Numerical Solution of the Boltzmann Based Lubrication Equation for the Air-Bearing Interface Between a Skewed Slider and a Disk With Discrete Data Tracks

2011 ◽  
Vol 133 (2) ◽  
Author(s):  
James White
Keyword(s):  
Numerical Solution ◽  
Data Storage ◽  
Knudsen Number ◽  
Numerical Solutions ◽  
Disk Surface ◽  
Air Bearing ◽  
Skew Angle ◽  
Lubrication Equation ◽  
Highly Nonlinear ◽  
Discrete Track

Discrete track recording (DTR) is a method for increasing the recording density of a data storage disk by use of a pattern arrangement of discrete tracks. The DTR track structure consists of a pattern of very narrow concentric raised areas and recessed areas underneath a magnetic recording layer. In order to design the air-bearing slider platform that houses the magnetic transducer for DTR application at very low fly heights, the influence of the disk surface topography as a surface roughness effect must be taken into account. This paper is focused on the numerical solution of the roughness averaged lubrication equation reported recently in the work of White (2010, “A Gas Lubrication Equation for High Knudsen Number Flows and Striated Rough Surfaces,” ASME J. Tribol., 132, p. 021701) and is specialized for the influence of discrete disk data tracks on the recording head slider-disk air-bearing interface subject to a nonzero skew angle formed between the slider longitudinal axis and the direction of disk motion. The generalized lubrication equation for a smooth surface bearing and appropriate for high Knudsen number analysis is quite nonlinear. And including the averaging process required for treatment of a nonsmooth disk surface, as well as the rotational transformation required to allow for a nonzero skew angle, increases further the nonlinearity and general complexity of the lubrication equation. Emphasis is placed on development of a numerical algorithm that is fast, accurate, and robust for air-bearing analysis of complex slider surfaces. The numerical solution procedure developed utilizes a time integration of the lubrication equation for both steady-state and dynamic analyses. The factored-implicit scheme, a form of the more general alternating-direction-implicit numerical approach, was chosen to deal with the two-dimensional and highly nonlinear aspects of the problem. Factoring produces tightly banded coefficient matrices and results in an algorithm that is second-order accurate in time while requiring only the solution of tridiagonal systems of linear equations in advancing the computation from one time level to the next. Numerical solutions are presented that demonstrate the performance of the computational scheme and illustrate the influence of some discrete track parameters on skewed air-bearing performance as compared with a flat surface data storage disk.

Surface Roughness Effects on Air Bearing Performance Over a Wide Range of Knudsen and Wave Numbers

2010 ◽  
Vol 132 (3) ◽  
Author(s):  
James White
Keyword(s):  
Surface Roughness ◽  
Data Storage ◽  
Knudsen Number ◽  
Poiseuille Flow ◽  
Air Bearing ◽  
Roughness Effects ◽  
Averaging Methods ◽  
Lubrication Equation ◽  
Wide Range ◽  
Averaging Techniques

Design of a near contact air bearing interface such as that created by a recording head slider and data storage disk requires consideration of a lubrication equation that is appropriate for high Knudsen number flows. The Poiseuille flow database reported by Fukui and Kaneko, 1990 [“A Database for Interpolation of Poiseuille Flow Rates for High Knudsen Number Lubrication Problems,” ASME J. Tribol., 112, pp. 78–83] is appropriate over a wide range of Knudsen numbers and is used throughout the data storage industry for analysis of the low flying recording head slider air bearing. However, at such low clearances, the topography of the air bearing surfaces also comes into question, making it important to consider both rarefaction and surface roughness effects in the air bearing design. In order to simplify the air bearing analysis of rough surfaces, averaging techniques for the lubrication equation have been developed for situations where the number of roughness elements (or waves) is either much greater or much less than the gas bearing number. Between these two extremes there are currently no roughness averaging methods available. Although some analytical and numerical studies have been reported for continuum and first-order slip conditions with simple geometries, little or no results have appeared that include both surface roughness and high Knudsen number flows outside the limited ranges where surface averaging techniques are used. In order to better understand the influence of transverse surface roughness over a wide range of Knudsen numbers and the relationship of key parameters involved, this paper describes a primarily analytical air bearing study of a wide, rough surface slider bearing using the Poiseuille flow database reported by Fukui and Kaneko. The work is focused outside the limited ranges where current surface averaging methods for the lubrication equation are expected to be valid.

