Rotor–stator contact dynamics using a non-ideal drive—Theoretical and experimental aspects

The possible contact between rotor and stator is considered a serious malfunction that may lead to catastrophic failure. Rotor rub is seen as a secondary phenomenon caused by a primary source, i.e. sudden mass unbalance, instabilities generated by aerodynamic and hydrodynamic forces in seals and bearings among others. The contact event gives rise to normal and friction forces exerted on the rotor at impact events. The friction force plays a significant role by transferring some rotational energy of the rotor to lateral motion. A mathematical model has been developed to capture this for a conventional backup annular guide setup. It is reasonable to superpose an impact condition to the rub, where the rotor spin energy can be fully transformed into rotor lateral movements. Using a nonideal drive, i.e. an electric motor without any kind of velocity feedback control, it is even possible to stop the rotor spin under rubbing conditions. All the rotational energy will be transformed in a kind of “self-excited” rotor lateral vibration with repeated impacts against the housing. This paper studies the impact motion of a rotor impacting a conventional backup annular guide for the case of dry and lubricated inner surface of the guide. For the dry surface case, the experimental and numerical analysis shows that the rotational energy is fully transformed into lateral motion and the rotor spin is stopped. Based on this study this paper proposes a new unconventional backup bearing design in order to reduce the rub related severity in friction and center the rotor at impact events. The analysis shows that the rotor at impacts is forced to the center of the backup bearing and the lateral motion is mitigated. As a result of this, the rotor spin is kept constant.     

DYNAMIC BEHAVIOR OF THE FULL ROTORSTOP RUBBING: NUMERICAL SIMULATION AND EXPERIMENTAL VERIFICATION

The dynamic behavior of a rotor rubbing, especially rubbing fully with a motion-limiting stop is investigated by numerical and experimental methods. In the dynamic simulation, the sinuous excitation force with low frequency excites the large whirl of the unbalanced rotor and thus causes the rubbing between the rotor and the stop. The simple Coulomb friction model and the multiple segments linear spring model are used to reveal the nature of the rubbing forces. The torque equation of the rotor is built to extract the rotating speed during partial and full rubbing. The stable partial rubbing motion demonstrates that the stop limits the violent vibration amplitude of the rotor effectively. The rubbing experiments confirm the idea of using the inner type of stop to suppress the violent backward whirl with low frequency. When the amplitude of the excitation force exceeds a certain value, the full rubbing occurs with serious continuous friction. During full rubbing, the center of the rotor moves counter-clockwise and whips in the amplitude exceeding the rotor/stop gap dramatically. Moreover, the whip frequency is much higher than the frequencies of the excitation and the unbalance force. And then the rotor rotation is broken quickly by the stop. The predicted dynamical behavior is verified by the rubbing experiments. The relation between the stop/bearing stiffness ratio and such dynamical behavior as the initiation of the rubbing, the over-limit ratio and the contact-ratio is discussed.     

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.     

Dynamics of a 3dof torsional system with a dry friction controlled path

A three-degrees of freedom semi-definite torsional system representing an automotive driveline is studied in presence of a torque converter clutch that manifests itself as a dry friction path. An analytical procedure based on the linear system theory is proposed first to establish the stick-to-slip boundaries. Smoothened and discontinuous Coulomb friction formulations are then applied to the nonlinear system, and the differential governing equations are numerically solved given harmonic torque excitation and a mean load. Time domain histories illustrating dry friction-induced stick–slip motions are predicted for different saturation torques and system parameters. Approximate analytical solutions based on distinct states are also developed and successfully compared with numerical studies. Analysis shows that the conditioning factor associated with the smoothened friction model (hyperbolic tangent) must be carefully selected. Then nonlinear frequency responses are constructed from cyclic time histories and the stick–slip boundaries predictions (as yielded by the linear system theory) are confirmed. In particular, the effect of secondary inertia is analytically and numerically investigated. Results show that the secondary inertia has a significant influence on the dynamic response. A quasi-discontinuous oscillation is found with the conventional bi-linear friction model in which the secondary inertia is ignored. Finally, our methods are successfully compared with two benchmark analytical and experimental studies, as available in the literature on two-degrees of freedom translational systems.     

An electric contact method to measure contact state between stator and rotor in a traveling wave ultrasonic motor

Performances of ultrasonic motor (USM) depend considerably on contact state between stator and rotor. To measure the contact state in a traveling wave ultrasonic motor (TWUSM), a special test method is necessary. This paper develops a new method named electric contact method to measure contact state of stator and rotor in traveling wave type USM. The effects of pre-load and exciting voltage (amplitude) of stator on contact state between stator and rotor are studied with this method. By a simulating tester of friction properties of TWUSM, the variations of stalling torque and no-load speed against the pre-load and the exciting voltage have been measured. The relative contact length that describes the contact characteristic of stator and rotor is proposed. The relation between the properties of TWUSM and the contact state of stator and rotor are presented. Additionally, according to a theoretical contact model of stator and rotor in TWUSM, the contact lengths at given conditions are calculated and compared with the experimental results.     

