# Research on the prediction method of unbalance responses of dual-rotor system based on surrogate models

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## Abstract

In this work, the prediction methods of unbalance responses based on the surrogate models were studied, where the simulation data of vibration responses of a dual-rotor system with four disks and five supportings were involved. Firstly, based on the Latin hypercube sampling and random sampling with uniform distribution of unbalance distribution of the fan disc and the hyper-compressor disc, the input variables of the training samples and the testing samples were respectively obtained. According to the sampling results, the multi-body dynamics simulations were conducted to extract the vibration responses at the corresponding measuring points as the output variables of the sample space. Then, the algorithms of multivariate adaptive regression splines (MARS), radial basis function (RBF) and Kriging, were selected to respectively construct the response-predicted models of the rotor system. Finally, predicted vibration responses were figured out by surrogate models and the prediction accuracies were verified by comparison with output parameters in the testing samples. The results showed that the prediction methods of unbalance responses based on MARS, RBF and Kriging enjoyed high prediction accuracies according to the standards, which were proved to be feasible in theoretically.

## Keywords

Dual-rotor system Surrogate model Latin hypercube sampling Dynamic simulation Accuracy test## 1 Introduction

In the application and development of aero-engine, unbalance force caused by unbalance mass in the multiple discs of the low-pressure fan section and the high-pressure compressor section will cause serious vibration, which is one of the most major vibration sources of aero-engine. However, the research on the unbalanced vibration response characteristics of aero-engine often faces the following problems and challenges. Firstly, complex structure and narrow internal space of the dual-rotor system make it difficult to place sensor. Secondly, it is the commonly used method in the field of engineering practice that obtaining a reliable response values in steady state of the system running through multiple start-stops, which costs a lot in time and economy.

At present, the researches of rotor unbalance are mainly the exploration of vibration characteristics. AL-Shudeifat et al. [1] numerically and experimentally found the change in the unbalance force angle with respect to the crack opening direction significantly altered the values of the critical whirl speeds and their corresponding peak whirl amplitudes in cracked rotor-bearing-disk systems for starting up operations. The research of Gao, P.’s revealed that the increasement of corresponding critical speeds and the vibration amplitudes of rotors happened, as the unbalances in LP and HP rotors mainly increased in a force model for the inter-shaft bearing with a local defect on the surface of the outer race or the inner race [2]. Cao et al. [3] analyzed quantitatively the effect of angular speed fluctuation on vibration responses of the unbalanced rotor, the result of which showed the speed fluctuation produced apparent frequency modulation, phase distortion and amplitude error of the unbalance. The research of Ref. [4] showed the sensitive intervals of unbalance vibration in the input side and output side are respectively in lower frequency and in higher frequency in a gas turbine rotor system. Zhang [5] proposed a non-whole beat correlation method to identify the unbalance responses, which was proved to be feasible and practicable from the numerical simulation and balancing experiment.

It is a fitting technology for the surrogate model that predicts the response value in the unknown situation using those in the known situation. Its essence is to approximately express the relationship between the input and output data through establishing the mapping between them, taking the fitting precision and the prediction precision as the constraints. With the application and development of surrogate model technology for more than 40 years, it has been quite mature in the applications of optimization design [6, 7, 8, 9, 10, 11, 12] and parameter identification [13, 14, 15] of complex engineering problems, replacing the high-precision model with heavy computation and solving the problem that analytical model cannot be established in some engineering fields. The introduction of surrogate model technology in the dual-rotor system, can establish the model vibration response of by the use of limited sample data to efficiently achieve accurate prediction for the unknown.

In the field of rotor dynamics, some progress in the application of surrogate model technology has been made. The research of the Ref. [16] illustrated the effectiveness of Kriging when predicting the critical speeds and the vibration amplitudes of a single flexible rotor modelled by analytic method. The research of the Ref. [17] used a polynomial surrogate method to effectively analysis steady-state response of cracked rotors with uncertain-but-bounded parameters by numerical simulations. Gu et al. [18] realized the identification of single-point unbalance parameters of the single-rotor system model, based on the PSO-SVR model, but there are not results for the multi-point and the dual-rotor. The research of the Ref. [19] presented a new method based on an improved Kriging surrogate model and evolutionary algorithm (IKSMEA), which was proved to effectively and accurately identify the structure parameters of a nonlinear rotor-bearing system by numerical studies and experimental validation.

