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This paper describes a theory for the approximated analytical determination of pressure distribution in the gap of a generic finite length journal bearing in isothermal laminar lubrication conditions as the position assumed by the journal axis varies in a reference frame fixed in the bearing. The same theory is used to set up lubricated finite length journal bearing models in order to obtain an analytical description of the fluid film force, which is characterized by considerable prediction accuracy in the whole field defining the length-diameter ratio. In order to assess the model’s prediction accuracy, the static characteristics of the journal bearing are analytically determined. The theoretical results, presented in the form of diagrams, are compared with the corresponding results obtained numerically.

A theoretical model for the normal instantaneous squeeze film force for a finite length cylinder is developed in this paper. The model assumes large unidirectional cylinder motion along a sleeve diameter. Based on the assumption of a parabolic flow field, a normal squeeze film model for an infinitely long cylinder is first obtained. Combining the infinitely long model with side-leakage factors, a finite length model is then obtained. The model shows that the instantaneous squeeze film force consists of three position-dependent nonlinear terms: namely a viscous term, an unsteady inertia term and a convective inertia term. From experimental measurements using water and a clearance to radius ratio of 0.032, the viscous term of the theoretical model should be corrected by a factor involving the instantaneous squeeze film Reynolds number and the absolute value of instantaneous eccentricity. The synthesized squeeze force waveforms obtained using the corrected equation with averaged weighting coefficients agree very well with the experimental waveforms for eccentricity ratios up to 0.9 and a wide frequency range. The corrected equation is suitable for the calculation of the normal instantaneous squeeze film force given the instantaneous position, velocity, and acceleration of the cylinder center.

This paper describes a theory for the approximated analytical determination of pressure distribution in the gap of a generic finite length journal bearing in isothermal laminar lubrication conditions as the position assumed by the journal axis varies in a reference frame fixed in the bearing. The same theory is used to set up lubricated finite length journal bearing models in order to obtain an analytical description of the fluid film force, which is characterized by considerable prediction accuracy in the whole field defining the length-diameter ratio. In order to assess the model’s prediction accuracy, the static characteristics of the journal bearing are analytically determined. The theoretical results, presented in the form of diagrams, are compared with the corresponding results obtained numerically.

A theoretical model for the normal instantaneous squeeze film force for a finite length cylinder is developed in this paper. The model assumes large unidirectional cylinder motion along a sleeve diameter. Based on the assumption of a parabolic flow field, a normal squeeze film model for an infinitely long cylinder is first obtained. Combining the infinitely long model with side-leakage factors, a finite length model is then obtained. The model shows that the instantaneous squeeze film force consists of three position-dependent nonlinear terms: namely a viscous term, an unsteady inertia term and a convective inertia term. From experimental measurements using water and a clearance to radius ratio of 0.032, the viscous term of the theoretical model should be corrected by a factor involving the instantaneous squeeze film Reynolds number and the absolute value of instantaneous eccentricity. The synthesized squeeze force waveforms obtained using the corrected equation with averaged weighting coefficients agree very well with the experimental waveforms for eccentricity ratios up to 0.9 and a wide frequency range. The corrected equation is suitable for the calculation of the normal instantaneous squeeze film force given the instantaneous position, velocity, and acceleration of the cylinder center.

The Collocation method is used to study the characteristics of a finite length Squeeze Film Damper (SFD) with variable clearance. The secant stiffness and damping coefficients of the SFD for centered circular orbits is compured by solving Reynolds equation for a finite bearing, using the collocation method. A significant disadvantage with a conventional SFD is the highly nonlinear variation of secant stiffness and damping coefficients with eccentricity ratio. It is shown in this paper that by suitably varying the parameters α and λ, which control the nature of clearance variation, the secant stiffness and damping coefficient variation with eccentricity; ratio can be altered to better suit specific design needs. Nonlinear transient analysis of a single journal in a finite SFD is carried out, and, periodic journal center orbits due to unbalance response for uncentered finite SFD’s are also obtained. The centered circular orbit response for different unbalance values and speeds is obtained. Results from the Collocation method show excellent agreement with finite difference results. The Collocation method is shown to be an efficient and viable technique that is an order of magnitude computationally faster than the finite difference method, for static and dynamic analysis of finite length SFD’s.

A rapid method for calculating the general nonlinear response of finite-length plain journal and squeeze film damper bearings is presented. The method incorporates a finite-length correction factor which modifies the nonlinear forces obtained from short bearing theory. The steady-state rotational, precessive squeeze, and radial squeeze forces obtained with the correction factor compare extremely well with the forces obtained from an analytic solution of Reynolds equation using a variational approach up to L/D of 1.25 and hence covers the most commonly encountered L/D ratios. The method is no more time consuming than the short bearing analysis and is especially suited to nonlinear transient analysis of flexible rotors.

