Non-synchronous Vibration Research Articles

Experimental investigation of non-synchronous vibration and its mechanism analysis by using a cantilever beam-like compressor cascade

With the increasement of unsteady load and the extensive utilization of light-weight material, compressor blade is encountering increasingly serious flow induced vibration problems. Among which, non-synchronous vibration (NSV) has received special attention due to its complex generation mechanism and significant impact on high cycle fatigue. To better understand the physical mechanism causing NSV in compressor, this study designed a cantilever beam-like compressor cascade and made a detailed investigation about its flow separation induced vibration by means of experimental and numerical methods. The experiments were conducted at 9 engine representative conditions and the data of both on-blade pressure and structural response were measured simultaneously. Results show that, the effect of Ma number on the dynamic response will increase with the increasement of incidence angle and there is a strong nonlinearity. The effect of incidence angle on the dynamic response shows obvious bifurcation phenomenon. The results of large eddy simulation (LES) further reveal that the obvious characteristics of the dynamic response are caused by the laminar separation bubble (LSB) of suction surface, and the periodic separation and reattachment of boundary layer provides initial excitation sources for NSV. The unsteady excitation acts on blades in the form of energy and then forces the blades to vibrate with low order modes. The vibration amplitudes are closely related to the separation points and increase as the separation points move forward.

Suppression of nonsynchronous vibration in a foil air bearing rotor system through piezoelectric actuation of the top foil—A theoretical investigation

It is well documented that the static and dynamic performance of foil air bearings (FABs) can be improved through the alteration of the radial clearance profile. One way of achieving such controllability is through piezoelectric actuation of the top foil. Such a piezoelectric foil air bearing (PFAB) avoids the dynamics complications introduced by alternative actuating mechanisms. So far, theoretical analysis of a PFAB pad in isolation has shown that top foil actuation can raise the static load capacity. This paper theoretically investigates the potential for improving the dynamic performance. A computational approach that uses Galerkin Reduction (GR) to model the air film and considers the detachment of the top foil from the bump foil is upgraded for multi-pad PFABs, thus facilitating parameter optimization. The investigation considers two conventional FAB rotor systems from the literature that exhibit strong sub-synchronous vibrations under high unbalance. It is found that the optimal voltage and placement of piezoelectric patches can result in total suppression of the strong sub-synchronous frequencies, as well as a rise in the onset of instability speed. The polarity of the voltage required for suppression of the sub-synchronous frequencies is found to be opposite to that required to raise the load capacity.

Information transfer between tip leakage vortex and blade aerodynamic force of a compressor cascade

The compressor blade exhibits intricate flow patterns near stall conditions, with the tip leakage vortex and the main flow shedding vortex from the blade's trailing edge identified as factors influencing the blade's aerodynamic force oscillation. The tip leakage vortex is recognized as the primary contributor to flow structures during non-synchronous vibration (NSV), although existing compressor NSV reduced order models often attribute the fluid oscillation to the passage flow shedding vortex. A tool capable of distinguishing between flow structures and blade aerodynamic force is crucial for resolving this discrepancy. Detached eddy simulations (DES) in a compressor cascade form the basis for gathering data on vortex structures and blade aerodynamic force. This study employs an information thesis method to elucidate the cause-and-effect relationship between the tip leakage vortex and blade aerodynamic force. Through a series of unsteady single passage simulations and experimental validation, the mechanisms leading to variations in blade aerodynamic force are demonstrated. Notably, this research utilizes Proper Orthogonal Decomposition (POD) mode time coefficients in span and axial directions to represent dynamic information on vortex structures. Additionally, transfer entropy is introduced as a metric to evaluate the causal interaction between turbulent flow of the tip leakage and the blade's aerodynamic force, marking a significant advancement. The analysis of information transfer entropy aids in identifying the vortex structure with a stronger relationship to aerodynamic force, a critical finding. By applying this approach, the study examines tip leakage vortex structures in various tip clearance sizes concerning aerodynamic force. The Tip3C and Tip5C tip geometries exhibit tip leakage vortex features with radial vortex characteristics and backflow, akin to structures associated with non-synchronous vibration. These tip vortex structures in the Tip3C and Tip5C tip geometry cascades demonstrate substantial correlations with aerodynamic force oscillation on the blade surface. Conversely, the Tip1C tip geometry cascade displays distinct tip leakage vortex features compared to Tip3C and Tip5C, with weaker ties to blade aerodynamic force oscillation. As simulation models progress, consideration of vortex structures in the passage and leakage vortex at the leading edge is imperative for accurate predictions of aerodynamic force oscillation response.

