## Abstract

The effect of mode shape and reduced frequency on the flutter stability of a linear low-speed compressor cascade was investigated experimentally, employing the aerodynamic influence coefficient (AIC) approach. This paper describes in detail the methodology, experimental setup, and measurement techniques. The paper presents experimentally determined influence coefficients, and discusses the findings with regard to the aeroelastic design parameter “plunge-to-twist incidence ratio” (PTIR), which combines reduced frequency and torsion axis setback in an attempt a design variable reduction. The influence of the vibrating blade on itself was always stabilizing, while other blades, mainly the first pressure side neighbor, vibrating at forward-traveling, low nodal diameter inter-blade phase angles (IBPAs), can destabilize the reference blade. As PTIR is increased, the phase of the complex modal force coefficients (AICs) of all blades is decreased, increasing the stabilizing effect of the vibrating blade on itself and overall cascade stability. Within the considered parameter range, the impact of reduced frequency is eliminated to a large extent if the incidence ratio is kept constant, requiring the torsion axis setback to be adjusted accordingly. The findings show that the plunge-to-twist incidence ratio is a meaningful aeroelastic design parameter that can explain the global effect of mode shape on flutter stability and should be considered early in the aeromechanical blade design process to ensure flutter-free blading.

## 1 Introduction

Modern aero-engine compressor designs strive for reduced weight and maximum efficiency. Thinner, lighter blades have reduced stiffness, and with blade-integrated disks (BLISK), structural damping is almost completely eliminated, making such designs susceptible to aeroelastic phenomena like flutter. Among various aeroelastic and aerodynamic parameters, most notably blade loading and Mach number, the mode shape is known to have a very strong influence on the flutter stability of a blade row [1–3]. The fundamental bending or “flap” (1F) mode shape of modern compressor blades typically contains both a bending^{2} and a torsion^{3} component, which vibrate in-phase and with a fixed amplitude ratio that is determined solely by the structural properties of the blade.

Such combined mode shapes have been studied numerically, and it has been demonstrated that flutter stability is increased by reducing the torsional component of the mode shape [3–6]. It has been shown by mathematical decomposition of the mode shape into pure bending and torsion mode shapes that cross-coupling terms in the aerodynamic work, particularly the twist motion induced unsteady lift acting on the bending motion, have a destabilizing effect. In contrast, the unsteady lift induced by bending motion, acting on the bending motion itself, has a strong stabilizing effect [4,6]. However, most of these studies are limited to a single or a few selected inter-blade phase angles (IBPAs) or nodal diameters, and/or few mode shapes, limiting the generality of the findings. Also, to the knowledge of the authors, no systematic experimental studies on the influence of mode shape on compressor flutter stability have been published.

The presented work is part of a joint experimental and numerical study on the effect of mode shape and reduced frequency on the flutter stability of a linear, subsonic compressor cascade, which was carried out at the Chair for Aero Engines at TU Berlin and the Rolls-Royce Vibration UTC at Imperial College London. Complementing a joint paper on the numerical investigation [7], this paper describes in more detail the experimental investigations and provides deeper insight into the stability trends based on an in-depth analysis of the aerodynamic influence coefficients (AICs).

### 1.1 Test Case.

The test case is a combined plunge-pitch mode shape of a compressor blade section, which oscillates in a solid body rotation about an axis at a distance $xF$ downstream from the leading edge (LE) or $zF$ from mid-chord, as depicted in Fig. 1. The geometric relation between the *small* twist angle $\u03d1$, torsion axis (TA) setback, and the LE, trailing edge (TE), and plunge displacement amplitudes is

Figure 2 shows a schematic of how the quasi-static plunge and twist incidences geometrically add up to an “effective” incidence in the complex plane, whose phase angle with respect to the modal displacement is defined by the incidence ratio.

In the presented experimental study, a systematic variation of this incidence ratio parameter was conducted over a range of reduced frequencies to assess its predictive capabilities.

### 1.2 Methodologies.

Forced motion flutter testing in combination with an AIC method is employed. Harmonic motion in the designated mode shape is prescribed on the central blade of the cascade, and the unsteady pressure response on the vibrating blade and the stationary blades is observed to obtain the aerodynamic influence coefficients.

