Frequency Sweep Modeling Method for the Rotor-Bearing System in Time Domain Based on Data-Driven Model

Abstract

In practice, the modeling and analysis of nonlinear rotor-bearing systems are difficult due to the nonlinearity and complexity. In the previous studies, finite element simulation and mathematical modeling methods are mostly adopted to conduct the analysis. However, due to the time-consuming problem in finite element simulation and the lack of sufficient prior knowledge in mathematical modeling, the traditional method is difficult to establish the representation model. In order to overcome this issue, in this study, a data-driven model referred to as the NARX (Nonlinear Auto-Regressive with exogenous inputs) model is introduced to conduct the modeling and analysis of the rotor-bearing system. The identification of the NARX model requires random excitation as the system input, while the input signal of the rotor system is harmonics. Therefore, a time-domain frequency sweep modeling method is proposed in this paper by introducing the rotating speed into the coefficients function expression of the NARX model, the system output can be predicted according to the given speed. Moreover, the representation model of the rotor-bearing system obtained by using the proposed method is validated under different rotating speeds, the results show the applicability of the proposed modeling approach. Finally, an experimental case of the rotor-bearing test rig is demonstrated to show the application in practice. Both the numerical and experimental studies illustrate the applicability of the proposed modeling method, which provides a reliable model for dynamic analysis and fault diagnosis of the rotor-bearing system.

1. Introduction

The rotor-bearing system is widely used in rotating equipment, such as aero-engine, steam turbine units, gas turbines, etc., and the nonlinear dynamic behaviors of the rotor-bearing system affect the performance of the rotating machine directly [1,2]. In the previous studies of dynamic characteristics of the rotor-bearing system, finite element simulation and mathematical modeling are the main research methods. For example, Andrzej et al. [3] and Liu et al. [4] studied the effects of structural parameters on the rotor system by using the nonlinear finite element simulation method. However, due to the time-consuming problems in finite element simulation, many scholars prefer to use a mathematical model to study nonlinear rotor dynamics. The reliable mathematical representation model directly affects the accuracy of the study results. Therefore, it is very important to further study the modeling method of the rotor-bearing system for the analysis and fault diagnosis.

The mathematical models can generally be categorized into the physical model and numerical model [5]. For the rotor-bearing system, the simplified physical models were usually obtained based on dynamics and contact mechanics. For example, Fonseca et al. [6] simplified the rotor-bearing system into a physical model based on mechanical-related theories. Pavlenko et al. [7] established a mathematical model for the multistage centrifugal compressor on non-linear bearing supports, the operating parameters are evaluated through the artificial neural network. Liaposhchenko et al. [8,9] proposed the method to improve the reliability of compressor using the inertial gas–dynamic separation of gas-dispersion flows. Villa et al. [10] established the mathematical model of a rotor-bearing system considering the kinematics of the balls and the Hertz contact. Hei et al. [11] established a 12-DOF mathematical model of rod fastening rotor, and the rods and connection surfaces of the disks are considered as the spring with cubic stiffness. Skylab et al. [12] investigated the dynamic behaviors of a flexible rotor supported on the gas foil journal bearings through deriving the mathematical model. Qin et al. [13,14] derived the analytical model of the bending stiffness of the bolted disk–drum joint structure, and then the mathematical model of the rotor system was obtained based on the finite element method. However, those mathematical models are derived based on some assumptions, and it is relatively simple to represent the dynamic characteristics of the rotor-bearing systems.

On the other hand, a numerical representation model of the rotor-bearing system can be established based on a data-driven modeling method without any prior knowledge, except for the input and output data sets [15,16,17]. In the past decades, the data-driven modeling method is widely used for constructing the mathematical model to reveal the dynamic characteristics of nonlinear systems [18,19,20,21,22]. There are many types of data-driven models. This paper focuses only on the NARX (Nonlinear Auto-Regressive with Exogenous Inputs) model, which was first introduced by Billings [23,24] in the 1980s. The NARX models have been successfully used for representing nonlinear systems and analyzing the dynamic behaviors for convenience [25,26,27]. Peng et al. [28] established the mathematical model of aluminum plate based on the NARX model, and a further fault diagnosis is carried out. Araújo et al. [29] derived the NARX model of Quanser Servor Base Unit, the relationship between the voltage applied to the motor and position was successfully established based on the model. In addition, the NARX-based modeling method can also be used in scenarios such as steel plate identification [30] and the modeling of global magnetic disturbance in near-Earth space [31]. Although the foregoing works provide insight into the modeling process and specific algorithm of identification of nonlinear systems, few reports, according to the authors’ knowledge, have been found on the modeling method of a rotor-bearing system based on the NARX model.