Finite-length roller contacts are found in apex seal-housing interfaces in rotary engines, as well as in many other mechanical systems, such as those at the interfaces of meshing gear teeth, cam followers, and roller bearings. Rollers are usually designed with crowns and modified ends to mitigate the negative effects of misalignment, thermal deformation, and edge-induced non-uniform displacement. A crowned-roller elastohydrodynamic lubrication model is enriched and employed to investigate seven housing surface profiles, including a sinusoidal recess, sinusoidal ends, the combination design of a sinusoidal recess and sinusoidal ends, multiple sinusoidal recesses, a concave shape, and a concave shape with sinusoidal ends or round corners, aiming to improve the tribological performance of the interface via increasing the minimum film thickness but reducing the average and maximum film thicknesses, and lowering friction coefficient. The modified discrete convolution and fast Fourier transform method with duplicated padding is utilized to deal with the potential free-end-surface effect. The simulation results suggest that the combinations of a concave shape and sinusoidal ends, or round corners, are the best designs because they can help distribute the lubricant most uniformly without causing a negative edge effect.

Expressions are derived which define dynamic coefficients for high-pressure annular seals typical of wear-ring and interstage seals employed in multistage centrifugal pumps. Completely developed turbulent flow is assumed in both the circumferential and axial directions, and is modeled by Hirs’ turbulent lubrication equations. Linear zeroth and first-order perturbation equations are developed by an expansion in the eccentricity ratio. The influence of inlet swirl is accounted for in the development of the circumferential flow. The zeroth-order momentum and continuity equations are solved exactly, while their first-order counterparts are reduced to three ordinary, complex, differential equations in the axial coordinate Z. The equations are integrated to satisfy the boundary conditions and define the pressure distribution due to seal motion. Integration of the pressure distribution defines the reaction force developed by the seal and the corresponding dynamic coefficients. Finite-length solutions for the coefficients are compared to two “short-seal” solutions.

The general Reynolds equation for self-acting, finite length gas-lubricated journal bearings is solved using an interior collocation method. The method assumes an approximate solution to the governing differential equation in the form of a series of trial functions, wn, with n unknown coefficients. The coefficients are evaluated by substituting the assumed solution in the governing differential equation, and forcing the residual (error) at n collocation points in the computational domain to be zero. The effectiveness of the collocation method is demonstrated by using the technique for the static and dynamic analysis of a journal supported by a plane gas bearing. The results from the collocation method agree very well with those obtained from a finite difference technique. Periodic orbit plots for a single journal in a finite bearing are presented for various journal unbalances and speeds. The collocation method is shown to be an order of magnitude computationally faster than the finite difference method. The method can be extended to other bearing types such as slider, hydrostatic and tilting pad bearings.

A linearization method is employed on the Reynolds equation to obtain the load and friction torque of a Rayleigh step-film scheme applied to a journal bearing of finite length. The analysis is broken into two parts; i.e., the stepped and ridge regions. The film thickness is considered constant in each region thereby simplifying the linearized Reynolds equation. In addition to the usual plain-bearing parameters (the compressibility number, eccentricity ratio, and length-to-diameter ratio), the load and friction force of a stepped bearing depend on the ratio of the stepped clearance to ridge clearance, the ratio of the width of ridge to the width of the pad, and the number of pads. The solution for the load and friction force for a single pad is obtained in the form of a highly convergent series. A number of equal pads are disposed around a journal and the total bearing load is computed for various eccentricities. Computation from the infinite series is lengthy and a digital computer was used in order to carry out the calculational work. This enabled optima to be selected for each system composed of different numbers of pads. The friction torque and hence the bearing power loss were computed for every case.

A method for treating a self-acting, gas-lubricated journal bearing of finite length is presented, which is a compromise between the analytic solution and numerical results. Analytic solutions have been obtained here which are essentially the higher approximations of the ph linearization. The obtained solutions compare closely with the numerical results and satisfy the condition given by Elrod and Burgdorfer.

