An Analytical Study on the Effects of Non-Uniform Vane Spacing on Forced Response Reduction in a Mistuned Blisk Rotor in a Multistage Axial Compressor

Rotor-forced response due to rotor–stator interactions is one of the major aeromechanics issues in axial compressors. Without careful design, it can quickly lead to premature blade failure due to high cycle fatigue. Common approaches to reduce rotor forced response are either to decrease the excitation force or to increase the damping. Stator with non-uniform vane spacing (NUVS) is an efficient way to reduce rotor-forced response by breaking down the symmetry of the flow field of a standard symmetric stator and, thus, spreading the strong excitation force at a single engine order (EO) to a series of weaker excitations at neighboring EOs over a wider speed range.

To apply a NUVS stator in a real engine, it is critical to know how much forced response reduction can be achieved for a certain NUVS design compared to the corresponding symmetric stator. The reduction factor, which is defined as one minus the ratio of the maximum forced response due to NUVS stator excitation to the maximum forced response due to the symmetric stator excitation, is commonly used to quantify the effectiveness of a NUVS stator. Kemp et al. [1] conducted some pioneering work to estimate the reduction factors for various NUVS designs and later verified the estimated values via some preliminary experimental testing by measuring the turbine blade stress under the excitation of different upstream asymmetric nozzle vane configurations. Their method, known as the classical estimation method, is based on the circumferential Fourier analysis of the approximated forcing function from the NUVS stator. Kaneko et al. [2] extended the classical estimation method by introducing the damping effect and solving the single-blade forced response under NUVS stator excitation using a time-domain method. Kaneko et al. [3] later considered the mistuning effect in their analytical study by using an equivalent spring-mass model of a bladed disk.

Mistuning is known to play an important role in rotor-forced response and has been studied extensively over the last few decades [4–11]. Mistuning is caused by small variations in blade structural properties, which change the structural coupling of the blades. The earlier modeling work with an equivalent spring-mass mistuning model was shown to be insufficient to capture the mistuned forced response behavior of a rotor with realistic geometry. The two well-known, state-of-the-art reduced order mistuning models are the fundamental mistuning model (FMM) developed by Feiner and Griffin [8] and the component mode mistuning (CMM) model developed by Lim et al. [9]. Both models have been validated and widely used to study the forced response of a rotor with real geometry. While the CMM model is known to be more general and accurate, the FMM model is more efficient and gives an accurate prediction of the blade-forced response for a rotor with relatively small mistuning levels. Most of the mistuned rotor-forced response studies were conducted with standard symmetric stator excitation.

Aerodynamic asymmetry is also known to affect the forced response and stability of a rotor. Aerodynamic asymmetry is caused by small variations in blade geometry, such as blade profile, stagger angle, and spacing. In high-speed turbomachinery, these small geometric variations could cause considerable changes in the aerodynamic coupling of the blades and, thus, the eigenfrequency and eigenmode of the rotor during operation. To deal with the asymmetry, the analytical models of the aerodynamic asymmetry were usually based on a simplified geometry, such as flat plate cascade [12–14]; while different computational cost-saving methods were used in the computational models, such as the symmetric group approach [15] and multi-scale influence coefficient method with the source term [16]. However, most of these studies focused on the stability of a blade row. Unlike mistuning, which always has a stabilization effect, these studies showed that aerodynamic asymmetry can both stabilize and destabilize a blade row depending on the asymmetric pattern and operating conditions. Although NUVS is one type of aerodynamic asymmetry, previous studies focused on the aeroelastic behaviors of the blade row with the aerodynamic asymmetry itself. The studies of the effect of aerodynamic asymmetry on the forced response of adjacent rotors are relatively few.

Several computational studies of the rotor-forced response due to NUVS stator excitation have been done in the past with various simplifications. To avoid the tremendous computational cost due to the asymmetry of the NUVS stator, many previous computational fluid dynamics (CFD) studies [17,18] quantified the forcing function reduction, not the actual reduction in blade vibratory response. Although some high-fidelity CFD–finite element method (FEM) simulations [19,20] have been attempted to study the blade-forced response under NUVS stator excitation, blade mistuning effects were not considered. Despite different clever simplifications, these time domain-based computational methods were still too slow to be utilized as a preliminary design tool.