It should be noted that a random signal is always selected as the system input for traditional NARX model identification, but such excitation cannot be produced in rotor system operation. To solve the aforementioned modeling problem of the rotor-bearing system, Ma et al. [32] proposed a new modeling method that applies a multi-harmonic signal generated through the speed-up process as the input signal to establish the NARX model. Although this method has successfully established the NARX model. which reflects the output characteristics of the rotor-bearing system in the simulation and experiment. However, different NARX models need to be established in different speed intervals, so a unified model structure cannot be obtained to characterize the rotor-bearing system. In order to solve this issue, a frequency sweep modeling method of the rotor-bearing system in the time domain is proposed in this paper. A case study is conducted based on a relatively complex rotor-bearing system, and an analytical relationship between the NARX model coefficients and the rotating speed is derived. Finally, an experimental case is also demonstrated to show the applicability of the modeling method. The work provides an important basis for the analytical study and fault diagnosis of the nonlinear rotor-bearing system.

This paper is organized as follows. Section 2 describes a nonlinear rotor-bearing system with a bolted joint structure, and the dynamic characteristics of the system are revealed by the traditional analytic method. Section 3 introduces the modeling framework, and then the proposed modeling method is introduced in detail. In Section 4, the rotor system described in Section 2 is demonstrated as a numerical illustrative case to show the effectiveness of the modeling method, and then a simple rotor-bearing system is presented as an experimental case. Finally, Section 5 conducts the conclusions of this paper.

2. Description of the Rotor-Bearing System

A rotor-bearing system with a bolted joint structure is shown in Figure 1a. A and B are the lumped mass points of both ball bearings, respectively. O_i (i = 1, 2) is the lumped mass point of the disk. The generalized coordinate of the rotor system is determined as $x=\left[x_A,y_A,x_{O1},y_{O1},{\theta }_{xO1},{\theta }_{yO1},x_{O2},y_{O2},{\theta }_{xO2},{\theta }_{yO2},x_B,y_B\right]$.

Figure 1b is the ball bearing which consists of the inner race, outer race, bearing cage, and rolling elements. The nonlinear force introduced by the position change of the balls and then introduced a strong nonlinear factor for the rotor system [34]. According to [33,35], the nonlinear forces of the bearing can be calculated by:

$$\left\{\begin{array}{c}{f}_x^b=-k_B{\displaystyle \sum _{j=1}^{N_b}{\left(x\cos {\theta }_j+y\sin {\theta }_j-\gamma \right)}^{m}H\left(x\cos {\theta }_j+y\sin {\theta }_j-\gamma \right)\cos {\theta }_j}\\ f_y^b=-k_B{\displaystyle \sum _{j=1}^{N_b}{\left(x\cos {\theta }_j+y\sin {\theta }_j-\gamma \right)}^{m}H\left(x\cos {\theta }_j+y\sin {\theta }_j-\gamma \right)\sin {\theta }_j}\end{array}\right.$$

(1)

where $k_B$ is the contact stiffness. ${\theta }_j$ is the angle location of the jth rolling ball, ${\theta }_j=2\pi \left(j-1\right)/N_b+{\omega }_{c}t$. $N_b$ is the number of balls. ${\omega }_c$ is the rotating speed of the bearing cage, ${\omega }_c=\omega \cdot r/(R+r)$. $R$ and r are the radii of the outer race and inner race, respectively. $\omega$ is the rotating speed of the rotor. ${\delta }_j$ is the deformation of the jth rolling ball. $\gamma$ is the radial clearance of the bearing. $H(\cdot )$ is the Heaviside function. According to the literature [32], when calculating the bearing force using Equation (1), m = 3/2 for the ball bearing.

The motion equations of the rotor-bearing system in Figure 1 can be defined as:

$$\mathit{M}\ddot{x}+(\mathit{C}+\mathsf{\Omega }\mathit{G})\dot{x}+\mathit{K}x=\mathit{F}$$

(2)

where $\mathit{M}$ is the mass matrix. $\mathit{C}$ is the damping matrix. $\mathsf{\Omega }$ represents the rotating speed of the system. $\mathit{G}$ is the gyroscopic matrix. $\mathit{K}$ is the stiffness matrix. $\mathit{F}$ is the force matrix.

The deducing process of model (2) can be found in [33], and the complete expression of the matrices is available in [33]. The parameters of the rotor system are listed in

Table 1. Parameters of the rotor system.

