Aeromechanical optimization of first rowcompressor test stand blades using a hybridmachine learning model of genetic algorithm,artificial neural networks and design ofexperiments
Mohammad Ghalandari, Alireza Ziamolki, Amir Mosavi, ShahaboddinShamshirband, Kwok-Wing Chau & Saeed Bornassi
To cite this article: Mohammad Ghalandari, Alireza Ziamolki, Amir Mosavi, ShahaboddinShamshirband, Kwok-Wing Chau & Saeed Bornassi (2019) Aeromechanical optimization of firstrow compressor test stand blades using a hybrid machine learning model of genetic algorithm,artificial neural networks and design of experiments, Engineering Applications of ComputationalFluid Mechanics, 13:1, 892-904, DOI: 10.1080/19942060.2019.1649196
To link to this article: https://doi.org/10.1080/19942060.2019.1649196
Aeromechanical optimization of first row compressor test stand blades using ahybrid machine learning model of genetic algorithm, artificial neural networksand design of experiments
Mohammad Ghalandaria, Alireza Ziamolkia, Amir Mosavib,c, Shahaboddin Shamshirband d,e, Kwok-Wing Chauf
and Saeed Bornassia
aResearch and Development Department, MAPNA Turbine Engineering and Manufacturing Company (TUGA), Karaj, Iran; bSchool of the BuiltEnvironment, Oxford Brookes University, Oxford, UK; cKando Kalman Faculty of Electrical Engineering, Institute of Automation, ObudaUniversity, Budapest, Hungary; dDepartment for Management of Science and Technology Development, Ton Duc Thang University, Ho ChiMinh City, Vietnam; eFaculty of Information Technology, Ton Duc Thang University, Ho Chi Minh City, Vietnam; fDepartment of Civil andEnvironmental Engineering, Hong Kong Polytechnic University, Hong Kong, People’s Republic of China
ABSTRACTIn this paper, optimization of the first blade of a new test rig is pursued using a hybrid model com-prising the genetic algorithm, artificial neural networks and design of experiments. Blade tuning isconducted using three-dimensional geometric parameters. Taper and sweep angle play importantroles in this optimization process. Compressor characteristics involvingmass flow and efficiency, andstress and eigenfrequencies of the blades are the main objectives of the evaluation. Owing to thedesign of blade attachments and their dynamic isolation from the disk, the vibrational behavior ofthe one blade is tuned based on the self-excited and forced vibration phenomenon. Using a semi-analytical MATLAB code instability, the conditions are satisfied. The code uses Whitehead’s theoryand force response theory to predict classical and stall flutter speeds. Forced vibrational instabilityis controlled using Campbell’s theory. The aerodynamics of the new blade geometry is determinedusing multistage computational fluid dynamics simulation. The numerical results show increasingperformance near the surge line and improvement in the working interval along with a 4% increasein mass flow. From the vibrational point of view, the reduced frequency increases by at least 5% inboth stall and classical regions, and force response constraints are satisfied.
ARTICLE HISTORYReceived 2 May 2019Accepted 24 July 2019
Aeromechanical considerations are the major issue indeveloping advanced turbomachinery blades. In additionto strength criteria, the main critical aeroelastic prob-lems, involving flutter, forced response and asynchronousvibration, can affect the design criteria of the blades.Forced response and asynchronous vibrations originatefrom aerodynamic sources, whereas flutter instability iscaused by the interaction of the motion of the blades andthe aerodynamic forces. All these phenomena, and espe-cially flutter, which has a limit-cycle oscillation besidesthe oscillatory aerodynamic loading, are arguably themost important factors in the design of durable blades.The stress induced by these excitations should be con-fined to the minimum possible level even under reso-nance conditions, where the amplitude of the vibrationcan increase significantly and is usually the main causeof high cycle fatigue failure.
