Finite Bearings Research Articles

Probing asymmetric uncertain effect on symmetric three-lobe journal bearing

The conventional design process of journal bearings is completely based on deterministic theoretical predictions. The deterministic approach ignores the probabilistic and random nature of input variables and consequently, poses an arbitrary tolerance or safety factor on the performance parameters. However, such arbitrary safety factors result in performance loss when the factor is very high or failure of the system when the factor is low. This paper presents a probabilistic approach for the design of a three-lobe journal bearing considering the uncertainty in geometric parameters such as clearance, preload, and offset factor. The Monte Carlo simulation (MCS) algorithm is used for the propagation of input uncertainties to the systems’ output. To increase the efficiency of computationally expensive MCS, the moving least square (MLS) method is used as a surrogate model. The training data of the surrogate model is obtained by solving the Reynolds equation, at selective design points, using the finite difference method. To explain the probabilistic response of the bearing for various applications and different operating conditions, the results are presented for long, short, and finite bearings at three different supply pressures considering three different eccentricity ratios. The results of the probabilistic analysis show a significant deviation of the system response from its deterministically predicted values and suggest safe design values for the reliable operation of a three-lobe bearing.

Three dimensional undrained bearing capacity analysis of laterally loaded pile in heterogeneous marine deposits

In this study, the ultimate undrained bearing capacity of a laterally loaded pile, embedded in heterogeneous marine deposits exhibiting linear and bi-linear undrained shear strength profiles was calculated by putting into practice the three-dimensional finite element lower bound limit analysis in conjunction with the second-order cone programming method. The lower bound finite element bearing capacity of a laterally loaded single pile can be found by formulating the element equilibrium, discontinuity equilibrium, boundary conditions, yield function, and optimization of the objective function through these constraints by second-order cone programming method. A parametric study is also conducted to evaluate the most effective factors including the dimensionless pile embedded length ratio, soil strength parameter, heterogeneity factors, transformation depth and soil–pile interface adhesion. The findings of the current study have been compared and validated with the results reported in the literature and presented in the form of tables and figures. The results show that that the effects of heterogeneity, linear and bi-linear variations of undrained shear strength with depth are more meaningful for soft marine clay in comparison to stiff clay and the curves of heterogeneous to homogeneous bearing capacity ratio reach to the unit value by increasing the dimensionless cohesion factor.

Performance of finite bearing under the combined influence of turbulent and non-Newtonian lubrication

The current numerical investigation is used to predict the dynamic performance of finite bearing considering the combined influence of turbulence regime and non-Newtonian flow. With appropriate assumptions, the Navier–Stokes and continuity equation have been revised by the linear turbulence and non-Newtonian fluid model together. The clearance space of the finite bearing has been determined by the finite element method adopting Galerkin’s procedure and robust iteration technique. The cylindrical coordinate forms of momentum and continuity equations are used for the flow field of the finite bearing assuming the turbulent regime and non-Newtonian lubricant. Dynamic performance parameters of a finite bearing have been computed regarding direct and cross-coupled fluid-film coefficients, equivalent stiffness coefficient, whirl ratio, and critical mass at different eccentricity ratios for distinct values of Reynolds numbers preferably up to 13,300 and varied values of non-linear factor of the non-Newtonian fluid model. The obtained results revealed the performance enhancement in the combined regime as compared to pure turbulence and non-Newtonian flow conditions. The stability of finite bearing is significantly improved under the proposed severe flow regime.

Lubrication problems solved by the boundary element method

A boundary element method (BEM) for the solution of lubrication problems on finite bearings is presented. The formulation requires the Reynolds equation to be transformed into a constant coefficient equation. Several film shapes that make the transformation possible are systematically obtained. Noticeably, they cover most practical cases. As an example of application, a numerical solution that only requires the discretization of the boundary is presented for a finite pad bearing.

Simulation of thermo-hydrodynamic behavior of journal bearings, lubricating with commercial oils of different performance

This paper tries to display complete thermo-hydrodynamic research in journal bearings, investigating the growing pressure and temperature rise inside the bearing during operation. The narrow and infinitely long bearings are extensively investigated including analytical solutions either for plain journal bearing, or journal bearing with an axial groove. In narrow bearing approximation, the temperature distribution inside the bearing is also demonstrated. As concerns the finite bearing, it is simulated using the Computational Fluid Dynamis (CFD) algorithm. It is worth noting that the heating effects that are included in this simulation assuming variable viscosity, as well as the bearing, are examined both with industrial lubricants and environmentally friendly oils. One of the most major findings is that, concerning the AWS-100, results in higher temperature values in all examined cases and a more variable temperature distribution at the oil film-bush interface, especially under the assumption of adiabatic bearing. Furthermore, the use of ISO VG 32 leads to a more uniform temperature distribution in the circumferential direction at the film-bush interface.