Effect of tip clearance on non-synchronous propagating flow disturbances of compressor rotors under high aerodynamic loading conditions

The complex tip flow instability and its induced non-synchronous vibration have become significant challenges, especially as aerodynamic loading continues to increase. This study investigates the effects of tip clearance on non-synchronous propagating flow disturbances of compressor rotors under high aerodynamic loading conditions by conducting full-annulus unsteady numerical simulations with three typical tip clearance values for a 1-1/2 stage transonic compressor. The non-synchronous aerodynamic excitation frequency, circumferential mode characteristics, and annular unstable flow structures are analyzed under near stall conditions. The results show that the total pressure ratio and normalized mass flow parameters first increase and then decrease as the tip clearance increases from 0.5%C (where C represents the tip chord length) to 2%C under high aerodynamic loading conditions, instead of constantly decreasing. For the 0.5%C tip clearance case, the traveling large-scale tornado-like separation vortices cause a low non-synchronous aerodynamic excitation frequency and severe pressure fluctuations. The periodic shedding and reattachment processes of the rotor blades separated by 2 – 3 pitches result in 19 dominant mode orders in the circumferential direction. As the tip clearance increases from 1%C to 2%C, the difference of tip flow structures in each blade passage is significantly weakened, and the dominant mode order of the disturbance is equal to the rotor blade-passing number. The pressure fluctuation is mainly caused by cross-channel tip leakage flow, and the aerodynamic excitation frequency exhibits evident broadband hump characteristics, which has been reported as a rotating instability phenomenon.

Self-induced non-synchronous resonance phenomena and stability in reduced aero-elastic system

The paper is aimed at the parametric dynamic analysis of the two degrees of freedom (DoF) phenomenological model of aero-elastic interaction between a friction-damped flexible mounted body and a flowing fluid. A coupled system of a linear structure oscillator and the van der Pol based wake oscillator is introduced. Firstly, the complex eigenvalue problem is used for the comprehensive analysis of frequency lock-in, veering and crossing areas from the structural mode stability point of view. Further, an in-depth analysis of the fully nonlinear model based on the numerical continuation method is presented. The analysis focuses on non-synchronous vibration in the resonance areas. Attention is paid to the investigation of transition mechanisms to frequency lock-in and frequency veering in the structural response. In experimental studies, the amplitude and frequency jumps are observed. Here, we show that the origin of these jump phenomena results from the mutual coexistence of two stable solutions, which form a hysteresis region, which can be found in experimentally gained data. Moreover, the influence of structure friction-damping on nonlinear response is evaluated. The proposed phenomenological model is validated concerning two different experimental-based investigations.

Down-film as a new non-frame porous material for sound absorption

Down-polyethylene film material has been introduced for the first time as an excellent non-frame sound absorber, showing a distinctively outstanding performance. It contains down fiber adjacent to each other without firm connection in between, forming a structure of elastic fiber network. The unique structure has broadband response to sound wave, showing non-synchronous vibration in low and middle frequency and synchronous vibration in middle and high frequency. The broadband resonance in middle and high frequency allows the structure to achieve complete sound absorption in resonance frequency band. Moreover, down-polyethylene film material possesses forced vibration, corresponding sound absorption coefficient has been obtained based on vibration theory. The down-film sound absorption material has the characteristics of light weight, soft, environment-friendly, and has excellent broadband sound absorption performance.

Non-synchronous vibration of rotor blade in a six-stage transonic compressor

This paper presents an experimental study on the Non-Synchronous Vibration (NSV) in a six-stage transonic compressor. The first part of the paper describes the NSV phenomenon of Rotor 1, which occurs when both Stator 1(S1) and Stator 2(S2) or S1 only are closed. Detailed measurements and analysis are carried out for the former case through the unsteady wall pressure and the Blade Strain (BS). The spinning mode theory used in the rotor/stator interaction noise is employed to explain the relation between the circumferential wave number of the aerodynamic disturbance and the Nodal Diameter (ND) of the blade vibration. The variations of the vibration amplitudes of different blades and the Inter-Blade Phase Angles (IBPAs) at different moments suggest that the evolution of NSV is a highly nonuniform phenomenon along the circumferential direction. In addition, the difference between the wall-pressure spectra generated by the NSV and the classic flutter has been discussed. In the second part, the variations of aerodynamic loading due to the adjustment of the staggers of the Inlet Guide Vane (IGV), S1 and S2 have been investigated. It is found that closing S1 only can result in a great fluctuation to the performance of the front stages, which might be detrimental to the flow organization and increase the risk of NSV. In contrast, the effect of closing S2 only on the performance of the first two stages appears to be slighter relatively.

