Influence of Regeneration-Induced Mistuning on the Aeroelasticity of Multistage Axial Compressors

Abstract

Current developments in turbomachinery favor blade designs which are characterized by low damping. The associated reduction of structural damping, however, creates new problems such as higher vibration amplitudes. Therefore, accurate predictions of these vibration amplitudes and of the fatigue life become increasingly important. Analyzing multistage machinery by only simulating isolated stages can be insufficient. This is particularly the case during repair processes when mistuning, such as that created by wear and repair, magnifies said amplitudes. In this chapter selected current results of the subproject C6 of the Collaborative Research Center (CRC) 871 are presented, building on the earlier research in the subprojects C3 and C6 focussing on mistuning and aeroelasticity, respectively. First, a simulation approach which accounts for structural and aeroelastic interstage coupling under conditions of mistuning is described, followed by an approach for incorporating large mistuning effects into a reduced order model of a single rotor stage. The former model is used to study the effect of intentional mistuning to decrease the sensitivity to additional mistuning due to wear and repair and to allow for smaller safety margins. The aeroelastic models are validated for a 1^1/2-stage axial compressor.

1 Introduction

The design of turbomachinery consistently faces the challenge of increasing the efficiency and, in particular for airplane applications, the power-to-weight-ratio. To achieve this goal, turbomachinery blading is increasingly manufactured as one piece with the rotor disk, as a so called blade integrated disk (blisk). However, eliminating the friction at the blade roots leads to significantly lower structural damping and therefore the blading faces higher vibration responses as well as higher sensitivity to flutter. Further amplitude amplifications are generated by deviations in the cyclic structure of the blisk, called mistuning. This can be a consequence of manufacturing tolerances or balancing procedures, but it also results from wear and regeneration, e.g. from blend and patch repairs. This demands an accurate prediction of the vibrational behavior of such blading in as-new, worn, and regenerated conditions. In particular, the prediction gets more challenging, when multistage effects are considered, where structural and aeroelastic coupling increases the complexity of the problem, as there might be no cyclic symmetry anymore and the model size also increases.

To investigate the influence of blend repairs on the aerodynamic damping a 1^1/2- stage transsonic axial compressor with a realistic nominal rotational speed of 17,100 rpm was designed and manufactured in the second period of funding by Keller et al. (2015); Keller (2021). The compressor features two exchangeable blisks with a blade shape, which, due to its complexity, can only be manufactured as a blisk. Compared to the reference blisk, a second blisk was modified with three blend repairs at different radial heights (Keller et al. 2017), see Fig. 1.1. Each blend repair extends over 17.3 mm of the blade height and is 1.2 mm deep. The modified blades are distributed uniformly over the circumference, with seven nominal blades between each pair of modified blades. Blade vibrations were then excited during operation using an acoustic excitation system (Meinzer and Seume 2020). Measurement of the blade vibrations was conducted using a commercial tip-timing system. Keller (2021) could show experimentally and numerically, that the investigated blend repairs have no significant influence on the aerodynamic damping of the first bending mode.

Fig. 1.1

Modified blades with blend repairs of the 1^1/2-stage axial compressor at 100% (left), 80% (middle) and 60% (right) blade height

Measurement data of this compressor is used to validate the aeroelastic numerical models in the following section. Afterwards, the models are extended towards multistage turbomachinery and applied to a 2^1/2-stage axial compressor test case. Thereafter, a reduced order model for large mistuning is presented.

2 Experimental Validation for a 1^1/2-stage Compressor

The aeroelastic numerical model for the multistage calculations was experimentally validated at TFD’s 1^1/2-stage axial compressor. Both flutter and forced response calculations were validated.

For the experimental evaluation of the aerodynamic damping the acoustic excitation system was used to excite different nodal diameters of the first (bending) mode family (Keller 2021). Vibration measurements during operation were taken using a commercial tip-timing system by Agilis. Numerical simulations using Computational Fluid Dynamics (CFD) were obtained using the flow solver TRACE by DLR. The results are shown in Fig. 1.2 left. The numerical results (CFD) mostly agree well with the experiment (Exp.).