Magnetohydrodynamic effects on finite and infinite slider bearings

This paper presents a numerical investigation of lubricating slider bearings with conducting couple stress fluids using externally applied magnetics fields. The modified two-dimensional magnetohydrodynamic couple stress Reynolds-type equation is obtained. This governing equation is resolved numerically by using finite difference scheme, which involves the Gauss–Seidel method to compute the bearing characteristics. Numerical results using different considered values of the couple stress and Hartman number are presented. These results demonstrate that the transverse magnetic field and couple stress effects are significant.

Shooting/continuation based bifurcation analysis of large order nonlinear rotordynamic systems

This study introduces an improved numerical algorithm that is capable of analyzing nonlinear vibrations and bifurcations of general, finite, large-order rotordynamic systems supported on nonlinear bearings. An industrial rotor generally consists of several sections and stages, but numerical shooting/continuation method has been applied to a simple Jeffcott type rotor instead of complex models due to the computational burden of the numerical procedure; it becomes significant when the rotor combined with nonlinear finite bearing models. Here, some mathematical/computational techniques such as a deflation algorithm and the parallel computing are suggested for acceleration along with the conventional treatment of model reduction scheme. An eight-stage compressor rotor supported by two identical five-pad tilting pad journal bearings (TPJB) is selected as a mechanical model to test the numerical incorporation of the algorithms. The rotor beam is modelled with 35 nodes, 140 DOF based on Euler beam theory, and the fluid reaction forces from the two TPJB are calculated using simplex, triangular type finite meshes on the pads. In the numerical procedure, the shooting/continuation combined with the acceleration schemes identifies the solution curves of periodic responses and determines their stability. The orbital motions of coexistent responses are obtained from the solution manifolds.

Shooting With Deflation Algorithm-Based Nonlinear Response and Neimark-Sacker Bifurcation and Chaos in Floating Ring Bearing Systems

The double-sided fluid film force on the inner and outer ring surfaces of a floating ring bearing (FRB) creates strong nonlinear response characteristics such as coexistence of multiple orbits, Hopf bifurcation, Neimark-Sacker (N-S) bifurcation, and chaos in operations. An improved autonomous shooting with deflation algorithm is applied to a rigid rotor supported by FRBs for numerically analyzing its nonlinear behavior. The method enhances computation efficiency by avoiding previously found solutions in the numerical-based search. The solution manifold for phase state and period is obtained using arc-length continuation. It was determined that the FRB-rotor system has multiple response states near Hopf and N-S bifurcation points, and the bifurcation scenario depends on the ratio of floating ring length and diameter (L/D). Since multiple responses coexist under the same operating conditions, simulation of jumps between two stable limit cycles from potential disturbance such as sudden base excitation is demonstrated. In addition, this paper investigates chaotic motions in the FRB-rotor system, utilizing four different approaches, strange attractor, Lyapunov exponent, frequency spectrum, and bifurcation diagram. A numerical case study for quenching the large amplitude motion by adding unbalance force is provided and the result shows synchronization, i.e., subsynchronous frequency components are suppressed. In this research, the fluid film forces on the FRB are determined by applying the finite element method while prior work has utilized a short bearing approximation. Simulation response comparisons between the short bearing and finite bearing models are discussed.

THE CHANGE OF FRICTION AND LOAD- CARRYING CAPACITY OF THE JOURNAL BEARING WITH THE CONSIDERATION OF THE OIL AGEING

This paper presents numerical calculations of the hydrodynamic pressure distribution, carrying capacity, and friction coefficient in the gap of a journal bearing. The analysed bearing is lubricated using motor oil. In this paper, oil ageing and temperature influence on viscosity are taken into account. Viscosity changes in the pressure and shear rate are not considered. These changes will be considered in other papers. For the hydrodynamic lubrication analysis, laminar flow of the lubrication fluid and non-isothermal lubrication model of the journal bearing were assumed. As the constitutive equation, the classical, Newtonian model was used. This model was extended by the viscosity changes in temperature and exploitation time. For the considerations, the cylindrical journal bearing with the finite length and smooth bearing, with the full angle of wrap were taken.

