AEROELASTIC INFLUENCE OF BLADE-SHAFT COUPLING IN A 1 1/2-STAGE AXIAL COMPRESSOR

The design process of turbomachinery, is often constraint by aeroelastic phenomena. The design choices are limited by possible structural failure, which can be caused by high vibration amplitudes. In particular, damping has an important impact on these phenomena. In the absence of friction, damping is mainly created by aerodynamics. In this paper, additional damping created by the shaft will be investigated. This becomes relevant when blade vibrations with nodal diameters 1 and -1 couple structurally with shaft vibrations. To investigate the blade-shaft coupling, a simulation process is set up based on a full structural dynamic model of the blisk-shaft assembly and a harmonic balance CFD model to account for the aeroelastic effects. In addition, mistuning identification is performed based on an experimental modal analysis at standstill. All results are incorporated into a structural reduced order model that calculates the vibrational behavior of the blading. These results are compared to damping determined during operation using an acoustic excitation system and measured forced frequency responses. The numerical results agree well with the experimental results, i.e. within the measurement uncertainty. Furthermore, the blade-shaft coupling results in significant changes of the eigenfrequencies and damping. As a consequence, damping increases by up to twelve times due to the coupling. This reduces amplitudes by a factor of nine for the mistuned blade responses. Consequently, higher structural safety factors can be achieved by taking the blade-shaft coupling into account so that the remaining potentials in the aerodynamic design could be better exploited.


Introduction
Recent design trends in turbomachinery applications increasingly require accurate predictions of the aerodynamics, but also of the aeroelasticity.The aeroelastic stability and vibra- * Address all correspondence to this author.tion amplitudes of rotor blades are highly dependent on the modal damping.It is widely known, that for blade-integrated discs (blisks) the main contribution to the overall damping is the aerodynamic damping, whereas for conventional blade-discassemblies contact friction at the blade root can create additional damping.However, this is only true for isolated disc-blisk assemblies.Even for blisks additionally damping mechanisms can arise, if the whole blade-disc-shaft assembly is considered.Structural coupling mechanisms between the blades and the shaft can lead to shaft vibrations and, therefore, to damping mechanisms, which are usually known from rotor dynamics.
The numerical modal analysis during the design process is usually only performed based on the isolated blade-discassembly or blisk, assuming a fixed support at the contact area with the shaft.It is assumed that the shaft is stiffer than the blisk.However, it has been known since many years, that this is not always the case.Already Okabe et al. (1991) [1] and Schaber et al. (1993) [2] developed simple reduced order models (ROM) to predict blade-shaft-coupled vibrations in gas and steam turbines.Those vibrations can be excited by harmonics or disturbances of the line frequency.Anegawa et al. (2008) [4] researched these coupling phenomena in detail.They stated that blade vibration patterns of distinct nodal diameters (ND) and blade mode shapes are likely to couple with certain shaft modes.ND0 blade vibrations can couple with torsional and axial shaft vibrations, whereas ND1 blade vibrations can couple with a translational and a bending mode of the shaft.The coupling is mainly dependent on the rotational speed, as the eigenmodes of shaft and blading originate from different frames of references.Therefore, blade modes in the stationary frame and shaft modes in the rotating frame of reference are highly rotational-speed dependent, even if stressstiffening and Coriolis effects are neglected.A coupling of blade and shaft vibration occurs, when their respective eigenfrequencies cross in the Campbell diagram.Anegawa et al. (2008) [4] and Anegawa et al. (2011) [5] carried out numerical and exper- Investigated domain of the axial compressor, adapted from [3] imental investigations for the coupling of shaft translational and bending modes with the blades.They investigated a very simplified structure with eight flat plates and artificially excited the rotor at the shaft.At the crossings of the shaft and blade eigenfrequencies the blades worked as a harmonic absorber, reducing the vibration amplitudes of the shaft.Grein et al. (2021) [6] investigated a combined commercial gas and steam turbine rotor.They showed measurement results where a drop of the vibration amplitude of the second shaft bending mode was observed, while the blade ND1 amplitudes peaked.Additionally, a reduced order model using a multi-frame approach with time integration and a Crack-Bampton reduction was used for numerical calculations.Using the model they could show, that the vibrational behavior was due to a coupling of shaft and blade vibrations.The vibrations were excited at the shaft by the anisotropic bearings.
In addition to these experimental applications, there are many papers on the development of reduced order models, which include the coupling of shaft and blades, see e. g. [7][8][9].These models are, however, mainly applied to academic structures with flat plates and without experimental validation.Furthermore, this research area is of high relevance for the modeling of inter-stage coupling via the shaft and fluid, see e. g. [10][11][12].
In summary, it is well known, that shaft and blade vibrations can couple during the operation of turbomachinery.Yet, the current research is mainly focused on the vibrational behavior of shaft-dominated modes ("shaft modes") and a harmonicabsorber effect of the blades at mostly academic structures.However, as the coupling of shaft and blades works in both directions, also an impact on blade-dominated modes ("blade modes") is expected.This can happen when the blades vibrate in nodal diameter -1, 0 or 1.These nodal diameters can be excited when specific blade/vane-count combinations are chosen, which are known to occur in industrial compressors as well, but are also more likely to flutter, as they are usually low damped.Especially under the influence of fluid-film bearings the shaft modes are usually highly damped, which means, that, besides the absorber effect, an additional damping effect due to blade-shaft-coupling can come into play for blade modes.
The goal of this work is to analyze the impact of the bladeshaft coupling for blade modes in a 1 1 /2-stage high-speed axial compressor.The aim is to quantify the influence of the coupling effect on the modal damping and the forced response amplitudes of the first three mode families.The paper is organized as follows: First, in sec. 2 the test rig and the experimental approach is presented.Afterwards, the numerical simulation procedure is shown in sec.3. The results will be discussed in sec.4. Lastly, in sec.5 conclusions are drawn.