A Gas Lubrication Equation for High Knudsen Number Flows and Striated Rough Surfaces

2010 ◽  
Vol 132 (2) ◽  
Author(s):  
James White
Keyword(s):  
Surface Roughness ◽  
Knudsen Number ◽  
Poiseuille Flow ◽  
Rough Surfaces ◽  
Multiple Scale ◽  
Scale Analysis ◽  
Recording Head ◽  
Lubrication Equation ◽  
Highly Nonlinear ◽  
Multiple Scale Analysis

This paper describes the derivation and numerical solution of a lubrication equation appropriate for high Knudsen number flows and certain types of striated rough surfaces. The derivation begins with the compressible form of the lubrication equation together with the nonlinear series form of the Poiseuille flow reported by Fukui and Kaneko (1990, “A Database for Interpolation of Poiseuille Flow Rates for High Knudsen Number Lubrication Problems,” ASME J. Tribol., 112, pp. 78–83.). A multiple-scale analysis is performed on the lubrication equation for a finite-width time-dependent bearing and is limited to either stationary-transverse or longitudinal striated surface roughness of very short length scale. The rough surface averaging that takes place within the multiple-scale analysis includes a fully coupled treatment of the Poiseuille flow. What results is an especially nonlinear lubrication equation with averaged surface roughness effects that is appropriate for high Knudsen number analysis. A rotational transformation is also introduced to provide the roughness averaged lubrication equation in a form that allows analysis of the skewed orientation of a recording head slider with roughness defined relative to the direction of disk motion but with the lubrication equation conveniently expressed in the coordinate system of the slider. A factored-implicit numerical algorithm is described that provides the solution of the roughness averaged lubrication equation. Even though the lubrication equation is highly nonlinear, the numerical scheme is crafted to be fully second-order, time-accurate, and noniterative for tracking the solution in time either to an asymptotic steady-state or in response to a dynamic event. Numerical solutions of several simple geometry bearings are presented that utilize parameters that are typical of the slider-disk interface of current hard disk drives. It is anticipated that the primary benefit of this work may be the ability to accurately and efficiently include the influence of discrete disk data tracks in the air bearing design of very low clearance recording head sliders.

Flying Characteristics of the “Zero-Load” Slider Bearing

1983 ◽  
Vol 105 (3) ◽  
pp. 484-490 ◽  
Author(s):  
J. W. White
Keyword(s):  
Data Storage ◽  
High Performance ◽  
Storage System ◽  
Disk Surface ◽  
Air Bearing ◽  
Slider Bearing ◽  
Storage And Retrieval ◽  
High Stiffness ◽  
Lubricating Film ◽  
Low Load

One of the most critical elements making up a disk storage system is the air bearing interface located between the magnetic transducer and the disk surface on which data is stored and retrieved. The air film provided between the slider which houses the transducer and the disk serves to eliminate contact and wear of the solid surfaces and in addition must be extremely thin (on the order of 1/4 micron) in order to achieve a high density of data storage. Two of the most sought after properties of this lubricating film are a low generated load and high fluid stiffness. Low load allows the slider to be in contact with the disk when it is started and stopped, while high stiffness tends to provide a nearly uniform air bearing clearance which is important for reliable and efficient data storage and retrieval. The Zero-Load slider incorporates both low load and high stiffness in a single bearing. In this paper, the flying characteristics of the Zero-Load slider are discussed and its potential is explored. Numerical simulations which are presented for both static and dynamic operation indicate that the Zero-Load slider should be a strong contender for application in high performance disk files.

Comparison of Moving and Stationary Surface Roughness Effects on Bearing Performance, With Emphasis on High Knudsen Number Flow

2012 ◽  
Vol 134 (3) ◽  
Author(s):  
James White
Keyword(s):  
Surface Roughness ◽  
Knudsen Number ◽  
Multiple Scale ◽  
Air Bearing ◽  
Roughness Effects ◽  
Lubrication Equation ◽  
Wide Range ◽  
Transverse Roughness ◽  
Number Flow ◽  
Gas Bearing

Low clearance gas bearing applications require an understanding of surface roughness effects at increased levels of Knudsen number. Because very little information has been reported on the relative air-bearing influence of roughness location, this paper is focused on a comparison of the effects of moving and stationary striated surface roughness under high Knudsen number conditions. First, an appropriate lubrication equation will be derived based on multiple-scale analysis that extends the work of White (2010, “A Gas Lubrication Equation for High Knudsen Number Flows and Striated Rough Surfaces,” ASME J. Tribol., 132, p. 021701). The resulting roughness averaged equation, applicable for both moving and stationary roughness over a wide range of Knudsen numbers, allows an arbitrary striated roughness orientation with regard to both (1) the direction of surface translation and (2) the bearing coordinates. Next, the derived lubrication equation is used to analyze and compare the influences produced by a stepped transverse roughness pattern located on the moving and the stationary bearing surface of a wedge bearing geometry of variable inclination. Computed results are obtained for both incompressible and compressible lubricants, but with an emphasis on high Knudsen number flow. Significant differences in air-bearing performance are found to occur for moving versus stationary roughness.