ВЛИЯНИЕ ДАВЛЕНИЯ НА ПРОНИЦАЕМОСТЬ ПОРИСТЫХ ЭЛЕМЕНТОВ

In the paper on the basis of nondimensional equations of motion of incompressible lubricant in a working gap and in porous layers of a bearing sleeve there is obtained a calculated model of a radial finite bearing with a double porous insert. A case is consi-dered when a permeability of porous layers on the boundary of gets one value. Lubricant is fed in a radial direction through a hole in a bearing body with subse-quent lubricant filtration through insert pores. As a result of the solution of the problem specified is obtained a field of rates and pressures in porous material and in a lubrication layer. There were determined analytical dependences for the constituents of a vector of a supporting force and a friction moment and also a load factor and a constant of friction were obtained. Besides, is defined a parameter characterizing a relative length of a porous constituent. It is proved that a rational running regime is achieved at the length of a porous constituent equal to 1/6 of a bearing length. The calculated specified models obtained allowed carrying out an analysis and comparison of theoretical results with experimental ones which defined a proximity of a new model to practice.

Warner–Sommerfeld Impedance and Mobility Maps

This note provides additional graphical representations and curve-fit expressions for characteristics of the Warner–Sommerfeld approximate solution of the Reynolds equation for cavitating journal finite bearings. In particular, curve-fit expressions for mobility components, end-leakage factor, and positive film pressure extent are provided in support of previously published graphical representations of the data.

Evaluation of inertia effect in finite hydrodynamic bearings with surface texturing using spectral element solver

Fluid inertia can have a major effect on the pressure distribution in hydrodynamic bearings with surface texturing. So far, however, the effect has been ignored in the simulation of finite bearings. In this work, we develop a spectral element solver for the Navier–Stokes equations specifically tailored to slider finite bearings textured with multiple spherical dimples. Using the solver, we studied the effect of inertia on the load-carrying capacity for 90 different bearing configurations. Our results show that the spatial arrangement of dimples has a significant impact on the inertia effect. For the Reynolds number 50 and the ratio 10 of dimple length to dimple depth, a change from full to partial texturing reduced the effect by roughly 85%.

A novel semi-analytical method for the dynamics of nonlinear rotor-bearing systems

A semi-analytical simulation of a rotor bearing system that consisted of a multi-segment continuous rotor and plain fluid film bearings with finite length is developed in this paper in order to investigate the system's response under the current proposal of simulation. The rotor is simulated using the continuous medium theory and the bearings with finite length follow a very recent analytical simulation that incorporates the analytical solution of the Reynolds equation for the plain, finite journal bearing. The boundary conditions combine the rotor's shearing force and the fluid film forces at the points where the bearings are located, which are expressed analytically with direct integration of the pressure distribution function.A case study of simulating a multi-segment shaft is used in order to compare the current simulation with corresponding simulations using continuous rotor mounted on linear bearings, or finite bearings numerically simulated using the finite difference method. The time response, the amplitude of the response and the phase during passage through resonance are evaluated for these three cases of bearing simulation using numerical procedures and the differences are notified and remarked.

Comparison Between a Meshless Method and the Finite Difference Method for Solving the Reynolds Equation in Finite Bearings

Meshless methods are an alternative procedure for solving partial differential equations in opposition to the numerical methods that require structured meshes. In this work, the meshless method with radial basis functions (MMRB) is compared to the finite difference method (FDM) for solving the Reynolds equation applied to lubricated finite bearing applications. The performance of these two methods is compared based on the precision of estimating the normal force applied to the sliding surface of the bearing. Different mesh families are tested for different bearing configurations. Results show that the MMRB is better than the FDM for nonrectangular geometries with coarser meshes. For rectangular domains without discontinuities, the FDM is still the best choice for solving the problem.

An exact analytical solution of the Reynolds equation for the finite journal bearing lubrication

The Reynolds equation for the pressure distribution of the lubricant in a journal bearing with finite length is solved analytically. Using the method of separation of variables in an additive and a multiplicative form, a set of particular solutions of the Reynolds equation is added in the general solution of the homogenous Reynolds equation thus a closed form expression for the definition of the lubricant pressure is presented. The Reynolds equation is split into four linear ordinary differential equations of second order with non-constant coefficients and together with the boundary conditions they form four Sturm–Liouville problems with the three of them to have direct forms of solution and one of them to be confronted using the method of power series. The mathematical procedure is presented up to the point that the application of the boundaries for the pressure distribution yields the final definition of the solution with the calculation of the constants. The current work gives in detail the mathematical path with the help of which the analytical solution is derived, and ends with the pressure evaluation and a comparison with past numerical solutions and an approximate analytical solution for a finite bearing. The resultant pressure distribution presents slight differences compared to this of the numerical solution and the approximate analytical solution in the values of maximum and minimum pressure but also in the domain of lower values of pressure.