The experimental studies of the blade-forced response reduction effects of NUVS stators were rare and less detailed [1,2]. Recently, experimental measurements have been acquired in Purdue three-stage axial research compressor (P3S) facility conducted by the same authors in Ref. [21]. Detailed strain gauge (SG) measurements of the rotor forced response were made for both symmetric and NUVS upstream stator excitations. The reduction factor predicted by the classical method [1] was far from the experimental measurement. This paper is the accompanying analytical study to the experimental study. The objective of this analytical study is to gain a deeper understanding of the experimental results and propose a more accurate (and still efficient) NUVS stator reduction factor estimation method by including both damping and mistuning effects.

This paper is organized as follows: Sec. 2 starts with a brief overview of the experimental work by summarizing the key experimental results and findings. The following analytical study in Sec. 3 presents a simple, single-blade analysis by including the damping effects missed in the classical estimation method. A steady-state linearized analysis procedure is proposed to calculate the blade-forced response under NUVS stator excitation. Case studies at different damping levels are conducted to study the effect of damping on multi-EO responses and reduction factors. To study the mistuning effect, Sec. 4 introduces the aeroelastic model for the mistuned blisk forced response under NUVS excitation. Based on the same mistuning pattern and NUVS configuration as the experimental study, several case studies are conducted to correlate with, and better explain, the corresponding experimental results. A statistical analysis with 100 randomly generated mistuning patterns is carried out to study the sensitivity of the forced response reduction effectiveness of a specific NUVS design on small changes in the mistuning pattern. Finally, key results and important findings are summarized in Sec. 5.

The experimental test campaign [21] was conducted in the P3S facility, which is aerodynamically representative of the last several stages of a modern high-pressure compressor. The three rotors are of blisk design. During the test, the embedded rotor 2 (R2) forced response was measured with SGs for two upstream stator 1 (S1) configurations: a symmetric 38-vane S1 and an asymmetric NUVS S1 with 18 vanes on one stator half and 20 vanes on the other stator half. The major relevant experimental results for the first torsion (1T) mode resonant response can be summarized as follows:

1. The primary 38EO-1T response for the symmetric S1 configuration was reduced and spread to a series of weaker adjacent EO responses over a much larger speed range for the NUVS S1 configuration, Fig. 1.

2. The blade-forced response curves were more complicated for the NUVS S1 configuration due to the overlap of the resonant speed range of adjacent EOs, as shown in Fig. 2. The simultaneous response of the adjacent EOs resulted in a beating pattern in the blade SG time history.

3. There was significant blade-to-blade variation in the blade-forced response and the corresponding reduction factor, even at the low non-intentional mistuning levels of R2.

4. Compared to the prediction of the classical estimation method, the experimental results showed large differences in both the value of reduction factors and the EO of the peak response.

The classical estimation method assumes that the blade-forced response is linearly proportional to the corresponding excitation strength at the corresponding EO. Without solving the equation of motion (EOM) for blade deflection, the reduction factor is directly estimated based on the relative strength of the maximum EO component in the forcing function of the NUVS stator and the symmetric stator. Thus, the lack of consideration of damping and mistuning in the classical estimation method is believed to cause large discrepancies between the prediction and experimental results. The following analytical studies add in the damping and mistuning effects one at a time, and then compare with experimental results to gain a deeper physical understanding of the forced response reduction effect of the NUVS stator.

To include damping effects while still retaining the simplicity of the classical estimation method, a single-blade analytical study is first performed. Mistuning effects will be considered in the next section.

Equations (1) and (4) can be solved in the time domain using a numerical method such as “ODE45” in matlab, or other time-domain analytical methods, such as the one used in Ref. [2]. However, following a similar approach to the forced response analysis system developed for standard, symmetric stator excitation [22], a steady-state linearized frequency domain analysis approach is used in this study. Unlike the time-domain methods, this frequency-domain approach is very efficient and can be readily incorporated into the mistuned whole-blisk aeroelastic model discussed later.

The first summation in Eq. (6) is in the time domain, which is the real summation, including the construction or destruction of the sinusoidal responses at all EOs. This is referred to as time summation in this paper. A much quicker, but less accurate way to approximate the amplitude of $x(t)$ is the summation of the amplitude of all EO response components, which is referred to as frequency summation. The frequency summation provides an upper bound for the maximum blade response envelope due to NUVS stator excitation, which in turn, results in a lower bound of the estimated reduction factor, R, for a conservative design.