Characterizing the Unsteady Flow Field in Low-Flow Turbine Operation

Abstract Current operational considerations require steam turbines to operate in a more flexible way, with more frequent and faster start-up and an increasing part-load operation. For very low mass flowrates, the interaction of highly separated flow with the high-speed rotor blades causes windage flow. This type of flow is characterized by increased temperature and highly unsteady flow, which forms vortex structures that rotate at a fraction of the rotor speed. If their magnitude is sufficiently high and the frequency is close to the blade eigenfrequency, non-synchronous vibration (NSV) can be induced. In this paper, low-flow turbine operation is investigated using a three-stage turbine rig that features an instrumentation concept focused on capturing aerodynamic and aeroelastic phenomena. Extensive steady probe, unsteady pressure, and tip-timing measurements are utilized. The experimental scope covers a wide range of operating points in terms of rotational speed and mass flowrates. Low-flow regimes are detected by a reversal in torque and increase in temperature. Unsteady measurements during transient operation identified large-scale vortical flow structures rotating along the circumference, so-called rotating instabilities (RIs). The onset, growth, and breakdown regimes of RI are characterized for different low-flow conditions. The quantitative characteristics of RI with regard to nodal diameter and rotational speed are derived by a cross-correlation of multiple unsteady sensors. The blade vibration measurements show a moderate structural response from unsteady aerodynamic excitation, indicating no significant NSV occurring in the present experimental setup. Later in the study, an acoustic excitation system has been applied to trigger a locked-in NSV without interrupting the coherent flow structures. From that, significant blade response has been observed, revealing a high degree of mistuning and damping of the rotor blading.

A New Axial Compressor Test Facility for the Investigation of Aerodynamic Damping Featuring an Electromagnetic Excitation System

Abstract Aerodynamic damping simulations involving detailed computational fluid dynamics (CFD) models are nowadays an integral part of turbomachinery design processes, thanks to advancements made in simulation tools and in computer hardware over the last decades. However, the test cases available in the open literature suitable for the validation and continued development of simulation tools is rather limited. On this background, a new test facility is introduced that allows acquiring relevant and high-quality aerodynamic damping test data on axial compressor rotor test objects in a controlled manner. The test facility is built as a closed loop and thus enables operation at variable pressure levels ranging from near-vacuum to overpressure and with different operating media. Overall damping test data are acquired at controlled synchronous and nonsynchronous vibration conditions. While vibrations of the blades at controlled nonsynchronous vibration conditions can be introduced in the test facility by means of a proprietary electromagnetic excitation system that is acting below the hub line, vibrations at controlled synchronous conditions can be introduced from a set of magnets integrated into the casing. Great care has been spent to create a clean test case without any intruding parts or openings in the test section. In addition, an axial compressor blisk is presented, which acts as test object to show the capabilities of the test facility as well as to allow for detailed investigations of aerodynamic damping. While the first part of the article discusses the setup and the instrumentation of the test facility to excite and measure blade vibration in a nonintrusive way, flow-field measurements and the validation accompanying steady-state Reynolds-averaged Navier-Stokes (RANS) simulations are shown in the second part. The third part of the article focuses on aerodynamic damping measurements for seven different modes at three different pressure levels in total. It can be shown that aerodynamic damping is by far the largest contributor to the overall damping.

