Cascade flutter driven by aerodynamic instability leads to severe structural destruction of turbine blades in aircraft engines. The development of a sophisticated methodology for detecting a precursor of cascade flutter is one of the most important topics in current aircraft engineering and related branches of nonlinear physics. A novel detection methodology combining symbolic dynamics, dynamical systems, and machine learning is proposed in this experimental study to detect a precursor of cascade flutter in a low-pressure turbine. Two important measures, the weighted permutation entropy in terms of symbolic dynamics and the determinism in recurrence plots in terms of dynamical systems theory, are estimated for the strain fluctuations on turbine blades to capture the significant changes in the dynamical state during a transition to cascade flutter. A feature space consisting of the two measures obtained by a support vector machine, can appropriately be classified into three dynamical states: a stable state, a transition state, and a cascade flutter state. The proposed methodology is valid for detecting a precursor of cascade flutter.

## I. INTRODUCTION

Nonlinear time series analyses based on the theories of chaos and fractals have created broad platforms of quantitative ways of evaluating the complexity in nonlinear systems, leading to a review of standard and conventional linear time series analysis such as power spectra.^{1} Recent advances have brought about proactive attempts to reveal a rich variety of complex dynamics emerging in many disciplines from physics to engineering.^{2} Nonlinear time series analyses are currently expected to mainly attain two aims in the fields of propulsion engineering and related fluid science: (i) in-depth physical understanding and interpretation of the deterministic nature hidden in complex dynamics and (ii) development of substitute detectors for capturing the onset of unstable dynamical states. Gotoda and co-workers have recently shown the potential utility of nonlinear time series analyses incorporating symbolic dynamics, statistical complexity, and complex networks for the characterization of highly nonlinear dynamic behaviors in various physical settings: flame front instability driven by the interaction of buoyancy and centrifugal force,^{3} radiative heat loss,^{4} and medium-scale turbulent fire,^{5–7} including the proposal of early detection methodologies for thermoacoustic combustion oscillations and lean blowout.^{8–10}

Unstable self-excited blade vibrations, referred to as flutter, arise in an aircraft engine as a result of the exponential growth of blade amplitudes owing to aeroelastic instability. The formation of flutter, in some cases, gives rise to unacceptable structural destruction of turbine blades. The physical mechanism underlying the limit cycle oscillations during flutter, which is strongly associated with structural damping and mistuning, has been extensively studied for various model aircraft turbines from the laboratory scale to the industrial scale.^{11–13} The onset of cascade flutter imposes a restriction on the development of advanced jet engines, and its early detection is a long-standing and challenging subject in current aerospace propulsion engineering. Nevertheless, the early detection of cascade flutter by nonlinear time series analyses has not been fully examined except in two recent experimental studies.^{14,15} These studies revealed the importance of recurrence quantification analysis and the Hurst exponent for the early detection of flutter on NACA0012 in a wind channel, including the presence of a multifractal structure during the dynamical state prior to the onset of flutter. In addition to nonlinear time series analyses, machine learning inspired by information processing in the brain has widespread applications in technologies covering all scientific disciplines. The importance of a support vector machine (SVM) in the optimization of a blade design has recently been discussed in Refs. 16 and 17. Our primary interest in this study is to explore the applicability of a methodology combining symbolic dynamics, dynamical systems, and machine learning for the early detection of cascade flutter.

Japan Aerospace Exploration Agency (JAXA) has recently launched a national project including the suppression of cascade flutter in a low-pressure turbine, namely, the advanced fan jet research (aFJR) project,^{16} with an aim of making significant progress in the development of environmentally compatible technology for prospective super high-bypass-ratio turbofan engines. The development of an early detection methodology for cascade flutter is one of the most important tasks in this project. In this paper, we propose a pragmatic methodology that integrates dynamical systems, symbolic dynamics, and machine learning to detect a precursor of cascade flutter in a model aircraft turbine.

## II. EXPERIMENTS AND NONLINEAR TIME SERIES ANALYSES

### A. Experiments

As an important case study on cascade flutter that occurs in a turbofan jet engine system, a low-pressure turbine test rig identical to that in a previous study^{17} is used in this study. All the experiments are conducted using the altitude test facility of JAXA (JAXA ATF).^{18} Figure 1 shows the ATF chamber, the cascade, and the setup of the strain gauge. The cascade is set between the upstream and downstream struts. The upstream total pressure is maintained at the atmospheric pressure, and the outlet is connected to a low-pressure test chamber. After the pressure in the low-pressure test chamber is reduced, the air flows into the duct through a suction wind tunnel. The low-pressure turbine consists of 80 stator vanes, and the blade aspect ratio is set to 8.8 to reduce the structural natural frequency. Here, the blade aspect ratio is the blade span length divided by the axial chord length. The locations of the blades are labeled from 1 to 80 in the clockwise direction. The mass flow rate of inlet air *Q*, which is normalized by the designed mass flow rate, is varied from 59.2% to 92.6% to induce cascade flutter. The error range for the measurement of the air flow rate is within $\xb1$1.5%. Details of the experimental conditions at JAXA ATF are described in Ref. 18. The strain fluctuations $\u03f5\u2032$ in a single direction are measured using a strain gauge and are acquired using a data logger (KEYENCE Co., NR-ST04). The strain gauge is located on the pressure side of each turbine blade near the tip shroud, which means that the total number of strain gauges is 80 in this study. The measurement direction of the strain gauges is circumferential in the right half of the blades (Nos. 1–40) and spanwise in the left half of the blades (Nos. 41–80). The sampling frequency of $\u03f5\u2032$ is 20 000 kHz. The strain rate of the strain gauge is $2.14\xb11.0$%, which ensures sufficient accuracy to detect the onset of cascade flutter.