Control of hysteretic instability in rotating machinery by elastic suspension systems subject to dry and viscous friction

Most of the undesired whirling motions of rotating machines can be efficiently reduced by supporting journal boxes elastically and controlling their movement by viscous dampers or by dry friction surfaces normal to the shaft axis, which rub against the frame. In the case of dry dampers, resonance ranges of the floating support configuration can be easily cut off by planning a motionless adhesive state of the friction surfaces. On the contrary, the dry friction contact must change automatically into sliding conditions when the fixed support resonances are to be feared. Moreover, the whirl amplitude can be restrained throughout the speed range by a proper choice of the suspension-to-shaft stiffness ratio and of the support-to-rotor mass ratio.This theoretical research deals firstly with the natural precession speeds and looks for Campbell plots in dependence on the shaft angular speed, for several rotor-suspension systems. Then, the steady response to unbalance is investigated, in terms of rotor and support orbits and of conical path of the rotor axis. In this search, the ranges of adhesive or sliding contact are identified in particular for system with dry friction damping. At last, the destabilizing influence of the shaft hysteresis in the supercritical regime is focalized and the counterbalancing effect of the other dissipative sources is verified. In the nonlinear case of dry friction dampers, the control of linear stability is fulfilled by a perturbation procedure, checking the magnitude of Floquet characteristic multipliers on the complex plane. Moreover, the nonlinear stability far from steady motion is tested by the direct numerical solution of the full motion equations. The comparison configuration of suspension systems with viscous dampers and no dry friction is examined through an analytical first approximation approach and closed-form results for stability thresholds are derived in particular for the symmetric case.     

Modeling of a rotor speed transient response with radial rubbing

A rotor-stator model of a turbogenerator is introduced in order to investigate speed transients with rotor-to-stator rubbing caused by an accidental blade-off imbalance. In order to assess the angular deceleration of the rotor due to rubbing, the angular position of its cross-section is considered as an unknown of the problem. Displacement fields are discretized through a finite element formulation. The highly nonlinear equations due to contact conditions are solved through an explicit prediction-correction time-marching procedure combined with the Lagrange multiplier approach dealing with a node-to-line contact strategy. The developed numerical tool is suitable for analyzing rotor-stator interactions in turbomachines as the system passes through critical speeds during an accidental shutdown. The sensitivity of the system response to modeling, physical and numerical parameters is investigated. The results highlight the significant role of the friction coefficient together with the diaphragm modeling, from rigid to fully flexible, in the interaction phenomenon. Rigid models have the advantage of simplicity and provide reasonable estimations of the overall response of the turbine. A flexible model, however, may be more computationally intensive but is more appropriate in order to accurately capture quantities of interest such as shaft eccentricity and bearing loads.     

Dynamic analysis of a BLDC motor with mechanical and electromagnetic interaction due to air gap variation

In this study, the dynamic behaviors of a BLDC motor are analyzed, when the motor undergoes mechanical and electromagnetic interaction due to an air gap variation between the stator and rotor. When considering the air gap variation caused by the translational motion of the rotor relative to the stator, the kinetic and potential energies, Rayleigh dissipation function, and the magnetic coenergy are expressed in terms of the rotor displacements and stator currents. With these energies and function, new equations of motion are derived using Lagrange’s equation. The equations for the proposed model are nonlinear equations in which the displacements and currents are coupled. The time responses for the displacements and currents are computed for the proposed and previous models. Furthermore, the effects of rotor eccentricity are also investigated. It is found that, when the air gap varies with time, the time responses for the proposed and previous models have small differences in the stator currents, electromagnetic torques, and rotating speeds. However, the time responses have large differences in the rotor displacements. Therefore, this paper claims that the proposed model describes the dynamic behaviors of the motor more accurately than the previous model. It is also shown that rotor eccentricity increases the stator current period and the electromagnetic torque, while it decreases the rotating speed of the rotor.     

