Stiffness and damping coefficients of an inclined journal bearingSteifigkeit und dämpfungskoeffiziente eines schrägen gleitlagers

In the determination of the dynamic behaviour of a rotating shaft, the fluid film stiffness and damping coefficients of the bearings play an important role. The general practice is to ignore the rotational stiffnesses and damping coefficients due to the tilt of the journal in the bearing. This paper presents the stiffness and damping coefficients of such journal bearings. Using the expression for film thickness, the modified Reynolds' Equation for the tilted finite journal bearing is set up. The solution of this equation for the film pressure is obtained by using Fedor's proportionality hypothesis. The results obtained are presented in the form of non-dimensional charts.     

Calculation of Journal Bearing Dynamic Characteristics Including Journal Misalignment and Bearing Structural Deformation©

A detailed journal bearing analysis for accurate evaluation of film dynamic characteristics is presented. The new formulation is based on a local perturbation of the oil film at each computational node that captures the important effects of journal misalignment and bearing structural deformation in rotor dynamics and engine NVH applications. The new algorithm is an extension to the classical approach of evaluating film dynamic characteristics based on journal eccentricity perturbation. The governing equations for the oil film pressure, stiffness, and damping are solved using a finite difference approach and their output is validated with numerical results from the literature.     

Unbalance response of rotor disks supported by fluid film bearings with a negative cross coupled stiffness using influence coefficient methodReponse de desequilibre d'un rotor multidisque sur paliers lisses par a methode des coefficients d'influence

The dynamic response of a multidisk rotor system supported by fluid-film bearing using the method of influence coefficient is presented. The stiffness and the damping coefficients have been considered as speed dependent. The two bearings, which support the rotor shaft, are dissimilar. The dynamic response is calculated by varying parameters such as spacing between the disks, the viscosity ratio of the oil of the bearings, the bearing clearance ratio and the relative mass of the rotor disks. The effect of damping on the rotor response and the critical speed is also presented.     

Influence of bearing damping on instability of asymmetric shafts, part IV: Effects of fluid-film bearings

The influence of fluid-film bearings on the parametric instability of asymmetric shafts is investigated. Since fluid-film bearings may have non-isotropic stiffness and damping, the system equations are time-varying coefficients in a rotating reference frame and the complete solution of transition curves includes coupled modes. Thus, the analytical modal method provided by our previous studies has been modified in this part for the determination of approximate solutions from a truncated Hill's determinant. In numerical examples, a plain cylindrical type and a tilting four-pad type are considered to estimate the effects of bearing parameters concerning static load, oil viscosity, and clearance on unstable regions.     

Dynamic Characteristics and Stability Analysis of a Wavy Thrust Bearing

A numerical procedure to analyze wavy thrust bearings is described. The numerical model is developed by assuming that two circular plates rotate relative to each other. The upper plate is assumed to be flat and rotating, whereas the lower plate is assumed to be stationary and wavy in surface geometry. A Reynolds-equation-based procedure is used to simulate the dynamics engendered by various wavy geometries and loading conditions. The equilibrium position of the journal results from the equilibrium between the forces generated by the fluid-film pressures and the externally applied loads. A numerical small perturbation technique is applied to calculate the linear stiffness and damping characteristics of the bearing at the equilibrium position. Using a three-degrees-of-freedom system with one axial and two rotational displacements, nine linear stiffness coefficients (three principal and six cross-coupled coefficients) and nine linear damping coefficients are calculated. These linear coefficients are then used to calculate the eigenvalues of the system by solving the homogeneous equations of motion. The stability of the bearing system is then expressed using the lowest logarithmic decrement obtained from these eigenvalues. Using this procedure, a parametric study is carried out to examine the effects of external load, location of the applied load, bearing number, and bearing wave amplitude on journal equilibrium position, bearing linear stiffness, damping characteristics, and bearing stability.     