### B. Mathematical framework of a nonlinear time series analysis

The weighted permutation entropy,^{19} which is an extended version of the original permutation entropy^{20} incorporating the amplitude information of time series data, is a useful measure quantifying the randomness of complex dynamics. To estimate the weighted permutation entropy, Eq. (1) is considered as the existence probability distribution of the permutation patterns $Pw(\pi j)(j=1,2,\u2026,d!)$,

where $i=0,1,\u2026,Nps(=N\u2212d+1)$ and $d$ is the number of discrete data points considered for permutation patterns. $d$ is set to 5 in this study. $1A(B)$ is the indicator function of set $A$ defined as $1A(B)=1(0)$ if $B\u2208A(B\u2209A)$, and $w(ti)$ expresses the weight value given by

The weighted permutation entropy $Sw$ is obtained by substituting $Pw(\pi j)$ into the information entropy,

Here, $Sw$ ranges from 0 to 1. $Sw=1$ corresponds to a completely random process, whereas $Sw=0$ corresponds to a monotonically increasing or decreasing process.

Recurrence plots (RPs)^{21} in terms of dynamical systems enable us to quantify the ordered and disordered pattern structures between the points on the trajectories in the phase space. For the estimation of the determinism, RPs consisting of $Rij=\Theta (r\u2212\Vert \u03f5i\u2212\u03f5j\Vert )$ are constructed, where $\Theta $ is the Heaviside function and $r$ is the threshold distance. Hirata and co-workers^{22,23} have recently proposed weighted delay coordinates as an extended version of Taken’s embedding theory,^{24} namely, infinite-dimensional delay coordinates, for constructing the phase space. In accordance with their previous studies,^{22,23} the phase space is constructed as $\u03f5i=(\u03f5\u2032(ti),\lambda \u03f5\u2032(ti+\tau ),\u2026,\lambda (d\u22121)\u03f5\u2032(ti+(d\u22121)\tau ))$, where $d$ is the embedding dimension and $\tau $ is the delay time of the phase space. In this study, $d$ and $\lambda $ are set to 4 and 0.8, respectively. The determinism^{21} $Dr$ is defined as the ratio of the number of black dots in RPs forming the diagonal line structures to the total number of black dots,

Here, $p(l)$ is the frequency distribution of the length $l$ of the black diagonal lines, and $lmin$ is the minimum allowable length of the diagonal line.

The standard support vector machine (SVM)^{25,26} is a class of machine learning for recognizing patterns and is a binary classifier for determining the boundaries of classes. The use of the SVM has attracted considerable attention in data science over the past few decades.^{27} The SVM is adopted in a two-dimensional plane consisting of the weighted permutation entropy and the determinism in RPs. Using the SVM, an optimal separating hyperplane in the feature space with the given training data set ${xi,yi}$ is determined, where $i=1,2,\u2026,n$, $xi$ is the input vector, and $yi$ is the class label, which takes a value of +1 or $\u2212$1. In this study, after dividing samples of ${xi,yi}$ into three clusters on the $Sw$-$Dr$ plane by the $k$-means clustering method,^{27} a SVM classifier is used for the derived cluster centers. To solve the nonlinear optimization problem of the decision function $f(x)$, the following Lagrangian function is considered:

where $a(=(a1,a2,\u2026,an)T)$ and $\mu (=(\mu 1,\mu 2,\u2026,\mu n)T)$ are the Lagrange multipliers. Under the stationary condition ($\u2202L/\u2202\omega =\u2202L/\u2202b=\u2202L/\u2202\xi =0$ and $\mu i\xi i=0$), the dual representation of the Lagrangian function $L~(a)$ is obtained as

s.t. $0\u2264ai\u2264\gamma $, $\u2211i=1naiyi=0$, and $k(xi,xj)$ is the radial basis function kernel. Finally, the output of the following $f(x)$ is obtained by maximizing Eq. (7),

## III. RESULTS AND DISCUSSION

The temporal evolution of $\u03f5\u2032$ at a representative location (blade no. 27) is shown in Fig. 2 for different $Q$, together with the power spectrum density. Aperiodic fluctuations are clearly observed during the stable state at $Q\u226481.5%$, whereas the dynamic behavior of aperiodic fluctuations undergoes a significant transition to limit cycle oscillations with large amplitudes at $Q=85.2%$. In this study, cascade flutter is observed in the subsonic regime with a Mach number of approximately 0.6. The cascade is designed to induce the first torsion (1T) mode along the torsion axis with increasing $Q$. Our preliminary study^{28} has confirmed that the dominant mode during limit cycle oscillations corresponds to the 1T mode, and the interblade phase angle is asymptotically almost constant value as $Q$ exceeds 85.2$%$. The onset of the subharmonic mode of the 1T mode ($\u223c$650 Hz) in the power spectrum at $Q=85.2%$ is associated with the vibrations of the pressure field around blades.

