Large Eccentricity Ratio Research Articles

Application of Multigrid for the Solution of Compressible Reynolds Equation of Gas Foil Bearings

The multigrid algorithm can reduce the iterative residuals quickly, but it is rarely used in research about gas foil bearings. This article adopts the Reynolds equation as the pressure governing equation of bump-type foil journal bearings to study the effect of different numerical methods on the execution time of the algorithm program, including the direct method, the iterative method, and the multigrid algorithm. The results indicate that the algorithm performs best, based on the nonlinear discretization scheme and multigrid, and the execution time is generally reduced by 50% compared to the traditional direct method under the same operating parameters. For the case of a large eccentricity ratio, the execution time can even be reduced by up to 68%. The multigrid algorithm is greatly affected by the cycle type parameter and the level numbers of the grid. V-cycle and larger grid-level numbers are recommended on the premise of program convergence. In summary, the multigrid algorithm is an excellent numerical solution that can balance computational accuracy and efficiency when solving the gas lubrication problem of foil bearings.

Study of Dynamic Characteristics of Water-Lubricated Journal Bearings Using Thermohydrodynamic Cavitating Lubrication Model

Cavitation is prone to occur in high-speed water-lubricated journal bearings. However, the effect of cavitation on the dynamic characteristics of high-speed water-lubricated journal bearings has not been studied thoroughly. This study focuses on this topic. A thermohydrodynamic (THD) cavitating lubrication model was established for the high-speed water-lubricated journal bearings, including the generalized Reynolds equation, the energy equation, and the force balance equation for bubbles. An equilibrium distribution was introduced to describe the cavitation effect, and parameters were obtained by fitting the measured volume distribution of bubbles. The dynamic characteristics of the water-lubricated journal bearings were calculated, considering the influence of thermal and cavitation effects. An experimental setup was developed to validate the proposed model. The simulated results show that two secondary temperature peaks appear in the divergence zone of the bearing due to the cavitation effect. The cavitation effect should be considered when establishing the model for high-speed water-lubricated journal bearings, especially if it is under a large eccentricity ratio.

Combined effects of surface roughness and physically adsorbed layer in hydrodynamic journal bearing with multiscale approach

AbstractNumerical calculations were made for the film pressure and carried load of the hydrodynamic journal bearing with very large eccentricity ratios, that is, very low clearances when the combined effects of surface roughness and physically adsorbed layer were considered. The shaft was rotating with perfectly smooth surface, and the sleeve was stationary with the nanoscale sinusoidal surface roughness. The fluid piezo‐viscous effect was also incorporated. Owing to the coexistence of the nanoscale adsorbed layer and the macroscopic intermediate continuum fluid film, the multiscale approach was used to simultaneously solve this sandwich film lubrication problem. It was found that owing to low bearing clearances the effect of the adsorbed layer normally very significantly increases the lubricant film pressure, depending on the fluid‐bearing surface interaction; While at the same time both the surface roughness effect and the fluid piezo‐viscous effect are particularly significant compared to those classically calculated, owing to the local more severe film squeezing and the resulting local higher pressures. By considering the effect of the adsorbed layer, the surface roughness more significantly increases the load‐carrying capacity of the journal bearing for very large eccentricity ratios, especially for a strong fluid‐bearing surface interaction. It is suggested that in modeling the hydrodynamic journal bearing for large eccentricity ratios, the combined effects of the surface roughness, physically adsorbed layer and fluid piezo‐viscous property should be considered.

Discussion on dynamic characteristics of long journal bearings considering surface roughness

The hydrodynamic bearing is the key part of many rotating machines, and dynamic characteristics of the bearing would affect the operation performance of machines directly. Thus, this study focused on dynamic characteristics of long journal bearings and tried to illustrate the influence of surface roughness on dynamic performances of long bearings. Applying Christensen stochastic model and long bearing approximation, the modified Reynolds equation was established. To solve the modified Reynolds equation, the numerical integration method was applied. It is found that the surface roughness would significantly weaken the stability threshold speed, normal stiffness coefficients ([Formula: see text], [Formula: see text]), and crossed dampness coefficients ([Formula: see text] and [Formula: see text]) for the long bearing with a large eccentricity ratio. In summary, the influences of surface roughness cannot be ignored during discussing dynamic characteristics of long journal bearings.

