Coupling dynamic problem of a completely free weightless thick plate in geostationary orbit

The coupling effects between the plane motion and the flexible vibration of the completely free plate bring a great challenge of the dynamic analysis on which. In this paper, the single variable simple plate theory is employed to establish the dynamic model of the completely free weightless thick plate describing the dynamic problem of some spatial plate-type structures in the geostationary orbit. Then, a 49-point multi-symplectic scheme is constructed to solve the proposed dynamic model in the symmetric form. Considering the initial conditions associated with the first six modes, the transverse displacements of the numbered nodes on the plate with different thickness ratios are presented respectively employing the 49-point multi-symplectic scheme, from which, the effect of the thickness ratio on the natural vibrational frequency of the plate is reproduced. From the numerical results of the transverse displacements, it can be found that, the existence as well as the distribution characteristics of the straight line of zero displacement in the initial conditions codetermine the motion pattern of the plate, which reveals the key role of the straight line of zero displacement in the rigid-flexible coupling dynamic behaviors (coupling behaviors between the translation, rotation and vibration) of the plate. To verify the validity and the precision of the numerical results reported in this work, the relative errors between the obtained dimensionless natural frequency with the reported dimensionless natural frequency in current documents are given. The main contributions of this work include that, the applicability of the single variable simple plate theory for the describing of the coupling dynamic problem of the completely free weightless thick plate is illustrated and the high-precision of the multi-symplectic method employed is verified, which provide an effective modeling method and an effective simulation approach for the vibration problem of the completely free weightless thick plate.