Mistuning blade-disc vibration characteristics in multi-field coupling load based on Kriging model Wenjun Yang1, Huiqun Yuan1, 2* and Hongyuan Zhang1 1School of Mechanical Engineering & Automation, Northeastern University, Shenyang, China 2College of Science, Northeastern University, Shenyang, China *Corresponding author: College of Science, Northeastern University, NO. 3-11, Wenhua Road, Heping District, Shenyang, P. R. China Abstract: In this research a new approach is put forward that mistuning vibration characteristics of aerodynamic and structural coupling are solved based on Kriging interpolation method. Take a compressor blade-disc system as the research object, 3D flow field and mistuned blade-disc models are established. Considered the interaction of stator-rotor blade rows, compressor flow characteristics are simulated with the numerical method. Through the combination of static frequency test, bisection method and numerical simulation, mistuning parameters are identified. By the interpolation method of Kriging model, load transfer of aerodynamic pressure and blade deformation is achieved successfully. Then the paper analyzes the distribution law of aerodynamic load on compressor blade surface. In addition, the effects of mistuning and aerodynamic pressure are discussed on vibration characteristics of blade-disc system. The results show load transfer of aerodynamic pressure and blade deformation has a higher precision based on Kriging model, it can meet the calculation requirement for aerodynamic and structural coupling dynamics. Unsteady aerodynamic pressure on blade surface fluctuates periodically, and dominant frequencies are manly at frequency doubling of stator-rotor interaction, especially at one time frequency (1×f0). In the interaction period T, variations of aerodynamic load on pressure and suction surfaces take the contrary trend, magnitude and pulsation amplitude on pressure surface are far greater than that on suction surface. By the effects of mistuning and aerodynamic pressure, it makes the vibration nonuniformity of blade-disc sectors increase. The maximal displacement fluctuates obviously, and the vibration of blade-disc is enhanced seriously. The research provides the theoretical basis for dynamics design of compressor blade-disc rotor system. Keywords: compressor rotor system; unsteady flow field; Kriging interpolation; aerodynamic pressure; mistuning vibration 1 INTRODUCTION

Compressor is one of the key components in aero-engine, the mistuning of blade-disc system is caused by many factors in the actual production process, such as manufacturing error, uneven material, uneven wear, subjective design for chatter suppression and so on. The phenomena of coupling vibration and vibration localization are induced by the mistuned blade-disc, and they seriously affect the operating stability of compressor blade-disc system. Currently engineers and scholars have taken more attention to the effects of mistuning factor. Ewins et al. [1] predicted the natural frequencies of a bladed disk with various packeting arrangements, and observed the phenomenon of mode localization by experiments. Hodge [2] applied the concept of vibration localization to structural dynamics, and researched the characteristic of mistuning vibration based on the lumped parameter model. Combining with modified perturbation method and modal analysis method, Wei et al. [3, 4] solved the natural frequency and vibration of mistuned blade-disc, and then steady response and vibration localization were discussed. Pierre and Murthy [5] presented the effects of small assembly mistuning on aeroelastic modes with aerodynamic coupling between blades. Judge et al. [6] developed an experiment to investigate

the effects of random blade mistuning on the forced dynamic response and mode localization phenomena. Yoo et al. [7] chosen a simplified model for mistuned cyclic structures, and analyzed the phenomena of vibration localization when undertaking external harmonic force.

In the research of blade-disc vibration, aerodynamic force is mostly simplified as the point load acted on the blade. Thus, it can't calculate the coupling effect of gas and structure. In order to predict the response results accurately, the effect of actual aerodynamic force is vital to be considered. Basu and Griffin [8] developed a model of limiting aerodynamic and structural coupling, and studied the effect of mistuning on bladed disk vibration. Silkowski et al. [9] presented a computational fluid dynamics tool for aeromechanical analyses, the forced response and flutter of mistuned blade-disc were demonstrated. Bleeg et al. [10] provided insight into how alternate blade mistuning affects aerodynamic coupling and flutter characteristics with a new reduced-order aeroelastic model. Sladojevic and Petrov et al. [11] exhibited the results of a study looking into changes in the forced response levels of bladed disc assemblies subject to both structural and aerodynamic mistuning. Miyakozawa et al. [12] discussed the effects of aerodynamic asymmetric perturbations on forced response of bladed disks, and found maximum blade amplitudes were significantly influenced by the perturbation of unsteady aerodynamic forces. Kielb et al. [13] studied the beneficial effect of mistuning on flutter, it is shown to be greatly inhibited by the inclusion of structural coupling. According to the current researches, several valuable results have been achieved with the simplified aerodynamic load. For simulating the aerodynamic action further, it is essential to establish the mistuned blade-disc system of aerodynamic and structural coupling more accurately.