Although this section is referred to as single-blade analysis, the EOM of a simple harmonic oscillator in Eq. (1) can also be used to model a tuned blisk vibrating at a specific system mode. For a tuned blisk, a system mode is the entire blisk vibrating at a specific nodal diameter (ND). Better than the time-domain approach, the damping ratio can be individually specified for each EO (based on the corresponding ND) with this linearized frequency-domain approach.

For the steady-state linearized analysis, the blade response is assumed to be fully developed for each revolution, and thus, every revolution is independent from each other. As discussed in Ref. [23], this is a good assumption when the sweep rate is slower than the critical sweep rate. All blade-forced response data in the experimental study [21] were collected at a sweep rate much lower than the critical sweep rate. This assumption enables us to solve the EOM for each revolution independently, and thus, the speed-dependent forcing function and aerodynamic damping can be easily incorporated into the analysis by specifying them at each revolution and speed.

The steady-state linearized analysis procedure to calculate the blade-forced response under NUVS stator excitation can be summarized in the following flowchart, Fig. 3. The mistuned blisk forced response can also be readily incorporated into the same analysis framework.

The major inputs and major steps with the number corresponding to the number in the flowchart are listed as follows:

1. Forcing function: Ideally, for NUVS stator excitation, the downstream rotor blade modal force calculation at a resonant crossing requires a full-wheel, unsteady CFD, and FEM modal analysis. However, as suggested in Ref. [17], the unsteady modal force that the blade experiences when passing by each stator half can be approximated by the corresponding symmetric stator CFD calculation. As a lower level of approximation, the blade modal force at primary VPF can be assumed to be linearly proportional to the corresponding circumferential component of the NUVS stator steady flow field, which can be calculated with steady CFD. At an even lower level, blade modal forcing at the primary VPF can be approximated by two sinusoidal waves with the frequency related to the corresponding vane number of each stator half, as shown in Eq. (5). For the 18–20 vanes NUVS S1 in the experimental study, the forcing function at primary VPF is shown in Fig. 4 in both the time and frequency domains.

2. Damping: The total damping of a rotor includes both structural damping and aerodynamic damping. While aerodynamic damping can be calculated using unsteady CFD and FEM, structural damping is usually measured or guessed with an empirical value.

3. Based on a steady-state linearized analysis, the blade response can be calculated for each EO component of the NUVS forcing function independently.

4. The final total blade response is the summation of all EO responses in the time domain. A summation of the amplitude of all EO responses in the frequency domain gives an approximated upper bound to the total blade response.

5. Mistuned blisk forced response calculation requires a mistuning model. As discussed in a later section, the FMM is used in this study. The modal frequency of each blade and tuned system modal frequencies at all possible NDs are the additional inputs for the FMM model.

To consider the blade-to-blade variation due to mistuning, three types of reduction factors can be defined:

1. Blade reduction factor, $Rblade$⁠, is the reduction factor for a specific blade

   $Rblade=1−xNUVSxsym$

   (7)
2. 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

   $Rrotor_avg=1−Avg(xNUVS)Avg(xsym)$

   (8)
3. 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

   $Rrotor_max=1−Max(xNUVS)Max(xsym)$

   (9)

The classical estimation method assumes the blade-forced response is linearly proportional to the corresponding excitation force strength at the corresponding EO, with no mistuning effect in consideration. Thus, this is only one reduction factor based on the relative strength of the forcing functions. According to the sinusoidal wave approximation of the NUVS forcing function in Fig. 4, the predicted peak blade response EO is 36EO or 40EO, and the corresponding reduction factor R = 0.5.

To correlate with the experimental study, the single-blade case study is conducted for P3S R2 1T response under both the symmetric 38-vane S1 excitation and the NUVS 18–20 vane S1 excitation. The blade 1T modal frequency is chosen to be 2721.8 Hz, which is the average blade frequency of the R2 blisk. A constant total critical damping ratio of 0.1% is chosen based on experimental results. As shown in Fig. 1, the 38EO-1T resonant crossing for the symmetric S1 occurs around 4298 rpm, while the resonant crossings for the NUVS S1 spread to a much wider speed range. To include all dominant NUVS stator exaction EOs (35, 36, and 37EO) and (39, 40, and 41EO) as shown in Fig. 4, a simulation of blade forced response with a speed sweep from 3800 rpm to 4800 rpm is carried out.