In this study, multi-measuring-point metamodellings of a typical dual-rotor system with double unbalance disks were established based on MARS, RBF and Kriging in sequence, and successfully predicted the vibration amplitudes under the unknown working conditions which were randomly sampled with uniform distribution, proving the application feasibility in this field. The researches applying the metamodelling methods for predicting the vibration response of the dual-rotor system are quite few, so this paper is a supplement. This is an exploratory practice of applying the relatively mature mathematical technology to the engineering field, which can provide the reference for the dynamics balance and design of dual-rotor aero-engine.

## 2 Object, approach and algorithms

### 2.1 Object

### 2.2 Approach

Design of experiment (DOE) is a scientific method to research the correlation between multiple factors and response variables [20]. Common DOE methods include Full Factorials Design, Orthogonal Experiment Design, Latin.

Then, unbalance parameters are set so that the dynamic simulations in ADAMS are carried out to obtain the vibration responses of multiple measuring-points under 51 working conditions. ADAMS software is the most excellent dynamic simulation software of mechanical system developed by MDI. It is one of the most authoritative and widely used dynamic analysis software of mechanical system in the world. It applies Lagrange method to calculate multi-body dynamics, which is a relative coordinate method. The form of its dynamic equation is the second order differential equations of Lagrange coordinate matrix, namely \(A\left( {q,t} \right)\ddot{q} = B\left( {q,\dot{q},t} \right)\). It was first proposed to solve the problem of spacecraft and has been widely used until now. Its advantages are that the number of equations is the least, the number of coordinates of the tree topology system is equal to the degree of freedom of the system, and the dynamics equation is easily converted into ordinary differential equations.

*R*

^{*,Y}for training and test sets can be separately found respectively in Tables 2, 3, 4 and 5, in which Y means output parameters (amplitudes) and * represents input parameters that A and B are respectively unbalance mass and angle phase of LPC while C and D are respectively unbalance mass and phase angle of HPC. From Tables 2 and 4, it can be seen that the amplitude under N1 is strongly correlated with unbalance mass of LPC, mediumly correlated with unbalance phase angle of LPC while weakly and weakly negatively correlated with unbalance mass and phase angle of HPC respectively; from Tables 3 and 5, it can be seen that the amplitude under N2 is strongly correlated with unbalance mass of HPC, mediumly correlated with unbalance phase angle of HPC while weakly and negatively correlated with unbalance mass and phase angle of LPC respectively.

The partial results based on testing points/μm

Rotate speed | Working condition | Measuring points | ||||||||
---|---|---|---|---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | ||

N1 | 1 | 1.04 | 38.07 | 0.09 | 1.80 | 1.99 | 0.44 | 0.76 | 1.84 | 2.02 |

2 | 0.80 | 29.29 | 0.07 | 1.38 | 1.52 | 0.36 | 0.59 | 1.41 | 1.55 | |

3 | 0.76 | 27.82 | 0.07 | 1.31 | 1.45 | 0.33 | 0.56 | 1.34 | 1.47 | |

N2 | 1 | 3.42 | 2.23 | 0.15 | 7.22 | 7.95 | 19.74 | 9.10 | 11.15 | 14.32 |

2 | 4.42 | 2.87 | 0.20 | 9.31 | 10.25 | 25.46 | 11.74 | 14.38 | 18.47 | |

3 | 6.29 | 4.10 | 0.28 | 13.27 | 14.60 | 36.28 | 16.72 | 20.49 | 26.32 |

The Pearson correlation coefficient *R*^{*,Y} for training sets under N1

| The position of measuring points | ||||||||
---|---|---|---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | |

| 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 0.9963 | 0.9996 | 0.9999 | 0.9999 |

| 0.4794 | 0.4800 | 0.4803 | 0.4791 | 0.4788 | 0.4881 | 0.4813 | 0.4791 | 0.4787 |

| 0.0177 | 0.0166 | 0.0159 | 0.0181 | 0.0181 | − 0.0006 | 0.0122 | 0.0192 | 0.0198 |

| − 0.0412 | − 0.0415 | − 0.0418 | − 0.0407 | − 0.0409 | − 0.0462 | − 0.0453 | − 0.0394 | − 0.0392 |

The Pearson correlation coefficient *R*^{*,Y} for training sets under N2

| The position of measuring points | ||||||||
---|---|---|---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | |

| 0.0166 | 0.0167 | 0.0163 | 0.0166 | 0.0166 | 0.0166 | 0.0164 | 0.0165 | 0.0165 |

| − 0.1283 | − 0.1283 | − 0.1286 | − 0.1283 | − 0.1283 | − 0.1283 | − 0.1284 | − 0.1284 | − 0.1284 |

| 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 |

| − 0.1631 | − 0.1631 | − 0.1631 | − 0.1631 | − 0.1630 | − 0.1631 | − 0.1630 | − 0.1632 | − 0.1632 |

