A reduced order model for coupled mode cascade flutter analysis

Hung HUANG, Xinki JIA, Ji REN, Boho CAO, Dingxi WANG,Xiuqun HUANG,*

a School of Power and Energy, Northwestern Polytechnical University, Xi’an 710072, China

b Yangtze River Delta Research Institute of Northwestern Polytechnical University, Taicang 215400, China

c Department of Aeronautics and Astronautics, Fudan University, Shanghai 200433, China

KEYWORDS Aerodynamic influence coefficients;Chirp signal;Coupled mode flutter;Eigenvalue problem;Reduced order model

Abstract A Reduced Order Model (ROM) based analysis method for turbomachinery cascade coupled mode flutter is presented in this paper. The unsteady aerodynamic model is established by a system identification technique combined with a set of Aerodynamic Influence Coefficients(AIC).Subsequently,the aerodynamic model is encoded into the state space and then coupled with the structural dynamic equations, resulting in a ROM of the cascade aeroelasticity. The cascade flutter can be determined by solving the eigenvalues of the ROM. Bending-torsional coupled mode flutter analysis for the Standard Configuration Eleven (SC11) cascade is used to validate the proposed method.

1. Introduction

Accurate aerodynamic damping prediction is critical for turbomachinery flutter analysis as the usage of blisk reduces mechanical damping to a negligible level. The industry has long been relying on the energy method to predict aerodynamic damping, because the fluid-structure decoupling analysis is computational efficient. The blade is prescribed with a harmonic motion with predetermined natural frequency and mode shape, and the induced unsteady flow is obtained through solving the unsteady Reynolds averaged Navier-Stokes equation either in time domain1,2or frequency domain3,4. The aerodynamic work over one vibration period is then evaluated to predict the aerodynamic damping. Negative aerodynamic damping indicates the presence of flutter risk.The fundamental assumption in the energy method is that the blade vibrates at its vacuum mode shape in the vicinity of the corresponding natural frequency. Therefore, it precludes the structural mode family coupling effects. With the trend of turbomachinery designs towards higher fuel efficiency,mass ratio of the blade to the air is reduced substantially in modern designs.For example,the civil aircraft engine uses light-weight composite materials for fan blade manufacture. The lower mass ratio and the smaller natural frequency separation make blades prone to coupled mode flutter. Bendiksen and Friedmann5-7proposed an eigenvalue method to investigate cascade bending-torsional coupled mode flutter, where Whitehead’s8unsteady aerodynamic theory, LINSUB, was used. It is an analytical aerodynamic model for two-dimensional, subsonic flows over a linear flat plate cascade. The LINSUB code is available in Ref. 9. They concluded that the flutter boundary was significantly altered due to the effects of bendingtorsional mode coupling. Clark10performed a coupled mode flutter analysis on an open rotor. Schuff and Chenaux11analyzed a low mass ratio fan blade using the p-k method.The difficulty for coupled-mode flutter analysis arises from the fact that the flutter frequency is not known a priori. To overcome this problem, the p-k method uses an iterative procedure between the reduced frequencies, k, which is used to compute the aerodynamic force and the eigenvalue, p, which is determined from the aeroelastic eigenvalue problem. Alternatively,the authors have attempted to establish aerodynamic models that are validated for a certain frequency range and are capable to predict coupled mode flutter12,13.The first method concerns an unsteady aerodynamic model, while the second one employs a linear interpolation between the two frequencies.

Liou and Yao14proposed a Reduced Order Model(ROM)based on the Volterra series to solve aeroelastic eigenvalue problem and determined the flutter boundary of NASA Rotor 67. Based on an efficient reduced order model, An et al.15developed a method to analyze nonlinear flutter and gust response of a large flexible wing. The AutoRegressive with eXogenous input model (ARX), a system identification technique, has been widely used to establish ROMs for aircraft wing flutter16,17. Su et al.18incorporated the AIC with the ARX model to analyze a single mode flutter for NASA Rotor 67 and Standard Configuration Four (SC4). Zhang et al.19used the ARX model and a fluid-structure coupling aeroelastic method to investigate the flutter characteristics of the tandem cascade. In this paper, ARX is generalized to study coupled mode cascade flutter.

2. Methodology

2.1. ARX model for cascade aerodynamics

In contrast to an isolated wing,the aerodynamic response of a cascade blade is affected not merely by the vibration of itself but also by neighboring blades.Since the pressure disturbances induced by vibration of a blade often decay rapidly in the circumferential direction,it suffices to include only a few adjacent blades rather than an entire annulus in a computational domain for underlying analysis.Fig.1 shows a schematic view of the computational domain for the AIC. It consists of(2n + 1) blade passages, the middle blade (No. 0) vibrates at a small amplitude, and the rest are stationary. The physical displacement of Blade 0, h0, can be defined as a linear combination of mode shape vectors as,

where Φ is the mode shape vector and u0is the generalized displacement vector. In the traditional AIC method, the blade is prescribed with a harmonic motion. However, in this study, a chirp motion is chosen to train the ARX model. The frequencies of Chirp change with the time t.The chirp signal is defined as sin((ω1+(ω2-ω1)t/T)t), where T is the time required to shift frequency from ω1to ω2.

