Force Of Bearing Research Articles

HALBACH STRUCTURES FOR PERMANENT MAGNETS BEARINGS

This paper is the third part of a series dealing with permanent magnet passive magnetic bearings. It presents analytical expressions of the axial force and stifiness in radial passive magnetic bearings made of ring permanent magnets with perpendicular polarizations: the inner ring polarization is perpendicular to the outer ring one. The main goal of this paper is to present a simple analytical model which can be easily implemented in Matlab or Mathematica so as to carry out parametric studies. This paper flrst compares the axial force and stifiness in bearings with axial, radial and perpendicular polarizations. Then, bearings made of stacked ring magnets with alternate polarizations are studied for the three kinds of polarizations, axial, radial and perpendicular. The latter correspond to Halbach structures. These calculations are useful for identifying the structures required for having great axial forces and the ones allowing to get great axial stifinesses.

Transverse earthquake response and design analysis of submerged floating tunnels with various shore connections

The study investigates the seismic response of the Submerged Floating Tunnel (SFT) with different types of connection between the ends of SFT and the shore, including the rigid connection, hinged connection, elastic bearing connection, bi-linear elastic bearing connection, and passive isolation connection. In addition, the influence of various design parameters of the different types of shore bearings on the dynamic behavior of SFT is studied, involving the elastic stiffness of the elastic bearing, the secondary stiffness and turning force of stiffness of the bi-linear elastic bearing, plastic stiffness and yield shear force of the passive isolation bearing. An effective design method for the parameters of the bearing is proposed. The transverse stiffness of the cable which represents the stiffness of the whole structure is introduced as a reference of the bearing design. The seismic response of SFT can be significantly reduced by choosing proper parameters of the bearing based on the principle of redistribution of stiffness.

Nonlinear Vibration Analysis for a Jeffcott Rotor with Seal and Air-Film Bearing Excitations

The nonlinear coupling vibration and bifurcation of a high-speed centrifugal compressor with a labyrinth seal and two air-film journal bearings are presented in this paper. The rotary shaft and disk are modeled as a rigid Jeffcott rotor. Muszynska's model is used to express the seal force with multiple parameters. For air-film journal bearings, the model proposed by Zhang et al. is adopted to express unsteady bearing forces. The Runge-Kutta method is used to numerically determine the vibration responses of the disk center and the bearings. Bifurcation diagrams for transverse motion of the rotor are presented with parameters of rotation speed and pressure drop of the seal. Multiple subharmonic, periodic, and quasiperiodic motions are presented with two seal-pressure drops. The bifurcation characteristics show inherent interactions between forces of the air-film bearings and the seal, presenting more complicated rotor dynamics than the one with either of the forces alone. Bifurcation diagrams are obtained with parameters of pressure drop and seal length determined for the sake of operation safety.

A New Analytic Approximation for the Hydrodynamic Forces in Finite-Length Journal Bearings

A new method for determining a closed-form expression for the hydrodynamic forces in finite-length plain journal bearings is introduced. The method is based on applying correction functions to the force models of the infinitely long (IL) or infinitely short (IS) bearing approximation. The correction functions are derived by modeling the ratio between the forces from the numerical integration of the two-dimensional Reynolds equation and the forces from either the IL or IS bearing approximation. Low-order polynomial models, dependent on the eccentricity ratio and aspect ratio, are used for the correction functions. A comparative computational study is presented for the steady-state behavior of the bearing system under static and unbalance loads. The results show the proposed models outperforming the standard limiting approximations as well as a model based on the finite-length impedance method.

ANALYTICAL AND NUMERICAL ANALYSIS OF FORCE AND STIFFNESS IN A DIAMAGNETIC BEARING

In this letter, a bearing consisting of a disk-shaped NdFeB permanent magnet levitated by a ferrite magnet with a diamagnetic stabilizer made of two bismuth blocks has been statically analyzed. The analysis, including levitation force and stiffness characteristics of this diamagnetic bearing, has been incorporated with the diamagnetic mirror image method based on the finite element and dipole approximation methods. Force equations of the levitated magnet are derived from the potential of the system in terms of magnetic, diamagnetic and gravitational interactions. The dipole approximation and finite element method were compared with each other. It is observed that while the dipole approximation (an analytical method) is successful in predicting the force and stiffness of the bearing, the finite element method, on the other hand, only estimates the levitation force. It is shown that the dipole approach has advantages over the finite element method for various perspectives such as calculation time and precision.