The Pearson correlation coefficient *R*^{*,Y} for testing sets under N1

| The position of measuring points | ||||||||
---|---|---|---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | |

| 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 0.9935 | 0.9996 | 0.9999 | 0.9999 |

| 0.1425 | 0.1408 | 0.1405 | 0.1420 | 0.1420 | 0.1268 | 0.1426 | 0.1398 | 0.1400 |

| 0.4180 | 0.4181 | 0.4183 | 0.4173 | 0.4175 | 0.4179 | 0.4172 | 0.4172 | 0.4168 |

| − 0.2616 | − 0.2584 | − 0.2568 | − 0.2620 | − 0.2620 | − 0.1981 | − 0.2460 | − 0.2648 | − 0.2661 |

The Pearson correlation coefficient *R*^{*,Y} for testing sets under N2

| The position of measuring points | ||||||||
---|---|---|---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | |

| 0.4180 | 0.4180 | 0.4180 | 0.4182 | 0.4181 | 0.4181 | 0.4181 | 0.4182 | 0.4180 |

| 0.4205 | 0.4206 | 0.4206 | 0.4207 | 0.4204 | 0.4205 | 0.4202 | 0.4209 | 0.4205 |

| 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 |

| − 0.0256 | − 0.0257 | − 0.0257 | − 0.0259 | − 0.0257 | − 0.0257 | − 0.0256 | − 0.0259 | − 0.0257 |

In this work, the mean square error (MSE) and error rate of the vibration response value are used as the error analysis standards of the surrogate models.

*N*is the number of testing points,

*y*

_{i}is the simulation result, and

*y*

_{i}′ is the corresponding predicted response values. The smaller the value of MSE is, the closer it is to 0, the higher precision of the model.

*μ*

_{simulate}and \(\mu_{predicted}^{*}\) respectively represent response value through simulation and prediction of the testing samples excited by the rotate frequency, the expression of error rate, \(\varepsilon_{\mu }^{*}\) is defined as,

### 2.3 Algorithms

*.*

**x***ε*is the predicted error, \(x = (x_{1} ,x_{2} , \cdots ,x_{p} )^{T}\) is input variables,

*p*is the number of training points, and the column number of matrix

*is the number of response variable*

**x***.*

**y***x*

_{j},

*j*∈ {1,

*…*,

*p*} with corresponding knots at all distinct relating values,

*x*

_{ij},

*i*∈ {

*1*,

*…*,

*n*}, where

*n*is the sample size, expressed by the set C in Eq. (4).

*and the target variable*

**x***f*(

*x*), the form of the MARS model approximating the function in Eq. (3) is defined as

*B*

_{m}(

*x*) represents a truncated linear function from set C or tensor product of more than one function, and

*M*is the number of truncated linear function in the current model [22, 23]. For multiple variable cases, multiplying an existing piecewise linear basis function (BF) with a truncated linear function involving a new variable where both components are nonzero can create the interaction terms which is nonzero only over the space of predictors in MARS [24].

*w*

_{ik}is the synaptic weight between the

*k*th neuron of the output layer of the second hidden layer and the

*i*th neuron of the output layer,

*c*

_{ki}is the

*i*th hidden center vector for the

*k*th pattern class of the first hidden layer, || · ||

_{2}is the Euclidean norm,

*M*denotes the number of neurons in the output layer and the second hidden layer, namely the patterns class number for the training samples set,

*Φ*

_{i}(·) is the kernel function guaranteeing the required accuracy. In this work, the Multiquadric is the choice, written as Eq. (7) [26],

*σ*

_{i}is the width of the receptive field.

*= (*

**x***x*

_{1},

*…*,

*x*

_{i},

*…*,

*x*

_{j},

*…*,

*x*

_{m})

^{T}, can be defined as Eq. (8) [27],

*g*

_{h}(

*x*), (

*h*= 1, 2,

*…*,

*p*) is the basis regression function; the second is a stochastic process (random function) whose mean is 0 and the covariance of random process

*z*(

*x*) is

*R*(

*c*,

*x*

_{i},

*x*

_{j}) is the correlation coefficient between

*z*(

*x*

_{i}) and

*z*(

*x*

_{j}) with parameter

*c*and

*σ*

^{2}is the process variance. The correlation function used in this study is the Gaussian, which is expressed as Eq. (10),

*x*

_{i}

^{n}is the nth component of

*x*

_{i}.

