Multi-objective rotor dynamics optimization of the plain bearing-rotor system

1. Introduction

Optimization is one of the key methods in the design of a rotor system. The natural frequencies of a rotor system were optimized by Reinhard Helfrich et al. [1], the positions and geometric sizes of the disks, the bearing stiffness and damping were considered as variables. A single-objective optimization problem was proposed by B. S. Yang et al. [2] who used Hybrid G. A. to lower the amplitude of a low-pressure rotor. Actually, besides considering the amplitude and critical speeds,the stabilityusually should be considered during the design of rotor system [3]. The stability can be changed by lots of optimization variables, so that the sealing of centrifugal pump rotors was optimized by Diewald W. et al. [4, 5], the stiffness of shaft was optimized by Choi B. G. et al. [6].

So far, the research on rotor dynamics optimization is mostly based on single-objective optimization, and multi-objective optimization problems were often transformed into single-objective optimization problems by penalty function method. But as for the design of actual rotor system, it is necessary to consider multiple factors at the same time, and it’s often impossible to find only one optimal solution set when multi-objective problem needs to be optimized, so an algorithm that can give a large number of optimal solution sets is valuable. In the meantime, the research on the optimization of rotor system stability is relatively rare. Therefore, the finite element model of the rotor system was established by using the beam elements with shear effect in this paper. After that, the stability and mass of the plain bearing-rotor system were optimized by non-dominated sorting multi-objective genetic algorithm (NSGA-II), this algorithm has the advantage of fast running and good solution convergence.

2. The rotor system model

The model of the plain bearing-rotor system is shown in Fig. 1, the dimension of the model is given in Table 1. The data of disk are given in Table 2. The material parameters of shaftare given in Table 3. The disk was simplified to mass point. The two bearings were located at nodes 11 and 15 respectively. The derivation of inertia matrix and gyro matrix of the disk were directly quoted the conclusions in [7], so do the stiffness matrix and damping matrix of the bearings. As for the plain bearing, we can get the dimensional stiffness and damping coefficient from [8].

The equation of motion of the system is:

where $R$ and $B$ represent the rotor and bearing respectively, ${\mathbf{M}}_R$ is the inertia matrix, ${\mathbf{K}}_R$ is the stiffness matrix of the rotor, ${\mathbf{K}}_B$ is the stiffness matrix of the bearing, ${\mathbf{C}}_B$ is the damping matrix, ${\mathbf{G}}_R$ is called the gyro matrix, $\mathbf{Q}$ is called the excitation force vector.

Fig. 1. The model of the rotor system

3. Optimization

For the plain bearing-rotor system in this paper, the optimization of its rotor dynamics can be described by:

where $\mathbf{X}$ is the set of design variables, $f_i$ is the objective functions,${\mathbf{X}}_{LB}$ and ${\mathbf{X}}_{UB}$ represent the boundary of the design variables.

3.1. The objective function

Before optimization, the relationship between rotational speed and the real parts of eigenvalues were given in Fig. 2. It can be seen from Fig. 2 that the real part of the natural frequency line W1 is greater than zero after the rotational speed is greater than 9000 rpm, which means the system is unstable; as for the natural frequency line W2, the unstable state will appear after the rotational speed is greater than 37500 rpm. The natural frequency lines W3 and W4 remain stable throughout the entire speed range. The above statement shows that the plain bearings bring stability problems to the rotor system. Therefore, the stability of the rotor system and the mass of the shaft were optimized in this paper, for the purpose of increasing the speed of each modal enters an unstable state, and reducing the mass of the shaft elements.

Fig. 2. Real parts of the eigenvalues before optimization

The objective functions can be defined as:

(3)

$\begin{array}{l}\min f_1=\left\{\begin{array}{l}\left(W_{max}-{\omega }_u\right)/W_{max}, {\omega }_u<W_{max},\\ 0,\mathrm{ }\mathrm{ }\mathrm{ }{\omega }_u\ge W_{max},\end{array}\right.\\ \mathrm{m}\mathrm{i}\mathrm{n}{ f}_2=m\left(\mathbf{X}\right),\end{array}$

where ${\omega }_u$ is the minimum rotational speed of each modal enters unstable state, $W_{max}=$ 50000 rpm is the maximum rotational speed of the rotor system, $m\left(\mathbf{X}\right)$ represents the mass of entire shaft elements.

4. Results

The relationship between the objective functions after optimization is shown in Fig. 3. $f_1$ is the stability of rotor system, $f_2$ is the mass of entire shaft elements. It can be seen that each individual result is better than original results, $f_1=$ 0.82 and $f_2=$ 5.4886.

Fig. 3. Optimization results for complex rotor system

Fig. 4. Real parts of eigenvalues for individual A

The real parts of eigenvalues for individual A in Pareto front is shown in Fig. 4. Individual A means that the stability is the best, whereas its mass is biggest. Individual A isn’t the most suitable choice in most cases for the designers, people still need to choose the only solution from the Pareto front. The simplest method is the reference point method [9]. There are many methods can be used to select suitable individual from the Pareto front. The reference point method is the only one of them. Deeply discussion of these methods is not the focus of this paper, so only the reference is listed here [10].

5. Conclusions

In this paper, the model of the rotor system was established by the finite element method, and the optimization of weight and stability for the rotor system was carried out by multi-objective genetic algorithm.

The research shows that:

1) Multiple optimal solutions can be given by NSGA-II algorithm, and there is no dominant relationship between any two individuals in these solutions. Therefore, a large number of alternatives can be provided through only one optimization in the rotor system design stage.

2) The stability of the rotor system with plain bearings and the weight of the shaft element had been successfully optimized, which improves the stability of the rotor system and greatly reduces the total mass of the rotor system.