Subharmonic resonance of a symmetric ball bearing–rotor system

The performance of a ball bearing–rotor system is often limited by the occurrence of subharmonic resonance with considerable vibration and noise. In order to comprehend the inherent mechanism and the feature of the subharmonic resonance, a symmetrical rotor system supported by ball bearings is studied with numerical analysis and experiment in this paper. A 6DOF rotordynamic model which includes the non-linearity of ball bearings, Hertzian contact forces and bearing internal clearance, and the bending vibration of rotor is presented and an experimental rig is offered for the research of the subharmonic resonance of the ball bearing–rotor system. The dynamic response is investigated with the aid of orbit and amplitude spectrum, and the non-linear system stability is analyzed using the Floquet theory. All of the predicted results coincide well with the experimental data to validate the proposed model. Numerical and experimental results show that the resonance frequency is provoked when the speed is in the vicinity of twice synchroresonance frequency, while the rotor system loses stability through a period-doubling bifurcation and a period-2 motion i.e. subharmonic resonance occurs. It is found that the occurrence of subharmonic resonance is due to the together influence of the non-linear factors, Hertzian contact forces and internal clearance of ball bearings. The effect of unbalance load on subharmonic resonance of the rotor system is minor, which is different from that of the sliding bearing–rotor system. However, the moment of couple has an impact influence on the subharmonic resonances of the ball bearing–rotor system. The numerical and experimental results indicate that the subharmonic resonance caused by ball bearings is a noticeable issue in the optimum design and failure diagnosis of a high-speed rotary machinery.     

Adaptive neural DSC for stochastic nonlinear constrained systems under arbitrary switchings

This paper focuses on the primary resonance analysis of a dual-rotor system having two rotor unbalance excitations of different rotating speeds and being connected by an inter-shaft ball bearing. Due to the complex excitation condition and the complicated nonlinear bearing forces of the inter-shaft bearing, the general analytical methods, e.g., the multiple scales method or the harmonic balance method, are failed to give the theoretical solutions. Thus, the harmonic balance–alternating frequency/time domain (HB–AFT) method is formulated to deal with this problem. The basic idea of the method is using the inverse discrete Fourier transform and the discrete Fourier transform, instead of the direct analytical relationship between the supposed solutions of the system and the nonlinear forces, to construct the harmonic expressions of the nonlinear forces, which is the so-called alternating frequency/time domain technique. By using the HB–AFT method, therefore, a Newton– Raphson iteration procedure can be performed to demonstrate the approximate solutions of the system. Accordingly, the frequency responses of the system affected by some critical parameters, such as rotating speed ratio, unbalances of both the inner and outer rotors, and clearance of the inter-shaft bearing, are analyzed, respectively. A variety of phenomena including double resonance peaks, biperiodic and quasi-periodic behaviors, and resonance hysteresis phenomenon are obtained, which are discussed in detail through diagrams for separated frequency responses with different frequency components and rotors’ orbits for different combinations of system parameters. Most prominently, for a relatively small unbalance of rotor as well as a relatively large clearance of the inter-shaft bearing, the resonance hysteresis phenomena are more obvious. The results obtained are also compared with the direct numerical simulation results, and the comparisons show good agreements. In addition, the methodology formulated in this paper is a general approach, which can be applied to other engineering systems with multi-frequency excitations.     

滚动轴承-偏置转子系统动力特性数值分析与实验研究

应用有限元法建立偏置转子的计算模型,采用考虑轴承Hertzian接触力和内间隙等非线性因素的二自由度滚动轴承模型,建立了滚动轴承-偏置转子系统的非线性动力学模型.通过数值仿真和实验研究分析了转子系统的非线性动力特性.实验数据和有限元模型计算结果是一致的,证实了所建立滚动轴承-转子系统非线性模型的合理性.发现由于滚动轴承非线性因素的影响,当转速达到系统共振转速的两倍附近时,激起了系统亚谐共振.     

