Abstract

A mathematical model of damped vibrations of three-layer plates formed by two rigid anisotropic layers and a soft middle isotropic viscoelastic polymer layer is proposed in this paper. The model is based on the Hamilton’s variational principle, the refined theory of first-order plates, the model of complex modules, and the principle of elastic-viscoelastic correspondence in the linear theory of viscoelasticity. The frequency-temperature dependence of the elastic-dissipative characteristics is considered negligible for rigid layer materials; however, this dependence is taken into account for the viscoelastic polymer of the soft layer. By minimizing the Hamilton functional we reduce the problem of damped vibrations of anisotropic structures to the algebraic problem of complex eigenvalues. The Rietz method using the Legendre polynomials as coordinate functions is applied to form the system of algebraic equations. The real solutions are found. When determining the complex natural frequencies of the plate, real natural frequencies obtained are used as their initial values, and then the complex frequencies are calculated by the third-order iteration method. The results of the study of the convergence of the numerical solution are discussed. The estimation of reliability of the mathematical model and the numerical solution method obtained by comparing the calculated and the experimental values of natural frequencies and loss factors is presented.

中文翻译：

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粘弹性三层复合板的耦合振动。1.问题的表述

摘要

本文提出了由两个刚性各向异性层和一个柔软的各向同性粘弹性聚合物中间层组成的三层板的阻尼振动的数学模型。该模型基于汉密尔顿的变分原理，一阶板的精细理论，复杂模块的模型以及粘弹性线性理论中的弹性-粘弹性对应原理。弹性耗散特性的频率-温度依赖性对于刚性层材料而言可以忽略不计；然而，对于软层的粘弹性聚合物考虑了这种依赖性。通过最小化汉密尔顿函数，我们将各向异性结构的阻尼振动问题简化为复特征值的代数问题。采用勒让德多项式作为坐标函数的里兹方法，形成了代数方程组。找到了真正的解决方案。在确定板的复数固有频率时，将获得的实际固有频率用作其初始值，然后通过三阶迭代法计算复数频率。讨论了数值解的收敛性研究结果。给出了数学模型的可靠性估计，并通过比较计算出的固有频率和损耗因子与实验值得到了数值解法。将获得的真实固有频率用作其初始值，然后通过三阶迭代方法计算复数频率。讨论了数值解的收敛性研究结果。给出了数学模型的可靠性估计，并通过比较计算出的固有频率和损耗因子与实验值得到了数值解法。将获得的真实固有频率用作其初始值，然后通过三阶迭代方法计算复数频率。讨论了数值解的收敛性研究结果。给出了数学模型的可靠性估计，并通过比较计算出的固有频率和损耗因子与实验值得到了数值解法。