Dynamic performance of a novel integral magnetorheological damper-rotor system

Magnetorheological dampers (MRDs), as active damping devices, show superior performances in vibration control in rotor systems. However, MRDs tend to cause nonlinear behaviors of rotor systems under a large eccentricity, and the direction of the output damping force cannot be controlled. To overcome these deficiencies of MRDs, this paper proposes an integral magnetorheological damper (IMRD). In the proposed IMRD, the damper combines the elastic support and the squeeze film as a whole, where both magnitude and direction of oil film force are adjustable. Firstly, by employing bilinear constitutive equations to represent the magnetorheological fluid (MRF), the implicit form Reynolds equations of IMRD are established. Then, by implementing the proposed IMRD model into a flexible rotor system, the nonlinear dynamic model of a flexible rotor system with IMRD is developed by finite element (FE) method. The mechanical characteristics of IMRD are revealed from the view of oil film pressure and dynamic coefficients, and the nonlinear dynamic responses of the rotor system with IMRD are investigated and compared with those of rotor system with traditional MRD. The results indicate that oil film pressure in each bearing region can be adjusted independently with the current, which can lead to changes in both the direction and magnitude of the oil film force. Moreover, different oil film bearing regions have different contributions to the stiffness and damping of IMRD. Under certain situations, IMRD can provide pure damping with negligible stiffness. Besides, the over-squeezed MRF with large journal eccentricity ratio tends to produce strong nonlinearity; whereas, the IMRD proposed here can significantly reduce the level of nonlinearity compared with traditional MRDs.

Effects of journal static eccentricity on dynamic responses of a rotor system under base motions using FDM inertia model

When an oil-lubricated rotor-bearing system is subjected to base motions, it will inevitably be affected by oil-film forces, base excitations, and inertial forces. The alignment state cannot be guaranteed over time, resulting in a static eccentricity. An inertia model for open-ended SFD incorporated three-dimensional fluid velocities and pressure distribution represented by the Reynolds number. In addition to viscous pressure, the inertial pressure distribution was also solved by finite difference method (FDM), and the oil-film forces were determined by numerical integration. Using transient modal integration method, dynamic responses of a rotor-SFD-support system considering oil-film inertia were simulated. The effects of oil-film inertia, static eccentricity, and base motions on transient responses and bifurcation behavior were discussed. The results indicate that for large eccentricity ratio, the oil-film inertia increases the reaction forces. For medium and large static eccentricity, the journal performs quasi-periodic motions. Under base harmonic translation, a series of sub- and super-harmonic frequencies kΩ±Ωz are stimulated along with the multiples of rotational frequency kΩ and base harmonic frequency Ωz. The bifurcation motion of journal with static eccentricity fluctuates seriously in resonance. Overall, nonlinear dynamic responses of rotor-damping-support systems subjected to base motions can be evaluated during dynamics design and vibration suppression.

Numerical analysis on the performance characteristics of a new gas journal bearing by using finite difference method

This paper proposes a novel gas journal bearing in which orifices are different in diameter and distribute unevenly. Finite Difference Method (FDM) combined with Linear Perturbation Method (LPM) is used to solve the unsteady-state Reynolds equation of the flow field in the bearing clearance. Moreover, four types of bearing structures are used to discuss the effects of orifices different in diameter and uneven distribution on the bearing performance. The results demonstrate that the new bearing has better static and dynamic performances compared with those of traditional bearing in which orifices are equal in diameter and distribute evenly. Moreover, thin gas film thickness, high supply pressure, and large eccentricity ratio are hopeful for improving load capacity of the new bearing. Furthermore, the stability of the novel bearing is improved if eccentricity ratio is 0.25–0.3.

Effect of Torsion on Seismic Response of Single-Story Structures: an Energy-Based Design Approach (EBD)

Torsion of structural frames is one of the main causes of damage to structures during earthquakes. Accordingly, this study investigated the effects of torsion on frame structures with different asymmetries in stiffness when subjected to earthquakes with natural translational vibration period of less than 1.00 s. In this study, an idealized single-story lateral load resisting system was considered, and an energy-based design approach was used to analyze and investigate the effect of the resulting eccentric center of rigidity on the seismic response of frames. First, the stability of the energy spectrum due to torsional behavior was verified by conducting a nonlinear dynamic analysis. A series of nonlinear dynamic analyses was then used to investigate the responses of torsionally flexible systems with different eccentricity ratios in terms of total energy input, plastic strain energy distribution, cumulative plastic deformation ratio, and displacement ratio which found to be sensitive in different translational vibration of system. The results demonstrate that the energy input into asymmetric systems is less than that input into symmetric systems under the same accumulated plastic deformation ratio. Moreover, the vertical members that are located farthest to center of rigidity (flexible-edge member) were observed to suffer greater damage than other vertical members, particularly under large eccentricity ratios.