Without mechanical damping, the stability of the blade depends only on the sign of $\phi 1$. For $\phi 1>0$, i.e., if the modal force is leading the displacement, the aerodynamic work is positive and the vibration is excited. For $\phi 1<0$, the vibration is damped.

Here, $c~M\sigma $ is the modal force coefficient for reference blade $0$ in a TWM, and $c~Maicm,0$ are the aerodynamic influence coefficients for the modal force induced by blade *m* on blade $0$.

Following the recommendations by Bölcs and Fransson [9], the aerodynamic work is normalized by $\u2212\pi \u22c5q\u22c5bc2\u22c5(h^LE/c)2$, defining the dimensionless damping parameter $\Xi $:

## 2 Experimental Investigation

### 2.1 Flutter Test Rig.

The experimental investigation was done on a linear cascade with 11 blades in an open-return low-speed wind tunnel (Fig. 3). The rectangular measurement section housing the cascade is 955 mm tall (equivalent to 12 blade passage heights) and 200 mm wide. Blade span is 197 mm, leaving 1.5 mm tip gap on either side. The cascade geometry is summarized in Fig. 4.

The cascade front has a fixed 45 deg angle to the inlet flow direction (i.e., the inlet flow angle $\beta $ with respect to the “machine axis” is fixed at 45 deg). The blades are mounted to the back wall of the measurement section at half-chord and can be rotated to change the mean angle of attack $\alpha 0$. In doing so, the mean stagger angle $\gamma 0$ is also changed according to $\gamma 0=\beta \u2212\alpha 0$. For the aeroelastic experiments, $\alpha 0=2deg$ ($\gamma 0=43deg$) was used (cf. Sec. 4.1).

The blade section is a slightly modified version of Standard Configuration 1 [9], which was adopted from previous experimental investigations at the German Aerospace Center (DLR) [10] and TU Berlin [11,12]. It features a NACA65-series thickness distribution with 6% maximum thickness (modified for finite thickness at the trailing edge) over a circular arc mean camber line with approx. 6 deg camber angle.

All blades in the cascade are equipped with transition strips on the suction surface (SS) at $17%$ chord to prevent flow separation.

The highest achievable flow velocity in the measurement section is around 38 m/s; during the experiments, $U0=30m/s$ was used. The corresponding Reynolds and Mach numbers are $Rec\u2248285,000$ and $Ma\u22480.085$, depending on the temperature, and the inlet turbulence intensity is $u\u2032/U0\u22480.2%$. A detailed account of the base flow properties is given in Ref. [11].

### 2.2 Vibrating Blade Module.

The vibrating blade module (VBM) allows implementation of the variable combined mode shape (Fig. 1) in the wind tunnel. A schematic of the VBM is shown in Fig. 5. It consists of a lever with a long slot and a mechanical drive system, which are mounted to a traversing plate on the outside of the wind tunnel side wall. This unit can be moved along linear bearings parallel to the flow to change the lever axis location relative to the vibrating blade ($zF$). The vibrating blade is fixed to a rectangular sliding block, which is inserted into the lever's slot through an aperture in the wind tunnel wall and can be clamped in position with four bolts. The drive system consists of an electric step motor with an eccentric, a scotch yoke, and a connecting rod. It creates a purely sinusoidal angular displacement of the lever with a fixed amplitude of $\u03d1^=1.5deg$. A digital laser triangulation distance sensor (Micro-Epsilon ILD1420) is used to track the displacement of the lever at a rate of 4 kHz, providing a reference for the phase of the unsteady pressures. The operating parameter of the VBM are summarized in Table 1.