In the analysis of coupling field dynamics, load transfer on coupling interface is the key problem [14, 15]. Grid node model is the basis of data storage and numerical analysis. Load transfer between different physical fields can be achieved by the relationship of coupling interface nodes. In general, the coupling interface grids do not match, so data fitting methods are needed to complete the load transfer. Goura et al. [16] illustrated an interpolation method for the exchange of displacement data between fluid and structural meshes. The method was a local method where it didn’t rely on the information from structural model, but the interpolated smoothness was not well. Then mesh moving techniques were introduced to solve the problem of data transfer. Baker and Cavallo [17] developed an unstructured mesh evolution method on domains that underwent substantial deformation. Dwight [18] and Stein [19, 20] et al. presented mesh moving techniques where the motion of the nodes was governed by the equations of elasticity. For the mesh moving technique, it can’t guarantee the high calculation efficiency and ensure the rebuilding grid in good quality. In the technology of data fitting, Kriging model is the best unbiased estimation model with minimum variance. Lloyd and Atkinson [21] examined the applicability of Kriging in assessing uncertainty. Xia and Wang et al. [22] proved that the Kriging estimator in the domain surrounded by measuring points could meet engineering requirements. It can be seen Kriging model has high enough credibility in data fitting, but there are few applications of Kriging model in coupling field dynamics.

In our research a new approach is put forward that mistuning vibration characteristics of aerodynamic and structural coupling are solved based on Kriging interpolation method. Take a compressor blade-disc system as the research object, 3D flow field and mistuned blade-disc models are established. Considered the interaction of stator-rotor blade rows, compressor flow characteristics are simulated with the numerical method. Through the combination of static frequency test, bisection method and numerical simulation, mistuning parameters are identified. By the interpolation method of Kriging model, load transfer of aerodynamic pressure and blade deformation is achieved successfully. Then the paper analyzes the distribution law of aerodynamic load on compressor blade surface. In addition, the effects of mistuning and aerodynamic pressure are discussed on vibration characteristics of blade-disc system. 2 PHYSICAL MODELS AND ANALYSIS PROCESS

2.1 Physical models

This research takes the compressor flow field and structural models as the calculation domains. Mistuned blade-disc system is selected as structural domain. For considering the interaction of stator-rotor blade rows, the channel of former stator and downstream rotor is selected as flow field domain. Here the number of stator blades is 42, and the number of rotor blades is 38. With the professional preprocessing tool of commercial CFD software, structured hexahedral grid is generated in flow field domain. Total number of elements is 884034, total number of nods is 1005952. After checking, it shows grid aspect ratio is less than 5, orthogonality is greater than 10, and extension ratio is less than 1000. So there confirms the quality of grid is good. Structural domain is preprocessed in Ansys software, parts of blade and disc are meshed with the element Solid 185 and element Solid 187 respectively. In the structural domain, total number of elements is 358348, total number of nodes is 1468672. Specific models are as shown in Fig. 1.

(a) (b) (c) Fig. 1 The models of computation: (a) Models of structure and flow field; (b) Mesh of rotor and stator

flow field; (c) Mesh of mistuned blade-disc system.

Compressor stator-rotor interface is processed with sliding mesh. RNG k-ε turbulence model and implicit coupling solving method are adopted [23, 24]. Compressor operating speed is 11383rpm. Inlet total pressure is 1.0atm, inlet temperature is 300K, and outlet static pressure is 1.08atm. Solid wall is no-slip and adiabatic, medium is compressible ideal air. Considering rotational speed and time consuming of unsteady calculation, physical time step is set for T/60, and it is 2.31×10-6s. Here T is stator-rotor interaction period, the frequency of stator-rotor interaction f0=1/T.

For the structural domain, blade material is titanium alloy TA11, the density is 4370kg/m3, Poisson's ratio is 0.3. Disc material is titanium alloy TC17, the density is 4680kg/m3, elasticity modulus is 112GPa, Poisson's ratio is 0.3. Nodes in the end of hub tube are all constrained when solving the structure domain.

In order to make the model much closer to engineering practice, the mistuning of blade-disc is considered. In our research, disturbance coefficient jP of blade elasticity modulus is introduced to simulate the mistuning of blade-disc, it can be expressed as:

0 1 , 1, 2, ,j jE E P j N (1)

Where jE is the mistuning elasticity modulus, 0E is the desired elasticity modulus, N is the number of blade-disc sector. It is assumed that the disc is tuned, only the mistuning of blade structure is considered. Disturbance coefficient jP reflects the equivalent elasticity of blade mistuning, although it can't absolutely describe the mistuning of blade property including geometry size, mass, etc.

