Bearing parameter identification of rotor-bearing system based on Kriging surrogate model and evolutionary algorithm

Journal of Sound and Vibration 332 (2013) 2659–2671

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Bearing parameter identification of rotor-bearing system based on Kriging surrogate model and evolutionary algorithm Fang Han, Xinglin Guo n, Haiyang Gao State Key Laboratory of Structural Analysis for Industrial Equipment, Dalian University of Technology, Dalian 116024, PR China

a r t i c l e i n f o

abstract

Article history: Received 14 May 2012 Received in revised form 23 December 2012 Accepted 23 December 2012 Handling Editor: H. Ouyang Available online 4 February 2013

Bearing dynamic parameters are important factors governing the vibration characteristics of rotating machinery, but they are usually unknown in the modeling. In this paper, an effective method is proposed to identify the bearing parameters and unbalance information of a rotor-bearing system based on the Kriging surrogate model and evolutionary algorithm (KSMEA). The initial Kriging surrogate model is constructed by the samples of various identification parameters (bearing parameters and magnitude of mass unbalance) and measured unbalance responses, which substitutes the original finite element model. It effectively reduces the computational expense of identification. In order to search for the global optimal solution exactly, one of the evolutionary algorithms, differential evolution (DE) algorithm is employed based on the constructed Kriging surrogate model. The effect on different numbers of samples is discussed to improve the accuracy of the Kriging surrogate model. Both numerical example and experimental results indicate that the proposed method can identify the bearing parameters and unbalance information of rotor-bearing system accurately and reliably. & 2012 Elsevier Ltd. All rights reserved.

1. Introduction In high-speed rotating machineries, such as turbines, compressors and engines, one of the most important factors governing the vibration characteristics of rotating machinery are bearing dynamic parameters. Therefore, the accurate identification of bearing parameters of rotor-bearing system has become increasingly important. Recently, several time domain and frequency domain techniques have been developed for experimental estimation of bearing dynamic parameters, such as impulse, random and synchronous/nonsynchronous unbalance excitation techniques. Tiwari [1,2] had identified residual unbalance and bearing dynamic parameters using impulse response measurement for multidegree-of-freedom (mdof) flexible rotor-bearing system. Two separate identification algorithms have been used to simultaneous estimate the residual unbalance and bearing dynamic parameters in a rigid rotor-bearing system [3]. M.S. De [4] presented a relatively simple feedback strategy for identifying the unbalance parameters of a rotating system with time-varying rotor speed via a pair of active radial control force. Lee and Hong [5] identified the bearing dynamic characteristics by using unbalance response measurements which was developed for rigid rotor systems supported by two anisotropic bearings. Oscar and Luis [6,7] identified the bearing support parameters from recorded transient rotor responses due to impact and imbalance.

n

Corresponding author. E-mail address: [email protected] (X. Guo).