Planetary gearbox with localised bearings and gears faults: simulation and time/frequency analysis

Planets bearings of planetary gear sets exhibit high rate of failure; detection of these faults which may result in catastrophic breakdowns have always been challenging. The objective of this paper is to investigate the planetary gears vibration properties in healthy and faulty conditions. To seek this goal a previously proposed lumped parameter model (LPM) of planetary gear trains is integrated with a more comprehensive bearing model. This modified LPM includes time varying gear mesh and bearing stiffness and also nonlinear bearing stiffness due to the assumption of Hertzian contact between the rollers/balls and races. The proposed model is completely general and accepts any inner/outer race bearing defect location and profile in addition to its original capacity of modelling cracks and spalls of gears; therefore, various combinations of gears and bearing defects are also applicable. The model is exploited to attain the dynamic response of the system in order to identify and analyze localized faults signatures for inner and outer races as well as rolling elements of planets bearings. Moreover, bearing defect frequencies of inner/outer race and ball/roller and also their sidebands are discussed thoroughly. Finally, frequency response of the system for different sizes of planets bearing faults are compared and statistical diagnostic algorithms are tested to investigate faults presence and growth.     

Effects of axial preload of ball bearing on the nonlinear dynamic characteristics of a rotor-bearing system

This research studies the effects of axial preload on nonlinear dynamic characteristics of a flexible rotor supported by angular contact ball bearings. A dynamic model of ball bearings is improved for modeling a five-degree-of-freedom rotor bearing system. The predicted results are in good agreement with prior experimental data, thus validating the proposed model. With or without considering unbalanced forces, the Floquet theory is employed to investigate the bifurcation and stability of system periodic solution. With the aid of Poincarè maps and frequency response, the unstable motion of system is analyzed in detail. Results show that the effects of axial preload applied to ball bearings on system dynamic characteristics are significant. The unstable periodic solution of a balanced rotor bearing system can be avoided when the applied axial preload is sufficient. The bifurcation margins of an unbalanced rotor bearing system enhance markedly as the axial preload increases and relates to system resonance speed.     

航空发动机整机耦合动力学模型及振动分析

面向航空发动机整机振动, 建立了航空发动机转子-滚动轴承-机匣耦合动力学模型. 该模型具有如下特点: (1)考虑转子、滚动轴承及机匣之间的耦合作用; (2)考虑了实际航空发动机的弹性支承及挤压油膜阻尼效应; (3)将转子考虑为等截面自由欧拉梁模型, 运用模态截断法进行分析; (4)考虑了滚动轴承间隙、非线性赫兹接触力以及变柔性VC(Varyingcompliance)振动; (5)考虑了转子与机匣之间的碰摩故障. 运用数值积分方法研究了航空发动机的整机振动规律, 包括: 滚动轴承VC振动分析、弹性支承刚度对耦合系统临界转速的影响、转轴模态截断阶数NM对系统响应的影响分析、挤压油膜阻尼器参数对系统响应的影响分析、突加不平衡的瞬态响应分析以及转静碰摩故障特性分析等.      

Suppression of super-harmonic resonance response using a linear vibration absorber

Super-harmonic resonances may appear in the forced response of a weakly nonlinear oscillator having cubic nonlinearity, when the forcing frequency is approximately equal to one-third of the linearized natural frequency. Under super-harmonic resonance conditions, the frequency-response curve of the amplitude of the free-oscillation terms may exhibit saddle-node bifurcations, jump and hysteresis phenomena. A linear vibration absorber is used to suppress the super-harmonic resonance response of a cubically nonlinear oscillator with external excitation. The absorber can be considered as a small mass-spring-damper oscillator and thus does not adversely affect the dynamic performance of the nonlinear primary oscillator. It is shown that such a vibration absorber is effective in suppressing the super-harmonic resonance response and eliminating saddle-node bifurcations and jump phenomena of the nonlinear oscillator. Numerical examples are given to illustrate the effectiveness of the absorber in attenuating the super-harmonic resonance response.     

Vibration based performance prediction of ball bearings caused by localized defects

This paper is focused on accurate performance prediction due to localized defects (like spalls) of microns level on the bearing components, which is essential to the design for high performance. In the mathematical formulation, the contacts between the rolling elements and the races are considered as nonlinear springs, whose stiffnesses are obtained by using Hertzian contact deformation theory. The formulation predicts the discrete spectra with the characteristic defect frequencies and their harmonics, which is helpful in prediction of system stability and to avoid severe (chaotic) vibrations in a rotor bearing system. The results are presented in the form of bifurcation diagrams, Fast Fourier Transformation (FFT) and Poincaré maps for individual defects of bearing components. The system also shows the three different categories of system behavior under nonlinear dynamic conditions.     

