Skip to main content

2022 | Book

Dynamics and Fault Diagnosis of Nonlinear Rotors and Impellers


About this book

This contributed volume presents recent developments in nonlinear dynamics applied to engineering. Specifically, the authors address stability and bifurcation in large-scale, complex rotor dynamic systems; periodic motions and their bifurcations in nonlinear circuit systems, fault diagnosis of complex engineering systems with nonlinear approaches, singularities in fluid-machinery and bifurcation analysis, nonlinear behaviors in rotor dynamic system with multi-mistuned blades, mode localization induced by mistuning in impellers with periodical and cyclic symmetry, and nonlinear behaviors in fluid-structure interaction and their control. These new results will maximize reader understand on the recent progress in nonlinear dynamics applied to large-scale, engineering systems in general and nonlinear rotors and impellers in particular.

Table of Contents

Chapter 1. Periodic Motions to Chaos in a Nonlinear Rotor System
In this chapter, a route of periodic motions to chaos in a nonlinear Jeffcott rotor system are studied through an implicit mapping method. The continuous nonlinear rotor system is discretized to construct implicit mappings. Through the mapping structures, semi-analytical solutions of periodic motions are obtained. Bifurcation trees of period-1 motion to chaos are presented. The stability and bifurcations of periodic motions are discussed through the eigenvalue analysis. Numerical simulations of periodic motions are completed with initial conditions from the analytical predictions. Phase trajectories, displacement orbits and velocity planes are presented for comparison of numerical and analytical results.
Yeyin Xu, Albert C. J. Luo
Chapter 2. Fault Diagnosis of Gear Rotor System Based on Collaborative Filtering Recommendation Method
The gear rotor system is the core component of high-end energy power equipment such as compressors, wind generators and other speed-increasing box devices, and failures are inevitable. However, the gear-rotor system often suffers from pitting, cracks and even broken teeth or their combination failures, and the vibration signals exhibited have weak, very strong random interference, nonlinear and non-stationary characteristics. In the fault diagnosis of gear rotor system based on vibration signal, the fault feature extraction and fault identification are particularly difficult.
Based on the strong characterization ability of multi-scale permutation entropy for time series signal complexity and multi-level wavelet energy entropy for non-stationary time-varying signal, the entropy change law of vibration signal, under different fault states of gear, are analyzed; In order to solve the problem of multi-class state data imbalance, an adaptive oversampling balance method is proposed; In order to solve the problem that the collaborative filtering recommendation algorithm cannot construct a scoring matrix as used in fault diagnosis, design an innovative design of the gear fault feature-state joint scoring matrix. Combining time-domain and time-frequency domain entropy, a new collaborative filtering recommended gear fault diagnosis model is constructed, and the model parameters are optimized by using gradient descent algorithm and alternating least square method; Finally, a collaborative filtering recommended gear fault diagnosis method based on adaptive minority oversampling technology and multi-scale and multi-domain entropy fusion is formed.
The results show that the diagnostic accuracy of collaborative filtering recommendation based on multi-scale permutation entropy has been improved from about 70% to about 90%, after adopting adaptive minority oversampling to balance data types, and collaborative filtering recommendation based on multi-layer wavelet energy entropy is accurate. Moreover, the rate has risen from around 75% to over 90%, and fusion of multi-scale permutation entropy and multi-layer wavelet energy entropy as the feature quantity for gear fault diagnosis, the diagnosis and recognition rate of unbalanced data can reach more than 90%, and the balanced data can reach more than 95%.
Guangbin Wang, Qingkai Han, Tengqiang Wang
Chapter 3. Study on Dynamic Behaviors of Rotor Model with Coupling Faults and Applications of TPOD Method
The transient proper orthogonal decomposition (TPOD) method is applied to order reduction in the rotor-bearing system with the coupling faults in this chapter. A 24 degrees of freedom (DOFs) rotor model supported by a pair of sliding bearings with both crack and rub-impact faults is established with the discrete modeling method. The complexity of dynamic behaviors of the rotor system with the coupling faults is analyzed via the comparison of the rotor system with the single fault (crack or rub-impact). The proper orthogonal mode (POM) energy method is proposed to confirm DOF number of the reduced model, and TPOD method is used in the coupling faults system to obtain the optimal order reduction model based on POM energy. The presented order reduction method is verified by comparing the bifurcation behaviors between the original and the reduced system. As one conclusion, it can be drawn that TPOD method could provide the optimal order reduction model to study the nonlinear dynamic behaviors of the complex rotor system with the coupling faults.
Kuan Lu, Yongfeng Yang, Jin Chen, Ruijuan Sang, Yushu Chen
Chapter 4. Nonlinear and Linear Phenomenon Investigation of Coupled Vibration of a Multi-disc Rotor Based on Multi-mistuned Blades Length or Multi-disordered Staggle Angle Blades