Test Rig
The research was carried out at the 1 1 /2-stage axial highspeed compressor of the Institute of Turbomachinery and Fluid Dynamics of Leibniz University Hannover.The test rig was specifically designed for aeroelastic investigations.Its design process was described by [13].Initially, the aim of the test rig was to determine the influence of blend repairs both numerically and experimentally.The results are shown in [14] and [15].The compressor configuration is shown in Fig. 1.It features 23 variable inlet guide vanes (IGV), a rotor manufactured as blisk with 24 blades and a stator with 27 rotatable vanes.To vary the excitation intensity at forced response, the IGV blades can be rotated from 0 • (IGV0) up to 20 • (IGV20) or −20 • .The hub-to-tip-ratio in the rotor row is approximately 0.6.At the design point the rotor rotates at 17 100 rpm with a relative circumferential Mach number of 0.9.The rotor becomes transsonic at approximately 12,000 rpm.
The rotor is supported by a non-locating oil film bearing at the upstream side of the compressor and a locating oil film bearing at the downstream side.It is worth to mention, that these bearing types introduce comparably high damping, which can be used in rotor dynamics to damp the shaft modes.At the downstream end of the rotor the shaft is connected to a gear shaft via a curved tooth coupling.
Flow field measurements at the inlet and downstream of each row can be performed using pneumatic 5-hole probes.In order to measure blade vibrations during operation an optical tip-timing system by AGILIS is used.The probes can be positioned at the leading edge (TT LE, for mode family 1 and 3, first bending and first torsion) or at the trailing edge (TT TE, for mode family 2, mixed torsional-bending mode), depending on the mode shape, see Fig. 2. In addition to aerodynamic excitation by the IGV and stator blades, vibrations can be excited artificially by means of an acoustic excitation system, which features eight speaker/excitation units around the circumference to excite the blades.The general design aspects and the application to determine aerodynamic damping during test-rig operation are described by [16].
Previous experimental aeroelastic investigations regarding forced response at this compressor are described by [17] and [18].Beside the excitation of the engine orders 23 (IGV) and 27 (stator), they intentionally excited deviating EO by imposing circumferentially varying IGV stagger angles.[18] additionally compared the results to numerical results based on time-domain and frequency-domain approaches.
To give an overview of the blade eigenmodes the Campbell diagram of the isolated blisk is shown in Fig. 2. Mode family 1 is the first blade-bending mode, mode family 2 is a mixed torsionalbending mode and mode family 3 is a torsional mode.In this work only the first three mode families will be considered.