Reduced frequency | $k={0.2,0.3,0.4,0.5,0.6}$ |
---|---|

PTIR | $\Phi \theta ={0.16,0.2,0.25,0.3,0.35,0.4}$ |

Lever length | $zF=40\u2026200mm$ |

Torsion axis setback | $xF/c=0.7667\u20261.833$ |

Vibration frequency | $f={6.4,9.6,12.7,15.9,19.1}Hz$ |

Vibration amplitude | $\u03d1^=1.5deg=0.0262rad$ |

Reduced frequency | $k={0.2,0.3,0.4,0.5,0.6}$ |
---|---|

PTIR | $\Phi \theta ={0.16,0.2,0.25,0.3,0.35,0.4}$ |

Lever length | $zF=40\u2026200mm$ |

Torsion axis setback | $xF/c=0.7667\u20261.833$ |

Vibration frequency | $f={6.4,9.6,12.7,15.9,19.1}Hz$ |

Vibration amplitude | $\u03d1^=1.5deg=0.0262rad$ |

The aerodynamic and inertial forces acting on the oscillating blade exert a torque about the longitudinal axis of the lever, thus torsional stiffness was a primary target for the mechanical design of the lever and its pivot axis, in order to minimize undesired flap motion of the blade. At the same time, the lever needs to be light weight to avoid excessive motor loads. To achieve this, the lever was designed as a half-open structure with strengthening ribs oriented at $\xb145deg$, and the pivot axis is supported by a bridge.

The instrumented blades, one of which was always used as vibrating blade, the other switched between the $\xb11,2,3$ positions in the cascade, consist of a carbon fiber laminate shell, glued with epoxy onto a milled steel blade root piece. Each blade features 20 pressure taps (10 pressure surface (PS), 10 SS), which are bunched toward the leading edge for better resolution of this aerodynamically critical region and staggered in the spanwise direction, to avoid downstream pressure taps being affected by the wakes of upstream taps. Metal pipes run inside the blade from the pressure taps to the blade root, where silicone tubes are attached to connect the blade to the pressure transducers. The remaining cavity inside the blade is filled with a plastic foam. An overview is given in Fig. 6. Tap holes, internal pipes, and silicone tubes have a nominal inner diameter of 0.5 mm. All tubes have a length of 40 cm, while the internal pipes have individual lengths between 10 and 17 cm.

#### 2.2.1 Vibrating Blade Module Validation.

To assess the fidelity of the mechanical mode shape implementation, experimental modal analysis (impact hammer testing) and measurements of the elastic deformation due to inertial forces of the blade-lever assembly were performed.

For the modal analysis, the measurement blade was installed in a fixed blade mount as well as in the VBM lever in the position furthest away from the axis. The eigenmodes with the lowest frequencies are first flap and first torsion. In both modes, the deformation occurs mostly in and around the blade root piece, while the carbon fiber shell remains mostly flat. The modal frequencies are summarized in Table 2. The lowest modal frequency of the measurement blade in the lever (1F) is more than three times greater than the highest oscillation frequency in the experiments, hence unwanted resonance can be ruled out.

Flap mode (1F) | Torsion mode (1T) | |
---|---|---|

Fixed blade mount | 80.9 Hz | 162.3 Hz |

Blade in lever | 67.5 Hz | 136.8 Hz |

Flap mode (1F) | Torsion mode (1T) | |
---|---|---|

Fixed blade mount | 80.9 Hz | 162.3 Hz |

Blade in lever | 67.5 Hz | 136.8 Hz |

To measure the elastic blade deformation due to inertial forces, the periodic displacement was measured on a nine-point grid on the vibrating blade PS, using the ILD1420 sensor, as schematically depicted in Fig. 7. All operating points of the VBM for the aeroelastic experiments were tested this way at operational frequency and at 1 Hz for reference.

It was found that the shell of the blade deflects in the form of a rigid body, which, in addition to the desired parallel rotation about the lever pivot axis performs a flap motion about the lever longitudinal axis and some twist motion about its half-chord axis (Fig. 8). Consequently, the true plunge-to-twist incidence ratio, defined in terms of LE and TE displacements as $\Phi \theta =(k/2)\u22c5((HLE+HLE)/(HLE\u2212HLE))$, is generally larger than the design value and increasing with spanwise distance from the blade root, as well as with vibration frequency. The excess plunge-to-twist ratio was found to be less than 4% for $k=0.6$ ($f=19.1Hz$) at the outer end of the pressure tap zone. Results for all VBM operating points are summarized in Table 3.

