## Abstract

Stators with nonuniform vane spacing (NUVS) are known to effectively reduce the forced response of adjacent rotors by spreading the primary engine order (EO) excitation at a certain speed to a series of weaker excitations at nearby EOs over a wider speed range. To be used in a real gas turbine engine, it is essential to know the forced response vibratory reduction that can be achieved for a specific NUVS configuration at different operating conditions. The classical estimation method to predict the blade response reduction is to quantify the reduction of the forcing function at a potential resonance crossing by conducting a circumferential Fourier analysis of the asymmetric flow field. However, besides excitation reduction, the blade forced response also depends on other factors, such as damping and blade-to-blade interactions due to mistuning. To study the blade forced response reduction in a realistic environment, a comprehensive experimental study was conducted in the Purdue three-stage axial research compressor at three different loading conditions. The vibratory response of the first torsion (1 T) mode forced response of rotor 2 was measured by strain gages (SGs) for two different upstream stator 1 configurations: the symmetric 38-vaned stator 1 configuration and the NUVS stator 1 configuration with 18–20 vane halves. As expected, the strong 38EO-1T response from the symmetric stator reduced to a series of weaker responses from 35EO to 41EO in the NUVS configuration. The overlap of adjacent EO responses caused considerable beating phenomena in the blade SG time–history data. There was substantial blade-to-blade variation in blade response for both symmetric and asymmetric stator 1 configurations due to the inherent nonintentional mistuning in the rotor 2 blisk. This, in turn, causes a large blade-to-blade variation in the reduction factors. While higher responding blades tend to have larger reduction factors, some blades show almost no forced response reduction when switching to the NUVS stator. In addition, the reduction factors for each blade and the maximum reduction factor for the whole blisk also change significantly with loading conditions. The measured reduction factors and the peak response EO are not well predicted by the classical estimation method. This indicates that both damping and mistuning effects need to be considered in the NUVS reduction factor prediction.

## 1 Introduction

In axial compressors, the forced response of a rotor due to the relative motion between rotors and stators is a significant aeromechanical problem causing blade high cycle fatigue. The current design trend of increasing power density leads to higher blade loading and reduced blade row spacing, making the potential forced response problem even worse. Stators with nonuniform vane spacing (NUVS) have been considered as a way to mitigate the rotor forced response problem, in addition to various damping enhancement methods. Physically, NUVS can effectively reduce the forced response of adjacent rotors by spreading the primary engine order (EO) excitation at a certain speed to a series of weaker excitations at neighboring EOs. Of the various types of NUVS designs, the half-half design, where the stator is divided into two stator halves, each with uniform vane spacing but slightly different vane counts, is considered to be the most practical design and had been applied to real gas turbine engines [1].

To utilize a NUVS stator in a real engine, it is essential to predict how much blade vibratory forced response can be reduced compared to the corresponding symmetric stator (i.e., reduction factor). The classical estimation method [2] predicts the blade forced response reduction by quantifying the reduction of the forcing functions at potential EOs through the circumferential Fourier analysis of the asymmetric potential field/wake generated by the NUVS stator. However, other important factors affecting blade forced response, such as damping and mistuning, are not considered in the classical method.

Mistuning plays an important role in blade forced response by causing high vibratory amplitude on certain blades because of mode localization. Extensive experimental, analytical, and computational studies have been done on mistuned rotor forced response [3–10]. However, most of the previous mistuning studies focus on the amplification factor evaluation due to symmetric stator excitation (i.e., the maximum mistuned rotor-to-tuned-rotor forced response ratio). The study of mistuned rotor response due to NUVS stator excitation is rare. Kaneko et al. [11] conducted early analytical work using an equivalent spring-mass model of a bladed disk.

Aerodynamic asymmetry, associated with small variations in the blade profile and blade spacing, is also known to affect the blade dynamic response. While mistuning changes the structural coupling among the blades, aerodynamic asymmetry changes the aerodynamic coupling among the blades. There have been a series of analytical studies [12–14], computational studies [15–17], and an experimental study [18] on the aeroelastic behavior of a blade row with aerodynamic asymmetry, such as nonuniform stagger angle and nonuniform blade spacing. However, most of the aerodynamic asymmetry studies focus on the forced response and the stability of the aerodynamic asymmetric blade row itself, not the forced response of neighboring rotor as studied in the article.

Due to asymmetry, a high-fidelity computational fluid dynamics (CFD)-finite element method simulation of mistuned rotor forced response due to NUVS stator excitation in a multistage environment can be very computationally intensive. As in the classical method, most of the previous CFD works [19,20] have been centered on quantifying the forcing function reduction, not the final blade response reduction. Niu et al. [21] and Monk et al. [22] managed to simulate the blade forced response under NUVS stator excitation through various simplifications in their aeroelastic models. However, no mistuning effects were included in either study.