Parameters	Values	Parameters	Values
$m_A$$,m_B$	$20 \mathrm{kg}$	$c_a$$,c_b$	$3000 \mathrm{N}\cdot \mathrm{m}/\mathrm{s}$
$m_{O1}$$,m_{O2}$	$40 \mathrm{kg}$	$c_{d1}$$,c_{d2}$	$3000 \mathrm{N}\cdot \mathrm{m}/\mathrm{s}$
$J_{d1}$$,J_{d2}$	$0.803\hspace{0.33em}\mathrm{kg}\cdot {\mathrm{m}}^2$	$c_{O1}$$,c_{O2}$	$3000\hspace{0.33em}\mathrm{N}\cdot \mathrm{m}/\mathrm{s}$
$J_{p1}$$,J_{p2}$	$1.6059\hspace{0.33em}\mathrm{kg}\cdot {\mathrm{m}}^2$	$k_1$	$8\times {10}^5\hspace{0.33em}\mathrm{N}$
$a_1$	$0.29 \mathrm{m}$	$k_2$	$8\times {10}^5\hspace{0.33em}\mathrm{N}$
$b_1$	$0.61 \mathrm{m}$	$k_b$	$2.5\times {10}^7\hspace{0.33em}\mathrm{N}/\mathrm{m}$
$a_2$	$0.31 \mathrm{m}$	$e$	0.6 mm
$b_2$	$0.59 \mathrm{m}$	$\beta$	$\mathsf{\pi }/4$
$k$	$2.5\times {10}^8\hspace{0.33em}\mathrm{N}/\mathrm{m}$	$k_{\theta }$	$2.5\times {10}^7\hspace{0.33em}\mathrm{N}\cdot \mathrm{m}$

, where m_A, m_B represent lumped masses of the rotor system at left and right bearings, respectively; m_O1, m_O2 are lumped masses at O₁ and O₂, respectively; J_di (i = 1, 2) represents the polar moments of the inertia of the disks; J_pi (i = 1, 2) represents the equatorial moment of inertia of the disks; c_a, c_b are damping coefficients at two bearings; c_d1, c_d2 are damping coefficients introduced by the bending effect of the shaft; c_O1, c_O2 are damping coefficients introduced by the rotation of the disks; k₁ and k₂ are the coupled stiffness of shaft AO₁ and BO₂, respectively; k_b is the stiffness due to the contact effect between two disks; k is the bending stiffness of the shaft; k_θ is the bending stiffness between adjacent disks; e is the eccentric distance of the disks; and β is the eccentric phase difference between adjacent disks. Parameters of the ball bearing are shown in

Table 2. The structural parameters of the ball bearing.

Radius of Outer Race R (mm)	Radius of Inner Race r (mm)	Radial Clearance 2r0 (μm)	Numbers of Ball Elements Nb	Contact Stiffness kB (N/m3/2)	BN
63.9	40.1	1	8	13.34 × 106	3.08

, where r₀ is the radial clearance of the bearing; BN is a parameter used for the identification of nonlinearity introduced by the bearing, where BN = (r/(r + R) × N_b).

The Runge–Kutta method [1] was used to solve the motion equations shown in Equation (2), and then the maximum amplitude response of the disk in x-direction changing with rotating speed can be obtained as shown in Figure 2. To visualize the nonlinear characteristics of the rotor system changing with speed, a waterfall diagram is displayed in Figure 3.

As shown in Figure 2, the critical speed of the rotor system is 144.5 rad/s, and then the system output characteristic is considered under different rotating speeds (low speed, critical speed, and overcritical speed), and the results of amplitude in the time domain and the corresponding frequency spectra of signals are calculated, as shown in Figure 4, Figure 5 and Figure 6.

It can be seen from Figure 3, Figure 4, Figure 5 and Figure 6 that the fundamental frequency f_r is the main frequency component in the speed interval from 50 to 320 rad/s. Meanwhile, the second harmonic frequency 2f_r emerges in the frequency spectrums. According to the literature [31], the second harmonics appear due to the effect of the nonlinear bearing force. Therefore, the aforementioned output characteristics of the rotor system should also be reflected in the NARX model.

3. The Frequency Sweep Modeling Method in the Time Domain

The main tasks for NARX model identification include common model structure identification, coefficient calculation, and model validation. Motivated by the system identification method for the common model structure [36] and inspired by the feature extraction method based on the information obtained through frequency sweep [37], this paper aims to develop a new frequency sweep modeling method in the time domain that can be used to select the most significant model terms corresponding to all the rotating speed of the rotor system. Then, the speed is introduced into the coefficients of the common model structure to predict the output of the underlying system just based on the operating speed.

