There is a drive towards minimising operating clearances within turbomachines in order to limit reverse leakage flows and hence improve their efficiency. This increases the likelihood of contact occurring between the blade and the casing, which can give rise to high amplitude vibration. Modelling this interaction represents a significant computational challenge. The non-linear contact precludes the use of well-established linear methods, and is also subject to uncertainties: the contact law is imprecisely known and the exact geometry of imperfections that trigger contact may be unknown.
In this paper a novel approach is presented that aims to account for the uncertainties associated with the non-linearity in a non-probabilistic way. The worst case is sought, by framing the system as a constrained anti-optimisation problem. The target to be maximised represents a metric of the output of interest. The degrees of freedom of the anti-optimisation are the non-linear input forces (considered as external loads), and the constraints are designed to capture what is thought to be known about the non-linear contact law and geometry.
A realistic three-dimensional model of a turbine blade is used to explore the approach, with contact considered at the leading and trailing edge. The blade dynamics are described in terms of a linear transfer function matrix and the target metric of interest is chosen to be the peak displacement of the contact points. The non-linearity is taken to result from an offset shaft, giving a sinusoidal clearance variation. The blade is driven at constant frequency and the scope of the study is limited to finding bounds on periodic solutions. A variety of constraint conditions are explored that describe aspects of the non-linearity. For example, only compressive forces are permitted (no tension from the contact), and the displacement must not exceed the clearance.
The method yields encouraging initial results: constraints can be identified that give efficient estimates of the upper bound response of the system as a function of drive frequency. The results are compared with a benchmark time-domain simulation and are found to correctly over-predict the response without being overly conservative. Broad trends are also in agreement with the benchmark solution. The proposed method appears to be a promising approach for efficiently accounting for uncertainties associated with the non-linearity and thus improving blade design.