For the identification of mistuning parameter jP , a method is presented based on the combination of static frequency test, bisection method and numerical simulation [25, 26]. By the method, the research predicts the mistuning response of practical structure. Relationship between mistuning elasticity modulus and blade first-order frequency is confirmed, it is the

linear relationship approximatively. Mathematical expression is obtained by linear fitting, the form is as follows:

111606 225126 , 1, 2, ,j jE j N (2)

Where j is the first-order bending static frequency of blade, it is obtained by the static

frequency test. According to the equation (2), the mistuning elasticity modulus jE can be achieved. 2.2 Control equations (1) Structural mechanics equation

Dynamic response of blade structure is generally expressed as: Mu Cu Ku F t (3)

In the equation, M is the matrix of mass, C is the matrix of damping, K is the matrix of stiffness, u is the displacement of blade structure, u is the velocity of blade structure, u is the accelerated speed of blade structure, F t is aerodynamic force acting on the blade surface.

According to the experience [27], blade aerodynamic force can be simplified as simple harmonic excitation. It is assumed that:

i i imax 0e e et tu u u (4)

i i imax 0e e et tF F F (5)

Where maxu is the amplitude of displacement, maxF is the amplitude of aerodynamic force, is phase angle of displacement, is phase angle of aerodynamic force, is the frequency of vibration.

Substituting equation (4) and equation (5) into equation (3), we obtain,

20 0iK M C u F (6)

By solving equation (6), dynamic response under the action of aerodynamic load can be achieved. Assuming that the displacement of solid wall in coupling interface is equal to the displacement of fluid surface, boundary condition is obtained through flow field calculation.

(2) Fluid dynamics equation The flow of fluid is controlled by the physical conservation laws. Basic conservation laws

include the law of mass conservation (continuity equation), the law of momentum conservation (Navier-Stokes equation), the law of energy conservation (energy equation) and the equation of constituent mass conservation (species equation). These equations reflect the nature of conservation in unit time and unit volume, although variable numbers of these equations are different. Assuming that is generic variable, the form of control equations can be expressed as:

di u di ra St

g d

(7)

Where

t

is the transient item, di u is the convective term, gi add r is the

diffusion term, and S is the source item. Compressor fluid domain needs to satisfy the continuity equation, Navier-Stokes equation,

energy equation and species equation. Boundary condition is that the displacement of solid wall in coupling interface is equal to the displacement of fluid surface, and it is obtained through flow field calculation. The movement of compressor internal fluid belongs to 3D unsteady, rotary and irregular motion, hence turbulence model must be considered.

This research chooses the RNG k-ε turbulence model [28, 29], control equations of turbulent energy and turbulent dissipation rate are as follows:

ti k

i j k j

kk ku G

t x x x

(8)

2

1 2t

i ki j j

u C G Ct x x x k k

(9)

Where is the air density, is the coefficient of fluid dynamic viscosity, k is the turbulent energy, is the rate of turbulent dissipation, t is the coefficient of turbulent

viscosity, 2

t

kC

. k and are respectively Prandtl numbers of turbulent energy k and

turbulent dissipation rate , where 1.0k , 1.3 . C is relate to turbulent viscosity

coefficient t , 1C and 2C are the empirical constants of turbulence model, here 0.09C ,

1 1.44C , 2 1.92C . 2.3 Analysis process of coupling field dynamics

In the analysis of mistuning vibration, main program is divided into five parts, including the CFD simulation of compressor flow field, the identification of blade mistuning parameter, load transfer of aerodynamic pressure based on Kriging model, the analysis of aerodynamic and structural coupling dynamics, mesh update of flow field based on assumption elastic method. Detail steps of the process are shown in Fig. 2.

Fig. 2 Analysis process of compressor aerodynamic and structural coupling dynamics

In the analysis, values of calculation parameters are presented in chapter 2.1. The process of simulation is as follows:

(1) CFD model of compressor flow field is built based on Gambit software. (2) CFD simulation of flow field is carried out based on Fluent software. Mesh file of flow

field is read into Fluent, the software will perform various checks and report the mesh quality. Then boundary conditions are set, including rotational speed, inlet total pressure and temperature, outlet static pressure, solid wall and fluid material, etc. In addition, sliding mesh method and RNG k-ε turbulence model are selected for analyzing unsteady flow field.

