Magnetic field effects on the nonlinear vibration of a rotor

The nonlinear vibration of a rotor operated in a magnetic field with geometric and inertia nonlinearity is investigated. An asymmetric magnetic flux density is generated,resulting in the production of a load on the rotor since the air-gap distribution between the rotor and the stator is not uniform. This electromagnetic load is a nonlinear function of the distance between the geometric centers of the rotor and the stator. The nonlinear equation of motion is obtained by the inclusion of the nonlinearity in the inertia, the curvature, and the electromagnetic load. After discretization of the governing partial differential equations by the Galerkin method, the multiple-scale perturbation method is used to derive the approximate solutions to the equations. In the numerical results, the effects of the electromagnetic parameter load, the damping coefficient, the amplitude of the initial displacement, the mass moment of inertia, and the rotation speed on the linear and nonlinear backward and forward frequencies are investigated. The results show that the magnetic field has significant effects on the nonlinear frequency of oscillation.     

Nonlinear dynamics of a Jeffcott rotor with torsional deformations and rotor-stator contact

The dynamics of a modified Jeffcott rotor is studied, including rotor torsional deformation and rotor-stator contact. Conditions are studied under which the rotor undergoes either forward synchronous whirling or self-excited backward whirling motions with continuous stator contact. For forward whirling, the effect on the response is investigated for two commonly used rotor-stator friction models, namely, the simple Coulomb friction and a generalized Coulomb law with cubic dependence on the relative slip velocity. For cases with and without the rotor torsional degree of freedom, analytical estimates and numerical bifurcation analyses are used to map out regions in the space of drive speed and a friction parameter, where rotor-stator contact exists. The nature of the bifurcations in which stability is lost are highlighted. For forward synchronous whirling fold, Hopf, lift-off, and period-doubling bifurcations are encountered. Additionally, for backward whirling, regions of transitions from pure sticking to stick-slip oscillations are numerically delineated.     

Interaction between torsional and lateral vibrations in flexible rotor systems with discontinuous friction

In this paper, we analyze the interaction between friction-induced vibrations and self-sustained lateral vibrations caused by a mass-unbalance in an experimental rotor dynamic setup. This study is performed on the level of both numerical and experimental bifurcation analyses. Numerical analyses show that two types of torsional vibrations can appear: friction-induced torsional vibrations and torsional vibrations due to the coupling between torsional and lateral dynamics in the system. Moreover, both the numerical and experimental results show that a higher level of mass-unbalance, which generally increases the lateral vibrations, can have a stabilizing effect on the torsional dynamics, i.e. friction-induced limit cycling can disappear. Both types of analysis provide insight in the fundamental mechanisms causing self-sustained oscillations in rotor systems with flexibility, mass-unbalance and discontinuous friction which support the design of such flexible rotor systems.     

The effects of Coulomb friction on the performance of centrifugal pendulum vibration absorbers

We investigate analytically and experimentally the effects of Coulomb friction on the performance of centrifugal pendulum vibration absorbers (CPVAs), which are used to reduce torsional vibrations in rotating machinery. The analysis is based on perturbation methods applied to the nonlinear equations of motion for a rotor subjected to an engine order applied torque and equipped with a circular path CPVA with viscous and Coulomb damping. The experimental work is based on quantifying parameters for the damping model using free vibration measurements with a viscous and Coulomb damping identification scheme that is enhanced to better handle measurement noise, and running tests for steady-state operation under a range of loading conditions. The level of Coulomb damping is varied by adjusting the friction of the absorber connection bearing. Good agreement is found between the analytical predictions and the experimental data. It is shown that the absorber sticks up to a level of excitation that allows it to release, after which the Coulomb damping acts in the expected manner, resulting in lowered response amplitudes. The results obtained are of general use in assessing absorber performance when dry friction is present in absorber suspensions.     

Theoretical development and closed-form solution of nonlinear vibrations of a directly excited nanotube-reinforced composite cantilevered beam

The nonlinear equations of motion of planar bending vibration of an inextensible viscoelastic carbon nanotube (CNT)-reinforced cantilevered beam are derived. The viscoelastic model in this analysis is taken to be the Kelvin–Voigt model. The Hamilton principle is employed to derive the nonlinear equations of motion of the cantilever beam vibrations. The nonlinear part of the equations of motion consists of cubic nonlinearity in inertia, damping, and stiffness terms. In order to study the response of the system, the method of multiple scales is applied to the nonlinear equations of motion. The solution of the equations of motion is derived for the case of primary resonance, considering that the beam is vibrating due to a direct excitation. Using the properties of a CNT-reinforced composite beam prototype, the results for the vibrations of the system are theoretically and experimentally obtained and compared.     

