It is interesting to note that over the entire range of this constant-speed characteristic, the efficiency values (not shown here) were approximately 50–55%. Therefore, these experiments represent an extremely off-design operating condition. However, power generation compressors are required to operate in this regime as they are powered on and off; therefore, even under these off-design conditions, it is important to understand the aerodynamic forcing and how it relates to blade vibration. In the following sections, the transient results will be discussed first, followed by a detailed analysis of the steady operating points.
3.1. Transient Results
The transient casing pressure characteristics are useful for investigating the physics of the aerodynamic forcing. In particular, the literature has shown that rotating disturbances and other unsteady flow physics relevant to blade vibration are recognizable in the auto-spectral density (ASD) results from a single casing pressure transducer [
2,
3,
9,
11,
27]. Motivated by this,
Figure 3 shows the auto-spectral density (ASD),
, from a single leading-edge casing pressure transducer during the transient experiment (black line from
Figure 2). The ASD was computed using moving windows of 0.002 s with 50% overlap, where each window was divided into 30 blocks for Welch’s method of block averaging [
28]. The amplitude of the ASD was plotted on a logarithmic scale (color contours) versus time on the abscissa and frequency on the ordinate. All frequency values were normalized by shaft rate to be expressed as an engine order (EO), and the frequency resolution was approximately 0.1 EO. In addition, time on the abscissa was shifted so that 0 s corresponded to the opening of the throttle valve at the end of the transient. The entire transient was 110 s, but only 70 s are shown for ease of visualization. The vertical white lines near −60 and −22 s represent a pause in data acquisition necessary to save the data files and restart the acquisition. Along the top of
Figure 3, the steady operating points that had the same corrected mass flow rate as certain times in the transient are labeled, which will be useful for comparison between steady and transient results.
The transient experiment began with the fully open throttle; thus, the initial part of the transient (−110 to −55 s) indicated very low levels of pressure fluctuations. The only exception was the blade-passing frequency (BPF) indicated by the dark red line at 32 EO. From −55 to −25 s, discrete bands of unsteady pressure fluctuations around half of the BPF (8–16 EO) were observed, similar to what Kameier and Neise called “rotating instability” [
2]. The spacing between the different bands was approximately 0.65 EO. This value approximates the circumferential propagation speed as a fraction of rotor speed [
2,
5]. It will be shown in the following section that each of these peaks corresponded to a different circumferential wavenumber. The frequencies of each band increased slightly as the mass flow was reduced, while the spacing between the different bands remained approximately constant. From −25 to −15 s, the mid-frequency content shifted to slightly higher frequencies, and the discrete bands were no longer as pronounced.
At approximately −10 s, low-frequency content appeared in the 0–8 EO range. Again, distinct bands were visible, separated by about 0.55 EO. It will be shown in the following section that each of these bands also corresponded to a different circumferential wavenumber. Similar bands were observed as an upper sideband to the BPF, at the same frequencies relative to the BPF as the low-frequency values. During this time, the mid-frequency amplitude decreased and became more broadband. At 0 s, the rig emergency valve was opened, rapidly decreasing unsteady pressure at all frequencies below the BPF.
The blade vibration amplitudes during the transient experiment are now considered.
Figure 4 shows the vibration amplitude versus time measured by NSMS probes during the transient experiment. For clarity, only the nodal diameters with the highest amplitude vibration are shown: for the 1B mode (
Figure 4a),
, and
, and for the 2B mode (
Figure 4b),
and 10. As with the pressure data, the numbers on the top denote the corresponding steady-state operating points. Additionally, time on the abscissa has been shifted such that 0 s corresponds to the opening of the emergency valve.
First, the instantaneous amplitudes for the 1B mode are considered (
Figure 4a). At the beginning of the transient, from −110 to −55 s, no significant 1B vibration was present. From −55 to −25 s, the amplitude of all 1B nodal diameters increased slightly to around 0.18 mm. From −25 to −15 s, the amplitude of all modes decreased to a negligible value. Then, from −10 to 0 s, the
amplitude increased sharply to a maximum value of around 0.62 mm. The
and
nodal diameters also increased in amplitude from −10 to 0 s but had a lower maximum amplitude of about 0.4 mm and 0.25 mm, respectively.
