On blade/casing rub problems in turbomachinery: An efficient delayed differential equation approach

This paper presents an alternative pathway in studying the ubiquitous blade/casing rub problem in turbomachinery. Bladed disks interfere with the stationary shroud (casing) for a variety of reasons, such as axial offsets, thermal expansions. Both components being compliant, time-varying interface characteristics, nonlinearities and uncertainties in the rub forces make this dynamics very complex to model and analyze. The main idea in this paper originates from the conjecture that this dynamics is inherently akin to internal machining operation which also deals with compliant cutters (blades) but relatively more rigid workpiece (casing). This analogy directs our attention to the fact that the blade/casing impingement dynamics manifests a ‘regenerative mechanism’ which is impregnated with time delays. The ensuing time-delayed system (TDS) can be stable, which is ideal. If it is unstable, however, the interference amplitudes between the blade and the casing grow, and the nonlinear effects become dominant. If the components survived the exercise, this evolution would reach a limit-cycle behavior. Existing literature indicates that this limit cycle mode is the common state of operation in most modern-day turbomachinery. Consequently, the state-of-the-art research effort is focused on minimizing its amplitude to alleviate the destructive levels of fatigue effect. In this article we consider a different perspective in looking at these problems, by proposing the conditions to achieve stable rub interference. For this, a recent mathematical tool of the authors’ group called the Cluster Treatment of Characteristic Roots (CTCR) is deployed. CTCR declares the complete stability outlook of such time-delayed systems in the space of the operational and design parameters. We show how this new capability can assist the design process of the blade-casing interface. Simulations, relevant stability observations and comparisons with a peer technique are provided for some case studies to demonstrate the capabilities of the approach.