Axisymmetric and Non-Axisymmetric Vibration of Thin Growing Viscoelastic Disc

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.