Abstract

Refined geometrically nonlinear equations of motion are derived for elongated rod-type plates which are made of composite materials. The equations are derived on the basis of the previously proposed relations of the consistent version of the geometrically nonlinear theory of elasticity at small deformations and the classical Bernoulli–Euler model. In the model, the axes of the chosen coordinate system do not coincide with the orthotropy axes of the plate material in a plane stress-strain state. It is shown that for a plate made of cross-ply reinforced composite material the derived equations, compiled even in the geometrically linear approximation, describe coupled bending-torsional oscillations. As an example of their application, numerical solutions of linear problems on free and forced bending-torsional oscillations of an anisotropic elongated plate fixed on a spherical hinge are found. It is assumed that such a supporting element of the plate is located at some small distance from the end cross-section and subjected to kinematic loading by setting the deflection and torsional angle according to the harmonic law of their variation in time with a given frequency. The model under consideration is intended to simulate natural processes and structures in applied engineering problems aimed at developing innovative oscillatory biomimetic propulsion systems.

中文翻译：

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固定在球形铰链上的各向异性细长板的自由和受迫弯曲扭转振动

摘要

针对复合材料制成的细长杆型板推导了精细的几何非线性运动方程。这些方程是根据先前提出的小变形几何非线性弹性理论和经典伯努利-欧拉模型的一致版本的关系推导出来的。在模型中，所选坐标系的轴与平面应力应变状态下板材料的正交各向异性轴不重合。结果表明，对于由交叉层增强复合材料制成的板，即使在几何线性近似中编译的导出方程也描述了耦合的弯曲扭转振荡。作为其应用的一个例子，找到了固定在球形铰链上的各向异性细长板的自由和受迫弯曲扭转振动线性问题的数值解。假设板的这种支撑元件位于距端部横截面一定距离处，并根据给定频率随时间变化的调和定律设置偏转角和扭转角，从而承受运动载荷。正在考虑的模型旨在模拟应用工程问题中的自然过程和结构，旨在开发创新的振荡仿生推进系统。