3.1. Modeling Framework of the Rotor-Bearing System

If all the data sets come from the same system, a common model structure is much more preferred to represent the system. However, the modeling method proposed by Ma et al. [32] needs to establish three NARX models in different operating speed ranges (low speed, critical speed, and overcritical speed) to represent the rotor system. Therefore, the common model structure is selected through the data sets obtained by frequency sweep in the time domain, and then the functional relationship between rotating speed and the model coefficients are derived. The NARX model that reflects the rotor system in the full operating speed range can thus be obtained. The single-input single-output discrete-time NARX model can be represented as [38,39]:

$$\begin{array}{l}y(t)=f(y(t-1),\dots y(t-n_y),u(t-1),\dots u(t-n_u))+e(t)\\ \hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}={\displaystyle \sum _{\overline{m}=1}^L{\displaystyle \sum _{p=0}^{\overline{m}}{\displaystyle \sum _{k_1=1}^N\cdots }{\displaystyle \sum _{k_{p+q}=1}^N\left[{\theta }_{p,q}(k_1,k_2,\dots ,k_{p+q}){\displaystyle \prod _{i=1}^{p}y(t-k_i){\displaystyle \prod _{i=p+1}^{p+q}u(t-k_i)}}\right]}}}+e(t)\end{array}$$

(3)

where $\overline{m}=p+q$. L is the highest order of model (3). $f(\cdot )$ is the unknown nonlinear function to be identified. u(t) and y(t) are system input and output signals, respectively. n_u and n_y are time lags of u(t) and y(t), respectively. e(t) is the noise sequence, where the noise is assumed to be white noise and not shown in the following framework to emphasize the feasibility of the modeling method. ${\theta }_{p,q}(k_1,k_2,\dots ,k_{p+q})$ are the coefficients of the model.

For the NARX models corresponding to dissimilar working conditions, the structure of the models may be non-uniform [5]. In order to construct the common model structure of the rotor-bearing system with the different cases of rotating speed, the modeling framework can be defined as:

$$\begin{array}{l}y(t)=F(y(t-1),\dots y(t-n_y),u(t-1),\dots u(t-n_u),\theta (\omega ))\\ \hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}={\displaystyle \sum _{\overline{m}=1}^L{\displaystyle \sum _{p=0}^{\overline{m}}{\displaystyle \sum _{k_1=1}^N\cdots }{\displaystyle \sum _{k_{p+q}=1}^N\left[{\theta }_{p,q}(k_1,k_2,\dots ,k_{p+q},\omega ){\displaystyle \prod _{i=1}^{p}y(t-k_i)}{\displaystyle \prod _{i=p+1}^{p+q}u(t-k_i)}\right]}}}\end{array}$$

(4)

where $F(\cdot )$ is the unknown common model structure to be identified. ${\theta }_{p,q}(k_1,k_2,\dots ,k_{p+q},\omega )$ represents the NARX model coefficients corresponding to the rotating speed of the rotor system.

The linear regression representation of the model (4) can be written as:

$$y_k(t)={\displaystyle \sum _{\overline{m}=1}^M{\theta }_{k,\overline{m}}\left(\omega \right)p_{k,\overline{m}}(t)},k=1,\dots ,K$$

(5)

where $p_{k,\overline{m}}(t)$ represents the selected common model terms. ${\theta }_{k,\overline{m}}\left(\omega \right)$ is coefficients concerning the kth rotating speed value; M is the number of total model terms, which can be calculated by:

$$M=\frac{(n+L)!}{n!L!}$$

(6)

where n = n_u + n_y. L is the highest order of the model.

Model (5) can also be written as a matrix form as:

$${\mathit{y}}_k={\mathit{P}}_k{\mathit{\theta }}_k$$

(7)

where ${\mathit{y}}_k$ represents the system output under operating speed ${\omega }_k$; ${\mathit{P}}_k$ is regression matrix formed by the candidate model terms, ${\mathit{P}}_k=[p_{k,1}(t),p_{k,2}(t),\dots p_{k,M}(t)]$; ${\mathit{\theta }}_k$ is the coefficient matrix, ${\mathit{\theta }}_k=[{\theta }_{k,1},{\theta }_{k,2},\dots ,{\theta }_{k,M}{]}^{\mathrm{T}}$.

3.2. Model Structure Detection

The common model structure can be obtained by using the AERR-based EFOR algorithm [15,35]. For the rotor-bearing system, assume that there are K input–output data sets $D_k={\left\{u_k(t),y_k(t)\right\}}_{t=1}^Z(k=1,2,\dots ,K)$ corresponding to K different operating speeds. Then, the candidate model terms matrix can be formed as:

$$\begin{array}{l}{\mathit{P}}_k=\left[{\mathit{P}}_{k,1},{\mathit{P}}_{k,2},\dots ,{\mathit{P}}_{k,M}\right]\\ \hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}=\left[\begin{array}{cccc}{p}_{k,1}(z)& p_{k,2}(z)& \cdots & p_{k,M}(z)\\ p_{k,1}(z+1)& p_{k,2}(z+1)& \cdots & p_{k,M}(z+1)\\ ⋮& ⋮& \ddots & ⋮\\ p_{k,1}(Z)& p_{k,2}(Z)& \cdots & p_{k,M}(Z)\end{array}\right]\end{array}$$

(8)

where $p_{k,m}\left(m=1,2,\dots ,M\right)$ is the column vector of candidate model terms. $p_{k,m}\left(t\right)\left(t=z,z+1,\dots ,Z\right)$ denotes the value at sampling time t.

