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Magnetic bearings are widely applied in High Temperature Gas-cooled Reactor (HTGR) and auxiliary bearings are important backup and safety components in AMB systems. The performance of auxiliary bearings significantly affects the reliability, safety, and serviceability of the AMB system, the rotating equipment, and the whole reactor. Research on the dynamic behavior during the touchdown process is crucial for analyzing the severity of the touchdown. In this paper, a data-based dynamic analysis method of the touchdown process is proposed. The dynamic model of the touchdown process is firstly established. In this model, some specific mechanical parameters are regarded as functions of deformation of auxiliary bearing and velocity of rotor firstly; furthermore, a machine learning method is utilized to model these function relationships. Based on the dynamic model and the Kalman filtering technique, the proposed method can offer estimation of the rotor motion state from noisy observations. In addition, the estimation precision is significantly improved compared with the method without learning. The proposed method is validated by the experimental data from touchdown experiments.

Magnetic bearings are widely applied in High Temperature Gas-cooled Reactor (HTGR), where the rotating machines are running under highly purified helium environment. Compared with conventional bearings, Active Magnetic Bearings (AMBs) possess several attractive advantages, such as no friction, no need of lubrication, and the ability of long-term high speed running. Some types of essential rotating equipment in HTGR are supported by AMBs, that is, the main helium circulator and the helium blower in the fuel circulation system. The reliability of this equipment, especially in accident condition, plays a crucial role in the safety of the whole nuclear plant.

In an AMB system, the rotating rotor is suspended by electromagnetic forces and there is no contact between the rotor and the stator. An AMB system is a complex mechatronic system and consists of a large number of structural, electronic components and sophisticated control software. AMB systems are usually equipped with a series of control, protection, backup, and safety components and software to ensure the reliability and the safety of the whole rotating equipment. The control and protection software of an AMB system is designed to suspend the rotor within a wide range of operating conditions. Some abnormal operation conditions, for example, long-term overload, unexpected impact, and slight failure of components, are sustainable and can hardly affect the suspension. However, in the worst case, the magnetic suspension will fail and the rotor will touch down. To avoid damage to the rotor and the stator during the touchdown process, the so-called auxiliary bearings are necessary for AMB systems. The auxiliary bearings bear the rotating rotor during the touchdown process. They are important backup and safety components in AMB systems and can be regarded as the “last stand” for the safety of an AMB system.

In the applications associated with HTGR, the performance of auxiliary bearings significantly affects the reliability, safety, and serviceability of the AMB system, the rotating equipment, and the whole reactor. Research on the dynamic behavior during the touchdown process is crucial for analyzing the severity of the touchdown. Evaluating the contact forces between the rotor and the auxiliary bearings plays a central role of dynamic analysis of a touchdown process. However, due to the restrictions in structure design, it is difficult to equip force sensors in engineering rotating equipment. Thus these forces can hardly be measured directly and should be estimated from the acceleration and velocity of the rotor. On the other hand, only the displacements of the rotor are recorded in AMB systems. As is well known, estimating the velocity and acceleration from noisy displacement data is hardly realizable. Therefore this paper focuses on the estimation of velocities and accelerations of the rotor based on the dynamic model and the measured data.

Many remarkable achievements have been accomplished in the literature to estimate the highly nonlinear dynamic process associated with rotor touchdown. In 1991, Ishii and Kirk [

On the other hand, rapidly developing machine learning techniques offer an attractive solution for data process. In the region of motion control of mechatronic systems, soft sensing [

In this paper, a novel touchdown process analysis method based on prior knowledge based learning technique is proposed. Similar to some above-mentioned literature, the presented work mainly utilizes a data-based design verification strategy. In other words, the proposed method depends on the actual data and will be implemented in the machine which is already designed and built and when touchdown incident or test has occurred.

More specifically, this paper is an extension of [

The proposed method can be utilized to evaluate the operational state and residual life of the auxiliary bearings [

Prior Knowledge Based Kernel Regression (PKBKR) [

Suppose that

In this paper, the output

In this paper, auxiliary bearing in the form of angular contact rolling-element bearing and vertically arranged AMB system is studied. A bearing of this type can bear both radial and axial load. In other words, it can be utilized as a radial-axial auxiliary bearing.

The touchdown process is quite complicated, which involves highly nonlinear and coupled interactions between the rotor, inner ring, balls, and outer ring of auxiliary bearings. Among these interactions, the impact between the rotor and the inner ring of auxiliary bearing plays a dominant role. Thus, for simplicity, in this paper the interaction between the rotor and a radial-axial auxiliary bearing is considered, the auxiliary bearing is regarded as a whole, and the dynamic characteristics of auxiliary bearing parts and interactions between these parts are presented by the contact model with variable mechanical parameters, which will be discussed in this section.

On the other hand, as discussed in [

The rotor motion can be described by its displacements in three directions, namely,

Figure

Sketch of auxiliary bearing and contact forces.

The dynamic behavior of the rotor can be described by the following equations [

In the touchdown process, the dynamic behaviors of the rotor can be divided into a few classes according to the contact situations between the rotor and the auxiliary bearing. In the following part we divide the contact forces into two parts. The subscripts

(

(a) No radial contact exists: the criterion of this case is

(b) Radial contact occurs: in this case, the radial force is given by the stiffness-damping model, and the tangent and axial forces are given by the friction model; namely,

(

(a) No axial contact exists: the criterion of this case is

(b) Axial contact occurs: in this case, the axial force is obtained by the stiffness-damping model, and the radial force is zero. The tangent force, estimated by theoretical derivations and experiments, is proportional to the axial positive pressure due to friction. It is important to notice the positive pressure is exactly

The model discussed in the above subsection involves some mechanical parameters, such as

Kalman filtering is an effective algorithm for state estimation based on system model and observations. Nonlinear extended Kalman filtering can be applied to deal with the nonlinear discrete-time system.