Following the flowchart in Fig. 3, the blade forced response for each EO excitation of unit strength is calculated first, then multiplied by the strength of each EO component of the NUVS stator forcing function, and summed together to get the maximum envelope of the total blade response. Figure 5 shows each EO's response and the maximum response envelope calculated with both the time-summation and frequency-summation methods. The blade response amplitude is normalized by the maximum response amplitude under the symmetric stator exaction. For the small critical damping ratio of 0.1%, the peak of the maximum response envelope is close to the peak of each corresponding EO response. The small difference is due to the contribution from adjacent EO responses, even though the overlap with adjacent EOs is small. As discussed earlier, the summation in the frequency domain shows a higher maximum blade response than the summation in time domain, but the difference is very small when the damping level is low.

To study the effect of damping, the simulated blade-forced response at a higher damping, with the critical damping ratio of 0.5%, is shown in Fig. 6. Compared to Fig. 5, higher damping broadens the normalized resonant response for each EO and thus, causes more overlapping with adjacent EOs. The overlapping increases the normalized total blade response considerably. The classical estimation method is based on a single max EO response, neglecting the overlap. As shown in Fig. 6, with adjacent EO overlapping in consideration, the peak NUVS response (the sum in time domain) reaches 0.66 of the symmetric stator response at 40EO. Thus, the reduction factor is 0.34, which is considerably lower than the classical method prediction of 0.5. At higher damping, the difference between time-domain summation and frequency-domain summation becomes larger. However, since frequency domain summation is always higher, it still provides a quicker and conservative estimation of reduction factor (i.e., the actual reduction is larger).

While damping affects the width of resonance response of each EO, vane number also plays an important role in the overlapping of adjacent EOs by affecting proximity of the peaks of two adjacent EOs. With higher vane numbers, the corresponding resonant EO becomes higher, the resonant speed difference between adjacent EOs $(Δrpm)$ becomes smaller, and thus, two adjacent EO responses are more likely to overlap. Note that $Δrpm=60ωnEO−60ωnEO+1=60ωnEO(EO+1)$⁠, and thus, when EO increases, $Δrpm$ decreases.

Similar to blade SG data in the experimental study [21], the overlap between adjacent EO responses shows up as a beating pattern in the blade response time history. The simulated total blade response at the 36EO-1T resonant speed (∼4536.6 rpm) is shown in both the time and frequency domains in Fig. 7. Even though the blade response is plotted at the 36EO-1T resonant speed, a considerable contribution from the adjacent 35EO and 37EO responses causes the beating pattern in the time domain.

The steady-state linearized approach provides a quick analytical tool for parametric study. Figure 8 shows the reduction factors at various critical damping ratios under the NUVS stator excitation. When damping goes to zero, each EO's resonant peak goes to infinity, the contribution from adjacent EO response is negligible, and thus, the reduction factor approaches 0.5, which is the same as the single EO based estimation using the classical method. However, when damping gets larger, each normalized peak becomes broader, and the overlap between adjacent EOs becomes larger. The contribution from adjacent EOs increases the total blade response and thus, reduces the reduction factor. In addition, at a lower damping ratio, the frequency-summation method approaches the time-summation method and, thus, provides a quick estimation of the reduction factor.

Besides damping, the experimental study shows there was a large blade-to-blade variation in the forced response curve and the corresponding reduction factor due to mistuning. In this section, the aeroelastic model for a mistuned blisk forced response under NUVS stator excitation is presented first. Case studies with both constant damping and aerodynamic damping (aerodynamic damping varies with nodal diameter) are conducted next to compare with the experimental study. Finally, a statistical analysis is done to show the sensitivity of the maximum-to-maximum reduction factor of a specific NUVS stator to a small change of the mistuning pattern.

The aerodynamic coupling effect among blades is considered in the aerodynamic influence coefficient matrix $[Am]$⁠, which is a diagonal matrix with each entry representing the aerodynamic force on the blisk causing by the blisk vibration at rth tuned system mode. The real part of $[Am]$ represents the aerodynamic stiffness, while the imaginary part represents the aerodynamic damping.

To put $bn$⁠, the strength of component $EOn$⁠, into forcing vector ${W^}$ in traveling wave coordinates, the corresponding system mode ND excited by the $EOn$ component needs to be known. For a blisk with NB blades, the relation is defined as $ND=EO±mNB$⁠, where m can be any integer such that ND is from 0 to NB−1.

To correlate with the experimental study, the case studies in this section are based on the same mistuning pattern as in the P3S R2 with a focus on the forced response reduction of the R2 1T mode excited by the NUVS S1. As shown in Fig. 3, the inputs for the mistuned blisk forced response model are: forcing function, blade and tuned system modal frequencies, structural damping, and aerodynamic damping.