Fig. 1.2

Experimental validation of the numerical setups for the 1^1/2-stage compressor. Left: Aerodynamic damping of first (bending) mode family at 16,245 rpm. Measurement data from Keller (2021). Right: Forced response of third (torsional) mode family at 6660 rpm. Data from Maroldt et al. (2022a)

For forced response measurements the inlet guide vanes (IGVs) were rotated from a stagger angle of 0° (IGV0) by angles up to 20° (IGV20). The excitation of the third mode family (first torsional mode) by the IGV (engine order 23) was the focus of the investigations. A brief summary of the results is shown in Fig. 1.2. The numerically predicted vibration amplitudes agree with the experimentally measured amplitudes between a stagger angle of 0° and 15°. However, at 20° stagger angle the vibration amplitude is overestimated due to the flow separations at the IGVs, which cannot be predicted accurately by the employed turbulence-models. With these results in mind, it is possible to conclude that the numerical setup is capable of predicting forced response at operating points with a sufficiently large stall margin, where no significant separated flow regimes occur.

The investigations were supported by flow field measurements using 5-hole probes to evaluate the simulation results regarding the IGVs’ wake shape and therefore the excitation intensity. Additionally, in order to mimic multistage effects a second engine order was imposed by circumferentially varying the IGVs’ stagger angles during forced response measurements. However, as observed for IGV20 also in this case large flow separations occur, which deteriorates the prediction quality. Details can be found in Amer et al. (2020) and Maroldt et al. (2022a).

3 Multistage Computations

The multistage calculations are performed in a simulation procedure, applying a reduced order model and a CFD harmonic balance approach. The procedure is shown in Fig. 1.3. The full structural model is initially created in Ansys Mechanical and then exported to the Matlab-based application RAMBO (Reduced Analysis of Mistuned Blisks in Operation). Based on a cyclic modal analysis of the separate stages and displacements, generated by certain interface basis functions, the degrees of freedom (DOFs) of the ROM are created. The deformations of the ROM’s DOFs are then used in flutter and forced response simulations to calculate the aeroelastic coefficients (aerodynamic damping and stiffness) and the aerodynamic forcing. The results are fed back to the ROM to calculate the system modes including the aeroelastic contributions and the frequency response. Additionally, mistuning can be added to the structure, for example to incorporate regeneration-induced variances or to conduct Monte-Carlo simulations to define limits of manufacturing tolerances. This approach was numerically applied by Maroldt et al. (2022b) to a 2^1/2-stage axial compressor, which is an extension of the existing 1^1/2-stage compressor. A summary of the approach and the results will be given below.

Fig. 1.3

Simulation sequence for the calculation of the vibrational behavior of multistage turbomachinery according to Maroldt et al. (2022b)

3.1 Structural Reduced Order Model

The goal of the reduced order model is the efficient calculation of the eigenfrequencies and forced response amplitudes of multistage rotors, including aeroelastic coupling and mistuning effects. Therefore, the final ROM must provide accurate results with short calculation times and small memory requirements. But the computational effort for the creation of the ROM cannot be prohibitive, especially considering the large model size due to the presence of multiple stages and the necessary CFD simulations. With these aspects in mind, the ROM is designed to need only sector-level structural calculations and to minimize the necessary aeroelastic calculations. As it is designed to capture only small mistuning the reduction basis is derived from the tuned structure.

The ROM is based on a combination of substructuring methods for the stages, while at the same time making use of the cyclic symmetry of each stage. Building on the Fourier Constraint Mode (FCM) method (Song et al. 2005), the interface reduction is improved to increase the efficiency of the reduction (Schwerdt et al. 2020). In both methods, each stage is transformed into cyclic coordinates. Each harmonic index of each stage is then treated as a single substructure and reduced using the Craig-Bampton reduction method (Craig et al. 1968) with interface reduction. The interfaces in this context are the surfaces connecting adjacent stages. The interior degrees of freedom are reduced with the classic fixed- interface modes, which are modes of each substructure with the interface DOFs held fixed. Constraint modes displace the interface DOF and contain the static deflection of the rest of the structure given the interface displacement.