## 3 Prediction and accuracy tests

Predicted amplitudes in N1 of 3 working conditions/μm

Working condition | Surrogate models | Measuring points | ||||||||
---|---|---|---|---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | ||

1 | MARS | 1.04 | 38.07 | 0.09 | 1.80 | 1.99 | 0.47 | 0.75 | 1.84 | 2.03 |

RBF | 1.02 | 37.25 | 0.09 | 1.76 | 1.95 | 0.43 | 0.74 | 1.80 | 1.98 | |

Kriging | 1.04 | 38.06 | 0.09 | 1.79 | 1.99 | 0.45 | 0.76 | 1.83 | 2.02 | |

2 | MARS | 0.80 | 29.28 | 0.07 | 1.38 | 1.53 | 0.34 | 0.58 | 1.42 | 1.56 |

RBF | 0.80 | 29.17 | 0.07 | 1.38 | 1.52 | 0.34 | 0.58 | 1.41 | 1.55 | |

Kriging | 0.80 | 29.28 | 0.07 | 1.38 | 1.53 | 0.35 | 0.59 | 1.41 | 1.55 | |

3 | MARS | 0.76 | 27.82 | 0.07 | 1.31 | 1.45 | 0.32 | 0.55 | 1.35 | 1.48 |

RBF | 0.76 | 27.93 | 0.07 | 1.32 | 1.46 | 0.33 | 0.56 | 1.35 | 1.49 | |

Kriging | 0.76 | 27.82 | 0.07 | 1.31 | 1.45 | 0.32 | 0.56 | 1.34 | 1.48 |

Predicted amplitudes in N2 of 3 working conditions

Working condition | Surrogate models | Measuring points | ||||||||
---|---|---|---|---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | ||

1 | MARS | 3.42 | 2.23 | 0.15 | 7.22 | 7.94 | 19.73 | 9.10 | 11.15 | 14.32 |

RBF | 3.44 | 2.24 | 0.16 | 7.26 | 7.99 | 19.85 | 9.16 | 11.22 | 14.40 | |

Kriging | 3.42 | 2.23 | 0.15 | 7.22 | 7.94 | 19.73 | 9.10 | 11.15 | 14.31 | |

2 | MARS | 4.42 | 2.87 | 0.20 | 9.31 | 10.25 | 25.46 | 11.74 | 14.38 | 18.47 |

RBF | 4.45 | 2.90 | 0.20 | 9.38 | 10.32 | 25.65 | 11.83 | 14.49 | 18.61 | |

Kriging | 4.42 | 2.87 | 0.20 | 9.31 | 10.25 | 25.46 | 11.74 | 14.38 | 18.47 | |

3 | MARS | 6.29 | 4.09 | 0.28 | 13.27 | 14.60 | 36.28 | 16.72 | 20.49 | 26.32 |

RBF | 6.23 | 4.06 | 0.28 | 13.15 | 14.46 | 35.94 | 16.57 | 20.31 | 26.07 | |

Kriging | 6.29 | 4.10 | 0.28 | 13.27 | 14.60 | 36.28 | 16.73 | 20.50 | 26.32 |

Average computing time of the surrogate models

Surrogate models | MARS | RBF | Kriging |
---|---|---|---|

Computing time/s | 0.01 | 0.001 | 0.1 |

The residual results based on testing points/μm

Rotate speed | Working condition | Measuring points | ||||||||
---|---|---|---|---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | ||