Spatial variations in $Sw$ are shown in Fig. 3 for different $Q$. Note that the strain fluctuations were not measured in the white region. $Sw$ for each blade remains nearly constant at $Q\u226474.1%$ and takes a value of approximately 0.8, showing that the strain fluctuations are nearly governed by a random process. $Sw$ starts to decrease at $Q=81.5%$, forming periodic fluctuations due to the onset of cascade flutter at some blades. This is due to the effect of the mistuning between the blades of our low-pressure turbine test model. The number of blades with a low $Sw$ increases as $Q$ exceeds 85.2%. A significant decrease in $Sw$ is observed in the region of the right-hand side of the annulus of the blades. These results clearly show that the weighted permutation entropy can detect a precursor of cascade flutter, capturing the partially propagating process of flutter during the transition and the subsequent limit cycle oscillations. Spatial variations in $Dr$ are shown in Fig. 4 for different $Q$. A gradual increase in $Dr$ is observed at some blades with increasing $Q$. At $Q=81.5%$, $Dr$ starts to increase significantly in the region of the right-hand side of the annulus of the blades. As $Q$ exceeds 88.9%, $Dr$ in the above region becomes almost unity, showing the notable formation of limit cycle oscillations due to the strong periodicity of cascade flutter. These results show that, in addition to the weighted permutation entropy, the determinism in RPs is also a useful measure for capturing a precursor of cascade flutter.

Figure 5 shows the variations in $Dr$ and $Sw$ in terms of $Q$ at blade no. 27. The location of ($Dr$, $Sw$) moves from left to right on the *D _{r}* -

*S*plane with increasing

_{w}*Q*. Note that a similar trend is obtained for different blade locations (Nos. 25, 26, and 28). This shows that the $Dr$ - $Sw$ plane appropriately captures the significant transition from the small-amplitude stable state to large-amplitude limit cycle oscillations with increasing mass flow rate of inlet air. The feature space of the $Dr$ - $Sw$ plane obtained by the SVM is shown in Fig. 6(A). Note that $\u03f5\u2032$ used in Fig. 5 is the learning data. The dynamical state can clearly be classified by the SVM into three regions on the feature space, namely, the stable state (blue), cascade flutter (red), and the transition from the stable state to cascade flutter (yellow). Here, $Q$ is gradually decreased over time so as to induce cascade flutter [see Fig. 6(B)]. Black plots are depicted on the feature space with increasing time. The dynamical state changes back and forth between the stable state and the transition state at approximately $26s\u2264t\u226430$ s, with cascade flutter finally occurring at approximately $t=31$ s. These results clearly show that the feature space can capture the subtle transition from the stable state to cascade flutter. The feature space at blade no. 27 is presented as the representative result, but in our preliminary test, the temporal evolution of the dynamical state at other blades with increasing $Q$ was also studied using the learning data obtained at a different blade location. On the basis of Figs. 3 and 4, $\u03f5\u2032$ at nos. 25–28 was taken to be the suitable learning data. Our preliminary test confirms that with appropriate classifications of the dynamical state, a significant change in the dynamical state can be reasonably detected. Two of the authors have more recently demonstrated the applicability of two methodologies combining the SVM with (i) complex networks

^{29}and (ii) statistical complexity

^{30}for detecting a precursor of thermoacoustic combustion oscillations in a laboratory-scale gas-turbine model combustor. The methodology combining the SVM and the two-dimensional plane consisting of the weighted permutation entropy and the determinism in RPs has potential use for the early detection of cascade flutter. To the best of our knowledge, a symbolic dynamics/dynamical systems/machine learning-based approach for the early detection of cascade flutter has not been proposed in previous studies on turbofan jet engines.

## IV. SUMMARY

An experimental study on the early detection of cascade flutter in a model aircraft turbine has been carried out using a method combining symbolic dynamics, dynamical systems, and machine learning. The dynamic behavior of aperiodic strain fluctuations during the stable state undergoes a significant transition to limit cycle oscillations with increasing mass flow rate of inlet air. Two important measures, the weighted permutation entropy and the determinism in RPs, are considered for the temporal evolution of strain fluctuations. They capture a significant change in the dynamic behavior of strain fluctuations during a transition and subsequent limit cycle oscillations. A feature space consisting of the weighted permutation entropy and the determinism, which is obtained by the SVM, enables the detection of a precursor of cascade flutter.

## REFERENCES

*Proceedings of the International Gas Turbine Congress 2019 Tokyo*(

*ASME Turbo Expo 2016*(

*Dynamical Systems of Turbulence*, Lecture Notes in Mathematics (Springer-Verlag, 1981), Vol. 898.

*Proceedings of the International Gas Turbine Congress 2019 Tokyo*(