To improve the efficiency in nonlinear dynamic calculation of finite-length tilting-pad journal bearings (TPJBs) under dynamic loads, an analytical method for hydrodynamic bearing forces, which considers the turbulence effect, is proposed using the method of separation of variables under the dynamic Gümbel boundary condition. No thermal effects are considered because this method is designed for the low viscosity case. The infinitely long bearing pressure is introduced as the circumferential pressure, and a general solution of the nonhomogeneous Reynolds equation is derived as the axial pressure. The turbulence model of Ng and Pan is characterized by a linear function of film thicknesses. A complete analytical expression of hydrodynamic bearing forces is derived. The analytical simulation shows slight differences and extremely low time expense in lubricating and dynamic performance compared to published data and finite difference method (FDM) simulation. The analytical method could be used to fast evaluate the nonlinear dynamic performance of a TPJB-rotor system in the low viscosity environment, supporting the nonlinear dynamic design of the system.

The effects of porosity on the performance of gas bearings are investigated. A thin-wall approximation is developed to simplify the coupling between the Reynolds equation in the fluid film and the three dimensional Laplace equation in the porous matrix. The Newton-Raphson method is used to linearize the modified Reynolds equation and the column method is employed to solve the system of linear equations in matrix forms. The solution of a solid-wall bearing is included as a special case. Bearing characteristics are presented in tables and charts for varied permeabilities, slip coefficients, porous layer thicknesses, eccentricity ratios, length-to-diameter ratios, and for a wide range of compressibility number.

Despite intuition, the squeeze-film bearing of finite dimensions produces much more load support per unit area than is found from a solution based on an infinite bearing. The load support is shown to be made up of two components—that due to the infinite journal plus an additive term due to the axial pressure distribution. Two methods of solution of the governing equation are applied: (a) A small-parameter analysis and (b) a numerical finite-difference technique. In the limit, the infinitely short journal is shown to have a load support 2.5 times that of the infinitely long journal.

The interaction and surface features between point contact surfaces composed of longitudinal roughness with infinite or finite length and transverse roughness were discussed based on a transient non-Newtonian thermal elastohydrodynamic lubrication (EHL) model. Each surface shape is greatly affected by the difference in elastic moduli, thermal conductivities, and velocities of both contact surfaces. There is a large difference in pressure behavior when the transverse roughness is in contact with the longitudinal roughness with finite length and when it is in contact with the longitudinal roughness with infinite length. In the contact between surfaces with infinitely long longitudinal and transverse roughness, the friction coefficient is lower when the surface with longitudinal roughness has a low thermal conductivity than when it has a high thermal conductivity. Furthermore, the pressure fluctuation is larger when the transverse roughness surface has a high thermal conductivity than when it has a low thermal conductivity.

Bearing impedance vectors are introduced for plain journal bearings which define the bearing reaction force components as a function of the bearing motion. Impedance descriptions are developed directly for the approximate Ocvirk (short) and Sommerfeld (long) bearing solutions. The impedance vector magnitude and the mobility vector magnitude of Booker are shown to be reciprocals. The transformation relationships between mobilities and impedance are derived and used to define impedance vectors for a number of existing mobility vectors including the finite-length mobility vectors developed by Moes. The attractiveness and utility of the impedance-vector formulation for transient simulation work is demonstrated by numerical examples for the Ocvirk “π”, and “2π” bearing impedances and the cavitating finite-length-bearing impedance. The examples presented demonstrate both bearing and squeeze-film damper application. A direct analytic method for deriving a complete set of (analytic) stiffness and damping coefficients from impedance descriptions is developed and demonstrated for the cavitating finite-length-bearing impedances. Analytic expressions are provided for all direct and cross-coupled stiffness and damping coefficients, and compared to previously developed numerical results. These coefficients are used for stability analysis of a rotor, supported in finite-length cavitating bearings. Onset-speed-of-instability results are presented as a function of the L/D ratio for a range of bearing numbers. Damping coefficients are also presented for finite-length squeeze-film dampers.

A rotating shaft vibrating in a squeeze film bearing and a tube in a heat exchanger oscillating with fluid-filled cylindrical supports both involve cylindrical squeeze films. Many theoretical and experimental results show that the squeeze film force consists of both a damping force and an inertia force. For relatively large amplitude motions or when the initial eccentricity is large, the time waveform of the squeeze force is significantly nonlinear. In order to predict the transient response of a rotor with squeeze film bearings or a heat exchanger tube subject to flow induced vibration, the nonlinear instantaneous squeeze force must be calculated. This paper presents a model for the instantaneous cylindrical squeeze film force for planar motion. The squeeze film model for a two-dimensional plate shows that there are three nonlinear terms included in the squeeze force. Based on this model, an equation for the short length, cylindrical squeeze film force for moderately large eccentricities is developed. The equation includes the three nonlinear terms: the viscous term, the unsteady inertia term, and the convective inertia term. All three terms are functions of instantaneous eccentricity. The equation predicts the nonlinear multi-harmonic and unsymmetrical time waveforms of the instantaneous squeeze film force for planar motions with both in-line and out-of-line initial eccentricities. The results are compared with experimentally measured squeeze force waveforms obtained with a length to diameter ratio of 0.75 and instantaneous eccentricities less than 0.75. The squeeze force waveforms for this finite length geometry can be reasonably predicted if correction coefficients, which account for the circumferential flow, are applied to the three nonlinear force terms. These coefficients are themselves functions of frequency, initial eccentricity and amplitude.