Relationship between Casing Pressure and Non-Synchronous Vibration in an Axial Compressor

The relationship between aerodynamic forcing and non-synchronous vibration (NSV) in axial compressors remains difficult to ascertain from experimental measurements. In this work, the relationship between casing pressure and blade vibration was investigated using experimental observations from a 1.5-stage axial compressor under off-design conditions. The wavenumber-dependent auto-spectral density (ASD) of casing pressure was introduced to aid in understanding the characteristics of pressure fluctuations that lead to the aeromechanical response. Specifically, the rotor blade’s natural frequencies and nodal diameters could be directly compared with the pressure spectra. This analysis indicated that the rotating disturbances coincided with the first bending (1B) and second bending (2B) vibration modes at certain frequencies and wavenumbers. The non-intrusive stress measurement system (NSMS) data showed elevated vibration amplitudes for the coincident nodal diameters. The amplitude of the wavenumber-dependent pressure spectra was projected onto the single-degree-of-freedom (SDOF) transfer function and was compared with the measured vibration amplitude. The results showed a near-linear relationship between the pressure and vibration data.

Experimental and numerical investigations on non-synchronous vibration and frequency lock-in of transonic compressor rotor blades

Non-synchronous vibration (NSV) problems in axial compressors are challenging and frequently result in high vibration stress or structural failure within the frequency lock-in region. A deeper understanding of the lock-in phenomenon in NSV is necessary to improve blade reliability and safety. In this study, an experiment was conducted on a multistage transonic axial compressor to obtain frequency lock-in characteristics. The enforced motion of the flexible blade simulation method was used to study the frequency lock-in mechanism between blade vibration and tip unstable flow. The results showed that the aerodynamic frequency characteristic changed from a broadband spectrum to a distinct frequency peak as the NSV onset. The nonlinear vibration pattern could maintain the limited cycle oscillation for 26 s. The vibration stress of the frequency locked-in case was approximately 19 times that of the frequency unlocked one. The pressure fluctuations generated by the large-scale radial separation vortex structures were stronger than those of the tip leakage flow, and the third-order harmonic frequency of the circumferential instability flow was close to the first-order bending model natural frequency. The periodic shedding and reattachment process of flow separation provided the initial non-synchronous aerodynamic excitation source. The frequency lock-in phenomenon occurred at a maximum vibration amplitude of 2 % and 3 % tip chord length. The tip unstable flow characteristics were consistent and entirely dominated by blade vibration.

UHBR Open-Test Case Fan ECL5/CATANA: Non-Linear Analysis of Non-Synchronous Blade Vibration at Part-Speed Conditions

Abstract The onset of non-synchronous vibrations has been predicted for a composite open-test-case UHBR fan geometry by time-linearized simulations at part-speed conditions. In the current study, time-accurate simulations with harmonic blade oscillations have been conducted to investigate the non-linear behavior. Results for different operating conditions show the evolution of aeroelastic stability. The lock-in of convected aerodynamic disturbances with structural vibration is analyzed for different vibration amplitudes and frequencies. The simulations show the development of multiple wave numbers, mode scattering, and a non-linear dependency on vibration amplitude. The results indicate the possibility of limit-cycle prediction and the necessity of strongly coupled methods for the assessment of non-synchronous vibration.

Quantification of Blade Vibration Amplitude in Turbomachinery

Experimental monitoring of blade vibration in turbomachinery is typically based on blade-mounted strain gauges. Their signals are used to derive vibration amplitudes which are compared to modal scope limits, including a safety factor. According to industrial guidelines, this factor is chosen conservatively to ensure safe operation of the machine. Within the experimental campaign with the open-test-case composite fan ECL5/CATANA, which is representative for modern lightweight Ultra High Bypass Ratio (UHBR) architectures, measurements close to the stability limit have been conducted. Investigation of phenomena like non-synchronous vibrations (NSV) and rotating stall require a close approach to the stability limit and hence demand for accurate (real-time) quantification of vibration amplitudes to ensure secure operation without exhaustive safety margins. Historically, short-time Fourier transforms of vibration sensors are used, but the complex nature of the mentioned coupled phenomena has an influence on amplitude accuracy, depending on evaluation parameters, as presented in a previous study using fast-response wall-pressure transducers. The present study investigates the sensitivity of blade vibration data to evaluation parameters for different spectral analysis methods and provides guidelines for fast and robust surveillance of critical vibration modes.