0022-460X/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jsv.2012.12.025

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In past few years, optimization techniques have been introduced in some fields of rotor dynamics, such as fault diagnosis, vibration control, structural parameter identification [8–10] and so on. As an alternative technique in parameter identification, one of the most traditional optimization algorithms is least square method [11,12]. However, in real rotating shaft system, it is impossible to avoid the causes of noises, the method of least square optimizer involves the gradient information and may not guarantee global optimal solution, especially for the problem with noise and some uncertain factor. By contrast, intelligent optimization techniques such as genetic algorithm [13], simulated annealing algorithm [14] and ant colony algorithm [15] achieved a better performance on searching global optimized solution in parameter identification field. To improve identification quality and searching ability of these optimization methods, many modified versions of intelligent algorithm have been reported [16–18], while higher time consuming may be the disadvantage for these intelligent algorithms, due to their iterative strategy and repeated analysis in simulation models during the optimization process. Compared with the traditional simulation methods, such as finite element method (FEM), second-order polynomial response surfaces method and so on, Kriging surrogate model provides an explicit function to represent the relationship between the inputs and outputs with a small initial training sample set in linear or nonlinear system [19]. As a good surrogate method, it reduces the computational expense effectively. Moreover it can also offer an optimization platform to make sure the global convergence. Therefore, Kriging surrogate model has drawn much attention and has been increasingly used in structural optimization problems [20], aerodynamic optimization [21]and engineering design [22]. At the same time, it is also combined with other methods to improve the optimization efficiency [23,24]. However, there are few works on bearing parameter identification of rotor-bearing system using the Kriging surrogate model. In this paper, an effective method based on the Kriging surrogate model and evolutionary algorithm is proposed to construct global approximation in parameter identification of a rotor-bearing system. Different numbers of samples are obtained by FEM to construct the initial Kriging surrogate model. To estimate the actual rotor parameters, Differential Evolution (DE) algorithm is employed [25]. After that, some new temporary optimal results are inserted into the initial sample set to update the Kriging surrogate model, until the Kriging surrogate model is sufficiently accurate and the optimization process converges. Through comparing, the proposed method is more robust to the noise and costs less time than other traditional identification methods. Both numerical and experimental results from the KSMEA are also presented to validate and assess the proposed method. 2. Unbalance response analysis Based on the theory of rotor dynamics, numerical example and experimental model with unbalance response are established using FEM to simulate the actual situation. The FE equations of motion are given below [26]. 2.1. Motion equations of rigid disk The rigid disk is modeled as a four-degree-of-freedom rigid body with the generalized coordinates, and two translations x,y of the mass center in the X,Y directions and two rotations yx, yy of the plane on the X, Y axes, they can be expressed as u1d ¼ ½x, yy T u2d ¼ ½y,yx T

(1)

According to the Lagrange equations, the motion equations of rigid disk are Md u€ 1d þ OJd u_ 2d ¼ Q 1d Md u€ 2d OJd u_ 1d ¼ Q 2d

(2)

where Md is the inertia matrix, O is the spin speed, Jd is the gyroscopic matrix, Q1d and Q2d are corresponding generalized forces. 2.2. Motion equations of elastic shaft The shaft element is modeled as an eight-degree-of-freedom element, two translations and two rotations at both ends of the element, they are u1s ¼ ½xA , yyA , xB , yyB T u2s ¼ ½yA , yxA , yB , yxB T

(3)

The motion equations of shaft element under constant spin speed O condition are given by Ms u€ 1s þ OJs u_ 2s þ Ks u1s ¼ Q 1s Ms u€ 2s OJs u_ 1s þ Ks u2s ¼ Q 2s

(4)

where Ms is the mass matrix including translational inertia and rotational inertia, Js is the gyroscopic matrix, Ks is the stiffness matrix, Q1s and Q2s are the force vectors acting on the shaft element.

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2.3. Motion equations of bearing The fluid film bearings are generally represented by the spring and damper, they present the nonlinear characteristics. In this paper, the bearings can be linearized using the assumption of a small vibration. The motion equations of bearings are represented by two stiffness coefficients and two damping coefficients in FEM, they can be expressed as Cb u_ 1b þ Kb u1b ¼ Q 1b Cb u_ 2b þKb u2b ¼ Q 2b

(5)

where Cb and Kb are the damping and stiffness matrices of the bearing element respectively. 2.4. Motion equations of system From Eq. (2), Eq. (4) and Eq. (5), the equations of each element are established. The equations of other elements are the same as above equations. Integrating the motion equations of disk, shaft and bearing element, the motion equations of rotor system in the fix frame are given as below M1 u€ 1 þ C1 u_ 2 þK1 u1 ¼ Q 1 M1 u€ 2 þ C2 u_ 1 þ K1 u2 ¼ Q 2