The experimental studies of blade forced response reduction under NUVS stator excitation were few and lacked detail. Kaneko et al. [1] presented some strain gage (SG) data of the rotor forced response under the upstream NUVS stator excitation in a three-stage scaled-model compressor. Kemp et al. [2] measured turbine blade stress reduction for different upstream asymmetric nozzle vane configurations in a turbojet engine. However, the experimental results presented in both studies were very brief. No detailed blade resonant response curves were given, and the mistuning effect was not discussed at all.

The objective of this study is to fill the gap by presenting and analyzing a detailed blade forced response dataset under both symmetric stator and NUVS stator excitations. To be relevant to a realistic environment, the experimental study was conducted on a naturally occurring, nonintentionally, mistuned rotor in a multistage axial compressor of mid-TRL (technology readiness level). The organization of this article starts with Sec. 2, where the experimental facility, instrumentation, and data acquisition and processing strategy are presented. Section 3 introduces the classical reduction factor estimation method, which is then used as a reference for the following experimental results discussion. Section 4 covers the detailed blade response SG data under both symmetric stator excitation and NUVS stator excitation, and the limitation of the classical method is discussed to explain the large discrepancy between the predicted and measured reduction factors. Finally, Sec. 5 presents the key findings of this experimental study.

## 2 Experimental Setup

The Purdue three-stage axial research compressor (P3S) was used for this NUVS experimental study. It is a scaled-up version of a highly loaded axial compressor of mid-TRL, aerodynamically representative of the last several stages of a modern high-pressure compressor, with engine-relevant Mach numbers and Reynolds numbers. More details about the P3S can be found in Refs. [23–25]. As shown in Fig. 1, the compressor consists of an inlet guide vane, three integrally bladed rotors (or blisks), and three shrouded stator rows. Stator 1 (S1) can be configured as either a symmetric stator with 19 vanes in each half or an asymmetric stator with 18 vanes in one half and 20 vanes in the other. Each stator is individually indexable and can be moved circumferentially in an approximate 16 deg range, which allows detailed S1 wake profile measurement using the downstream seven-element stagnation pressure rakes.

To study the effectiveness of NUVS on forced response reduction, the rotor 2 (R2) forced response was measured under the excitation of two different configurations of the upstream stator 1: a uniform spaced symmetric 38-vanes stator 1 and an asymmetric 18–20 vanes NUVS stator 1 (vane spacing is uniform on each stator half). The NUVS stator was designed to have the same vane number as the symmetric stator and just slightly different uniform vane spacing on each stator half, in order to minimize the impact on aerodynamic performance of the compressor [26].

The Campbell diagrams of rotor 2 for both symmetric and NUVS stator 1 excitation are shown in Fig. 2. The focus of this study is on the first torsion mode (1 T) resonant response of rotor 2. During the test, rotor 2 forced response was measured simultaneously by strain gages and an eight-probe Agilis nonintrusive stress measurement system (NSMS), for a speed sweep from 3200 rpm to 4800 rpm at a sweep rate of 3.125 rpm/s. This sweep rate was chosen since it is much lower than the critical sweep rate of R2, which is around 15 rpm/s calculated using the criterion given in Ref. [27]. This low sweep rate ensures that blade forced response reaches the steady state for each revolution and, thus, minimizes the transient response in the collected data.

There were three strain gages (Micro-measurement ED-DY-031CF-350) per blade attached to eight blades of the 33-bladed rotor 2. The strain gage locations were optimized using the commercial software gagemap for the first five vibrational modes of R2. Generally speaking, for a specific mode, the SG should be placed on a blade where the modal strain is high for better sensitivity, but where the gradient of modal strain is low to reduce effects associated with placement error. The SG2 location was optimized to capture the 1 T mode response, and thus, the SG2 data were the main experimental data analyzed in this study. Prior to the experiments, a detailed benchtop modal analysis was conducted. The nonintentional mistuning level of rotor 2 was characterized by measuring each blade's modal frequencies with a standard impact test. The SG data were transferred out of the rotor using an Aerodyn slip ring system, conditioned using a Precision Filter 28144 signal conditioning system and collected at a 100 kHz sampling rate using an NI-PXI data acquisition system.