The flutter phenomenon as a root cause of failurein turbomachinery blades is still under investigation bymany researchers. Early studies on flutter speed pre-diction in turbomachinery and evaluation of its conse-quences on blade failure were conducted by Whitehead(1965, 1966, 1973). His presented method was based onthe simple aeroelastic model for cascades which wereexposed to a subsonic airflow. The vortex panel the-ory and non-penetrating flow boundary conditions sup-port the analytical solution of lift andmoment of the cas-cade in flexural and torsional oscillating modes. Thetheory has been established based on Theodorsen andMutchler’s (1935) aerodynamic model, which predictsclassical flutter for a single oscillating airfoil. The incom-pressibility of inviscid flow and the low angle of attackof flat airfoils are other fundamental assumptions of themodel. Later, Mikolajczak, Arnoldi, Snyder, and Star-gardter (1975), Lubomski (1980), Kielb and Kaza (1983)
ENGINEERING APPLICATIONS OF COMPUTATIONAL FLUID MECHANICS 893
and Srinivasan (1997) extended the Whitehead theoryand suggested different limits for classical and stall flutterinstability.
From a design point of view, shrouds, clappers anddampers are coupling components that are commonlyapplied to reduce or postpone the disturbance effects ofthese dynamic phenomena. In addition, the elastic axis
Figure 1. Newly presented compressor test stand.
position of the blades can influence aerostructural prob-lems. Therefore, geometric parameters such as the sweepangle and taper ratio can play a positive role in aerostruc-tural modifications of the blades (Pathak, Kushari, &Venkatesan, 2008).
Nowadays, the blade shape is often designed usingan optimization tool. Multidisciplinary design optimiza-tion (MDO) is a powerful method commonly applied tothe design and optimization of turbomachinery compo-nents (Ashihara, Goto, Guo, & Okamoto, 2004; Deng,Shao, Fu, Luan, & Feng, 2018; Iwaniuk, Wiśniowski, &Żółtak, 2016; Xiaodong & Xiuli, 2015; Zhang, Gou, Li,Wang, & Yue, 2016). Demeulenaere, Ligout, and Hirsch(2004) took advantage of the MDO approach to createthe optimum shape of the disk and blade turbine com-ponents. Shen, Cao, and Yang (2009), using MDO andthermoelastic–plastic analysis, presented an algorithm toextract the shape of the blades and disk. Luo, Song, Li, andFeng (2009) introduced a simplified three-dimensional(3D) transonic blade based on MDO. The finite ele-ment method (FEM) and computational fluid dynamics(CFD) (Faizollahzadeh Ardabili et al., 2018) are robustand efficient methods which have facilitated the design
Figure 2. Classical design flow of the blade.
894 M. GHALANDARI ET AL.
process (Ghalandari, Mirzadeh Koohshahi, Mohama-dian, Shamshirband, & Chau, 2019).
In this paper, optimization of the first rotary blades ina new axial compressor test stand (Figure 1) with a com-mon design flow consideration (Figure 2) is presented.The compressor has five stages, each with 3.5 pressureand 9 temperature ratios, and operates in the speed rangeof 9500–16,500 rpm.
Because of the occurrence of flutter of the main bladesin the first stage, which is a challenging part of thisproject, aeromechanical assessments in both stall andclassical regions are performed; and using theMDO tech-nique, CFD and FEM, the best shape of the blades isextracted. Taper ratio and sweep are the main geometricparameters in the presented optimization process, whichis performed on the US National Advisory Commit-tee for Aeronautics (NACA) 65 basic airfoil. Therefore,our structural considerations were flutter speed preven-tion and setting a reasonable margin for forced responseand steady-state stress. The aerodynamic objectives wereto optimize the characteristics of the compressor andincrease the performance of each stage.