(6)

where M1, C1, C2 and K1 are global mass matrix, damping matrix and stiffness matrix, respectively. M1 ¼Md þMs, K1 ¼ Kb þ Ks, C1 ¼Cb þ OJd þ OJs, C2 ¼Cb  OJd  OJs, Q1 and Q2 are the unbalance forces in fixed frame coordinates. Solving the equations, the unbalance responses are obtained. 3. Optimization procedure based on the Kriging surrogate model Kriging is a spatial statistical technique. It was used in mining and geostatistical applications at the beginning [27] and modeled experimental data in multidimensional spaces as a tool [28].The application of the Kriging surrogate model in parameter identification bases on its interpolation accuracy. It can play a role in two aspects in the process of parameter identification, one is that Kriging surrogate model is used to construct the relationship between input and output which is acted as a substitute of the FE model; the other is that Kriging surrogate model serves an identification tool to search for the identification solutions. To construct the Kriging surrogate model, we assume that xn, yn denote n samples of various parameters and the unbalance responses of rotor system respectively, the expressions as follows: 3 2 13 2 1 x1 x12 x13    x1N x 7 6 27 6 x21 x22 x23    x2N 7 6x 7 6 7 6 7 6 6 3 3 3 3 3 7 x1 x2 x3    xN 7 X¼6 (7) 7 6x 7¼6 6 7 6 7 6 ^ ^ ^ ^ ^ 7 4^ 5 4 5 xn1 xn2 xn3    xnN xn 2

2

3

y1 ðx1 Þ 6 6 y2 7 6 y ðx2 Þ 6 7 6 1 6 37 6 7 Y¼6 y1 ðx3 Þ 6y 7¼6 6^ 7 6 6 4 5 4 ^ yn y1 ðxn Þ y1

y2 ðx1 Þ

y3 ðx1 Þ



y2 ðx2 Þ

y3 ðx2 Þ



y2 ðx3 Þ

y3 ðx3 Þ



^

^

&

y2 ðxn Þ

y3 ðxn Þ



yq ðx1 Þ

3

7 yq ðx2 Þ 7 7 7 yq ðx3 Þ 7 7 ^ 7 5 yq ðxn Þ

(8)

The responses yn are modeled by T

yl ðxi Þ ¼ f ðxi Þbl þ zl ðxi Þ

i ¼ 1,2,. . .,n

l ¼ 1,2,. . .,q

(9)

where f(xi) is a linear combination vector of p chosen function; bl is regression parameter; zl (xi) is assumed to be a Gaussian stationary process with zero mean; the covariance matrix of zl(xi) is given as Cov½zl ðxi Þ,zl ðxj Þ ¼ s2l Rðy,xi ,xj Þ

i,j ¼ 1,2,. . .,n

l ¼ 1,2,. . .,q i

2 l

(10)

j

where s is the process variance for the lth component of the responses and R(y,x ,x ) is the Gaussian correlation function with the parameter y, which is defined as the following form: Ri,j ðy,xi ,xj Þ ¼ expðy:xi xj :Þ

(11)

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Giving the responses at an untested sample xnew based on the model in Eq. (9), considering a linear predictor of Y, new y^ l ðxnew Þ ¼ cT Yl

l ¼ 1,2,. . .,q

(12)

The prediction error is T y^ l ðxnew Þyl ðxnew Þ ¼ cT Yl yl ðxnew Þ ¼ cT ðFbl Zl Þðf ðxnew Þbl þ zl ðxnew ÞÞ ¼ cT Zl zl ðxnew Þ þðFT cfðxnew ÞÞT bl

2 6 F¼6 4

f 1 ðx1 Þ



f p ðx1 Þ

^

&

^

n

f 1 ðx Þ



n

f p ðx Þ

(13)

3 7 7 5

Zl ¼ ½zl ðx1 Þ,zl ðx2 Þ,. . .,zl ðxn ÞT

(14)

To make sure the process of simulation has no deviation, we demand that FT cfðxnew Þ ¼ 0

(15)

