A Deterministic Limit for Rotordynamic Instability: Deriving the Critical Whirl Radius via Shaft Restitution

The prediction of catastrophic synchronous whirl and rotordynamic instability in high-speed turbines relies heavily on the classical Jeffcott rotor model and Campbell diagram eigenvalue analysis. While these frequency-domain frameworks effectively estimate critical speeds based on aggregate mass and stiffness matrices, they fail to deterministically define the exact spatial boundary where a localized radial bow triggers a cascading transition into destructive mechanical runaway. This paper introduces a strict continuum framework for macroscopic rotordynamic scaling. By modeling the rotating shaft as a dynamic kinematic balance between the spatial capacity for kinetic energy dispersion (structural restitution) and the localized rate of centrifugal forcing caused by mass unbalance, we derive a universal critical whirl radius (Rwhirl). We demonstrate that turbine disintegration is not merely a velocity-frequency threshold, but an exact deterministic limit where localized centrifugal energy injection strictly overpowers the advective kinematic dispersion capacity of the rotor-bearing span. This approach provides a precise, spatially defined boundary for mechanical failure. We propose a blueprint for Active Whirl Suppression (AWS) using distributed eddy-current spatial telemetry, enabling real-time deflection prediction and mitigation before catastrophic stator contact.