Short Bearing Research Articles

Whirl and whip instabilities in rotor-bearing system considering a nonlinear force model

Linear models and synchronous response are generally adequate to describe and analyze rotors supported by hydrodynamic bearings. Hence, stiffness and damping coefficients can provide a good model for a wide range of situations. However, in some cases, this approach does not suffice to describe the dynamic behavior of the rotor-bearing system. Moreover, unstable motion occurs due to precessional orbits in the rotor-bearing system. This instability is called “oil whirl” or “oil whip”. The oil whirl phenomenon occurs when the journal bearings are lightly loaded and the shaft is whirling at a frequency close to one-half of rotor angular speed. When the angular speed of the rotor reaches approximately twice the natural frequency (first critical speed), the oil whip phenomenon occurs and remains even if the rotor angular speed increases. Its frequency and vibration mode correspond to the first critical speed. The main purpose of this paper is to validate a complete nonlinear solution to simulate the fluid-induced instability during run-up and run-down. A flexible rotor with a central disk under unbalanced excitation is modeled. A nonlinear hydrodynamic model is considered for short bearing and laminar flow. The effects of unbalance, journal-bearing parameters and rotor arrangement (vertical or horizontal) on the instability threshold are verified. The model simulations are compared with measurements at a real vertical power plant and a horizontal test rig.

Theoretical and nonlinear behavior analysis of a flexible rotor supported by a relative short herringbone-grooved gas journal-bearing system

This paper considers the bifurcation and nonlinear behavior of a flexible rotor supported by a relative short herringbone-grooved gas journal bearing system. A numerical method is employed to a time-dependent mathematical model. A finite difference method with successive over relation method is employed to solve the Reynolds’ equation. The system state trajectory, Poincaré maps, power spectra, and bifurcation diagrams are used to analyze the dynamic behavior of the rotor and journal centers in the horizontal and vertical directions under different operating conditions. The analysis reveals a complex dynamic behavior comprising periodic and quasi-periodic response of the rotor and journal centers. It further shown the dynamic behavior of this type of system varies with changes in bearing number and rotor mass. The results of this study contribute to a better understanding of the nonlinear dynamics of herringbone-grooved gas journal bearing systems.

Bifurcation analysis of a relative short spherical aerodynamic journal bearing system

This paper studies the nonlinear dynamic behavior and bifurcation of a rigid rotor supported by relative short spherical aerodynamic journal bearings. The modified Reynolds equation is solved by a hybrid numerical method combined with the differential transformation method and the finite difference method. The analytical results reveal a complex dynamic behavior including periodic, sub-harmonic, and quasi-periodic responses of the rotor center. Furthermore, the results reveal the changes which take place in the dynamic behavior of the bearing system as the rotor mass and bearing number increase. The current analytical results are found to be in good agreement with those of other numerical methods. Therefore, the proposed method provides an effective means of gaining insights into the nonlinear dynamics of relative short spherical aerodynamic rotor-bearing systems.

Approximate closed‐form solution for the dynamical analysis of short bearings with couple stress fluid

AbstractThe film forces in journal bearings may cause self‐exciting oscillations under particular operating conditions; if the amplitude of these oil‐whirl oscillations becomes too large, it may involve unsafe machinery dynamics. Thus, there is a need to understand the dynamic characteristics of the journal bearings evaluating the performances of the couple stress lubricants used to minimise the friction losses in steady operating conditions.The purpose of this paper was to illustrate a method to formulate with closed‐form solutions the non‐steady fluid film forces, and the stiffness and damping coefficients for ‘short’ liquid‐lubricated couple stress journal bearings, assuming the micro‐continuum Stokes' model.It is shown that the model allows the advantage of minimising the computational time required for the non‐linear dynamic analysis of couple stress journal bearings without any significant loss of accuracy, while the analytical form of the solution involves a better readability of the parameter effects on the system unsteady behaviour. Copyright © 2007 John Wiley & Sons, Ltd.