Analytical study of nonlinear synchronous full annular rub motion of flexible rotor–stator system and its dynamic stability

The nonlinear synchronous full annular rub motion of a flexible rotor induced by the mass unbalance and the contact-rub force with rigid and flexible stator is studied analytically. The nonlinear property is due to the dry friction force between stator and rotor. The exact solutions of the synchronous full annular rub motion and its run speed regions are obtained. The stability of the synchronous full annular rub motion is discussed analytically. The stability criterion and the stability regions of the synchronous full annular rub motion are obtained. A simplified approximate criterion formula for dynamic stability is also derived under the conditions of large impact stiffness, small damping and small friction. The simplified criterion formula can be used conveniently in engineering and matches the real situations of industry.     

Friction force model and rolling dynamics for a ball on a rough surface with presliding displacement taken into account

A model of sliding and spinning friction forces for a ball in the form of finite relations obtained by integrating the tangential stresses over the contact area whose parameters are determined by Hertz’s theory for the “ball-rough horizontal surface” tribological conjunction pair is supplemented with a model of rolling friction torques. The combined model is peculiar in that the presliding displacement effect in rolling and spinning friction torques is taken into account. It is shown that the ball motions in the presliding displacement zone are of quasilinear character and, under shock perturbations, have the form of damping vibrations in the three orientation angles. The numerical parameters of the rolling and spinning friction model are experimentally determined for the presliding displacement zones, while the sliding friction parameters and partly the spinning friction parameters are calculated. Mathematical modeling permits one to discover new properties of the ball, namely, its deceleration in rolling, the onset of damping vibrations at the beginning and end of motion, and the transient process parameters.     

有限长大间隙环流中同心转子动特性系数研究

基于作者建立的大间隙环流中转子运动理论模型,用摄动法推导了有限长大间隙环流流场非线性控制方程的零阶和一阶摄动方程,研究了摄动方程的数值求解方法,并用该数值方法深入研究了有限长大间隙环流中同心转子的动特性系数以及壁面粗糙度、入口压力、长径比和入口预旋等参数的影响,研究结果表明,系统参数对有限长大间隙环流中同心转子动特性系数的影响是流体惯性效应、旋流效应、摩擦耗散效应和Lomakin效应综合影响的因素。     

Transient response of single-disk rotor on anisotropic supports passing through critical speed

This paper clarifies the nonlinear and nonstationary characteristics of transient vibration of a single-disk rotor system on anisotropic supports passing through critical speed under limited energy supply. By analytical dynamics and Bogoliubov-Mitropolskii asymptotic method, the governing equations of motion of the system are reduced to a set of first order differential equations capable of numerical integral solution. The influences of gyroscopic effect of the rotor, external and internal damping as well as the initial phase angle of static unbalance on transient response are discussed.     

超声马达转子摩擦材料厚度对驱动性能的影响研究

制备了一系列不同厚度的摩擦材料 ,利用超声马达摩擦特性模拟试验装置 ,研究了摩擦材料厚度对超声马达空载转速和堵转力矩的影响 .基于一个简化的行波超声马达定子和转子接触模型 ,用有限元法计算了定子和转子接触变形随摩擦材料厚度的变化规律 ,提出了定子和转子具有合理接触变形的摩擦材料厚度范围 .根据试验和理论计算结果确定了摩擦材料最佳厚度 ,为超声马达摩擦材料厚度设计提供了理论依据     

Stability analysis of the whirl motion of a breathing cracked rotor with asymmetric rotational damping

The stability of the whirl motion of a breathing cracked rotor with the distinction of stationary damping and the asymmetric rotational damping is studied. By Lagrange’s principal, the motion equations are formed in rotational frame such that the multi-asymmetric system, i.e., asymmetric rotational damping and asymmetric time-periodic varying stiffness, is simplified to be a system with anisotropic damping and anisotropic time-periodical varying stiffness in rotational operation. Based on the multiple scales solution of the simplified whirling equation in moving frame, root locus method for stability analysis is proposed. Different from the former stability estimation method, the corresponding Campbell diagram, decay rate plot, and root locus plot of the fifth-order approach are derived to prove the effects of both crack depth and damping effects. The numerical results of the instabilizing effects of the crack depth are well agreeing with the previous studies. In addition, the destabilizing influence of the rotational damping on the breathing cracked rotor is presented for the first time.     