2. Case study
Sweep angle, which is created by dislocation of the cross-section along the chord of the airfoil, plays an impor-tant role in the performance of axial turbomachinery.In general, the 3D influences of the blade with a sweepeffect are enhancements in the efficiency and range ofthe operation. Aerodynamic studies have shown that thisparameter can increase the pressure ratio value, limit thesecondary loss effects, decrease the loading near the lead-ing edge and reduce blockage in the tip area of the blades.The sweep angle can also change the shockwave structureand improve the loss effects in transonic systems. Theadvantages of the forward swept blade are highlighted,so that it has direct effects on aerodynamic parametersinvolving leading edge loading, incidence angle effectsand leakage (Pathak et al., 2008). From the mechanicalpoint of view, sweep angle can also change the elastic axisposition and alter the dynamic behavior. Investigation ofthe presented main blades with 80mm length (Figure 3)and material of 26NiCrMov145 (Table 1) shows that thesweep parameter has an effect of about 1.5%, especially onthe second mode (first torsional mode) (Table 2); aeroe-lastic failure of turbomachinery blades, as in our test case,usually occurs in this mode.
Taper parameters such as sweep angle have a greatimpact on performance and other aerodynamic charac-teristics. Taper parameters can increase the frequency ofthe blades (Kaza & Kielb, 1985) but have negative effectson the compressor characteristics (Figure 4).
Figure 3. Main blade geometry.
Table 1. Cord length and twist angle distribution of the NACA 65airfoil along the main blade span.
Table 2. Sweep effects on the second and third modefrequencies.
ModeMain blade
frequency (Hz)
Sweepbackward bladefrequency (Hz)
Sweepforward bladefrequency (Hz)
First torsion 1405.2 1427.3 1381.3
3. Description of optimization process
MDO (Shen et al., 2009), as a relatively new conven-tional method, and other in-house modules can producea axial turbomachinery blades with good performanceand an acceptable range of mechanical criteria. By com-bining a genetic algorithm (GA) and an artificial neuralnetwork (ANN) with low cost and high-speed calcula-tions, based on the MDO method, the design process ofthe main blade is accomplished. These design activities
ENGINEERING APPLICATIONS OF COMPUTATIONAL FLUID MECHANICS 895
Figure 4. Aerodynamic effects of tapering.
include adaptive geometric parameterization, an evolu-tionary optimization algorithm and a popular surrogatemodel using highly efficient 3D aeromechanical software.This process starts with the creation of a database basedon the design of experiments (DOE)method, main bladespecifications and ANN models. Bezier control pointsaffect the distributions of thickness and sweep anglefor the generation of new geometries. By generating adatabase, the interactions and accuracy of the selectedbounds for each variable are evaluated.
Using the trained ANN, the results of the new geom-etry are generated, some special estimations of the ANNare numerically modeled and the target results are addedto the database. This procedure is conducted until theresults of the ANN converge with the numerical results.Finally, design software is used to assess the objec-tive function under the given boundary conditions withregard to mechanical constraints.
4. Geometric parameterization
The geometry is parameterized with a conventionalBezier method using an in-house tool. The geometry isdefined by a certain number of parameterized airfoil sec-tions with a distribution of camber lines and thicknesses.Each two-dimensional (2D) section can be parameter-ized by 15Bezier control points, five control points for thecamber line and 10 control points for the thickness dis-tribution. The definition of changes in the control pointsdepends on the geometric constraints and properties ofthe boundary conditions; in this design, some controlpoints in the blade’s leading and trailing edges are fixedlocally. The variation in the control points of each sectiondepend on the other sections, and this can reduce thedisturbances occurring on the generated blade’s surface.The stacking line is parameterized by five control points;this can enable the sweep angle of the blade geometry to
Figure 5. Optimized design flow of the blade. CFD =computational fluid dynamics.
896 M. GHALANDARI ET AL.
be changed. In the spanwise direction, the definition ofmovable control points differs. The selection of appro-priate control points and their margin of variation, fromminimum tomaximum values, is very important becausethis selection should be made so that the geometry canobtain the best position with the least amount of dis-sonant geometry. The selection of inappropriate controlpoints with unsuitable margins can lead to more com-putational time being required, with errors in the designprocedure.
