Non-contact test set-up for aeroelasticity in a rotating turbomachine combining a novel acoustic excitation system with tip-timing

Sound can be described as a pressure perturbation whose amplitude is in most cases small compared to ambient pressure. Based on this assumption, sound propagation in a continuum can be described by the acoustic wave equation:

$$\begin{equation} \frac{1}{a^{2}}\frac{\partial ^{2}{p^{\prime }}}{\partial {t}^{2}}-\frac{\partial ^2 p^{\prime }}{\partial x_i^2}=Q. \end{equation} \tag{ 1 }$$

The left-hand side characterizes the change in time and space of a small pressure perturbation p' and defines the wave dynamics. It depends on the speed of sound a, the second derivative of p' with respect to time t and on the divergence of the sound pressure gradient ∂p'/∂x_i. Q on the right-hand side represents all acoustic sources in the considered volume. In turbomachines, the sound field is bounded by the hub, defined by the radius r_hub, and the shroud, defined by the radius r_shroud, of the flow channel. Thus, the boundary conditions differ from sound propagation in the free field. Usually, an annular channel with acoustically hard walls is used to approximate the flow path of turbomachines. In order to describe the complex sound field in an annular channel, a modal approach can be used. In this case, the sound field theoretically consists of different modes, which can be characterized by the order in azimuthal (m) and in radial (n) directions. m defines the number of sound pressure cycles in the circumferential direction. The radial mode order n represents the number of sound pressure nodes (zero pressure zones) in the radial sound pressure distribution. Finally, the overall sound field in an annular channel can be described by the superposition of all propagating modes. Some examples of instantaneous pressure distributions of low-order modes in an axial plane of an annular channel are illustrated in figure 1.

Zoom In Zoom Out Reset image size

To describe the sound propagation of a defined acoustic mode in an annular channel with a uniform air flow velocity W, the homogeneous convective wave equation

$$\begin{equation} \frac{1}{a^2}\frac{\textrm {D}^2 p^{\prime }}{\textrm {D}t^2}-\frac{\partial ^2 p^{\prime }}{\partial z^2}-\frac{1}{r}\frac{\partial }{\partial r}\left(r\frac{\partial p^{\prime }}{\partial r}\right)-\frac{1}{r^2}\frac{\partial ^2 p^{\prime }}{\partial \varphi ^2}=0 \end{equation} \tag{ 2 }$$

with

$$\begin{equation} \frac{\textrm {D}}{\textrm {D}t}=\frac{\partial }{\partial t}+W\frac{\partial }{\partial z} \end{equation} \tag{ 3 }$$

can be solved by using a harmonic separation approach. The resulting equation

$$\begin{eqnarray} &&p^{\prime }_{mn}(r,\varphi ,z,t)\nonumber\\ &&=\big(A_{mn}^+\,\mathrm{e}^{\mathrm{i}k_{mn}^+ z}+A_{mn}^-\,\mathrm{e}^{\mathrm{i}k_{mn}^- z}\big)f_{mn}\left(\xi _{mn}\frac{r}{r_{\rm shroud}}\right)\,\mathrm{e}^{\mathrm{i}m\varphi }\,\mathrm{e}^{\mathrm{i}\omega t}\textrm {,} \end{eqnarray} \tag{ 4 }$$

given by Munjal (1987), defines the dependence of the sound pressure p' on the geometric position in the annular channel and its dependence on time. In this case, ω denotes the angular frequency of the oscillating acoustic pressure. The acoustic pressure distribution in the radial direction is described by a linear combination of the first and second kinds of Bessel function:

$$\begin{eqnarray} &&f\left(\xi _{mn}\frac{r}{r_{\rm shroud}}\right)=J_m\left(\xi _{mn}\frac{r}{r_{\rm shroud}}\right)+Q_m\cdot Y_m\left(\xi _{mn}\frac{r}{r_{\rm shroud}}\right). \nonumber\\ \end{eqnarray} \tag{ 5 }$$

Further variables of equation (4), like the acoustic mode amplitudes $A_{mn}^+$ and $A_{mn}^-$ as well as the axial wave number $k_{mn}^{\pm }$ defined by

$$\begin{eqnarray} &&k_{mn}^{\pm }\,{=}\,\frac{k}{1-Ma^2_z}\cdot \left[{-}Ma_z\pm \sqrt{1\! -\! (1\! -\! Ma_z^2)\cdot \left(\frac{\xi _{mn}}{k\cdot r_{\rm shroud}}\right)^2}\right] \nonumber\\ \end{eqnarray} \tag{ 6 }$$

are mode dependent. Thus, they must be determined for each mode and each frequency participating in the sound field. The axial wave number $k_{mn}^{\pm }$ depends on the wave number k = ω/a, the axial Mach number Ma_z and the eigenvalues ξ_mn.