To study the forced response reduction associated with the NUVS stator at different operating conditions, the same experiments were repeated at three different loading conditions (i.e., high loading (HL), peak efficiency (PE), and low loading (LL)). The loading conditions were set by slowly closing the throttle to properly load the compressor to match the desired corrected mass flowrate and total pressure ratio (TPR) at 68%Nc (3400 rpm). Table 1 summarizes the corrected mass flowrate, $M\u02d9c$, and TPR at each loading condition for both symmetric and asymmetric stator 1 configurations.

## 3 Reduction Factor and the Classical Estimation Method

*R*, which quantifies the reduction of blade resonant response excited by the NUVS stator $(\sigma NUVS)$ compared to the blade resonant response excited by the symmetric stator $(\sigma sym)$ for a specific resonant crossing:

*n*th circumferential component, which has a circumferential wave number

*n*.

For a symmetric stator, where *n _{1}* =

*n*and

_{2}*a*=

_{1}*a*=

_{2}*a*, there is only one circumferential component with a circumferential wave number equal to

*n*+

_{1}*n*after decomposition. Thus, the excitation frequency is at only one corresponding EO, which is EO =

_{2}*n*=

*n*+

_{1}*n*with strength

_{2}*a*. For a NUVS stator, there are many more circumferential components due to the asymmetry. As shown in Fig. 3, for the 18–20 vanes NUVS stator 1 used in this experimental study, the 38EO excitation (as experienced for the symmetric stator 1) has spread to a series of the neighboring EO components, with the peaks at 36EO and 40EO, twice the vane counts of each stator half.

*x*is blade deflection, $\zeta $ is the critical damping ratio, $\omega n$ is the blade natural frequency, and $f(t)$ is the forcing function.

As shown in Fig. 3, the maximum EO components are 36EO and 40EO, both having strength at half of the symmetric 38 EO response. Thus, the estimated reduction factor *R* = 1−0.5/1 = 0.5 for the NUVS used in the experimental study. This value will be used as a reference to compare with the experimentally measured reduction factors in this study.

## 4 Experimental Results and Discussion

In this section, the experimental results of a single representative blade are given first. The experimental results of all strain-gaged blades are presented and analyzed next. Finally, the limitations of the classical estimation method are discussed to explain the large differences between the predictions and experimental results.

### 4.1 Single Representative Blade Results.

The raw SG data for a representative blade, blade 33, are shown in Fig. 4 for both the symmetric 38-vane S1 excitation and the NUVS 18–20 S1 excitation. Blade 33 was chosen as a representative blade because its maximum response is close to the average of all the strain-gaged blades for both symmetric and asymmetric S1 configuration.

In Fig. 4, the 44EO-1T resonance is due to excitation from the downstream 44-vaned stator 2. As expected, a similar response is observed for both configurations. On the other hand, after switching to the NUVS stator, the strong 38EO-1T response from the symmetric 38S1 excitation spreads into a series weaker responses in a 41-40-39EO-1T cluster and 37-36-35EO-1T cluster over a wider speed range. Although the SG2 location was not optimized for the first chordwise bending (1CWB) mode (and thus, its sensitivity to 1CWB is low), the 76EO-1CWB response (excited by the second harmonic of the symmetric 38-vane S1) can still be discerned in the symmetric 38-vane S1 SG signal. After switching to the NUVS excitation, the 76EO-1CWB response also spreads to a larger rpm range and blends in with the other 1 T responses. Overall, the maximum blade response amplitude for the NUVS configuration reduces to about 72% of that for the symmetric 38-vane S1 configuration, i.e., a reduction factor of 0.28. However, this blade reduction factor value is much lower than the classical prediction of 0.5.

Next, the SG2 data are analyzed in more detail using a spectrogram, which provides additional vibrational frequency information during the sweep. Using short-time Fourier transforms, with a window size of 0.1 s and 50% window overlap, the spectrograms of the forced response of blade 33 under the symmetric S1 and NUVS S1 excitation are shown in Figs. 5 and 6, respectively. The log scale is used here to highlight each EO's response, even the relatively weak ones. For the symmetric 38-vane S1 case, the three resonant responses in Fig. 4 show up clearly as the three corresponding resonant crossings in Fig. 5. The resonant speed range of each crossing is well separated. On the other hand, the spectrogram becomes crowded for the NUVS S1 excitation, as shown in Fig. 6. The single 38EO-1T resonant crossing in the symmetric stator case becomes a series of adjacent resonant crossings ranging from 35EO to 41EO due to the asymmetry of the NUVS stator. Similarly, the single 76EO-1CWB resonant crossing also spreads to a series of resonant crossings over a larger speed range.

