# Stator-Rotor Contact Force Estimation of Rotating Machine

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## Abstract

**:**

## 1. Introduction

## 2. Stator–Rotor Contact Force Modeling

#### 2.1. Modeling of Rotating Shaft

#### 2.2. Modeling of Stator–Rotor Contact

## 3. State and Contact Force Estimation

## 4. Results

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Finite Element Modeling

## References

- Jacquet-Richardet, G.; Torkhani, M.; Cartraud, P.; Thouverez, F.; Baranger, T.N.; Herran, M.; Gibert, C.; Baguet, S.; Almeida, P.; Peletan, L. Rotor to stator contacts in turbomachines. Review and application. Mech. Syst. Signal Process.
**2013**, 40, 401–420. [Google Scholar] [CrossRef] [Green Version] - Ma, H.; Zhao, Q.; Zhao, X.; Han, Q.; Wen, B. Dynamic characteristics analysis of a rotor–stator system under different rubbing forms. Appl. Math. Model.
**2015**, 39, 2392–2408. [Google Scholar] [CrossRef] - Chamroon, C.; Cole, M.O.; Wongratanaphisan, T. An active vibration control strategy to prevent nonlinearly coupled rotor–stator whirl responses in multimode rotor-dynamic systems. IEEE Trans. Control Syst. Technol.
**2013**, 22, 1122–1129. [Google Scholar] [CrossRef] [Green Version] - Jiang, J.; Shang, Z.; Hong, L. Characteristics of dry friction backward whirl—A self-excited oscillation in rotor-to-stator contact systems. Sci. China Technol. Sci.
**2010**, 53, 674–683. [Google Scholar] [CrossRef] - Jiang, J. Determination of the global responses characteristics of a piecewise smooth dynamical system with contact. Nonlinear Dyn.
**2009**, 57, 351–361. [Google Scholar] [CrossRef] [Green Version] - Varney, P.; Green, I. Nonlinear phenomena, bifurcations, and routes to chaos in an asymmetrically supported rotor–stator contact system. J. Sound Vib.
**2015**, 336, 207–226. [Google Scholar] [CrossRef] - Prabith, K.; Krishna, I.P. The numerical modeling of rotor–stator rubbing in rotating machinery: A comprehensive review. Nonlinear Dyn.
**2020**, 101, 1317–1363. [Google Scholar] [CrossRef] - Ehehalt, U.; Alber, O.; Markert, R.; Wegener, G. Experimental observations on rotor-to-stator contact. J. Sound Vib.
**2019**, 446, 453–467. [Google Scholar] [CrossRef] - Chipato, E.T.; Shaw, A.D.; Friswell, M.I. Nonlinear rotordynamics of a MDOF rotor–stator contact system subjected to frictional and gravitational effects. Mech. Syst. Signal Process.
**2021**, 159, 107776. [Google Scholar] [CrossRef] - von Groll, G.T.; Ewins, D.J. A mechanism of low subharmonic response in rotor/stator contact—Measurements and simulations. J. Vib. Acoust.
**2002**, 124, 350–358. [Google Scholar] [CrossRef] - Varney, P.; Green, I. Rotordynamic analysis of rotor–stator rub using rough surface contact. J. Vib. Acoust.
**2016**, 138. [Google Scholar] [CrossRef] - Kuseyri, İ.S. Robust control and unbalance compensation of rotor/active magnetic bearing systems. J. Vib. Control
**2012**, 18, 817–832. [Google Scholar] [CrossRef] - Kitanidis, P.K. Unbiased minimum-variance linear state estimation. Automatica
**1987**, 23, 775–778. [Google Scholar] [CrossRef] - Gillijns, S.; De Moor, B. Unbiased minimum-variance input and state estimation for linear discrete-time systems. Automatica
**2007**, 43, 111–116. [Google Scholar] [CrossRef] - Friedland, B. Treatment of bias in recursive filtering. IEEE Trans. Autom. Control
**1969**, 14, 359–367. [Google Scholar] [CrossRef] - Hmida, F.B.; Khémiri, K.; Ragot, J.; Gossa, M. Three-stage Kalman filter for state and fault estimation of linear stochastic systems with unknown inputs. J. Frankl. Inst.
**2012**, 349, 2369–2388. [Google Scholar] [CrossRef] - Ding, B.; Zhang, T.; Fang, H. On the equivalence between the unbiased minimum-variance estimation and the infinity augmented kalman filter. Int. J. Control
**2020**, 93, 2995–3002. [Google Scholar] [CrossRef] - Klein, B. Grundlagen und Anwendungen der Finite-Element-Methode im Maschinen-und Fahrzeugbau; Springer Vieweg: Wiesbaden, Germany, 2012. [Google Scholar]
- Yu, J.; Goldman, P.; Bently, D.; Muzynska, A. Rotor/seal experimental and analytical study on full annular rub. J. Eng. Gas Turbines Power
**2002**, 124, 340–350. [Google Scholar] [CrossRef] - Kim, Y.B.; Noah, S. Quasi-periodic response and stability analysis for a non-linear Jeffcott rotor. J. Sound Vib.
**1996**, 190, 239–253. [Google Scholar] [CrossRef] - Kwakernaak, H.; Sivan, R. Linear Optimal Control Systems; John Wiley & Sons: Hoboken, NJ, USA, 1972; Volume 1. [Google Scholar]