An important criterion for sound propagation in annular channels is the cut-off frequency f_cutoff. This parameter indicates whether acoustic modes are damped or propagable. It can be deduced by the interpretation of the exponents of the exponential functions in the axial propagation part of equation (4). The cut-off frequency is defined by

$$\begin{equation} f_{\rm cutoff}=\frac{\xi _{mn}\cdot a}{2\pi \cdot r_{\rm shroud}}\sqrt{1-Ma_z^2} \end{equation} \tag{ 7 }$$

where ξ_mn are mode-dependent eigenvalues for the radial pressure distribution in the case of acoustically hard walls:

$$\begin{eqnarray} \frac{\partial }{\partial r} \left[J_m\left(\frac{\xi _{mn}}{r_{\rm shroud}}r\right)+Q_m\cdot Y_m\left(\frac{\xi _{mn}}{r_{\rm shroud}}r\right)\right]_{r=r_{\rm hub}}&=0 \end{eqnarray} \tag{ 8 }$$

$$\begin{eqnarray} \frac{\partial }{\partial r} \left[J_m\left(\frac{\xi _{mn}}{r_{\rm shroud}}r\right)+Q_m\cdot Y_m\left(\frac{\xi _{mn}}{r_{\rm shroud}}r\right)\right]_{r=r_{\rm shroud}}&=0 . \end{eqnarray} \tag{ 9 }$$

In conclusion, the cut-off frequency depends on the order of the considered mode given by m and n, the speed of sound a, the outer radius of the considered annular channel r_shroud, the axial Mach number Ma_z and the hub-to-shroud ratio r_hub/r_shroud. All modes propagating with a frequency below the cut-off frequency are damped in the far field.

Finally, the overall sound field within an annular channel can be calculated for a defined frequency by the superposition of all participating modes:

$$\begin{eqnarray} &&p^{\prime }(r,\varphi ,z,t) \nonumber\\ =\sum _{m={-}\infty }^{\infty } \sum _{n=0}^{\infty }&\left(A_{mn}^+\,\mathrm{e}^{\mathrm{i}k_{mn}^+ z} + A_{mn}^-\,\mathrm{e}^{\mathrm{i}k_{mn}^- z}\right)f_{mn}\left(\xi _{mn}\frac{r}{r_{\rm shroud}}\right)\cdot \nonumber\\ &\mathrm{e}^{\mathrm{i}m\varphi }\,\mathrm{e}^{\mathrm{i}\omega t} \end{eqnarray} \tag{ 10 }$$

A more detailed description of the governing equations for sound propagation in annular channels is given, for example, by Ghiladi (1981). In order to take into account the swirl of the turbomachinery flow, equations (6) and (7) must be changed slightly according to Lohmann (1977).

On one hand, the theory of acoustics discussed in this subsection is used for evaluating the sound propagation inside the acoustic excitation units (AEUs) (see section 3). On the other hand, it is used to determine the propagation behavior of the acoustic modes generated by the acoustic excitation system (AES) and exciting blade vibrations (see subsection 5.2). In both cases, the calculation of the cut-off frequencies of the considered acoustic modes is necessary.

The main aeroelastic effects in turbomachines are forced response and flutter. For both phenomena, the aerodynamic damping has a significant impact. In the case of forced response, the resulting vibration amplitude is inversely proportional to the overall damping in the system. In the case of flutter, the aeroelastic instability results from negative aerodynamic damping. One of the main drivers for aerodynamic damping in turbomachines is the interblade phase angle σ, which defines the phase shift between the vibrations of adjacent blades in turbomachinery cascades. This parameter is based on the cyclic symmetry of turbomachinery cascades. Due to the cyclic nature, blade vibrations propagate in turbomachines as so-called 'traveling waves'. These vibration modes can be characterized by the interblade phase angle or the ND. In analogy to the azimuthal order of acoustic modes, the latter parameter defines the number of oscillation cycles of the blade vibration amplitude in the circumferential direction. The dependence between the interblade phase angle and the ND is given by