Due to these additional resonant crossings, there are considerable multi-EO responses occurring within the same speed range. The first type is a typical multimode response, where the 1CWB and 1 T modes are excited at the same speed by two very different EO excitation lines, e.g., the vertical rectangle illustrated in Fig. 6. The second type of multi-EO response is caused by the overlapping of the adjacent EO blade responses for the same mode, e.g., the horizontal rectangle illustrated in Fig. 6. This overlapping causes the blade to vibrate at the same mode but at two, or even more, closely spaced frequencies due to the simultaneous adjacent EO excitations.

Focusing on the 1 T response excited by the NUVS stator, the maximum response envelope of the SG data of blade 33 is shown in Fig. 7, along with each individual EO response from 35EO to 41EO. Based on the once-per-revolution signal, the SG data were cut into each revolution, and then a Fourier transform was used to extract each individual EO response.

First, the relative strength of each EO response is very different from the prediction by the classical method in Fig. 3. The highest response for the whole sweep occurs at 35EO, not the 36EO or 40EO as predicted by the classical estimation method. Also, for the 39-40-41EO-1T response cluster, the high response occurs at 41EO, not 40EO.

Second, both types of multi-EO responses occur in Fig. 7 as the difference between the peaks of the maximum response envelope and the peaks of each individual EO response. For example, the peak around 4127 rpm corresponds to the additional strong 72EO-1CWB response occurring at the same speed as the relatively weak 40EO-1T response. The overlap of the 35EO-1T response and the 36EO-1T response causes the peak in the maximum response envelope around 4638 rpm. Overall, the overlap of the 1 T response with the adjacent EOs is shown in both the 39-40-41 EO-1T response cluster and 35-36-37 EO-1T response cluster.

The overlapping of the 35EO-1T response and the 36EO-1T response is examined in more detail in Fig. 8. Zooming into the red-arrow pointed peak response at 4638 rpm in Fig. 7, a close-up view of two revolutions of blade response time–history and the corresponding EO components in the frequency domain are shown in Fig. 8. The strong response at both 35EO and 36EO shown in the frequency domain causes the beating pattern in the time–history plot. Additionally, although weaker, the frequency domain also shows a noticeable 64EO-1CWB response, which further complicates the blade response time–history.

The complicated multi-EO blade response excited by the NUVS stator poses a great challenge for the accurate postprocessing of the NSMS data. During the experimental campaign, the blade forced response data were collected simultaneously using an eight-probe Agilis NSMS system along with the SG measurements. Like any blade tip timing system, the NSMS data collected were far undersampled (i.e., eight data points per revolution in this study). It requires prior knowledge of the blade vibration and extensive postprocessing to extract the blade vibration information. The Agilis NSMS system uses a type of sine-fitting algorithm to calculate the blade deflection during forced response, similar to the processing method used by Hood Technology [28] and Rolls-Royce [29]. The sine-fitting algorithm assumes that blades vibrate at a certain EO (or a few EOs) and use a sine wave (or sine waves) at the corresponding frequency (or frequencies) to fit the blade NSMS data measured at each probe for each revolution.

As an inverse problem, the circumferential locations of the probes need to be optimized for the expected EO (or a few EOs) to minimize the condition number of the fitting process [30]. The current NSMS probe locations were fixed and optimized in the past for the 44EO-1T and 38EO-1T blade responses [31]. Since the condition number of the current NSMS probe locations is too high for the multi-EO responses observed in the SG data for the NUVS stator excitation, there are considerable errors in the processed NSMS data for the NUVS S1 configuration. Thus, only the SG data are presented and discussed in this article. There are other types of NSMS postprocessing methods that could be more suitable to measure the blade response due to NUVS stator excitation. For example, the multi-revolution-based resonant curve fitting method [32] is known to be more robust and less sensitive to probe locations.

#### 4.1.1 Loading Effect.

Since the NUVS configuration is fixed after application in a real gas turbine engine, it is desirable to have it sufficiently reduce the downstream rotor forced response at all possible operating conditions. To study this effect, the experimental study was conducted at three different loading conditions as defined in Sec. 2: HL, PE, and LL.

The primary excitation for the rotor 2 1 T resonance is the upstream stator 1 wakes. The stator vane traverse capability of the P3S facility provides a good way to quantify the S1 wake profile and, thus, characterize the excitation forcing function. However, due to the asymmetry of the NUVS stator, it requires a whole circumferential survey to fully characterize the NUVS wakes. Since the stator vanes in the P3S could only traverse about 16 deg, the full NUVS wake profile was not able to be collected. Instead, the symmetric 38S1 wake profile was acquired to provide some insight on how the excitation forcing function changes with the loading condition. Using seven-element stagnation pressure rakes at station 4 (see Fig. 1), the wake profiles of the symmetric 38-vane S1 at different spans were measured. The 88% span wake profile at three loading conditions is shown in Fig. 9. As expected, at high loading, the wake becomes wider and deeper, and the amplitude of the corresponding primary VPF component becomes larger. There is also a small phase shift as shown in the shift of wake center at different loading conditions.