The forcing function approximation and EO decomposition are done using Eq. (5), the same as in the single-blade analysis and the classical estimation method. The blade modal frequencies were measured experimentally using a standard impact test method. The tuned system modal frequencies are calculated using a standard modal analysis in ansys mechanical with cyclic symmetry. Following the suggestion of Ref. [26], the tuned system modal frequencies used in this analysis are shifted such that the tuned system modal frequency at ND = ± 16 is equal to the averaged blade frequency. This is done to better match the single family of modes assumption in the FMM mistuning model and to reduce the discrepancy due to errors in the FEM modal analysis and/or experimental impact test. The blade frequencies and shifted tuned system modal frequencies are shown in Fig. 9. The eight strain-gauged blades in the experimental study were B03, B06, B09, B15, B23, B25, B30, and B33 and are marked with stars in Fig. 9.

Structural damping can be measured or empirically chosen. The aerodynamic damping is the imaginary part of the aerodynamic influence coefficient matrix $[Am]$ and is calculated based on the well-known 2D flat plate unsteady aerodynamic cascade model, LINSUB [27]. Based on the R2 flow field, geometry, and modal shape, the calculated aerodynamic damping at the 38EO-1T resonant crossing (∼4300 rpm) is shown in Fig. 10. As expected, there is a large variation in the damping value depending on the ND of the blisk vibration.

Accurate forcing function and aerodynamic damping calculations at different operating conditions require high-fidelity, unsteady CFD and FEM modal analysis. Nevertheless, the simplified sinusoidal wave types of forcing and low-fidelity LINSUB-calculated aerodynamic damping are used in this case study, since the goal of this paper is to provide insight into the ND varied damping effects on a mistuned blisk response due to NUVS excitation, not for an accurate prediction of the experimental results. However, for the steady-state linearized approach proposed in this paper, the rotor speed and loading dependent forcing function, and aerodynamic damping can be easily incorporated in Eq. (10), as discussed earlier.

The ND varied aerodynamic damping is important for the mistuned blisk forced response since a single EO forcing function excites not only the primary ND response $(ND=EO±mNB)$ but also all other NDs due to the structural coupling effect introduced by the fully populated mistuning matrix $[A^]$⁠. To analyze this effect, two different dampings are chosen for the case studies. The first is using the ND varied aerodynamic damping calculated by LINSUB without structural damping (which is usually very small for a blisk rotor). The second is using a constant total damping of 0.1%. The critical damping ratio of 0.1% is chosen because it is the level of damping measured for most of the blades in the 1T forced response measured in the experimental study. It is also close to the average of the LINSUB-calculated aerodynamic damping for the primary NDs excited by NUVS (i.e., ND of −2 to −8 for excitations from −35EO to −41EO on the 33-bladed R2). This allows for a fair comparison between the constant damping case with the aerodynamic damping case.

Before the detailed mistuned blisk NUVS response analysis, a case study of a tuned blisk response with constant damping of 0.1% is conducted as a special case to validate the whole-blisk aeroelastic model in Eq. (10) with the single-blade analytical response calculated in Eq. (1). All tuned system modes and blade modal frequencies are specified as the average of the R2 blade frequencies. The simulated maximum blade response envelope under NUVS stator excitation is shown in Fig. 11 for both the whole-blisk analysis (solid) and the single-blade analysis (dots). The perfect agreement provides a special case validation for the whole-blisk aeroelastic model derived in Eq. (10).

The same as in the experimental study [21], Blade 33 is chosen as the representative blade for the mistuned blisk response case studies. Before analyzing the total blade response (i.e., summation of all EO responses), the effect of mistuning on each EO response is studied first. For the constant damping case, the unit strength EO response for the tuned blisk (dash line) and mistuned blisk (solid line) are compared in Fig. 12. The amplitude is normalized by the maximum response for the symmetric 38-vane stator case. As expected, with the same unit excitation strength and the same damping, each EO's resonant response curve is the same, with only a shift in the peak resonant speed for the tuned blisk case. On the other hand, for the mistuned blisk case, the blade responses are different for different EOs. With the same unit excitation strength and the same damping for each EO response, the differences come entirely from the mistuning effect. The different blade resonant response curves for different EO (multiple peaks and different amplitudes) are due to different couplings of multiple close-spaced system modes caused by the fully populated mistuning matrix $[A^]$⁠. Mistuning can cause the vibrational energy to concentrate on a few blades of a blisk (known as mode localization), which could be the reason for the significantly higher response at 35EO shown in Fig. 12. Also, compared to the tuned blisk response, mistuning also broadens the resonant speed range of each EO considerably. This makes the issue of overlapping between adjacent EO responses more serious for a mistuned blisk.