Under this condition, the mean squared error (MSE) of the predictor can be derived by using Eq.(10)

jðxnew Þ ¼ E½ðy^ new ðxnew Þynew ðxnew ÞÞ2  ¼ E½ðcT Zl zl Þ2  ¼ E½z2l þ cT Zl ZTl c2cT Zl zl  ¼ sl 2 ð1 þ cT Rc2cT rÞ

(16)

rðxÞ ¼ ½Rðy,x1 ,xnew Þ, Rðy,x2 ,xnew Þ,. . .,Rðy,xn ,xnew ÞT

(17)

where

Introducing Lagrange multiplier for minimizing j with respect to c and the constraint of Eq. (15) is Lðc, kÞ ¼ s2l ð1 þ cT Rc2cT rÞkT ðFT cfðxnew ÞÞ

(18)

The gradient of Eq. (18) with respect to c is L0c ðc, kÞ ¼ 2s2l ðRcrÞFk

(19)

from the first order necessary condition for optimality, we get the following condition:      R F c r ¼ T 0 k~ F f

(20)

where we have defined

k~ ¼ 

k 2s2

The solution can be obtained from Eq. (20)

k~ ¼ ðFT R1 FÞ1 ðFT R1 rfðxnew ÞÞc ¼ R1 ðrFk~ Þ

(21)

Substituting the above equation into Eq. (12) gives y^ ðxnew Þ ¼ RT ðrFðFT R1 FÞ1 ðFT R1 rfðxnew ÞÞÞT Yl

l ¼ 1,2,. . .,q

(22)

According to above process, the new relationship between bearing parameters and the unbalance response values can be obtained, this surrogate model as a black box can be used to estimate the unknown information and reflect global and local statistical properties [29]. We can identify the unknown parameters depending on the Kriging surrogate model. After constructing the initial Kriging surrogate model, we evolve the model gradually until its quality is precise enough, and this process can help us to find out the accurate identification parameters of rotor system. According to the evolutionary algorithms, a certain number of evolutionary generations are used as the termination condition. It is disadvantage that the necessary number of evolutionary generations for convergence is unknown. However, the evolution process of the Kriging surrogate model can provide the squared multiple correlation (SMC) as an extra termination condition, SMC is also used to make sure the quality of the Kriging surrogate model. At the same time, the simple different function (SDF) is another termination criterion in this work q P

SMC ¼ 1

l¼1 q P l¼1

SDF ¼

q X l¼1

½y^ l ðxnew ÞylðFEÞ ðxnew Þ2 (23)

½yðFEÞ ðxnew ÞyðTureÞ ðxnew Þ2 l l

9y^ l ðxnew ÞyðFEÞ ðxnew Þ9 l

(24)

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where y^ l and yðFEÞ are the lth component of the response vector of the surrogate model and the results calculated by FEM l are the true values of the rotor unbalance responses. respectively; yðTureÞ l

4. Identification procedure Fig. 1 shows the general identification procedure for determining the unknown parameters, such as bearing parameters, unbalance parameters of rotor-bearing system and so on. At first, the initial Kriging surrogate model is generally described by samples of various rotor parameters X and their responses vector Y. At second, based on the constructed Kriging surrogate model, we start the identification process as follows: 8 find xnew > > > q  > X <  ^ new ÞYtrue Þ ¼ y^ ðxnew Þytrue  minðYðx l (25) l > l¼1 > > > : subject to lb oxnew o ub where Y^ is the simulation response vector, Ytrue represents the measured response vector. lb and up denote the lower and upper bounds. To search for the global optimal solution effectively, the evolutionary algorithm is employed based on the Kriging surrogate model. Comparing to other evolutionary algorithms, DE algorithm uses a greedy algorithm and less stochastic approach in problem solving. DE algorithm combines simple arithmetical operators with the classical operators of recombination, mutation and selection to evolve from a randomly generated starting population to a final solution. It is similar to genetic algorithm, but the mechanisms of mutation and crossover are differences between the two algorithms. In DE algorithm, mutation is an operation that adds a differential vector to a population vector of individuals [30]. For each target vector zi,g, a mutant vector is produced according to the following equation: vi,g ¼ zr1,g þ Fðzr2,g zr3,g Þ