The instantaneous amplitudes for the 2B mode are shown in
Figure 4b. Overall, the 2B mode had lower amplitudes than the 1B mode, as is visible by noting the different ordinate scales between
Figure 4a,b. For the 2B mode, the nodal diameter with the highest instantaneous vibration amplitude changed as the mass flow rate was reduced throughout the transient. At the beginning of the transient experiment, all 2B vibration amplitudes were low. Then, around −55 s, the 2B,
nodal diameter grew in amplitude to a value around 0.08 mm. From −48 to −42 s, the
nodal diameter grew above the
response and reached a maximum amplitude of approximately 0.15 mm. Then, from −40 to −30 s, the
nodal diameter had the highest instantaneous amplitude. However, this value was lower than the
response of the previous time segment, reaching a maximum amplitude of approximately 0.12 mm for the
nodal diameter. Subsequently, from −30 to −20 s, the
nodal diameter had the highest instantaneous amplitude, reaching a maximum value of 0.21 mm, the highest of all the 2B mode amplitudes throughout the transient. Finally, towards the end of the transient, from −20 to 0 s, all 2B nodal diameters decreased in amplitude as the mass flow rate was further reduced.
The vibration data can now be considered in the context of the pressure data presented in
Figure 3. In both
Figure 3 and
Figure 4, there were two regions of interest: (1) −55 to −15 s and (2) −10 to 0 s. From −55 to −15 s, the unsteady pressure showed distinct bands around half of the BPF. In this same time segment, the 1B vibration amplitudes increased slightly for all nodal diameters, and the 2B vibration amplitudes increased significantly. Additionally, which 2B nodal diameter had the highest amplitude at a given instant changed throughout this time segment. In the region from −10 to 0 s, additional unsteady pressure fluctuations appeared at low frequencies (0–7 EO), which also showed distinct bands indicative of rotating disturbances. In this time segment, there was a sharp increase in the 1B,
vibration amplitude, and some increase in the 1B
and −8 amplitudes. There was a decrease in the 2B vibration amplitudes during this time period. Qualitatively, for both modes, the vibration amplitudes observed in
Figure 4 scaled with the amplitude of the pressure disturbances observed in
Figure 3.
3.2. Steady Results
The relationship between the unsteady casing pressure and blade vibration was further investigated using steady operating conditions. Eleven steady operating conditions were obtained, and approximately 20 s of data were acquired at each operating condition. The results from a single steady operating point are presented first, followed by results from all remaining operating points.
Vibration temporal characteristics are useful for understanding forcing mechanisms [
11,
18]. However, the vibration data shown in
Figure 4 had temporal characteristics that were affected by both the unsteady aerodynamics and the transient boundary conditions of the experiment, i.e., the throttle closure rate. For example, it was unclear whether the growth of the 1B,
vibration amplitude in
Figure 4a was due to a forced response, where the magnitude of the aerodynamic forcing grew as the throttle was closed, or whether the vibration response was an exponential growth characteristic of a linear instability (i.e., flutter). Therefore, the temporal characteristics at steady operating conditions provide greater insight into the underlying physics.
Figure 5 shows a time segment of the instantaneous vibration amplitude for (a) the 1B,
mode at OP11 and (b) the 2B,
mode at OP3. The time series for each nodal diameter was achieved via a least-squares fit of the NSMS data using standard techniques in Agilis’ commercial software [
26]. For both modes, the instantaneous amplitude grew and decayed in an intermittent, stochastic manner. This intermittency was characterized by statistics such as the vibration amplitude’s mean and standard deviation, where the mean was visualized by the horizontal black line in
Figure 5. For example, the 1B,
mode in
Figure 5a had a mean amplitude of 0.34 mm and a standard deviation of 0.165 mm. The 2B,
mode in
Figure 5b had a mean value of 0.078 mm and a standard deviation of 0.0385 mm. For both modes, the coefficient of variation, defined as the standard deviation divided by the mean, was approximately 0.47. The coefficient of variation was approximately 0.47 for all nodal diameters at all steady operating points, indicating that similar, intermittent vibration amplitudes characterized each nodal diameter and operating point. It is known theoretically that an SDOF system forced by white noise has a vibration response amplitude that oscillates stochastically [
19,
23,
24]. Extending this to the steady vibration data in
Figure 5, the observed stochastic amplitude modulation was likely indicative of stochastic aerodynamic forcing.