N1 | 4 | 0.38 | 13.91 | 0.03 | 0.66 | 0.73 | 0.16 | 0.28 | 0.67 | 0.74 |

5 | 1.14 | 41.72 | 0.10 | 1.97 | 2.18 | 0.49 | 0.84 | 2.01 | 2.21 | |

6 | 0.55 | 20.14 | 0.05 | 0.95 | 1.06 | 0.22 | 0.40 | 0.98 | 1.08 | |

7 | 1.07 | 38.80 | 0.09 | 1.83 | 2.03 | 0.43 | 0.76 | 1.88 | 2.08 | |

8 | 0.96 | 34.78 | 0.08 | 1.65 | 1.82 | 0.38 | 0.68 | 1.69 | 1.87 | |

9 | 0.70 | 25.63 | 0.06 | 1.21 | 1.34 | 0.31 | 0.51 | 1.23 | 1.36 | |

10 | 1.22 | 44.65 | 0.11 | 2.11 | 2.33 | 0.52 | 0.89 | 2.15 | 2.38 | |

11 | 0.60 | 21.97 | 0.05 | 1.03 | 1.14 | 0.26 | 0.44 | 1.06 | 1.17 | |

N2 | 4 | 1.88 | 1.22 | 0.08 | 3.96 | 4.36 | 10.83 | 4.99 | 6.12 | 1.88 |

5 | 4.20 | 2.73 | 0.19 | 8.85 | 9.74 | 24.19 | 11.15 | 13.66 | 4.20 | |

6 | 3.04 | 1.98 | 0.14 | 6.40 | 7.05 | 17.50 | 8.08 | 9.89 | 3.04 | |

7 | 6.73 | 4.38 | 0.30 | 14.20 | 15.62 | 38.80 | 17.89 | 21.93 | 6.73 | |

8 | 4.14 | 2.69 | 0.19 | 8.73 | 9.61 | 23.87 | 11.01 | 13.48 | 4.14 | |

9 | 6.62 | 4.31 | 0.30 | 13.97 | 15.37 | 38.19 | 17.61 | 21.57 | 6.62 | |

10 | 4.64 | 3.02 | 0.21 | 9.78 | 10.76 | 26.74 | 12.33 | 15.11 | 4.64 | |

11 | 2.76 | 1.80 | 0.12 | 5.82 | 6.40 | 15.91 | 7.34 | 8.99 | 2.76 |

Predicted amplitudes in N1 of residual working conditions/μm

Working condition | Surrogate models | Measuring points | ||||||||
---|---|---|---|---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | ||

