 ' : m 
+ B [(ajaR) (GI RUE Os ie,a,y"| = 0 
(3.10) 
A [-e,c¢' - (10 + e,7)9| 
3 
1/ . UT) 
+ B [-e,€¢a,08) Ws (t0 Eee yi ats rear ] = 0. 
The condition of existence of solutions in this set of equations is the setting equal to zero 
of the determinant of the coefficients of A and B. Again we will note that when e, and e, 
are equal to zero this condition will reduce to the one obtained for a rigid plate. This is as 
it should be since e, and e, vanishing means zero compliance; the plate cannot be deformed. 
It may be readily seen that the matrix of the coefficients of A and B in Eq. (3.10) is 
given by 
1-e,C -e,7 p' (aaRy¥3 0 we, ay” 
+ .. “@eim) 
SENG Clann 9) p wy OF Meta wa 
To obtain the eigenvalues of the differential equation, we must set the determinant of this 
matrix equal to zero. 
Since the second matrix in Eq. (3.11) is singular, we cannot divide through by it, but if 
the prefactor on the first term is not singular, we may multiply through by its inverse so that 
Eq. (3.11) is transformed to 
