Internal Clearance Research Articles

An approach for predicting the internal behaviour of ball bearings under high moment load

This paper presents a modelling approach for predicting the internal dynamic behaviour of ball bearings under high moment loads. This type of loading is a specific feature of helicopter main gear boxes because of special design rules and the high structural flexibility of such systems. The ball bearing model proposed here is not limited to planar systems and incorporates several different phenomena such as contact deformation, elastohydrodynamic contact, internal clearance and cage run-out. The cage–race interaction is treated as a hybrid short journal model, which ensures continuity of the contact force at the transition from the hydrodynamic to metal-to-metal regime. In the dynamic analysis of such a severely loaded bearing, the dependence of the shaft-to-inner-ring force on inner race position cannot be neglected. An equivalent viscoelastic hinge joint has been developed, it produces an additional force that represents the overall rigidity of the system. The stiffness parameters of the joint are identified using global finite element simulations. A ball bearing loaded with two different moments is chosen as an example. Relevant results concerning the internal dynamic behaviour are given. The predicted cage trajectory has been compared to experimental observations, and good agreement has been found.

Analysis of grease contamination influence on the internal radial clearance of ball bearings by thermographic inspection

One of the most important factors influencing ball bearings service life is its internal radial clearance. However, this parameter is also very complex because it depends on applied radial load and ball bearings dimensions, surface finish and manufacturing materials. Thermal condition of ball bearings also significantly affects internal radial clearance. Despite many researches performed in order to find out relevant facts about different aspects of ball bearings thermal behaviour, only few of them are dealing with the real working conditions, where high concentration of solid contaminant particles is present. That?s why the main goal of research presented in this paper was to establish statistically significant correlation between ball bearings temperatures, their working time and concentration of contaminant particles in their grease. Because of especially difficult working conditions, the typical conveyor idlers bearings were selected as representative test samples and appropriate solid particles from open pit coal mines were used as artificial contaminants. Applied experimental methodology included thermographic inspection, as well as usage of custom designed test rig for ball bearings service life testing. Finally, by obtained experimental data processing in advanced software, statistically significant mathematical correlation between mentioned bearings characteristics was determined and applied in commonly used internal radial clearance equation. That is the most important contribution of performed research - the new equation and methodology for ball bearings internal clearance determination which could be used for eventual improvement of existing bearings service life equations.

A Concept for the Consideration of Dimensional and Geometrical Deviations in the Evaluation of the Internal Clearance of Roller Bearings

Internal clearance is a functional key characteristic of bearings. The bearing clearance can be evaluated using rigid body movements of the inner ring. Due to the deviations of dimension and roundness the contact areas between the bearing components cannot be predicted. Hence contact detection is necessary. For this purpose the herein presented concept is based on a raytrace algorithm. The concept can be used for the simulation of the bearing clearance for deviated bearing components. This enables the evaluation of the influence of dimensional and form deviations on the bearing clearance.

Dynamics Analysis of Unbalanced Motorized Spindles Supported on Ball Bearings

This paper presents an improved dynamic model for unbalanced high speed motorized spindles. The proposed model includes a Hertz contact force model which takes into the internal clearance and an unbalanced electromagnetic force model based on the energy of the air magnetic field. The nonlinear characteristic of the model is analysed by Lyapunov stability theory and numerical analysis to study the dynamic properties of the spindle system. Finally, a dynamic operating test is carried out on a DX100A-24000/20-type motorized spindle. The good agreement between the numerical solutions and the experimental data indicates that the proposed model is capable of accurately predicting the dynamic properties of motorized spindles. The influence of the unbalanced magnetic force on the system is studied, and the sensitivities of the system parameters to the critical speed of the system are obtained. These conclusions are useful for the dynamic design of high speed motorized spindles.