$$\begin{equation} \textrm {ND}=\frac{\sigma \cdot N_{\rm blade}}{2\pi }, \end{equation} \tag{ 11 }$$

where N_blade represents the number of blades of the considered blade row. The crucial importance of σ is illustrated by an example for aeroelastic stability shown in figure 2. A typical s-curve can be observed. Depending on the interblade phase angle of the blade row vibration, the aeroelastic stability will be positive or negative. In the latter case, the system will be prone to self-excited vibrations, e.g. flutter, which results in increasing blade vibrations leading to a substantial increase of the dynamic stress in the blade material. Due to the importance of σ, the possibility of exciting different interblade phase angles is an important requirement for the experimental investigation of aeroelastic effects.

Zoom In Zoom Out Reset image size

Assuming well-separated eigenfrequencies for a turbomachinery blade, the blade vibration behavior can be described by a modal approach. The equation of motion for a one degree of freedom model representing, for example, the first bending mode of a turbomachinery blade is given by

$$\begin{equation} \left(m+A_m\right)\ddot{x}+\left(d+A_d\right)\dot{x}+\left(c+A_c\right)x=\hat{F}\cos {\Omega t}, \end{equation} \tag{ 12 }$$

where m denotes the mass, c the mechanical stiffness and d the sum of the material and the structural damping of the turbomachinery blade. Additionally, the aerodynamic mass A_m, the aerodynamic stiffness A_c and the aerodynamic damping A_d influence the eigenfrequency and the eigenmode of the considered system. The excitation of this system is given by the right-hand side. In the case of acoustic excitation, the mechanical system is forced to vibrations by a periodical force caused by the acoustic pressure. Therefore, the connection between the acoustic field generated by the AES and the blade dynamics can be found in the forcing function on the right-hand side of equation (12). The amplitude $\hat{F}$ is then given by the product of the acoustic pressure acting on the blade multiplied by the blade surface. Furthermore, the excitation frequency Ω is defined by the frequency of the acoustic pressure. If this frequency equals the eigenfrequency of the considered system, the resonance condition is fulfilled and blade vibrations can be observed.

In order to prove the feasibility of acoustic excitation in turbomachines, the LSAC rotor blades of the second stage are excited acoustically at different rotational speeds. Figure 5 shows the blade responses of the second rotor row of the LSAC caused by acoustic forcing at different rotational speeds. The peak-to-peak blade vibration amplitudes are determined by a least square model fit (LSMF). The most important input parameter for the LSMF analysis is the blade vibration frequency. This parameter is determined by a fast Fourier transformation (FFT) of individual blade responses for all cases presented in this paper. To compensate for the difference between the real blade vibration amplitude and the vibration amplitude observed by tip-timing, all tip-timing amplitudes are processed by using the local chord angle at the instrumented axial position of the rotor blade. All LSMF amplitudes are normalized by the axial chord length of the rotor blade.

Zoom In Zoom Out Reset image size

For the PoP, the acoustic excitation is activated approximately 10 s after the measurement of the vibration amplitude is started and operated for a period of 10 s. Cascade responses can be clearly observed for each rotational speed. In this case, the same amplitude and frequency is set for each excitation unit. While the amplitude of the voltage supply is controlling the generated sound pressure, the frequency represents the acoustic excitation frequency and has to be set with respect to the blade eigenfrequencies. The excitation units operate in phase, i.e., there is no phase shift between the sine signals of adjacent excitation units. The chosen amplitude of the excitation unit voltage supply results in a sound pressure level output between 144 and 158 dB for each excitation unit. These values are determined in an anechoic chamber with a microphone placed directly in front of the AEUs. The defined distance between the microphone and the horn outlet is representative of the distance between the horn outlet and the blade tips in the LSAC. Caused by the characteristic transfer behavior of the compression drivers and the horn geometry, the generated sound pressure level strongly depends on the operating frequency of the compression drivers. The frequency which must be set in order to excite the blades depends on the rotational speed, the eigenfrequency of the blades and the acoustic field caused by the acoustic excitation. A detailed explanation of the relationship between the acoustic excitation frequency and the resulting blade vibration frequency is given in the second part of this section. The observed blade vibration frequencies f_blade and the frequencies of the AES f_AES during the PoP are given in figure 5.