Application of the Hopf bifurcation theory to limit cycle prediction of short journal bearings with isotropic roughness effects

On the basis of the Christensen stochastic theory, the effects of isotropic surface roughness upon the bifurcation behaviour of a short journal-bearing system are investigated. By applying the Hopf bifurcation theorem to the non-linear equations of motion of rough journal bearings, the steady-state performance, the linear characteristics, and the weakly non-linear bifurcationphenomenaarepresented. For the short bearing with length-to-diameter ratio l = 0. 5, the onset of oil-whirl rough bearing system can manifest a bifurcation behaviour exhibiting subcritical limit cycles or super-critical limit cycles for running speeds near the bifurcation point. For a particular system parameter, the effects of isotropic surface roughness are found to enlarge the size of sub-critical limit cycles and super-critical limit cycles when the Hopf bifurcation occurs. On the whole, the roughness effects of isotropic surface patterns upon the Hopf bifurcation phenomena of the short-bearing system are more pronounced for a smaller system parameter ( p = 0. 4 in the sub-critical region and Sp = 0. 05 in the super-critical region) and a higher roughness parameter (Λ = 0. 4).

Bifurcation Analysis of a Flexible Rotor Supported by Two Fluid-Film Journal Bearings

The effects of rotor stiffness on the bifurcation regions of a flexible rotor supported by two identical fluid-film journal bearings are presented. It is shown that the rotor stiffness has a pronounced influence on the bifurcation characteristics at the instability threshold speed. For short bearings, two bifurcation regions exist if the dimensionless rotor stiffness K¯⩾4.3. On the other hand, three bifurcation regions exist if the dimensionless rotor stiffness K¯<4.3. Information is presented that allows one to easily predict both the instability threshold speed and its bifurcation type of a rotor-bearing system with any specific set of operating parameters. The results presented have been verified by laboratory experiments as well as several published results in the open literature. Several examples are presented to illustrate the application of the results for design purposes.

Bifurcation and nonlinear dynamic analysis of a flexible rotor supported by relative short gas journal bearings

Abstract This paper studies the bifurcation and nonlinear behaviors of a flexible rotor supported by relative short gas film bearings. A time-dependent mathematical model for gas journal bearings is presented. The finite difference method with successive over relation method is employed to solve the Reynolds’ equation. The system state trajectory, Poincare maps, power spectra, and bifurcation diagrams are used to analyze the dynamic behavior of the rotor and journal center in the horizontal and vertical directions under different operating conditions. The analysis reveals a complex dynamic behavior comprising periodic and subharmonic response of the rotor and journal center. This paper shows how the dynamic behavior of this type of system varies with changes in rotor mass and rotational velocity. The results of this study contribute to a further understanding of the nonlinear dynamics of gas film rotor-bearing systems.

Reduction of Noise Effects for In Situ Balancing of Rotors

Precision balancing any given rotor for smooth running at one or more fixed speeds requires that unbalance-originated deflections and/or forces be measured. When the rotor is balanced in situ, the standard procedure is to conduct trial runs to obtain all of the requisite information. These include at least one run with the rotor in its uncorrected state and then, for each balancing plane on the rotor, at least one run with a trial balance mass fitted to that plane. Synchronous components of deflection and/or force may be measured at numerous transducers on the machine. If the machine has B short bearings, then at any one rotor speed, the dimension of the space spanned by all of the complex vectors of synchronous stator response from the trial runs should be (at most) B. In practice, the dimension of that space is very often larger. This paper firstly demonstrates how minimal adjustments can be applied to the (complex) measured synchronous stator vibration vectors to force those vectors into a space of dimension B. Standard least-square methods can then be applied to discover a suitable set of unbalance corrections. The paper shows that in virtually all cases where the noise is high, applying this procedure of projecting the data from a given rotational speed into a space of appropriate dimension before computing the least-squares calculation is beneficial. Reductions in the balancing cost function (the scalar quantity minimized by the least squares calculation) by factors of 2 to 4 are typically obtained for realistic levels of measurement noise. At very low levels of noise, the procedure is neither beneficial nor harmful. There is a strong argument that it should always be applied.