(3) Combining static frequency test, bisection method and numerical simulation, mistuning parameter of blade-disc system is identified.

(4) Finite element model of mistuned blade-disc is built based on Ansys software. In Ansys preprocessor, parts of blade and disc are meshed with the element Solid185 and element Solid187 respectively. Meanwhile structure materials are defined, including density, mistuning elasticity modulus, Poisson's ratio and so on.

(5) Load transfer of aerodynamic pressure on blade surface is achived from flow field to structure field based on Kriging model.

(6) The analysis of aerodynamic and structural coupling dynamics is carried out based on Ansys software. In the boundary condition, nodes in the end of hub tube are all constrained. Aerodynamic pressure achieved by Kriging interpolation and the load of centrifugal force are applied. Then aerodynamic and structural coupling dynamics of mistuned blade-disc system are solved.

(7) Blade deformation is transferred from structure field to flow field based on Kriging model.

(8) Mesh update of flow field is achieved based on assumption elastic method. Node displacement on blade surface is applied as the initial condition, and the nodes of no displacement in the flow channel are constrained. The grid of flow field is assumed as elastic body. With the static analysis based on Ansys software, each node displacement of flow channel is obtained. According to the output of each node displacement, original 3D coordinates of grid nodes are modified, and mesh update of flow field is achieved.

(9) Iterative simulation is repeated, and until the maximum value of displacement deviation between two iterative simulations is less than the given precision e ( 610e ), the iterative calculation is terminated.

(10) Postprocessing and output of the analysis results. 3 LOAD TRANSFER BASED ON KRIGING METHOD 3.1 Data fitting theory of Kriging model

Load transfer on coupling interface is the main problem of aerodynamic and structural coupling analysis. Based on Kriging model in this research, aerodynamic pressure and blade deformation are successfully transferred between flow field and blade structure.

Originated in geostatistics, Kriging model is an unbiased estimation model with minimum variance [30, 31]. It can estimate the load value of unknown point more accurately. In Kriging model, the relationship between function value and independent variable is expressed with polynomial and random distribution. The form is as follows:

,y x F x z x (10)

Where ,F x is the regression model, 1 2,T

pF x f x f x f x f x .

z x is the statistics random process when average value is 0 and variance is 2 . Covariance

matrix of z x indicates the local outlier degree, the form is as follows:

2cov , ,i j i jz x z x R R x x

(11)

Where R

denotes the correlation matrix, it is n n order symmetric positive definite

diagonal matrix, here n is the number of sample points. ,i jR x x is the correlation function

between two arbitrary sample points ix and jx , the basic form of correlation function is expressed as:

1, ,

m k ki j k i jk

R x x R x x

(12)

Where m is the number of design variables, k denotes the unknown correlation parameter, k ki jx x denotes the k order element distance between ix and jx .

By derivation, predicted value y x is obtained at interpolation point x :

1ˆ ˆˆ Ty x r x R y f (13)

Where y is n -dimensional column vector, it contains visual response value of each design point. f is n -dimensional column vector, it can be simplified as the unit column vector when f x is constant in equation (10). Tr x is n -dimensional column vector, it denotes the

correlation between observation point x and sample points 1 2, , , nx x x , the form is as follows:

1 2, , , , , ,TT

nr x R x x R x x R x x (14)

According to the following formula, is estimated:

11 1ˆ T Tf R f f R y (15)

When f x is constant, is simplified as a scalar. The variance estimator 2 is as follows:

1

2ˆ ˆ

ˆ

T

y f R y f

n

(16)

The maximum likelihood estimation of correlation parameter k in formula (12) is obtained

by formula (17), here 2 and R are both the function of k , 0k .

2ˆln ln

2

n R (17)

By solving the above formula which is the k -dimensional unconstrained nonlinearity optimization problem, the optimal fitting Kriging model can be achieved. 3.2 Comparison and verification of load transfer results

In the analysis of aerodynamic and structural coupling dynamics, there are mainly two types of loads including aerodynamic pressure and blade deformation. The position of nodes on blade surface is determined by x, y and z coordinates in three dimensional space. Each node has the data information of aerodynamic pressure and blade deformation. Hence, it involves four dimensional interpolation problem for a single type of load transfer. In order to identify the precision of load transfer based on Kriging model, response surface methodology (RSM) is adopted for comparison. RSM is the comprehensive experiment technology based on statistics, and it is a method to solve the relationship between input variable and output response in complex system [32, 33]. For data fitting applications, RSM has a strong adaptability.