5. Calculation
Generated geometries are evaluated on design and off-design operating points based on the sensed bound-ary conditions. Geometric parameters, consisting of thestack-line position defining the sweep angle and thick-ness in UG software, are considered as design variables tomaximize the stage characteristics. Aeromechanical con-siderations also play a key role in the design of blades.Vibratory constraints are introduced based on the flut-ter and forced response criteria. The margins of thefirst three frequencies of the blade are set to be at least3–5% in the lower and upper bands of the Campbelldiagram. This condition generate forbidden ranges forspeed: 9000–10,000, 10,000–11,000, . . . 16,000–17,000.In addition, the flutter criterion should be satisfied inboth operational (classical theory) and stall regions basedon the recommended reduced frequencies (Lubomski,1980). Thus, using the above-mentioned parameters, theobjective function (OF) created by mass flow and effi-ciency can be written as:
OF = ð1ηo
ηb+ ð2
τo
τb(1)
where ðk is defined as a weighting coefficient with equalvalues that indicate the importance of each parameter,ηo/ηb is the efficiency ratio, and τo/τb is the mass ratio.
The optimization process is conducted based on theflowchart in Figure 5. Owing to the proper range ofthe main blade efficiency (ηb) in both design and off-design working intervals, the aerodynamic constraintsare adjusted to have a much higher weighting value on
Figure 7. Fitness value versus number of generations.
Figure 6. Network training and validation of efficiency calculation.
ENGINEERING APPLICATIONS OF COMPUTATIONAL FLUID MECHANICS 897
mass flow. The idea takes advantage of rough meta-models. The ANN is trained using 3D CFD analyses,which were created by a DOE technique. The type ofANN employed in this study is a feed-forward back-propagation network with 15 hidden layers and one out-put neuron. Based on experience, the acceptable range ofthe geometric parameters in training the network is setto be within a 5% deviation of the main blade parame-ters. The optimization cycle is started with the presentedapproximated function values (Equation 1) by the ANN,
and the GA is used to find the optimized point of theapproximated function and validated with 3D CFD solu-tions to update the ANNmodel.
6. Results and discussion
The results of the optimization procedure are presentedin this section. Based on the DOE matrix selected here,64 experiments were performed, of the Latin hypercubetype. In the first step, based on 70% of the experiments,
Figure 9. Aerodynamic comparison of the main and optimized blades.
898 M. GHALANDARI ET AL.
the neural network was trained; then, the remaining 30%of the datawas evaluated by the validation of the network.The ANN results for efficiency show 99.9% precisionin the training procedure and 0.09% deviations in datavalidations of the ANN (Figure 6).
In the optimization process, each time the GA is exe-cuted and the network is used as a tool for evaluating
the produced geometry, two elite geometries are com-pared to 3D solutions to estimate the accuracy of thenetwork results, and the results of the 3D solution andthe network estimation are compared. In the event ofan error of more than 0.5%, the 3D solution results areadded to the database and the network is retrained. Afterreducing the error to less than 0.5%, the 3D calculations
Figure 10. Comparison of main and optimized blade characteristics.
ENGINEERING APPLICATIONS OF COMPUTATIONAL FLUID MECHANICS 899
are completely eliminated from the optimization cycleand the algorithm continues with optimization using theneural network only. Optimization is followed up by a50-generation GA, which exists in each generation of 100members (Figure 7). The fitness value started at nearly 2and reached a value less than 0.2. The results are assessedby CFD and FEM in the optimization loop and compared
Table 3. Structural mesh study.
FEM calculation
Mesh Mesh element Mode 1 (Hz)
1 10,000 369.52 40,000 354.13 90,000 353.2
Note: FEM = finite element method.
with the main blade characteristics to obtain the bestgeometry of the blade.
