Analytical solutions of the problem on free vibrations of an orthotropic cuboid with free faces, based on the use of trigonometric functions as base ones in all three directions, are constructed. It is established that the solutions obtained describe the purely shearless vibration modes occurring in the body considered at particular ratios between its faces and physicomechanical characteristics of its material. It is shown that, on passing to the zero harmonics in one or two directions, or at zero values of six or all Poisson ratios, the solutions constructed are reduced to the equations corresponding to the classical theory of plates or rods. It is established that, at certain geometrical and physicomechanical parameters, the body considered can have natural frequencies much lower than the known ones.

中文翻译：

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具有自由面的正交各向异性长方体的自由振动 3D 问题的解析解

基于使用三角函数作为所有三个方向的基函数，构造了具有自由面的正交各向异性长方体的自由振动问题的解析解。已经确定，所获得的解决方案描述了在物体表面与其材料的物理机械特性之间的特定比率下发生在物体中的纯无剪切振动模式。结果表明，当传递到一个或两个方向上的零谐波时，或者在六个或所有泊松比为零值时，所构造的解被简化为对应于经典板或棒理论的方程。已经确定，在某些几何和物理机械参数下，所考虑的物体的自然频率可能远低于已知频率。