To facilitate the coupling of stages with non-matching interface meshes, the interface reduction is performed a-priori using predefined interface basis functions. These basis functions are matched to the circular or ring-shaped geometry of the interstage interfaces. In circumferential direction a Fourier basis is used. This matches the FCM method, where the interface is split into rings of nodes, which are reduced individually using Fourier bases. Note that the DOFs of each node corresponding to the three displacement directions are reduced individually. The approach is extended in the improved Polynomial Fourier Constraint Modes method by using polynomial basis functions for the interface reduction in the radial direction, as shown in Fig. 1.5. This makes the reduction basis totally independent from the FE mesh resolution, although the maximum Fourier harmonic to include and the polynomial degree is limited theoretically by the FE mesh to ensure the number of interface DOFs does not increase by introducing the transformation using the basis functions.

To assemble the substructures, the corresponding interface DOFs are matched and displacement compatibility is enforced. Here, the different number of sectors of adjacent stages must be taken into account. This is due to the aliasing of the higher Fourier harmonics, which may belong to different cyclic symmetry harmonics in the stages. A simple example of this fact is a rotor with two stages with ten and 20 blades. The interface harmonic with ten nodal diameters corresponds to alternating displacements of the twenty sectors of the second stage, but is aliased to harmonic index zero in the first stage, because each spatial harmonic period encompasses exactly a single sector.

To increase the computational efficiency, real-valued cyclic modes and interface harmonics are used instead of complex traveling wave coordinates. This is due to the influence of mistuning on the matrix structure. While the different coordinates lead to the same computational effort for cyclic modal analyses, introducing mistuning leads to fully populated matrices of the same size in either case, which means more effort is required when using complex arithmetic.

Different DOFs of the ROM of the 2^1/2-stage compressor are pictured in Fig. 1.4. In the top row, examples of the fixed interface modes are shown, while two interface DOFs are shown on the bottom. Here, the main advantage of the Fourier basis is obvious: The lowest Fourier harmonics are most important, as they are coupled with large blade deflections in the constraint modes, which also highlights their importance for the subsequent aeroelastic simulations. The constraint modes of the higher harmonics are localized and can be omitted from the reduced order model. The same holds true for the polynomial coefficients. Additionally, the influence of the choice of the interface location is evident. Here, the interface is close to the first stage. This leads to large blade deflections of the first stage’s blades in the interface DOFs.

Fig. 1.4

Degrees of freedom of the ROM. Top: fixed interface modes. Bottom: interface DOFs

Fig. 1.5

Polynomial Fourier Constraint Modes (PFCM) interface basis functions

To demonstrate the effect of the additional polynomial basis functions on the efficiency of the interface reduction, results from Schwerdt et al. (2020) are shown in Fig. 1.6. Here the maximum eigenfrequency error of the first mode family is plotted against the number of DOFs for a simplified two stage rotor. Only the interface is reduced to isolate the effect of the interface reduction. For each graph the maximum interface polynomial degree is fixed (from 0 to 4) and the number of Fourier harmonics increases from left to right. The interface between the stage is ring shaped and consists of five concentric rings of nodes in the FE model. Therefore, a polynomial degree of four represents the maximum possible, and yields the same results as the FCM method. The results show that including the highest polynomial degree is optimal only if a large number of Fourier harmonics is included as well, which corresponds to little reduction of the interface DOFs. For reasonable reduction levels, a balanced truncation of the polynomial and Fourier basis is optimal, and best- case accuracy improvements of more than a factor of 100 are possible using PFCM compared to FCM for this system. For further discussion on this topic, including the possibility to truncate the individual DOFs instead of using a fixed global limit of the polynomial degree and maximum Fourier harmonic, see Maroldt et al. (2022c).

Fig. 1.6

Eigenfrequency error of the first mode family depending on the amount of interface reduction for a simplified two-stage geometry (Schwerdt et al. 2020). Only the interface is reduced (©SEM)

As with all substructuring-based ROMs the reduction basis contains more degrees of freedom than necessary. Therefore, it can be further reduced using a modal analysis of the reduced model to get the modes of the tuned system (Bladh et al. 2001). Here it is beneficial due to the presence of multiple stages, but for single stage rotors the tuned system modes can be calculated directly, and substructuring is usually not advantageous if it is used only to calculate the tuned modes. To include mistuning into the ROM multiple approaches are possible depending on the required accuracy and targeted calculation times. This is similar for all model order reduction methods based only on the tuned structure. First, the mistuning can be projected directly onto the reduced basis. This is the most accurate but most expensive option. Second, the structural changes can be limited to projection modes selected beforehand. This allows for some calculations to be performed in advance and thus reduces the computational effort. The most popular of these methods is the CMM (Component Mode Mistuning) method (Lim et al. 2007), where the changes of blade-alone modes of each blade are used as mistuning parameters.