6.1. Aerodynamic analysis
The aerodynamic study takes advantage of the CFD anal-ysis and is based on the loosely coupled method. Spe-cific aerodynamic loading of the blades is calculated bydefinition of the wall boundary conditions and fluiddomain analysis. The boundary condition consists of theinlet and outlet boundaries and cyclic symmetry con-ditions. CFD calculations are conducted based on theK − ε model and the efficiency value is controlled within± 0.5% of the main working interval. A mesh study isconducted to prove the adequacy of convergence valuesfor the CFD calculation (Figure 8).
Figure 11. Blade root and hub stress level.
Figure 12. Campbell diagram of the blade.
900 M. GHALANDARI ET AL.
Figure 9 illustrates the velocity domain around theairfoil in the tip of the blade. Based on the CFD calcu-lation, separate construction of the optimized blade givesa smaller area than the main blade.
From the results (Figure 10), an increased mass flowvalue of at least 4% in both design and off-design oper-ational conditions is highlighted. In addition, the effi-ciency values show a 0.12% decrease in design and 0.08%increase in off-design compared with the main blade.
6.2. Structural analysis
Static and dynamic structural analyses were conductedusing prestressmodal analysis in the optimization frame-work (Shen et al., 2009). Mesh convergence valueswith the SOLID186 element for FEM structural analysisof the fundamental frequency used 40,000 element values(Table 3).
Based on the safe-life design theory, a safety factorequal to 1.5 versus the strength of the blades is con-sidered. Here, the condition of maximum stress, whichgenerally occurs in the hub area, does not represent alimiting factor in the optimization procedure because
Table 4. Optimized and main blade frequencies at 9000 rpm.
Main blade Optimized blade
Mode 1 Mode 2 Mode 3 Mode 1 Mode 2 Mode 3
Frequency (Hz) 412.2 1513.2 1841.4 438.5 1543.1 1868.6
Table 5. Stall flutter investigation.
Reduced frequency Mode
0.527 1 Main blade1.5 20.54 1 Optimized blade1.56 2
the maximum von Mises stress is always below 500MPa(Figure 11).
An appropriate interval speed to satisfy the forcedresponse, as constructed by the Campbell diagram, needsto be checked based on all the criteria mentioned in theset-up phase. As illustrated in Figure 12, the intervalspeed is increased. The amount of increase for the firstthree frequency modes is about 1.9% (Table 4). There-fore, the forced response investigation prove that theoptimized blade is free from resonance conditions inthe mentioned operating range, with regard to AmericanPetroleum Institute (API) recommendations (Yaozeng,Wenxiang, Yi, & Liya, 2016).
In addition, the vibrational constraint, which formsthe main critical aspect of this study, is based on flut-ter speed prevention in both the working interval andstall regions, and is usually defined based on the reducedfrequency as (Whitehead, 1965, 1966, 1973):
λ = bCω
V∞(2)
where λ is reduced frequency, C is chord distance, ω isfrequency, and V∞ is the infinite velocity of the bladeupstream. The stall flutter can be postponed based on theexperimental recommendations. Indeed, the suggestedempirical reduced frequency values for torsional andflexural modes should have values larger than 1.5 and0.35, respectively (Kielb & Kaza, 1983; Lubomski, 1980;Mikolajczak et al., 1975; Smith & Yeh, 1963).
However, themain blade did not satisfy the stall fluttercriteria for torsional modes, whereas the optimized blademet the reduced frequency criteria (Table 5).
Classical flutter instability is predicted by a semi-analytical in-house code based on Whitehead’s theory(Whitehead, 1965, 1966, 1973). To minimize the cost ofthe computations, the instability condition is evaluatedfor a 2D section located at 75% blade length (Figure 13).
Figure 13. Typical section of 75% airfoil.
ENGINEERING APPLICATIONS OF COMPUTATIONAL FLUID MECHANICS 901
Figure 14. Comparison of the eigenvalue part of the presented model and for ζ = 0 (Kielb & Kaza, 1983).