Performing a time-accurate CFD analysis, we can obtain the unsteady aerodynamic modal force y on the (2n + 1)blades.Given a discrete time series representation of the aeroelastic system inputs and outputs, an ARX model can be constructed to model the linear unsteady aerodynamic response,that is

In this study, we consider the bending and torsion coupled flutter for a cascade. Thus, u0(m) is the generalized displacement vector for the plunging and pitching motion at time instance m; y(m) is the modal force vector consisting of two modes at time instance m for all (2n + 1) passages; and naand nbare user-specified output and input lag orders, respectively.Matrices,A～iwith a dimension of 2(2n+1)×2(2n+1)and B～iwith a dimension of 2(2n+1)×2,can be estimated by a least square method. Note that the steady modal force has been removed to ensure zero input and zero output in this model.

2.2. State space model for cascade aerodynamics

The state space form of the established ARX model can be written as,

2.3. State space model for cascade aeroelastic

Neglecting disk blade coupling and structural damping, the entire annulus cascade structural dynamic equation can be written as M¨ζ+Kζ=fa(t) (8)

where M and K denote the mass and stiffness matrices,respectively, ζ is the physical displacement vector, fais the aerodynamic force vector.

Introducing the structural state space vectors xs(t)=[ζ(t),ζ˙(t)], its state space form can be written as

The aeroelastic stability is determined by the eigenvalues of the matrix in Eq. (10). The real part of the eigenvalue denotes the aerodynamic damping,and the imaginary part denotes the aeroelastic system frequency.

3. Results

In this section,the proposed method is applied to investigate a coupled mode flutter for the linear turbine cascade ─Standard Configuration Eleven (SC11)20. This turbine chord is 77.8 mm, the blade gap is 56.55 mm, and the stagger angle is 40.85°. We consider the transonic off-design flow condition with the inlet flow angle of 34° and the exit isentropic Mach number of 0.99. The steady state solution is computed by ANSYS CFX with the Shear Stress Transport (SST) turbulence model. Fig. 2 shows the computed steady state Mach number contours. The high incidence angle results in a flow separation bubble at the cascade leading edge,and a weak normal shock is formed near 80% chord on the suction side.

Finally,the eigenvalue problem in the state space is solved.The number of eigenvalues of the system matrix in Eq. (10) is equal to the dimension of the system matrix, which is quite large. However, most of its eigenvalues are spurious. Fortunately, it can be straightforward to identify the meaningful ones of which the imaginary parts are close to the blade natural frequencies. Shown in Fig. 7 are the filtered eigenvalues.The plunge modes are all stable, and the pitch modes have some eigenvalues with negative real part (negative damping ratio), which indicate aeroelastic instability. That can also be confirmed from the curve of aerodynamic damping versus IBPA (Inter-Blade Phase Angle) in Fig. 8. The associated eigenvector represents the generalized displacement for each blade and vibration mode. Fig. 9 shows the generalized displacement of the least stable mode, where the indices of 1 to 20 belong to the plunge mode and 21 to 40 belong to the pitch mode. Note that the least stable plunge mode is Nodal Diameter (ND) of 1, and the least stable pitch mode is nodal diameter of 5. It also implies that mode couplings effects are negligible for this case.

In order to show a case in which mode coupling is more prominent, the pitch mode frequency is reduced to 250 Hz,and the flutter analysis process is repeated. Presumably, the ARX aerodynamic model can be reused as the frequency still falls in the frequency range of the chirp signal. The filtered eigenvalue map is shown in Fig. 10. The two branch modes are still separated and do not coalesce. The aerodynamic damping ratio for the pitch mode moves downward, and the least stable nodal diameter shifts to 7.Meanwhile,the aerodynamic damping for the plunge mode remains almost unchanged,as shown in Fig.11.The generalized displacements of the least stable plunge and pitch modes are shown in Fig.12.The plunge mode eigenvector is nodal diameter of 1 with negligible pitch mode component coupled. In contrast, the pitch mode eigenvector shows a substantial amount of plunge mode coupled. The ratio between the plunge and pitch amplitude is 0.20, and the two modes have the same nodal diameter of 7.Actually, the pitch and plunge mode with different nodal diameters will not be coupled.

4. Conclusions

To overcome the inability of the energy method for predicting coupled mode flutter,a reduced order aeroelastic model is proposed in this paper. A system identification technique is used to establish an unsteady aerodynamic model for two structural mode motions in a cascade. The flutter stability is determined by an aeroelastic eigenvalue problem formed by incorporating the established aerodynamic ROM with a linear structural dynamic model in the state space form. When the plunge and the pitch frequency separation of SC11 is large, the cascade behaves in the classical single-mode flutter manner.However, reducing the two frequency gap results in an apparent modal coupling effect despite no evidence of frequency coalescence.It is also found that the pitch and plunge mode with different nodal diameters would not be coupled, e.g., a plunge mode of ND 2 can’t couple with a pitch mode of ND 3. Further research on the coupled mode flutter mechanisms would be beneficial, e.g. the effects of natural frequency separation,cascade solidity and the phase angle difference between the plunge and pitch mode.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

This work was supported by the National Science and Technology Major Project, China (No. 2017-II-0009-0023), the Aeronautical Science Foundation of China (No.2020Z039053004) and the Fundamental Research Funds for the Central Universities, China (No. 3102019OQD701). We also appreciate fruitful discussions with Prof.Weiwei ZHANG of Northwestern Polytechnical University on the ARX model.