Aerodynamic pressure and blade deformation on blade surface have been obtained through the numerical simulation. Then 50 nodes are chosen randomly from blade surface as the interpolation samples, other nodes are chosen as the given points. Methods of Kriging model and RSM are respectively applied to calculate the load values. Blade surface loads of numerical simulation are defined as the accurate values, relative errors of interpolation results are obtained, as shown in Fig. 3-Fig. 4.

             

(a) (b) Fig. 3 The comparison of load transfer accuracy on pressure surface: (a) Aerodynamic pressure, (b) Blade

deformation

               

(a) (b) Fig. 4 The comparison of load transfer accuracy on suction surface: (a) Aerodynamic pressure, (b) Blade

deformation

Fig. 3 and Fig. 4 exhibit the comparison results of load transfer accuracy on pressure and suction surfaces respectively. It is observed that relative error of Kriging model is within 5%, no matter whether it is the transfer accuracy of aerodynamic pressure or blade deformation. Compared with RSM, Kriging model has a higher precision to load transfer. Hence, Kriging model is adopted in this research. 3.3 Load transfer of aerodynamic pressure and blade deformation

According to original aerodynamic pressure and blade deformation on the coupling interface, Kriging model is established with the dacefit function. Predictor function is adopted, aerodynamic pressure and blade deformation are successfully transferred between flow field and blade structure based on Kriging model. k is the relevant parameter of Kriging model, and it is optimized with the optimization algorithm to improve the transfer precision of aerodynamic pressure and blade deformation on the coupling interface.

(1) Aerodynamic pressure transfer from flow field to blade structure Aerodynamic pressure of flow field on the coupling interface has been obtained by the CFD

simulation. Based on Kriging model, transfer program of aerodynamic pressure is compiled. Through load transfer of aerodynamic pressure, aerodynamic pressure of structure field on the coupling interface is achieved. The process is shown in Fig. 5.

Fig. 5 Program process of aerodynamic pressure transfer

Load transfer of aerodynamic pressure on the coupling interface is achieved based on Kriging model. The results of flow field and structure field are shown in Fig. 6-Fig. 7.

40

0300

260

220

18060 80 100 120 0.8

0.9

1.0

1.1

1.2

×105

x/mm

z/m

m

P/Pa

y/mm

40

0300

260

220

18060 80 100 120 0.8

0.9

1.0

1.1

1.2

×105

x/mm

z/m

m

P/Pa

y/mm

(a) (b) Fig. 6 Aerodynamic pressure on blade pressure surface: (a) Flow field, (b) Structure field

40

0300

260

220

18060 80 100 120

0.7

0.8

0.9

1.0

1.1×105

x/mm

z/m

m

P/Pa

y/mm

z/m

m

(a) (b) Fig. 7 Aerodynamic pressure on blade suction surface: (a) Flow field, (b) Structure field

Fig. 6-Fig. 7 display the distributions of aerodynamic pressure on the coupling interface before and after interpolation. With the simulation of compressor folw field, aerodynamic pressure on blade surface is obtained as shown in Fig. 6(a)-Fig. 7(a). Based on blade surface aerodynamic pressure of folw field and Kriging interpolation method, blade surface aerodynamic pressure of structure field is achieved as shown in Fig. 6(b)-Fig. 7(b). Meanwhile interpolation results are written into text file according to the node order, they are called and loaded in the analysis of aerodynamic and structural coupling. (2) Blade deformation transfer from structure field to flow field

Node displacement of structure field on the coupling interface has been obtained by the analysis of aerodynamic and structural coupling. Based on Kriging model, transfer program of blade deformation is compiled. Through load transfer of blade deformation, node displacement of flow field on the coupling interface is achieved. The process is shown in Fig. 8.

Fig. 8 Program process of structure deformation transfer

Load transfer of blade deformation on the coupling interface is achieved based on Kriging model. The results of structure field and flow field are shown in Fig. 9-Fig. 10.

40

0300

260

220

180 60 80 100 1201.0

1.5

2.0

2.5

3.0

x/mm

S/m

y/mm

×10-3

(a) (b) Fig. 9 Node displacement on blade suction surface: (a) Structure field, (b) Flow field

(a) (b) Fig. 10 Node displacement on blade pressure surface: (a) Structure field, (b) Flow field

Fig. 9-Fig. 10 display the distributions of node displacement on the coupling interface of before and after interpolation. With the analysis of aerodynamic and structural coupling, node displacement on blade surface is obtained as shown in Fig. 9(a)-Fig. 10(a). Based on blade surface node displacement of structure field and Kriging interpolation method, blade surface node displacement of flow field is achieved as shown in Fig. 9(b)-Fig. 10(b). Meanwhile interpolation results are written into text file according to the node order, they are called and loaded in the mesh update of flow field based on assumption elastic method.

