Nonlinear dynamic modeling of surface defects in rolling element bearing systems

In this paper an analytical model is proposed to study the nonlinear dynamic behavior of rolling element bearing systems including surface defects. Various surface defects due to local imperfections on raceways and rolling elements are introduced to the proposed model. The contact force of each rolling element described according to nonlinear Hertzian contact deformation and the effect of internal radial clearance has been taken into account. Mathematical expressions were derived for inner race, outer race and rolling element local defects. To overcome the strong nonlinearity of the governing equations of motion, a modified Newmark time integration technique was used to solve the equations of motion numerically. The results were obtained in the form of time series, frequency responses and phase trajectories. The validity of the proposed model verified by comparison of frequency components of the system response with those obtained from experiments. The classical Floquet theory has been applied to the proposed model to investigate the linear stability of the defective bearing rotor systems as the parameters of the system changes. The peak-to-peak frequency response of the system for each case is obtained and the basic routes to periodic, quasi-periodic and chaotic motions for different internal radial clearances are determined. The current study provides a powerful tool for design and health monitoring of machine systems.     

Improved model of a ball bearing for the simulation of vibration signals due to faults during run-up

In this paper an improved bearing model is developed in order to investigate the vibrations of a ball bearing during run-up. The numerical bearing model was developed with the assumptions that the inner race has only 2 DOF and that the outer race is deformable in the radial direction, and is modelled with finite elements. The centrifugal load effect and the radial clearance are taken into account. The contact force for the balls is described by a nonlinear Hertzian contact deformation. Various surface defects due to local deformations are introduced into the developed model. The detailed geometry of the local defects is modelled as an impressed ellipsoid on the races and as a flattened sphere for the rolling balls. With the developed bearing model the transmission path of the bearing housing can be taken into account, since the outer ring can be coupled with the FE model of the housing. The obtained equations of motion were solved numerically with a modified Newmark time-integration method for the increasing rotational frequency of the shaft. The simulated vibrational response of the bearing with different local faults was used to test the suitability of the envelope analysis technique and the continuous wavelet transformation was used for the bearing-fault identification and classification.     

Nonlinear dynamics of planetary gears using analytical and finite element models

Vibration-induced gear noise and dynamic loads remain key concerns in many transmission applications that use planetary gears. Tooth separations at large vibrations introduce nonlinearity in geared systems. The present work examines the complex, nonlinear dynamic behavior of spur planetary gears using two models: (i) a lumped-parameter model, and (ii) a finite element model. The two-dimensional (2D) lumped-parameter model represents the gears as lumped inertias, the gear meshes as nonlinear springs with tooth contact loss and periodically varying stiffness due to changing tooth contact conditions, and the supports as linear springs. The 2D finite element model is developed from a unique finite element-contact analysis solver specialized for gear dynamics. Mesh stiffness variation excitation, corner contact, and gear tooth contact loss are all intrinsically considered in the finite element analysis. The dynamics of planetary gears show a rich spectrum of nonlinear phenomena. Nonlinear jumps, chaotic motions, and period-doubling bifurcations occur when the mesh frequency or any of its higher harmonics are near a natural frequency of the system. Responses from the dynamic analysis using analytical and finite element models are successfully compared qualitatively and quantitatively. These comparisons validate the effectiveness of the lumped-parameter model to simulate the dynamics of planetary gears. Mesh phasing rules to suppress rotational and translational vibrations in planetary gears are valid even when nonlinearity from tooth contact loss occurs. These mesh phasing rules, however, are not valid in the chaotic and period-doubling regions.     

Continuous-scale mathematical morphology-based optimal scale band demodulation of impulsive feature for bearing defect diagnosis

Integer-scale structuring element is usually used in the traditional mathematical morphology (MM) for signal processing. When applied for impulsive feature demodulation of vibration signal of rolling element bearings, the integer-scale MM (ISMM) may lead to low resolution result and thus undermines its defect diagnosis capability. For this reason, this paper proposes a continuous-scale MM (CSMM) scheme by interpolation and re-sampling to improve scale resolution for more reliable fault signature extraction. Based on the frequency domain kurtosis criterion, a narrowband merging operation is employed to locate the optimal scale band that best reflects the impulsive feature from the CSMM analysis results. The demodulated components in the optimal scale band are employed to detect the existence of the bearing fault. The proposed optimal CSMM demodulation technique is evaluated using both simulated and experimental bearing vibration signals. The results show that, the CSMM is capable of generating demodulation signals with higher resolution, and the optimal scale band demodulation based on the CSMM can reliably extract impulsive features for bearing defect diagnosis.     