シール・冷却空気系統・クリアランスの開発の現状と将来展望(<特集>発電用ガスタービン技術の変遷と将来展望)

シール・冷却空気系統・クリアランスの開発の現状と将来展望(<特集>発電用ガスタービン技術の変遷と将来展望)

複列円すいころ軸受の転動体荷重に与える内部すきまの影響

複列円すいころ軸受の転動体荷重に与える内部すきまの影響

Effects of bearing characteristics on load distribution and sharing of pitch reducer for wind turbine

The most important design factors in the life of a pitch reducer for a wind turbine are the load distribution on the contact surface of the gear teeth and load sharing among the planet gears. The effects of the output shaft bearing of the pitch reducer of a wind turbine on the load distribution and load sharing were analyzed in this study. Using commercial software, four pitch reducer models with different clearances and types of output shaft bearings were constructed, and the resulting load distribution and load sharing characteristics were analyzed. The results show that using a combination of tapered roller bearings reduces the gear’s face load factor and mesh load factor, thus improving the load distribution and load sharing characteristics, better than a combination of spherical roller bearings (SRB) and cylindrical roller bearings (CRB). A smaller radial internal clearance was found to be better when the SRB–CRB combination is used. To analyze the load distribution and load sharing of the planetary gear train, it was essential to model all the elements that constituted the pitch reducer in detail and then analyze the system characteristics, taking into consideration the interactions among the elements.

Phosphate Kinetics During Weekly Cycle of Hemodialysis Sessions: Application of Mathematical Modeling.

Both hyperphosphatemia and hypophosphatemia are associated with increased morbidity and mortality among patients on dialysis. The control of serum phosphate concentration is a considerable clinical problem. Our study aimed to improve understanding of phosphate kinetics in patients on dialysis using mathematical modeling. Three consecutive hemodialysis sessions with breaks of 2-2-3 days were monitored in 25 patients. Phosphate concentration was measured every hour and 45 min after the end of dialysis in blood serum and every 30 min in dialysate during each session. Volume of fluid compartments and body composition were assessed by bioimpedance. The pseudo one-compartment model was applied to describe the profile of phosphate in blood serum during intra- and interdialytic periods of 1-week cycle of three hemodialysis sessions. Model parameters, such as phosphate internal clearance (KM ) and the rate of phosphate mobilization (RM ), were correlated with the reduction of serum phosphate concentration during dialysis (Cpost /Cpre ) and with equivalent continuous clearance (ECC) for phosphate. KM correlated negatively with predialysis serum phosphate concentration. There was significant positive correlation between RM and age. Postdialysis volume of phosphate central compartment was lower than, but correlated to, extracellular water volume. Parameters of the pseudo one-compartment model, phosphate internal clearance, and the rate of phosphate inflow to the central compartment (the one accessible for dialysis) from other phosphate body reservoirs correlated with the indices of dialysis adequacy, such as reduction of serum phosphate and ECC. The pseudo one-compartment model can be successfully extended from a single hemodialysis to the standard weekly cycle of sessions and the model parameters strongly correlate with the adequacy parameters of dialytic removal of phosphate.

Bifurcations and hysteresis of varying compliance vibrations in the primary parametric resonance for a ball bearing

This paper investigates the bifurcation and resonant hysteresis of varying compliance (VC) vibrations in a rigid-rotor ball bearing system with Hertzian contact and radial internal bearing clearance. With the aid of arc-length continuation, the harmonic balance and alternating frequency time (HB-AFT) technique is employed to trace the branches of periodic responses, for which the stability is also investigated using Floquet theory. It is found that the soft resonant hysteresis resulted from the Hertzian contact resonance dominates the primary parametric resonance. In addition, this paper also demonstrates that the mechanism of the period doubling in the case of small bearing clearances is arisen from one to two parametrical internal resonance in the primary resonant area, which can lead nonlinear responses such as quasi-period and chaotic motions to the system, and the evolvements of these complex behaviors are also explored.