(26)

In Eq. (26), F is the mutation scaling factor which is a positive number, it is usually taken in the range of [0.5,1], where i,r1,r2,r3A{1,2,y,n} are randomly chosen numbers and iar1ar2ar3. n is the number of population. g represents the number of generation. Crossover operation produces trial vector. Crossover is employed to generate a trial vector by replacing certain parameters of the target vector with the corresponding parameters of a randomly generated mutant vector ( vj,i,g if ðrand r CÞ or j ¼ jrand uj,i,g ¼ (27) zj,i,g otherwise In Eq. (27), C is the crossover rate, 0.8 rCr1. j¼ 1,2,yd, where d is the number of parameters to be optimized.

Fig. 1. Flowchart of rotor parameters identification process.

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Selection is the procedure of producing better offspring. DE algorithm uses a greedy algorithm to select the better vector which replaces the target vector in the next generation ( ui,g if f ðui,g Þ rf ðxi,g Þ zi,g þ 1 ¼ (28) zi,g otherwise After completed some generations of mutation, crossover and selection operations, the optimal solution is obtained to rebuild the Kriging surrogate model. The Kriging surrogate model is updated by this point-adding process until it is sufficiently accurate according to SMC or SDF criterion. 5. Simulations To demonstrate the accuracy and reliability of the present method, a numerical study of a rotor-bearing system with three disks is carried out [31] and Fig. 2 shows the FE model with thirteen grids. An unbalance mass is added on disk 2 with magnitude of 200 g mm. The unbalance responses at the 2nd node are selected as simulation measured responses. Details of the rotor-bearing model are given in Table 1. x¼[kxx,kyy,cxx,cyy,u] are chosen as identification parameters, where kxx and kyy are the horizontal and vertical stiffness coefficients, cxx and cyy are the horizontal and vertical damping coefficients, respectively. u is the magnitude of mass unbalance of disk. A certain amount of samples of identification parameters are firstly generated by Latin Hypercube design [32]. The corresponding responses on the 2nd node with the rotating speeds from 200 to 1500 rpm are calculated by EFM. They constructed the initial Kriging surrogate model. After formed the initial Kriging surrogate model, the DE algorithm is used to estimate the optimal results based the surrogate model. Meanwhile the accuracy of the current surrogate model is inspected according to the given termination condition (SMC40.999 or SDFo1.0e 11) to decide whether the model is precise enough to stop the work. In practical situations, noise is always existent and it cannot be eliminated completely. To take care of the inherent noise which presents in measured signals and examine the robustness of the proposed method, 10% Gaussian noise is added in the simulation responses.

Fig. 2. A rotor-bearing model with three disks.

Table 1 The parameters in rotor-bearing model. Disk

Disk 1

Disk 2

Disk 3

Thickness (m) Inner radius (m) Outer radius (m)

0.05 0.05 0.12

0.05 0.05 0.20

0.06 0.05 0.20

Shaft

L1

L2

L3

L4

Length (m) Radius (m)

0.2 0.05

0.3 0.05

0.5 0.05

0.3 0.05

Bearing stiffness and damping Stiffness (MN/m) Damping (kN  s/m)