任意斜裂纹转子的耦合振动研究

对包含不同类型裂纹(横裂纹、横-斜裂纹以及任意斜裂纹)的转子的耦合振动进行研究,以揭示裂纹转子在不同方向上刚度参数的变化规律及其交叉耦合机理,特别是由此引发的振动特征. 对于包含不同类型裂纹的转子轴段,采用六自由度Timoshenko梁单元模型对其进行单元建模,并基于应变能理论推导计算柔度参数和刚度矩阵. 在此基础上, 采用纽马克-$\beta$数值算法求解裂纹转子的运动方程,获得裂纹转子在单故障或多故障激励(不平衡激励、扭转激励或不平衡激励加扭转激励)作用下的耦合振动响应,进而分析耦合振动谱特征. 与横裂纹和横-斜裂纹相比,任意斜裂纹使转子刚度矩阵的交叉耦合效应更显著,导致转子发生更强烈的弯-扭耦合甚至是纵-弯-扭耦合振动.无论是在不平衡激励还是扭转激励作用下, 弯曲振动与扭转振动幅度都更大. 而且,包含不同类型裂纹的转子的耦合振动特征频率,例如旋转基频与二倍频、扭转激励频率及其边带成分的幅值,对裂纹面方向角具有不同的敏感性. 所得的这些研究结果,可以为转子裂纹的特征参数辨识与诊断提供理论依据.      

Non-linear dynamic of rotor–stator system with non-linear bearing clearanceDynamique non-linéaire d'un ensemble rotor–stator comportant un roulement non-linéaire avec jeu

The study deals with a rotor–stator contact inducing vibration in rotating machinery. A numerical rotor–stator system, including a non-linear bearing with Hertz contact and clearance is considered. To determine the non-linear responses of this system, non-linear dynamic equations can be integrated numerically. However, this procedure is both time consuming and costly to perform. The aim of this Note is to apply the Alternate Frequency/Time Method and the ‘path following continuation’ in order to obtain the non-linear responses to this problem. Then the orbits of rotor and stator responses at various speeds are investigated. To cite this article: J.-J. Sinou, F. Thouverez, C. R. Mecanique 332 (2004).     

The effect of tangential elasticity of the contact layer between stator and rotor in travelling wave ultrasonic motors

In travelling wave ultrasonic motors the elliptical motion of material points of the stator drives the rotor due to frictional mechanisms. The motor characteristic strongly depends on the mechanical properties of the components stator, rotor and contact layer. In order to predict the motor behaviour, a model for the contact between stator and rotor has been developed. The goal of the present paper is to point out the importance of the tangential elasticity of the contact layer which is responsible for the formation of stick zones and also for the amount of friction losses and overall efficiency. Therefore a comparison with a model with a contact layer rigid in tangential direction is given. Based on a visco-elastic foundation model for the contact layer, torque-speed curves as well as torque-efficiency curves are computed. Experimental investigations for identification of parameters, check of assumptions and model validation are carried out. Finally, the model is used to show the results of parameter variations for normal force, vibration amplitude and modulus of elasticity of the contact layer.     

Equivalent linearizations for practical hysteretic systems

Experimental testing of a friction damped base isolation system has indicated a need for a new model of friction damping and for an appropriate equivalent linearization technique. The model for the damping adopted is a combination of viscous damping, constant Coulomb friction and linear Coulomb friction.This model is incorporated into the equation of motion for a single-degree-of-freedom system and the exact solutions are given for free vibrations and for steady-state vibrations excited by a harmonic force. The exact solution is taken as a basis for an equivalent linearization technique that can be used in conjunction with conventional design spectra for a practical design of such a system.     

Optimization of travelling wave ultrasonic motors using a three-dimensional analysis of the contact mechanism at the stator–rotor interface

The present paper investigates optimization rules and new design methodologies dealing with the contact mechanics in rotative travelling wave ultrasonic motors (TWUM). The proposed approaches focus on the design of the rotor, including the friction layer that is usually deposited onto its lower surface, while the stator is supposed to be preliminary designed. Contact aspects such as the transmission of mechanical power as well as the wear mechanism of the friction layer are investigated, according to the analysis of the stator–rotor contact mechanics in both hoop and radial directions. Considering a classical wear criterion in a preliminary step, a contact ratio, that allows the mechanical power to be optimized, is pointed out in the hoop direction. In a further step, the contact conditions in the radial direction are improved through the elastic fitting of the stator and rotor radial deflexions, therefore allowing the material's wear to be decreased. Some experimental tests, that have been recently performed, give a comparison of wear marks, which occur onto optimal and non-optimal rotor geometries. A first mechanism synthesis is finally proposed in such a way to allow the mechanical architecture of the rotor (including the friction layer) to be automatically designed according to a given set of mechanical constraints.     