It can be seen that the acoustic forcing results in a blade excitation at a defined frequency for the results presented in figure 5. Due to mistuning, the amplitude of blade vibration varies from one blade to another. In this case, mistuning means that the eigenfrequencies differ from one rotor blade to another. Hence, the excitation frequency does not match the blade eigenfrequency of all blades. The observed maximum blade response changes as the rotational speed increases. The highest vibration amplitude occurs at 2000 rpm at blade 9. One reason for mistuning and the different levels of excitable amplitude is the design of the LSAC rotor. The rotor blades are connected to the disc by a retaining ring, which is axially screwed onto the disc and shown in figure 6. In this type of attachment, the boundary conditions and the mechanical damping differ from blade to blade. This latter key parameter impacts the level of forced response of the rotor blades and it changes with rotational speed due to the change of centrifugal force. An impact of not calibrating the AEUs on the amplitude distribution can be excluded because each measured blade vibration amplitude is averaged over one revolution. Since the AEUs are fixed to the casing, all blades are passing through the same acoustic field and experiencing the same excitation force during each revolution.

Zoom In Zoom Out Reset image size

In summary, exciting rotating turbomachinery blades using acoustics is feasible. This PoP is the basis for the application of acoustic excitation to subsequent aeroelastic studies in turbomachines.

Comparing the acoustic excitation frequency to the blade vibration frequency, a factor of two or even higher can be observed. This shift in frequency cannot be explained by the rotational speed of the rotor blades and the resulting Doppler effect alone. In fact, an additional velocity component must be considered caused by the developing acoustic field. Since the LSAC is not instrumented for a detailed analysis of the acoustic field, the blade vibration measurements are used to develop a deeper understanding of the dominant modes of the acoustic field caused by the excitation system. In theory, the vibration behavior of the rotor row, which is characterized by the number of NDs, represents an image of the excitation pattern. This means that the number of NDs can be directly related to the azimuthal order of the dominant acoustic mode. Therefore, the blade responses presented in figure 5 are analyzed by traveling wave analysis. The results of the traveling wave analysis are shown in figure 7 for different rotational speeds. All traveling wave results are given in terms of the amplitude of the displacement at the tip of the excited rotor blade normalized by the axial chord length.

Zoom In Zoom Out Reset image size

For all rotational speeds, the content of traveling wave mode amplitudes is described by plotting the time-dependent amplitude of three different NDs. The first plotted traveling wave mode is the main component with the highest measured vibration amplitude. Due to the fact that the rotor cascade is mistuned, the vibration energy is not concentrated in a single traveling wave mode. To characterize the vibration behavior more completely, the upper and the lower limits of the amplitudes of the remaining traveling wave modes are plotted as well. For all rotational speeds, the system vibration behavior can be described by a backward traveling wave mode. The ND of the dominant traveling wave mode is −14 for all rotational speeds except the low speed case (1028 rpm). At 1028 rpm, the dominant traveling wave mode changes from ND = −14 to ND = −6. To obtain a more reliable data base for analyzing the relationship between the acoustic excitation frequency, the acoustic field and the system vibration behavior, a sweep of the acoustic excitation frequency is performed between 300 and 2000 Hz at 2000 rpm. In doing so, several blade responses are observed. Due to the well-separated eigenfrequencies of the investigated rotor, the excited modes are identified by the frequency of vibration, which is determined by the FFT of the recorded signals. In this case, only first bending mode responses are analyzed further. The main results of the traveling wave analysis of these responses to the AES are summarized in table 3, in addition to the cases presented in figure 7. In this table, the ND and the corresponding averaged amplitude represent the dominant traveling wave mode.

The comparison between the different blade responses at 2000 rpm shows that the ND of the excited mode can be changed by increasing the excitation frequency. Since the response of the blades is defined by the excitation pattern, it is presumed that the azimuthal order of the acoustic mode which is responsible for excitation can be determined by the ND of the vibration mode. In order to prove this hypothesis, the identified azimuthal orders are used to calculate the frequency of the acoustic pressure that is observed by the rotating blades. According to Petry (2011), the frequency acting on rotating turbomachinery blades can be calculated by means of the difference between the phase velocity of the exciting acoustic mode and the rotational speed of the blades. The phase velocity of an acoustic mode is given by

$$\begin{equation} \omega _{\rm phase}=\frac{\frac{2\pi }{m}}{T}=\frac{\frac{2\pi }{m}}{\frac{1}{f_{{\rm AES}}}}=\frac{2\pi f_{{\rm AES}}}{m} \end{equation} \tag{ 13 }$$