Stability analyses of inclined rotor bearing system based on non-linear oil film force models

Stability characteristics of inclined rotor bearing system are investigated by numerical and experimental research. The dynamic equation is established based on finite element method. To describe the oil film force of journal bearing, Capone’s short bearing model and a variational approximate finite bearing model are employed separately and compared. Different inclination states are considered as different radial loads on journal bearing. Newmark method employed to solve the dynamic equation and stability characteristics are discussed. The numerical results show that the short and finite bearing models have differences on describing the oil force induced oil whirl and whip. Compared with the short bearing model results, unstable zones given by finite bearing model have larger breadth and larger whirl/whip amplitude. With the development of rotor inclination, the unstable threshold decreases and the unstable zone become wider. Influences of parameters on unstable thresholds are also affected by rotor inclination. Comparisons of numerical and experimental results show that the finite bearing model results are more close to the test results.

Numerical Analysis Method and Bifurcation of Gas-Lubricated Journal Bearing-Rotor System with Gyroscopic Effect

Based on the nonlinear theory, the unbalanced responses of the gas-lubricated journal bearing-rotor system are investigated. A time-dependent mathematical model is established to describe the pressure distribution of gas-lubricated journal bearing with nonlinearity. The rigid rotor with gyroscopic effect supported by self-acting gas journal bearing with three axial grooves is modeled. The differential transformation method is employed to solve the time-dependent gas-lubricated Reynolds equation, and the dynamic motion equation is solved by Newmark-β method. The unbalanced responses of the rotor system supported by finite gas-lubricated journal bearings are analyzed by bifurcation diagram, orbit diagram, Poincaré map. The numerical results reveal periodic, period-4 motion of nonlinear behaviors of the system.

스러스트 볼 베어링이 적용된 왕복동형 압축기의 마찰손실 해석

In this paper, a study on the frictional losses and dynamic behaviors of a reciprocating compression mechanism used in small refrigeration compressor is performed. In the problem formulation of the compressor dynamics, the viscous frictional force between piston and cylinder wall is considered in order to determine the coupled dynamic behaviors of piston and crankshaft supported on a thrust ball bearing. The solutions of the equations of motion of the reciprocating mechanism along with the time dependent Reynolds equations for the lubricating film between piston and cylinder wall and lubricant films of the journal bearings are obtained simultaneously. The hydrodynamic forces of journal bearings are calculated using finite bearing model and G<TEX>$\hat{u}$</TEX>m-bel boundary condition. And, a Newton-Raphson procedure was employed in solving the nonlinear equations of piston and crankshaft with a thrust ball bearing. The results explored the effects of design parameters on the frictional losses and dynamic stability of the compression mechanism.

Novel approach to solve the dynamical porous journal bearing problem

Analytical pressure solutions are a simple and robust way to model plain journal bearings in rotordynamics, but cannot be extended to porous journal bearings. Using the method of weighted residuals—viz. Galerkin's method with global shape functions—a novel and fast approach to solve the dynamical porous journal bearing problem is proposed. This approach allows for the influence of rough surfaces on the hydrodynamic pressure implemented through flow-factors, journal misalignment and calculation of stiffness and damping coefficients for finite bearings. The proposed method will be verified using analytical expressions and new results will be shown for porous journal bearings including the influence of rough surfaces.

Vibration Modal Analysis of the Active Magnetic Bearing System

Based on the principle of axial active magnetic bearing rotor-system with single degree of freedom, magnetic rotor-system vibration mode was described. A 3D finite element model of active magnetic bearing rotor-system was established by ANSYS11. Initial six order natural frequencies and vibration mode were calculated by the module of modal analysis of ANSYS11, and compared with the results of system identification. Analysis shows that there was a certain difference between the results, but the difference was small. The analysis result indicates that it is an effective method to apply FEA in the design of the active magnetic bearing. It provided the foundation for structure optimal design of the system and theoretical basis for flexible system design.