4 | MARS | 0.38 | 13.91 | 0.03 | 0.66 | 0.73 | 0.16 | 0.28 | 0.67 | 0.74 |

RBF | 0.41 | 14.80 | 0.04 | 0.70 | 0.77 | 0.17 | 0.30 | 0.71 | 0.78 | |

Kriging | 0.38 | 13.92 | 0.03 | 0.65 | 0.73 | 0.16 | 0.28 | 0.67 | 0.74 | |

5 | MARS | 1.14 | 41.72 | 0.10 | 1.97 | 2.18 | 0.48 | 0.84 | 2.02 | 2.23 |

RBF | 1.13 | 41.08 | 0.10 | 1.94 | 2.14 | 0.47 | 0.82 | 1.98 | 2.18 | |

Kriging | 1.14 | 41.72 | 0.10 | 1.97 | 2.18 | 0.48 | 0.84 | 2.01 | 2.22 | |

6 | MARS | 0.55 | 20.14 | 0.05 | 0.95 | 1.06 | 0.22 | 0.40 | 0.97 | 1.08 |

RBF | 0.54 | 19.87 | 0.05 | 0.94 | 1.04 | 0.23 | 0.40 | 0.96 | 1.06 | |

Kriging | 0.55 | 20.14 | 0.05 | 0.95 | 1.05 | 0.23 | 0.40 | 0.97 | 1.08 | |

7 | MARS | 1.06 | 38.80 | 0.09 | 1.83 | 2.03 | 0.49 | 0.78 | 1.88 | 2.07 |

RBF | 1.03 | 37.71 | 0.09 | 1.78 | 1.97 | 0.44 | 0.75 | 1.82 | 2.00 | |

Kriging | 1.06 | 38.80 | 0.09 | 1.82 | 2.03 | 0.46 | 0.77 | 1.88 | 2.07 | |

8 | MARS | 0.95 | 34.77 | 0.08 | 1.64 | 1.82 | 0.33 | 0.70 | 1.68 | 1.85 |

RBF | 0.94 | 34.28 | 0.08 | 1.62 | 1.79 | 0.39 | 0.68 | 1.65 | 1.82 | |

Kriging | 0.95 | 34.76 | 0.08 | 1.64 | 1.82 | 0.38 | 0.69 | 1.69 | 1.86 | |

9 | MARS | 0.70 | 25.62 | 0.06 | 1.21 | 1.34 | 0.29 | 0.51 | 1.24 | 1.37 |

RBF | 0.70 | 25.63 | 0.06 | 1.21 | 1.34 | 0.29 | 0.51 | 1.25 | 1.37 | |

Kriging | 0.70 | 25.62 | 0.06 | 1.21 | 1.34 | 0.30 | 0.51 | 1.24 | 1.36 | |

10 | MARS | 1.22 | 44.65 | 0.11 | 2.11 | 2.33 | 0.52 | 0.89 | 2.16 | 2.38 |

RBF | 1.20 | 43.69 | 0.10 | 2.06 | 2.28 | 0.50 | 0.87 | 2.11 | 2.33 | |

Kriging | 1.22 | 44.65 | 0.11 | 2.11 | 2.33 | 0.52 | 0.89 | 2.16 | 2.38 | |

11 | MARS | 0.60 | 21.96 | 0.05 | 1.04 | 1.15 | 0.24 | 0.44 | 1.06 | 1.17 |

RBF | 0.60 | 21.78 | 0.05 | 1.03 | 1.14 | 0.25 | 0.43 | 1.05 | 1.16 | |

Kriging | 0.60 | 21.96 | 0.05 | 1.03 | 1.15 | 0.25 | 0.44 | 1.06 | 1.17 |

Predicted amplitudes in N2 of residual working conditions/μm

Working condition | Surrogate models | Measuring points | ||||||||
---|---|---|---|---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | ||