For the purpose of minimizing the error, the matrix ${\mathit{P}}_k$ needs to be orthogonalized first [5], by using the Gram–Schmidt algorithm [40], and then model (7) can be rewritten as:

$${\mathit{y}}_k={\mathit{W}}_k{\mathit{G}}_k$$

(9)

where ${\mathit{W}}_k$ formed by the orthogonal vectors ${\mathit{w}}_{\mathit{k}\mathit{,}\mathit{1}},{\mathit{w}}_{\mathit{k}\mathit{,}\mathit{2}},\dots {\mathit{w}}_{\mathit{k},\mathit{M}}$; ${\mathit{G}}_k$ is the coefficient matrix corresponding to ${\mathit{W}}_k$, ${\mathit{G}}_k=[g_{k,1},g_{k,2},\dots ,g_{k,M}{]}^{\mathrm{T}}$. ${\mathit{w}}_{\mathit{k},\mathit{m}}$ and $g_{k,m}$ can be derived by Equations (8) and (9), respectively:

$$w_{k,m}=p_{k,m}-{\displaystyle \sum _{i=1}^{m-1}\frac{〈p_{k,m},w_{k,i}〉}{〈w_{k,i},w_{k,i}〉}}{w}_{k,i}, m=1, 2,\dots , M$$

(10)

$$g_{k,m}=\frac{〈y_k,w_{k,m}\left(t\right)〉}{〈w_{k,m}\left(t\right),w_{k,m}\left(t\right)〉}, m=1, 2,\dots , M$$

(11)

where <·,·> represents the inner product.

The AERR-based EFOR algorithm selects the common model terms based on the AERR criterion in a stepwise manner, which is measured by the Average Error Reduction Ratio (AERR) [36]:

$${\mathrm{AERR}}_m=\frac{1}{K}{\displaystyle \sum _{k=1}^K\frac{{\left(g_{k,m}\right)}^2〈w_{k,m}\left(t\right),w_{k,m}\left(t\right)〉}{〈w_{k,m}\left(t\right),w_{k,m}\left(t\right)〉}}\times 100\%=\frac{1}{K}{\displaystyle \sum _{k=1}^K\frac{{〈y_k,w_{k,m}〉}^2}{〈y_k,y_k〉〈w_{k,m},w_{k,m}〉}}\times 100\%$$

(12)

where the subscript m denotes the mth candidate orthogonal model terms.

The model term with the maximum AERR value is selected as the ith model term at the ith searching step; the ith AERR is defined as:

$${\mathrm{AERR}}_i=\max \left\{{\mathrm{AERR}}_m\right\}$$

(13)

A threshold $\sigma$ is specified, and the searching process will terminate while the Error-to-Signal Ratio (ESR) is less than $\sigma$. The ESR is defined as:

$$\mathrm{ESR}=100-{\displaystyle \sum _{i=1}^{M_0}{\mathrm{AERR}}_i}\le \sigma$$

(14)

where M₀ denotes the M₀th step of the searching process.

The linearized representation of orthogonalized parametrical dynamical model can be finally obtained as:

$$y_k={\displaystyle \sum _{m_0=1}^{M_0}{\overline{g}}_{k,m_0}{\overline{w}}_{m_0}}, m{0}_{ }=1,2,\dots , M_0$$

(15)

where ${\overline{w}}_{m_0}$ is the selected orthogonalized model terms; ${\overline{g}}_{k,m_0}$ is the corresponding coefficients.

3.3. Derivation of the Coefficients Function with the Operating Speed

Based on Equation (15) and the Inverse Gram–Schmidt algorithm [41], the NARX model with respect to the kth rotating speed value can be obtained as:

$$y_k={\displaystyle \sum _{m_0=1}^{M_0}{\overline{\theta }}_{k,m_0}{\overline{p}}_{m_0}}, m{0}_{ }=1,2,\dots ,M_0$$

(16)

where ${\overline{p}}_{m_0}$ is the selected terms from the original candidate dictionary $P_k$; ${\overline{\theta }}_{k,m_0}$ is the corresponding coefficients.