To study vibration for each nodal diameter and operating point, the stochastic vibration response was characterized by the temporal mean, as was demonstrated by the black line in
Figure 5.
Figure 6 shows the mean vibration amplitude versus the nodal diameter for each of the 11 steady operating points. For clarity, only the range of nodal diameters with the highest amplitude vibration is shown: (a) 1B,
through −5 and (b) 2B,
to 11.
For the 1B mode in
Figure 6a, all nodal diameters from
to
increased in amplitude monotonically as the compressor mass flow rate was reduced from OP1 to OP11. The only exception was that the vibration amplitudes for OP3 were slightly higher than OP4 for nodal diameters −6 to −10. At OP5, the
and
nodal diameters had similar amplitudes around a mean value of 0.1 mm. At OP6, the
nodal diameter had the highest amplitude, and from OP7 to OP11,
grew monotonically as the mass flow rate was reduced, reaching a maximum of 0.35 mm at OP11. Nodal diameters
and
also had moderately high amplitudes and demonstrated a clear monotonic increase with the operating point.
For the 2B mode in
Figure 6b, the shape of the curve (i.e., which nodal diameter had the highest amplitude vibration) differed for the various operating points. At OP3,
had the highest amplitude of 0.08 mm, while
and 10 also had elevated amplitudes. Then, at OP4,
had the highest amplitude vibration, around 0.065 mm, which was lower than the
amplitude from OP3. As the mass flow rate was further reduced, all 2B vibration amplitudes decreased slightly. Still, the shape of the curve remained similar to OP4, with nodal diameters
or 6 having the highest amplitude vibration. By OP11, the amplitude was around 0.04 mm for all nodal diameters, approximately half of the maximum amplitude at OP3.
The results from the steady experiments were consistent with the transient results from
Figure 4. Direct comparison of amplitudes was not feasible because of the intermittency:
Figure 4 shows instantaneous amplitudes while
Figure 6 shows mean amplitudes. However, in both the transient and steady results, the 1B,
mode had the highest amplitude vibration at low mass flow rates. Additionally, the 2B amplitudes were elevated at moderate mass flow rates (in the middle of the part-speed characteristic), but the nodal diameter that had the highest amplitude vibration changed as the mass flow rate was reduced. For example, OP3 corresponded to approximately −43 s in the transient, when the 2B,
mode had the highest amplitude vibration. In addition, OP4 corresponded to −18 s in the transient, when the 2B,
mode was higher than the other nodal diameters. Steady-state OP3 and OP4 were significantly different in terms of mass flow rate. The transient results in
Figure 4 indicated elevated amplitudes for
at mass flow rates between OP3 and OP4. Thus, it could be speculated that, had steady-state measurements been acquired at this intermediate-mass flow rate,
Figure 6b would have shown a continuous trend from
to 8 to 7 as the mass flow rate was reduced.
The casing pressure data at the steady operating conditions are now considered. Since data from 12 pressure transducers were available, it was helpful to analyze both the frequency and circumferential wavenumber characteristics of the casing pressure. The pressure fluctuations were first expressed as a function of the discrete circumferential wavenumber,
k, by computing the Fourier Series coefficients as:
where
was the signal from the
pressure transducer in the circumferential array of
equally spaced transducers. To analyze frequency content, the auto-spectral density function was then computed from each of these (complex-valued) Fourier coefficients,
, defined as [
22,
23,
29]
where
was the (temporal) discrete Fourier transform of the complex conjugate of
. The sign convention was chosen such that a forward traveling wave in the lab reference frame corresponded to positive
k and positive
f. As in the previous section, Welch’s method [
28] of block averaging was employed. Thus, the wavenumber-dependent ASD,
, characterized the casing pressure’s average frequency and circumferential wavenumber attributes.
