008618266359149
Abstract: The dynamics models for rolling bearings is reviewed, and the dynamics models for bearing － rotor system and its application are discussed. The dynamics analysis software for bearings is introduced. The existing problems are discussed about dynamics research of bearings, and the development tendency is prospected．
Key words: rolling bearing; dynamics model; rotor; local defect
With the development of the mechanical industry, the requirements for precision, performance, lifespan, and reliability of rolling bearings are becoming increasingly high. Among them, the study of dynamic performance of bearings is particularly important. Therefore, the research progress of bearing dynamic models will be discussed in the following text.
1 Research on Dynamic Model of Bearing
1.1 Development History
The mechanical model of bearing has gone through three development stages: Statics analysis model, quasi Statics analysis model and dynamic analysis model. In the early days, it was difficult to accurately predict and describe the bearing motion state by establishing Statics analysis model only based on the ideal motion state and simple motion relationship. The quasi Statics analysis model is relatively perfect, which can effectively predict the rolling element speed, bearing fatigue life, bearing deformation and stiffness and other motion parameters, and can basically meet the engineering needs, but it can not analyze the transient instability of bearings, nor can it completely describe the dynamic performance of bearings. The dynamic analysis model can not only effectively analyze the working state of the bearing's load and speed over time, as well as the stability of the rolling element and cage, but also more accurately describe the dynamic and steady-state motion of the bearing. Therefore, research on dynamic analysis methods has received attention in recent years.
The dynamic performance analysis of high-speed bearings began in the 1960s. It was the first time to discuss and analyze the pseudo Statics of bearings. Based on the traditional Statics analysis, it proposed the control theory of race raceway. It was assumed that the motion state of the rolling body was pure rolling, no spin motion. At the same time, the effects of gyroscopic torque and eccentricity were considered in the force analysis of bearings under high-speed rotation, but the role of elastic lubrication was not considered in the model, Unable to accurately predict internal sliding of bearings. With the development of elastohydrodynamic theory, based on the pseudo Statics model, considering the role of lubrication and inertia force, a bearing pseudo dynamic analysis model is established, which can analyze the stability parameters such as bearing deformation, rolling body rotation and revolution speed, but still cannot analyze the dynamic performance such as bearing transient instability and the impact of time-varying parameters. Therefore, the bearing dynamic model is studied and explored on this basis. The first dynamic model of a 4-degree of freedom ball bearing was established to appropriately simplify the interaction forces between various components of the ball bearing. Subsequently, a 6-degree of freedom bearing cage model was established to groundbreaking analyze the dynamic changes between the cage and the rolling element. The fourth order Runge Kutta method was used to calculate the instantaneous displacement, rotational speed, and internal sliding of the bearing between the rolling element and the cage, However, the model did not consider the issue of oil film lubrication and the elasticity of the cage. Considering the changes in bearing load and slip on the cage under lubrication, a 3-degree-of-freedom dynamic model of the bearing cage was established. However, due to insufficient research on the theory of elastohydrodynamic lubrication at that time, the model needed to be improved. The article comprehensively analyzes various factors that affect the dynamic performance of bearings, systematically studies the motion and force state of rolling elements, establishes a 6-degree of freedom bearing dynamic model, and establishes motion differential equations for each component of the bearing. This model is suitable for various types of bearings, mainly used to analyze the instantaneous dynamic characteristics of bearings during speed and load changes.
1.2 Typical Models
The intermediate bearing of the bearing dual rotor system is simplified as spring and damper, and the critical speed vibration mode and unbalance response of the dual rotor system are studied by using the Transfer-matrix method. The bearing model established in reference [11] is shown in Figure 1. In the figure, e is the eccentricity; Fu is the unbalanced force; Nb is the number of rolling elements; ω C is the rotational speed of the cage; T is the time; W is a constant force in the vertical direction.
Figure 1 Bearing Model
This model assumes that the rolling elements are uniformly distributed inside the bearing for pure rolling motion, without considering the effect of lubricating oil film. Through analysis, it was found that
In the formula: Fx is the composite spring force of the x-direction component; Fy is the composite spring force in the y-direction component; K is the Hertz contact stiffness; θi is the angular position at the i-th rolling element; γ O is the radial clearance of the bearing.