Hydrodynamic lubrication analysis of water-lubricated bearings with partial microgroove considering wall slip

In the present works, the Reynolds equation considering the partial wall slip is used to study the hydrodynamic lubrication performance of Partial Microgroove Water-lubricated Bearings (PMWB). Based on the developed model, the hydrodynamic performances of partial microgroove water-lubricated bearings with three microgroove distributions are systematically compared. The results show that herringbone-groove bearing yields the maximum load capacity, followed by spiral-groove bearings and straight-groove bearings. Furthermore, numerical results also show that wall slip can enhance the load capacity for all three partial microgroove distributions. In addition, the effects of the groove depth and the groove angle, eccentricity ratio and slip arrangement on the lubrication performance of the microgroove bearings are studied. The simulation results indicate that the eccentricity ratio significantly affects the lubrication performance of partial microgroove water-lubricated bearings considering wall slip, while the groove depth and the angle only slightly affect the lubrication performance. Moreover, the slip arrangement is less sensitive to the depth and angle of groove than eccentricity ratio. The parametric study demonstrates that the load capacity of PMWB will decline at a relatively large eccentricity ratio.

Lubrication Characteristics of Water-Lubricated Rubber Bearings With Partial Wear

Abstract Water-lubricated rubber bearings are widely used in the propulsion shafting of military craft and ships. These bearings may wear down after a period of service, and consequently, their lubrication characteristics will change, affecting the operation of the shaft. Hydrodynamic lubrication characteristics of water-lubricated rubber bearings with partial wear at the bottom are studied by finite difference method. The steady-state characteristics of water-lubricated rubber bearings with wear and elastic deformation of rubber liner considered are solved with successive over-relaxation iteration method, and the dynamic characteristics are further calculated with finite perturbation method. The results show that water film thickness and distribution of water film pressure is significantly changed by wear. With the same eccentricity ratio, the maximum water film pressure, load capacity, attitude angle, and friction force are reduced by wear, but the friction coefficient is increased by wear. Under the same load, the minimum water film thickness and the maximum water film pressure are slightly affected by wear, the eccentricity ratio increases with the increase of wear, and the attitude angle and the friction coefficient decrease with the increase of wear. For large eccentricity ratios, the direct stiffness coefficients and direct damping coefficients are decreased by wear. The maximum allowable wear depth is approximately 1/6 of the bearing clearance.

Dynamic coefficients and stability analysis of a water-lubricated hydrostatic bearing by solving the uncoupled Reynolds equation

This paper concerns the dynamic characteristics of incompressible laminar flow hydrostatic bearings. An improved method based on the mathematical perturbation technique for calculating the stiffness and damping coefficients of hydrostatic bearings is presented; it raises the calculation efficiency by uncoupling the recess flow continuity equations from the solving of the Reynolds equation. The dynamic coefficients of a water-lubricated four-recess hydrostatic bearing are calculated with both the improved method and the conventional method. The comparisons between these two methods validate the effectiveness and correctness of the improved method. Furthermore, the dynamic response and stability characteristics of a rotor supported by the hydrostatic bearing are calculated and compared with experimental results. First, the dynamic responses calculated with the linear and nonlinear bearing dynamic coefficients models show that the linear model is inaccurate if the rotor whirls in a large eccentricity ratio range, owing to the nonlinearity of stiffness and damping. Second, according to the numerical and experimental results, the nonlinearity of bearing stiffness and damping coefficients could induce double rotational frequency component in the unbalance response frequency spectrum. Finally, the numerical results indicate that the stability threshold speed of the bearing increases with eccentricity ratio. The improved method proposed can be used to evaluate the dynamic performance of hydrostatic bearings efficiently, and the bearing characteristics presented could contribute to a better understanding of the performance of water-lubricated hydrostatic bearings.