横向荷载作用下Winkler地基上有限长梁3次超谐共振分析

基于Winkler地基模型和Euler-Bernoulli梁理论,建立了Winkler地基上有限长梁的非线性运动方程。运用Galerkin方法对运动方程进行一阶模态截断,得到了离散的非线性振动方程,然后利用多尺度法求得了该系统3次超谐共振的幅频响应方程及其位移的一阶近似解。为揭示弹性地基上有限长梁的3次超谐共振响应的特性,分别分析了长细比、弹性模量、基床系数、阻尼、密度等主要参数对该系统3次超谐共振幅频响应曲线的影响,并通过与非共振硬激励情况的对比分析了3次超谐共振对系统实际动力反应的影响。研究结果表明:3次超谐共振响应曲线有跳跃和滞后现象;增大阻尼和基床系数均对3次超谐共振的发生有抑制作用;增大外激励幅值,系统3次超谐共振区域增大;3次超谐共振将增大系统的稳态动力响应幅值和加速度。     

Super-Harmonic and Internal Resonances of a Suspended Cable with Nearly Commensurable Natural Frequencies

In this paper, super-harmonic and internal resonance characteristics ofa viscously damped cable with nearly commensurable natural frequenciesare investigated by use of a novel method. The proposed frequency-domainsolution method is based on the combined use of a three-dimensionalnonlinear finite element approach and the incremental harmonic balancetechnique. It is an accurate algorithm in the sense that it accommodatesmulti-harmonic components and no mode-based model reduction is utilizedin the solution process. The alternating frequency/amplitude-controlledalgorithm enables complete solution to the frequency-response curvesincluding unstable branches, sub- and super-harmonic resonance andinternal resonance. A suspended cable paradigm under internal resonancecondition is studied using the proposed method. Nonlinear response andmodal interaction characteristics of the cable at different frequencyregions are identified from analysis of response profiles and harmoniccomponent features. The super-harmonic and internal resonance responsesare respectively characterized based on the harmonic distribution. Underan in-plane harmonic excitation, the two-to-one internal resonancebetween the in-plane and out-of-plane modes and the super-harmonicresonance around the second symmetric in-plane mode are revealed. Strongnonlinear interaction among different modes in the parameter spaceranging from primary resonance to super-harmonic resonance is observed.     

Forced vibration analysis of the milling process with structural nonlinearity,internal resonance,tool wear and process damping effects

In this paper, forced vibration analysis of an extended dynamic model of the milling process is investigated, in the presence of internal resonance. Regenerative chatter, structural nonlinearity, tool wear and process damping effects are included in the proposed model. Taking into account the average and first order expansion of Fourier series for cutting force components; their closed form expressions are derived. Moreover, in the presence of large vibration amplitudes, the loss of contact effect is included in this model. Analytical approximate response of the nonlinear system is constructed through the multiple-scales approach. Dynamics of the system is studied for two cases of primary and super-harmonic resonance, associated with the internal resonance. Under steady state motion, the effects of structural nonlinearity, cutting force coefficients, tool wear length and process damping are investigated on the frequency response functions of the system. In addition, existence of multiple solutions, jump phenomenon and energy transfer between vibration modes are presented and compared for tow cases of primary and super-harmonic resonances.     

直立码头前船波浪力耦合计算模型

建立了外域用Boussinesq方程、内域用刚体运动方程的直立码头前二维船剖面波浪力的时 域计算耦合模型,内域与外域在交界面的匹配条件是流量连续和压力相等. 进行了相关模型 实验,并把计算结果与实验结果进行了对比. 推导了船体与水底和直立码头之间间隙内流体 运动的自振频率,研究了间隙内流体运动的共振现象.     