Aeroelastic Measurements
During forced response measurements the rotational speed was once slowly ramped up and once ramped down to pass through each resonance crossing of each investigated mode.Afterwards, typical post-processing routines were applied with the corresponding software Offline by AGILIS, using zeroing, a high-pass filter, and the low-pass like Savitzky-Golay filter [19].The measured vibration amplitudes were then scaled to the maximum blade amplitudes, based on the numerically obtained mode shapes (FEM cycl., see Fig. 3.1).
Furthermore, damping measurements were performed during operation of the compressor using the acoustic excitation system.As done by [18] the rotational speed was kept constant and the blades were excited with a frequency sweep of 1 Hz/s and a targeted nodal diameter.The non-synchronous responses were then extracted and a single-degree of freedom oscillator was fitted to the targeted nodal diameter response.Based on the fit the eigenfrequency and damping could be determined.More details regarding the AES and the approach to measure the damping can be found in [16,[20][21][22].

Operating Points
For forced response measurements multiple operating points where investigated, at which the first three mode families are in resonance with the EO of the IGV.During measurements, the compressor's outlet throttle was closed by 10% at all operating points to achieve enough distance to the stall line and get stable flow conditions.Multiple operating points were investigated, whereas the IGV were fixed to the angles given below.The operating points, which will be analyzed in detail are shown in Tab.
1. Most forced response measurements were taken at the resonance crossing of EO23 and mode family 3 (M3) with IGV stagger angles between 0 and 20 • .The measurements where taken to investigate the impact of different excitation intensities, whereas higher angles lead to higher vibration amplitudes but also flow separations.For mode family 2 (M2) the crossing with EO23 was only investigated for IGV20, as the excitation intensity was not high enough at lower IGV stagger angles.To validate the numerical approach a dominant response in a nodal diameter differing from -1, 0, and 1 was needed.Consequently, the resonance crossing of EO18 with M3 was investigated, which nominally excites ND6.The engine order was artificially imposed as described by [17] varying the IGV stagger angle in such way, that a sinusoidal-like shape of the stagger angles with a periodicity of 18 is formed around the circumference (Sine18).Furthermore, damping measurements were performed at 7 500 rpm for M3 and at 15 660 rpm for M1.

Numerical Models
The calculation of forced response is based on multiple submodels.As shown in Fig. 3 the results of the sub-models are then fed into a structural reduced order model as modal coefficients.As a final result the reduced order model can calculate the mistuned frequency response functions of the blading.For the investigations at hand one of two FE-models is used: either a cyclic model of one blisk sector (FEM cycl.) or a model of the whole rotor including main and gear shaft.By comparing the results, the influence of the blade-shaft coupling can be quantified.

FE-Models
All FE-models are set-up and solved using ANSYS Mechanical 19.3.The FEM cycl.model is shown in Fig. 4. The structure is fixed at the interfaces to the shaft.Cyclic symmetry boundary conditions are defined in circumferential direction to calculate all nodal diameters.The mesh contains approximately 89 500 nodes.
The FEM full model is shown in Fig. 5.It includes the whole rotor with main and gear shaft.The bearings are modeled using rotational-speed dependent stiffness coefficients provided  FIGURE 6. Solved frequencies in the harmonic balance calculations, adapted from [3] by the manufacturer.However, due to the rotational asymmetry of the rotor the stiffness matrix of the bearings had to be adapted to be skew-symmetric by averaging the main diagonal and the off-diagonal entries.The coupling between main and gear shaft is modeled as joint which behaves rigidly for radial displacements and rotation in circumferential directions.All other degrees of freedom are set as free.The mesh contains approx. 1 mio.nodes.
The calculations with both FE-models were conducted for every operating point considered.Before the modal analysis is performed the structure is pre-stressed by the rotational forces as well as static temperature and pressure on the blade from the steady CFD calculation.Additionally, for FEM full the bearing oil temperature from the experiment is used.The modal analysis includes gyroscopic effects.