To include aeroelastic effects, CFD simulations are performed which are detailed in the following Sect. 1.3.2. The results are incorporated into the ROM as stiffness and damping terms representing the linearized aeroelastic contributions to the system dynamics. These aerodynamic coefficients are calculated for all DOFs of the ROM. And by calculating the aerodynamic effect of a displacement of one DOF on another, aeroelastic intermodal and interstage coupling is included. Due to the comparatively large computational effort of the CFD calculations, it is paramount to reduce the number of DOFs as much as possible while keeping the monoharmonic nature of the displacements of each DOF. To achieve this, the benefits of a secondary interface reduction were investigated in Maroldt et al. (2022c).

3.2 Aeroelastic Model

The computation of the aeroelastic coefficients follows the procedure described in Willeke et al. (2017) for single-stage simulations. However, compared to the calculation on a single or isolated stage-basis in multistage calculations two challenges arise: First, aeroacoustic scattering of the exciting engine order occurs, leading to the excitation of multiple nodal diameters. These effects need to be considered in the forcing calculations. Second, the unsteady pressure distributions on a blade, generated by a vibrating blade of another stage lead to additional aerodynamic damping and stiffness. Therefore, the computational domain of the flutter simulations needs to incorporate at least the neighboring stages, which highly increases the computational effort.

To solve this problem efficiently, a harmonic balance approach, implemented in the flow solver TRACE, is used. The method allows calculating one passage using phase- shifted periodic boundary conditions. Calculations were performed using the k-log (ω) (Müller and Morsbach 2018) version of the Menter-SST (Menter et al. 2003) turbulence model with the Kato & Launder (Kato and Launder 1993) stagnation anomaly fix. This is the same setup as used for the calculations of the 1^1/2-stage compressor described and validated in Sect. 1.2. Both, aerodynamic excitation and aerodynamic coefficients, are calculated based on the aerodynamic work done on rotor 1 and 2 for the DOFs of each nodal diameter. In each flutter calculation one DOF is set to vibrate at the frequency of interest and the aerodynamic work done on all DOFs of the same nodal diameter is calculated. To account for interstage coupling the disturbance created by the vibration is calculated in the stator and both rotor domains. For example, a vibrating DOF of the first rotor can create unsteady pressure on the second rotor and therefore couples DOFs of both stages. The investigated vibrational response is excited by EO23, which is created by the IGV. To address mode scattering in the forcing calculation, all acoustic cut-on modes are calculated, see Fig. 1.7. The modes are generated by scattering of the EO23, which is scattered at rotor 1 to a forward traveling wave with nodal diameter 1, and at rotor 2 to a forward traveling wave with nodal diameter 8.

Fig. 1.7

Calculated circumferential modes m in the harmonic balance forcing and flutter calculation. Used with permission of American Society of Mechanical Engineers ASME, from Maroldt et al. (2022b); permission conveyed through Copyright Clearance Center, Inc

3.3 Tuned Results

The results on stage and nodal diameter basis are shown in Fig. 1.8. The frequency response is separated into the excitation of stage 1 and stage 2 by ND1 and ND8. The response is dominated by the excitation of stage 1 with ND1, since it is located just downstream of the exciting IGV. Modes localized in stage 1 (approx. 110 Hz), stage 2 (approx. 132 Hz) and in both stages (e.g. approx. 124 Hz) are visible. Especially the latter mode is created by structural interstage coupling and would not appear in an isolated stage modeling. Modes which are at least partly localized in the second stage also respond, when exciting the second stage with ND1. In contrast, only one mode, solely localized in the second stage, is visible when exciting stage 2 with ND8. The reason for this is that stage-coupled modes usually occur at lower nodal diameters. Lastly, the response, when exciting stage 1 with ND8 only plays a minor role. The overall response, created by the superposition of the individual responses, is shown in Fig. 1.9. Due to the superposition of multiple nodal diameters created by the mode scattering in the forcing calculation, variations of the individual blade amplitudes are visible, leading to an amplitude magnification.