Comparison to aerodynamic pressure and blade deformation before and after interpolation, it is shown that aerodynamic pressure and blade deformation between flow field and structure field coincide quite well. Hence, it further illustrates that load transfer based on Kriging model can meet the calculation requirement for aerodynamic and structural coupling dynamics.

4 RESULTS AND DISCUSSION 4.1 Characteristics of unsteady aerodynamic load on blade surface

In order to study the characteristics of blade surface load, aerodynamic pressure on rotor blade is monitored. At the working speed of 11383rpm, stator-rotor interaction frequency ( f0 ) is approximately 7200Hz. In Fig. 11, variation curves of aerodynamic pressure on blade surface are presented.

0 1 2 3 4 5

0.72

0.96

1.20

1.44

×10-3

BP/a

tm

t/s

A: Pressure surface B: Suction surface

A

         0.00

0.06

0.0 0.5 1.0 1.5 2.0 2.5 3.00.00

0.06

2×f0

A/a

tm

1×f0 Pressure surface

Suction surface

2×f0

1×f0

A/a

tm

f/Hz

×104

(a) (b) Fig. 11 Variation curves of aerodynamic pressure on blade surface: (a) Time domain curves, (b)

Amplitude-frequency curves

As shown in Fig. 11, when compressor flow field reaches the steady state, aerodynamic pressure on blade surface is convergent. Fig. 11(a) gives the time domain curves of aerodynamic pressure, it is shown curves begin to fluctuate periodically after 0.003s. Hence, aerodynamic load is the unsteady pulsation pressure. What’s more, the convergent value of aerodynamic pressure on pressure surface (1.09 atm) is far greater than the convergent value on suction surface (0.86 atm). And the pulsation amplitude of aerodynamic load on pressure surface is obviously higher than the pulsation amplitude on suction surface.

In Fig. 11(b), the amplitude-frequency curves of aerodynamic load are plotted. As can be found that peak frequencies of pressure and suction surfaces are consistent, and both are mainly at frequency doubling of stator-rotor interaction. Moreover, one time frequency (1× f0) component takes a dominant position, the amplitudes of double frequency (2×f0) and higher order frequency are smaller. Hence, aerodynamic load on blade surface is mainly affected by the stator-rotor interaction. Furthermore, peak value of aerodynamic load on pressure surface is far greater than peak value on suction surface, it illustrates unsteady flow on pressure surface is even stronger.

According to above analysis, the characteristics of aerodynamic pressure on blade surface have been basically mastered. In order to obtain the distribution details further, it exhibits variation curves of aerodynamic pressure on blade surface at the interaction period T, as shown in Fig. 12.

1.04

1.08

1.12

1.16

T0.75T0.5T0.25T0

P/a

tm

t/s

0.84

0.86

0.88

0.90

T0.75T0.5T0.25T0

P/a

tm

t/s

(a) (b) Fig. 12 Variation curves of blade surface aerodynamic pressure at the interaction period T: (a) Pressure

surface, (b) Suction surface

In Fig. 12, aerodynamic pressure on pressure surface reaches the minimal value at 5T/8, and reaches the maximum value at T. While aerodynamic pressure on suction surface reaches the maximum value at T/2, and reaches the minimal value at T. Except the moments of extreme

points have a little difference, variations of aerodynamic pressure take the contrary trend on pressure and suction surfaces. 4.2 Effects of mistuning and aerodynamic pressure on vibration characteristics

Considering the effect of aerodynamic pressure, vibration characteristics of tuned and mistuned blade-disc system are solved. Then the effect laws of mistuning and aerodynamic pressure are discussed on the vibration characteristics of blade-disc system. The distributions of displacement and von mises stress are obtained, as is shown in Tab. 1-Tab. 2. Here maxS denotes

the maximum displacement, max denotes the maximum von mises stress. 1A and 1B are respectively the parameter values of tuned and mistuned blade-disc system in the condition of ignoring aerodynamic pressure. 2A and 2B are respectively the parameter values of tuned and mistuned blade-disc system in the condition of considering aerodynamic pressure.