kxx ¼ 50, kyy ¼ 70, kxy ¼ kyx ¼0 cxx ¼0.5, cyy ¼ 0.7, cxy ¼cyx ¼0

Mechanical properties Young’s modulus (GPa) Density (kg/m3) Poisson’s ratio

200 7800 0.3

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Table 2 shows the identification results without noise and good agreements between the actual and estimated parameters based on Kriging surrogate model are obtained. To investigate the effect of sampling size on the precision of the identification results, different numbers of samples for 40, 70, 100 and 200 are considered, and the required updating steps of Kriging surrogate model are listed in Fig. 3. It seems that the initial sampling size has little effect on the precision of the identification parameters in rotor-bearing system when ignoring the noise. The required updating steps are small and the objective function converges in a short time. Before reaching the termination condition, the values of objective function have little fluctuation. Fig. 4 compares the unbalance response (UR) values predicted by the final Kriging surrogate model and the values by FE model. It proves that the Kriging surrogate model is accurate enough as a substitute for FE model. The identification results using the proposed method with 10% Gaussian noise are given in Table 3. For four kinds of sampling size, the stiffness coefficients and magnitude of mass unbalance are identified well. With the samples for 70,100 and 200, the errors are less than 1% comparing with the reference values. But the higher errors are obtained in damping coefficients. Analyzing the reasons, one is that damping coefficients of bearings are strongly affected by the unbalance responses especially near the resonant peaks; the other is that damping coefficients are sensitive to the noise and the samples. Table 3 shows that the computational time increases with increasing of sampling size, and for 70,100 and 200 samples, the errors of identification results are much better than 40 samples. In this section, the other grid density is also carried out. The rotor-bearing system is divided into twenty-one grids, the unbalance response in Fig. 5 is nearly same as that obtained by thirteen grids. Moreover, the identification results listed in Table 4 prove that a larger grid density dose not improve the identification precision but increases the computational time, suggesting less effect of the grid density on identification results. According to above discussion, the appropriate samples should be selected to construct the initial Kriging surrogate model. Fig. 6 shows the comparison between the original and identification unbalance responses with 10% Gaussian noise. From Fig. 7, the unbalance response (UR) values predicted by the final Kriging surrogate model also correspond to the values by FE model.

Table 2 Identification results for different samples without noise. Design variables

kxx (MN/m) kyy (MN/m) cxx (kN  s/mm) cyy (kN  s/mm) u (g  mm) Surrogate model updating step

Reference values

50 70 0.5 0.7 200

Identified values 40 Samples

70 Samples

100 Samples

200 Samples

50 70 0.5 0.7 200 9

50 70 0.5 0.7 200 9

50 70 0.5 0.7 200 11

50 70 0.5 0.7 200 12

Fig. 3. Updating step of Kriging surrogate model (k).

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Fig. 4. UR values predicted by the final Kriging surrogate model and FE model for 100 samples.

Table 3 Identification results of different samples with 10% Gaussion noise for the rotor-bearing system divided into thirteen grids. Design variables

Reference values

kxx (MN/m) kyy (MN/m) cxx (kN  s/mm) cyy (kN  s/mm) u (g  mm) Surrogate model updating step Time (s)

50 70 0.5 0.7 200

Identified values (% error) 40 Samples

70 Samples

100 Samples

200 Samples

50.3 (0.6) 70.8 (1.1) 1 (100) 1 (98.6) 201 (0.5) 7 196.21

50.1 (0.2) 70.5 (0.7) 0.482 (3.6) 0.467 (33.3) 201(0.5) 6 231.32

50.1 (0.2) 70.5 (0.7) 0.482 (3.6) 0.467 (33.3) 201 (0.5) 6 271.27

50.6 (1.2) 70.6 (0.9) 0.38 (24) 0.01(99) 201 (0.5) 5 321.99

Fig. 5. Unbalance response calculated by the FE model divided into thirteen grids and twenty-one grids.