Gastric pull-up reconstruction combined with free jejunal transfer (FJT) following total pharyngolaryngo-oesophagectomy (PLE)

IntroductionReconstruction following total pharyngolaryngo-oesophagectomy (PLE) still challenges surgeons because of the extreme length of removed tissue. Gastric pull-up reconstruction, one of the most common reconstructive methods after PLE, has many complications such as anastomotic fistula and gastric necrosis caused by the high anastomotic tension. However, modifications of gastric pull-up reconstruction aiming to reducing the high anastomotic tension have been less reported compared with other aspects with this technique. Here we report a modified gastric pull-up reconstruction combined with free jejunal transfer (FTJ) to reduce the anastomosis tension, and thus to reduce the risk of complications after PLE. MethodsPatients underwent a standard surgical procedure including total pharyngolaryngo-oesophagectomy and bilateral internal jugular lymph nodal clearance. A free jejunal graft about 10 cm was harvested and placed in the appropriate position between mobilized stomach and oropharynx. The anastomosis between the free jejunal graft and the gastric tube was created through a stapler. Vascular anastomosis was made between the jejunal artery and the transverse cervical artery, and between the jejunal vein and the internal jugular vein. Hand suturing technique was used in the anastomosis between jejunum and pharynx. ResultsNone of the patients suffered from any complications such as anastomotic fistula. Both patients resumed early postoperative oral intake. So far, they remain free of tumor recurrence and are in good health for 46 and 18 months, respectively. ConclusionConsidering the tumor status and the patient condition, the gastric pull-up reconstruction combined with FJT after PLE could be a reliable choice.

Dynamic analysis of a slider-crank mechanism with two types of non-ideal revolute joints

Rolling element bearings and journal bearings are typical components comprising a revolute joint in a mechanical system. In the traditional dynamic analysis of multi-body systems, these two types of bearings are generally treated as ideal revolute joints and are modelled by kinematic constraints. The non-linearity of such bearings including internal radial clearance, local contact deformation, friction, lubrication and other phenomena associated with real joints are routinely ignored. In this work, an approach to the dynamic modelling of a multi-body system with these two types of non-ideal revolute joints is outlined. These non-ideal joints were modelled by their force interaction. In rolling element bearings, the frictionless multi-point contacts and bearing kinematics are considered. In the case of a journal bearing, the normal contact force and tangential dry friction force are considered. A planar slider-crank mechanism with multiple non-ideal revolute joints including a rolling ball bearing and two journal bearings is presented as a numerical example to demonstrate application of this approach. The variations of slider kinetic characteristics and the bearing reaction force are used to expound the dynamic characteristics of the mechanism when multiple non-ideal revolute joints are considered.

Global nonlinear dynamic characteristics of rod-fastening rotor supported by ball bearings

Flexible rod-fastening rotor-ball bearing system (FRRBS) is usually adopted in aircraft engines. In this paper, a simplified model for the FRRBS is proposed, and the global nonlinear dynamic characteristics of FRRBS are investigated. The FRRBS has two kinds of special structural features: rods and interfaces. The circumferentially distributed rods are modelled as a constant stiffness matrix with an additional moment (when unequal pre-tightening occurs) without mass effect. The interfaces between disks are modelled as stiffness matrices with normal and tangential contact stiffness, which are determined by pre-tightening force. Then, the shaft is discretized by Timoshenko elements and supported by ball bearings with radial internal clearance. Periodic motions and bifurcation sets are obtained by shooting method. The result shows that the insufficient pre-tightening brings about abnormal vibrations through quasi-periodic instability in high speed region, and the unequal pre-tightening leads to severe vibrations in stable periodic operation state. Moreover, the mass eccentricity would be the most nonlinear source for the nonlinear instabilities. Generally, this paper presents a method for modelling a FRRBS to study the nonlinear dynamic characteristics of the FRRBS.

