The applications of POD method in dual rotor-bearing systems with coupling misalignment

Highlights

• The dual rotor-bearing system with misalignment is established.

• POD method is the first time apply to dimension reduction of dual rotor-bearing system.

• Efficiency and accuracy of POD method is verified.

• Nonlinear characteristics of dual rotor-bearing system with misalignment are revealed.

Abstract

In this paper, the proper orthogonal decomposition (POD) method is first time apply to the dimension reduction of dual rotor-bearing experiment rig which is similar to an aero-engine rotor. The basic theory of the POD method and the application in dynamical system are introduced. The supporting bearing nonlinearity and the coupling misalignment of high and low pressure rotors are considered, and the dynamical model is established by finite element method based on dual rotor experiment rig. The frequency behaviors of dual rotor-bearing coupling misalignment response are discussed via comparing the numerical and experiment results to verify the efficiency of the model. The POD method is used for dimension reduction of rotor-bearing system, and the dynamical behaviors of the reduced-order model (ROM) are compared with the full order system (FOM) and experiment to verify the higher computational efficiency and accuracy of the POD method. These results in this paper can provide engineering guidance to actual dual rotor-bearing system with coupling misalignment fault.

Introduction

The dual rotor-bearing system is one of the core components of many kinds of advanced aero-engines which has the features of complicated structures, complex nonlinearities, thus, it is an intrinsic nonlinear dynamical system with the fiendishly complex dynamics. These complex nonlinear vibrations of dual rotor system are the sources of many faults of the aero-engine, such as the rotor/stator rub-impact [1], [2], [3], [4], blade fatigue and damages [5], [6], bearing faults [7], [8], and other mechanical failures. Thus it is important to investigate the dynamic characteristics of the dual rotor-bearing system to improve the performance and reliability of the whole aero-engine. However, the degrees of freedom (DOFs) of the dual rotor system is usually high and large-scale, consequently it is difficult to quickly analyze the dynamic characteristics of the actual system, control and optimize the structure parameters, and evaluate the reliability, etc.

Actually, many scholars have revealed the dynamic mechanisms and the complicated nonlinear vibration phenomena of the rotor-bearing system in the literatures in the past decades. A group of papers [9], [10], [11], [12], [13], [14], [15], [16] have been conducted on the researches of linear dynamic modeling or a simple nonlinear rotor modeling, which focus on the critical speed, stability and unbalance response, etc. The corresponding modeling approaches mainly include the lumped-mass method (Jeffcott model), analytical methods, transfer matrix method and finite element method (FEM)[14]. Jeffcott rotor modeling and many analytical modeling methods are only applicable to qualitative theoretical analysis, and for a real rotor system, the transfer matrix method and FEM are more appropriate for the dynamic design [17]. The later two methods, however, also have some weaknesses to deal with the real rotor-bearing system of an aero-engine because of the complicated structures and coupling of multi-nonlinearities and multi-physics[18]. The transfer matrix method does not work due to these problems. A large-scale finite element model can be obtained, but a numerical simulation of this model is usually a very high computational cost and computational resources, moreover, the numerical simulations often provide a little understanding in a certain parameters. To solve this problem, the basic idea is to reduce the dimension of the large-scale dynamical model by using some model order reduction (MOR) methods [18], [19], [20], [21], [22]. So the dimension reduction of the complex dual rotor-bearing system has become one of the central issues in the rotor dynamics of the aero-engine.

The common dimension reduction methods include center manifold method [23], Lyapunov-Schmidt (L-S) method [24], Galerkin method [25], mode synthesis method [26], POD method [27], [28], [29] and other dimension reduction methods [30]. These methods were summarized by Rega[19], Wagner[20] and Lu[22]in their applied studies of nonlinear dynamics. Benner et al. [31] summarized a survey of projection-based model dimension reduction methods for parametric dynamical systems and provided the outlooks as some new challenges.

Many researchers concentrated on model order reduction of rotor structure of aero-engine, and the main method is mode synthesis method[18]. The mode synthesis method divided the complex structure system into sub-structures, neglected the high order mode of each sub-structure, and constructed the system synthesized the low order mode of each sub-structure so that to reduce the DOFs of the system[26], [32], [33], [34]. Although the mode synthesis method is appropriate for large-scale complex structure problem, this method is a linear order reduction method and neglects the effects of high order and local modes of the system, which can lead to larger error of order reduction model. The precision is limited and order reduction efficiency is not obvious for larger deformation and strongly coupled nonlinear systems [18].

POD method is an effective and powerful method for data analysis and model dimension reduction aimed at obtaining low-order modes of the original system [35]. Lu [22] reviewed the POD method and provided the classification method of POD, and the outlooks of POD were also discussed. Karmer and Willcox [36] presented a structure-exploiting nonlinear model reduction method for systems with general nonlinearities. POD method was applied to the FitzHugh-Nagumo benchmark problem and to a tubular reactor model with Arrhenius reaction terms. The POD method was used to develop a low-dimensional parametrization of these quantities of interest, and the proposed method combines the parametrization with machine learning methods to learn the map between the input parameters and the POD expansion coefficients [37].

In many cases, the POD method has been applied for dimension reduction of rotor system via numerical simulation. Yu et al. [38] has found that the accurate ROM of a high-dimensional nonlinear rotor-bearing system can be obtained by extracting the POMs from the transient signal. The transient POD (TPOD) method was proposed in Ref. [28], and applied to the rotor system model supported by sliding bearing. Jin [21] proposed a new adaptive POD method called the interpolation Grassmann manifold to address the weakness of local property of the interpolation tangent-space of Grassmann manifold method in a wider parametric region. Ref. [39] used the POD method to reduce the 6-DOFs rotor system model to a 1 DOF rotor system model, and the bifurcation behaviors of the reduced system model was studied in details. The POD method can obtain the principal components of the complex, nonlinear system and minimized the DOF of the system, but the POD method is never applied to dual rotor system. Meanwhile, few researchers pay attention to the experiment study of POD method.

The aim of this paper is to generalize the POD method to the dual rotor-bearing experiment rig. In Section 2, the basic theory of the POD method is introduced. The dual rotor-bearing system model with coupling misalignment based on finite element method and the corresponding reduced-order model is presented in Section 3. In Section 4, the dynamical response behaviors of the dual rotor-bearing system with coupling misalignment are analyzed, and POD method is used for dimension reduction of the complex dual rotor system, the reduced-order system is compared with the original system and experiment to verify the higher computational efficiency and accuracy of POD method. Finally, the conclusions and outlooks are drawn in section 5.