Identification of Journal Bearing Force Coefficients under High Dynamic Loading Centered Static Operation

Lightly damped rotor bearing systems experience large amplitudes of vibration when traversing critical speeds. Bearing linearized force coefficients, strictly valid for minute motions about an equilibrium position, may not be reliable for design or troubleshooting in rotordynamics predictive analyses. Experiments assessing the dynamic forced response of a plain journal bearing undergoing large orbital motions due to single-frequency excitation forces were conducted in a test rig. The short test bearing of slenderness ratio L/D = 0.25 has a nominal radial clearance of 0.127 mm (5 mils). Tests were conducted at three rotor speeds (900, 1800, and 2700 rpm), three feed pressures (1, 3, and 6 psig), and three excitation frequencies (15, 30, and 45 Hz). Baseline bearing motions due to shaft runout are recorded and subtracted in the parameter identification procedure. The forces exerted on the bearing induce large orbital motions with peak amplitudes exceeding 50% of the nominal bearing clearance. Identified cross-coupled stiffness and direct damping coefficients fall within value bands predicted by the π and 2π models of the fluid film, even for the largest amplitudes of motion. The bearing whirl frequency ratio approaches the typical 50% value at the highest speed tested. Excitation frequency has a marked influence of the test direct dynamic stiffness coefficients with added mass coefficients at least twice as large as predicted values.

Nonlinear dynamic and bifurcation analysis of short aerodynamic journal bearings

A numerical analysis of a rigid rotor supported by relative short aerodynamic journal bearings is presented for nonlinear dynamic behaviors and bifurcation. The compressible Reynolds' equation is solved by the finite differences method and the successive over relation method and a time-dependent mathematical model for aerodynamic journal bearings is studied. A comparison of the results for the system state trajectory, Poincaré maps, power spectra, and bifurcation diagrams is made and dynamic behavior of the rotor center in the horizontal and vertical directions under different operating conditions is analyzed. The analysis shows how the existence of a complex dynamic behavior comprises periodic and subharmonic response of the rotor center. This paper shows how the dynamic behavior of this type of system varies with changes in bearing number and squeeze number.

Comparison of the Dynamic Behavior and Lubrication Characteristics of a Reciprocating Compressor Crankshaft in Both Finite and Short Bearing Models©

The compression force of refrigerant gas, the viscous and inertial force of the piston, and the centrifugal force of balancer weight induce rotating whirl of the crankshaft in a small reciprocating compressor. It is necessary to develop an analytical model for the accurate prediction of dynamic behavior of the compressor mechanism having coupled characteristics between the piston and crankshaft. The reciprocating compression mechanism is dynamically modeled by considering the viscous frictional force of a piston and the variation in the contact length of the piston-cylinder system, and then numerical analysis is performed for the coupled dynamic behavior of the piston and crankshaft. For the accurate predictions of the dynamic behavior and characteristics of lubrication of the crankshaft-journal bearing system, a finite bearing model is adopted. In addition, the dynamic trajectory and characteristics of lubrication of the crankshaft such as power consumption and oil leakage are compared between the finite bearing model and the short bearing approximation. The influences of the variation in the radial clearance of the journal bearings, lubricant viscosity, and mass and mass moment of inertia of the piston and connecting rod on the dynamic behavior and characteristics of lubrication such as power consumption and oil leakage are investigated.

Bartsia alpina L.