The wavenumber-dependent ASD of casing pressure can now be considered in tandem with the vibration eigenmodes. This was motivated by studies that have shown that it is important to compare both the frequency and the circumferential wavenumber of the aerodynamics to the natural frequency and nodal diameter of the vibration mode [
3,
8,
14]. Vibration modes can be overlaid on a contour plot of
using the relationship [
14,
19]
where
p indicates the structural nodal diameter,
k indicates the aerodynamic circumferential wavenumber, and
c represents a Fourier aliasing constant due to a continuous pressure field being projected onto
N discrete blades (Equation (
1)). The frequency
indicates the vibration’s natural frequency in the lab reference frame, while
is the natural frequency defined in the rotor’s reference frame, both in units of EO. Under this part-speed operating condition, the 1B mode had a natural frequency of
EO, and the 2B mode had a natural frequency of
EO. These frequencies were predicted with finite-element analysis and were consistent with the values observed experimentally in the NSMS data. Physically, Equation (
7) arises from the change in the reference frame between the vibration modes expressed in the rotor’s reference frame and pressure data measured in the lab reference frame. Additionally, the aliasing can occur because aerodynamic wavenumbers,
k, can range from
to
∞ due to the continuous pressure field, while structural nodal diameters are limited to
N due to the discrete number of blades.
Contours of the casing pressure spectra,
, are shown in
Figure 7 with circumferential wavenumber
k on the ordinate and frequency on the abscissa, where the frequency resolution was approximately 0.01 EO. Since there were 12 pressure transducers, only 12 independent wavenumbers could be computed; however, the data were repeated on the ordinate for visualization of the higher, aliased wavenumbers. This repetition allowed for visual identification of diagonal trends in
, the slope of which could be interpreted as the propagation speed of rotating disturbances [
3]. The red circles in
Figure 7 indicate the 1B vibration mode for various nodal diameters, while the red squares indicate the 2B vibration mode. Nodal diameters corresponding to the highest amplitude response (
Figure 4 and
Figure 6) are indicated by the red arrows. Since speed was constant, the vibration modes had the same frequencies across all steady operating points. However, the pressure data changed relative to these vibration modes as the mass flow rate was reduced from OP1 to OP11, as shown by
Figure 7a–f.
Operating point 2, shown in
Figure 7a, resulted in low-amplitude, broadband pressure fluctuations from around 10–25 EO. Several distinct groups of diagonal lines were visible, with a slope ranging from 0.5 to 0.8. These diagonal regions of local maxima were interpreted to be rotating pressure fluctuations traveling circumferentially at a fraction of the shaft rate.
At OP3, in
Figure 7b, the pressure local maxima formed a single diagonal with a slope of 0.62. This diagonal region spanned frequencies from 8 to 15 EO and wavenumbers from 10 to 32, forming a “stairstep pattern” that the literature has identified as a key characteristic of pre-stall rotating disturbances, or “rotating instabilities” [
1,
2,
5,
10,
13]. These higher wavenumbers,
k = 10–32, were assumed to be physically relevant compared to the fundamental
k = 0–11 based on the fact that rotating disturbances are known to propagate at a speed around half of the rotor speed [
2,
5,
6,
7,
8,
9].
In
Figure 7b, the diagonal of pressure local maxima intersected the 1B vibration mode (red circles) at aerodynamic wavenumbers of
to 10, corresponding to structural nodal diameters −7 to −10, respectively. Additionally, the local maxima at higher frequencies intersected the 2B vibration mode (squares) at aerodynamic wavenumbers
to 25. These wavenumbers corresponded to structural nodal diameters +11 to +7, respectively, given a Fourier aliasing constant of
with 32 rotor blades in Equation (
7). As shown in
Figure 6, OP3 showed elevated vibration amplitudes for the nodal diameters intersected by the rotating pressure disturbances. In particular, the 2B mode had the highest amplitude vibration for nodal diameter +9, which was directly intersected by the diagonal region of rotating pressure fluctuations at wavenumber
in
Figure 7b.
