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.

中文翻译：

---

对地静止轨道上完全自由失重厚板的耦合动力学问题


平面运动与完全自由板的柔性振动之间的耦合效应对其中的动力学分析提出了很大的挑战。本文采用单变量简单板理论建立了完全自由失重厚板的动力学模型，描述了对地静止轨道上一些空间板型结构的动力学问题。然后，构建一个 49 点多对称方案，以对称形式求解所提出的动力学模型;考虑到与前六种模式相关的初始条件，采用 49 点多对称方案分别给出了不同厚度比的板上编号节点的横向位移，从中再现了厚度比对板固有振动频率的影响。从横向位移的数值计算结果可以发现，零位移直线在初始条件下的存在和分布特征共同决定了板的运动模式，这揭示了零位移直线在刚柔耦合动力学行为中的关键作用（平移、 旋转和振动）。为了验证本文中报告的数值结果的有效性和精度，给出了当前文档中获得的无量纲固有频率与报告的无量纲固有频率之间的相对误差。 这项工作的主要贡献在于，阐述了单变量简单板理论描述完全自由失重厚板耦合动力学问题的适用性，验证了所采用的多对称方法的高精度，为完全自由失重厚板的振动问题提供了一种有效的建模方法和有效的仿真方法。