Traveling wave ultrasonic motor: coupling effects in free stator

Generally a stator of traveling wave ultrasonic motor (TWUM) consists of piezoelectric transducers (annular plate or rods) coupled by the way of a metallic ring. These transducers divided into halves are excited independently by two electrical signals with different phases of about 90 degrees. So an elastic traveling wave propagates along the circumference of the ring and a rotor pressed on this vibrating surface is then driven by the stator via contact forces. Many difficulties appear in developing TWUM because the contact between the stator and the rotor via a frictional material is very important. However that may be, the first stage consists in obtaining a vibrating stator with optimum characteristics with two symmetrical phases. The aim of this paper is to discuss some coupling effects in a free stator through an enhanced equivalent circuit model. A simple experimental method based on impedance measurements is performed to estimate the coupling characteristics at a low driving voltage. This paper reports results obtained with the free stator of the well known piezoelectric ultrasonic motor "USR60" by Shinsei Co. Ltd. Since the stator behaves as an elastic body, interactions between the two electrical inputs might be described by the introduction of a coupling oscillator. The comparison of experimental and theoretical results leads to validate the new equivalent circuit of the free stator. The presence of coupling impedance could imply a change of electrical supply condition to optimize the TWUM efficiency. The effects of unbalanced features for each electrical input and the applicability of the proposed model to actual operating condition are discussed in the paper.     

Mode-splitting and quasi-degeneracies in circular plate vibration problems: The example of free vibrations of the stator of a traveling wave ultrasonic motor

In systems with rotational symmetry, bending modes occur in doubly-degenerate pairs with two independent vibration modes for each repeated natural frequency. In circular plates, the standing waves of two such degenerate bending modes can be superposed with a 1/4 period separation in time to yield a traveling wave response. This is the principle of a traveling wave ultrasonic motor (TWUM), in which a traveling bending wave in a stator drives the rotor through a friction contact. The stator contains teeth to increase the speed at the contact region, and these affect the rotational symmetry of the plate. When systems with rotational symmetry are modified either in their geometry, or by spatially varying their properties or boundary conditions, some mode-pairs split into singlet modes having distinct frequencies. In addition, coupling between some pairs of distinct unperturbed modes also causes quasi-degeneracies in the perturbed modes, which leads their frequency curves to approach and veer away in some regions of the parameter space. This paper discusses the effects of tooth geometry on the behavior of plate modes under free vibration. It investigates mode splitting and quasi-degeneracies and derives analytic expressions to predict these phenomena, using variational methods and a degenerate perturbation scheme for the solution to the plate’s discrete eigenvalue problem; these expressions are confirmed by solving the discrete eigenvalue problem of the plate with teeth.     

Balancing of machinery with a flexible variable-speed rotor

The balancing procedure of machines composed of a flexible rotating part (rotor) and a non-rotating part (stator) mounted on suspensions is presented. The rotating part runs at a variable speed of rotation and is mounted on bearings with variable-speed-dependent characteristics. Assuming that the unbalance masses are relatively well defined, such as in the case of a crank-shaft, the procedure is based on a numerical approach using rotordynamics theory coupled with the Finite Element and Influence Coefficient Methods. An academic rotor/stator model illustrates the procedure. Moreover, the industrial application concerns a refrigerant rotary compressor whose experimental investigation permits validating the model. Assuming that the balancing planes are located on the rotor, it is shown that reducing the vibration level of both rotor and stator requires a balancing procedure using target planes on the rotor and on the stator. In the case of the rotary compressor, this avoids rotor-to-stator rubs and minimizes vibration transmission through pipes and grommets.     

A traveling wave ultrasonic motor of high torque

A traveling wave ultrasonic motor of high torque with a new configuration is proposed in this paper. In the new design, a part of the motor serves as the stator. The rotor is the vibrator consisting of a toothed metal ring with piezoelectric ceramic bonded, which generates ultrasonic vibration. The rotor is in contact with the shell of motor and is driven by the friction between the rotor and the stator. This configuration not only removes the rotor in a conventional type of traveling wave ultrasonic motor but also changes the interaction between the rotor and the stator of the motor so that it improves the output performance of the motor. Although an electric brush is added to the ultrasonic motor, it is easy to be fabricated because of the low speed of motor. The finite element method was used to compute the vibration modes of an ultrasonic motor with a diameter of 100mm to optimize the design of the motor. A 9th mode was chosen as the operation mode with a resonance frequency about 25 kHz. According to the design, a prototype was fabricated. Its performance was measured. The rotation speed-torque curves for various frequencies were obtained. The result shows that its stall torque is greater than 4 Nm within a range of 400 Hz. This ultrasonic motor was used to drive the window glass of a mobile car and the result was satisfactory. In the further the research on the friction material between the stator and the rotor is under way to improve the efficiency of the ultrasonic motor.     

Stochastic theory of ultrathin lubricant film melting in the stick-slip regime

Melting of an ultrathin lubricant film under friction between atomically smooth surfaces is studied in terms of the Lorentz model. Additive noise associated with shear stresses and strains, as well as with film temperature, is introduced, and a phase diagram is constructed where the noise intensity of the film temperature and the temperature of rubbing surfaces define the domains of sliding, dry, and stick-slip friction. Conditions are found under which stick-slip friction proceeds in the intermittent regime, which is characteristic of selforganized criticality. The stress self-similar distribution, which is provided by temperature fluctuations, is represented with allowance for nonlinear relaxation of stresses and fractional feedbacks in the Lorentz system. Such a fractional scheme is used to construct a phase diagram separating out different types of friction. Based on the study of the fractional Fokker-Planck equation, the conclusion is drawn that stick-slip friction corresponds to the subdiffusion process.     