Vibration analysis of a rotating system due to the effect of ball bearing waviness

This research presents an analytical model to investigate vibration due to ball bearing waviness in a rotating system supported by two or more ball bearings, taking account of the centrifugal force and gyroscopic moment of the ball. The waviness of rolling elements is modelled by the sinusoidal function, and it is incorporated into the position vectors of the race curvature center. The Hertzian contact theory is applied to calculate the elastic deflection and non-linear contact force, while the rotor has translational and angular motions. Both the centrifugal force and gyroscopic moment of the ball and the waviness of the rolling elements are included in the kinematic constraints and force equilibrium equations of a ball to derive the non-linear governing equations of the rotor, which are solved by using the Runge–Kutta–Fehlberg algorithm to determine the new position of the rotor. The proposed model is validated by the comparison of the results of the prior researchers. This research shows that the centrifugal force and gyroscopic moment of the ball plays the important role in determining the bearing frequencies, i.e., the principal frequencies, their harmonics and the sideband frequencies resulting from the waviness of the rolling elements of ball bearing. It also shows that the bearing vibration frequencies are generated by the waviness interaction not only between the rolling elements of one ball bearing, but also between those of two or more ball bearings constrained by the rotor.     

灰色极限学习机在滚动轴承故障预测中的应用

针对较强噪声环境下的滚动轴承故障预测问题,为提高轴承故障预测的精度,提出并研究了一种新的滚动轴承预测技术。采用将灰色模型和极限学习机（ELM）相结合的方法,针对轴承运行状态值的非线性特点,先将样本数据进行灰色处理,解决数据的随机性和波动性问题,然后代入学习速度快,泛化精度高的ELM神经网络进行训练。在训练完毕后,对未来的轴承运行状态数据进行分析,将其与轴承设备的理论诊断标准相比较以达到故障预测的目的。     

ANALYSES OF DYNAMIC RESPONSE OF VEHICLE AND TRACK COUPLING SYSTEM WITH RANDOM IRREGULARITY OF TRACK VERTICAL PROFILE

A dynamic computational model for the vehicle and track coupling system is developed by means of finite element method in this paper. In numerical implementation, the vehicle and track coupling system is divided into two parts; lower structure and upper structure. The vehicle as the upper structure in the coupling system is a whole locomotive or rolling stock with two layers of spring and damping system in which vertical and rolling motion for vehicle and bogie are involved. The lower structure in the coupling system is a railway track where rails are considered as beams with finite length rested on a double layer continuous elastic foundation. The two parts are solved independently with an iterative scheme. Coupling the vehicle system and railway track is realized through interaction forces between the wheels and the rail, where the irregularity of the track vertical profile considered as stationary ergodic Gaussian random processes and simulated by trigonometry series is included. The amplitudes of vibrations, their velocities and the accelerations generated in the vehicle and rail and the interaction forces between the vehicle and the rail due to the random irregularity of the track vertical profile and different line grades and train speeds have been analyzed numerically by this model. Analyses of system responses are performed in time and frequency domains.     

利用声信号对滚动轴承进行故障诊断的研究

旋转机械在运行过程中产生的声信号包含了滚动轴承的运行状态信息,且可采用非接触式测量,本文应用它对滚动轴承进行故障诊断。基于morlet小波变换的包络分析对采集的声信号进行降噪及包络处理,然后变换到频域,提取出特征频率并经过转换后作为线性神经网路的输入向量,辨识滚动轴承的状态。实验表明,本方法对滚动轴承故障诊断是有效的。     

Vibration of a liquid with a free surface in a spinning spherical tank

The problem of forced fluid vibrations in a partially filled spinning spherical tank is solved numerically by using the finite element method. The governing equations include Coriolis acceleration and spatially homogeneous vorticity. An exponential instability is detected in the present simulation for fill ratios below 0·5 and centrifugal acceleration to thrust ratios less than 1·7. This fictitious instability appears in the model as a result of the homogeneous vortex assumption since the free slosh equations are neutrally stable in the Liapunov sense.     