Φ (H_{LE}, H_{TE}) | Design incidence ratio | |||||
---|---|---|---|---|---|---|

Φ = 0.16 | Φ = 0.2 | Φ = 0.25 | Φ = 0.3 | Φ = 0.35 | Φ = 0.4 | |

k = 0.2 | 0.160 | 0.200 | 0.251 | |||

k = 0.3 | 0.161 | 0.201 | 0.251 | 0.301 | 0.352 | 0.403 |

k = 0.4 | 0.162 | 0.203 | 0.254 | 0.304 | 0.353 | 0.404 |

k = 0.5 | 0.164 | 0.205 | 0.255 | 0.307 | 0.357 | 0.408 |

k = 0.6 | 0.166 | 0.208 | 0.259 | 0.310 | 0.361 | 0.412 |

Φ (H_{LE}, H_{TE}) | Design incidence ratio | |||||
---|---|---|---|---|---|---|

Φ = 0.16 | Φ = 0.2 | Φ = 0.25 | Φ = 0.3 | Φ = 0.35 | Φ = 0.4 | |

k = 0.2 | 0.160 | 0.200 | 0.251 | |||

k = 0.3 | 0.161 | 0.201 | 0.251 | 0.301 | 0.352 | 0.403 |

k = 0.4 | 0.162 | 0.203 | 0.254 | 0.304 | 0.353 | 0.404 |

k = 0.5 | 0.164 | 0.205 | 0.255 | 0.307 | 0.357 | 0.408 |

k = 0.6 | 0.166 | 0.208 | 0.259 | 0.310 | 0.361 | 0.412 |

Because of the staggered alignment of the pressure taps, each pressure tap experiences a slightly different true mode shape, so that it is not possible to exactly specify the true incidence ratio of an aeroelastic measurement or apply a correction factor. This mode shape error was therefore left unaccounted in the evaluation of the aeroelastic measurement campaigns.

### 2.3 Pressure Measurement System.

For blade surface pressure measurements, a set of 20 recessed mounted piezo-resistive differential pressure transducers (Endevco 8510B-1) is connected to one instrumented blade at a time. The reference pressure ports of the transducers are exposed to the ambient pressure. The transducers have a range of $\xb11PSI$ gauge ($\xb17kPa$). To allow accurate reading of the much smaller steady and unsteady blade surface pressures, they were calibrated against precision pressure cells with 25 Pa, 100 Pa, and 1000 Pa ranges. The transducers exhibit linear behavior within the $\xb11kPa$ range, as shown in Fig. 9.

Dynamic transfer functions of the pressure lines were calibrated on-site in a pressure chamber (Fig. 10), exposing either the PS or SS pressure taps to a sinusoidal pressure oscillation of $\xb1200Pa$ with adjustable frequency (1–30 Hz) and comparing the readings on the recessed mounted transducers to a reference transducer on the chamber side wall. Complex transfer functions accounting for amplitude attenuation and phase lag in the pressure lines are obtained by comparing the first harmonic Fourier coefficients. The calibrated transfer functions for one of the measurement blades are shown in Fig. 11.

### 2.4 Data Acquisition, Processing, and Evaluation.

A DEWETRON data acquisition system was employed for synchronized recording of the 20 blade surface pressure transducers, the displacement sensor, and sensors to monitor inlet flow conditions. Sampling rate of 7.5 kHz and a 3 kHz Butterworth low-pass filter were used. For each aeroelastic measurement, roughly 1800 oscillation periods were recorded. To separate the aerodynamic response to the blade vibration from flow turbulence and other noise, the unsteady pressure measurement data were evaluated in the frequency domain. The following steps were taken in data evaluation:

Based on the noise-free displacement sensor signal, crop signals to integer number of oscillation periods to minimize frequency spread in the discrete Fourier transform (DFT).

DFT of the cropped displacement and pressure signals pick complex Fourier coefficients $\u03d1~1H$ and $p~1H$ corresponding to blade oscillation frequency $fosc$.

Compensate output delay (3 scan cycles) of the digital laser sensor on displacement first harmonic normalize to unit magnitude.

Divide pressure signal first harmonic by pressure line transfer function to compensate phase lag and amplitude attenuation, convert transducer voltage to pressure.

Divide $p~1H$ by $\u03d1~1H$ to get the unsteady pressure phase angle relative to the displacement.

*q*and the dimensionless leading edge displacement amplitude to form the unsteady pressure coefficient:

Figure 12 depicts amplitude spectra of the pressure signals. It shows that the first and second harmonic of the pressure response to the blade oscillation (markers) are very clearly identified, despite the turbulent background noise. Two other sharp peaks at first and second engine order of the wind tunnel's radial blower are also visible.