and is defined by the wavelength of the acoustic wave in the azimuthal direction divided by the time of oscillation of the acoustic wave T. Therefore, high phase velocities can be observed for low azimuthal orders. Using equation (13) to calculate the phase velocity, the frequency experienced by the rotor blades can be calculated as

$$\begin{eqnarray} \omega _{{\rm AES,blade}}=&\frac{2\pi f_{{\rm AES,blade}}}{m}=\omega _{\rm phase}-\omega _{\rm shaft} \end{eqnarray} \tag{ 14 }$$

$$\begin{eqnarray} f_{{\rm AES,blade}}=&\left(\frac{\omega _{\rm phase}-\omega _{\rm shaft}}{2\pi }\right)\cdot m. \end{eqnarray} \tag{ 15 }$$

The results for the transformation of the acoustic excitation frequency to the coordinate system of the rotor are summarized in table 4 for all cases. In fact, there are several acoustic modes, each characterized by the azimuthal order, which are able to excite a defined ND. For the 1028 rpm case, a ND = −6 response is observed. Depending on the direction of rotation of the acoustic mode, an azimuthal order of −6 or 24 can be linked to this ND. These values are determined under consideration of the number of LSAC rotor blades N_blade leading to

$$\begin{eqnarray} &&\textrm {ND}=m \leftrightarrow m=\textrm {ND}\quad \hbox { for } \left|m\right|\le \frac{N_{\rm blade}}{2} \end{eqnarray} \tag{ 16 }$$

$$\begin{eqnarray} &&\textrm {ND}=-(N_{\rm blade}-m) \leftrightarrow m=\textrm {ND}+N_{\rm blade} \nonumber\\ &&\!\qquad\qquad\qquad\qquad\quad\hbox {for } \frac{N_{\rm blade}}{2}< m \le N_{\rm blade} \end{eqnarray} \tag{ 17 }$$

$$\begin{eqnarray} &&\textrm {ND}=m-N_{\rm blade} \leftrightarrow m=\textrm {ND}+N_{\rm blade} \nonumber\\ &&\!\qquad\qquad\qquad\qquad\quad\hbox {for } N_{\rm blade}< m \le \frac{3 N_{\rm blade}}{2} . \end{eqnarray} \tag{ 18 }$$

Equations (17) and (18) are based on the vibrational behavior of turbomachinery cascades characterized by aliased traveling wave responses for m > N_blade/2. Since the blade vibration frequency is lower than the acoustic excitation frequency, the rotational direction of the acoustic mode and the rotor must be the same (equation (15)). Consequently, ω_phase and therefore m must be a positive number. This is why m = −6 can be excluded. In addition to m = 24, higher harmonic azimuthal modes, e.g. $m=54,\textrm { }84,\textrm { }114,\ldots \,$, are also able to excite an ND of −6 but are of minor interest because the acoustic energy and therefore the potential to excite blade vibrations decreases with increasing azimuthal order. The comparison between the calculated (f_{AES, blade}) and the measured blade vibration frequencies (f_blade) shows good agreement for all cases (see table 4). Therefore, the determined azimuthal orders of the acoustic modes responsible for exciting the blades are assumed to be valid. Based on these findings, the basic principle of the presented acoustic excitation method is explained below.

The eight AEUs, which are assumed as circumferentially equally spaced monopole sources, excite an acoustic field. This field contains acoustic modes whose azimuthal order is given by an integer multiple of the number of AEUs. To excite the rotating blades, the acoustic excitation frequency must be adapted such that f_{AES, blade} matches the blade eigenfrequency. The frequency shift between f_AES and f_{AES, blade} is caused by the relative velocity between the rotation of the acoustic mode responsible for blade excitation and the rotational velocity of the rotor.

A further conclusion can be drawn from the cut-off frequencies. Even under consideration of the circumferential Mach number of the flow, none of the acoustic modes responsible for blade excitation are propagable. Nevertheless, these modes are able to excite blade vibrations locally.

The radial mode order of the exciting acoustic modes is not regarded, since the radial order of the acoustic modes does not have any impact on the frequency shift between f_AES and f_{AES, blade}. Thus, n is needed for a complete characterization of acoustic modes as described in section 2, but it is of minor interest for this study.

The importance of being able to excite different NDs for aeroelastic investigations is outlined in subsection 2.2. The different blade responses presented for the 2000 rpm case indicate that the excited ND can be varied to some extent by changing the acoustic excitation frequency. Another possibility for varying the excited ND is to apply a phase shift between the different AEUs as shown below.