Investigation of inlet flow influenced on the first rotor flow instability at blade non-synchronous vibration state

Multistage axial compressor first stage rotor blades have occurred non-synchronous vibration (NSV). An experiment, including fluid and structure measurements, is adopted at the NSV occurred conditions to conduct a detailed investigation about NSV and rotating instability. The blade vibration stress was obtained from the strain gauges. High-frequency response two-hole pressure probes captured the flow characteristics at the inlet and outlet of the first stage rotor. NSV has a complex aerodynamic disturbance source. Wavelet and frequency spectrum analyses of pressure and blade stress results were utilized to determine the relationship between the blade vibration and the flow field. The inlet flow angle has been adjusted through the inlet guide vane (IGV) to reduce the vibration intensity. This paper revealed the aerodynamic origin behind this adjustment of IGV angle operation. The yaw angle and the axial direction March number at the first stage rotor inlet at different inlet guide vane (IGV) angles revealed the aerodynamic effect. Three operation points, including the maximum stress point and near surge point, are analyzed to broaden the cognition boundary of the NSV. The time and space characteristics of the rotating instabilities have been studied with the azimuthal mode analysis method. This research confirmed that the enlarged inlet incidence angle and the tip flow worsened caused the first stage rotor non-synchronous vibration in this multistage axial compressor. These experiments help find a straight relationship between the NSV and rotating instability.

Effects of tip clearance on non-synchronous aerodynamic excitation of compressor rotor blades

The unstable tip flow of compressor rotor may cause the aeroelastic problems, inducing the non-synchronous vibration with high amplitude level, which has a significant effect on blade high cycle fatigue and safety. In order to investigate the effects of tip clearance on non-synchronous aerodynamic excitation of compressor rotor blades, the full annular numerical simulations were conducted for the unsteady flow field of a 1.5-stage compressor under near stall conditions. The frequency characteristics of non-synchronous aerodynamic excitation and tip unstable flow structure were analysed for different tip clearances (0.5%C, 1.0%C, 2.0%C, and 3.0%C). The results showed that the frequency and amplitude are strongly influenced by the tip clearance size. For 0.5%C tip clearance, the main frequency of non-synchronous aerodynamic excitation is the low-frequency aerodynamic excitation frequency, which is mainly caused by tornado-like vortex in the tip suction side, the dominant aerodynamic disturbance mode number is 19. For 2.0%C and 3.0%C tip clearance, the dominant aerodynamic disturbance mode number is 47, and the main aerodynamic excitation frequency is high frequency non-synchronous aerodynamic excitation frequency, mainly caused by the shedding vortices propagating circumferentially at the blade tip, exhibiting the feature of “rotating instability”.

The Influence of Speed Ratio on the Nonlinear Dynamics of a Magnetic Suspended Dual-Rotor System with a Fixed-Point Rubbing

Magnetic suspended dual-rotor systems (MSDS) provide the potential to significantly improve the performance of aero-engines by eliminating the wear and lubrication system, and solve vibration control issues effectively. However, the nonlinear dynamics of MSDS with rubbing is rarely investigated. In this work, the nonlinear support characteristics of active magnetic bearings (AMBs) are described by the equivalent magnetic circuit method, the impact force is characterized by the Lankarani–Nikravesh model, and the nonlinear dynamic model is established using the finite element method. On this basis, the influence of speed ratio on the nonlinear dynamics is investigated. Simulation results show that the fundamental sub-synchronous vibration of period n is the dominant motion of MSDS, where n is determined by the speed ratio. The frequency components of sub-synchronous vibrations of period k are integer multiples of the minimum dimensionless frequency component 1/k, where k is a positive integral multiple of n. Quasi-periodic and chaotic vibrations are more likely to occur near critical speeds, and their main frequency components can be expressed as a variety of combined frequency components of the rotating frequency difference and its fractional frequency. To reduce the severity of fluctuating stresses stemming from complicated non-synchronous vibrations, speed ratios, corresponding to smaller n and AMB control parameters attenuating vibration amplitude or avoiding critical speeds, are suggested.