光滑粒子流体动力学有限元法接触算法研究

 耦合光滑粒子流体动力学（SPH）方法和有限元法（FEM）,是研究冲击动力学问题的一种有效途径。为解决SPH粒子和有限单元间的接触问题,提出了SPH-FEM接触算法。该算法是在有限元节点处设置背景粒子,采用SPH接触算法的思想,计算施加在SPH粒子和有限元节点上的接触力,并且以外力的形式分别加入到SPH动量方程和有限元动力学方程中。利用SPH-FEM接触算法,对两杆撞击以及圆柱形钢弹正冲击钢板发生的冲塞破坏过程进行了三维数值模拟,靶板采用含损伤的Johnson-Cook模型和Grüneisen状态方程,模拟结果与实验结果吻合较好。     

The suppression of selected acoustic frequencies in MRI

A study is made of certain dominant frequencies in the acoustic noise spectrum of the magnetic resonance imaging system. Motivated by both spring and string ideas, we investigate whether the contributions to the sound from certain frequencies can be canceled by the appropriate gradient pulse sequence design. From both simulations and experiments, vibrations resulting from an impulsive force associated with a ramping up of a gradient pulse are shown to be cancelled immediately upon the application of another impulsive force coming from the subsequent appropriately timed ramping down of that pulse. A general approach to suppression of multiple-frequency contributions involving a series of gradient pulses with variable timings is given for the cancellations between pairs of impulsive forces. Various examples are confirmed through string simulations, MRI experiments, and linear response theory. This also provides a foundation to explain some results in previous papers on this subject. The method suggests that a variety of pulse profiles and timing combinations can be used to attenuate important contributions to the acoustic spectrum.     

基于电磁力近似算法的驱动电机NVH快速仿真方法

电动汽车驱动电机产生的电磁噪声是汽车NVH关注的重点问题,对其进行全转速段多工况NVH仿真通常需要耗费大量时间和计算资源。该文通过有限单元分析揭示了电机气隙电磁力随转速变化的规律,并根据这一规律提出了基于外特性曲线的电磁力时间缩放及插值的近似算法。文章采用电磁力到结构网格的映射算法对结构振动有限元模型进行激励力加载,使用声场有限元方法计算电机的辐射噪声,最终实现了车用驱动电机的电磁振动及噪声的快速仿真。使用该方法对车用电机进行全转速段振动噪声仿真,可大大压缩多工况电磁场有限元分析所需的计算时间,提升仿真效率。     

A finite element method for analysis of vibration induced by maglev trains

This paper developed a finite element method to perform the maglev train–bridge–soil interaction analysis with rail irregularities. An efficient proportional integral (PI) scheme with only a simple equation is used to control the force of the maglev wheel, which is modeled as a contact node moving along a number of target nodes. The moving maglev vehicles are modeled as a combination of spring-damper elements, lumped mass and rigid links. The Newmark method with the Newton–Raphson method is then used to solve the nonlinear dynamic equation. The major advantage is that all the proposed procedures are standard in the finite element method. The analytic solution of maglev vehicles passing a Timoshenko beam was used to validate the current finite element method with good agreements. Moreover, a very large-scale finite element analysis using the proposed scheme was also tested in this paper.     

Vibro-acoustic propagation of gear dynamics in a gear-bearing-housing system

This work developed a computational process to predict noise radiation from gearboxes. It developed a system-level vibro-acoustic model of an actual gearbox, including gears, bearings, shafts, and housing structure, and compared the results to experiments. The meshing action of gear teeth causes vibrations to propagate through shafts and bearings to the housing radiating noise. The vibration excitation from the gear mesh and the system response were predicted using finite element and lumped-parameter models. From these results, the radiated noise was calculated using a boundary element model of the housing. Experimental vibration and noise measurements from the gearbox confirmed the computational predictions. The developed tool was used to investigate the influence of standard rolling element and modified journal bearings on gearbox radiated noise.     

Model for the vibration produced by a single point defect in a rolling element bearing

A model is developed to describe the vibration produced by a single point defect on the inner race of a rolling element bearing under constant radial load. The model incorporates the effects of bearing geometry, shaft speed, bearing load distribution, transfer function and the exponential decay of vibration. A comparison of predicted and measured demodulated vibration spectra confirms the satisfactory performance of the model.     