The results from KSMEA are compared with DE and other two well know intelligent algorithm without Kriging, Particle Swarm Optimization (PSO) and Genetic Algorithm (GA), the control parameters for these optimization algorithms are shown in Table 5. Table 6 lists the compared results and it indicates that the Kriging surrogate model uses less

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Table 4 Identification results ofdifferent samples with 10% Gaussion noise for the rotor-bearing system divided into twenty-one grids. Design variables

Reference values

kxx (MN/m) kyy (MN/m) cxx (kN  s/mm) cyy (kN  s/mm) u (g  mm) Surrogate model updating step Time (s)

50 70 0.5 0.7 200

Identified values (% error) 40 Samples

70 Samples

100 Samples

200 Samples

50.3 (0.6) 71.8 (2.6) 1 (100) 0.01 (99) 201 (0.5) 8 230.28

50.04 (0.08) 71.4(2) 0.874 (74.8) 0.01 (99) 201(0.5) 8 298.25

50.5 (1) 71.3 (1.9) 0.687 (37.4) 0.353 (49.6) 201 (0.5) 7 331.53

50.3 (0.6) 70.8 (1.1) 0.658 (31.6) 0.01(99) 201 (0.5) 6 385.99

Fig. 6. Original and identified unbalance response.

Fig. 7. UR values predicted by the final Kriging surrogate model and FE model for 100 samples.

computational time but achieves better identification results. Since the calculation of unbalance responses of rotor-bearing system is substituted by the Kriging surrogate model, it reduces the computational burden and saves the computational time.

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Table 5 Control parameters for three optimization algorithms. DEA

PSO

Mutation factor Crossover factor Number of population

0.8 0.95 100

GA

Inertia constant Cooperative factor Cognitive learning factor Number of population

0.5 2 0.5 100

Crossover probability Mutation probability Length of chromosome Number of population

0.5 0.1 52 100

Table 6 Identification results for different methods with 10% Gaussion noise. Design variables

kxx (MN/m) kyy (MN/m) cxx (kN  s/mm) cyy (kN  s/mm) u (g  mm) Time (s)

Reference values

50 70 0.5 0.7 200

DC motor

Identification methods (% error) KSMEA

DEA

PSO

GA

50.1 (0.2) 70.5 (0.7) 0.482 (3.6) 0.467 (33.3) 201 (0.5) 271.27

50.2 (0.4) 70.7 (1) 0.999 (99.8) 0.01 (98.6) 201 (0.5) 3086.2

50.1 (0.2) 70.7 (1) 1.0 (100) 0.01 (98.6) 201 (0.5) 3460.5

49.7 (0.6) 69.4 (0.86) 0.133 (73.4) 0.564 (19.4) 201 (0.5) 2959.8

Spring-bearing Flexible coupling

Disk

Sensor

Shaft

Spring-bearing

Fig. 8. (a) Laboratory test rig, (b) schematic diagram of test tig.

6. Experimental verification 6.1. Description of the test rig The test rig shown in Fig. 8 is a RK-4 rotor system by BENTLY NEVADA. A controllable DC motor is connected to one side of the shaft. The disk is installed in the middle of the shaft and each side of the rotor system has a ball bearing which is simplified two springs and two dampers. The shaft vibration in the vertical and horizontal directions is measured by two sensors, and the measured signals are processed by the ME’scope 4.0 software. The parameters of the Rotor-Kit system are listed in Table 7 and the equivalent FE model is given in Fig. 9.

6.2. Experimental results and discussion In this case, x ¼[kxx,kyy,cxx,cyy,u] are also the identification parameters, where kxx and kyy are the horizontal and vertical stiffness coefficients, cxx and cyy are the horizontal and vertical damping coefficients, respectively. u is the magnitude of mass unbalance of disk. The initial Kriging surrogate model is constructed by the unbalance responses on the 6th node

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Table 7 Parameters of test rig. Parts

Parameters

Specifications

Shaft

Length (mm) Overhung length (mm) Diameter (mm) Density (kg/m3) Young’s modulus (MPa) Poisson ratio Mass (kg) Magnitude of unbalance (g  mm) Horizontal stiffness (N/m) Vertical stiffness (N/m)

320 120 10 7850 210 0.3 0.583 80 2.8e7 3.2e7

Disk Bearing

Fig. 9. A FE model of rotor-bearing model with one disk.