At OP4, shown in
Figure 7c, the local maxima shifted to higher frequencies and wavenumbers, but a coherent diagonal region indicative of rotating disturbances was still present. The highest amplitude pressure fluctuations occurred from 12 to 20 EO and
to 32. This shift to higher frequencies at OP4 compared to OP3 was consistent with the transient data in
Figure 3: the mid-frequency pressure fluctuations shifted to higher frequencies as the mass flow rate of the compressor was reduced. Although
Figure 3 shows less pronounced “banding” from 12 to 20 EO at −18 s, the steady results in
Figure 7c indicate that the stair-step pattern between frequency and wavenumber was still present at this lower mass flow operating point. Considering the vibration modes, the shift of the pressure fluctuations to higher frequencies caused the intersection with the 2B mode to occur primarily at nodal diameter 7 (
). As shown in
Figure 6, OP4 had the highest 2B vibration amplitude at
. Furthermore, both the pressure and vibration data showed lower amplitudes at OP4 than at OP3. Additionally, OP4 in
Figure 7c indicated low-amplitude, low-frequency, and low-wavenumber content with a slope around 0.7. These were interpreted as low-frequency rotating pressure fluctuations distinct from the mid-frequency disturbances.
At OP5, shown in
Figure 7d, the amplitude of the low-frequency (0–5 EO) pressure fluctuations increased, and the amplitude of the mid-frequency (12–20 EO) pressure fluctuations decreased slightly such that the two regions had a similar amplitude. The slope of the local maxima in the 0–5 EO region decreased to a value of approximately 0.55. The decreased speed of the low-frequency fluctuations resulted in an intersection with the 1B vibration mode, predominantly at nodal diameter −7. In
Figure 6a, OP5 showed elevated vibration amplitudes for the 1B mode at nodal diameters −7 and −6. Additionally, for the mid-frequency pressure fluctuations in
Figure 7d, the slope of the maxima from 12 to 20 EO remained approximately the same, but the amplitude of the maxima decreased compared to OP4. Thus, the intersection with the 2B vibration mode occurred at the same nodal diameters, but, as seen in
Figure 6b, the 2B vibration amplitudes were lower at OP5 compared to OP4.
At OP6, shown in
Figure 7e, the low-frequency rotating disturbances grew in amplitude. Still, the range of frequencies, wavenumbers, and the slope of the local maxima remained the same. In fact, from OP 7 to 11 (only OP11 is shown here in
Figure 7f), the
characteristics remained the same, but the amplitude of the low-frequency pressure disturbances grew monotonically with the operating point (i.e., decreasing mass flow rate). Comparatively, the 1B vibration data in
Figure 6a indicated that at OP6, the
nodal diameter began to have the highest amplitude. Then, from OP6 to OP11, the 1B vibration amplitudes grew monotonically, particularly for nodal diameters −6, −7, and −8, with the highest amplitudes occurring for
. In contrast, the mid-frequency rotating pressure disturbances decreased in amplitude monotonically from OP6 to OP11, and
Figure 6b shows that the 2B vibration amplitudes for nodal diameters 6 through 9 also decreased monotonically.
Overall, the qualitative relationship between casing pressure and vibration amplitudes was evident by comparing
Figure 7 to
Figure 6. For both the 1B and 2B vibration modes, elevated vibration amplitudes for certain nodal diameters occurred when the diagonal regions of high-amplitude pressure fluctuations were coincident with the vibration mode in both frequency and wavenumber (i.e., “intersections” in
Figure 7). These coherent diagonal regions of pressure fluctuations were physically indicative of pre-stall rotating disturbances. Hernley showed that pre-stall disturbances’ frequency and wavenumber characteristics were affected by physical parameters such as their propagation speed, size, and frequency of occurrence [
19]. Thus, the speed, size, and frequency of occurrence of pre-stall disturbances affect which, if any, vibration modes are forced on resonance. Furthermore, comparing
Figure 7 to
Figure 6 showed that when the pre-stall disturbances forced a mode on resonance, the amplitude of the vibration scaled with the amplitude of the pressure fluctuations. A more precise comparison of pressure and vibration amplitudes is discussed in the following section.