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NARX model-based dynamic parametrical model identification of the rotor system with bolted joint

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Abstract

This article presents a novel system identification method for the dynamic parametrical model of the rotor-bearing system based on the Nonlinear Auto-Regressive with exogenous inputs (NARX) model, where the physical parameter of the system appears as coefficients in the model. Customarily, the NARX model-based modeling techniques require random signals as input, which leads to the rotor-bearing system that cannot be modeled using such techniques. To solve this issue, an improved system identification method, defined as the frequency sweep system identification approach is proposed in this paper. Firstly, the frequency domain version modeling framework with a physical parameter is derived based on the traditional modeling framework of the dynamic parametrical model. And then, the candidate model term dictionary corresponding to the frequency domain version modeling framework is derived. The Predicted Residual Sums of Squares based Extended Forward Orthogonal Regression algorithm is applied to identify the dynamic parametrical model of the rotor system. The model obtained by using the proposed method is validated based on the Model Predicted Output method. Finally, an experimental case of the rotor-bearing test rig is demonstrated to show the feasibility of the proposed method for real-world scenarios. Both the numerical and experimental studies illustrating the feasibility of the proposed modeling method, which provides a reliable model for time-domain response prediction and dynamic analysis of the rotor system.

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Abbreviations

\({\mathbf{C}}_{L}^{b}\), \({\mathbf{C}}_{R}^{b}\) :

Damping matrices of the left and right shafts

\({\mathbf{G}}_{k}\) :

Matrix of coefficients corresponding to \({\mathbf{W}}_{k}\)

\({\mathbf{G}}_{J}^{e}\) :

Gyroscopic matrix of the jointed element

\(J_{p1}\), \(J_{p2}\) :

Polar moments of inertia of disk 1 and disk 2

\(J_{d1}\), \(J_{d2}\) :

Equatorial moment of inertia of disk 1 and disk 2

\(k_{B}\) :

Contact stiffness of ball bearing

\({\mathbf{K}}_{J}^{e}\) :

Stiffness matrix of the jointed element

\({\mathbf{K}}_{L}^{b}\), \({\mathbf{K}}_{R}^{b}\) :

Stiffness matrices of the left and right shafts

\(m_{{1}}\), \(m_{{2}}\) :

Lumped mass of disk 1 and disk 2

\(M\) :

Total number of potential model terms

\({\mathbf{M}}_{J}^{e}\) :

Mass matrix of the jointed element

\({\mathbf{M}}_{L}^{b}\), \({\mathbf{M}}_{R}^{b}\) :

Mass matrices of the left and right shafts

\(N_{b}\) :

Number of balls in the bearing

\(n_{u}\), \(n_{y}\) :

Maximum time delay of system input and output

\({\mathbf{P}}_{k}\) :

Time-domain version model term dictionary

\({\tilde{\mathbf{P}}}_{k}\) :

Frequency domain version model term dictionary

\({\hat{\mathbf{P}}}_{k}\) :

Frequency domain version model term dictionary used for identification

\({\mathbf{Q}}_{L}^{s}\), \({\mathbf{Q}}_{R}^{s}\) :

External force matrices acting on both sides of the system

\(u,\;w\) :

Lateral displacement

\({\mathbf{W}}_{k}\) :

Orthogonal matrix of \({\hat{\mathbf{P}}}_{k}\)

\({\tilde{\mathbf{Y}}}_{k}\) :

System output in frequency-domain

\(j\omega\) :

Harmonic components in the frequency spectrum

\(\Omega\) :

Rotating speed of the rotor system

\(\Phi_{0}\) :

Relative rotation angle at the transition point

\(\Phi\) :

Relative rotation angle between disk1 and disk2

\({{\varvec{\upxi}}}\) :

Physical parameter vector

\(\it {{\varvec{\uptheta}}}_{k}\) :

Coefficient matrix corresponding to the kth physical parameter value

\(\theta_{j}\) :

Angle location of the jth rolling ball

\(\theta ,\;\varphi\) :