4 | MARS | 1.88 | 1.22 | 0.08 | 3.96 | 4.36 | 10.82 | 4.99 | 6.11 | 7.85 |

RBF | 2.04 | 1.33 | 0.09 | 4.30 | 4.73 | 11.76 | 5.42 | 6.64 | 8.53 | |

Kriging | 1.88 | 1.22 | 0.09 | 3.96 | 4.36 | 10.83 | 4.99 | 6.11 | 7.86 | |

5 | MARS | 4.20 | 2.73 | 0.19 | 8.85 | 9.74 | 24.19 | 11.15 | 13.66 | 17.55 |

RBF | 4.21 | 2.74 | 0.19 | 8.88 | 9.77 | 24.28 | 11.19 | 13.72 | 17.61 | |

Kriging | 4.19 | 2.73 | 0.19 | 8.85 | 9.73 | 24.18 | 11.15 | 13.66 | 17.54 | |

6 | MARS | 3.04 | 1.98 | 0.14 | 6.40 | 7.05 | 17.51 | 8.07 | 9.89 | 12.70 |

RBF | 3.02 | 1.96 | 0.14 | 6.37 | 7.00 | 17.40 | 8.02 | 9.83 | 12.62 | |

Kriging | 3.04 | 1.98 | 0.14 | 6.40 | 7.04 | 17.51 | 8.07 | 9.89 | 12.70 | |

7 | MARS | 6.73 | 4.38 | 0.30 | 14.20 | 15.62 | 38.82 | 17.89 | 21.93 | 28.16 |

RBF | 6.39 | 4.16 | 0.29 | 13.49 | 14.84 | 36.87 | 17.00 | 20.83 | 26.75 | |

Kriging | 6.73 | 4.38 | 0.30 | 14.20 | 15.62 | 38.82 | 17.90 | 21.93 | 28.16 | |

8 | MARS | 4.14 | 2.69 | 0.19 | 8.73 | 9.61 | 23.87 | 11.01 | 13.48 | 17.32 |

RBF | 4.16 | 2.71 | 0.19 | 8.77 | 9.65 | 23.97 | 11.05 | 13.54 | 17.39 | |

Kriging | 4.14 | 2.69 | 0.19 | 8.73 | 9.60 | 23.86 | 11.00 | 13.48 | 17.31 | |

9 | MARS | 6.62 | 4.31 | 0.30 | 13.97 | 15.37 | 38.18 | 17.60 | 21.57 | 27.70 |

RBF | 6.40 | 4.16 | 0.29 | 13.50 | 14.85 | 36.89 | 17.01 | 20.84 | 26.76 | |

Kriging | 6.62 | 4.31 | 0.30 | 13.97 | 15.37 | 38.19 | 17.61 | 21.57 | 27.70 | |

10 | MARS | 4.64 | 3.02 | 0.21 | 9.78 | 10.76 | 26.73 | 12.33 | 15.10 | 19.39 |

RBF | 4.63 | 3.01 | 0.21 | 9.77 | 10.75 | 26.70 | 12.31 | 15.08 | 19.36 | |

Kriging | 4.64 | 3.02 | 0.21 | 9.78 | 10.76 | 26.73 | 12.32 | 15.10 | 19.39 | |

11 | MARS | 2.76 | 1.80 | 0.12 | 5.82 | 6.41 | 15.91 | 7.34 | 8.99 | 11.55 |

RBF | 2.74 | 1.78 | 0.12 | 5.78 | 6.35 | 15.79 | 7.28 | 8.92 | 11.45 | |

Kriging | 2.76 | 1.80 | 0.12 | 5.82 | 6.40 | 15.91 | 7.34 | 8.99 | 11.54 |

From Fig. 15, the MSE values of the RBF model at measuring point 2 of the LP rotor are much higher than those of MARS and Kriging under N1, so the prediction accuracy of the RBF model here is much lower; at the other measuring points of the LP rotor, MSE values of all models are less than 5e-02, with high prediction accuracy. At the measuring points of the HP rotor, MSE values of all models are less than 3e-02, so the prediction accuracy is also quite high. In general, the magnitude order from the models at measurement points of the HP section is: RBF > MARS > Kriging, so the order of precision is: Kriging > MARS > RBF.

It can be seen from Fig. 16 that, at all measuring points under N2, the MSE values of MARS and Kriging are much smaller than those of RBF, with higher prediction accuracies.

## 4 Conclusions

- 1.
The predicted unbalance response values of these three algorithms basically converge to the corresponding simulation results under N1 and N2;

- 2.
The three algorithms are high-efficiency, among which RBF is the fastest, while Kriging is the slowest by contrast.

- 3.
Compared with those under N2, the error rates under N1 are significantly bigger, so the precision order is the opposite. The reason of the phenomenon is that the vibration amplitudes under N1 are significantly smaller than those under N2, so the former is more difficult to accurately predict.

- 4.
When the maximum error rate of 5% is taken as an evaluation standard, the predicted results of Kriging’s are the most precise, and under N1 and N2 the MARS and the RBF respectively have better performances by contrast.

- 5.
The MSE of testing points taken as an evaluation standard, the three surrogate models all have high prediction accuracies; the MSE values of RBF are obviously larger than those of the others, which accuracy is the lowest by contrast. Under N1, the Kriging’s accuracy is a little better than that of the MARS, but under N2, the two are much the same.

## Notes

### Acknowledgments

This work was supported by National Natural Science Foundation of China (Grant No. 51705064).

### Compliance with ethical standards

### Conflict of interest

The authors declare that they have no conflict of interest.