Fig. 1.8

Frequency response of stage 1 and stage 2 when exciting an individual stage with a certain nodal diameter

Fig. 1.9

Overall frequency response of the individual blades of stage 1 and stage 2. Used with permission of American Society of Mechanical Engineers ASME, from Maroldt et al. (2022b); permission conveyed through Copyright Clearance Center, Inc

3.4 Mistuned Results

In addition to the forced response calculations using the tuned rotor, Monte-CarloSimulations were done to assess the influence of the multistage aeroelastic coupling on the mistuning amplitude magnification (Maroldt et al. 2022b). The results are pictured in Fig. 1.10. Here, the amplitude magnification due to mistuning are com- pared for two different damping cases. First, the full aeroelastic damping including interstage coupling (fullA) is used. In the simulations to produce the second graph, only the aerodynamic coefficients within each stage were kept, while the coefficients responsible for the coupling between stages were set to zero (singleStageA). The structural damping was omitted in both cases, due to its comparatively small magnitude for blisks. For the investigated resonance crossing of this rotor and the chosen mistuning level, the mistuning sensitivity is underestimated approximately by ten percent when omitting the aeroelastic interstage coupling. When looking at the resulting stresses and therefore, the estimated fatigue life, it is useful to perform evaluations based on the amplitude frequency strength (af-strength) (Hanschke et al. 2017), which is the ratio of actual and allowable loading. The results with full aero-coupling show lower amplitudes, which is reflected by an increase of the af-strength of 17% (Maroldt et al. 2022b).

Fig. 1.10

Amplitude magnification of the first stage including full aeroelastic coupling (fullA) and aeroelastic coupling only within each stage (singleStageA). Both cases use 5000 identical samples of Gaussian stiffness mistuning of the sectors with a standard deviation of 0.002. Used with permission of American Society of Mechanical Engineers ASME, from Maroldt et al. (2022b); permission conveyed through Copyright Clearance Center, Inc

Similar to the way intentional mistuning can mitigate the effects of unwanted random mistuning for single stages, it can be applied to rotors with multiple stages. For modes with participation of more than one stage, it is possible to introduce the intentional mistuning in one stage to reduce the vibration amplitudes in another stage. This is analogous to the possibility of affecting the blade vibration with intentional mistuning of the disk (Schwerdt et al. 2019), which was previously researched in the subproject C3 of the CRC 871. This is demonstrated for the 2^1/2-stage compressor in Fig. 1.11, where the splitting of the double mode due to intentional mistuning as well as the distribution of expected amplitude magnifications with and without intentional mistuning (MT) are shown. In this case the median of the amplitude magnification can be reduced from 1.202 to 1.076.

Fig. 1.11

Left: Frequency response of the 2^1/2-stage compressor with intentional mistuning of the second stage. Right: Amplitude magnification of the first stage with and without intentional mistuning (MT) of the second stage

4 Reduced Order Modelling for Large Mistuning

Reduced order models based on the tuned structure, such as the one discussed in Sect. 1.3.1, are most efficient for structures with small mistuning. Different methods are needed if accurate modeling of larger structure changes is necessary. This may be the case when investigating large damages, or the milling process during a patch repair of a blisk. In these cases, the large structural change is confined to a single blade.