Tab. 1 Mistuning effect on blade-disc system vibration

ParametersIgnoring aerodynamic pressure Considering aerodynamic pressure

A1 B1 Deviation A2 B2 Deviation

maxS /mm 0.580 0.630 8.62% 0.607 0.660 8.73%

m ax /MPa 444 447 0.68% 442 445 0.68%

Where deviation is defined as /ii iB A A , 1, 2i . As shown in Tab. 1, maximum displacement in the mistuned blade-disc is obviously larger than that in the tuned blade-disc, the maximum deviation is 8.73%. While maximum von mises stress in the mistuned blade-disc has little different from that in the tuned blade-disc, the deviation is only 0.68%. It indicates the mistuning makes blade-disc vibration more heavily, but von mises stress is little affected by the mistuning.

Tab. 2 Aerodynamic pressure effect on blade-disc system vibration

ParametersTuned blade-disc system Mistuned blade-disc system

A1 A2 Deviation B1 B2 Deviation

maxS /mm 0.580 0.607 4.66% 0.630 0.660 4.76%

m ax /MPa 444 442 0.45% 447 445 0.45%

In Tab. 2, deviation is defined as 12 1/A A A or 12 1/B B B . When the effect of aerodynamic pressure is considered, maximum displacement appears a certain increase, the maximum deviations is 4.76%. While maximum von mises stress changes very little at the effect of aerodynamic pressure, the deviation is only 0.45%. Hence the effect of aerodynamic pressure makes blade-disc vibration much greater, this is similar to the effect of mistuning on blade-disc vibration.

In Fig. 13- Fig. 14, the distributions of maximum displacement and von mises stress are exhibited. Here N denotes the number of blade-disc sector.

0 5 10 15 20 25 30 35 400.50

0.56

0.62

0.68

S max

/mm

N

    

0 5 10 15 20 25 30 35 400.50

0.56

0.62

0.68

S max

/mm

N

   

0 5 10 15 20 25 30 35 400.50

0.56

0.62

0.68

S max

/mm

N

 

(a) (b) (c)  Fig. 13 Maximal displacement distribution of blade-disc sectors: (a) Without mistuning and aerodynamic

pressure, (b) Action of aerodynamic pressure, (c) Action of mistuning and aerodynamic pressure

0 5 10 15 20 25 30 35 40300

350

400

450

500

σ max

/MP

a

N

    

0 5 10 15 20 25 30 35 40300

350

400

450

500

σ max

/MP

a

N

   

0 5 10 15 20 25 30 35 40300

350

400

450

500

σ max

/MP

a

N

(a) (b) (c) Fig. 14 Maximal stress distribution of blade-disc sectors: (a) Without mistuning and aerodynamic pressure, (b) Action of aerodynamic pressure, (c) Action of mistuning and aerodynamic pressure

According to Fig. 13-Fig. 14, it is shown maximal displacement distribution of blade-disc sectors fluctuates heavily after adding mistuning and aerodynamic pressure respectively. While maximum von mises stress doesn't have obvious changes. It reflects the sectors of blade-disc vibrate more unevenly at the effect of mistuning and aerodynamic pressure. For further research on the effect law, Tab. 3 is made at four different types of loads. Ⅰ denotes the load condition of ignoring mistuning and aerodynamic pressure, Ⅱ denotes the load condition of considering mistuning separately, Ⅲ denotes the load condition of considering aerodynamic pressure separately, Ⅳ is at the coupling effect of mistuning and aerodynamic pressure. Here C denotes the value of parameter at differet conditions, deviation is defined as /NC C C Ⅰ Ⅰ ,

.N Ⅱ,Ⅲ,Ⅳ

Tab. 3 Mistuning and aerodynamic pressure effect on blade-disc system vibration

Types of loads maxS m ax

Value/mm Deviation Value /MPa Deviation

Ⅰ 0.580 — 444 — Ⅱ 0.630 8.62% 447 0.68% Ⅲ 0.607 4.66% 442 0.45% Ⅳ 0.660 13.79% 445 0.23%

In Tab. 3, it is observed that maximal displacement is affected by mistuning and aerodynamic pressure more obviously than maximum von mises stress. Compared to the load condition of ignoring mistuning and aerodynamic pressure, the maximal deviation of displacement is 13.79%, and it is at the coupling effect of mistuning and aerodynamic pressure. However, the maximal deviation of von mises stress is only 0.68%. 4.3 Quasi static vibration characteristics at the action of aerodynamic pressure

In the analysis of flow field, aerodynamic load on blade surface is monitored. Then the variation curves of aerodynamic pressure on suction and pressure surfaces are obtained, as is shown in Fig. 15.