CFD Model
The harmonic balance approach implemented in the CFD solver TRACE 9.3 by the German Aerospace Center (DLR) [23] was applied to perform unsteady simulations and calculate aerodynamic stiffness, aerodynamic damping and the modal aerodynamic excitation forces.The approach solves the flow field at a selected base frequency and multiples/harmonics of it in so called harmonic sets.All frequencies of each harmonic set are coupled nonlinearly.A scaled total pressure and total temperature profile based on 5-hole measurements and a Prandtl-probe was set at the inlet.The turbulent intensity was set to 3% with a length scale of 0.1 mm.At the outlet the measured mass flow rate was prescribed as boundary condition.The CFD model and domain is similar to the model described in [18], but the k-ω turbulence model [24] with stagnation point fix [25] and γ-Re θ t transition model [26] was applied, as the turbulence model captures the separated flow regimes more accurate in this case.The mesh is relatively fine to capture transitional effects on the IGV and its thin wake.It consists of 5.8 million nodes and 87 cells per (acoustic) wavelength at the highest relevant frequency, which results in very small dissipation of acoustic waves.
For each operating point 25 simulations were conducted: 1 aerodynamic excitation simulation and 24 aerodynamic damping/stiffness calculations.The latter corresponds to one calculation per nodal diameter which is necessary for the consideration of mistuning, which lifts up the orthogonality of the system modes regarding other nodal diameters.The solved frequencies in harmonic balance are shown in Fig. 6.Aerodynamic excitation was calculated by resolving the vane passing frequency of the IGV in the rotor domain with 5 harmonics, as shown in the sensitivity study by [18].Additionally, mode scattering can lead to a cut-on acoustic mode with blade passing frequency and a low circumferential mode order m (B = 24 blades, EO23: ).This acoustic mode can be reflected at the neighboring rows and is, thus, included in the calculation [12].The resulting modal aerodynamic excitation forces are then calculated based on the aerodynamic work [12].
Furthermore, the aerodynamic damping and stiffness coefficients are calculated by mapping the complex mode shapes calculated by the FEM cycl.model to the blade.The mode shape is scaled to a maximum vibration amplitude of 100 µm.Also in this case acoustic reflections of the neighboring rows are included.For subsonic operating points the flow field can be assumed to be linear and the flow field is resolved with one harmonic.For transsonic operating points a shock can introduce nonlinearities and the flow field is resolved with three harmonics.
Besides mass, momentum, and energy conservation also the two turbulence equations are solved in frequency domain [27].In order to prevent instabilities by the Gibbs-phenomenon, Lanczos filtering is applied to the turbulent kinetic energy and the specific dissipation rate [28].

Mistuning Identification
In order to calculate accurate vibration responses of the real structure the mistuning needs to be included.Mistuning is identified using the advanced Fundamental Mistuning Model (FMM) based identification (ID) method [29].The model works on the basis of isolated mode families, which is valid for the mode families investigated, see Fig. 2. As input the eigenfrequencies and mode shapes of the (mistuned) system modes are used.The eigenmodes in the ROM are described by one degree of freedom per nodal diameter mode.
An experimental modal analysis of the main rotor including blisk and shaft is performed to characterize the system modes.The experimental setup is shown in Fig. 7.The structure is excited at the disc using an automated modal hammer.The 3-dimensional vibration response is measured with three PSV-500 laser vibrometers mounted on a KUKA robot arm (RoboVib by Polytec).Measurements were taken at 572 measurement points on the blisk and shaft.For the modal identification the least-squares rational function estimation method implemented in Matlab was applied to all measurement points at the leading edge of the blade tips (one point per blade).

Reduced Order Model
The reduced order model is based on the FMM with extension for aerodynamic coupling [30] and is formulated in traveling-wave coordinates.The mistuning is therefore projected into a basis of complex nominal modes from either the FEM full or the FEM cycl.Thus, each entry of ⃗ x = (x ND=−12 , x ND=−1 , x ND=0 , x ND=1 , ..., x ND=11 ) T describes the modal deflections of the tuned complex traveling-wave modes.The system of equation of movement is shown in Fig. 3.The ROM includes the aerodynamic damping D• aero , stiffness K• aero , and excitation ⃗ f • aero from the harmonic balance calculations.Additionally, the tuned eigenfrequencies of the traveling-wave modes ω • ND , calculated by either the FEM full or the FEM cycl.model, are included.As mentioned in sec.3.1 gyroscopic effects are included in the FE-models and therefore in the eigenfrequency calculation.This means, that also the resulting frequency splitting of backward and forward traveling modes is considered.In case of the FEM full model also the additional damping introduced by the bearings D• bearing is considered.All mentioned matrices are diagonal as the calculations of the sub-models were performed for a tuned system.Mistuning is introduced by the non-diagonal mistuning matrix Ã, which is a result of the FMM ID.The system of equations can either be solved in frequency domain, assuming quasi-steady conditions, or in time domain, when transient vibration behavior is present.In the latter case Matlab's ode45 function was applied to solve the equations by a combined fourth and fifth-order Runge-Kutta method.