**Figure 1.**Finite element modeling of elastic rotor. (

**a**) Shaft rotating around the z axis with angle $\phi $ and angular velocity $\mathsf{\Omega}$. Rotating shaft modeled by eight elastic beam elements with nine nodes. Displacements of node two and eight in x and y directions and rotating angle $\phi $ measured. (

**b**) Each node i is described by displacements ${x}_{i}$, ${y}_{i}$ and angles ${\theta}_{{x}_{i}}$, ${\theta}_{{y}_{i}}$.

**Figure 2.**Radial displacement and stator–rotor contact. (

**a**) Radial displacement ${d}_{5}$ of node five and rotation of node five around z axis with angular velocity ${\mathsf{\Omega}}^{\ast}$. Both effects are caused by unbalanced mass ${m}_{u}$, which generates unbalance force ${F}_{u}$. (

**b**) Rotor–stator contact with normal force ${F}_{s}$ and force of friction ${F}_{f}$. Body-fixed coordinate system with axis ${x}^{\ast}$, ${y}^{\ast}$ rotating with shaft. Velocity ${v}_{c,{x}^{\ast}}$ is velocity of rotating shaft at contact point projected on the ${x}^{\ast}$ axis.

**Figure 3.**Visualization of forward whirl. (

**a**) Rotating angles $\phi \left(t\right)={\int}_{{t}_{0}}^{t}\mathsf{\Omega}d\tau $ and ${\phi}^{\ast}\left(t\right)={\int}_{{t}_{0}}^{t}{\mathsf{\Omega}}^{\ast}d\tau $ with $\mathsf{\Omega}$, ${\mathsf{\Omega}}^{\ast}$ defined by Figure 2a. (

**b**) Time span in which radial displacement exceeds clearance for the first time. Visualization of true and estimated radial displacements.

**Figure 4.**Visualization of true and estimated contact forces during forward whirl. Components x and y with respect to inertial coordinate system. (

**a**) y−component. (

**b**) x−component.

**Figure 5.**Visualization of backward whirl. (

**a**) Rotating angles $\phi \left(t\right)={\int}_{{t}_{0}}^{t}\mathsf{\Omega}d\tau $ and ${\phi}^{\ast}\left(t\right)={\int}_{{t}_{0}}^{t}{\mathsf{\Omega}}^{\ast}d\tau $ with $\mathsf{\Omega}$, ${\mathsf{\Omega}}^{\ast}$ defined by Figure 2a. (

**b**) Time span in which radial displacement exceeds clearance for the first time. Visualization of true and estimated radial displacements.