Force and Stiffness of Passive Magnetic Bearings Using Permanent Magnets. Part 2: Radial Magnetization

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> This paper deals with the calculation of the force and the stiffness between two ring permanent magnets whose polarization is radial. Such a configuration corresponds to a passive magnetic bearing. The magnetic force exerted between ring permanent magnets is determined by using the Coulombian model. The expressions obtained are semianalytical and we show that it is not possible to find an exact analytical expression of the force between two ring permanent magnets. Then, thanks to these semianalytical calculations, the ring dimensions are optimized in order to have a great force or a great stiffness. Moreover, we show that the relative position of the rings for which the force is the strongest depends on the air gap dimension. This result is new because the curvature effect is taken into account in this paper. We can say that such semianalytical expressions are more precise than the numerical evaluation of the magnetic forces obtained with the finite-element method. Moreover, semianalytical expressions have a low computational cost whereas the finite-element method has a high one. Thereby, as shown in this paper, such calculations allow an easy optimization of quadripolar lenses or devices using permanent magnets. </para>

Force and Stiffness of Passive Magnetic Bearings Using Permanent Magnets. Part 1: Axial Magnetization

This paper deals with the calculation of the force and the stiffness between two ring permanent magnets whose polarization is axial. Such a configuration corresponds to a passive magnetic bearing. All the calculations are determined by using the Coulombian model. The paper also discusses the optimal ring dimensions that provide a large force or stiffness between the rings. Such properties are commonly sought in passive magnetic bearings. We propose a three-dimensional method for optimizing these parameters. An important result is established in this paper: the exact relative position of the rings for which the force is the strongest depends on the air gap dimension. As the expressions in this paper give this exact relative position, manufacturers can easily optimize their passive magnetic bearings. This result is new because the curvature effect is taken into account. Such semianalytical expressions are more precise than the numerical evaluation of the magnetic forces obtained with the finite-element method. In addition, semianalytical expressions have a low computational cost whereas the finite-element method is computation-intensive. Such calculations make it easy to optimize quadripolar lenses and other devices utilizing permanent magnets.

Using magnetic force in the gas-static bearings of high-speed spindles

A gas-magnetic bearing for a spindle is considered. Test data illustrate the characteristics of the proposed bearing.

Vibration of the Rigid Rotor Supported by a Repulsive Magnetic Bearing (Influences of Magnetic Anisotropies of Magnets)

This study focuses on the vibration of the rotor supported by a repulsive-type passive magnetic bearing. It evaluates the restoring force of a repulsive magnetic bearing numerically by considering the effects of magnetic anisotropies of the inner and the outer magnets. The study mainly investigates the occurrences of linear and nonlinear parametric characteristics, and the excitation forces for each pattern of magnetic anisotropies. Moreover, it clarifies the effects of each magnetic anisotropy pattern on the occurrences of various resonance phenomena theoretically and experimentally.

Unsteady oil film forces in porous bearings: analysis of permeability effect on the rotor linear stability

The film forces in porous journal bearings are capable of sustaining self-exciting oscillations under certain operating conditions; if their amplitude becomes too large, it may endanger safe operation. Thus, there is a need to understand the dynamic characteristics of porous journal bearings identifying the dominant parameters.

Nonlinear Bearing Force and Torque Model for a Toothless Self-Bearing Servomotor

We report developments in the modeling of bearing force and torque for a toothless self-bearing servomotor (TSBS), a permanent-magnet synchronous machine that utilizes the Lorentz force for levitation and torque simultaneously. The contactless nature and toothless construction of the device results in smooth torque production, making the TSBS ideal for precision pointing and slewing applications. We present, for the first time, nonlinear analytical expressions for force and torque based on first principle modeling. The parameters are identified from system data using linear least squares. The resulting nonlinear model aligns very well with finite-element analysis predictions. This approach has advantages over previous modeling efforts, which were purely analytical and always yielded linearized force and torque expressions.

Three-Dimentional Seismic Response Analysis of Semi-Submersible-Type Isolation Structure Based on Expanded Mild Slope Equations

Three-dimentional expanded mild-slope equation models were applied to seismic response analysis of semi-submersibletype isolation structure. Against horizontal vibration, damping effects caused by wave generation calculated by the 3Dmodel was relatively weak, and the peak of magnification factor indicated larger value than the results of 2D-model. We also estimated axial forces of rubber bearings under vertical and horizontal external vibration, and found that not only heaving but also rocking motion increases axial forces.

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 &amp; Sons, Ltd.

Method of calculating the magnetic flux density and forces in contact-free magnetic bearings

To mount electrical machines intended to work a very long time without maintenance, there are offered magnetic bearings. This article describes the physical fundamentals of calculating the magnetic flux density and the forces in magnetic bearings, in consideration of radial and axial displacement of the parts of the bearing, and the derived program. Subsequently the conclusions for the construction of magnetic bearings are summed up.

Force-Deformation Behavior of Isolation Bearings

The isolation bearings are widely used in earthquake prone areas to protect the structure from seismic forces. The isolation bearing consists of an isolator to increase the natural period of the structure away from the high-energy periods of the earthquake, and a damper to absorb energy in order to reduce the seismic force. The most common isolation bearings used are lead–rubber bearings. They combine the function of isolation and energy dissipation in a single compact unit, giving structural support, horizontal flexibility, damping, and a centering force in a single unit. The relation between the horizontal force and horizontal displacement of the isolation bearings is nonlinear; to calculate the stiffness and the damping constant, which correspond to effective design displacement, the nonlinear behavior is expressed by bilinear behavior. This technical note presents new relations to calculate yield force, horizontal displacement, and damping.