Results
First, a comparison with the experiment (RoboVib) is made in standstill to validate the FEM full model.Figure 8 shows the experimentally determined mode shape of the second shaft bending at an eigenfrequency of 875.7 Hz (top) and the numerically determined mode shape at an eigenfrequency at 859.9 Hz.The results of the measurements were interpolated on the FE-mesh.Despite some artifacts from the interpolation process the mode shapes agree quite well.There are little differences in the vibration nodes at the shaft visible.Those differences are the reason for the slightly differing eigenfrequencies (approximately 2%) and can be explained by the differing support of the shaft: in the experiment the rotor laid on a wooden block, whereas there was no fixed support defined in the FE-model.Additionally, it can be seen, that the blading is simultaneously vibrating in the M1 ND1 mode.
To compare experiment and FEM, Fig. 9 shows the eigenfrequencies of the blade (dominated) modes of M3.As the FEmodels are based on a tuned system, the tuned eigenfrequencies calculated by the FMM ID are compared with the numerical results.For nodal diameters above 2 the results agree well.However, for nodal diameters 0 to 2 differences between FEM cycl.and FEM full can be observed, whereas the results of FEM full matches the FMM ID tuned results.This is due to an increased disc vibration for lower nodal diameters, which depends on the disk's support.In the FEM cycl.model the disc is fixed, which leads to increased stiffness and eigenfrequencies.At ND1 the differences between both FE-models are lower, which is not the case for mode family 1 and 2 (not shown).It is assumed, that the decrease in stiffness due to the missing fixed support is counteracted by a nearby shaft-mode.The maximum sector mistuning calculated by the FMM ID is 0.7% (mode family 1), 0,6% (mode family 2), and 0,25% (mode family 3).All in all, the FEM full model is very well able to reproduce the vibrational behavior of blades and shaft.In Fig. 10 a Campbell diagram calculated with the FEM full model is shown.The calculation was conducted in the rotating frame of reference.Following [4,5] the diagram includes ND1 modes with positive frequencies and ND-1 modes with negative eigenfrequencies.The blade modes show slightly increasing absolute eigenfrequencies, due to stress-stiffening.However, the shaft-mode eigenfrequencies are decreasing approximately proportional to the rotational speed, as they are modes from the stationary frame of reference.The Coriolis effect seems to be rather small.At eigenfrequency crossings blade-shaft coupling is expected.This is for example the case at approx. 5 800 rpm for M2 ND1 and at 15 700 rpm for M1 ND1.At 5 800 rpm veering also occurs, which indicates a stronger coupling.
To provide a better understanding of the flow field a Mach number contour at 90% channel height is shown in Fig. 11.A thin wake is visible, which leads to small excitation forces, however, the wake becomes larger when the IGV is rotated.Additionally, at design speed (17 100 rpm) a strong shock structure is visible at the suction side of the rotor which is expected to have a large impact on the aerodynamic damping.It is noted, that the stator was rotated by 10 • at this operating point.

Damping
Not only to correctly predict vibration amplitudes but also to predict possible flutter instabilities, it is essential to correctly predict the modal damping.In case of a blade-shaft coupling the impact can be directly noticed when looking at the damping, as the additional bearing damping is expected to increase the overall damping.