Figure 15. Comparison of main and optimized classical flutter speed estimation in three reduced frequencies.
The aeroelastic analysis is verified by a 2D typicalbeam, known as NASA Test Rotor 12, the parameters ofwhich are given in Kielb and Kaza (1983). As shown inFigure 14, there is good agreement between the resultsof the current estimation of torsional mode in k = 0.642and those predicted by Kielb and Kaza (1983).
The allowable limit of reduced frequency avoidance(Kf = 1/γ ) based on the classical flutter experiments isdefined in range 0.4–0.75 (Lubomski, 1980; Whitehead,1973). Calculations show that the classical flutter speedcondition in the optimized blade is improved (Figure 15).
Figure 16 illustrates the real parts of the complex con-jugate form of each interblade phase angle of the bladesin a row of the compressor. As shown in this figure, flut-ter instability happened at k = 0.488 reduced frequency.The results also imply that a 5.7% improvement in the tor-sional mode in the optimized blade (Figure 17), Table 6)is achieved (Table 7).
Figure 16. Variation of damping versus reduced frequency.IBPA = interblade phase angle.
902 M. GHALANDARI ET AL.
Figure 17. Optimized blade shape of the first row compressortest rig.
Table 6. Cord length and twist angle distribution of the NACA 65airfoil along the main blade span.
Blade span (r/R) Chord variation (%) Dislocation of the sections (mm)
An optimization procedure based on theMDO approachin the first stages of the design of the a new compressortest rig was defined. Blade geometry was parameterizedto have the best aeromechanical performance. Limita-tions on the stacking line position, or in other words,taper and sweep, play an important role during optimiza-tion.Using 3D simulations, the 3D shape of the optimizedblade was generated. Aerodynamic performance, stresslevel and aeroelastic behavior of the blade were set asthe aeromechanical limitations. After optimization, theresults showed that the coupled behavior of the bladeand aerodynamic performance improved by about 5.7%and established the best shape of the blade geometry(Figure 9). Consideration of the nonlinearity of the cou-pled domain in the design process and its effects on the
quality of the design optimizations can be suggested forfuture investigations.
Disclosure statement
No potential conflict of interest was reported by the authors.
Funding
This research has been supported by the R&D Department ofMAPNA Group.
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Appendix
The linear form of the aerostructural equation is represented asfollows (Whitehead, 1965, 1966, 1973):[
m mbxθ
mbxθ IP
]{hθ
}+
[ch 00 cθ
] {hθ
}+
[kh 00 kθ
]{hθ
}
={LM
}(3)
where h is the transverse displacement, θ is the torsional dis-placement, IP is the mass moment of inertia, m is the mass ofthe system, mbxθ is the elastic axis distance from the centerof mass, ch is flexural damping, cθ is torsional damping, kh isflexural stiffness, kθ is torsional stiffness, and L and M are thelift and moment of the system, respectively. The aerodynamicformulations emerged fromTheodorsen andMutchler’s (1935)unsteady theory, which is presented for a cascademodelwith aninfinite number of blades. Small amplitude and constant phasebetween the blades are the main assumptions of the presentedmodel, and the lift and moment distribution around the elasticaxis is formulated as:
L = πρb3ω2N−1∑r=0
[lhhr
hθr
b+ lhθrθθr + lwhr
]ei(ωt+βrs)
M = πρb4ω2N−1∑r=0
[lθhr
hθr
b+ lθθrθθr + lwθr
]ei(ωt+βrs) (4)
where ρ is air density,U is infinite velocity, and the coefficientslwhr and lwθr are aerodynamic force functions which highlightthe weak impacts of upstream flow. The other aerodynamiccoefficients involving lhhr , lhθr , lθhr , lθθr , lwhr and lwθr can beintroduced as:
lhhr = 2ikCFq
lhθr = 2k2
(CFθ − iληCFq)
lθhr = 4ik
(CMq − ηCFq)
lθθr = 4k2
(CMθ − ηCFθ − iληCMq + iλη2CFq)
lwhr = −2wr
k2U(eiληCFw)
lwθr = −4wr
k2U(eiληCMw − ηeiληCFw)
λ = 2k, η = 1 + a2
, k = bwU
wherewr is the amplitude of the sinusoidal wake. The coeffi-cients CFq, CFθ , CMq, CMθ , CMw and CFw are related to reducedfrequency (k), interblade phase angle (βr), pitch distance tochord ratio (s/c), damping ratio ζ of both torsional and flexuralmodes, and location of the elastic axis (a). Based on the sinu-soidal displacement of each degree of freedom, the state spaceof the above equation (4) for the Sth blade can be rewritten as:
−[
ms msbxθsmsbxθs IPs
]ω2
{hsθs
}
+[(1 + 2iζhs)msω
2hs 0
0 (1 + 2iζθs)IPsω2θs
]{hsθs
}
=
⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩
πρb3ω2N−1∑r=0
[lhhr
hθr
b+ lhθrθθr + lwhr
]ei(ωt+βrs)
πρb4ω2N−1∑r=0
[lθhr
hθr
b+ lθθrθθr + lwθr
]ei(ωt+βrs)
⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭(5)
By multiplying both sides of Equation (5) into the stiffnessmatrix, the Equation (5) can be written in the following form:
− μs
⎡⎢⎢⎢⎣
1μsω
2hs(1 + 2iζhs)
xθs
μsω2hs(1 + 2iζhs)
xθs
μsr2θsω2θs(1 + 2iζθs)
r2θsμsr2θsω
2θs(1 + 2iζθs)
⎤⎥⎥⎥⎦
{hsbθs
}
+ 1ω2
[1 00 1
]{hsbθs
}
=
⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩
N−1∑r=0
[lhhr
hθr
b+ lhθrθθr + lwhr
]ei(ωt+βrs)
μsω2hs(1 + 2iζhs)
N−1∑r=0
[lθhr
hθr
b+ lθθrθθr + lwθr
]ei(ωt+βrs)
μsr2θsω2θs(1 + 2iζθs)
⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭(6)
where
μs = ms
πρb2, rθs = IPs
msb2, xθs = Sθ s
msb
904 M. GHALANDARI ET AL.
Using the non-dimensional form of the following parameters,the equation of motion can be obtained as:
γhs = ωhsω0
, γθs = ωθs
ω0, γ =
(ω0
ω
)2
− μs
⎡⎢⎢⎢⎣
1μsγ
2hs(1 + 2iζhs)
xθs
μsγ2hs(1 + 2iζhs)
xθs
μsr2θ2γ2θs(1 + 2iζθs)
r2θsμsr2θ2γ
2θs(1 + 2iζθs)
⎤⎥⎥⎥⎦
{hsbθs
}
+[γ 2 00 γ 2
]{hsbθs
}
=
⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩
N−1∑r=0
[lhhr
hθr
b+ lhθrθθr + lwhr
]ei(ωt+βrs)
μsγ2hs(1 + 2iζhs)
N−1∑r=0
[lθhr
hθr
b+ lθθrθθr + lwθr
]ei(ωt+βrs)
μsr2θ2γ2θs(1 + 2iζθs)
⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭(7)
By definition of {X}eiωt = [E]{Y}eiωt , which relates theSth mistuned blade motion to the summation of all possi-
ble interblade phase modes({
hs/bθs
}eiωt = ∑N−1
r=0
{hai/bθai
}
ei(ωt+βrs)), the matrix form of Equation (7) for a row of blades
is obtained as:
([P] − [I]γ ){Y} = −[E]−1[G][E]{[AD]} (8)
The stability condition of the aeroelastic system can be deter-mined by calculation of the eigenvalue problem of the right-hand side of Equation (8). Positive values of the real part of thefrequencies are specified as the flutter instability condition atdifferent velocities.