Scrophulariaceae Tribe Pedicularieae (Rhinantheae) Bartsia L. [subdivided into Section Bartsia by Molau (1990) and his description of the species is incorporated in the following account]. A tufted perennial hemiparasitic herb with a short woody subterranean rhizome bearing, often abundantly at the nodes, adventitious roots with sparse root hairs. Haustoria develop on those adventitious roots in contact with neighbouring roots of host plants. The aerial shoots annual, erect or ascending, 8-30(-40) cm tall, unbranched or with a few short lateral branches, the stems pilose to hirsute with retrorse, white eglandular hairs. Leaves 10-25 x 615 mm, remote, decussate, sessile, herbaceous or subcoriaceous, ovate, apex obtuse to subacute, rounded to truncate at base, glabrous and rugose above, pilose to hirsute beneath, margins crenate-serrate. Inflorescence relatively loose, terminal with (1-)4-8(-11) floral nodes; bracts like the leaves but decreasing in size upwards, dull purple, exceeding the calyx; flowers 17-22 mm long, shortly pedicellate. Calyx green, often suffused with purple, glandular-hirsute with violet hairs, 610 mm long, tubular-campanulate, 4-cleft; the lobes narrowly triangular, obtuse to acute, entire, about as long as the tube or shorter. Corolla 15-20 mm, with a cylindrical tube and 2-lipped limb, violet or deep purple; upper lip entire or emarginate, longer than the lower; lower lip 3-lobed. Stamens 4, didynamous; anthers hairy, equally mucronate at the base. Style 1420 mm long, pale violet, straight, decurved at apex; the stigma capitate ? exserted. Capsule narrowly ovoid, about twice as long as the calyx, hirsute, held ? erect in fruit, dehiscing into two valves, many-seeded. Seeds winged, 0.7-1.6 mm long, with mass 0.08-0.19 mg. Bartsia alpina shows rather little variation. Several forms with deviating corolla pigmentation appear occasionally throughout the distributional range but are of no taxonomical value: the yellow-flowered var. jensenii Lange described from Greenland, f. ochroleuca Blytt from Norway, and the rose-coloured var. pallida Wormskj. ex Lange from Greenland, also observed at Abisko, northern Sweden. None of these has been reported from the British Isles. Most variation is in shoot size and leaf dimensions

Effect of velocity slip in a narrow rough porous journal bearing

The paper examines the effect of velocity slip in a thin-walled infinitely short rough porous journal bearing operating under steady conditions in a hydrodynamic regime. The analysis extends earlier work [1] in which the tangential velocity at the surface of the porous material was neglected. The problem is solved analytically together with associated boundary conditions. It is found that there exists a strong interaction between roughness and slip effects. A comparison with the case of an infinitely long journal bearing [2] shows that there are significant qualitative and quantitative differences in load capacity and coefficient of friction. However, the slip-induced variations in friction force are similar to those for an infinitely long journal bearing.

水圧システム用斜板式ピストンポンプの摺動部設計法について

This paper presents a tribological design on swash plate type piston pumps for tap water hydraulic systems. Although this is a study about infinite short bearings, the overall structure can be applied to more complex treatments. The main structures and parameters included are as follows : effect of a valve plate with hydrostatic pads, effect of a journal bearing supporting a cylinder block, shaft spring effects, solid contact due to inclination of bearing/seal parts, and effect of compressibility of PEEK.

Surface roughness effects in a short porous journal bearing with a couple stress fluid

In this paper, a theoretical study of the effect of surface roughness in hydrodynamic lubrication of a porous journal bearing with couplestress fluid as lubricant is made. The modified Reynolds equations accounting for the couple stresses and randomized surface roughness structure are mathematically derived. The Christensen stochastic theory of hydrodynamic lubrication of rough surfaces is used to study the effects of surface roughness on the static characteristics of a short porous journal bearing with couplestress fluid as lubricant. Further, it is assumed that, the roughness asperity heights are small compared to the film thickness. It is observed that, the effects of surface roughness on the bearing characteristics are more pronounced for couplestress fluids as compared to the Newtonian fluids.