**Figure 6.**Visualization of true and estimated contact forces during backward whirl. Components x and y with respect to inertial coordinate system. (

**a**) y−component. (

**b**) x−component.

Description | Parameter | Unit | Value |
---|---|---|---|

Length of beam elements | ${l}_{i}\phantom{\rule{4.pt}{0ex}}\mathrm{with}\phantom{\rule{4.pt}{0ex}}i\in \mathcal{A}$ | m | 437.5 |

Cross section area | ${\mathsf{\Sigma}}_{i}\phantom{\rule{4.pt}{0ex}}\mathrm{with}\phantom{\rule{4.pt}{0ex}}i\in \mathcal{B}$ | ${\mathrm{mm}}^{2}$ | 159,043 |

${\mathsf{\Sigma}}_{j}\phantom{\rule{4.pt}{0ex}}\mathrm{with}\phantom{\rule{4.pt}{0ex}}j\in \mathcal{C}$ | ${\mathrm{mm}}^{2}$ | 31,416 | |

Material density | $\rho $ | $\mathrm{kg}/{\mathrm{m}}^{3}$ | 7850 |

Shaft mass | m | kg | 863 |

Unbalance mass | ${m}_{u}$ | kg | 20 |

Displacement of unbalance mass | $\gamma $ | mm | 35 |

Young modulus | E | $\mathrm{kg}/\mathrm{mms}{}^{2}$ | $190\times {10}^{4}$ |

Second moment of area | ${J}_{i}\phantom{\rule{4.pt}{0ex}}\mathrm{with}\phantom{\rule{4.pt}{0ex}}i\in \mathcal{B}$ | ${\mathrm{mm}}^{4}$ | $2.01\times {10}^{9}$ |

${J}_{j}\phantom{\rule{4.pt}{0ex}}\mathrm{with}\phantom{\rule{4.pt}{0ex}}j\in \mathcal{C}$ | ${\mathrm{mm}}^{4}$ | $7.85\times {10}^{7}$ | |

Spring coefficient (bearing) | ${k}_{b}$ | $\mathrm{N}/\mathrm{m}$ | $10\times {10}^{8}$ |

Damper coefficient (bearing) | ${d}_{b}$ | $\mathrm{Ns}/\mathrm{m}$ | 1000 |

Damping coefficients (shaft) | $\alpha $ | - | 0.002 |

$\beta $ | - | 0.002 | |

Clearance | ${r}_{0}$ | mm | 1.55 |

Mean ${\mathit{p}}_{\mathit{x},\mathit{k}}$ | Mean ${\mathit{p}}_{\mathit{y},\mathit{k}}$ | Median ${\mathit{p}}_{\mathit{x},\mathit{k}}$ | Median ${\mathit{p}}_{\mathit{y},\mathit{k}}$ | |
---|---|---|---|---|

Forward whirl | 219.54 | 257.34 | 21.03 | 21.01 |

Backward whirl | 78.89 | 80.51 | 15.89 | 16.36 |

Forward whirl * | 24.30 | 24.21 | 16.90 | 17.09 |

Backward whirl * | 22.44 | 23.05 | 12.26 | 13.06 |

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**MDPI and ACS Style**

Spiller, M.; Söffker, D.
Stator-Rotor Contact Force Estimation of Rotating Machine. *Automation* **2021**, *2*, 83-97.
https://doi.org/10.3390/automation2030005

**AMA Style**

Spiller M, Söffker D.
Stator-Rotor Contact Force Estimation of Rotating Machine. *Automation*. 2021; 2(3):83-97.
https://doi.org/10.3390/automation2030005

**Chicago/Turabian Style**

Spiller, Mark, and Dirk Söffker.
2021. "Stator-Rotor Contact Force Estimation of Rotating Machine" *Automation* 2, no. 3: 83-97.
https://doi.org/10.3390/automation2030005