Investigations of the static and dynamic characteristics of water-lubricated hydrodynamic journal bearing considering turbulent, thermohydrodynamic and misaligned effects

Abstract A misaligned thermohydrodynamic (THD) model considering turbulent-flow is presented to predict the performance of water-lubricated hydrodynamic journal bearing (WHJB). The load capacity, frictional power loss, flow rate, mid-plane pressure distribution, translational, tilting and coupled translational-tilting dynamic coefficients versus the rotary speed, eccentricity ratio and tilting angle are analyzed; quantitative comparisons between models considering and without considering turbulent, THD and misaligned effects are given. Results show that the turbulence improves the load capacity while the THD effect decreases it; the THD effect reduces the frictional power loss significantly while the turbulent effect is weak; at large rotary speed and eccentricity ratio, the THD and turbulent effects on the dynamic coefficients should be fully considered at both of aligned and misaligned conditions.

Dynamics of a rotor system coupled with water-lubricated rubber bearings

Dynamic behaviour is significantly important in the design of large rotor systems supported on water-lubricated rubber bearings. In this study, the mathematical model of elastohydrodynamic lubrication of the bearing is established based on the theory of hydrodynamic lubrication after considering the elastic deformation of rubber, and the dynamic characteristics of water-lubricated rubber bearings are analysed under small perturbation conditions according to the load increment method and the finite difference method. Next, the differential equation of rotor systems coupled with the water-lubricated rubber bearing is deduced using Lagrange’s approach, and its critical speeds, stability, and unbalanced responses are analysed in detail. The numerical results show that several parameters, such as the eccentricity, length–diameter ratio, and clearance of bearing and the rotating speed of the rotor, have a great impact on the dynamic performance of water-lubricated rubber bearings, and this influence cannot be ignored, especially in the case of large eccentricity ratios. The dynamic characteristics of rotor systems guided by water-lubricated rubber bearings reveal that the critical speeds are much lower than the ones under the rigid supports because of the elastic deformation, and they also indicate that the rotor system supported on water-lubricated rubber bearings has a weaker stability. In addition, the steady-state responses of the rotor system are analysed when the mass unbalance of the propeller exists, and the effect of the thickness of the rubber liner is also considered.

Thermo-hydrodynamic analysis of large-eccentricity hydrodynamic bearings with texture on journal surface

To improve the performance of a heavy-loaded and high-speed hydrodynamic bearing, partial texture is designed on the journal surface. A three-dimensional thermo-hydrodynamic analysis model is developed based on the computational fluid dynamics method. Thermal conduction of bearing bush and journal, heat convection on the air-bearing surface, viscosity–temperature effect, and cavitation effect are taken into account. Performance of textured and un-textured high-speed journal bearing under different eccentricity ratios and different external loads is compared. The results indicated that partial texture on journal surface can play a positive role for large eccentricity ratio conditions in this study, and the textured bearing has a higher load carrying capacity, a lower maximum oil pressure, and a lower oil temperature rise than the un-textured bearing based on an optimal design of texture.

Dynamic Characteristics of Spiral-Grooved Opposed-Hemisphere Gas Bearings

The traditional eight-coefficient bearing model only considers the translational motion of the bearings and neglects the tilting motion and coupling effects between them. In this paper, the dynamic characteristics of the spiral-grooved opposed-hemisphere gas bearing considering five degrees-of-freedom are studied, and 50 dynamic coefficients including the translational, tilting, and coupling components are completely calculated. The Reynolds equations and their perturbed equations are solved by the finite element method to obtain the dynamic stiffness and damping coefficients. The effects of the tilting motion on the dynamic coefficients and response are analyzed, respectively. The results show that the coupling coefficients between the translational and tilting motions, which have been neglected in most previous studies, are significant at large eccentricity ratio. But these coupling coefficients have little effect on the dynamic response. On the other hand, the influences of the tilting motion on the synchronous response and natural frequency are remarkable and will decrease the stability of the rotor bearing system.

Numerical analysis of spiral grooved opposed-hemisphere gas bearings: A parametric study

Although the spiral grooved opposed-hemisphere gas bearings have been widely applied in supporting the spin axis of inertial grade gyroscopes, the accurate prediction of the performance characteristics of the bearings is still very difficult because of their structural complexity. In this paper, the static characteristics of the bearings considering the coupling effects between the journal and two hemisphere bearings are obtained theoretically. The Reynolds equations modeling the journal and hemisphere bearings are solved simultaneously by the finite element method satisfying the continuous boundary conditions between these three bearings. The fluctuations of the bearing load and reaction torque caused by rotating grooves are shown and can be neglected at large number of grooves. The results show that neglecting the coupling effects between the journal and two hemisphere bearings will result in large discrepancies at high rotational speeds and large eccentricity ratios. And the hydrodynamic effect can be divided into two regions for spiral grooved gas bearings: the groove effect domain region and the converged wedge effect domain region. A parametric study provides efficient guidance for the optimal design of the opposed-hemisphere gas bearing.