Angular displacement about x and y axes

\(\it {{\varvec{\updelta}}}_{J}^{e}\) :

Displacement vector of the bolted joint structure

\(\it {{\varvec{\updelta}}}\) :

Generalized coordinates of the whole system

\({\text{DTFT}}[ \cdot ]\) :

Discrete-time Fourier transform

\(H( \cdot )\) :

Heaviside function

EFOR:

Extended forward orthogonal regression

PRESS:

Predicted residual sums of squares

CMS:

Common model structure

EFOR:

Extended forward orthogonal regression

PRESS:

Predicted residual sums of squares

CMS:

Common model structure

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Acknowledgements

This project is supported by the National Natural Science Foundation of China (Grant No. 11872148, 11572082); the Key research and development projects in Guangdong Province (No.2020B1515120015); the Fundamental Research Funds for the Central Universities of China (Grant No. N2003012, N160312001, N170308028, N180703018).

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Authors and Affiliations

  1. School of Mechanical Engineering and Automation, Northeastern University, Shenyang, 110819, People’s Republic of China

    Yuqi Li, Zhong Luo, Baolong Shi & Fengxia He

  2. Foshan Graduate School of Northeastern University, Foshan, 528312, People’s Republic of China

    Zhong Luo

  3. Key Laboratory of Vibration and Control of Aero-Propulsion System Ministry of Education, Northeastern University, Shenyang, 110819, People’s Republic of China

    Yuqi Li, Zhong Luo, Baolong Shi & Fengxia He

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  1. Yuqi Li
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Correspondence to Zhong Luo.

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Appendix A

Appendix A

The matrices represented in Eq. (27) are as follows:

$${\mathbf{M}}_{J}^{e} { = }\left[ {\begin{array}{*{20}c} {m_{1} } & {} & {} & {} & {} & {} & {} & {} \\ 0 & {m_{2} } & {} & {} & {} & {} & {} & {} \\ 0 & 0 & {J_{d1} } & {} & {} & {{\text{sym}}} & {} & {} \\ 0 & 0 & 0 & {J_{d2} } & {} & {} & {} & {} \\ 0 & 0 & 0 & 0 & {m_{1} } & {} & {} & {} \\ 0 & 0 & 0 & 0 & 0 & {m_{2} } & {} & {} \\ 0 & 0 & 0 & 0 & 0 & 0 & {J_{d1} } & {} \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & {J_{d2} } \\ \end{array} } \right]$$
(37)
$${\mathbf{K}}_{J}^{e} = \left[ {\begin{array}{*{20}c} {k_{s} } & {} & {} & {} & {} & {} & {} & {} \\ 0 & {k_{s} } & {} & {} & {} & {} & {} & {} \\ 0 & 0 & {k_{\theta } } & {} & {} & {{\text{sym}}} & {} & {} \\ 0 & 0 & 0 & {k_{\theta } } & {} & {} & {} & {} \\ { - k_{s} } & 0 & 0 & 0 & {k_{s} } & {} & {} & {} \\ 0 & { - k_{s} } & 0 & 0 & 0 & {k_{s} } & {} & {} \\ 0 & 0 & { - k_{\theta } } & 0 & 0 & 0 & {k_{\theta } } & {} \\ 0 & 0 & 0 & { - k_{\theta } } & 0 & 0 & 0 & {k_{\theta } } \\ \end{array} } \right]$$
(38)
$${\mathbf{G}}_{J}^{e} { = }\left[ {\begin{array}{*{20}c} 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & { - J_{p1} } & 0 & 0 & 0 & 0 \\ 0 & 0 & {J_{p1} } & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & { - J_{p2} } \\ 0 & 0 & 0 & 0 & 0 & 0 & {J_{p2} } & 0 \\ \end{array} } \right]$$
(39)

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Li, Y., Luo, Z., Shi, B. et al. NARX model-based dynamic parametrical model identification of the rotor system with bolted joint. Arch Appl Mech 91, 2581–2599 (2021). https://doi.org/10.1007/s00419-021-01906-4

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