The Inverse Gram–Schmidt algorithm is introduced in detail as follows:

$$\left\{\begin{array}{l}{\overline{\theta }}_{k,M_0}={\overline{g}}_{k,M_0}\hfill \\ {\overline{\theta }}_{k,M_0-1}=\overline{g}k,M_0-1-a_{M_0-1,M_0}\overline{\theta }{k}_{,}{M}_0\hfill \\ {\overline{\theta }}_{k,M_0-2}={\overline{g}}_{k,M_0-2}-a_{M_0-2,M_0-1}{\overline{\theta }}_{k,M_0-1}-a_{M_0-2,M_0}{\overline{\theta }}_{k,M_0}\hfill \\ ⋮\hfill \\ {\overline{\theta }}_{k,m}={\overline{g}}_{k,m}-{\displaystyle \sum _{i=m+1}^{M_0}{a}_{m,i}{\overline{\theta }}_{k,i}}\hfill \end{array}\right.$$

(17)

where

$$a_{m,i}=\frac{〈{\overline{p}}_{k,i},{\overline{w}}_{k,m}〉}{〈{\overline{w}}_{k,m},{\overline{w}}_{k,m}〉},1\le m\le M_0-1$$

(18)

The relationship between the rotating speed $\omega$ ($\omega =[{\omega }_1,{\omega }_2,\dots ,{\omega }_K]$) and the coefficients presented in Equation (17) can be revealed by the polynomial function [15]:

$${\overline{\theta }}_{k,m_0}\left(\omega \right)={\displaystyle \sum _{j_1=0}^J\cdots {\displaystyle \sum _{j_K=0}^J{\beta }_{j_1,\dots ,j_S}{\omega }_1^{j_1}\cdots {\omega }_K^{j_K}}}$$

(19)

where J is the degree of Function (19); ${\beta }_{j_1,\dots ,j_S}$ is the coefficients, which can be calculated through the Least Squares method [42].

As can be seen from Equation (16), ${\overline{\theta }}_{k,m_0}$ is the coefficient of the NARX model with respect to the kth rotating speed value. Therefore, the identified model has multiple sets of coefficients corresponding to the rotating speed, which can be expressed by the polynomial function as shown in Equation (19). ${\beta }_{j_1,\dots ,j_S}$ is the coefficient of the polynomial function, which can be estimated by using the Least Squares algorithm according to the ${\overline{\theta }}_{k,m_0}$ and rotating speed $\omega$ ($\omega =[{\omega }_1,{\omega }_2,\dots ,{\omega }_K]$).

It should be mentioned that the response signals corresponding to the frequency sweep process should be collected first for the application of the proposed modeling approach for the rotor system. The common model structure with respect to all the rotating speeds can be found through using the system identification method described in Section 3.2. Finally, the speed is introduced into the coefficients of the common model structure. The system output of the rotor system can be obtained based on the identified NARX model with a given rotating speed. By comparing the predicted output with the real output of the rotor system, the feasibility of the identified model and proposed modeling method can then be validated. The modeling process is shown in Figure 7.

4. Validation of the Proposed Modeling Method

In order to verify the applicability and efficiency of the proposed modeling method, in this section, the rotor-bearing system described in Section 2 is illustrative as a numerical case, and a rotor-bearing test rig is demonstrated as an experimental case. The MPO (Model Predicted Output) method [32,41] is used to verify the results, which indicates that the NARX model can be obtained by using the proposed modeling method can reflect the system output accurately.

4.1. Case 1: Simulation Data Sets of the Rotor-Bearing System

Considering the rotor-bearing system shown in Figure 1, where the unbalanced force $m_{e}e{\omega }^2\cos \left(\omega \cdot t\right)$ is the system input signal, where m_e is the mass of the rotor. According to the modeling method described in Section 3, the signal with excitation frequency $\omega \in \left[81.7:12.6:276.5\right]$ rad/s is taken as system input for frequency sweep. Then, 19 data sets of 81921 sample points were generated as input signals, and the corresponding output signals were obtained by solving the Equation (2). The maximum time lags for the input and output signals are three, and the degree of the nonlinearity function was three. The results of the identification by using the proposed rotor-bearing system modeling method are shown in

Table 3. The modeling results of the rotor-bearing system in Figure 1.