A general method for impact dynamic analysis of a planar multi-body system with a rolling ball bearing joint

Impact affects the dynamic characteristics of mechanical multi-body systems and damages those rotating parts, such as the joint rolling element bearings, which are high-precision, defect intolerant components. Based on multi-body dynamic theory, Hertzian contact theory, and a continuous contact model, this study proposed a modelling method that can describe the dynamic behaviour of planar mechanical multi-body systems containing a rolling ball bearing joint under impact. In this method, the rigid bodies and bearing joint were connected according to their joint force constraints; the impact constraint between the multi-body system and the target rigid body was constructed using a continuous contact force model. Based on this method, the reflection relationship between the external impacts of the mechanical multi-body system and the variation law governing the dynamic load on the rolling bearing joint were revealed. Subsequently, an impact multi-body system, which was composed of a sliding–crank mechanism containing a rolling ball bearing joint and the target rigid body with an elastic support, was analysed to explore the dynamic response of such a complex discontinuous dynamic system andthe relevant relationship governing the dynamic load on the rolling bearing joint. In addition, a multi-body dynamic simulation software was used to build a virtual prototype of the impact slider–crank system. Compared with the theoretical model, the prototype had an additional deep groove ball bearing. That is to say, the prototype model took account of the specific geometric structural characteristics and the complex contact relationship of the inner and outer races, rolling balls, and bearing cage. Finally, the effectiveness of the theoretical method proposed in this study was verified by comparative analysis of the results. The results suggested that the external impact of a mechanical multi-body system was prone to induce sudden changes in the equivalent reaction force on its bearing joint and the dynamic load carried on its rolling balls. This study provided an effective method for exploring the distribution characteristics of dynamic loads on rolling ball bearing joints under working impact load conditions. Moreover, it offered support for the parameter optimisation of geometric structure, performance evaluation, and dynamic design of the rolling ball bearings.     

分数阶达芬振子的超谐与亚谐联合共振

研究了含分数阶微分项的达芬(Duffing)振子的超谐与亚谐联合共振.采用平均法得到了系统的一阶近似解析解,提出了超、亚谐联合共振时等效线性阻尼和等效线性刚度的概念.建立了联合共振定常解幅频曲线的解析表达式,并对联合共振幅频响应的近似解析解和数值解进行了比较,二者吻合良好,证明了求解过程及近似解析解的正确性.然后,将等效线性阻尼和等效线性刚度的概念与传统整数阶系统进行比较,证明分数阶微分项不仅起着阻尼的作用同时还起着刚度的作用.最后,通过数值仿真研究了不同的分数阶微分项系数和阶次对联合共振幅频曲线多值性和跳跃现象的影响,并与单一频率下超谐共振或亚谐共振进行了对比.研究发现,分数阶微分项系数与阶次不仅影响着系统的响应幅值、共振频率,同时还对系统的周期解个数、发生区域面积、发生先后等有重要影响.并且,在不同的基本参数下该系统分别表现出单独超谐共振、单独亚谐共振以及超谐共振和亚谐共振同时存在的现象.这些结果对系统动力学特性的研究具有重要意义.     

Effect of External Excitations on a Nonlinear System with Time Delay

The trivial equilibrium of a two-degree-of-freedom autonomous system may become unstable via a Hopf bifurcation of multiplicity two and give rise to oscillatory bifurcating solutions, due to presence of a time delay in the linear and nonlinear terms. The effect of external excitations on the dynamic behaviour of the corresponding non-autonomous system, after the Hopf bifurcation, is investigated based on the behaviour of solutions to the four-dimensional system of ordinary differential equations. The interaction between the Hopf bifurcating solutions and the high level excitations may induce a non-resonant or secondary resonance response, depending on the ratio of the frequency of bifurcating periodic motion to the frequency of external excitation. The first-order approximate periodic solutions for the non-resonant and super-harmonic resonance response are found to be in good agreement with those obtained by direct numerical integration of the delay differential equation. It is found that the non-resonant response may be either periodic or quasi-periodic. It is shown that the super-harmonic resonance response may exhibit periodic and quasi-periodic motions as well as a co-existence of two or three stable motions.     

A numerical modelling of stator–rotor interaction in a turbine stage with oscillating blades