Biomechanical analysis of stress distribution in the temporomandibular joint

The positions of the head of the mandible, the articular disc and the outline of the temporal surface are digitised from sagittal MRT-scans of the mandibular joint of a 32-year-old subject in five different positions of occlusion. The stress distribution in the joint is calculated on the basis of these data. For each position of the condyle, the momentary point of rotation in the head of the mandible and the tangent attached to the temporal surface are determined. The line connecting these two points indicates the direction of the resulting bearing force. Furthermore, the extension of the area available to the force transmission is estimated. By means of these parameters, the stress distribution is calculated independently of the position. The analyses show that the mandibular joint is slightly eccentrically loaded in all positions. The increase in stress is in all cases oriented caudo-ventrally. The results are verified in an anatomical specimen of the articular tuberculum. The trabecular structures as well as the subchondral bone-lamella of the articular tuberculum are functionally adapted to the analysed stress situations.

Experimental Identification of Bearing Dynamic Force Coefficients in A Flexible Rotor—Further Developments

Bearing force coefficients are a necessary component in the analysis of linear stability and response of rotating dynamic systems. Often, these bearing parameters are predicted using limited or restrictive flow models, or operating conditions in the actual machine differ largely from the assumed conditions during the analysis. In these instances, the identification of actual support properties represents a means to verify the rotordynamic predictions. The current work presents an identification procedure that is suitable for implementation in the field and that relies on measurements of rotor synchronous response to calibrated imbalance. The method is extended to the typical case when the displacement measurements occur away from the bearing locations in flexible rotor systems. Measurements and identification are performed on a test rotor supported on a pair of identical two-lobe fluid film bearings and for increasing values of imbalance over a speed range from 1,000 to 4,000 rpm. Identification using increased imbalances reveals the linear region of response in which the procedure renders reliable results. Also, a signal noise study shows that the method is robust to random external disturbances with a noise-to-signal ratio of up to 10%. Presented at the STLE Annual Meeting, in Calgary, Alberta, Canada Review led by Waldek Dmochowski

Experimental Contribution to High-Precision Characterization of Magnetic Forces in Active Magnetic Bearings

Parameter identification procedures and model validation are major steps toward intelligent machines supported by active magnetic bearings (AMB). The ability of measuring the electromagnetic bearing forces, or deriving them from measuring the magnetic flux, strongly contributes to the model validation and leads to novel approaches in identifying crucial rotor parameters. This is the main focus of this paper, where an intelligent AMB is being developed with the aim of aiding the accurate identification of damping and stiffness coefficients of active lubricated journal bearings. The main contribution of the work is the characterization of magnetic forces by using two different experimental approaches. Such approaches are investigated and described in detail. A special test rig is designed where the four pole AMB is able to generate forces up to 1900N. The high-precision characterization of the magnetic forces is conducted using different experimental tests: (i) by using hall sensors mounted directly on the poles (precise measurements of the magnetic flux) and by an auxiliary system, composed of strain gages and flexible beams attached to the rotor, (ii) by measuring the input current and bearing gap variations, monitoring the bearing input signals. Advantages and drawbacks of the different methodologies are critically discussed. The linearity ranges are experimentally determined and the characterization of magnetic forces with a high accuracy of &lt;1% is achieved (percent error is normalized with respect to the instantaneous measured force obtained from the strain gauges signals).

Analytical solution for two‐dimensional Reynolds equation for porous journal bearings

PurposeAims to explore many aspects of the dynamic behaviour of a rotor on bearings, such as the fluid film force. In order to find in an analytical model an effective solution to provide a more readable description of the interaction between journal and bearing.Design/methodology/approachThe present theoretical analysis is carried out to determine the analytical approximate formulation of the fluid film force of porous finite journal bearings. This paper focuses on an approximate mathematical model for the one‐dimensional lubrication equation about the infinitely long porous journal bearing, while the Warner's method is adopted to describe the oil film pressure film in the finite bearing domain.FindingsThe proposed model allows a quick analytical determination of the fluid film force for different values of the permeability coefficient, L/D ratio, eccentricity ratio. The results are in good agreement with those achieved by means of the FDM based solution.Originality/valueThe results show good agreement over the typical parameter ranges of the operation on this kind of tribological components.

Three Dimensional Vibration Characteristics of the Permanent Magnet-HTSC Magnetic Bearing

The three dimensional vibration of the rotor in a HTSC-permanent magnet bearing system is studied. We have developed the magnetic bearing system which can revolve up to 12,000rpm, and three dimensional vibration of the rotor is measured with laser displacement sensors. To consider the rotor vibration under the mechanical resonance state, static lateral and vertical pinning force of the magnetic bearing is measured. From the results, resonance frequency is given. There are two factors of mechanical resonance caused by the magnetic bearing. One is lateral equivalent spring and the other is vertical one. Influence of the resonance caused by the lateral spring is large, and that by the vertical one is small. Three dimensional vibration of the rotor position around the mechanical resonance frequency is measured. Because revolution of the rotor increases lateral force to the center, resonance frequency given from the free revolution experiment becomes larger than that from pinning force measurement.