Step	Term	Coefficients Corresponding to the Excitation Frequency					AERR (%)
		$\mathit{\omega } = 81.7\hspace{0.33em}\mathbf{rad}/\mathbf{s}$	$\mathit{\omega } = 94.3\hspace{0.33em}\mathbf{rad}/\mathbf{s}$	$\cdots$	$\mathit{\omega } = 263.9\hspace{0.33em}\mathbf{rad}/\mathbf{s}$	$\mathit{\omega } = 276.5\hspace{0.33em}\mathbf{rad}/\mathbf{s}$
1	y (t − 1)	2.49 × 10−8	2.37 × 10−8	$\cdots$	−1.03 × 10−9	1.44 × 10−9	99.9
2	y (t − 2)	2.49 × 10−8	2.37 × 10−8		−1.02 × 10−9	1.439 × 10−9	3.71 × 10−3
3	y (t − 3)	2.5 × 10−8	2.38 × 10−8		−1.02 × 10−9	1.435 × 10−9	1 × 10−9
4	u (t − 3)	1.12 × 10−4	1.1 × 10−4		7.2 × 10−6	1.28 × 10−5	1.02 × 10−13
5	u (t − 1)	−1.11 × 10−4	−1.09 × 10−4		−7.37 × 10−6	−1.3 × 10−5	1.97 × 10−14
6	u (t − 2)	2.79 × 10−7	1.396 × 10−7		−8.1 × 10−8	−8.04 × 10−8	4.82 × 10−14
7	u (t − 3)2	−8.95 × 10−8	1.71 × 10−7		−4.32 × 10−8	2.12 × 10−8	3.92 × 10−14
8	1	−2.97 × 10−7	2.12 × 10−8		−7.59 × 10−8	4.04 × 10−8	4.22 × 10−14
9	u (t − 1)2	−2.09 × 10−7	5.37 × 10−7		−1.22 × 10−7	5.82 × 10−8	4.21 × 10−14
10	u (t − 1) u (t − 2)	2.99 × 10−7	−7.08 × 10−7		1.65 × 10−7	−7.9 × 10−8	4.21 × 10−14
Total							99.996

.

As can be seen from Table 3, the NARX model used to represent the rotor-bearing system can be written as:

$$y(t)={\theta }_1(\omega )y(t-1)+{\theta }_2(\omega )y(t-2)\hspace{0.33em}+\cdots +{\theta }_8(\omega )\cdot 1+{\theta }_9(\omega )u{(t-1)}^2\hspace{0.33em}+{\theta }_{10}(\omega )u(t-1)u(t-2)$$

(20)

where ${\theta }_i(\omega )\left(i=1,\dots ,10\right)$ is the coefficients corresponding to the operating speed.

As illustrated in Table 3 and Equation (20), the dynamic characteristics of the rotor system under different operating speeds can be expressed by the same NARX model terms. However, the coefficients corresponding to different rotating speeds are different. In order to obtain a parsimonious model to represent the rotor-bearing system, for this numerical case, the coefficients are written as a function corresponding to the rotating speed. Then, the system output can be predicted according to the expression of the unbalanced force, the coefficient function, and the model terms, as long as the rotation speed is given. In this case, the coefficient function is expressed as a third-order function corresponding to the rotational speed as follows:

$${\theta }_i(\omega )={\beta }_{i,j}+{\beta }_{i,j}\omega +{\beta }_{i,j}{\omega }^2+{\beta }_{i,j}{\omega }^3$$

(21)

where ${\beta }_{j_1,\dots ,j_S}$ is the coefficients of the coefficient function in Equation (21), which can be obtained by using the Least Squares algorithm according to the identification results listed in Table 3 and $\omega$. The results for the parameters ${\beta }_{m,i}$ are shown in

Table 4. Results for the parameters β_m,i in Equation (21).

i	j
	0	1	2	3
1	−4 × 10−9	5.2 × 10−9	−2.77 × 10−10	3.72 × 10−12
2	−3.2 × 10−9	5.13 × 10−9	−2.74 × 10−10	3.7 × 10−12
3	−2.48 × 10−9	5.05 × 10−9	−2.72 × 10−10	3.66 × 10−12
4	−3.9 × 10−5	2.4 × 10−5	−1.16 × 10−6	1.47 × 10−8
5	4.48 × 10−5	−2.47 × 10−5	1.18 × 10−6	−1.48 × 10−8
6	2.4 × 10−6	−2.38 × 10−7	7.2 × 10−9	−6.98 × 10−11
7	−5.3 × 10−6	6.4 × 10−7	−2.27 × 10−8	2.48 × 10−10
8	−6.7 × 10−6	7.88 × 10−7	−2.79 × 10−8	3.09 × 10−10
9	−1.53 × 10−5	1.85 × 10−6	−6.52 × 10−8	7.12 × 10−10
10	2.06 × 10−5	−2.5 × 10−6	8.79 × 10−8	−9.59 × 10−10

.

In order to test the NARX model in the time and frequency domain, the rotating speed $\omega =100.5,\hspace{0.33em}138.2,\hspace{0.33em}213.6\hspace{0.33em}\mathrm{rad}/\mathrm{s}$ are selected to generate the harmonic input signals and the coefficients of the NARX model. The validation is performed with the MPO criterion [32,41]; the results are presented in Figure 8, Figure 9 and Figure 10.

It can be observed from Figure 8, Figure 9 and Figure 10 that the predicting output of the NARX model is basically consistent with the real results of the underlying system, which means that the modeling method proposed in this paper can provide a parsimonious and reliable model for output prediction of the rotor-bearing system.

4.2. Case 2: Real Data Sets of the Rotor-Bearing System

As shown in Figure 11a, a rotor-bearing test rig is taken as the experimental subject for the real data validation. The eddy current displacement sensor is located between the disk and the bearing to obtain the output signals, and the Labview test system is applied to collect the data sets, as shown in Figure 11b.