The second order nonlinear differential equation of the bearing rotor coupling system is established as
In the formula, m represents the mass of the rotor and bearing inner race supported by the bearing; C is damping; ω is the rotor speed.
Significant contributions have been made to the study of bearing dynamics models, with a relatively complete theoretical system. However, issues such as oil damping, material encounter damping, and collision contact have not yet been considered. Moreover, the model is very complex and requires a large amount of calculation. When using formulas, many conditions and range requirements need to be considered, which is not continuous enough and difficult to widely apply in practical engineering. The established model establishes a 6-degree of freedom bearing cage dynamic model, and combines it with the established model to optimize the parameters such as clearance and friction between the cage and the rolling element, and improve the dynamic stability of the cage. The simplified model of the established bearing is shown in Figure 2. In the figure, xi and xo represent the sliding speeds of the rolling element relative to the inner and outer rings, respectively; Xc is the speed of the Centroid of the rolling element; θ B is the rotational speed of the rolling element; Pi and Po are the positive loads of the rolling element and the inner and outer rings, respectively; Fi and Fo are the sliding friction between the rolling element and the inner and outer rings respectively; FS is the collision force between the rolling element and the cage; μ Is the friction coefficient; RB is the radius of the rolling element; Kb is the stiffness characteristic of the rolling element; Cb is the damping coefficient of the rolling element.
Figure 2 Simplified Model of Bearings
This model assumes that the mass center of the rolling body slides on the center line of the cage track, which is simplified as a viscous sliding damping, and the interaction between the rolling body and the cage is simplified as a damping spring system under the normal force of the contact surface and the sliding friction force.
The sliding friction force of the bearing is
In the formula, kt is the friction coefficient between the rolling element and the ring; XROLL is the tangential velocity component of the rolling element.
The force balance equation is
In the formula, mB is the mass of the rolling element; X ¨ c is the acceleration of the Centroid of the rolling element.
The torque balance equation is
In the formula: JB is the mass moment of inertia of the rolling element; θ ¨ B is the Angular acceleration of the rolling element.
This model can be used for bearing fluid lubrication analysis, but there is less analysis of lubrication and force between various components inside the bearing, making the model relatively simple. Introduce the method for establishing a model of cylindrical roller bearings with degrees of freedom, and establish collision and wear models of rollers, rings, and cages with 6 and 3 degrees of freedom. Propose the general method and steps for establishing a dynamic model of bearings. A dynamic model of deep groove ball bearing with degree of freedom is proposed, in which Hertz contact deformation, elastohydrodynamics, radial clearance of bearing and other factors are considered, as well as the influence of bearing defects, such as the waviness of inner and outer rings and local defects. A general method for establishing a dynamic model of a planar multibody system supported by deep groove ball bearings with radial clearance parameters is proposed. Nonlinear dynamic forces are introduced into the model, taking into account the contact stiffness between the ball and the groove, as well as the geometric and material deformation effects of the contact body. The established deep groove ball bearing model is applied to the crank slider mechanism for verification, providing useful reference for the study of bearing dynamic models.
1.3 Dynamic Model of Defective Bearings
During the manufacturing and use of bearings, various defects are inevitable. Although they may not affect their use, they can have a certain impact on the vibration response and mechanical performance of the system during operation. Analyze the contact surface damage caused by bearing slip and the dynamic characteristics of the interaction between bearing components, and establish a bearing dynamic model. Establish a dynamic model of a bearing rotor multi body system, taking into account the contact stiffness between various components of the bearing, and study the vibration frequency of defective bearings. Model bearings with radial clearance and local defects in the inner and outer rings, and use the model to analyze and test the defective bearings. The load distribution, oil film characteristics, structural elasticity and Sliding friction of bearings with local surface defects are studied, and a 3-dof coupling model of bearings is established to analyze the excitation response of local defects to bearings. Establish a dynamic model of deep groove ball bearings with multiple surface defects on the inner and outer ring surfaces, use the Runge Kutta method to solve the coupled motion control equation, and analyze the instantaneous vibration frequency of the cage, ring, and ball with surface defects. This model has been widely used in analyzing the vibration of bearing defects. By improving the two-dimensional vibration model of ball bearings and considering centrifugal load, radial clearance and nonlinear Hertz contact problems, the Newmark time domain integration method is used to solve the motion differential Equation solving of the bearing rotor system, and the effects of surface defects and local deformation on bearings are studied. To obtain the vibration response of defective bearings, a 3-degree of freedom mass spring damping system dynamic model is proposed for diagnosing various defects in bearings. This model assumes that the oil film lubrication effect of bearings is linear and can be used to diagnose local defects on inner, outer rings, and rolling elements. The simplified model of bearing dynamics established is shown in Figure 3. In the figure, KOR, KOF, KIF, KIR, COF, and CIF respectively represent the stiffness and damping between the various components of the bearing; MIR and MOR are the masses of the inner and outer rings, respectively; MB is the mass of the rolling element.