According to the previous researches in the authors’ Lab, some papers only used assumed mode (AM) method to investigate the rotor system with mono-flexible-disk and mono-blade that have mistuned or disordered. Calling for special attention to this work is that there are two methods adopted together, including the FEM and the assumed mode (AM) method. This chapter refined the disk-transverse, blade-bending and shaft-torsion coupling vibrations phenomenon of a multi-disc system with multi-mistune or disorder blades. The first one is primary, and the others play subsidiary roles. Some beneficial and interesting results are shown in this chapter. First of all, the authors used the AMM and FEM (ANSYS) to study and compare the change regulations of the natural frequencies (NF) and mode shapes (MS) in the rotor system. The second is that the authors pointed out how multi mistune or disorder blades affect the coupling vibration phenomenon of the system. Numerical calculation results also showed that the number of the mistune or disorder blades as well as the symmetry of the mistune or disorder blades would affect the natural frequencies. Above all, the author give the reasons why the finite element method is not suitable for analyzing these complex and changeable phenomenon of coupling vibration in a multi-disc system with multimistune or disorder blades. Lastly, the rotation effect is also explored in this chapter, and the authors found that mistuned effect would become complex and unstable.
Yi-Jui Chiu, Ya-Zheng Zhao, Xiao-Yun Li, Chia-Hao Yang, Guo-Fei Yu, Cheng-Wei Ye
Chapter 5. Lateral-Torsional-Coupled Model Based Dynamic Analyses of Spur Gears Under Time-Varying External Load Conditions with Surface Wear
To study the coupling influences of time-varying external load and gear wear on dynamics of spur gear systems, a lateral-rotational-coupled dynamic model is developed. The influence of gear wear is incorporated into the dynamical model through the Fourier formulations of meshing stiffness and static transmission error. An example gear system is used to predict dynamic characteristics with and without surface wear under constant, as well as time-varying external load conditions. The influences of wear and that of time-varying load on dynamics of the example system in time domain and frequency domain are studied at different speeds. Finally, the quantitative relationships between the dynamic factor and operating parameters as well as wear depth are revealed.
Jun Zhang, Jian Wang, Xike Li, Ligang Yao, Xianzeng Liu
Chapter 6. Experimental and Numerical Studies on Compressor Nonlinear Behaviors with Inlet Distortion and Their Interaction
It is well known that the aero-engine stability is greatly reduced by the non-uniform inflow conditions. It is important for designers to evaluate the distortion influence on the compressor stall line. This chapter presents an investigation of the interaction between a low-speed compressor and baffle inlet distortion. The circumferential distortions and the compressor response were measured and analyzed through 360° measurements and full annulus Detached-Eddy simulation. Compared with the pressure element of the inlet distortion, the velocity element played a more direct role in rotor performance. Due to the rotor characteristics with distortion, three novel distortion coefficients are presented to evaluate the distortion features. The rotor stall process with distortion is well captured through the unsteady simulation combined with experimental casing pressure. Differing from normal stall process, rotor stall process with distortion has the complex fractal and non-linear characteristics. And the inlet distortion degree is well reduced by the downstream rotor. This study indicates that numerical studies that would not be realized if only the interaction between rotor and inlet distortion was analyzed.
Jun Li, Youtian Zhou, Guoxing Song
Chapter 7. Study on Mode Localization Induced by Material and Aerodynamic Mistunings in Impellers with Periodical and Cyclic Symmetry
The structure of ideal impellers of large-scale centrifugal compressors possess typical cyclic symmetry, which can be divided into exactly the same fan-shaped substructure, with respect to both material attributes and geometric shapes. Different from non-cyclic symmetrical structure systems, the ideal impellers own many unique dynamic properties. Nevertheless, in practical operation process, impellers are always mistuned due to manufacturing errors, running wears, adherence of working medium and corrosion etc., defined as mistuning. On the other hand, the impellers are sensitive to vibration mode localization induced by mistuning. Therefore, the existing results have shown that the vibration mode localization induced by the increasing diameter of impellers is responsible for the frequent happening of the catastrophic accidents, including high-cycle fatigue fracture of centrifugal impellers and instability of rotors.
Under such background, the mode localization induced by material and aerodynamic mistunings in centrifugal impellers with cyclic symmetry are studied in this study.
First, the finite element available for analyzing material mistuning and mathematical model for aerodynamic mistuning are given in detail. The material properties, such as Young’s modulus, Possion ratio, density, are considered in the mistuning, which have influences on the mode localization of impellers of centrifugal compressor, and probability distribution function is introduced to model the stochastic sequence of mistuning. Further, such function and model are applied to the element, and the mistuning of material and the elastic matrix etc. are then obtained. More, it is found that the non-uniform of flow field in each passage could lead to mistuning of aerodynamics, and a perturbation is introduced at the inlet of impeller to simulate the difference between each passage.
Then, the results obtained are extended to software, ANSYS, and some programs are developed based on APDL. Further, the mode localization of centrifugal impeller introduced by material mistuning is studied numerically in detail. In particular, the influence of material mistuning on the mode localization is investigated, and the differences in natural frequencies between perfect and mistuning impellers are analyzed further. Following that, some important results related to material mistuning are obtained.