0 1 2 3 4 5 6

0.70

1.05

1.40

1.75 A: Pressure surface B: Suction surface

×10-3

121110987654

32

P/a

tm

t/s

1A

B

Fig. 15 Aerodynamic pressure curves of blade pressure and suction surfaces

Here 12 time points are selected in the variation curves, vibration characteristics are further discussed at the effect of mistuning and aerodynamic pressure, specific details are shown in Fig. 15 and Tab. 4.

Tab. 4 Iteration number and time points Number Iteration step Time /s 1 1 2.31185e-6 2 12 2.77422e-6 3 116 2.68175e-4 4 297 6.86619e-4 5 377 8.71567e-4 6 617 0.00143 7 778 0.0018 8 936 0.00216 9 1156 0.00267 10 1533 0.00354 11 1859 0.0043 12 2239 0.00518

Based on the method of Kriging interpolation, load transfer on coupling interface is achieved successfully. Through the analysis of aerodynamic and structural coupling dynamics, vibration characteristics of tuned and mistuned blade-disc are obtained. Variation curves are plotted according to maximal displacement and von mises stress at each time point, as is shown in Fig. 16-Fig. 17.

0 1 2 3 4 5 60.5

0.7

0.9

1.1

1.3

S max

/mm

t/s

Tuned blade-disc Mistuned blade-disc

×10-3

        

0 1 2 3 4 5 6460

470

480

490

500

×10-3

t/s

σ max

/MPa

Tuned balde-disc Mistuned balde-disc

 

     (a) (b) Fig. 16 Quasi-static curves of blade-disc system under the action of aerodynamic pressure: (a) Maximal

displacement, (b) Maximal von mises stress

0 1 2 3 4 50

5

10

15

Max=14.85%

Dev

iati

on /%

t/s

Smax

×10-3

                 

0 1 2 3 4 50

1

2

3

4

5

Max=4.26%

×10-3

Dev

iati

on /%

t/s

max

 

(a) (b) Fig. 17 Deviation curves of blade-disc system under the action of aerodynamic pressure: (a) The maximal

displacement, (b) The maximal von mises stress

In Fig. 16, quasi-static curves of tuned and mistuned blade-disc are presented. Under the action of aerodynamic pressure, maximal displacement and von mises stress of mistuned blade-disc are obviously greater than that of tuned blade-disc. It illustrates the mistuning is one of main sources of blade-disc vibration. According to the deviation curves in Fig. 17, it is shown maximal deviations of displacement and von mises stress are 14.85% and 4.26% respectively. The deviation of maximal displacement is far greater than the deviation of maximal von mises stress, so it can draw a conclusion that the effect of mistuning on the displacement is much stronger than the effect on von mises stress. 5 CONCLUSIONS

In this research a new approach is put forward that mistuning vibration characteristics of aerodynamic and structural coupling are solved based on Kriging interpolation method. Take a compressor blade-disc system as the research object, 3D flow field and mistuned blade-disc models are established. Considered the interaction of stator-rotor blade rows, compressor flow characteristics are simulated with the numerical method. Through the combination of static frequency test, bisection method and numerical simulation, mistuning parameters are identified. By the interpolation method of Kriging model, load transfer of aerodynamic pressure and blade deformation is achieved successfully. Then the paper analyzes the distribution law of aerodynamic load on compressor blade surface. In addition, the effects of mistuning and aerodynamic pressure are discussed on vibration characteristics of blade-disc system. The results show:

1) Compared with the result of RSM, load transfer based on Kriging model has a higher precision. Aerodynamic pressure and blade deformation between flow field and structure field coincide quite well before and after interpolation, it can meet the calculation requirement for aerodynamic and structural coupling dynamics.

2) Aerodynamic load on compressor blade surface is unsteady pulsation pressure, and dominant fluctuation frequencies of aerodynamic pressure are manly at frequency doubling of stator-rotor interaction, especially at one time frequency (1×f0). In the interaction period T, variations of aerodynamic load on pressure and suction surfaces take the contrary trend, magnitude and pulsation amplitude on pressure surface are far greater than that on suction surface.

3) By the effects of mistuning and aerodynamic pressure, it makes the vibration nonuniformity of blade-disc sectors increase. The maximal displacement fluctuates obviously, and the vibration of blade-disc is enhanced seriously. In addition, maximal displacement is affected by mistuning and aerodynamic pressure more obviously than maximum von mises stress. ACKNOWLEDGMENT

The work is supported by the National Science Foundation of China (Grant No. 51275081) and the National Science Foundation of China (Grant No. 51335003). REFERENCES [1] Ewins DJ, Imregun M. Vibration Modes of Packeted Bladed Disks. ASME Journal of Vibration,

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