Tribological properties of dry,fluid, and boundary friction

A friction pair is studied under lubricant-free dry friction, hydrodynamic, and boundary lubricant conditions. It is shown that, in dry friction, the number of harmonics in the time dependence of the coordinate of the lower rubbing block decreases with increasing frequency of an applied periodic action until the interacting surfaces stick when a critical frequency is exceeded. The surfaces then move together. The behavior of a friction pair with a lubricant made of a Newtonian fluid, pseudoplastic fluid, or dilatant non-Newtonian fluid is analyzed in the hydrodynamic case. It is found that a pseudoplastic fluid or a boundary lubricant leads a intermittent (stick-slip) friction mode, which is one of the main causes of fracture of rubbing parts, over a wide parametric range.     

轴向间距对压气机失速特性的影响

采取在静子叶片表面埋入微型压力传感器的方法,对叶尖、叶中和叶根三个截面上的动态压力进行测量,实验研究了六个轴向间距下压气机的旋转失速特性。实验结果表明:轴向间距对压气机失速点的流量系数影响很大,转静子轴向间距减少,压气机失速推迟;压气机刚进入旋转失速的模态与轴向间距有关,轴向间距较大时,压气机首先进入多团全叶高旋转失速,当轴向间距为21％CR时,压气机直接进入单团全叶高旋转失速;轴向间距不同,近失速点的压力扰动也存在差别。     

RELIABILITY SENSITIVITY FOR ROTOR-STATOR SYSTEMS WITH RUBBING

On the basis of the dynamic equations of the Jeffcott rotor-stator model with imbalance, the reliability sensitivity of the rotor-stator systems with rubbing is examined. A statistical fourth moment method is developed to determine the first four moments of system response and state function. The distribution function of the system state function is approximately determined by the standard normal distribution functions using the Edgeworth series technique. The reliability and reliability sensitivity are obtained and the effect on reliability and reliability sensitivity of shaft stiffness and damping, stator stiffness and damping, radial clearance and stator radial stiffness is studied. Numerical results are also presented and discussed.     

Global behavior of gear system using mixed cell mapping

In some mechanical nonlinear systems, the transient motion will be undergoing a very long process and the attractor-basin boundaries are so complicated that some difficulties occur in analyzing the system global behavior. To solve this problem a mixed cell mapping method based on the point mapping and the principle of simple cell mapping is developed. The algorithm of the mixed cell mapping is studied. A dynamic model of a gear pair is established with the backlash, damping, transmission error and the time-varying stiffness taken into consideration. The global behaviors of this system are analyzed. The coexistence of the system attractors and the respective attractor-basin of each attractor with different parameters are obtained, thus laying a theoretical basis for improvement of the dynamic behaviors of gear system.     

A finite volume method and experimental study of a stator of a piezoelectric traveling wave rotary ultrasonic motor

Piezoelectric traveling wave rotary ultrasonic motors are motors that generate torque by using the friction force between a piezoelectric composite ring (or disk-shaped stator) and a metallic ring (or disk-shaped rotor) when a traveling wave is excited in the stator. The motor speed is proportional to the amplitude of the traveling wave and, in order to obtain large amplitudes, the stator is excited at frequencies close to its resonance frequency. This paper presents a non-empirical partial differential equations model for the stator, which is discretized using the finite volume method. The fundamental frequency of the discretized model is computed and compared to the experimentally-measured operating frequency of the stator of Shinsei USR60 piezoelectric motor.     

非设计工况下动静叶相互干扰的非定常流动特性

本文对非设计工况下动静叶栅相互干扰的非定常流动特性进行数值研究。非设计工况主要考虑大攻角下的分离 流动,动静叶干扰的非定常流动的数值求解在N—S方程的基础上采用分区计算的方法来完成。数值分析结果揭示了在分 离流动本身的非定常特性与动静叶栅相互干扰的非定常特性的双重影响下的流场情况。     

Global behavior of gear system using mixed cell mapping

In some mechanical nonlinear systems, the transient motion will be undergoing a very long process and the attractor-basin boundaries are so complicated that some difficulties occur in analyzing the system global behavior. To solve this problem a mixed cell mapping method based on the point mapping and the principle of simple celll mapping is developed. The algorithm of the mixed cell mapping is studied. A dynamic model of a gear pair is established with the backlash, damping, transmission error and the time-varying stiffness taken into consideration. The global behaviors of this system are analyzed. The coexistence of the system attractors and the respective attractor-basin of each attractor with different parameters are obtained, thus laying a theoretical basis for improvement of the dynamic behaviors of gear system.