A general model of the axle vibration in piezoelectric motors

In the present work a general model of the vibrational behavior of the axle of a piezoelectric motor is proposed. In this motor, a cylinder-shaped permanent magnet, which act as a rotor, is pressed in contact with an end of a steel axle by means of the magnetic forces. The other end of the axle is fitted at the center of a rotating traveling wave generator. A piezoelectric membrane, vibrating in a flexural anti-symmetrical mode, or a thick disk, vibrating in a radial anti-symmetrical mode, can be exploited as traveling wave generators. In the first case a bending moment, in the second case a transverse force is applied to the axle. In both cases, if the driving frequency coincides with a resonance frequency of the axle, the axle acts as a resonant displacement amplifier; a continuous slipping takes place between the axle and the rotor, and a torque is transmitted to the rotor. The proposed model is able to describe the axle vibrational behavior when it is excited by a bending moment, by a transverse force, and also when these two excitations are simultaneously applied. The axle is modeled as a four-port system and all its transfer functions, as well as the transversal displacement along the axle at each frequency can be easily computed. Computed results have been compared with experimental measurements carried out on two motor prototypes that exploit as traveling wave generators a membrane and a disk, respectively. A good agreement was obtained by properly taking into account the loading effect of the generator on the axle.     

Soil influence on unbalance response and stability of a simple rotor-foundation system

The equations of motion are set up for a simple rotor (Jeffcott or Laval rotor) on a rigid foundation mass resting on an elastic half space (soil). The unbalance response and the stability limit against self-excited vibrations caused by the internal damping of the rotating shaft are calculated. The numerical results presented as response diagrams and stability graphs show that the damping effect of the soil on the system, due to radiation of energy, may have a very positive influence on the smooth running of the rotors.     

Influence of load torque on stability of rotor driven by flexible shaft

The influence of the load torque on the stability of a symmetric rotor, driven by a flexible shaft, is studied. Both linear and angular displacements of the rotor are considered. The analysis—which is approximate, and with the deflection, the damping and the load torque assumed to be small—shows that the main destabilizing effect of the load torque is due to the transverse moments acting while the rotor is inclined. The reaction pattern at the ends of the shaft, determined by means of the Kirchhoff equations, indicates that the semitangential mode of loading (conservative) is operative.     

Transverse vibration of a rotor system driven by a Cardan joint

The transverse vibration of a rotor system driven by a Cardan joint is analyzed and the effect of the transmitted torque on the dynamic stability of the system evaluated. As a result of the analysis, the following facts are proved: when the driving shaft and driven shaft (rotor shaft) are included, both parametric and self-excited vibrations arise due to transmitted torque; asymmetrical stiffness of the rotor supports has the effect of stabilizing this self-excited vibration.     

Coupled longitudinal and bending vibrations of a rotating shaft with an open crack

The coupling of longitudinal and bending vibrations of a rotating shaft, due to an open transverse surface crack is investigated. The assumption of the open crack leads to a system with behaviour similar to that of a rotor with dissimilar moments of inertia along two perpendicular directions. The local flexibility due to the presence of the crack can be represented by way of a 6×6 matrix for six degrees of freedom in a short shaft element which includes the crack. This matrix has off-diagonal terms which cause coupling along the directions which are indicated by these terms. Here shear is not considered and three degrees of freedom are used: bending in the two main directions and extension. This leads to a 3×3 stiffness matrix with coupling terms. The undamped free and forced coupled vibration are first considered. The coupling is investigated and the effects of unbalance and gravity are examined. Then damped coupled vibration is considered for free and forced vibration. The existence of coupling between longitudinal and bending vibration due to the crack is a very useful property which, together with the sub-critical resonance due to crack, can form a basis for crack identification in rotating shafts. New and interesting phenomena of coupled transverse and longitudinal motion are presented and discussed.     