To apply Kalman filtering the state space description should be established firstly. Define the state variable as

The derivative matrix of

In the last subsection, the system is described by the following time-invariant continuous-time state space description:

The first-order derivative matrix of

Then extended Kalman filtering can be applied to compute the rotor displacement estimation

Suppose that all mechanical parameters are given and a measured data set

Experiments are performed to validate the proposed method. The data in the experiments are attained from 21 touchdown experiments on the backup helium circulator system of HTR-10. A detailed description of this system can be found in [

Summation of touchdown experiments.

Number of experiments | Touchdown speed (rpm) | Brake? | Additional axial load? | Auxiliary bearing type |
---|---|---|---|---|

1 | 5000 | No | No | Ceramic ball |

2 | 5000 | No | No | Ceramic ball |

3 | 5000 | No | No | Ceramic ball |

4 | 5000 | No | No | Ceramic ball |

5 | 5000 | No | No | Ceramic ball |

6 | 5000 | Yes | No | Ceramic ball |

7 | 5000 | Yes | No | Ceramic ball |

8 | 5000 | Yes | Yes | Ceramic ball |

9 | 5000 | Yes | Yes | Ceramic ball |

10 | 5000 | Yes | Yes | Ceramic ball |

11 | 5000 | Yes | Yes | Ceramic ball |

12 | 5000 | Yes | Yes | Ceramic ball |

13 | 5000 | Yes | No | Ceramic ball |

14 | 5000 | Yes | No | Ceramic ball |

15 | 5000 | Yes | Yes | Ceramic ball |

16 | 5000 | Yes | Yes | Ceramic ball |

17 | 5000 | Yes | Yes | Ceramic ball |

18 | 5000 | Yes | Yes | Ceramic ball |

19 | 5000 | Yes | Yes | Ceramic ball |

20 | 5000 | Yes | Yes | Ceramic ball |

21 | 5000 | No | No | Steel ball |

In the analysis, the data in the first 0.15 seconds after the touchdown is used. The proposed method is utilized to estimate the state of the system. Moreover, the results in [

Firstly, the results of rotor displacement estimation in experiment #11 are shown in Figures

Measured and estimated displacements.

Measured and estimated displacements.

Measured and estimated displacements.

Moreover, the displacement estimation errors of all 21 experiments are shown in Figure

Displacement estimation errors. All directions and all 21 experiments.

As shown in Figure

It is obvious that the proposed method makes an excellent estimation of rotor displacement and outperforms the method without learning, especially in the estimation of

The main reason is that, in the original dynamic model, some mechanical parameters are assumed to be constant, but in practice these parameters may be affected by the motion of rotor and auxiliary bearing. By introducing learning method, these parameters are allowed to vary with respect to rotor motion and the quality of the dynamic model is significantly improved. Moreover, the relationship between these parameters and the rotor motion is determined by the measured data; thus precise estimations are achieved. On the other hand, as for the precision of radial displacement estimation, both methods (with and without learning) are similar, shown in Figure

The estimated contact force in experiment #11 is shown in Figure

Estimated force. Experiment #11.

The experiment results have shown the validity of the proposed method in estimating the motion state of the rotor during the touchdown process.

Furthermore, in order to visualize the force estimations in all 21 experiments, the peak values of axial and radial forces are extracted. The peak values are grouped according to the number of impacts. The statistical distributions are illustrated in the form of boxplot, shown in Figures

Estimated axial force. All 21 experiments.

Estimated normal force. All 21 experiments.

Magnetic bearings are widely applied in High Temperature Gas-cooled Reactor (HTGR) and auxiliary bearings are important backup and safety components in AMB systems. In the applications associated with HTGR, the performance of auxiliary bearings significantly affects the reliability, safety, and serviceability of the AMB system, the rotating equipment, and the whole reactor. The dynamic model of the touchdown process is firstly established. In this model, some mechanical parameters are regarded as functions of deformation of auxiliary bearing and velocity of the rotor and a machine learning technique is utilized to model these function relationships. Based on the dynamic model and Kalman filtering technique, the proposed method can offer estimations of rotor’s motion state from noisy observations and estimation precision is significantly improved compared with the method without learning. The proposed method is validated by the experimental data from touchdown experiments.

The proposed method in this paper provides a novel data processing method for the touchdown process and is a foundation for further researches.

The future work includes the following:

(

(

(

Contact angle between the rotor and the auxiliary bearing

Second-order derivatives of displacements, namely, the accelerations of the rotor

Derivatives of displacements, namely, the velocities of the rotor

Vector of auxiliary bearing radial deformation and rotor radial velocity

Vector of auxiliary bearing axial deformation and rotor axial velocity

Friction coefficient between the rotor and the auxiliary bearing

Rotational speed of the rotor

Signum function

Rotational angle of the rotor

Rotational angle of the rotor at the instant of the shutdown of magnetic bearings

Radial and axial damping coefficient

Eccentricity between the mass center and the rotational center of the rotor

Radial, tangential, and axial contact forces between the rotor and the auxiliary bearing

Contact forces caused by radial contact

Contact forces caused by axial contact

Acceleration due to gravity

Radial and axial stiffness

Coefficient between tangential force and axial positive pressure

Mass of the rotor

Kernel function

Radial displacement of the rotor

Nominal radial and axial clearance between the rotor and the auxiliary bearing at operation position

Time elapsed since the shutdown of magnetic bearings

Sample sets

Displacements of the rotor in three directions

Coefficients in learning models.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

This paper is financially supported by the National Science and Technology Major Project of China (2011ZX069) and Project 61305065 and Project 51275261 supported by NSFC.