This investigation deals with the stability characteristics of oil filled porous journal bearings of finite length and with velocity slip. The stability curves are drawn for different slip parameters, eccentricity ratios, slenderness ratios, and in the absence of any experimental data, the theoretical results for the solid bearings obtained by this analysis have been compared with the available results of solid bearing.

The paper treats on a comparative basis the development of the flow and pressure maps in deep, and respectively, shallow hydrostatic bearing pockets. The numerical simulation uses a dimensionless formulation of the Navier-Stokes equations written for a body fitted coordinates system, and applied through a collocated grid. The present work is a continuation of the simulations performed by Braun et al. (1993, 1994a, 1994b) to cases where the length and diameter of the restrictor feedline is of consequence to the flow in both the shallow and the deep pocket. The model includes the coupling between the pocket flow and a finite length feedline flow, on one hand, and the pocket and the adjacent lands on the other hand. Geometrically, all pockets have the same projected footprint, lands length, and capillary feedline. This numerical study uses the Reynolds number Re based on the runner velocity (laminar range only), and the inlet jet strength F as the dynamic similarity parameters, while the dimensionless clearance C is used as a geometric parameter. The flow structures, and the physical reasons underlying the causes of the pressure variation inside the deep and shallow pockets [either longitudinally (radially) or transversally (circum-ferentially)], are discussed quantitatively and comparatively. It is further shown that the transversal pressure distributions under the runner are highly dependent on whether the flow is dominated by the rotation of the runner (hydrodynamic effects), or by the strength of the hydrostatic jet (F). Finally the longitudinal pressure curves in the depth of the pocket, and restrictor are presented and the nature of their variation discussed. The back step and Rayleigh step effects are also discussed on a comparative basis for the two types of pockets, and in conjunction with the Couette, and the jet dominated flow. The modification of the central vortical zones of the deep pockets into oblong vortical zones for the shallow pockets is presented, and its consequences are discussed.

The paper treats on a comparative basis the development of the flow and pressure maps in deep, and respectively, shallow hydrostatic bearing pockets. The numerical simulation uses a dimensionless formulation of the Navier-Stokes equations written for a body fitted coordinates system, and applied through a collocated grid. The present work is a continuation of the simulations performed by Braun et al. (1993, 1994a, 1994b) to cases where the length and diameter of the restrictor feedline is of consequence to the flow in both the shallow and the deep pocket. The model includes the coupling between the pocket flow and a finite length feedline flow, on one hand, and the pocket and the adjacent lands on the other hand. Geometrically, all pockets have the same projected footprint, lands length, and capillary feedline. This numerical study uses the Reynolds number Re based on the runner velocity (laminar range only), and the inlet jet strength F as the dynamic similarity parameters, while the dimensionless clearance C is used as a geometric parameter. The flow structures, and the physical reasons underlying the causes of the pressure variation inside the deep and shallow pockets [either longitudinally (radially) or transversally (circum-ferentially)], are discussed quantitatively and comparatively. It is further shown that the transversal pressure distributions under the runner are highly dependent on whether the flow is dominated by the rotation of the runner (hydrodynamic effects), or by the strength of the hydrostatic jet (F). Finally the longitudinal pressure curves in the depth of the pocket, and restrictor are presented and the nature of their variation discussed. The back step and Rayleigh step effects are also discussed on a comparative basis for the two types of pockets, and in conjunction with the Couette, and the jet dominated flow. The modification of the central vortical zones of the deep pockets into oblong vortical zones for the shallow pockets is presented, and its consequences are discussed.

This paper presents the results of an analytical study of the performance of a cylindrical journal bearing of finite length as influenced by undulations intentionally produced on the surface. With the aid of a digital computer, the analysis has been applied to some common cases to obtain relevant numerical solutions. Compared with journal bearings having perfectly smooth surfaces, wavy bearings may well run at lower values of journal eccentricities and attitude angles. Wavy bearings may thus operate with higher safety. It is herein also established that, with load criterion as parameter, the higher the wave amplitude ac and the number of waves along the bearing circumference nc, the lower would be the eccentricity ratio. Moreover, when running at same eccentricity, higher values of ac and nc show a tendency of the journal center to move closer to the load line, thus leading to lower attitude angles.