Dynamic stiffness and damping coefficients of aerodynamic tilting-pad journal bearings

The dynamic gas–film forces of aerodynamic bearing often can be characterized by eight linear stiffness and damping coefficients. How to theoretically predict these coefficients is a very difficult issue for tilting-pad gas bearing design because of its structural complexity. The current study presents a novel and universal theoretical analysis method for calculating the dynamic stiffness and damping coefficients of aerodynamic tilting-pad bearing. The gas–film pressure within the bearing is expressed in the form of dimensionless compressible gas-lubricated Reynolds equation, which is solved by means of the finite element method. With the assumption that the journal and the pads are disturbed with the same frequency, the dynamic coefficients of tilting-pad gas bearing are computed by using the partial derivative method and the equivalent coefficient method. Finally, the investigations are conducted about the effects of bearing number, perturbation frequency of the journal and the pads, eccentricity ratios, preload and length-to-diameter ratio of the bearing on the dynamic coefficients of aerodynamic tilting-pad journal bearing. The numerical results indicate that the dynamic stiffness and damping coefficients of tilting-pad gas bearing are closely related with these factors. The proposed analytical method provides a valuable means of predicting dynamic performances of tilting-pad gas bearing. The solution can be used for the purpose of prediction of dynamic behavior of the rotor systems supported by aerodynamic tilting-pad bearings.     

Elastohydrodynamically Lubricated Ball Bearings with Couple-Stress Fluids,Part II: Dynamic Analysis and Application

Stiffness and damping coefficients of isothermal elastohydrodynamically lubricated point-contact problems are evaluated numerically with couple-stress fluids. A set of equations under steady-state and dynamic conditions is derived from the modified Reynolds equation using a linearized perturbation method. This paper is the second part of the present study; the modified Reynolds equation derived from the Stokes micro-continuum theory is used in the previous article. Dynamic pressures are found after solving the set of perturbed equations using the previously obtained steady-state pressure from the modified Reynolds equation. The stiffness and damping coefficients of the film are determined using the dynamic pressures. Then the overall stiffness and damping matrices of the ball bearing are obtained from load distribution, coordinate transformation, and compatibility relations. The bearing coefficients are introduced into a rotor system to simulate the response. It has been observed that the influence of couple-stress fluids on the dynamics of a rotor supported on lubricated ball bearings is marginal; hence, Newtonian theory can be used instead for simplicity. However, with increasing content of polymer additives in lubricant, for an accurate analysis the effect of couple stresses in a fluid should not be neglected.     

Dynamic behavior of fluid‐structure coupling of hydrostatic spindle under effect of oil film slip

To better understand the dynamic characteristics of a hydrostatic spindle in fluid‐structure coupling, the impact of oil film slip on the 4 dynamic stiffness and damping coefficients of the spindle is studied. On the basis of modified Reynolds equation, which considers the microscale velocity slip effect, rotation error of the spindle is calculated. To solve the rotor axis orbit under the existence of eccentric quality, 4 dynamic stiffness and damping coefficients of the oil film, which describe the dynamics of a rotor axis orbit, are calculated by using load increment method and the pressure perturbation method. The research results show that velocity slip caused a certain impact on dynamics of bearing stiffness and damping performance. The experiment of the measuring path of the shaft verifies the correct and effect of the orbit of shaft center model.     

Steady state and dynamic characteristics of axial grooved journal bearings

The steady state and dynamic characteristics including whirl instability of oil journal bearings with single axial groove located at the top of the bearing and then at some angular interval from the top from which oil is supplied at constant pressure are obtained theoretically. The Reynolds equation is solved numerically by finite difference method satisfying the appropriate boundary conditions. The dynamic behaviour in terms of stiffness and damping coefficients of fluid film and stability are found using a first-order perturbation method for each location of the groove. It has been shown that both load capacity, end flow is maximum when the feeding groove is at 12° location and thereafter the load capacity falls, stability improves for smaller groove angle and groove length at higher value of eccentricity ratio and speed. The stiffness and damping coefficient magnitude is found to be higher for the bearing with smaller groove angle and groove length, the difference between the hydrodynamic and hydrostatic load increases at 12° groove location.     