Finally, the modal analyses of the impellers with steady aerodynamic mistuning are carried out using ANSYS and the developed program. The fluid-structure interaction is introduced to analyze the non-uniform flow field, with cycle periodic boundary conditions. In the study, the stress stiffening effect, which is induced by steady aerodynamic forces and rarely considered in the study before, is regarded as a main factor to mode localization. For impellers working under high pressure conditions, more attentions are paid to the aerodynamic mistuning, as it may lead to rather violent mode localization easily.
Jiazhong Zhang, Yan Liu, Pengliang Wang
Chapter 8. Fluid-Structure Interactions of a Perimeter-Reinforced Membrane Wing in Laminar Shear Flow
Effect of shear inflow on the aerodynamic performance and dynamic response of a perimeter-reinforced (PR) membrane wing in two-dimensional laminar flow are investigated, using fluid-structure interaction (FSI). The FSI solution procedure developed for the study combines the modified characteristic-based split finite element method, dual-time stepping method, segment spring analogy technique, Galerkin finite element method, generalized-α method and loosely-coupled partitioned method, the stability and accuracy of which are verified by evaluating the fluid-membrane interaction of a PR membrane wing in the uniform flow against benchmark solutions. The effect of the velocity gradient in the free stream is studied in detail by analyzing the time-averaged aerodynamic load, flow-induced membrane vibration and instantaneous flow field. The numerical results show that, at very small angle of attack, the shear flow makes the membrane wing deflect downward and generates negative lift and lift-to-drag ratio, which is dangerous to the membrane-wing based Micro Air Vehicles; at moderate and high angles of attack, the shear flow usually helps to suppress the leading- and trailing-edge vortices in the flow, which reduces the amplitude of flow-induced vibration of the membrane wing.
Xu Sun, C. Steve Suh, Bo Yu
Chapter 9. Periodic Motions and Bifurcations in a Double Pendulum
In this chapter, period-1 to period-4 motions and an independent period-3 motion of a periodically forced double-pendulum are predicted through a discrete implicit mapping method. The corresponding stability and bifurcation of periodic motions are determined through eigenvalue analysis. Numerical simulations of the periodic motions in the double-pendulum system is completed for verification of analytical predictions. The harmonic terms effects on periodic motions were presented through the harmonic amplitude spectrums. For such a study, the pendulum is not expanded through the Taylor series expansion, and many higher-order harmonic terms are involved to make periodic motions complicated, which cannot be obtained from the perturbation analysis.
Chuan Guo, Albert C. J. Luo
Chapter 10. Analytical Periodic Motions for a First-Order Nonlinear Circuit System Under Different Excitations
In this chapter, the generalized harmonic method is used to obtain the analytical solutions of first-order nonlinear circuits under different excitations. At first, the direct current, harmonic amplitude and harmonic phase are obtained under different excitation frequencies, with the change of excitation amplitude in finite Fourier series. Then, the stability and bifurcation of the nonlinear system under different excitation forces are studied by the eigenvalues of the system matrix. Further, the spectrums and waveforms are compared with the results obtained from the numerical model, the circuit simulation model and the circuit. The results show that the waveforms and spectrums obtained from numerical model are in agreement well with those of circuit simulation model. However, there exist slight errors in both waveform and spectrum between the two models and the circuit, due to the nonlinear properties of the elements in the circuit and intrinsic voltage parameters of the chips. As a conclusion, the generalized harmonic balance method could accurately describe the output waveforms, amplitude and phase of nonlinear circuits. Moreover, the method can provide theoretical support for precise design of nonlinear circuits.
Yan Liu, Kai Ma, Hao He, Jun Xiao
Chapter 11. Model Reduction on Approximate Inertial Manifolds for NS Equations through Multilevel Finite Element Method and Hierarchical Basis
A numerical method is proposed to approach the Approximate Inertial Manifolds (AIMs) in unsteady incompressible Navier-Stokes equations, using multilevel finite element method with hierarchical basis functions. Following AIMS, the unknown variables, velocity and pressure in the governing equations, are divided into two components, namely low modes and high modes. Then, the couplings between low modes and high modes, which are not accounted by standard Galerkin method, are considered by AIMs, to improve the accuracy of the numerical results. Further, the multilevel finite element method with hierarchical basis functions is introduced to approach low modes and high modes in an efficient way. As an example, the flow around airfoil NACA0012 at different angles of attack has been simulated by the method presented, and the comparisons show that there is a good agreement between the present method and experimental results. In particular, the proposed method takes less computing time than the traditional method. As a conclusion, the present method is efficient in numerical analysis of fluid dynamics, especially in computing time.
M. Nauman Aslam, Jiazhong Zhang, Nannan Dang, Riaz Ahmad
Dynamics and Fault Diagnosis of Nonlinear Rotors and Impellers
Prof. Jiazhong Zhang
Copyright Year
Electronic ISBN
Print ISBN

Premium Partners