After multiplying the forcing function EO strengths and summing all EO responses, the total blade response and each EO's contribution are shown in Fig. 13 for the constant damping case. The results of the corresponding case study with aerodynamic damping are given in Fig. 14. Compared with the constant damping case in Fig. 13, the aerodynamic damping causes additional complication in the total blade response curve. Overall, the resonant peaks become sharper due to the mistuning coupling of multiple system modes, of which some have lower damping as shown in the aerodynamic damping curve in Fig. 10. Compared to the constant damping case in Fig. 13, although the highest response EO is still 35EO overall, the 39EO response becomes the highest in the 39-40-41EO cluster. Between the constant damping case and the aerodynamic damping case, there are more differences in the 39-40-41EO cluster than in the 35-36-37EO cluster because aerodynamic damping at ND = − 2, −3, −4 (excited by the 35-36-37EO cluster) are close to the constant damping ratio 0.1%, while the aerodynamic damping at ND = − 6, −7, −8 (excited by the 39-40-41EO cluster) has larger variation and larger deviation from the constant damping ratio of 0.1%.

Compared to the experimental Blade 33 response curve shown in Fig. 2, the highest responding EO is predicted correctly at 35EO for both the constant damping and aerodynamic damping case studies. Even though the highest excitation EO components of the approximated NUVS stator forcing function are 18 × 2 = 36EOs and 20 × 2 = 40EOs as shown in Fig. 4, because of the strong unit strength EO response at 35EO (as shown in Fig. 12), the highest total blade response still occurs at 35EO. While the prediction in the 35-36-37EO cluster is good in general, the relative strength of the 39-40-41EO cluster predicted in the analytical case studies (Figs. 13 and 14) is higher than the experimental measurements (Fig. 2). In the analytical studies, a constant forcing function over the whole speed range of the sweep is applied. However, the forcing function generally increases with speed. Comparing the analytical blade response curve in Fig. 14 with the experimental blade response curve in Fig. 2, a better agreement could be expected if an increasing forcing function is multiplied by the analytical blade response curve. Nevertheless, the analytical response curves in both case studies start to resemble the experimentally measured blade response curve in Fig. 2, despite the steady-state linearized analysis assumptions, simplified sinusoidal wave approximation of the NUVS stator forcing function, the low order model of mistuning, and the low-fidelity aerodynamic damping input.

Finally, similar to the experimental study, there is also a large blade-to-blade variation in the blade-forced response due to NUVS stator excitations in the analytical case studies because of mistuning. For example, in Fig. 15, the resonant response of the representative Blade 33 is compared with the resonant response of the highest responding Blade 18 and lowest responding Blade11 for the analytical case study with aerodynamic damping. There are significant differences in peak response EOs, peak amplitudes, and the overall shape of the response curves.

Due to mistuning, three different reduction factors have been defined in Eqs. (7)–(9). In this section, these reduction factors are evaluated for both analytical case studies and compared with the experimental results for the strain-gauged blades.

First, the analytical reduction factor of each blade is shown in Fig. 16 for the constant damping case, and in Fig. 17 for the aerodynamic damping case, along with the max blade response amplitude under both the symmetric stator excitation and the NUVS stator excitation.

Comparing Fig. 16 with Fig. 17, the ND varied aerodynamic damping plays an important role in blade forced response not only for the NUVS stator but also for the symmetric stator. This, in turn, leads to a large difference in blade reduction factor between the constant damping case and the aerodynamic damping case. As discussed before, this difference is caused by the system modes coupling effect of the mistuning matrix because there is a large variation of the aerodynamic damping at different ND, as shown in Fig. 10.

Similar to the experimental results, there are large blade-to-blade variations in maximum response and reduction factor due to mistuning in both the constant damping case (Fig. 16) and the aerodynamic damping case (Fig. 17). Some blades, especially the relatively low responders under the symmetric stator excitation, experience an increased amplitude after switching to the NUVS stator, like Blade 20 in Fig. 16 and Blades 5, 8, 18, 20, and 26 in Fig. 17. This results in the negative reduction factor for these blades. On the other hand, the amplitudes of most of the high responders under the symmetric stator excitation are reduced a lot leading to large reduction factor (larger than the classical prediction of 0.5), for example, Blade 2 and Blade 30 in both case studies. Due to this large reduction, the highest responding blade under the symmetric stator excitation is no longer the highest responding blade under NUVS excitation. This change in the highest responding blade was also observed and discussed in the experimental study [21]. Thus, it is the max-to-max reduction factor that really matters when analyzing the effectiveness of a NUVS stator for a mistuned blisk.