An analytical analysis of journal bearing performance under conditions of film rupture by separation and by cavitation is performed. The ruptured region is considered to have an infinite number of cavities. The boundary condition of Reynolds’ equation at the trailing edge is influenced by the bearing’s operating characteristics and the method of oil admission. A variational solution is given in order to extend the applicability of the boundary conditions to bearings of finite length.

Both theoretical and experimental thermohydrodynamic problem of a finite length journal bearing is studied. The analysis takes into acount heat transfer between the film and both the shaft and the bush. Cavitation and lubricant recirculation are also taken into account. The experimental program is conducted on an original device to study the performance of a plain bearing. The pressure and the temperature distribution on bearing wall are measured along with the eccentricity ratio and the flows rate for different speeds and loads. The effect on the eccentricity ratio of differential dilatation is underlined. Agreement between theoretical results and experimental data is satisfactory.

A rotating shaft vibrating in a squeeze film bearing and a tube in a heat exchanger oscillating with fluid-filled cylindrical supports both involve cylindrical squeeze films. Many theoretical and experimental results show that the squeeze film force consists of both a damping force and an inertia force. For relatively large amplitude motions or when the initial eccentricity is large, the time waveform of the squeeze force is significantly nonlinear. In order to predict the transient response of a rotor with squeeze film bearings or a heat exchanger tube subject to flow induced vibration, the nonlinear instantaneous squeeze force must be calculated. This paper presents a model for the instantaneous cylindrical squeeze film force for planar motion. The squeeze film model for a two-dimensional plate shows that there are three nonlinear terms included in the squeeze force. Based on this model, an equation for the short length, cylindrical squeeze film force for moderately large eccentricities is developed. The equation includes the three nonlinear terms: the viscous term, the unsteady inertia term, and the convective inertia term. All three terms are functions of instantaneous eccentricity. The equation predicts the nonlinear multi-harmonic and unsymmetrical time waveforms of the instantaneous squeeze film force for planar motions with both in-line and out-of-line initial eccentricities. The results are compared with experimentally measured squeeze force waveforms obtained with a length to diameter ratio of 0.75 and instantaneous eccentricities less than 0.75. The squeeze force waveforms for this finite length geometry can be reasonably predicted if correction coefficients, which account for the circumferential flow, are applied to the three nonlinear force terms. These coefficients are themselves functions of frequency, initial eccentricity and amplitude.

A theoretical analysis has been carried out to determine the stability characteristics of externally pressurized porous oil journal bearings of finite length considering the tangential velocity slip at the bearing-film interface. The stability curves have been drawn for different slip parameters, eccentricity ratios, slenderness ratios, bearing speed parameters etc.

Systems of finite elements are organized using matrix notation for finite length bearings. Most fluid film bearings have surface areas which can be divided into a grid of elements whose nodes are labeled in matrix form. The resulting equations for nodal pressures are block tridiagonal and the solution is easily obtained with direct methods. Analysis of both general slider and journal bearings is included. The choice of how the film is divided into elements can significantly affect the error involved in the numerical solution and some criteria are developed for optimizing the division scheme. In the analysis of a square squeeze pad of uniform thickness, choosing the diagonal sides of elements nearly perpendicular to the pressure gradient direction gives an error in the calculated load carrying capacity of over two times that obtained by aligning element diagonal sides approximately with the pressure gradient direction. For rotating bearings, varying the grid spacing in the circumferential direction directly as the film thickness and properly choosing diagonal alignment can significantly reduce computer time.

An approximate solution for the pressure distribution around a finite length, herringbone grooved journal bearing is derived in analytical form. Some results for the bearing stiffness and attitude angle are presented and compared with experimental results and other theories. The effect of changing the number of grooves is investigated. The validity of the approximate method used is checked by applying it to a simple step slider bearing for which exact solutions are available. The theoretical development is extended to cover the dynamic loading case. Theoretical solutions for the amplitudes of vibration for different frequencies are compared with experimental results.

This paper deals with the forced cooling problem of a slider bearing with wedge film of finite length, where most of the heat generated in the lubricant film is removed by a coolant which flows under the surface of the bearing pad. Analysis was made on the generalized Reynolds’ equation, including viscosity variations with temperature throughout the film and the energy equation. Simultaneous solutions of these equations seemed to be supported by experiments. From the analysis, calculations were made on the heat flow rate into the coolant, the temperature difference between slider and pad surfaces, bearing characteristics using the representative viscosity, and the mean heat transfer coefficient of the wedge film.