## 3 Computational Fluid Dynamics Simulation Setup

To complement the experiments and gain an understanding of the aerodynamics of the finite cascade, 2D computational fluid dynamics (CFD) simulations of the base flow (Reynolds-averaged Navier–Stokes (RANS)) and of the aeroelastic experiments (unsteady RANS) were done with ansys cfx v19.2, using the *k*–*ω* shear stress transport (SST) turbulence model (fully turbulent, no transition modeling). The computational domain is a mid-span section of the wind tunnels measurement section. The block-structured mesh is one element thick and uses approximately $3.4\xd7105$ elements. Average first element $y+$ on the blades is $\u22480.7$, and maximum $y+$ (at the LE) $<10$. Automatic wall functions were used.

At the outlet, an area-averaged static pressure of 0 Pa and at the inlet a constant total pressure of $533.15Pa$ (both relative to a reference pressure of 1 bar) were prescribed, so that the mass flow averaged inlet velocity was $30m/s$. Inlet total temperature was $298.15K$ and turbulence intensity was 1%.

For aeroelastic simulations, 22 vibration cycles were simulated, each resolved with 50 time steps.

## 4 Results and Discussion

### 4.1 Base Flow Aerodynamics.

The CFD simulations reveal a strong effect of the top and bottom walls on the pressure and velocity field around the cascade at non-zero angles of attack ($\alpha 0$). Because the straight walls prevent the cascade from deflecting the flow as it naturally would, they impose a pressure gradient between both ends of the cascade, which increases with $\alpha 0$, as seen in Fig. 13 for $\alpha 0=2deg$.

Accordingly, with increasing $\alpha 0$, the blade loading becomes more non-uniform, with the highest loading next to the walls and the smallest loading on blade −2, as depicted in Fig. 14. It is also noteworthy that for positive $\alpha 0$, the blade loading in the finite cascade is substantially higher than in an infinite cascade, whereas for $\alpha 0=0deg$, where the blade section lift is very small (and in fact slightly negative due to cascading effects, despite the positive camber), the blade loadings are very similar.

Time-averaged blade pressure measurements in the experimental cascade at 0 and 2 deg $\alpha 0$ without blade motion are in good agreement with the CFD results. The measurements confirm the CFD predictions regarding the non-uniform blade loading and the pressure gradient along the cascade (visible in the decline in TE pressure from blade to blade). Compared to the simulations, the measured *c _{p}* profiles are offset to higher values. This is due to viscous pressure losses on the wind tunnel side walls between cascade and outlet, which are not captured in the 2D CFD, in combination with the definition $cp=(p\u2212pamb)/q$ used here, which compares the local static pressure to ambient pressure, rather than inlet static pressure, in accordance with the measurement procedure. Also, in the measurements at $\alpha 0=2deg$, the blade loading in the front portion is substantially reduced by secondary flows not captured in the 2D CFD. In a 3D simulation, the pressure distributions in the aft portion match well and the discrepancies in the front diminish (Fig. 15).

### 4.2 Unsteady Pressures, Aerodynamic Influence Coefficients, and Cascade Stability.

Unsteady blade surface pressure distributions from CFD and measurements are in very close agreement (Fig. 16). In terms of the shapes, the results found here also resemble those presented by Buffum and Fleeter [13].

On the central (vibrating) blade 0, the unsteady pressure amplitudes on the pressure and suction surface are almost identical, but the phases are offset by almost exactly 180 deg over most of the blade chord. The unsteady pressure difference between the surfaces, which creates the unsteady lift and moment about the torsion axis, is therefore almost twice as large as the unsteady pressure amplitude on either surface. Most of the unsteady lift/moment is generated near the leading edge, within the first 10–15% chord, as reported by Carta and St. Hilaire [14]. Remarkably, the phase distributions on both surfaces of blade 0 are sloped upwards, meaning that the change in unsteady pressure occurs first at the trailing edge and is then propagated upstream toward the leading edge. This phenomenon has also been pointed out in Ref. [14]. The slope in the phase distribution is more pronounced at higher reduced frequencies.