The corresponding 38EO-1T blade forced response at three loading conditions is compared in Fig. 10, for the symmetric 38-vane S1 configuration. There is a considerable difference in the resonant response curves at different loading conditions, and it is not simply a scaling effect based on the primary 38EO excitation strength in the S1 wake, as shown in the right plot of Fig. 9. Besides the excitation forcing function, aerodynamic damping could also change significantly at different loading conditions. Since aerodynamic damping is the dominant damping source for the R2 blisk studied in this article, the change in the aerodynamic damping could be the reason for the large difference in the forced response pattern at different loadings observed in Fig. 10. Also, the mistuning effect is evident from the complicated multipeak patterns in the response curves.

For the NUVS S1 configuration, the blade forced responses at different loading conditions are compared in Fig. 11. Due to the asymmetry, the response curve is more complicated, and the differences at different loadings are even larger than the symmetric 38-vanes S1 case. There are significant changes in the peak EO location, the relative strength of each EO response, and the overall response pattern at different loading conditions. Similar to the symmetric 38-vane S1 case, the aerodynamic damping change at different loading conditions plays an important role in altering the blade forced response. Besides the loading condition, the aerodynamic damping is known to vary significantly with blisk vibration nodal diameter (ND). For a rotor with NB (number of blades) blades, the primary response is $ND=EO\xb1mNB$, where *m* can be any integer such that ND is from *0 to* NB*−*1. Thus, for the series of the adjacent EO excitations caused by the NUVS stator, the corresponding primary blade vibration ND is different, and thus, the corresponding aerodynamic dampings are different. This could further enhance the sensitivity of the forced response on loading for the NUVS stator configuration. For all loading conditions, the primary 38EO-1T responses for the symmetric stator configuration have spread to a series of weaker responses from 35EO to 41EO for the NUVS stator. The large change in the blade forced response for both the symmetric stator and the NUVS stator at different loading conditions results in a large variation in reduction factors, which are 0.28 at HL, 0.49 at PE, and 0.01 at LL. Both the 28% reduction at HL and almost no reduction at LL are far from the classical estimation of 50% reduction factor.

In summary, the experimental results, based on the representative blade 33, show that after switching to the NUVS S1 from the symmetric 38-vane S1: (1) The 38EO-1T response spreads to a wider speed range with weaker response at multiple nearby EOs. However, the measured blade reduction factor is 0.28, which is much lower than the classical prediction of 0.5. (2) The peak EO response occurs at 35EO at HL, which is different from the classical prediction at twice the vane number of each stator half, i.e., 36EO and 40EO. (3) There are considerable multi-EO responses from adjacent EOs and also from the simultaneous 1 T and 1CWB response at the same speed. This is not considered in the classical estimation method. (4) The same NUVS design could have very different reduction factors at different loading conditions, whereas the classical prediction depends on the NUVS vane position only, not on the operating condition. These findings, based on the nominal blade B33, were observed for all strain-gaged blades.

### 4.2 Results From All Strain-Gaged Blades.

Due to the large discrepancy between the classical prediction and the experimental results for the representative blade 33, all eight strain-gaged blades are analyzed in this section for a better understanding of the forced response reduction effect of NUVS stators.

To study the forced response of different blades, the mistuning effect has to be considered. The rotor 2 blisk contains some naturally occurring, nonintentional mistuning. The mistuning pattern was characterized using a standard impact test procedure. For the 1 T mode, each blade's modal frequency is shown in Fig. 12. The strain-gaged blades are marked with a star: B03, B06, B09, B15, B23, B25, B30, and B33. These blades were chosen to have a good representation of the blade frequency mistuning range and are also roughly uniformly distributed over the entire blisk. The mean value of the 1 T blade frequencies is 2721.8 Hz, and the standard deviation is 9.4 Hz, which is only ∼0.34% of the mean value.