Figure 3 Simplified Model of Bearing Dynamics
2. Research and application of dynamic models for bearing rotor systems
Bearings are no longer considered as separate research entities, but rather are studied together with the interaction between the rotor, which can provide a more realistic and reliable description of bearing performance. The bearing rotor system is further divided into bearing single rotor system and bearing multi rotor system. In the bearing single rotor system, the bearings mainly play a supporting role, while in the bearing multi rotor system, the bearings also play a connecting role. For example, in the dual rotor system of an aircraft engine, the low-pressure rotor and high-pressure rotor are connected by bearings in the rotor system.
2.1 Dynamic model of bearing single rotor system
The bearing rotor system supported by deep groove ball bearing is simplified, and the influence on the performance of the bearing rotor system when the support system is respectively rigid support and flexible support is compared and analyzed, and the influence of non-linear Hertz Contact force, centrifugal load of the ball, angular Contact force and axial force on the model is considered. A bearing dynamics model is proposed to study the dynamic characteristics of a bearing rotor system, mainly focusing on the internal clearance and waviness of the bearing. The focus is on discussing the effects of waviness, radial clearance, and preload on the speed of the cage. On the basis of previous research, the influence of rotor imbalance force on the bearing rotor system is further considered. The bearing element is simplified as a mass spring model, and the contact between the rolling element and the inner and outer rings is considered as a nonlinear elastic contact. Its stiffness is obtained using Hertz contact deformation theory, and a structural vibration analysis model of a high-speed rotor supported by bearings is established. When establishing a dynamic model, it is assumed that the plastic deformation of the bearing is small enough to be negligible, and only the influence of elastic deformation under Hertz theory is considered. Assuming that the angular velocity of the cage is constant, the inner, outer rings, and rotor move in the same plane, and all components and rotors of the bearing are rigid bodies. The simplified model of the bearing mass spring system established is shown in Figure 4. In the figure, R represents the outer radius; R is the inner circle radius; Min and mout are the masses of the inner and outer rings, respectively; Rin and rout are the positions of the mass centers of the inner and outer rings, respectively; ρ J is the radial position of the rolling element; θ J is the angular position of the rolling element; θ X is the angular position of the contact point between the rolling element and the bearing inner ring; χ J is the deviation angle between the center of the jth rolling element and the center of the bearing inner race; (kin) j and (Kout) j are the stiffness of the rolling element and the inner and outer rings, respectively.
Figure 4 Simplified Model of Mass Spring System for Bearings
The waviness is an important factor affecting the performance of bearings, and the established dynamic model for the waviness of the inner and outer rings of bearings is shown in Figure 5. In the figure, π 0 is the original amplitude of the waviness; π P is the maximum amplitude of the waviness; J is the number of rolling elements within the turning angle range; ω C is the angular velocity of the inner circle; ω Y is the common angular velocity of the rolling element.
Figure 5 Dynamic model of bearing inner and outer ring waviness
The amplitude of the waviness of the inner and outer rings of the bearing is
In the formula: L is the arc length; λ Is the wavelength.