Correction for the mistuning impact
In order to compare the damping of the tuned traveling-wave modes, the experimental damping Λ exp,mistuned was corrected by the mistuning influence.The correction was done using an simple approach based on the mistuned damping, determined by fitting a single-degree of freedom oscillator to the numerically calculated mistuned vibration response Λ num,mistuned , see Fig. 12.The experimental tuned damping was then calculated using the This implies, the experimental results are corrected by numerical results; however, the correction allows the comparison of physical meaningful damping values, which is not necessarily the case when fitting a single-degree of freedom oscillator to a mistuned response.The approach is expected to be accurate for small mistuning and accurate numerical predictions, as will be demonstrated below.Anyway, the impact of the correction was rather small: the corrected damping (logarithmic decrement, log.dec.) decreased in all cases due to mistuning-induced frequency FIGURE 12. Numerically and experimentally (acoustic excitation) determined frequency response of M3 ND8 at 7 500 rpm.All nodal diameters with amplitudes larger than the measurement uncertainty are shown.ND5 and ND2 as well as ND10 and ND-3 could not be separated clearly in the measurement data and are displayed as superposition, adapted from [3] splitting.The average decrease was approx.0.2 percentage points and the maximum decrease 0.5 percentage points.

Mode family 3
Figure 13 displays a comparison of numerically and experimental damping of M3 at 7 500 rpm.In order to reduce the operating time of the compressor not all nodal diameters were measured.All overall damping values calculated by the numerical models are within the measurement uncertainty of the experiment (95% confidence interval).This also includes the jump in damping below ND-7 and proves a high accuracy of the CFD model.However, it can also be noted that the impact of the bearing damping is small and not significant, as the aerodynamic damping alone already lies within the measurement uncertainty for ND-1 and ND1.

Mode family 1
A comparison of numerically and experimentally determined modal damping is shown in Fig. 14.Due to frequencylimitations of the AES (approx.500 Hz -6700 Hz in stationary frame of reference) ND-5 to ND0 could not be excited in the experiment.The numerical results again agree with the experimental results within the measurement uncertainties.However, in this case for ND1 numerical results and experiment only Λ exp,tuned , corrected by the mistuning impact, the aerodynamic damping, calculated by CFD, and the bearing damping, calculated by FEM full, of M1 at 15 660 rpm, adapted from [3] agree when including the bearing damping, which is created by the blade-shaft coupling.The bearing damping is approx.two times as high as the aerodynamic damping in this case.Also, for ND-1 the overall damping almost doubles due to the impact of the bearing.The physical mechanism can be derived from Fig. 15, where the mode shape of the M1 ND1 system mode of FEM full is shown.The blade vibrations couple with a higher-order shaft bending mode.The deflections at the shaft are comparably low, however, as the rotor is supported by fluid-film oil bearings, high damping is introduced already by small deflections.For the damping mechanism only the deflections at the bearings are relevant, which are approximately 1% of the maximum amplitude.
As the coupling mechanism is highly dependent on the rotational speed in Fig. 16 the damping of M1 ND1 is plotted against the rotational speed.For all rotational speeds the numerically calculated overall damping agrees with the experiments in terms of the measurement uncertainty.The aerodynamic damping stays 16.Damping of M1 ND1 for varying rotational speed.The grey markers show measurements, for which no calculations were performed and which were taken after reassembling the test rig, adapted from [3] approximately constant over the rotational speed, whereas the bearing damping highly changes.It can be seen, that the highest damping occurs at 15 660 rpm, which was investigated in Fig. 14.As shown in Fig. 16 close to this rotational speed the eigenfrequency of a shaft and a blade mode cross, leading to increased coupling and, therefore, increased damping.When the rotational speed is deviated, the bearing damping decreases quickly.Additionally, in grey measurement results after reassembling are shown, proofing, that the effect is reproducible.
In Fig. 17 the damping of M1 ND-1 is plotted against the rotational speed.In this case the bearing damping stays approximately constant.This is because the (absolute) eigenfrequencies of the shaft modes in the rotational frame of reference increase with a similar slope as the blade modes due to stress-stiffening, preventing any crossings.For this nodal diameter only limited experimental data is available due to the frequency limitations of the AES.

Forced Response
Forced response measurements were performed at various operating points.Selected results will be presented in detail below.At the end a summary over all results will be given.