## References

- 1.AL-Shudeifat MA, Al Hosani H, Saeed AS, Balawi S (2019) Effect of unbalance force vector orientation on the whirl response of cracked rotors. J Vib Acoust 141(2):021001. https://doi.org/10.1115/1.4041462 CrossRefGoogle Scholar
- 2.Gao P, Hou L, Yang R, Chen Y (2019) Local defect modelling and nonlinear dynamic analysis for the inter-shaft bearing in a dual-rotor system. Appl Math Model 68:29–47MathSciNetCrossRefGoogle Scholar
- 3.Cao H, He D, Xi S, Chen X (2018) Vibration signal correction of unbalanced rotor due to angular speed fluctuation. Mech Syst Signal Process 107:202–220CrossRefGoogle Scholar
- 4.Yang W, Liang M, Wang L, Yuan H (2018) Research on unbalance response characteristics of gas turbine blade-disk rotor system. J VibroEng 20(4):1676–1690CrossRefGoogle Scholar
- 5.Zhang ZX, Wang LZ, Jin ZJ, Zhang Q, Li XL (2013) Non-whole beat correlation method for the identification of an unbalance response of a dual-rotor system with a slight rotating speed difference. Mech Syst Signal Process 39(1–2):452–460CrossRefGoogle Scholar
- 6.Blanning RW (1975) The construction and implementation of metamodels. Simulation 24(6):177–184. https://doi.org/10.1177/003754977502400606 CrossRefzbMATHGoogle Scholar
- 7.Motosh N (1976) Determination of joint stiffness in bolted connections. J Eng Ind 98(3):858–861CrossRefGoogle Scholar
- 8.Sacks J, Welch WJ, Mitchell TJ, Wynn HP (1989) Design and analysis of computer experiments. Stat Sci 4:409–423MathSciNetCrossRefGoogle Scholar
- 9.Simpson TW, Poplinski JD, Koch PN, Allen JK (2001) Metamodels for computer-based engineering design: survey and recommendations. Eng Comput 17(2):129–150CrossRefGoogle Scholar
- 10.Li W, Padula S (2005) Approximation methods for conceptual design of complex systems. In: Chui C, Neaumtu M, Schumaker L (eds) Approximation theory XI: Gatlinburg 2004. Nashboro Press, Brentwood, pp 241–278Google Scholar
- 11.Li M, Sadoughi M, Hu C, Hu Z, Eshghi AT, Lee S (2019) High-dimensional reliability-based design optimization involving highly nonlinear constraints and computationally expensive simulations. J Mech Des 141(5):051402CrossRefGoogle Scholar
- 12.Raponi E, Bujny M, Olhofer M, Aulig N, Boria S, Duddeck F (2019) Kriging-assisted topology optimization of crash structures. Comput Methods Appl Mech Eng 348:730–752MathSciNetCrossRefGoogle Scholar
- 13.Hou Y, Zhao Q, Sapanathan T, Dumon A, Rachik M (2019) Parameter identifiability of ductile fracture criterion for DP steels using bi-level reduced surrogate model. Eng Fail Anal 100:300–311CrossRefGoogle Scholar
- 14.Ovissipour M, Rai R, Nitin N (2019) DNA-based surrogate indicator for sanitation verification and predict inactivation of Escherichia coli O157: H7 using vibrational spectroscopy (FTIR). Food Control 100:67–77CrossRefGoogle Scholar
- 15.Sangireddy SAR, Bhatia A, Garg V (2019) Development of a surrogate model by extracting top characteristic feature vectors for building energy prediction. J Build Eng 23:38–52CrossRefGoogle Scholar
- 16.Sinou JJ, Nechak L, Besset S (2018) Kriging metamodeling in rotordynamics: application for predicting critical speeds and vibrations of a flexible rotor. ComplexityGoogle Scholar
- 17.Fu C, Ren X, Yang Y, Lu K, Qin W (2019) Steady-state response analysis of cracked rotors with uncertain-but-bounded parameters using a polynomial surrogate method. Commun Nonlinear Sci Numer Simul 68:240–256MathSciNetCrossRefGoogle Scholar
- 18.Yujiong Gu, Chen Dongchao, Jing Xu, He Chengbing (2015) Quantitative identification of unbalance in a rotor system based on surrogate model. J Chin Soc Power Eng 35(12):982–987. https://doi.org/10.3969/j.issn.1674-7607.2015.12.006 CrossRefGoogle Scholar
- 19.Han F, Guo X, Mo C, Gao H, Hou P (2017) Parameter identification of nonlinear rotor-bearing system based on improved kriging surrogate model. J Vib Control 23(5):794–807CrossRefGoogle Scholar
- 20.Shelke D, Sakurada K (2016) Simulation of gear microgeometry for gear whine noise reduction in passenger car. In Simulation and testing for vehicle technology. Springer, Cham, pp 137–146. https://doi.org/10.1007/978-3-319-32345-9_11 Google Scholar
- 21.Zhang Wei, Song Xueguan, Shi Maolin, Huo J (2019) The lightweight design of the boom of mechanical excavator based on surrogate model technology machinery. J Des Manuf 4:1–4. https://doi.org/10.19356/j.cnki.1001-3997.2019.04.001 CrossRefGoogle Scholar
- 22.Friedman JH (1991) Multivariate adaptive regression splines. Ann Stat 19(1):1–67MathSciNetCrossRefGoogle Scholar
- 23.Friedman JH, Silverman BW (1989) Flexible parsimonious smoothing and additive modeling. Technometrics 31(1):3–21MathSciNetCrossRefGoogle Scholar
- 24.Koc EK, Bozdogan H (2015) Model selection in multivariate adaptive regression splines (MARS) using information complexity as the fitness function. Mach Learn 101(1–3):35–58MathSciNetzbMATHGoogle Scholar
- 25.Shen W, Guo X, Wu C, Wu D (2011) Forecasting stock indices using radial basis function neural networks optimized by artificial fish swarm algorithm. Knowl-Based Syst 24(3):378–385CrossRefGoogle Scholar
- 26.Seydaoğlu M (2019) A meshless method for burgers’ equation using multiquadric radial basis functions with a Lie-group integrator. Mathematics 7(2):113CrossRefGoogle Scholar
- 27.Sun Z, Wang J, Li R, Tong C (2017) LIF: a new Kriging based learning function and its application to structural reliability analysis. Reliab Eng Syst Saf 157:152–165CrossRefGoogle Scholar
- 28.Kleijnen JP (2009) Kriging metamodeling in simulation: a review. Eur J Oper Res 192(3):707–716MathSciNetCrossRefGoogle Scholar