Design sensitivity analysis of dynamic responses for a BLDC motor with mechanical and electromagnetic interactions

This paper presents a design sensitivity analysis of dynamic responses of a BLDC motor with mechanical and electromagnetic interactions. Based on the equations of motion which consider mechanical and electromagnetic interactions of the motor, the sensitivity equations for the dynamic responses were derived by applying the direct differential method. From the sensitivity equation along with the equations of motion, the time responses for the sensitivity analysis were obtained by using the Newmark time integration method. The sensitivities of the motor performances such as the electromagnetic torque, rotating speed, and vibration level were analyzed for the six design parameters of rotor mass, shaft/bearing stiffness, rotor eccentricity, winding resistance, coil turn number, and residual magnetic flux density. Furthermore, to achieve a higher torque, higher speed, and lower vibration level, a new BLDC motor was designed by applying the multi-objective function method. It was found that all three performances are sensitive to the design parameters in the order of the coil turn number, magnetic flux density, rotor mass, winding resistance, rotor eccentricity, and stiffness. It was also found that the torque and vibration level are more sensitive to the parameters than the rotating speed. Finally, by applying the sensitivity analysis results, a new optimized design of the motor resulted in better performances. The newly designed motor showed an improved torque, rotating speed, and vibration level.     

Detailed ball bearing model for magnetic suspension auxiliary service

Catcher bearings (CBs) provide backup protection for rotating machines with active magnetic bearings (AMBs). The CBs are required in the event of an AMB failure or high transient loads. Numerical simulations of a rotor drop on CBs in flywheel energy storage system are conducted with a detailed CB model which includes a Hertzian load–deflection relationship between mechanical contacts, speed-and-preload-dependent bearing stiffness due to centrifugal force, and a Palmgren's drag friction torque. The transient simulation results show the rotor shaft response variations with the design parameters: shaft/bearing friction coefficients, axial preload, support damping of damper liner, and side loads from magnetic bearings. The results reveal that friction coefficients, support damping, and side loads are critical parameters to satisfy CB design objectives and prevent backward (super) whirl.     

Vibration of cracked shafts in bending

Cracked rotating shafts exhibit a certain particular dynamic response due to the local flexibility of the cracked section. In this response, most of the features of the response of a shaft with dissimilar moments of inertia can be identified. Moreover, the non-linear behavior of the closing crack introduces the characteristics of non-linear systems. For many practical applications, the system can be considered bi-linear and analytical methods can be applied. A de Laval rotor with an open crack is investigated by way of application of the theory of shafts with dissimilar moments of inertia. Furthermore, analytical solutions are obtained for the closing crack under the assumption of large static deflections, a situation common in turbomachinery. Finally, a solution is developed for the case in which the local flexibility function is found experimentally.     

DYNAMIC RESPONSE AND STABILITY ANALYSIS OF AN AUTOMATIC BALL BALANCER FOR A FLEXIBLE ROTOR

Dynamic stability and time responses are studied for an automatic ball balancer of a rotor with a flexible shaft. The Stodola-Green rotor model, of which the shaft is flexible, is selected for analysis. This rotor model is able to include the influence of rigid-body rotations due to the shaft flexibility on dynamic responses. Applying Lagrange's equation to the rotor with the ball balancer, the non-linear equations of motion are derived. Based on the linearized equations, the stability of the ball balancer around the balanced equilibrium position is analyzed. On the other hand, the time responses computed from the non-linear equations are investigated. This study shows that the automatic ball balancer can achieve the balancing of a rotor with a flexible shaft if the system parameters of the balancer satisfy the stability conditions for the balanced equilibrium position.     

超导磁悬浮支承系统干扰力矩及漂移误差分析

在超导磁悬浮支承系统中, 如果被悬浮的超导球形转子是一个理想的球体, 并且是表现出完全的迈斯纳态, 那么由于球体的对称性, 就不会产生干扰力矩. 但实际的情况并非如此, 一般情况下, 超导球形转子总是存在加工制造误差, 且在高速旋转时总是存在离心变形, 因此转子的表面并不是理想的球面, 当超导转子悬浮在磁场中时, 沿转子表面法线方向的磁悬浮力, 不是完全通过转子质心, 将会产生磁支承干扰力矩, 从而引起转子的漂移误差. 本文从超导转子磁支承干扰力矩的物理机理出发, 对干扰力矩及其引起的漂移误差进行了分析, 包括转子非球形产生的一次干扰力矩、转子非球形与失中度和装配误差产生的二次干扰力矩, 并推导出了磁支承干扰力矩引起的漂移率计算公式, 代入转子参数计算出各种干扰力矩引起的漂移率大小, 为转子漂移测试和系统误差补偿提供了参考, 对于转子的结构优化设计具有指导意义.     