螺旋槽气体轴承动态特性及失稳分析

针对高速动静压气体轴承气膜的复杂非线性动力学行为,以球面螺旋槽动静压气体轴承为研究对象,建立润滑分析数学模型;采用有限差分法与导数积分法进行求解,得到动态扰动压力分布及动态特性系数,并研究切向供气条件下螺旋槽参数、径向偏心率、供气压力、转速对气膜刚度阻尼系数的影响规律;建立线性稳定性计算模型,预测气膜涡动失稳转速,分析运行参数对失稳转速的影响。结果表明:气膜阻尼是一种抑制涡动的因素,气膜的稳定性取决于气膜刚度与阻尼的协同作用;气膜刚度阻尼随着槽宽比、槽深比、螺旋角的增大,整体上呈先增大后减小的趋势;刚度随转速的升高而增大,阻尼则随转速的升高而减小;径向偏心率和供气压力越大,气膜刚度和阻尼越大;在一定范围内,提高供气压力、增大径向偏心率能够提高系统失稳转速;合理地选取轴承结构参数和运行参数,能够优化轴承动态特性,保证气体轴承较高的运行稳定性。     

新型动静压转台油膜力及动态特性研究

为研究新型动静压转台的油膜力分布与动特性变化情况,在充分考虑转台轴向速度产生的挤压流量的同时,基于流量平衡建立转台油膜力的数学模型,利用偏导数法推导转台轴向油膜刚度和阻尼的计算式。在固定负载和和恒定的供油压力下,分析转速对转台的静动特性的影响。分析结果表明:在固定负载和恒定的供油压力下,随着转速的增加,转台的油膜厚度逐渐减小,动压区的承载力逐渐增大,转台总泄漏量逐渐增加,转台轴向油膜刚度和阻尼均逐渐增大。所建的模型充分考虑了转台轴向速度产生的挤压流量,因而仿真计算更符合实际工况,为新型动静压转台的实际生产和性能研究提供理论支持。     

On the Dynamics of Lubricated Cylindrical Roller Bearings,Part I: Evaluation of Stiffness and Damping Characteristics

The stiffness and damping coefficients of a single roller-to-race contact of lubricated cylindrical roller bearings are numerically evaluated using a linearized perturbation method for both elastohydrodynamic lubrication (EHL) finite and infinite contact theories. A steady-state pressure equation is solved by a multilevel method and the elastic deformation is evaluated with the multilevel multi-integration method. Dynamic pressures are obtained by solving a set of perturbed pressure equations and are used to calculate the stiffness and damping coefficients. The influence of various nondimensional parameters (load parameter, speed parameter, material parameter, edge radius, and geometrical parameter) on the stiffness and damping are studied. The results show that a finite line contact gives higher values of stiffness and damping coefficients compared to an infinite contact, particularly at higher load and lower speed values. Based on the numerically evaluated data, curve-fitted relations for the stiffness and damping coefficients of a single roller-to-race contact are developed that can be used in the dynamic analysis of rotor–bearing systems.     

Parametric Study of Spiral Groove Gas Face Seals

A finite element analysis for the isothermal flow in spiral groove gas face seals is detailed along with a successive approximation method for the iterative solution of the nonlinear Reynolds equation. Zeroth- and first-order pressure fields are calculated for evaluation of the seal opening force and leakage and the axial stiffness and damping force coefficients, respectively. A parametric study shows the static and dynamic force behavior of a baseline SGFS operating with a large pressure ratio. The recommended geometric parameters presented ensure large static stiffness and damping force coefficients while still allowing for low seal leakage rates. A reduction in the power loss and a significant increase in the seal static stiffness coefficient are unique features of thin seal dams.     

Stability of rotors supported in gas bearings with bushes mounted in air rings

When during the operation of rotors supported in gas bearings their rotational velocity reaches a sufficiently high value, the loss of steady-state stability occurs. This instability is caused by the loss of damping properties of the gas film, which leads to self-excited vibrations. These vibrations are the basic obstacle to a widespread application of gas bearings.The phenomenon of self-excited vibrations can be avoided by introducing an elastic supporting structure between the bearing bushes and the casing, characterised by properly selected stiffness and damping coefficients. In practice such a structure can have the form of an externally pressurised gas ring.In this paper we demonstrate, on the basis of selected examples, which ranges of the values of stiffness and damping coefficients of the gas ring make it possible to retain steady-state stability at practically any rotational velocity of the rotor. We also show a design of the ring structure, especially of its feeding system, which ensures the required values of stiffness and damping coefficients (with regard to the stability). Our investigations have been carried out by means of a numerical simulation method with the use of a mathematical model of the gas bearing, verified already many times.     