The vibration produced by multiple point defects in a rolling element bearing

A model for the high-frequency vibration produced by a single point defect on the inner race of a rolling element bearing under radial load is extended to describe the vibration produced by multiple points defects. The model incorporates the effects of bearing geometry, speed, load distribution, transfer function and the decay of vibration. A comparison of predicted and measured spectra for a bearing with two point defects confirms satisfactory performance of the model.     

A hybrid finite element–energy finite element method for mid-frequency vibrations of built-up structures under multi-distributed loadings

In this study, a hybrid method combining finite element analysis and energy finite element analysis is developed to predict the vibrations of built-up structures in mid-frequency, and the associated general formulations are derived. The interactions between the structural components are modeled using a global matrix of the system and the reverberant blocked force. To validate the proposed method, three examples of different built-up structural systems, subjected to different types of excitations, are analyzed and discussed. These types of excitations are single point, multipoint and distributed forces on stiff member or flexible member. The results predicted by the presented method show good agreements with the dense finite element model, and the detailed local energies of the whole system are acquired under the different regional loadings. These results indicate that the proposed method can be utilized for the prediction of vibrations in the mid-frequency range.     

An improved multiscale noise tuning of stochastic resonance for identifying multiple transient faults in rolling element bearings

Stochastic resonance (SR), a noise-assisted tool, has been proved to be very powerful in weak signal detection. The multiscale noise tuning SR (MSTSR), which breaks the restriction of the requirement of small parameters and white noise in classical SR, has been applied to identify the characteristic frequency of a bearing. However, the multiscale noise tuning (MST), which is originally based on discrete wavelet transform (DWT), limits the signal-to-noise ratio (SNR) improvement of SR and the performance in identifying multiple bearing faults. In this paper, the wavelet packet transform (WPT) is developed and incorporated into the MSTSR method to overcome its shortcomings and to further enhance its capability in multiple faults detection of bearings. The WPT-based MST can achieve a finer tuning of multiscale noise and aims at detecting multiple target frequencies separately. By introducing WPT into the MST of SR, this paper proposes an improved SR method particularly suited for the identification of multiple transient faults in rolling element bearings. Simulated and practical bearing signals carrying multiple characteristic frequencies are employed to validate the performance improvement of the proposed method as compared to the original DWT-based MSTSR method. The results confirm the good capability of the proposed method in multi-fault diagnosis of rolling element bearings.     

加速度场下陀螺光纤环形变的影响分析

通过航天器大机动时直线加速场下陀螺光纤环的有限元分析,得出光纤环在加速场下的形变机理。基于各向异性复合材料理论,采用细观力学有限元方法,对光纤环代表性体积单元施加周期边界条件,分析得出光纤环的等效材料参数。再利用有限元程序ANSYS建立光纤环组件的空间有限元模型,施加加速度场,并建立接触单元分析形变过程中的结构接触耦合影响。分析结果表明:在加速场的作用下,光纤环形变的主要原因包括光纤环组件中的U型槽和顶盖形变后与光纤环的接触耦合作用,以及U型槽和顶盖结构的形变引起光纤环和光纤环本体结构的形变。     

On the sound radiation of a rolling tyre

The sound radiation from rolling tyres is still not very well understood. Although details such as horn effect or directivity during rolling have been investigated, it is not clear which vibrational modes of the tyre structure are responsible for the radiated sound power. In this work an advanced tyre model based on Wave Guide Finite Elements is used in connection with a contact model validated in previous work. With these tools the tyre vibrations during rolling on an ISO surface are simulated. Starting from the calculated contact forces in time the amplitudes of the modes excited during rolling are determined as function of frequency. A boundary element model also validated in previous work is applied to predict the sound pressure level on a reference surface around a tyre placed on rigid ground as function of the modal composition of the tyre vibrations. Taking into account different modes when calculating the vibrational field as input into the boundary element calculations, it is possible to identify individual modes or groups of modes of special relevance for the radiated sound power. The results show that mainly low-order modes with relative low amplitudes but high radiation efficiency in the frequency range around 1 kHz are responsible for the radiated sound power at these frequencies, while those modes which are most strongly excited in that frequency range during rolling are irrelevant for the radiated sound power. This fact is very essential when focusing on the design of quieter tyres.