Dynamics of a Rigid Unbalanced Rotor with Nonlinear Elastic Restoring Forces. Part I: Theoretical Analysis

In a previous paper, the dynamic behaviour of a Jeffcott rotor was studied in the presence of pure static unbalance and nonlinear elastic restoring forces. The present paper extends the analysis to a rigid rotor with an axial length such as to make the transverse moment of inertia greater than the axial one. As in the previous investigation, the elastic restoring forces are assumed to be nonlinear and the effects of couple unbalance are also included but, unlike the Jeffcott rotor, the system exhibits six degrees-of-freedom. The Lagrangian coordinates were fixed so as to coincide with the three coordinates of the centre of mass of the rotor and the three angular coordinates needed in order to express the rotor's rotations with respect to a reference frame having its origin in the centre of mass. The precession motions of such a rotor turn out to be cylindrical at low angular speeds and exhibit a conical aspect when operating at higher speeds. The motion equations of the rotor were written with reference to a system that was subsequently adopted for the experimental analysis. The particular feature of this system was the use of a steel wire (piano wire) for the rotor shaft, suitably constrained and with the possibility of regulating the tension of the wire itself, in order to increase or reduce the nonlinear character of the system. The numerical analysis performed with integration of the motion equations made it possible to point out that chaotic solutions were manifested only when the tension in the wire was given the lowest values – i.e. when the system was strongly nonlinear – in the presence of considerable damping and rotor unbalance values that were so high as to lose any practical significance. Under conditions commonly shared by analogous real systems characterised by poor damping, where the contribution to nonlinearity is almost entirely due to elastic restoring forces, the analysis pointed out that precession motions may be manifested with a periodic character, whether synchronous or not, or a quasi-periodic character, but in no case is the solution chaotic.     

Rotor whirl damping by dry friction suspension systems

An efficient and automatic attenuation technique for the whirling motion of rotating machinery can be achieved by supporting the journal boxes elastically and providing them with suitable rubbing surfaces subject to dry friction normal to the shaft axis. The critical flexural speeds are easily cut off and the whirl amplitude is minimized throughout the frequency range. Confining the usual operative angular speed of the rotor in the range of adhesive contact between the dry friction surfaces, there is no significant increase of power dissipation or heat production as a whole due to this type of suspension system, whose task is just to suppress the resonant peaks when passing the critical speeds. Moreover, the wear of the rubbing surfaces can be easily compensated by use of suitable spring loading systems for the friction contact. The dry friction damping is also compared with an equivalent viscous damping, where the equivalence has to be understood in terms of work dissipated per single revolution of the rotor. As for other conventional cases, the shaft hysteresis is found to exert a destabilizing effect above the first critical speed, which however can be compensated by the other dissipation sources. The system stability is here studied perturbing the periodic motion and applying the Floquet theory.     

密封环挠性安装形式对干气密封动态追随性的影响

基于气体润滑理论,并通过小扰动法建立了螺旋槽干气密封微扰膜压控制方程,在高速高压条件下获得了气膜动态特性系数;基于动力学相关知识,在考虑转轴轴向振动的情况下,利用气膜轴向动态刚度和阻尼系数分别求解了静环挠性安装、动环挠性安装和两环均挠性安装的干气密封挠性环运动方程.在不同轴向激励振幅、激励频率、挠性环质量、弹簧刚度和辅助密封圈阻尼下分别研究了三种典型结构干气密封动态追随性并进行了对比分析.结果表明:当轴向激励频率较高或挠性环质量较大时,静环挠性安装干气密封在刚受到外界激励时膜厚突变相对严重,动态追随性较差;在轴向激励频率较低且挠性环质量较小时,静环挠性安装干气密封相比动环挠性安装干气密封表现出更好的动态追随性;在三种密封环挠性安装形式中,两环均挠性安装干气密封动态追随性最好,且具有绝对优势.