For the NUVS S1 excitation, the maximum response envelope for all strain-gaged blades at HL is shown in Fig. 13. There is a large blade–blade variation in the blade response curve during the sweep. The highest response amplitude, peak EO location, and the overall response pattern are all very different from blade to blade. The corresponding maximum response envelope for all strain-gaged blades under the symmetric 38-vane S1 excitation at HL is shown in Fig. 14. Similarly, there is a large blade-to-blade variation in the blade response curve. This indicates a strong mistuning effect for both the symmetric stator and NUVS stator excitation, even at the small nonintentional mistuning level of this blisk. A small change in the blade modal frequency could lead to a considerable change in the blade forced response for both symmetric stator and NUVS stator excitations, and, thus, the considerable change of the corresponding reduction factor.

To consider the blade-to-blade variation due to mistuning, three different reduction factors are defined as follows:

- Blade reduction factor, $Rblade$, is the reduction factor for a specific blade:(5)$Rblade=1\u2212xNUVSxsym$
- Rotor average-to-average reduction factor, $Rrotor_avg$, is based on the ratio of the average response of all blades of a mistuned rotor excited by a NUVS stator and the average response of all blades of the mistuned rotor excited by the corresponding symmetric stator:(6)$Rrotor_avg=1\u2212Avg(xNUVS)Avg(xsym)$
- Rotor maximum-to-maximum reduction factor, $Rrotor_max$, is based on the ratio of the maximum response of all blades of a mistuned rotor excited by a NUVS stator and the maximum response of all blades of the mistuned rotor excited by the corresponding symmetric stator:(7)$Rrotor_max=1\u2212Max(xNUVS)Max(xsym)$

Note that the maximum responding blade is not necessarily the same between the symmetric stator and the NUVS stator excitations.

In practice, the maximum response of a rotor is the one of the greatest concern. Thus, the max-to-max reduction factor should be considered the most important parameter to evaluate a NUVS stator design.

Table 2 summarizes the peak amplitude and peak EOs of all strain-gaged blades for both the symmetric S1 and NUVS S1 excitation at HL, along with the three types of reduction factors.

Symmetric S1 | NUVS S1 | Reduction factor | |||
---|---|---|---|---|---|

Peak amp (ksi) | Peak EO | Peak amp (ksi) | Peak EO | ||

B03 | 13.3 | 38EO | 7.9 | 35EO | 0.40 |

B06 | 18.1 | 38EO | 6.8 | 37EO | 0.62 |

B09 | 19.4 | 38EO | 15.2 | 37EO | 0.22 |

B15 | 31.9 | 38EO | 12.7 | 37EO | 0.60 |

B23 | 14.0 | 38EO | 10.1 | 37EO | 0.27 |

B25 | 9.5 | 38EO | 9.4 | 35EO | 0.01 |

B30 | 14.5 | 38EO | 8.0 | 37EO | 0.44 |

B33 | 16.1 | 38EO | 11.6 | 35EO | 0.28 |

Max | 31.9 | 38EO | 15.2 | 37EO | 0.52 |

Avg | 17.1 | — | 10.2 | — | 0.40 |

Symmetric S1 | NUVS S1 | Reduction factor | |||
---|---|---|---|---|---|

Peak amp (ksi) | Peak EO | Peak amp (ksi) | Peak EO | ||

B03 | 13.3 | 38EO | 7.9 | 35EO | 0.40 |

B06 | 18.1 | 38EO | 6.8 | 37EO | 0.62 |

B09 | 19.4 | 38EO | 15.2 | 37EO | 0.22 |

B15 | 31.9 | 38EO | 12.7 | 37EO | 0.60 |

B23 | 14.0 | 38EO | 10.1 | 37EO | 0.27 |

B25 | 9.5 | 38EO | 9.4 | 35EO | 0.01 |

B30 | 14.5 | 38EO | 8.0 | 37EO | 0.44 |

B33 | 16.1 | 38EO | 11.6 | 35EO | 0.28 |

Max | 31.9 | 38EO | 15.2 | 37EO | 0.52 |

Avg | 17.1 | — | 10.2 | — | 0.40 |

First, for both symmetric and NUVS S1 excitation, there is a large blade-to-blade variation in the peak response amplitude and, thus, the corresponding blade reduction factor. In general, the high-responding blades tend to have high reduction factors, such as B15. The low-responding blades tend to have low reduction factors. For example, B25 has reduction factor of nearly zero. Note that the highest responding blade has shifted from B15 for the symmetric S1 configuration to B09 for the NUVS S1 configuration. For all eight strain-gaged blades, the NUVS peak response EO occurs at either the 35EO or 37EO, not the 36EO or 40EO predicted by the classical estimation method.