Turning maneuver caused response in an aircraft rotor-ball bearing system

This paper focuses on the nonlinear response caused by aircraft turning flight in a rotor-ball bearing system with internal clearances and Hertzian contact forces. The equations of motion are formulated by using Lagrange equation, where the aircraft turning flight is modeled as a maneuver load. Numerical analyses are carried out to detect the response of the system under maneuver load. The sub-harmonic resonance and irregular motions subjected to the maneuver load are obtained and presented by bifurcation diagrams, amplitude spectrum diagrams, Poincare maps and phase portraits. Two typical patterns that the maneuver load inducing response are found: one is the alternating of period 2 motion and period 1 motion for $$\omega =1{,}350\,\hbox {rad/s}$$ ; the other is the alternating of quasi-periodic motion and period 2 motion for $$\omega =1{,}470\,\hbox {rad/s}$$ . Moreover, mass eccentricity shows a significant effect on the sub-harmonic resonance of the rotor-ball bearing system under turning flight, which is different from that without considering maneuvering flight condition (Bai and Zhang in Int J Nonlinear Mech 50:1–10, 2013). The results obtained in this paper will contribute an understanding of the nonlinear dynamic behaviors of aircraft rotor systems in maneuvering flight.

Dynamic analysis of double-row self-aligning ball bearings due to applied loads, internal clearance, surface waviness and number of balls

In this paper, a three degrees of freedom (dof) model was established for a double-row self-aligning ball bearing (SABB) system, and was applied to study the dynamic behavior of the system during starting process and constant speed rotating process. A mathematical model was developed concerning stiffness and damping characteristics of the bearing, as well as three-dimensional applied load, rotor centrifugal force, etc. Balls and races were all considered as nonlinear springs, and the contact force between ball and race was calculated based on classic Hertzian elastic contact deformation theory and deformation compatibility theory. The changes of each ball׳s contact force and loaded angle of each row were taken into account. In order to solve the nonlinear dynamical equilibrium equations of the system, these equations were rewritten as differential equations and the fourth order Runge–Kutta method was used to solve the equations iteratively. In order to verify accuracy of the dynamical model and correctness of the numerical solution method, a kind of SABB-BRF30 was chosen for case studies. The effects of several important governing parameters, such as radial and axial applied loads, normal internal, inner and outer races waviness, and number of balls were investigated. These parametric studies led to a complete characterization of the shaft-bearing system vibration transmission. The research provided a theoretical reference for new type bearing design, shaft-bearing system kinetic analysis, optimal design, etc.

Rolling Bearing Fault Diagnosis Based on Physical Model and One-Class Support Vector Machine

This paper aims at diagnosing the fault of rolling bearings and establishes the system of dynamics model with the consideration of rolling bearing with nonlinear bearing force, the radial clearance, and other nonlinear factors, using Runge-Kutla such as Hertzian elastic contactforce and internal radial clearance, which are solved by the Runge-Kutta method. Using simulated data of the normal state, a self-adaptive alarm method for bearing condition based on one-class support vector machine is proposed. Test samples were diagnosed with a recognition accuracy over 90%. The present method is further applied to the vibration monitoring of rolling bearings. The alarms under the actual abnormal condition meet the demand of bearings monitoring.

Strength analysis of the polyoxymethylene cycloidal gears of the gerotor pump

The aim of the article is to present strength analysis of a POM gear system used in the gerotor pump, as well as to determine the design rules in terms of strength. A computational model of the gear system in the gerotor pump has been developed and subjected to the FEM strength analysis carried out. The research has shown that the stress and deformations of the gear system are influenced by design factors, namely by the tooth profile, the gear-shaft connection type, as well as by the mechanical and hydraulic loads working on the system.It has been concluded that the load-carrying capacity limit for the gears ought to be defined based on the POM gears deformation and thus generated additional internal clearances in the gerotor pump which cause uncontrolled leakage in the pump and limit the discharge process. It has also been concluded that the POM gears can be loaded with pressure p=6–8MPa.As a result of the analysis, the principles of designing the gerotor pump gear system have been specified. According to the principles, a model of the gerotor pump has been produced. The model has been presented in the further paper.

