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we have that Finally, substituting Eq. (12) into (11) and dividing by co 2 we obtain the standard equation (l/co 2 ) Xl = (13) Assuming that kn" 1 is available, the only matrix inversion required in Eq. (13) is for the product (Ti2 r miiTi 2 ) which, for three-dimensional structures, is of the order (6 X 6). This compares with two inversions in Eq. (9), one for (Ti 2 r mi 2 + m 22 ) of order (6 X 6) in R and another for (I -T 12 R) of order (n X ri), where n is the number of degrees of freedom in Xi. It follows therefore that, in general, if the introduction of a few massless node points in the structure is an acceptable idealization so that m 22 = 0 and mi 2 = 0 (zero inertia in the direction of x 2 ) then it would be preferable to compute the nonzero frequencies of a free-free system from Eq. (13).

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Coupling between lateral and longitudinal oscillations of a launch vehicle can occur as either mass or spring offset. While, in general, mass coupling usually acts through the inertia of the total vehicle, spring coupling acts only through the asymmetrically mounted mass. The Saturn A S-502 space vehicle was analyzed for both types of coupling, with the POGO phenomenon acting as the longitudinal excitation force. Although the mass coupling influence was small, a large lateral acceleration resulted from a spring-type coupling between the ascent and descent stages of the LEM during the latter part of S-IC burn. A response analysis of the vehicle using time-varying coefficients was made and compared with observed flight conditions. The results of this study confirm the necessity of including dynamic crosscoupling mechanisms for adequate prediction of vehicle response. The resulting simulation closely duplicated flight results. The important vehicle parameters involved in creating the lateral/longitudinal coupling phenomenon were studied.