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An improved transfer-matrix method on steady-state response analysis of the complex rotor-bearing system

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Abstract

This paper proposes an improved transfer-matrix method (TMM) for investigating the steady-state response of complex rotor-bearing systems. The internal damping of the shafts is considered to enhance the accuracy of the method. A transfer matrix of ball bearings with clearance and Hertzian contact is established. In addition, the incremental harmonic balance method is combined with the TMM to obtain the steady-state response of the rotor systems and simplify the investigation processes. The simulation results verify the superiority of this method in programming, reducing the order of system matrix, and determining stability. Thus, this method is efficient in investigating the nonlinear dynamic characteristics of large-scale rotor-bearing systems.

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Acknowledgement

This work was supported by the National Science Foundation of China [Grant Numbers 11872148, 11572082]; the Fundamental Research Funds for the Central Universities of China [Grant Numbers N170308028, N160312001]; and the Excellent Talents Support Program in Institutions of Higher Learning in Liaoning Province of China [Grant Number LJQ2015038].

Author information

Authors and Affiliations

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

    Zhong Luo & Haopeng Liu

  2. Key Laboratory of Vibration and Control of Aero-Propulsion System Ministry of Education, Northeastern University, Shenyang, 110819, China

    Zhong Luo & Haopeng Liu

  3. College of Intelligence Science and Technology, National University of Defense Technology, Changsha, 410073, China

    Zifang Bian

  4. Department of Automatic Control and System Engineering, University of Sheffield, Sheffield, S13JD, UK

    Yunpeng Zhu

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  1. Zhong Luo
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  2. Zifang Bian
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  3. Yunpeng Zhu
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  4. Haopeng Liu
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Correspondence to Zhong Luo or Yunpeng Zhu.

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Appendix 1

Appendix 1

$$ {\user2{T}}_{o} = \left| {\begin{array}{*{20}l} {\begin{array}{*{20}l} {\frac{{\partial F_{x0} }}{{\partial a_{0} }}} & {\frac{{\partial F_{x0} }}{{\partial c_{0} }}} & 0 & 0 \\ {\frac{{\partial F_{y0} }}{{\partial a_{0} }}} & {\frac{{\partial F_{y0} }}{{\partial c_{0} }}} & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ \end{array} } & \cdots & {\begin{array}{*{20}l} {\frac{{\partial F_{x0} }}{{\partial a_{j} }}} & {\frac{{\partial F_{x0} }}{{\partial c_{j} }}} & {\frac{{\partial F_{x0} }}{{\partial b_{j} }}} & {\frac{{\partial F_{x0} }}{{\partial d_{j} }}} \\ {\frac{{\partial F_{y0} }}{{\partial a_{j} }}} & {\frac{{\partial F_{y0} }}{{\partial c_{j} }}} & {\frac{{\partial F_{y0} }}{{\partial b_{j} }}} & {\frac{{\partial F_{y0} }}{{\partial d_{j} }}} \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ \end{array} } & {{\user2{O}}_{4 \times 4} } & \cdots & {{\user2{O}}_{4 \times 2} } \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ {\begin{array}{*{20}l} {\begin{array}{*{20}l} {\frac{{\partial F_{xck} }}{{\partial a_{0} }}} & {\frac{{\partial F_{xck} }}{{\partial a_{0} }}} \\ {\frac{{\partial F_{yck} }}{{\partial a_{0} }}} & {\frac{{\partial F_{yck} }}{{\partial c_{0} }}} \\ {\frac{{\partial F_{xsk} }}{{\partial a_{0} }}} & {\frac{{\partial F_{xsk} }}{{\partial c_{0} }}} \\ {\frac{{\partial F_{ysk} }}{{\partial a_{0} }}} & {\frac{{\partial F_{ysk} }}{{\partial c_{0} }}} \\ \end{array} } & {{\user2{O}}_{4 \times 2} } \\ \end{array} } & \cdots & {\begin{array}{*{20}l} {\frac{{\partial F_{xck} }}{{\partial a_{j} }}} & {\frac{{\partial F_{xck} }}{{\partial c_{j} }}} & {\frac{{\partial F_{xck} }}{{\partial b_{j} }}} & {\frac{{\partial F_{xck} }}{{\partial d_{j} }}} \\ {\frac{{\partial F_{yck} }}{{\partial a_{j} }}} & {\frac{{\partial F_{yck} }}{{\partial c_{j} }}} & {\frac{{\partial F_{yck} }}{{\partial b_{j} }}} & {\frac{{\partial F_{yck} }}{{\partial d_{j} }}} \\ {\frac{{\partial F_{xsk} }}{{\partial a_{j} }}} & {\frac{{\partial F_{xsk} }}{{\partial c_{j} }}} & {\frac{{\partial F_{xsk} }}{{\partial b_{j} }}} & {\frac{{\partial F_{xsk} }}{{\partial d_{j} }}} \\ {\frac{{\partial F_{ysk} }}{{\partial a_{j} }}} & {\frac{{\partial F_{ysk} }}{{\partial c_{j} }}} & {\frac{{\partial F_{ysk} }}{{\partial b_{j} }}} & {\frac{{\partial F_{ysk} }}{{\partial d_{j} }}} \\ \end{array} } & {{\user2{O}}_{4 \times 4} } & \cdots & {{\user2{O}}_{4 \times 2} } \\ {{\user2{O}}_{4 \times 4} } & \cdots & {{\user2{O}}_{4 \times 4} } & {{\user2{I}}_{4 \times 4} } & \cdots & {{\user2{O}}_{4 \times 2} } \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ {{\user2{O}}_{2 \times 4} } & \cdots & {{\user2{O}}_{2 \times 4} } & {{\user2{O}}_{2 \times 4} } & \cdots & {{\user2{I}}_{2 \times 2} } \\ \end{array} } \right| $$
(A-1)