Second, the max-to-max reduction factor measured is 0.52, which is higher than the classical prediction of 0.5. The avg-to-avg reduction factor measured is 0.40, which is lower than the classical prediction. The blade reduction factor varies a lot and can be far from the classical estimation. Note that only 8 of the 33 blades were measured. Although providing a good indication, the max-to-max and avg-to-avg reduction factor listed in Table 2 do not define the whole rotor blisk.

#### 4.2.1 Loading Effect.

To study the effectiveness of NUVS on forced response reduction at different operation conditions, the peak response EOs and reduction factors for all strain-gaged blades at the three loading conditions are summarized in Table 3. At different loading conditions, the reduction factor for each blade varies significantly. From high loading to low loading, the max-to-max reduction factor follows roughly a linearly decreasing trend, but the avg-to-avg reduction factor does not show any clear trend.

NUVS peak EO | Reduction factor | |||||
---|---|---|---|---|---|---|

HL | PE | LL | HL | PE | LL | |

B03 | 35EO | 37EO | 40EO | 0.4 | 0.64 | 0.51 |

B06 | 37EO | 40EO | 40EO | 0.62 | 0.19 | 0.51 |

B09 | 37EO | 37EO | 39EO | 0.22 | 0.06 | 0.52 |

B15 | 37EO | 37EO | 41EO | 0.6 | 0.45 | 0.55 |

B23 | 37EO | 40EO | 40EO | 0.27 | 0.54 | 0.05 |

B25 | 35EO | 35EO | 40EO | 0.01 | 0.32 | 0.22 |

B30 | 37EO | 40EO | 40EO | 0.44 | 0.65 | 0.23 |

B33 | 35EO | 35EO | 40EO | 0.28 | 0.49 | 0.01 |

Max | 37EO | 37EO | 40EO | 0.52 | 0.45 | 0.39 |

Avg | — | — | — | 0.4 | 0.45 | 0.37 |

NUVS peak EO | Reduction factor | |||||
---|---|---|---|---|---|---|

HL | PE | LL | HL | PE | LL | |

B03 | 35EO | 37EO | 40EO | 0.4 | 0.64 | 0.51 |

B06 | 37EO | 40EO | 40EO | 0.62 | 0.19 | 0.51 |

B09 | 37EO | 37EO | 39EO | 0.22 | 0.06 | 0.52 |

B15 | 37EO | 37EO | 41EO | 0.6 | 0.45 | 0.55 |

B23 | 37EO | 40EO | 40EO | 0.27 | 0.54 | 0.05 |

B25 | 35EO | 35EO | 40EO | 0.01 | 0.32 | 0.22 |

B30 | 37EO | 40EO | 40EO | 0.44 | 0.65 | 0.23 |

B33 | 35EO | 35EO | 40EO | 0.28 | 0.49 | 0.01 |

Max | 37EO | 37EO | 40EO | 0.52 | 0.45 | 0.39 |

Avg | — | — | — | 0.4 | 0.45 | 0.37 |

At HL, the peak response EOs of all strain-gaged blades reside in the 35-36-37EO cluster, while at LL, the peak response EOs reside in the 39-40-41EO cluster. As a transition, at the PE condition, the peak response EOs of all strain-gaged blades occurs in both clusters. Note that higher speeds usually lead to stronger excitation for all loading conditions. Thus, the lower the EO excitation, the higher the forcing function strength is, because lower EO resonant crossings occur at higher speeds. However, at LL, the measured peak EO response occurs at the higher EO and thus in the lower speed cluster, the 39-40-41EO cluster. The speed-dependent aerodynamic damping could be the reason for this apparent contradiction. For the blisk rotor used here, aerodynamic damping is the dominant damping source, and it can change significantly with speed and loading conditions in a nonlinear way.

Overall, mistuning together with the loading and speed-dependent aerodynamic damping and forcing is believed to contribute to the large variation in the peak response EO and reduction factor at different loading conditions shown in Table 3. In addition, the pressure waves due to multiage interaction, known as Tyler-Sofrin modes [33] or spinning modes [34], could also alter aerodynamic damping and forcing at different operating conditions. These pressure waves have been studied and quantified using casing unsteady pressure measurement in multistage axial compressors before [35,36].

### 4.3 Limitations of the Classical Estimation Method.

To understand the large discrepancy between the measured reduction factor and the predicted reduction factor, the major limitations of the classical estimation method are discussed in this section.

