Data-Driven Modeling of Tracked Order Vibration in Turbofan Engine

Aircraft engines are one of the most heavily instrumented parts of an aircraft, and the data from various types of instrumentation across these engines are continuously monitored both offline and online for potential anomalies. Vibration monitoring in aircraft engines is traditionally performed using an order tracking methodology. Currently, there are no representative and efficient physics-based models with the adequate fidelity to perform vibration predictions in aircraft engines, given various parametric dependencies existing among different attributes such as temperature, pressure, and external conditions. This gap in research is primarily attributed to the limited understanding of mutual interactions of different variables and the nonlinear nature of engine vibrations. The objective of the current study is three-fold: (i) to present a preliminary investigation of tracked order vibrations in aircraft engines and statistically analyze them in the context of their operating environment, (ii) to develop data-driven modeling methodology to approximate a dynamical system from input-output data, and (iii) to leverage these data-driven modeling methodologies to develop highly accurate models for tracked order vibration in a turbo-fan engine valid over a wide range of operating conditions.

Off-the-shelf data-driven modeling techniques, such as machine learning methods (eg., regression, neural networks), have several drawbacks including lack of interpretability and limited scope, when applying them to a complex multiscale multi-physical dynamical system. Moreover, for dynamical systems with external forcing, the identified model should not only be suitable for a specific forcing function, but should also generally approximate the input-output behavior of the data source. The author proposes a novel methodology known as Wavelet-based Dynamic Mode Decomposition (WDMD). The methodology entails using wavelets in conjunction with input-output dynamic mode decomposition (ioDMD). Similar to time-delay embedded DMD (Delay-DMD), WDMD builds on the ioDMD framework without the restrictive assumption of full state measurements. The author demonstrates the present methodology's applicability by modeling the input-output response of an Euler-Bernoulli finite element beam model, followed by an experimental investigation.

As a first step towards modeling the tracked order vibration amplitudes of turbofan engines, the interdependencies and cross-correlation structure between various thermo-mechanical variables and tracked order vibration are analyzed. The order amplitudes are further contextualized in terms of their operating regime, and exploratory data analyses are performed to quantify the variability within each operating condition (OC). The understanding of complex correlation structures is leveraged and subsequently utilized to model tracked order vibrations. Switching linear dynamical system (SLDS) models are developed using individual data-driven models constructed using WDMD, and its performance in approximating the dynamics of the $1st$ order amplitudes are compared with the state-of-the-art time-delay embedded dynamic mode decomposition (Delay-DMD) and Lasso regression.

A parametric approach is proposed to improve the model further by leveraging previously developed WDMD and Delay-DMD methods and a parametric interpolation scheme. In particular, a recently developed pole-residue interpolation scheme is adopted to interpolate between several linear, data-driven reduced-order models (ROMs), constructed using WDMD and Delay-DMD surrogates, at known parameter samples. The parametric modeling approach is demonstrated by modeling the transverse vibration of an axially loaded finite element (FE) beam, where the axial loading is the parameter. Finally, a parametric modeling strategy for tracked order amplitudes is presented by constructing locally valid ROMs at different parametric samples corresponding to each pass-off test. The performance of the parametric-ROM is quantified and compared with the previous frameworks.

This work was supported by the Rolls-Royce Fellowship, sponsored by the College of Engineering, Virginia Tech.