Fig. 10. Measured and identified unbalance response for 200 samples.

with the rotating speeds from 1500 to 2500 rpm. We also consider the 40, 70, 100 and 200 sampling sizes,Fig. 10 shows the unbalance responses calculated by identification results and the measured responses. The identification results of the experiment are given in Table 8. Comparing to the different numbers of samples, errors of the identification results and the Kriging surrogate model updating steps are much different. Good choices of the samples are 100 and 200, which can construct the exact Kriging surrogate model between the identification parameters and unbalance responses. The stiffness and damping coefficients and the magnitude of mass unbalance are obtained with less error and little updating steps. Fig. 11 proved that the UR values predicted by the final Kriging surrogate model correspond to the values measured by the experiment. From this discussion, the proposed method can identify the bearing parameters and the magnitude of mass unbalance effectively and reliably, meanwhile this method is also robust to noisy data and modeling errors.

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Table 8 Identification results for the spring-bearing system. Design variables

Reference values

kxx (MN/m) kyy (MN/m) cxx (kN  s/mm) cyy (kN  s/mm) u (g  mm) Surrogate model updating step

28 32 _ _ 80

Identified values (% error) 40 samples

70 samples

100 samples

200 samples

29.3 (4.65) 100 (212.5) 0.1 0.1 80 (0) 50

27.5 (1.79) 36.5 (14.1) 0.25 5.58 80 (0) 47

28.4 (1.43) 29.1 (9.06) 1.59 10 80 (0) 25

28.3 (1.0) 30.5 (4.7) 1.23 5.07 80 (0) 18

Fig. 11. UR values predicted by the final Kriging surrogate model and measured values for 200 samples.

7. Conclusions This paper presents an effective method based on Kriging surrogate model and evolutionary algorithm to identify the bearing parameters and unbalance information in rotor-bearing system. The developed algorithm has been tested with numerical example and experimental application. Different numbers of samples for 40, 70, 100 and 200 are tested to construct the initial Kriging surrogate model. It proves that the initial sampling sizes have certain effect on the precision of the bearing parameters and magnitude of mass unbalance when 10% Gaussion noise is considered in the numerical application but quite agreement with the reference values without noise. In addition, the appropriate sampling sizes reduce updating sizes of Kriging surrogate model and consume less time. The comparison of KSMEA with DE, PSO and GA without Kriging surrogate model indicates that KSMEA has prominent advantages in parameter identification of rotorbearing system. It is robust to the noise comparing to other traditional identification techniques and cost less computational expense. At the end, experimental rotor-bearing system which has a spring-bearing is used to validate this method. Though the modeling errors exist, the identification parameters are good agreement with the reference values when the sampling sizes are 100 and 200. Nevertheless it is envisaged that this method should be widely used in the parameters identification of the large, more complicated machines in future work. References [1] R. Tiwari, V. Chakravarthy, Simultaneous identification of residual unbalances and bearing dynamic parameters from impulse responses of rotorbearing systems, Mechanical Systems and Signal Processing 20 (2006) 1590–1614. [2] R. Tiwari, A.W. Lees, M.I. Friswell, Identification of speed-dependent bearing parameters, Journal of Sound and Vibration 254 (2002) 967–986. [3] R. Tiwari, V. Chakravarthy, Simultaneous estimation of the residual unbalance and bearing dynamic parameters from the experimental data in a rotor-bearing system, Mechanism and Machine Theory 44 (2009) 792–812. [4] M.S.De Queiroz, An active identification method of rotor unbalance parameters, Journal of Vibration and Control 15 (2009) 1365–1374. [5] C.W. Lee, S.W. Hong, Identification of bearing dynamic coefficients by unbalance response measurements, Journal of Mechanical Engineering Science 15 (1989) 1365–1374. [6] O.C. De Santiago, L. San Andre´s, Field methods for identification of bearing support parameters—part I: identification from transient rotor dynamic response due to impacts, Journal of Engineering for Gas Turbines and Power 129 (2007) 205–212.

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