New breathing functions for the transverse breathing crack of the cracked rotor system: Approach for critical and subcritical harmonic analysis

The actual breathing mechanism of the transverse breathing crack in the cracked rotor system that appears due to the shaft weight is addressed here. As a result, the correct time-varying area moments of inertia for the cracked element cross-section during shaft rotation are also determined. Hence, two new breathing functions are identified to represent the actual breathing effect on the cracked element stiffness matrix. The new breathing functions are used in formulating the time-varying finite element stiffness matrix of the cracked element. The finite element equations of motion are then formulated for the cracked rotor system and solved via harmonic balance method for response, whirl orbits and the shift in the critical and subcritical speeds. The analytical results of this approach are compared with some previously published results obtained using approximate formulas for the breathing mechanism. The comparison shows that the previously used breathing function is a weak model for the breathing mechanism in the cracked rotor even for small crack depths. The new breathing functions give more accurate results for the dynamic behavior of the cracked rotor system for a wide range of the crack depths. The current approach is found to be efficient for crack detection since the critical and subcritical shaft speeds, the unique vibration signature in the neighborhood of the subcritical speeds and the sensitivity to the unbalance force direction all together can be utilized to detect the breathing crack before further damage occurs.     

Performance evaluation of traveling wave ultrasonic motor based on a model with visco-elastic friction layer on stator

A new visco-elastic contact model of traveling wave ultrasonic motor (TWUSM) is proposed. In this model, the rotor is assumed to be rigid body and the friction material on stator teeth surface to be visco-elastic body. Both load characteristics of TWUSM, such as rotation speed, torque and efficiency, and effects of interface parameters between stator and rotor on output characteristic of TWUSM can be calculated and simulated numerically by using MATLAB method based on this model. This model is compared with that one of compliant slider and rigid stator. The results show that this model can obtain bigger stall torque. The simulated results are compared with test results, and found that their load characteristics have good agreement.     

Order domain analysis of speed-dependent friction-induced torque in a brake experiment

A friction-induced forced vibration problem, as excited by the geometric distortions of the brake rotor, is studied in this article. The focus is on the order domain analysis, as the speed-dependent behavior of friction torque is not well understood. First, a new laboratory experiment is constructed to simulate vehicle brake judder in a scientific and yet controlled manner. The variations in pressure and torque are measured as the rotor slows down, and the order domain tracking is used to construct shaft torque vs. speed diagrams. A quasi-linear model of the laboratory experiment is then developed to obtain an analytical solution and to estimate the torque envelope function. A nonlinear model of the laboratory experiment (with a clearance) is also investigated to examine the resonant amplitude growth. Finally, predictions are successfully compared with measurements. Several contributions emerge over the prior literature. In particular, the experimental data clearly show that multiple-orders of the rotor surface distortion profile excite the friction-induced torque, and a clearance in the torsional system controls the resonant amplitude regime. New analytical and numerical solutions provide much insight into the speed-dependent resonant amplitude growth process.     

INFLUENCE OF A LEVELNESS DEFECT IN A THRUST BEARING ON THE DYNAMIC BEHAVIOUR OF AN ELASTIC SHAFT

This paper examines the non-linear dynamic behaviour of a flexible shaft. The shaft is mounted on two journal bearings and the axial load is supported by a defective hydrodynamic thrust bearing at one end. The defect is a levelness defect of the rotor. The thrust bearing behaviour must be considered to be non-linear because of the effects of the defect. The shaft is modelled with typical beam finite elements including effects such as the gyroscopic effects. A modal technique is used to reduce the number of degrees of freedom. Results show that the thrust bearing defects introduce supplementary critical speeds. The linear approach is unable to show the supplementary critical speeds which are obtained only by using non-linear analysis.     

Analysis of a non-contact magnetoelastic torque transducer

Results are presented for the performance of a magnetoelastic torque transducer that converts a torque-induced strain in a non-magnetic shaft into changes in a measurable magnetic field. The magnetic field is generated by a thin magnetostrictive layer that is coated onto the circumference of the shaft. The layer is magnetized and has an initial residual strain. The magnetization within the layer rotates in response to changes in the strain which occur when the shaft is torqued. The magnetic field produced by the layer changes with the magnetization and this can be sensed by a magnetometer to monitor the torque on the shaft. In this paper, a phenomenological theory is developed for predicting the performance of the transducer. The theory can be used to predict the magnetic field distribution of the transducer as a function of the physical properties of the magnetic coating, its residual strain, and the applied torque. It enables rapid parametric analysis of transducer performance, which is useful for the development and optimization of novel non-contact torque sensors.     