The control signal U_i for the AEU i is given by

$$\begin{equation} U_i=A_i \sin {\left(2\pi f_{{\rm AES}}\cdot t+\varphi _i\right)}; \quad i=0,1,\ldots N_{{\rm AES}}-1. \end{equation} \tag{ 19 }$$

All results shown up to this point are obtained by operating the AEUs in phase (φ_i = 0). The phase-lag control of the circumferentially distributed acoustic sources for exciting different acoustic modes is well known from aeroacoustics. Tapken (2010), for example, presented an acoustic system consisting of three actuator rings with eight circumferentially distributed compression drivers each. In his work, the undesired excitation of so-called 'spillover-modes' was investigated. A phase shift was applied to the control signal of the compression drivers to excite acoustic modes with a defined azimuthal and radial order in the inlet of a ventilator test rig. Subsequently, the resulting sound field was measured and analyzed. The approach of phase-lag control to influence the acoustic field shall be used below to change the ND of the excited vibration mode. By applying a phase shift between the different AEUs, a rotation of the acoustic field can be achieved. Presuming cyclic symmetry, the following phase shifts depending on the number of acoustic sources N_AES can be chosen:

$$\begin{equation} \varphi _i(j)= j\cdot \frac{2\pi \cdot {\rm i}}{N_{{\rm AES}}}; \quad i,j=0,1,\ldots N_{{\rm AES}}-1. \end{equation} \tag{ 20 }$$

The different phase shifts are characterized by the order j. The instantaneous pressure distributions in an axial plane are illustrated in figure 8 for different phase shifts. In this figure, the pressure field illustrated in the flow channel refers to the case of an infinite number of AEUs. Since there are only eight AEUs for this study, the instantaneous sound pressure amplitudes radiated by the AEUs are schematically illustrated by eight circumferentially distributed discrete points. The resulting pressure distributions can be characterized by the azimuthal order m_AES in analogy to the order m of acoustic modes. The direction of rotation of the sound pressure maxima and sound pressure minima caused by the phase-lag control and the direction of blade rotation is the same for 0 < j < N_AES/2. Due to the criterion of Nyquist, the direction of phase-lag induced rotation changes for j > N_AES/2 indicated by the negative sign of the azimuthal order m_AES. m_AES = 0 and m_AES = 4 are real modes where the direction of rotation cannot be explicitly determined. The first mode m_AES = 0 corresponds to the in-phase control where no additional rotation is induced. For m_AES = 4, the rotation of the pressure maxima and minima can be interpreted in both directions. In this study, a clockwise direction of rotation is assumed.

Zoom In Zoom Out Reset image size

In order to prove the approach of rotating the acoustic field to change the excited ND, acoustic excitation at different phase shifts is investigated in the LSAC at 2000 rpm. The results of this test series are summarized in table 5.

The experimental results confirm the phase-lag control as a suitable approach for varying the excited ND. It is assumed that the rotation of the acoustic field is superimposed on the exciting acoustic mode. Since the direction of the phase-lag induced rotation and the blade rotation is the same, the azimuthal order of the exciting acoustic mode is reduced for m_AES > 0. In the case of j > N_AES/2, the induced direction of rotation of the acoustic field changes and the azimuthal order of the exciting acoustic mode is increased. Consequently, the azimuthal order m of the acoustic mode which is responsible for blade excitation changes from 16 to 8. Due to the undefined rotational direction of the mode m_AES = 4, the order m of the acoustic mode responsible for excitation cannot be distinguished. In accordance with the assumption of a clockwise direction of rotation, m = 16 is listed in table 5 for m_AES = 4. The good agreement between the measured blade vibration frequencies and the transformed acoustic excitation frequencies f_{AES, blade}, calculated with an azimuthal order of m − m_AES, supports the theory of superposition. The averaged traveling wave amplitudes for phase-lag excitation are of the same order compared to in-phase excitation. Therefore, no constraint in terms of excitable amplitude is caused by the phase-lag control of the AEUs.

The results presented in table 5 are based on the excitation of the acoustic modes characterized by m = 8 and m = 16. By applying the phase-shift-induced rotation to acoustic modes with $m=\textrm {8, 16, 24 and 32}$, the vibration behavior of the LSAC rotor can be completely characterized, as shown by the excitable NDs listed in table 6.