Table 1 summarizes different types of reduction factors for all blades in both case studies, including the results calculated using the frequency-summation method as a quick estimation tool discussed in the single blade analysis section. For both cases, at a relatively low total damping ratio around 0.1% (typical for blisk rotors), the quick frequency-summation method gives very similar results to the standard time-summation method.

Since only eight blades of Rotor 2 were strained gauged in experimental study, the predicted reduction factors for both analytical studies are compared with the experimental results in Table 2 for the strain-gauged blades only. The analytical results show a very good overall agreement with the experimental results for the blade reduction factors range, max-to-max reduction factor, and average-to-average reduction factor, especially for the case study with aerodynamic damping.

From Table 2, both the maximum blade reduction factor and the max-to-max reduction factor are higher than the classical prediction of 0.5. Based on the single-blade analysis, the reduction factor should always be lower than 0.5 due to the increased blade response caused by the overlap in the adjacent EO responses. Also, mistuning broadens each EO's resonant speed range, making the overlap issue even worse. Thus, it seems that the reduction factor for the mistuned blisk should be even lower than 0.5 due to the enhanced overlapping issue. However, both the analytical results and experimental results in Table 2 show the opposite. This is probably due to the mode localization effect of the mistuning, where most of the vibrational energy concentrates on a few blades with much higher vibrational amplitude than the rest of the blades. The mode localization pattern can be very different for the NUVS stator excitation and symmetric stator excitation since the excitation ND, peak resonant speed, and the corresponding aerodynamic damping are different. The highest responding blades under symmetric stator excitation are not necessarily the highest responding blades under NUVS stator excitation. This could result in a max-to-max reduction factor higher than 0.5. At the same time, it is possible that the mode localization effect due to mistuning is higher for NUVS stator excitation than the symmetric stator excitation, and thus, the max-to-max reduction factor can be lower than expected.

The analysis above is based on one specific mistuned blisk, the P3S R2 with non-intentional mistuning. The mean of the blade 1 T frequencies is 2721.8 Hz, with a standard deviation of 9.4 Hz, which is 0.34% of the mean value. Since small non-intentional mistuning always exists in a rotor blisk, a statistical analysis is desired to evaluate how a small change in mistuning pattern impacts the effectiveness of forced response reduction for a certain NUVS design. Thus, 100 randomly generated mistuned blisks, with the same mean and standard deviation of the blade 1T frequencies as those of P3S R2, are simulated. Thanks to the efficient steady-state linearized analysis approach used in this study, the blade forced response of a specific mistuned blisk excited by both symmetric and NUVS stators can be calculated within 1 min on a standard PC for each mistuned blisk.

Since max-to-max reduction factor is the most important parameter, this is shown in Fig. 18. There is a large variation of the max-to-max reduction factor for the randomly generated mistuned blisks, even though the standard deviation of the mistuned blade frequencies is only 0.34% of the mean value. Compared to the aerodynamic damping cases, the cases with constant damping have a higher average and lower standard deviation of the max-to-max reduction factor. The distribution of the reduction factor roughly follows a normal distribution for both cases.

Table 3 summarizes the maximum, minimum, standard deviation, and mean of the max-to-max reduction factor for the 100 randomly generated mistuned blisks. For the aerodynamic damping case, the small non-intentional mistuning can cause the max-to-max reduction factor to be as low as 0.23 and as high as 0.57, for the same NUVS design. Such a large variation shows that the classical estimation method, or even the single-blade analysis method, is not sufficient to give an accurate reduction factor estimation. Mistuning effects must be included. Again, the quick summation of the amplitudes of all EO response components compares well with the standard summation in the time-domain method. For the statistical analysis, the aerodynamic damping case shows similar results as the constant damping case, with only a slightly lower average and larger range in the max-to-max reduction factor. However, for a specific mistuned blisk, the max-to-max reduction factor can be very different for the two damping cases as shown in Fig. 18.