Research on the Static and Dynamic Characteristics of Misaligned Journal Bearing Considering the Turbulent and Thermohydrodynamic Effects

Journal misalignment usually exists in journal bearings that affect nearly all the bearings static and dynamic characteristics including minimum oil film thickness, maximum oil film pressure, maximum oil film temperature, oil film stiffness, and damping. The main point in this study is to provide a comprehensive analysis on the oil film pressure, oil film temperature, oil film thickness, load-carrying capacity, oil film stiffness, and damping of journal bearing with different misalignment ratios and appropriately considering the turbulent and thermo effects based on solving the generalized Reynolds equation and energy equation. The results indicate that the oil thermo effects have a significant effect on the lubrication of misaligned journal bearings under large eccentricity ratio. The turbulent will obviously affect the lubrication of misaligned journal bearings when the eccentricity or misalignment ratio is large. In the present design of the journal bearing, the load and speed become higher and higher, and the eccentricity and misalignment ratio are usually large in the operating conditions. Therefore, it is necessary to take the effects of journal misalignment, turbulent, and thermal effect into account in the design and analysis of journal bearings.

On the rheology and microstructure of immiscible polymer blends in eccentric Taylor–Couette flow

Various consumer products contain droplets that are dispersed in a matrix. The rheological behavior and microstructural characteristics of these products are important determinants of product quality. This paper computationally investigates the rheological behavior and microstructural dynamics of multiple droplets dispersed in a polymeric matrix fluid in a steady eccentric Taylor–Couette flow field. A large eccentricity ratio was chosen for the cylinder system in order to allow flow recirculation to occur. A thermodynamically consistent model was used to describe the dynamics of the blend, including breakup and coalescence. The qualitative behavior of the droplets was found to be governed by the relative rotation and eccentricity of the cylinders. In addition, the matrix viscoelasticity was observed to have a large quantitative impact on the droplet characteristics. The findings of this research encourage us to experimentally assess the predictive capability of the model in the eccentric Taylor–Couette system.

Numerical solution of an extended White–Metzner model for eccentric Taylor–Couette flow

In this study, we have developed a new numerical approach to solve differential-type viscoelastic fluid models for a commonly used benchmark problem, namely, the steady Taylor—Couette flow between eccentric cylinders. The proposed numerical approach is special in that the nonlinear system of discretized algebraic flow equations is solved iteratively using a Newton–Krylov method along with an inverse-based incomplete lower-upper preconditioner. The numerical approach has been validated by solving the benchmark problem for the upper-convected Maxwell model at a large Deborah number. Excellent agreement with the numerical data reported in the literature has been found. In addition, a parameter study was performed for an extended White–Metzner model. A large eccentricity ratio was chosen for the cylinder system in order to allow flow recirculation to occur. We detected several interesting phenomena caused by the large eccentricity ratio of the cylinder system and by the viscoelastic nature of the fluid. Encouraged by the results of this study, we intend to investigate other polymeric fluids having a more complex microstructure in an eccentric annular flow field.

The Identifications of Milling Force Coefficients and Eccentricity Based on Multi-objective Optimization in Frequency-domain

Based on the assumption of orthogonal cutting,cutter geometry,and the strict model of the cutting load,a model of milling force is developed with the effects of eccentricity of cutter.Through a Fourier transform,the expression of milling force in frequency domain is obtained.By comparing the harmonic component of simulation results with the harmonic component of measurement results,the multi-function of residual harmonics are obtained by using matrix calculation in frequency domain.Then,a method based on multi-objective optimization in frequency domain is improved for the cutting force coefficients and eccentricity of tool identification.Experimental studies are also carried out to verify the accuracy of the cutting force coefficients and eccentricity of tool obtained by the proposed method.Results indicate the error of predicted results obtained by the proposed method is in the range of 2% to 6%.The efficiency and accuracy of the proposed method are also verified by comparing with the results presented in reference,especially for the milling processes with large cutter eccentricity ratio.