## Parametric Resonances in an Annular Disc, with a Rotating System of Distributed Mass and Elasticity; and Effects of Friction and Damping

### JE Mottershead, H Ouyang (Liverpool University), MP Cartmell (Edinburgh University) & MI Friswell (University of Wales Swansea)

### Proceedings of the Royal Society of London, Series A: Mathematical, Physical and Engineering Sciences, Vol. 453, No. 1956, 1997, pp. 1-19

### Abstract

The parametric resonances that occur in a stationary annular disc under the action of a distributed mass-spring-damper system which rotates, with friction, around the disc at subcritical speeds is the main focus of this article. The distributed system occupies an annular sector and the stiffness, mass, damping and friction are each expanded in a Fourier series. When the distributed system is rotated then a set of stiffness, mass, damping and friction terms are obtained that vary both spatially and temporally. Finite element discretisation is applied to the disc and the distributed system and the parametric resonances are determined by a multiple time-scales analysis of the finite element equations.
The equations that define the transition curves are generally complex, although they are strictly real when damping and friction are omitted. Numerical results show that friction has a destablising effect over the entire subcritical speed range. Disc damping and damping in the rotating system both tend to suppress the vibrations of the disc when the speeds are subcritical. The effects of the distributed mass and stiffness are found to be almost neutral at subcritical speeds, but active in the supercritical range where the findings of other researchers are available for comparison and found to be in agreement.

### Paper Availability

This material has been published in the *Proceedings of the Royal Society of London, Series A: Mathematical, Physical and Engineering Sciences,* Vol. 453, No. 1956, 1997, pp. 1-19, the only definitive repository of the content that has been certified and accepted after peer review. Copyright and all rights therein are retained by The Royal Society. This material may not be copied or reposted without explicit permission.

Link to paper using doi:10.1098/rspa.1997.0001

Proceedings of the Royal Society of London, Series A

The Royal Society of London