The Transient POD Method Based on Minimum Error of Bifurcation Parameter

Kuan Lu,Haopeng Zhang,Yulin Jin,Shibo Zhao,Yushu Chen,Kangyu Zhang,Chao Fu

doi:10.3390/math9040392

Abstract

An invariable order reduction model cannot be obtained by the adaptive proper orthogonal decomposition (POD) method in parametric domain, there exists uniqueness of the model with different conditions. In this paper, the transient POD method based on the minimum error of bifurcation parameter is proposed and the order reduction conditions in the parametric domain are provided. The order reduction model equivalence of optimal sampling length is discussed. The POD method was applied for order reduction of a high-dimensional rotor system supported by sliding bearings in a certain speed range. The effects of speed, initial conditions, sampling length, and mode number on parametric domain order reduction are discussed. The existence of sampling length was verified, and two- and three-degrees-of-freedom (DOF) invariable order reduction models were obtained by proper orthogonal modes (POM) on the basis of optimal sampling length.

Highlights

Many actual engineering systems are complex, high-dimensional, nonlinear and uncertain, e.g., aero-engine, steam turbine, etc [1–4]
If P(x0, α, ts) is the function of the system parameters, initial conditions, and sampling length, what sampling parameters can make Equation (12) small enough? Let us analyze this problem by defining the truncation error function (TEF), the steps can be expressed as follows: Response of the original system can be approximated as Equation (16) when the parameters satisfy x0 ∈ R2n, α ∈ Rl, ts ∈ R+ based on the proper orthogonal decomposition (POD) method m
POD proper orthogonal modes (POM) function φk x0, α, topt m k=1 obtained via optimal sampling length can approximate to the original system in the entire parametric domain if Em(t, x0, α, topt) < ε, ε is sufficiently small

Summary

Introduction

Many actual engineering systems are complex, high-dimensional, nonlinear and uncertain, e.g., aero-engine, steam turbine, etc [1–4]. Model order reduction (MOR) should be applied to reduce the high-dimensional system and the reduced system model is used to replace the original system. A series of MORs have been proposed to study high-dimensional engineering systems [7–11], and the methods have been summarized by the researchers in their applied studies of nonlinear dynamics [12–14]. Lu [9] proposed the transient POD method and applied it to the high-dimensional and nonlinear rotor-bearing system model. The key problem of the POD method is obtaining an invariable order reduction model of a high-dimensional system, and the order reduction model can reserve similar bifurcation behaviors of the original system in the parameter domain. On the basis of the order reduction model obtained by POD method, theoretical analysis of the high-dimensional system can be carried out.

Order Reduction Conditions of Parametric Domain

Order Reduction Model Equivalence of Optimal Sampling Length

Modeling of Rotor-Bearing System with Looseness Fault

Analysis of Dynamics and Order Reduction Efficiency

Conclusions and Outlooks