螺旋槽小孔节流动静压气体轴承动态特性分析*

为进一步改善小孔节流动静压气体轴承的稳定性,对螺旋槽小孔节流动静压气体轴承的动态特性进行了研究。建立不定常工况下的动态雷诺方程,采用偏导数积分法求解动态特性系数。研究有无螺旋槽、涡动比、转速、供气压力以及槽宽和槽深对轴承动态特性的影响规律。结果表明:螺旋槽可以显著提高轴承的动态特性,增加轴承的稳定性;随涡动比的增大,直接刚度系数增加,交叉刚度系数和各阻尼系数都减小;随转速的增大,各刚度系数增加,而各阻尼系数减小;随供气压力的增大,各刚度和阻尼系数均增加;随槽宽的增大,直接刚度系数和阻尼系数呈先增加后减小趋势,交叉刚度系数和阻尼系数变化较小;随槽深的增大,直接刚度系数增加,交叉刚度系数和各阻尼系数先增加后减小。     

动压气体轴承的动态刚度和动态阻尼系数

采用偏导数法求解动压气体润滑Reynolds方程,给出动压气体轴承动态刚度和动态阻尼系数普遍适应的计算方法.进行有限元数值仿真,计算动压气体轴承的动态刚度和动态阻尼系数,并重点研究轴颈扰动频率和不同轴颈扰动频率下的轴承静态载荷及轴承数对这些系数的影响.从理论上解释动压气体轴承的动态刚度和动态阻尼系数与轴颈扰动频率的密切相关性.     

Application of Hopf bifurcation theory to rotor-bearing systems with consideration of turbulent effects

This paper shows that the steady state and the dynamic characteristics of a rotor-bearing system with turbulent effects can be conveniently evaluated by applying the Hopf bifurcation theory to the equations of motion of a rigid rotor symmetrically supported by two identical fluid-film journal bearings. Results are presented for stiffness and damping coefficients and stability characteristics of a journal bearing, which take turbulence into consideration.     

ON THE AXIAL TRANSIENT RESPONSE ANALYSIS OF THE SECTOR-SHAPED HYDRODYNAMIC THRUST BEARING-ROTOR SYSTEM IN A TURBOEXPANDER

NOTATIONS(The letters in brackets denote the dimension to be as 1)C──Damping coefficient of oil filmD──Domain of a thrust pad in the thrust bearingF──Axial external force (static load)ΔF──Increment of axial external forcesFr──Resultant of oil film forcesF1,F2──Oil film forces of thrust bearings 1 and 2ΔFr──Increment of oil film forces resultantk──Stiffness coefficient of oil filmm──Mass of rotorp()──Oil film pressure,=pδ20/μωr21p1(1),p2(2)──Oil film pressu…     

基于高效深切磨床轴承主轴系统的有限元分析

针对实验室研制的高效深切磨床动静压轴承主轴系统,首先利用计算流体力学软件CFX,对动静压轴承的油膜进行稳态分析,得到油膜上的压力分布。通过在不同的小位移和速度扰动的情况下轴承油膜承载力的变化规律,依此来确定轴承油膜的刚度和阻尼系数。在ANSYS中,采用Ma-trix27单元来模拟动静压轴承油膜的刚度和阻尼作用,并对主轴进行模态分析。结果表明,在微小的位移扰动和速度扰动的情况下,油膜的弹性力和阻尼力近似为线性的,主轴模态有限元计算结果与实测结果基本一致,证明了该分析方法的正确性。     

A Method for Identification of Bearing Force Coefficients and Its Application to a Squeeze Film Damper with a Bubbly Lubricant

A general formulation of the instrumental variable filter (IVF) method for parameter identification of a n-DOF (Degrees Of Freedom) mechanical linear system is presented. The IVF is a frequency domain method and an iterative variation of the least-squares approximation to the system flexibilities. Weight functions constructed with the estimated flexibilities are introduced to reduce the effect of noise in the measurements, thus improving the estimation of dynamic force coefficients. The IVF method is applied in conjunction to impact force excitations to estimate the mass, stiffness, and damping coefficients of a test rotor supported on a squeeze film damper (SFD) operating with a bubbly lubricant. The amount of air in the lubricant is varied from nil to 100 percent to simulate increasing degrees of severity of air entrainment into the damper film lands. The experimental results and parameter estimation technique show that the SFD damping force coefficients increase as the air volume fraction in the mixture increases to about 50 percent in volume content. The damping coefficients decrease rapidly for mixtures with larger air concentrations. The unexpected increase in direct damping coefficients indicates the complexity of the SFD bubbly flow field and warrants further experimental verification.