Mode family 3
In Fig. 18 the frequency responses in traveling-wave coordinates at the resonance crossing of EO23 (IGV15) and mode family 3 are shown.Besides the nominal responding nodal diameter ND1 from EO23, other nodal diameters with similar frequencies respond due to mistuning.However, ND1 still is the dominant nodal diameters as the mistuning is still comparably small.The frequency responses of the FEM full model are shown in Fig. 18 left.They are very close to the experiment.The numerically calculated peaks are shifted by approximately 2 Hz, which can be caused by small temperature differences.Beside this shift all responses except ND3 agree with the experiment in terms of the measurement uncertainty.The small deviations for ND3 could be due to changes of the mistuning due to the coupling with the rotor, which cannot be accounted for in the FMM.In Fig. 18 right a comparison to FEM cycl. is made.The maximum ND1-amplitudes of the ROM exceed the experiment by 20%, as in this case the bearing damping is not included.Additionally, when looking at the bottom of the figure, it can be seen that the mistuning-induced responses of the remaining nodal diameters are not predicted correctly.This is presumably because of the deviations of the eigenfrequencies, which could already be observed in Fig. 9 for low nodal diameters.

Mode family 2
Figure 19 shows the frequency responses of mode family 2 at the resonance crossing with EO23 (IGV20) at approximately 5800 rpm.This resonance crossing could only be identified in the experiment with the highest excitation intensity, induced by IGV20.shown in Fig. 10 the operating point is close a crossing of a and a shaft mode.Again, nominal nodal diameter ND1 shows the dominant response in experiment with an maximum amplitude of 22 µm.Due to the higher disc-impact on the system modes resonance peaks of ND2 and ND-2 can be identified below and of ND3 and ND-3 above the resonance peak of ND1.When comparing the experimental with the ROM full results, differences can be observed for the secondary nodal diameters.The amplitudes of ND1 are again predicted within the measurement uncertainty, if the frequency shift is neglected.However, the calculated response of ROM cycl. in Figure 19 right highly exceeds the experimental results, reaching amplitudes up to 150 µ.Additionally, frequency splitting of ND1 and ND-1 can be observed, which was neither present in the experimental nor in the ROM full results.This over-prediction of 700% shows the effect the blade-shaft coupling can have.The coupling leads to an increase of the damping by approx.2.7 percentage points.As the aerodynamic damping of mode family 2 is low (0,17% to 0,79%, ND1: 0.2%) the additional bearing damping has a big impact on the vibration amplitudes.Additionally, the increase in damping also reduces the mistuning-induced frequency splitting as the resonance peak widens.

Mode family 2 -transient
The ROM full is able to correctly predict the vibration amplitudes of the nominal ND1, which are influenced by the increased damping due to the blade-shaft coupling.However, the secondary nodal diameters are overpredicted by the ROM.Especially the maximum amplitude of ND 2 is predicted to be 3.5 times higher than in the experiment.It is assumed, that this is caused by transient effects [31].These can arise, since the damping of mode family 2 is very low and the minimal gradient of the rotational speed of the test rig was limited.Therefore, the ROM was solved in time domain, prescribing the rotor frequency, measured by the tip-timing system, as base frequency for the excitation.The results are shown in Fig. 20.While the vibration amplitudes of ND1 stay the same, as this nodal diameter is more damped by the bearings, the amplitudes of the remaining nodal diameters decrease.Additionally, the calculated frequency responses show a modulation, which is typical for transient resonance passing.By including the transient effects into the calculation, the results of all nodal diameters except ND2 agree with the experiment (when removing the frequency shift).For ND2 it is expected, that the maximum vibration amplitude was not captured by the tip-timing system as the peak is very narrow and post-processing routines had to be applied to the signal.

Summary of all operating points
A summary of all investigated operating points is given in Fig. 21.The averaged peak blade amplitudes at resonance are shown.For mode family 3 the amplitudes increase with increasing IGV angle (IGV0 to IGV20).In all cases the ROM full predicts vibration amplitudes within the measurement uncertainty, while the ROM exceeds the experimental results in all cases except IGV15.It is presumed that in this case the flow starts to separate at the IGV and the separation is delayed in the CFD, see [18].Interest- ingly, at excitation of M3 by EO18 (Sine18) ROM full and ROM cycl. the amplitudes are almost identical.This is because the nominal nodal diameter is ND6 where no shaft-blade coupling occurs.This proves, that the ROM cycl., which is mainly based on state-of-the-art approaches, is capable to correctly predict the forced response of none shaft-coupled modes.For M2 IGV20 again the overprediction by ROM cycl. is visible.Additionally, it can be seen, that also for transient calculations the ROM cycl. the amplitudes are too high, although the transient effects have a bigger impact due to the missing bearing damping.