Analysis of dynamic behaviour of rotor-bearing system

This paper deals with a non-linear mathematical model for simulation and analysis of a dynamic behaviour of a rotor-bearing system. The model is used for the simulation of plane motion of a centre of rotor’s cross-section which makes the model convenient for the analysis of vibrations generated in a rolling bearing as well as for the analysis of accuracy of the revolution of rotor supported on a rolling bearing. The model takes into account the following quantities: internal radial clearance, external radial load and unbalanced load. Differential equations of a motion are derived using Lagrange’s equations. The contacts between the balls and the rings are considered to be non-linear with a stiffness derived by the Hertzian theory of an elastic contact. The proposed model enables direct determination of local contact deformations which significantly reduces needed computations and CPU time. The results obtained by the model are used to reconstruct phase-space trajectories and Poincaré maps and to calculate the largest Lyapunov exponent in order to establish the stability of rotor-bearing system motion. A computer program is developed based on the mathematical model for the simulation and analysis of a dynamic behaviour of a rotor-bearing system.

Calculation and Analysis of Submersible Well Pump Clearance Leakage Loss

The 200QJ80-22 submersible well pump was investigated to predict the effects of leakage change of the submersible well pump on external characteristics. The integral model of impeller, guide vane and wear-ring was established and the numerical simulation for three-dimensional whole flow field was carried out. The results show that the predicted results have a good agreement with the experimental data. The result of the numerical simulation shows that the head and efficiency of pump decrease with the size of clearance increasing. Especially when the size of clearance is larger than 0.7 mm, the decreasing becomes more obvious. The leakage calculated through numerical simulation were compared with the values obtained according to empirical formula and the results indicates internal clearance leakage at design point is close to the value calculated through empirical formula in smaller wear-ring clearance, but the simulated values of leakage have a small gap with calculated value obtained from empirical formula. Simultaneously, the Second stage leakage decrease significantly compared to the first stage leakage with the flow rate increasing, while the values got by the empirical formula have a small gap with simulated values obtained.

Research on the Mechanical Properties of “Z” Type Double-Decker Ball Bearings

The mechanical model of a "Z" type double-decker ball bearing under the action of radial load is established in this paper on the basis of the Hertz contact theory. According to the security contact angle theory, the influences of inner and outer bearings' internal clearances on the bearing's static load carrying capacity, radial deformation, radial stiffness, and load distribution of balls are analyzed. This model is verified in both stationary and rotational loading experiments. Moreover, the simulation results show that the static load carrying capacity of Z type bearing is smaller than that of either inner bearing or outer bearing that is contributed to compose the Z type bearing. The static load carrying capacity of a Z type bearing reduces with the increase of the inner and outer bearings' internal clearance. These simulation results also indicate that the contact angle of the maximum loaded ball in the outer bearing easily exceeds its security contact angle compared with the inner bearing, which, as the main factor, may cause the Z type bearing to overload and to fail. In this sense, the investigated Z type bearings are unfit to apply to situations with heavy load, high speed, or high precision.

Evaluation of the Influence of a Planned Interference Fit on the Expected Fatigue Life of a Conjugate Cam Mechanism- A Case Study

Due to dynamic effects, clearances, manufacturing and assembly errors in form-closed cam mechanisms, the follower jump can also occur. For conjugate cam mechanisms a technique to avoid the follower jump without the use of a spring involves making the conjugate cam profiles bigger than the kinematical ones by adding an external offset. This strategy produces an interference fit between the conjugate cam profiles and the follower train. This paper presents an ordered procedure to study the influence that the planned interference fit has on the evaluation of the contact forces, the expected fatigue life of the rollers, contact pressures and the lubrication conditions. The study is based on a conjugate cam mechanism with translational roller followers used in a real automatic process for manufacturing muselets. A three-degree-of-freedom dynamic model is proposed and the Hertzian theory for general profiles is used to model the nonlinear contact stiffness between the cams and the crowned rollers. The dynamic model predicts that it is difficult to obtain conjugate cam mechanisms with an infinite expected fatigue life of the rollers just by considering typical achievable manufacturing errors or clearances, and as happens in reality, a set-up process is highly recommended. The procedure is also tested with measured manufacturing errors on a coordinate measure machine—CMM—and with measured radial internal clearances for the rollers measured by an experimental apparatus. Also, to evaluate lubrication conditions, surface finishing measurements have been taken of the cams and the rollers with a surface profiler.