Mathematical Structure of Modal Interactions in a Spinning Disk-Stationary Load System

Abstract

In a previous paper (Chen and Bogy, 1992) we studied the effects of various load parameters, such as friction force, transverse mass, damping, stiffness and the analogous pitching parameters, of a stationary load system in contact with the spinning disk on the natural frequencies and stability of the system when the original eigenvalues of interest are well separated. This paper is a follow-up investigation to deal with the situations in which two eigenvalues of the freely spinning disk are almost equal (degenerate) and strong modal interactions occur when the load parameters are introduced. After comparing an eigenfunction expansion with the finite element numerical results, we find that for each of the transverse and pitching load parameters, a properly chosen two-mode approximation can exhibit all the important features of the eigenvalue changes. Based on this two-mode approximation we study the mathematical structure of the eigenvalues in the neighborhood of degenerate points in the natural frequency-rotation speed plane. In the case of friction force, however, it is found that at least a four-mode approximation is required to reproduce the eigenvalue structure. The observations and analyses presented provide physical insight into the modal interactions induced by various load parameters in a spinning disk-stationary load system.

Impact of time-dependent nonaxisymmetric velocity perturbations on dynamo action of von Kármán-like flows

“…By analyzing the involved growth rates and frequencies in the phase-locked regime and in the modulated regime, we identify the transition between them as a spectral exceptional point where eigenvalues and eigenfunctions of two modes coincide [29,30]. The observed behavior is in close analogy with typical (resonant) mechanical systems subject to periodic forcing, like, e.g., spinning disk systems [31], or to the behavior observed in the stability study of water waves [32] and we will see, by analyzing the solution of a simple Mathieu equation, that the observed spectral structure is quite generic for systems under the influence of periodic forcing. Comparable effects have also been found in mean-field dynamos of αω type that were designed to explain the bisymmetric field pattern observed in spiral galaxies.…”

Stabilizing and destabilizing perturbations of PT -symmetric indefinitely damped systems

“…As was established in Kirillov [54,55], destabilization of the positive energy modes of a Hamiltonian system by indefinite damping or non-conservative positional forces is a basic mechanism leading to the onset of oscillatory instabilities induced by friction applied to rotating or moving continua [21] such as the singing wine glass [19,20] or a squealing disc brake [56,57], see Chen & Bogy [58], Ono et al [59], Yang & Hutton [60] and Spelsberg-Korspeter et al [61] for further mechanical examples, and see Kirillov et al [62] for an example of a similar effect in magnetohydrodynamics. …”

Sensitivity of Sub-critical Mode-coupling Instabilities in Non-conservative Rotating Continua to Stiffness and Damping Modifications

“…It is straightforward to see that the polynomial (35) becomes biquadratic in case when the following conditions are fulfilled…”

“…Typical rotor dynamical applications are related to stability of high-speed machinery such as turbine shafts and wheels, circular saws, disks of computer data storage devices, to name a few [34][35][36][37][38][39][40][41]. In such applications gyroscopic forces are significant and it is natural to consider non-conservative and dissipative forces as a perturbation of a conservative gyroscopic system.…”

“…It is known that variation in the mass and stiffness distribution may cause the so-called mass-and stiffness-instabilities for such speeds [37][38][39][40][41][42]. It is remarkable that these instabilities manifest themselves as instability bubbles [43] of complex eigenvalues that originate due to unfolding of double eigenvalues at some crossings of eigencurves of the Campbell diagram of the non-modified system [30][31][32][33][34][35][36][37][38][39][40][41][42][44][45][46][47][48]. The mass and stiffness instabilities can easily be identified with the interaction of the so-called waves of positive and negative energy -the instability mechanism that is well-known in hydrodynamics and that is typical for Hamiltonian systems such as conservative gyroscopic ones [42,43].…”