In real flows unsteady phenomena connected with the circumferential non-uniformity of the main flow and those caused by oscillations of blades are observed only jointly. An understanding of the physics of the mutual interaction between gas flow and oscillating blades and the development of predictive capabilities are essential for improved overall efficiency, durability and reliability. In the study presented, the algorithm proposed involves the coupled solution of 3D unsteady flow through a turbine stage and the dynamics problem for rotor-blade motion by the action of aerodynamic forces, without separating the outer and inner flow fluctuations. The partially integrated method involves the solution of the fluid and structural equations separately, but information is exchanged at each time step, so that solution from one domain is used as a boundary condition for the other domain. 3-D transonic gas flow through the stator and rotor blades in relative motion with periodicity on the whole annulus is described by the unsteady Euler conservation equations, which are integrated using the explicit monotonous finite volume difference scheme of Godunov–Kolgan. The structural analysis uses the modal approach and a 3-D finite element model of a blade. The blade motion is assumed to be constituted as a linear combination of the first natural modes of blade oscillations, with the modal coefficients depending on time. A calculation has been done for the last stage of the steam turbine, under design and off-design regimes. The numerical results for unsteady aerodynamic forces due to stator–rotor interaction are compared with results obtained while taking into account blade oscillations. The mutual influence of both outer flow non-uniformity and blade oscillations has been investigated. It is shown that the amplitude-frequency spectrum of blade oscillations contains the high-frequency harmonics, corresponding to the rotor moving past one stator blade pitch, and low-frequency harmonics caused by blade oscillations and flow non-uniformity downstream from the blade row; moreover, the spectrum involves the harmonics which are not multiples of the rotation frequency.     

A new model for spall-rolling-element interaction

Evaluation of the spall size of a radially loaded rolling-element bearing is required for the assessment of the bearing damage severity and estimation of its remaining useful life. A new multi-body, nonlinear dynamic model of the interaction between the rolling-element and a spalled outer race is presented. The study focuses on the physics of rolling-element contact in a broader range of spall sizes than has been investigated to date, with intermittent contact between rolling-element and the outer race, which is a relevant defect size for diagnostics and prognosticsquery. The analysis is performed in several time intervals according to periods of rolling-element/race contact and periods when the rolling-element is not connected with the outer race. An explicit expression of the spall size as function of the time-to-impact has been developed by considering radial load, shaft speed, and gravity. The expression of the spall size was used in a sensitivity study of the effect of parameters such as geometry and radial load. The results obtained from the new model are in good agreement with a well-established general bearing model. The acceleration of the outer ring during the rolling-element/spall interaction with intermittent rolling-element race connection is a novel contribution, which allows verification of the model from direct observations using vibration sensors mounted on the structure     

Nonlinear response and stability of a spindle system supported by ball bearings

In this effort, the nonlinear responses and stability of a spindle system supported by ball bearings are presented. The dynamics of this system is described by a set of second order differential equations with a nonlinear piecewise smooth force. The Floquet theory is applied to investigate the stability of the periodic solution. Due to the loss of contact between the raceways and balls in the ball bearing, the bending of the frequency response curves switch to the left at the weak resonance region, which is similar to the frequency response curves of a system with a soft spring. With the decrease of the bearing clearance, the bending of the frequency response curves switch to the right, which is similar to the frequency response curves of a system with a hard spring. Increase of the frequency ratio, the bending of frequency response curves transforms from left to right. The route to chaos through a period doubling process is also observed in this spindle-bearing system.     

Output frequency properties of nonlinear systems

Nonlinear systems usually have complicated output frequencies. For the class of Volterra systems, some interesting properties of the output frequencies are studied in this paper. These properties show theoretically the periodicity of the output super-harmonic and inter-modulation frequencies and clearly demonstrate the mechanism of the interaction between different output harmonics incurred by different input nonlinearities in system output spectrum. These new results have significance in the analysis and design of nonlinear systems and nonlinear filters in order to achieve a specific output spectrum in a desired frequency band by taking advantage of nonlinearities. Examples and discussions are given to illustrate these new results.     

Detection of a slant crack in a rotor bearing system during shut-down

Abstract

The transient response of a slant-cracked rotor system during shut-down has been analyzed while it is decelerating through the critical speed. Vibration response has been simulated by using finite element method (FEM) for flexural vibrations. An unbalance force and a harmonically-varying torque is applied on the rotor system. Subharmonic frequency components at an interval frequency corresponding to torsional frequency were found in the frequency domain (Fast Fourier Transform (FFT) plot). These peaks are centered on the critical speed of the rotor system when a slant crack is present and can be used to detect the crack. The fundamental mode shapes of the rotor bearing systems are wavelet transformed to identify the crack location. A peak is observed at the crack location in the spatial variation of the wavelet-transformed fundamental mode shape. A few suggestions for the detection of slant cracks in the rotor during shut-down are described.