As stated in Sec. 2, the classical estimation method is based on the circumferential Fourier transform of the vane position-dependent sinusoidal forcing functions with uniform amplitudes, Eq. (2). The only input parameter is the vane positions of the NUVS stator. Without solving the EOM, Eq. (3), it inherently assumes that the maximum blade response is linearly proportional to the maximum EO components of the forcing functions. Thus, this approach has several limitations:

It ignores the damping difference for different EO excitations. Aerodynamic damping is the dominant damping source for a blisk rotor, and it can change significantly with blisk vibration ND, which depends on the excitation EO. Thus, the assumption of constant damping over all EO excitations in the classical estimation method may cause larger error in assessing the blisk response under NUVS excitation.

The classical estimation method is based on the strength of each individual forcing EO component. It does not consider the overlap of the resonant speed range of adjacent EO responses, as seen in these experimental results. The multiple adjacent EO responses increase the maximum blade response level and thus makes the reduction effect less than that predicted by the classical method. The multi-EO-1T-1CWB forced response present in the experimental results has a similar effect to reduce the reduction factor, but it is not considered in the classical estimation method.

The classical estimation method does not account for the change of the excitation forcing and damping (especially the aerodynamic damping) at different speeds and different loading conditions, both of which are known to vary significantly with operating condition.

The classical estimation method approximates the forcing function of the NUVS stator as a sinusoidal wave with peaks corresponding to the vane position. For NUVS excitation, the blade modal force time–history is complicated since a blade passing a different stator half is excited at a different frequency. However, at the primary vane passing frequencies, the complicated unsteady modal force can always be decomposed to a series of sinusoidal waves. Thus, while the sinusoidal wave approximation used in the classical estimation method cannot give accurate blade forced response amplitude predictions, it may be still good enough for the relative comparison between the symmetric stator and different NUVS designs.

The most important drawback of the classical estimation method is that it does not consider the blade-to-blade interaction in a blisk. The mistuning effect is totally ignored. As shown in Sec. 4, there are significant blade-to-blade variations in the forced response due to mistuning for both the symmetric and NUVS stator excitation.

## 5 Conclusions

In this article, a comprehensive experimental study of the blade forced response reduction due to NUVS excitation was conducted in P3S. The blade forced response of the embedded blisk rotor 2 was measured using strain gages for two configurations of the upstream stator: the symmetric 38-vane stator 1 and the asymmetric NUVS 18–20 stator 1. The major experimental findings are as follows:

The primary 38EO-1T response for the symmetric stator 1 configuration reduces and spreads to a series of weaker adjacent EO responses over a much larger speed range for the NUVS stator 1 configuration. The additional resonant crossings cause the overlap of resonant speed ranges. Considerable multi-EO responses from adjacent EOs, and also from simultaneous 1 T and 1CWB response at the same speed, are observed in the SG blade response data.

There is significant blade-to-blade variation in rotor 2 blade forced response for both symmetric and asymmetric stator 1 configurations, even at the low, nonintentional mistuning level of the rotor 2 blisk. (The standard deviation of the rotor 2 blade 1 T modal frequency is only 0.34% of the mean value.) The corresponding reduction factor of each blade varies greatly. In general, the high-responding blades tend to have high reduction factors, and the low-responding blades tend to have low reduction factors. The highest responding blade for the symmetric stator 1 configuration is no longer the highest responding blade for the NUVS stator 1 configuration.

For the same NUVS stator 1 design, both the blade forced response pattern and the corresponding reduction factors change greatly with the loading conditions.

The experimental results were compared with the prediction from the classical estimation method. Both the value of reduction factors and the EO of the peak response are very different between the measurement and prediction. The experimental results suggest that the mistuning, speed, and loading dependency of the aerodynamic damping and forcing function play a significant role in the forced response reduction effect of the NUVS stator. The classical estimation method needs to be extended to include at least mistuning and aerodynamic damping to give a reasonable prediction of the NUVS stator performance in a realistic engine operating environment.

## Acknowledgement

This project was funded by the GUIde VI and GUIde VII Consortia, and this support is gratefully acknowledged. The authors would like to thank Amanda Beach for helping with the experiments and thanks also go to Dr. Willem Rex from MTU for many helpful discussions.

## Conflict of Interest

There are no conflicts of interest.

## Data Availability Statement

Data provided by a third party listed in Acknowledgment.

## Nomenclature

- $x$ =
blade vibratory deflection

*R*=reduction factor

- 1T =
first torsion mode

- 1CWB =
first chordwise bending mode

- EO =
engine order

- EOM =
equation of motion

- ND =
nodal diameter

- NUVS =
nonuniform vane spacing

- NSMS =
nonintrusive stress measurement system

- SG =
strain gage

- TRL =
technology readiness level

- VPF =
vane pass frequency

- $\sigma $ =
blade vibratory stress

- $\zeta $ =
critical damping ratio

- $\omega n$ =
blade natural frequency