Abstract—

Axisymmetric and non-axisymmetric in-plane vibrations of thin radially growing disc are considered in the frame of the Kelvin–Voigt model of linear viscoelasticity. The main focus of the research is on asymptotic behavior of solutions to the model. Mixed boundary value problems are formulated and converted into standard form by means of time dependent coordinate transformation. The boundary value problems obtained are transformed into infinite systems of ordinary differential equations using the Galerkin-Kantorovich method. It is shown that both axisymmetric and non-axisymmetric problems can be considered in terms of the same mathematical model. The simplified single mode model of the growing disc vibration is formulated and analysed. In the frame of this model the exact solutions are obtained in some interesting special cases. In the general case, solutions are derived using the Wentzel–Kramers–Brillouin (WKB) method. Depending on the exponent of radial growth, the vibration amplitude of the disc demonstrates qualitatively different behavior: if the exponent is less than 0.5 the vibration amplitude decays, if it is more than 0.5 the amplitude increases without bound no matter what size of damping coefficient is assumed. If the exponent equals 0.5 the vibration behavior depends on damping, namely there exist the critical damping coefficient and if the damping coefficient of a particular mode is less than this coefficient, the vibration amplitude grows without bound. Otherwise, if the damping coefficient of the mode is larger than the critical damping coefficient the vibration amplitude decays to zero. In the case where damping coefficient equals to the critical damping one, the amplitude of vibration remains constant. All of the qualitative results obtained for the single mode model are numerically tested and verified using a truncated multimodal system.

中文翻译：

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成长中的粘弹性盘的轴对称和非轴对称振动

摘要-

在线性粘弹性的Kelvin-Voigt模型框架中考虑了径向增长的薄圆盘的轴对称和非轴对称面内振动。研究的主要重点是模型解的渐近行为。制定了混合边值问题，并通过与时间有关的坐标变换将其转换为标准形式。使用Galerkin-Kantorovich方法将获得的边值问题转化为常微分方程的无限系统。结果表明，可以根据相同的数学模型同时考虑轴对称和非轴对称问题。建立并分析了盘片不断振动的简化单模模型。在此模型的框架中，在一些有趣的特殊情况下可以获得确切的解决方案。在一般情况下，解决方案是使用Wentzel–Kramers–Brillouin（WKB）方法得出的。视径向增长的指数而定，圆盘的振动幅度在质量上表现出不同的行为：如果指数小于0.5，则振动幅度衰减；如果指数大于0.5，则无论阻尼系数大小如何，振幅都会无限制地增大。假定。如果指数等于0.5，则振动行为取决于阻尼，即存在临界阻尼系数，并且如果特定模式的阻尼系数小于该系数，则振动幅度会无限制地增大。否则，如果模式的阻尼系数大于临界阻尼系数，则振动幅度将衰减为零。在阻尼系数等于临界阻尼一的情况下，振动幅度保持恒定。使用截短的多峰系统对单模模型获得的所有定性结果进行了数值测试和验证。