Theoretical characteristics of hydrodynamic journal bearings lubricated with soybean‐based oil

AbstractVegetable‐based oils are not only biodegradable but also environmentally advantageous, and the range of lubrication applications offered by them continues to grow. Recently, vegetable‐based oils have been combined with synthetic esters to produce modified vegetable‐based oils.This paper presents an investigation of the theoretical characteristics of hydrodynamic journal bearings lubricated with non‐Newtonian soybean‐based oil. The soybean‐based oil was mixed with synthetic esters and silicone oil. The relationship between the shear stress and shear strain rate of the oil was obtained experimentally. The time‐dependent modified Reynolds equation including non‐Newtonian effects was formulated for short circular journal bearings. The perturbation technique was applied to the Reynolds equation to obtain zero‐ and first‐order pressure equations. The finite difference method was used to calculate the pressure distribution numerically. The static and dynamic characteristics, such as pressure distribution, Sommerfeld number, attitude angle, and spring and damping coefficients, were obtained numerically. It was found that the nonlinear factors of the non‐Newtonian soybean‐based oil strongly affected the performance characteristics of the journal bearings.

Analysis of Infinitely Short MHD Journal Bearings with Toroidal Magnetic Field

Using the approximation of short journal bearings, analytical solutions of the magnetohydrodynamic form of Reynold's bearing equation are derived. The basis of the approach are the MHD basic equations. The ordinary hydrodynamic assumptions of lubrication are applied. The flow is assumed to be isothermal. Induced magnetic fields are neglected. The Lorentz force is the only body force acting on the fluid. A constant and uniform magnetic field is applied in circumferential direction. The electric field is oriented in radial direction.

Squeeze film lubrication of a short porous journal bearing with couple stress fluids

This paper presents the theoretical investigations of the rheological effects of the couple stress fluids on the static and dynamic behaviour of the pure squeeze films in the porous journal bearings. The present study predicts the effects of percolation of the polar additives (microstructures) into the porous matrix on the performance of squeeze films in the porous journal bearings. The most general modified Reynolds-type equation is derived for a porous journal bearing with no journal rotation. The analysis takes into account of the tangential velocity slip at the porous interface by using the BJ-slip condition. The cases of a short porous journal bearing under a constant applied load and that under an alternating load are analyzed. As compared to the Newtonian lubricants, the lubricants which sustain the couple stresses yield an increase in the load carrying capacity. Under a cyclic load the couple stress fluids provide a reduction in the journal velocity and an increase in the minimum permissible height of squeeze films.

Bifurcation in a Flexible Rotor Supported by Short Journal Bearings With Nonlinear Suspension

The hydrodynamic journal bearings-rotor system is a common machine element that is strongly nonlinear due to oil film forces. In this study, bifurcation of the response of an unbalanced bearings-rotor system with nonlinear suspension is investigated based on the assumption of an incompressible lubricant to gether with the short bearing approximation. Theoretical bifurcation analysis of bearing center motion is carried out by means of numerical investigation, the procedure here being numerical integration. The six ma jor parameters characterizing the theoretical model of the system are considered with nondimensional speed. Jump phenomena, subharmonic and quasi-periodic vibrations of the bearing center trajectory, are found in this system. It is concluded that the bearing center trajectory had undesirable nonsynchronous vibrations. Under standing the bifurcation behavior of these parameters provides theoretical and practical ideas for controlling bearings-rotor systems and optimizing their operation.

Chaos and bifurcation analysis of a flexible rotor supported by short journal bearings with non-linear suspension

The bifurcation and chaos of the unbalanced response of a bearing-rotor system with non-linear suspension are investigated on the basis of the assumption of an incompressible lubricant together with short bearing approximation. Numerical results show that, owing to the non-linear factors, the trajectory of the journal centre demonstrates steady state symmetric motion even when the trajectory of the bearing centre is in a state of disorder. Poincaré maps, bifurcation diagrams and power spectra are used to analyse the behaviour of the bearing centre in the horizontal and vertical directions under different operating conditions. A unidirectional bifurcation phenomenon is detected in the bearing-rotor system in this study. The fractal dimension concept is used to determine whether the system is in a state of chaotic motion. Numerical results show that the dimension of the bearing centre trajectory is fractal and greater than 2 in some operating conditions. This indicates that the bearing centre is in the state of chaotic motion at these operating conditions.