Note that a non-intentional blisk mistuning pattern is usually random and changing over the life cycle due to usage. This statistical analysis shows that a small change in mistuning pattern can significantly change the max-to-max reduction factor. Thus, a statistical analysis including mistuning effects is needed to evaluate and optimize a NUVS design for real engine operating environments. The steady-state linearized analysis approach coupled with the FMM mistuning model provides a quick and accurate method to conduct such a statistical analysis.

As shown in Fig. 3, accurate prediction of blade forced response under NUVS stator excitation requires accurate inputs of forcing function and damping, both of which depend on speed and operating condition. Another potentially important effect to consider is multistage interactions. The stator and rotor blade rows are coupled together by the pressure and vortical waves between them, which are known as spinning modes [28]. Besides the primary excitations from the wake and potential fields of the adjacent stators, the reflected pressure waves (also known as Tyler–Sofrin modes [29]) due to multistage interactions provide additional unsteady loading on the rotor and alter the total forcing function and aerodynamic damping of the rotor. These reflected pressure waves have been studied and characterized using casing unsteady pressure measurement previously in multistage axial compressors [30,31]. The frequency and ND of the pressure waves depend on the vane/blade count of the neighboring stator/rotor rows. Thus, the pressure waves reflected from the NUVS stator are different from the pressure waves reflected from the symmetric stator. However, this secondary effect can still be included in the steady-state linearized analysis procedures proposed in this paper as long as the multistage interaction has been carefully considered during the forcing function and aerodynamic damping calculation, since each EO response at each revolution is treated independently in this steady-state linearized approach.

The transient blade response is not considered in this study since the sweep rate in the experimental study [21] was far less than the critical speed. However, when the sweep rate is high and total damping is low, the blade forced response under each stator half and at each revolution may not be fully developed, and thus, the steady-state assumption fails. Time-domain methods are more suitable to consider the transient effect, although significantly more computational resources are required.

In this paper, a detailed analytical study of the blade-forced response reduction due to NUVS excitation is carried out to accompany the experimental study performed in the Purdue three-stage axial research compressor [21]. A steady-state linearized analysis approach is proposed to calculate mistuned blisk response under NUVS excitation. Based on the same R2 mistuning pattern and NUVS stator configuration present in the experimental study, a series of case studies are performed to gain better understanding of the experimental results. The key results are summarized as follows:

• Single blade case studies show that damping plays an important role in increasing the normalized total blade response by increasing the overlap between the resonant response of adjacent EOs. The multi-EO response also shows up as a beating pattern in the blade response SG time-history data of the experimental study

• Mistuning can broaden the resonant speed range of each EO response and make the overlap between the resonant response of adjacent EOs even worse.

• Compared to the case with constant damping, aerodynamic damping that changes with ND causes considerable changes in the forced response of the mistuned blisk under both symmetric and NUVS stators excitation.

• The mistuned blisk case studies show many features observed in the SG data of the experimental study [21], such as: (1) large blade-to-blade variation in blade forced response and the corresponding reduction factor; (2) the highest response EO is different from the highest excitation EO for many blades; (3) the highest responding blade changes to another blade after switching from the symmetric stator to the NUVS stator configuration.

• The statistical analysis shows that a small change in mistuning pattern can significantly change the forced response reduction effect for a specific NUVS stator, even at low, non-intentional mistuning levels.

These analytical case studies show both mistuning and damping (especially the ND varied aerodynamic damping) are critical to recreating the important features observed in the experimental results of rotor forced response under NUVS excitation. The efficient steady-state linearized analysis approach proposed in this paper runs quickly, making it good for parametric and statistical studies. It has great potential to be used as a preliminary analysis tool for NUVS stator designs in terms of determining the effectiveness in blade forced response reduction.

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

There are no conflicts of interest.

Data provided by a third party are listed in Acknowledgment.

$x$ =

blade vibratory deflection

R =

reduction factor

$bn$ =

strength of nth EO component

$[A^]$ =

structural mistuning matrix

$[Am]$ =

aerodynamic influence coefficient matrix

1T =

first torsion mode

EO =

engine order

EOM =

equation of motion

FMM =

fundamental mistuning model

NB =

number of blades

ND =

nodal diameter

NUVS =

non-uniform vane spacing

VPF =

vane pass frequency

$ζ$ =

critical damping ratio

$η$ =

frequency ratio

$[Λ0]$ =

structural coupling matrix

$ϕ$ =

initial phase

$ωn$ =

blade natural frequency

$Ω$ =

rotor speed