Tip flow on rotating instability on an axial compressor with different tip clearances

Rotating instability (RI) as an unsteady phenomenon usually occurs at off-design conditions in compressors, featuring a broadband hump in the pressure spectra below the blade passing frequency (BPF). RI phenomenon could not only cause deterioration to the performance, but also lead to tip clearance noise or non-synchronous vibration to the compressor. In this paper a detailed experiment was carried out in a low-pressure single-rotor compressor. By replacing the casing the relative tip clearance size was changed and the time-resolved wall pressure with a distinct RI signature was thus measured under several different conditions through the Kulite pressure transducers both circumferentially and chordwise installed. The temporal-spatial distribution of RI signature evolution with the flow coefficient is interrogated, and its propagation characteristics are also investigated through the cross-correlation method. Moreover, the quasi-instantaneous contour measured by the chordwise transducers shows no distinct flow structure which could be connected to the RI signature, i.e., the prestall disturbance. Therefore, a quasi-SFT method is developed to identify different disturbance counts and the corresponding relationship between the disturbance count per revolution and the azimuthal mode order could thus be strictly established. The prestall disturbance is found to originate from the mid-chord positions as RI initiates and travel upstream as the reduction of flow coefficient, and eventually reaches the leading edge as rotating stall occurs. Furthermore, the location of prestall disturbances is proven to be independent on the blade position. These findings describe the tip flow field when the RI mode orders are relatively high, and provide new insights for the shear layer instability theory as RI mechanism.

Self-excited vibration suppression of a spline-shafting system using a nonlinear energy sink

The rotating instability and corresponding non-synchronous vibrations seriously affect the long-term operations of modern high-speed rotating machinery. However, there is still a lack of effective control methods to prevent the unstable vibration of the rotor system. This study focuses on the effective suppression of large-amplitude self-excited vibrations in spline-shafting systems using a nonlinear energy sink (NES). A nonlinear dynamic model for a spline-shafting system incorporating the non-smooth friction interface is developed. The corresponding friction-induced vibration and the instability mechanism are explored through the time-frequency analysis. The NES is then employed for the self-excited vibration suppression and the optimum parameters of the NES are obtained through the rotor vibration response surface. Several numerical analyses are conducted to investigate the suppression performance of the NES under the condition of different load torque. Lastly, the mitigation of the rotor self-excited vibration is realized and the mechanism of NES suppressing self-excited vibration by strong modulated response is elucidated. The obtained results show that the NES can not only effectively reduce the vibration of the rotor system in the vicinity of resonance, but also alleviate the large amplitude of self-excited vibration simultaneously, which may provide helpful guidance for the dynamic design of complex rotor systems.

Modal Identification of Bladed Disks by Time–Frequency Analysis of the Nonsynchronous Response

Abstract In some circumstances, it is impossible to exploit resonance crossings to identify the modal properties of rotor disks. In these cases, the identification process must rely on nonsynchronous vibrations and becomes challenging for two reasons. First, the signals are weak (compared to the levels measured during resonance crossings) and random, thus an averaging procedure is necessary. Second, the dynamical system is time-variant due to the variation of the rotor speed. This paper presents a modal identification procedure formulated in the framework of the time–frequency analysis. A region of the time–frequency plane is stretched to map the system into a fictitious linear time-invariant (LTI) system. Then, the power spectral density function (PSD) of the response is computed by an averaging procedure. Finally, the modal properties are estimated through an output-only modal identification algorithm. The procedure is applied to simulated and experimental data regarding a bladed disk of a steam turbine.

A Fluid–Structure Interaction Tool Using a Van der Pol-Based Reduced-Order Model for Buffet and Nonsynchronous Vibrations

Abstract This paper builds upon a two-degree-of-freedom Van der Pol oscillator-based reduced-order model for studying the mechanisms around nonsynchronous vibrations (NSV) in turbomachinery. One degree tracks the fluid motion utilizing a combination of a traditional Van der Pol oscillator and a Duffing oscillator; the other degree of freedom is a mass on a spring and a damper, in this case, a cylinder. Thus, this model can be considered one of fluid–structure interaction. The cubic stiffening from the Duffing oscillator proved to improve the match to experimental data. Using this model to study the time history of the fluid and the structure oscillation, additional parameters are extracted to understand the underlying mechanisms of frequency lock-in and limit cycle oscillation. First, the phase shift between the vortex shedding and the structural motion is calculated when it locks-in and then unlocks. Second, the work done per cycle is analyzed from the contributions of the mass, spring, and damping forces to determine the dominant contributor when locking-in versus unlocking. Third, the phase portrait is plotted on a Poincaré map to further study the locked-in versus unlocked responses. This model is then validated against not only experimental data but also computational simulation results and previous reduced-order models. The finalized model can now serve as a preliminary design tool for turbomachinery applications. For more realistic and accurate modeling, the third degree of freedom in the form of an airfoil pitching motion will be added in a separate paper as well.