一种新型垂直轴风力机的CFD分析

Savonius风力机由于作用在叶片的凸面上的"负阻力"和来流不稳定性导致效率低,该"负阻力"造成一个负转矩降低了输出功率,并且在某些来流角度下导致原装的Savonius风力机启动困难。本文提出一种采用控制叶片数目和叶片不同弯曲度的控制来提高风能的利用率,并用CFD方法模拟不同转子结构的内流特征,为消除"负阻力",提出一种采用导叶或挡风板的结构实现引流,并数值分析了该结构的流动特征。模拟结果显示,新结构的多叶片Savonius风力机的转矩和启动特性有明显改善,且功率系数也比普通的Savonius风力机高。     

Minimizing the torque ripple of variable capacitance electrostatic micromotors

In this paper, an approach for minimizing the torque ripple of a variable capacitance electrostatic micromotor is presented. To this end, a practical design for a variable capacitance electrostatic micromotor has been analyzed using a two-dimensional, numerical, finite element method. The procedure presented here for minimizing the torque ripple consists of two stages: optimization of the excitation sequence and optimization of the geometric parameters. Several excitation sequences have been studied for minimizing the torque ripple, and the optimal excitation sequence has been found for supplying the micromotor. Geometric parameters are optimized to minimize torque ripple. The geometric optimization procedure is based on the successive sampling of geometric parameters. These parameters include stator tooth width, rotor tooth width and slot radius.     

Measurement of vibrational energy flow in a plate with high energy flow boundary crossing using electronic speckle pattern interferometry

The measurement of vibrational energy flow is an important tool in understanding the vibrational behaviour of structures. In the past, because of transducer constraints, the measurement of vibrational energy flow was mostly restricted to single point measurements. However, recent developments in advanced laser measurement techniques, such as electronic speckle pattern interferometry (ESPI), have gained interest in applying two-dimensional, multi-point measurement techniques to the estimation of vibrational energy flow. This paper addresses the measurement of vibrational energy flow in a plate by using an ESPI based vibrational energy flow measurement technique. A radially symmetric bending wave plate vibration model is introduced and theoretical expressions for energy-based quantities are derived. To assess the accuracy of the measurement method, these theoretical quantities are compared to synthetic results derived from the ESPI energy flow measurement technique. The ESPI measurement technique is also applied to an experimental ‘infinite’ plate. Thus, a specially designed experimental apparatus was constructed so as to minimise undesired wave reflections in the plate and, thus, achieve a high energy flow boundary crossing at the edges of the plate. To reduce the effect of optical noise contamination on the ESPI measured out-of-plane plate displacement data, optimal filters were applied prior to the vibrational energy flow computation. To appraise the accuracy of the experimental method, measured vibrational power on the plate is compared with measured vibrational input power. A difference of less than 1 dB between both quantities indicates that vibrational energy flow within a rectangular plate that contains radially symmetric wave propagation can be measured to a good degree of accuracy if appropriate filtering is applied.     

An analytical study of rotor dynamics coupled with thermal effect for a continuous rotor shaft

This paper presents an analytical analysis of a continuous rotor shaft subjected to universal temperature gradients. To this end, an analytical model is derived to investigate the generic thermal vibrations of rotor structures. The analytical solutions are obtained in a rotating frame and include parameters related with both the thermal environment and the rotor dynamic structures. This provides an insight into the mechanisms for the rotor thermal vibration. Furthermore, numerical results based on the analytical solutions are given. An index denoting the temperature gradients is proposed for the occasions with nonlinear cross-sectional temperature distributions. Finally, the factors influencing the thermal vibrations are analyzed. The results show that the thermal vibration is affected by many factors including the shaft size, rotational speeds, heating locations, critical speed, etc. Moreover, it is investigated how the convection coefficient and the heat conductivity influence the thermal vibrations in order to provide an insight into the management of thermal vibrations from the perspective of thermal aspects.