R and P in Sect. 2.2 can be expressed into Fourier series, as follows:

$$ R = R_{0} + \sum\limits_{i = 1}^{N} {R_{ic} \cos (i\varOmega t) + R_{is} \sin (i\varOmega t)} $$
(A-2)
$$ P = P_{0} + \sum\limits_{i = 1}^{N} {P_{ic} \cos (i\varOmega t) + P_{is} \sin (i\varOmega t)} $$
(A-3)
$$ {\user2{T}}_{i} = \left| {\begin{array}{*{20}l} {\begin{array}{*{20}l} {\frac{{\partial F_{x0} }}{{\partial a_{0} }}} & {\frac{{\partial F_{x0} }}{{\partial c_{0} }}} & { - 1} & 0 \\ {\frac{{\partial F_{y0} }}{{\partial a_{0} }}} & {\frac{{\partial F_{y0} }}{{\partial c_{0} }}} & 0 & { - 1} \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ \end{array} } & \cdots & {\begin{array}{*{20}l} {\frac{{\partial F_{x0} }}{{\partial a_{j} }}} & {\frac{{\partial F_{x0} }}{{\partial c_{j} }}} & {\frac{{\partial F_{x0} }}{{\partial b_{j} }}} & {\frac{{\partial F_{x0} }}{{\partial d_{j} }}} \\ {\frac{{\partial F_{y0} }}{{\partial a_{j} }}} & {\frac{{\partial F_{y0} }}{{\partial c_{j} }}} & {\frac{{\partial F_{y0} }}{{\partial b_{j} }}} & {\frac{{\partial F_{y0} }}{{\partial d_{j} }}} \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ \end{array} } & {{\user2{O}}_{4 \times 4} } & \cdots & {\begin{array}{*{20}l} 0{R_{0} } \\ 0{P_{0} } \\ 00 \\ 00 \\ \end{array} } \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ {\begin{array}{*{20}l} {\begin{array}{*{20}l} {\frac{{\partial F_{xck} }}{{\partial a_{0} }}} & {\frac{{\partial F_{xck} }}{{\partial a_{0} }}} \\ {\frac{{\partial F_{yck} }}{{\partial a_{0} }}} & {\frac{{\partial F_{yck} }}{{\partial c_{0} }}} \\ {\frac{{\partial F_{xsk} }}{{\partial a_{0} }}} & {\frac{{\partial F_{xsk} }}{{\partial c_{0} }}} \\ {\frac{{\partial F_{ysk} }}{{\partial a_{0} }}} & {\frac{{\partial F_{ysk} }}{{\partial c_{0} }}} \\ \end{array} } & {{\user2{O}}_{4 \times 2} } \\ \end{array} } & \cdots & {\begin{array}{*{20}l} {\frac{{\partial F_{xck} }}{{\partial a_{j} }}} & {\frac{{\partial F_{xck} }}{{\partial c_{j} }}} & {\frac{{\partial F_{xck} }}{{\partial b_{j} }}} & {\frac{{\partial F_{xck} }}{{\partial d_{j} }}} \\ {\frac{{\partial F_{yck} }}{{\partial a_{j} }}} & {\frac{{\partial F_{yck} }}{{\partial c_{j} }}} & {\frac{{\partial F_{yck} }}{{\partial b_{j} }}} & {\frac{{\partial F_{yck} }}{{\partial d_{j} }}} \\ {\frac{{\partial F_{xsk} }}{{\partial a_{j} }}} & {\frac{{\partial F_{xsk} }}{{\partial c_{j} }}} & {\frac{{\partial F_{xsk} }}{{\partial b_{j} }}} & {\frac{{\partial F_{xsk} }}{{\partial d_{j} }}} \\ {\frac{{\partial F_{ysk} }}{{\partial a_{j} }}} & {\frac{{\partial F_{ysk} }}{{\partial c_{j} }}} & {\frac{{\partial F_{ysk} }}{{\partial b_{j} }}} & {\frac{{\partial F_{ysk} }}{{\partial d_{j} }}} \\ \end{array} } & { - {\user2{I}}_{4 \times 4} } & \cdots & {\begin{array}{*{20}l} 0{R_{ic} } \\ 0{P_{ic} } \\ 0{R_{is} } \\ 0{P_{is} } \\ \end{array} } \\ {{\user2{O}}_{4 \times 4} } & \cdots & {{\user2{O}}_{4 \times 4} } & {{\user2{I}}_{4 \times 4} } & \cdots & {{\user2{O}}_{4 \times 2} } \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ {{\user2{O}}_{2 \times 4} } & \cdots & {{\user2{O}}_{2 \times 4} } & {{\user2{O}}_{2 \times 4} } & \cdots & {{\user2{I}}_{2 \times 2} } \\ \end{array} } \right| $$
(A-4)

The state vector of the ball-bearing transfer matrix is presented as follows:

$$ \left[ {\begin{array}{*{20}l} {\Delta x_{0} }{\Delta y_{0} }{\Delta Q_{x,0} }{\Delta Q_{y,0} } \cdots {\Delta x_{ic} }{\Delta y_{ic} }{\Delta x_{is} }{\Delta y_{is} }{\Delta Q_{x,ic} }{\Delta Q_{y,ic} }{\Delta Q_{x,is} }{\Delta Q_{y,is} } \cdots {\Delta \varOmega }1 \\ \end{array} } \right]^{\text{T}} $$

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Luo, Z., Bian, Z., Zhu, Y. et al. An improved transfer-matrix method on steady-state response analysis of the complex rotor-bearing system. Nonlinear Dyn 102, 101–113 (2020). https://doi.org/10.1007/s11071-020-05952-5

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