Dynamic Instability of A Spinning Thick Disk Under Nonconservative Traction

This paper presents the dynamic stability analysis of a spinning, thick, annular disk loaded by a circumferentially distributed frictional traction. Considering the automotive/aircraft disc brake as a representative system for this study, the brake rotor is modeled as a spinning annular disk in the context of the Mindlin's thick plate theory to ensure a more accurate estimation of the eigenvalues when the disk involves high circumferential vibration modes that are often observed in unstable disc brake rotors. The frictional traction is decomposed into in-plane and transverse components. The in-plane component is equilibrated by membrane stresses while the transverse component is a nonconservative follower-type force that is the source of the dynamic instability of the disk. The corresponding nonconservative eigenvalue problem is formulated based on the modal expansion theory for traveling waves and direct discretization technique. The brake pad or stator is modeled as a second order viscoelastic subgrade that reacts to both transverse and shearing motion of the disc brake rotor. The instability behavior of the disk is investigated by examining the resulting complex eigenvalues under various combinations of the system parameters such as frictional traction, geometry of the disk, and viscoelastic properties of the pad lining. In order to assess the effects of shear deformations and rotary inertia of the disk model under such loading conditions, results are compared with those from the classical Kirchhoff–Love's thin disk model. Based on the phase angle information of the eigenvalues, it is found that there exists a critical value of friction, slenderness ratio of the disk, and the size of pad lining at which the transverse amplitude growth rates of high circumferential vibration modes drastically increase. Unless damping is present in the disc brake rotor or pad lining, vibration modes with repeated eigenvalues are found to be always unstable either by means of divergence or flutter even for a very small nonconservative traction load.