Conclusions
In this paper the influence of blade-shaft coupling on the vibrational behavior of nodal diameter 1 and -1 modes were investigated.The results show, that the coupling can have a relevant impact on the vibration response.Blade-shaft coupling can occur rotor-dynamic damping can then be partly transferred to the blade modes.This effect in particular becomes relevant when shaft modes are highly damped, for example due to fluid-film and the blades modes are lowly damped.Therefore, blade-shaft coupling has the potential to decrease vibration amwhen the stator count excites a ND-1 or a ND1 mode.It is expected that the results behave similarly for axial ND0 modes, for the locating bearing introduces as well.Additionally, the risk of flutter is reduced, as lower (absolute) nodal diameters, like 0, 1, and -1, are more often prone to flutter than higher (absolute) nodal diameters.It is expected, that the introduced concepts can be applied to turbines as well, as the physical mechanism is known and not expected to change, although, the eigenfrequencies of blades and shaft could differ and lead to different crossings in the Campbell diagram.
At the test case investigated the damping increased by a factor of up to thirteen at a crossing of mode family 2 and a shaft mode.As this crossing is close to a resonance crossing highly reduced vibration amplitudes also occurred at forced response.It was demonstrated that a model, that includes the blade-shaft coupling, mistuning, aerodynamic coupling, and transient effects, can correctly predict the blade vibration amplitudes in the 1 1 /2-stage axial compressor investigated.For a nodal diameter of 6, blade-shaft coupling was no longer observed and therefore a cyclic FE-model is sufficient to predict the vibrational behavior.In further studies the approach should be applied to different test rigs and also to higher mode-orders, as mode family 4 and 5.
As the taken approach is computationally expensive, for the application in design processes of multi-stage machines it is necessary to decrease the computational cost in future.This could be done by developing or applying existing reduced order models which e. g. use a Craig-Bampton reduction of the rotor structure.Furthermore, the necessity to use such prediction tools could be evaluated based on preliminary calculations of the individual shaft and blade eigenfrequencies, which are usually calculated as part of the design process.Close to crossings of shaft and blade eigenfrequencies more elaborated models need to be applied.
FIGURE 1. Investigated domain of the axial compressor, adapted from[3]

FIGURE 8 .
FIGURE 8. Mode shape of the second shaft bending mode from experiment (top) and FEM full (bottom), adapted from[3]

FIGURE 9 .
FIGURE 9. Mode shape from experimental modal analysis (top) and the FEM full model, adapted from[3]

FIGURE 10 .
FIGURE 10.Extended Campbell diagram for ND1 and ND-1 calculated by FEM full in the rotating frame of reference with the EO23 line.Positive eigenfrequencies indicate a forward travelling wave (ND1) and negative eigenfrequencies a backward travelling wave (ND-1), adapted from[3]

FIGURE 13 .FIGURE 14 .
FIGURE 13.Comparison of experimentally determined dampingΛ exp,tuned , corrected by the mistuning impact, the aerodynamic damping, calculated by CFD, and the bearing damping, calculated by FEM full, of M3 at 7 500 rpm, adapted from[3]

FIGURE 17 .
FIGURE 17.Damping of M1 ND-1 for varying rotational speed.The grey markers show measurements, for which no calculations were performed and which were taken after reassembling the test rig, adapted from[3]

FIGURE 18 .FIGURE 19 .
FIGURE 18.Traveling-wave frequency responses of ROM full (left) and ROM cycl.(right) compared to the experimental results at the resonance crossing of M3 with EO23 (IGV15).The colored areas indicate the measurement uncertainty (95% confidence interval).The lower plots show a zoom of the secondary nodal diameters without measurement uncertainties, adapted from[3]

FIGURE 20 .FIGURE 21 .
FIGURE 20.Traveling-wave frequency responses of transient ROM full compared to the experimental results at the resonance crossing of M2 with EO23 (IGV20).The colored areas indicate the measurement uncertainty (95% confidence interval), adapted from[3]

TABLE 1 .
Primarily investigated operating points