On the suction side neighbors of the vibrating blade (blades −1 and −2), the unsteady pressure amplitudes are also largest near the leading edge, however, the amplitudes are notably larger on the pressure surface which is facing the vibrating blade. Accordingly, the pressure fluctuations on the pressure surfaces of blades −1 and −2 are in sync with the suction surface of blade 0. On blade +1, the immediate pressure side neighbor of the vibrating blade, the largest unsteady pressure amplitudes occur on the aft portion of the suction side, which creates a flow passage with the pressure surface of blade 0.

Figure 17 gives an overview over the AICs at different incidence ratios and reduced frequencies in terms of relative magnitude and phase angle (with respect to the displacement). Narrow bars with solid filling represent measurements, wide bars with transparent filling CFD. The AIC magnitudes are normalized by the blade 0 AIC for each case to eliminate the effect of the somewhat arbitrary modal force normalization and instead highlight the neighbor blades' potential for destabilizing the reference blade. The relative magnitude of the blade 0 AIC is unity for all cases by definition, so the ordinate axis limit is set to 0.5 to provide a better resolution of the other AICs. As with the unsteady pressure distributions, measurement and CFD match very well and show consistent trends in the AICs with regard to incidence ratio and frequency parameters.

As incidence ratio is increased, the phase angle of all AICs is reduced. The change in phase with PTIR is not equal for all AICs: the phases of blades −1 and −2 change more than others.

Relative AIC magnitudes change very little with incidence ratio, except for blade −1, whose relative influence increases with PTIR. Considering Fig. 16, we assume that this is because the influence of a vibrating blade on its next pressure side neighbor results mostly from pressure fluctuations on the aft portion of the suction surface, which are more sensitive to plunge motion, whereas all other AICs are dominated by pressure fluctuations near the leading edge, which are more dependent on effective incidence change, regardless of whether this incidence results from plunge or pitch motion.

As reduced frequency is increased (at const. $\Phi \theta $), the phase of the blade 0 AIC remains almost constant, but the phase of other blades (especially $\xb12$ and $\xb13$) is reduced. Also, the relative magnitudes of the $\xb12$ and $\xb13$ AICs is decreased. We conclude that at higher reduced frequencies, the unsteady pressure propagation from the vibrating blade to further away blades is damped and delayed.

The significance of these trends for cascade stability becomes apparent in Figs. 18 and 19, where phasors for the AICs and the locus curves of the TWM modal force coefficients are plotted in the complex plane. As incidence ratio is increased, phasors for all blades are rotated clockwise. Consequently, the imaginary part of $c~Maic0$, which contributes the stabilizing effect of the vibrating blade on itself, is increased, and the TWM locus curve, which is obtained by revolving the other AICs around the tip of $c~Maic0$ according to Eq. (9), is gradually moved out of the unstable zone. Note that the shape and size (relative to $c~Maic0$) of the TWM locus curve remain relatively constant (Fig. 18).

In contrast, as reduced frequency is increased while keeping $\Phi \theta $ constant, the phase angle of $c~Maic0$ and with it the stabilizing effect of blade 0 on itself remain nearly constant; however, a reduced destabilizing contribution from the other blade pairs, especially $\xb12$ and $\xb13$, flattens the shape of the TWM locus curve and improves cascade stability (Fig. 19). This is partly due to the decreased relative AIC magnitudes of these blades and also (less apparent, but equally important) due to their relative phase angles. These stability trends are also visible in the damping curves in Fig. 20.

The flutter stability of the cascade is summarized in a stability map (Fig. 21), which shows a contour of minimum aerodamping (at any IBPA) in the reduced frequency ($k$) versus torsion axis setback ($xF/c$) space, based on CFD simulations. The damping is minimal and negative in the bottom left corner and increases monotonically with *k* and $xF/c$. The stability limit, defined by zero aerodamping, is marked for TWM synthesis from 7 and 11 AICs (CFD) and 7 experimentally obtained AICs. For reduced frequencies above 0.4, the stability limits are well approximated by lines of constant PTIR, but for lower *k*, the limits shift to increasingly larger PTIR, presumably due to the greater destabilizing effect of further away blade pairs.

The stability limits for 7 and 11 AICs from CFD are almost identical, indicating good convergence of the AIC method. We see that the experimental cascade is slightly less stable than the CFD, which can be explained by the slightly less negative phase of the blade 0 AIC and the consistently smaller relative magnitude of the blade −1 AIC (since it is out of phase with blade +1 in the IBPA-domain, this AIC reduces the destabilizing contribution of the $\xb11$ pair).