2.2 Dynamic model of bearing dual rotor system
As an intermediate bearing connecting the inner and outer rotors in the twin rotor system of Air Link engines, its motion state and stress are different from those of ordinary support bearings. On the basis of bearing dynamics and rotor dynamics, considering the coupling characteristics at the intermediate bearing, a nonlinear dynamic model of the dual rotor Hertz bearing coupling system is established using Newmark - β The integration method and Newton Raphson Iterative method method are used to solve the nonlinear dynamic differential equation of the bearing rotor system. The influence of rotor speed, the clearance of intermediate bearing, the number of rollers and the structural parameters of support bearing on the stability of the rotor system is analyzed. Deeply study the dual rotor model of aeroengine, take the bearing as an important part of aeroengine rotor system, establish its dynamic model, fully consider the bearing clearance, nonlinear Hertz Contact force between rolling body and raceway, variable flexible vibration and other factors, make contributions to the research of bearing dual rotor coupling dynamics, and establish the dynamic model of bearing rotor casing coupling, A series of achievements have been made in the dynamic model of aviation engine bearings. A dynamic analysis model of a 4-DOF aero-engine main shaft dual rotor system is established. The dual rotor in the system is supported by two sets of angular contact ball bearings and two sets of deep groove ball bearings. When establishing the model, factors such as Contact force, nonlinear displacement and elastic deformation of the bearing are considered, and compared with the established 3-DOF model and the established 5-DOF dynamic model, Explain the influence of the degrees of freedom of the bearing rotor system on the nonlinear dynamic model and system dynamic simulation results. When studying the nonlinear response of a bearing rotor system supported by ball bearings, a dynamic model of a ball bearing with a floating ring squeeze film damper is studied. This model assumes that the inner ring is fixed on the shaft and the outer ring is connected to the squeeze film damper or the squeeze film damper with a floating ring. Different bearing rotor models based on these two oil film damping models are compared and analyzed.
3. Bearing dynamics analysis software
The use of computer simulation technology for analyzing and researching bearing dynamics is very effective. The use of simulation technology for bearing performance analysis began in the late 1950s and has made significant progress to this day.
At present, significant progress has been made in the transient dynamic analysis of the interaction forces between three-dimensional bearing components. FAG, NSK, and SKF all provide verification packages for internal bearing development and design. For example, BEAST 3D analysis software [44] is a bearing dynamics simulation software developed by SKF, which can analyze the forces and their movements on the cage, roller deflection, and friction on the ball, and verify the reliability of the software with examples. SKF collaborated with PELAB to develop the BEAST bearing simulation software package, which uses a fully three-dimensional model with 6 degrees of freedom for each rolling element. This software package can conduct dynamic simulation experiments for most types of bearings, making bearing dynamic simulation design a reality. The dynamic analysis model of oil lubricated ball bearing is established by referring to the analysis model of SHABERTH of SKF American Technology Center. The dynamic analysis program of high-speed ceramic ball bearing is developed by using VB and Fortran 77 language, and the dynamic performance parameters of bearing, such as deformation, stiffness, life, heating and Friction torque, can be calculated.
4. Existing problems and development trends
Elastohydrodynamics must be considered in the research of bearing dynamics. In the early stage, the research and development of bearing dynamics were restricted due to the immature theory of elastohydrodynamics. Simplifying the elastohydrodynamic model makes the model more compact and simple, and the calculation speed is faster. It can be widely used in complex mechanical systems with elastohydrodynamic problems, which will have a great impact on the research of bearing dynamics.
With the development of rotating equipment towards high speed and heavy load, the coupled vibration, nonlinear dynamics, and fault diagnosis of bearings and high-speed and heavy load rotor systems are all research topics worth paying attention to in the future.
At present, the bearing dynamics simulation results are far from the actual ones. The domestic simulation analysis software is mainly aimed at the quasi-static analysis model. It can refer to some mature simulation software development experience, and use the intelligent optimization Theory of computation, parallel design, virtual reality technology and other contemporary cutting-edge technologies to make the simulation results more accurate.
More about KYOCM Thrust Ball Bearing:
Thrust ball bearings are designed to take axial (thrust) loads at high speeds, but they cannot take any radial loads. These bearings feature bearing washers with raceway grooves in which the balls move.
Thrust ball bearings are classified into “flat seat” and “aligning seat” types, based on the shape of the housing washer (outer ring seat), and as single-direction or double-direction. Spherical and aligning seat washers help to provide tolerance for mounting errors.
2023-07-11