Concerned with the safety and the working condition, the rotor-bearing test rig is operated under low-speed cases. The cos $\left(\omega t\right)$, $\omega \in \left[113:1.3:126\right]\hspace{0.33em}\mathrm{rad}/\mathrm{s}$, is defined as the system input, and the signals of the system response in vertical are considered as the system output. The NARX model of the test rig is established based on the frequency sweep modeling method, and a three-order model is identified as shown in

Table 5. The modeling results of the rotor test rig in Figure 11.

Step	Term	Coefficients Corresponding to the Excitation Frequency					AERR (%)
		$\mathit{\omega } = 113\hspace{0.33em}\mathbf{rad}/\mathbf{s}$	$\mathit{\omega } = 114.3\hspace{0.33em}\mathbf{rad}/\mathbf{s}$	$\cdots$	$\mathit{\omega } = 124.4\hspace{0.33em}\mathbf{rad}/\mathbf{s}$	$\mathit{\omega } = 126\hspace{0.33em}\mathbf{rad}/\mathbf{s}$
1	y (t − 1)	2.384	2.398	$\cdots$	2.358	2.336	99.43
2	y (t−2)	−1.974	−1.990		−1.927	−1.899	0.56
3	y (t−3)	0.562	0.567		0.534	0.525	1.12 × 10−5
4	u (t − 1)	0.089	0.015		−0.162	−0.221	2.18 × 10−6
5	u (t − 3)	−0.114	−0.040		0.135	0.201	1.59 × 10−6
6	u (t − 1) y (t − 3)	0.008	0.008		−0.009	−0.005	2.8 × 10−7
7	1	0.0003	0.0002		0.0001	0.0001	4.8 × 10−7
8	y (t − 3)3	−0.041	−0.031		−0.024	−0.029	3.34 × 10−7
9	y (t − 1) y (t − 3)	−0.295	−0.130		−0.138	−0.166	3.02 × 10−7
10	y (t − 2)2	0.286	0.128		0.124	0.156	5.5 × 10−8
Total							99.99

.

For the experimental validation case, the coefficient function is also expressed as a third-order function as shown by Equation (21). The coefficients ${\beta }_{m,i}$ can be obtained by the Least Squares algorithm through identification results shown in Table 5. The calculation results of ${\beta }_{m,i}$ are listed in

Table 6. Results for the parameters β_m,i.

i	j
	0	1	2	3
1	1.9470	0.0770	−0.0041	0.0001
2	−0.5630	−0.2287	0.0117	−0.0002
3	−0.4807	0.1650	−0.0083	0.0001
4	1.0970	−0.0914	0.0026	0.00004
5	−1.0948	0.0980	−0.0034	0.0001
6	−5.1413	0.7221	−0.0332	0.0005
7	−0.0064	0.0010	0.00005	0.1 × 10−5
8	−0.9153	0.1162	−0.0051	0.0001
9	−27.6713	3.7498	−0.1687	0.0025
10	22.1378	−2.9794	0.1335	−0.0020

.

The input signal with rotating speed $\omega =118.1\hspace{0.33em}\mathrm{rad}/\mathrm{s}$ is applied to verify the NARX model using the MPO validation criterion. The predicting output is compared with the real output in time and frequency domains, the results are shown in Figure 12. As Figure 12 implies the NARX model of the rotor-bearing system shows satisfying accuracy. These comparisons indicate that the proposed frequency sweep modeling method of the rotor-bearing system in the time domain can reflect the dynamic characteristics. Moreover, the identified NARX model can provide a most parsimonious and reliable model for the analysis and further fault diagnosis of the rotor-bearing system.

5. Conclusions

In this paper, a frequency sweep modeling method for the rotor-bearing system in the time domain is proposed based on one of the data-driven models, which is referred to as the NARX model. Different from other modeling methods of the rotor system based on system identification, it is a common model structure-detection-based model terms selection method used to characterize the underlying system. Using a pre-defined power exponential polynomial function and introducing the operating speed into the function to describe the NARX model coefficients, the most parsimonious model can be obtained to reproduce the rotor-bearing system under different operating speeds. Furthermore, the proposed modeling method enriches the rotor-bearing system modeling method and provides a reliable model for the analysis and fault diagnosis, which is of practical significance.

Both the numerical and experimental studies are conducted to show the efficiency and accuracy of the proposed modeling method by using the MPO validation criterion, the results indicate that the system output predicted by the identified model can accurately reflect the dynamic characteristics of interest in both the time and frequency domains. However, in the experiment case, it is difficult to collect the system output in a wide working speed range due to the unstable working condition of the test rig, which makes the experimental results insufficient. Consequently, more experimental cases will be discussed in future studies.