In literature, multiple methods are available to deal with these cases with large mistuning. Here, it is important to distinguish between methods that are restricted by their reliance on the structure of the finite element mesh, and those which can capture arbitrary changes of the blades. Due to the constraints on the FE-mesh the former methods (see for example Lupini and Epureanu (2019); Sinha (2009)) can be more efficient, and should therefore be preferred if applicable. But this section deals with the latter methods focusing on large arbitrary geometric mistuning. There, different variants of generally applicable substructuring algorithms have been known for decades (Craig et al. 1968). Specializing on bladed disks with large mistuning, the PRIME (Pristine Rogue Interface Modal Expansion) method (Madden et al. 2012) takes advantage of the cyclic symmetry of the base system. This method was used as a reference when evaluating the newly developed method PRISM (Schwerdt et al. 2021), which is discussed in this section. Only single stage rotors are considered here, although the methods can be extended to multistage applications (Kurstak and D’Souza 2018). The DOFs of the rotor are partitioned into those of nominal (or pristine) sectors, those belonging to modified (or rogue) sectors and the DOFs of the shared interface between these sector types, shown in Fig. 1.12. For the PRIME method, cyclic modal analyses are performed for full blisks consisting of only pristine and rogue sectors, respectively. The full blisk is then assembled in the reduced order space where the interface DOFs are reduced using the pristine modes. The key insight is to use independent sets of DOFs for the pristine sectors, the rogue sector, and the interface to reduce the influence of non-matching displacement subspaces of the interface in the different cyclic modes. The resulting ROM is reduced further to avoid unnecessary DOFs and numerical issues with a badly conditioned reduction basis.

Fig. 1.12

Mistuned compressor blisk split into pristine, interface and rogue degrees of freedom. Used with permission of American Society of Mechanical Engineers ASME, from Schwerdt et al. (2021); permission conveyed through Copyright Clearance Center, Inc

The newly developed PRISM method works by building the eponymous partially reduced intermediate system model of the whole blisk. First, the cyclic modes of the rotor with only nominal sectors are calculated. These are used to reduce the nominal sectors while the full FE-model of the modified sector is kept. To assemble the intermediate system model, the interface DOFs of the modified sector are reduced using the modes of the nominal sector. Thereby, displacement compatibility is ensured between the reduced nominal sectors and the non-reduced modified sector in the partially reduced intermediate system model. A modal analysis of this model results in the final reduction basis for the whole blisk.

The key advantage of the PRISM method is the reduced calculation time for the modal analysis of the intermediate system model compared to a full cyclic modal analysis of the modified sector type. This reduces the computational effort for the reduction procedure compared to the PRIME method, with a calculation time reduction of 88, and 45% when including the time taken for the modal analysis of the nominal sector type, as reported in Schwerdt et al. (2021). It should be noted however, that smaller gains are expected when using a fully iterative modal analysis solver, as opposed to the block Lanczos used in Schwerdt et al. (2021), or when calculating more modes. This is due to the time taken for the decomposition of the stiffness matrices for each harmonic index being the dominant part of the calculation time. In addition to the reduced time for the model reduction, compared to PRIME the accuracy is improved as well. By using the modified sector without reduction in the intermediate model, the modes of the complete system are captured slightly better compared to using a limited reduction basis of cyclic modes. This is shown in Fig. 1.13, where the accuracy in predicting the eigenfrequencies of both methods is compared using ten modes per sector for the blisk shown in Fig. 1.12.

Fig. 1.13

Relative eigenfrequency errors of PRIME and PRISM ROMs. Used with permission of American Society of Mechanical Engineers ASME, from Schwerdt et al. (2021); permission conveyed through Copyright Clearance Center, Inc

5 Conclusions

In this chapter efficient and accurate reduced order models (ROMs) for the calculation of the vibrational behavior of realistic turbomachinery was presented. The models account for structural mistuning due to realistic geometries, which are shaped by wear and regeneration. If a regeneration process of one blade leads to large geometric changes, the developed approach PRISM is able to calculate the influence on the vibrational behavior with slightly better accuracy in approximately half the time compared to the state of the art method for the analyzed example.

Furthermore, a ROM for the calculation of multistage turbomachinery including structural and aeroelastic interstage coupling was developed. It was shown that the interstage coupling can influence the vibration amplitudes in multistage turbomachinery. For the test case investigated, the predicted amplitude frequency-strength increases by 17% when interstage coupling is included, allowing a more accurate fatigue life prediction. However, the coupling effects depend on various parameters such as number of stages, stiffness of the rotor, and the eigenfrequencies of the isolated stages. Therefore, the influence of interstage coupling may be less severe in other cases.

Since for multistage calculations a large amount of aeroelastic flutter calculations needs to be performed, the ROM was extended by an additional a posteriori interface reduction to reduce the number of aeroelastic simulations necessary for a given accuracy level of the ROM. On this basis more complex investigations, such as using speed-dependent aeroelastic coefficients, can be conducted in the future.