### 4.3 Summary of Findings.

The influence of the vibrating blade on itself is always stabilizing, while other blades (mainly the first pressure side neighbor) vibrating at forward-traveling, low nodal diameter IBPAs can destabilize the reference blade. As PTIR is increased, the phase of the complex modal force coefficients (AICs) of all blades is decreased, increasing the stabilizing effect of the vibrating blade on itself and overall cascade stability.

Within the considered parameter range, the impact of reduced frequency is eliminated to a large extent if the incidence ratio is kept constant, requiring the torsion axis setback to be adjusted accordingly. However, second-order effects of *k* are apparent, predominantly at the lower end of the frequency range, and on the second and third neighbors of the reference blade.

The flutter stability limit for the cascade was found near an incidence ratio of $\u22480.3$, much lower than the hypothesized value of 1.0, which was based on empirical findings for aero-engine fans. It is assumed that this discrepancy is due to vastly different flow conditions and the absence of 3D-flow and acoustic effects in the present test case. This calls for the inclusion of additional flow parameters such as Mach number, aerodynamic loading, blade geometry, and stagger angle, which are being addressed in an ongoing follow-up research project.

## 5 Conclusion

The results presented in this work underline the significance of PTIR as an aeroelastic design parameter that should be considered early in the aeromechanical design process to ensure flutter-free blade design. While it is, by itself, not sufficient to accurately predict the stability limit of a given blade design, it provides a physical understanding of the global flutter stability trends of combined 1F mode shapes, which can be exploited. The results confirm previous findings from other researchers that aeroelastic stability of flap mode shapes can be improved by increasing the frequency parameter and/or by reducing the twist component (increasing the plunge component) of the mode shape. Moreover, the results of this research indicate that it may even be feasible to improve flutter stability by sacrificing blade stiffness (reduced frequency) in order to increase the plunge component of the mode shape, on the condition that this increases the plunge-to-twist incidence ratio. Validation for blade sections and flow conditions more representative of modern transonic fans and compressors is needed.

## Footnotes

Bending in the sense that the blade sections vibrate perpendicular to the blade chord, i.e., perform a plunging motion in the blade-to-blade plane.

Torsion in the sense that the blade twists along its half-chord axis, i.e., the blade sections oscillate in a pitching motion about half-chord.

## Acknowledgment

The authors would like to thank Dr. Bernhard Mück of Rolls-Royce Deutschland Ltd. for his support.

## Funding Data

This research was funded by an industrial collective research program (IGF/CORNET no. 223-EN, FVV project no. 1331). It was supported by the Federal Ministry for Economic Affairs and Energy (BMWi) through the AiF (German Federation of Industrial Research Associations eV).

## Conflict of Interest

There are no conflicts of interest.

## Data Availability Statement

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

## Nomenclature

- $b$ =
blade span

- $c$ =
blade chord

- $k$ =
reduced frequency

- $q$ =
free-stream dynamic pressure (compressible) $q=pt,0\u2212p0$

- $c~M$ =
unsteady moment/modal force coefficient

- $MTA$ =
aerodynamic moment about torsion axis

- $U0$ =
free-stream velocity

- $Waero$ =
aerodynamic work per vibration cycle

- $cp,c~p$ =
steady and unsteady coefficient of pressure

- $h$, $hLE$, $hTE$ =
plunge, leading and trailing edge displacement

- $p$, $pt$, $pamb$ =
static pressure, stagnation pressure, ambient pressure

- $xF,zF$ =
torsion axis setback from LE, mid-chord

- AIC =
aerodynamic influence coefficient

- PS, SS =
pressure surface, suction surface

- TWM =
traveling wave mode

- $\alpha 0$ =
angle of attack (un-deflected blade)

- $\beta $ =
inlet flow angle

- $\gamma 0$ =
stagger angle (un-deflected blade)

- $\theta p$, $\theta t$ =
plunge and twist incidence (quasi-steady)

- $\u03d1$ =
twist/modal displacement angle

- $\sigma $, IBPA =
inter-blade phase angle

- $\Phi \theta $, PTIR =
plunge-to-twist incidence ratio