Optimal Design of HDD Air-Lubricated Slider Bearings for Improving Dynamic Characteristics and Operating Performance

2000 ◽  
Vol 123 (3) ◽  
pp. 541-547 ◽  
Author(s):  
Tae-Sik Kang ◽  
Dong-Hoon Choi ◽  
Tae-Gun Jeong
Keyword(s):  
Steady State ◽  
Dynamic Characteristics ◽  
Optimization Technique ◽  
Operating Performance ◽  
Flying Height ◽  
Air Bearing ◽  
Skew Angle ◽  
Weighting Method ◽  
Lubrication Equation ◽  
Target Values

Flying attitudes of the slider, which are flying height, pitch, and roll, are affected by air-flow velocity, skew angle, and manufacturing tolerances. In the traditional design process of air-bearing surfaces, we have considered only the steady state flying attitude over the recording band. To reduce the flying height variation during track seek as well as in steady state, we design a new shape for air-bearing surfaces. An optimization technique is used to improve the dynamic characteristics and operating performance of the new air-bearing surface shapes. The quasistatic approach is used in the numerical simulation of the track seek operation because the skew angle effect dominates the inertial effect even at high seek velocities. The perturbation method is applied to the lubrication equation to obtain the air-bearing stiffness. We employ the method of modified feasible directions and use the weighting method to solve the multicriteria optimization problem. The optimally designed sliders show enhanced flying and dynamic characteristics. The steady state flying heights are closer to the target values and the flying height variations during track seek operation are smaller than those for the original ones. The pitch and roll angles are kept within suitable ranges over the recording band during track seek operation as well as in steady state. The air-bearing stiffnesses of the optimally designed sliders are larger than those of the original ones.

scholarly journals A fractional order epidemic model for the simulation of outbreaks of Ebola

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Weiqiu Pan ◽  
Tianzeng Li ◽  
Safdar Ali
Keyword(s):  
Numerical Solution ◽  
Root Mean Square ◽  
Fractional Order ◽  
Relative Error ◽  
Numerical Solutions ◽  
Real Data ◽  
Mean Square ◽  
Order Model ◽  
The Real ◽  
Fractional Orders

AbstractThe Ebola outbreak in 2014 caused many infections and deaths. Some literature works have proposed some models to study Ebola virus, such as SIR, SIS, SEIR, etc. It is proved that the fractional order model can describe epidemic dynamics better than the integer order model. In this paper, we propose a fractional order Ebola system and analyze the nonnegative solution, the basic reproduction number $R_{0}$ R 0 , and the stabilities of equilibrium points for the system firstly. In many studies, the numerical solutions of some models cannot fit very well with the real data. Thus, to show the dynamics of the Ebola epidemic, the Gorenflo–Mainardi–Moretti–Paradisi scheme (GMMP) is taken to get the numerical solution of the SEIR fractional order Ebola system and the modified grid approximation method (MGAM) is used to acquire the parameters of the SEIR fractional order Ebola system. We consider that the GMMP method may lead to absurd numerical solutions, so its stability and convergence are given. Then, the new fractional orders, parameters, and the root-mean-square relative error $g(U^{*})=0.4146$ g ( U ∗ ) = 0.4146 are obtained. With the new fractional orders and parameters, the numerical solution of the SEIR fractional order Ebola system is closer to the real data than those models in other literature works. Meanwhile, we find that most of the fractional order Ebola systems have the same order. Hence, the fractional order Ebola system with different orders using the Caputo derivatives is also studied. We also adopt the MGAM algorithm to obtain the new orders, parameters, and the root-mean-square relative error which is $g(U^{*})=0.2744$ g ( U ∗ ) = 0.2744 . With the new parameters and orders, the fractional order Ebola systems with different orders fit very well with the real data.

Analysis of Bouncing Vibrations of a 2-DOF Tripad Contact Slider Model With Air Bearing Pads Over a Harmonic Wavy Disk Surface

1999 ◽  
Vol 121 (4) ◽  
pp. 939-947 ◽  
Author(s):  
Kyosuke Ono ◽  
Kan Takahashi
Keyword(s):  
Contact Stiffness ◽  
Disk Surface ◽  
Design Parameters ◽  
Air Bearing ◽  
Static Contact ◽  
Tracking Ability ◽  
The Coefficient Of Friction ◽  
Bearing Stiffness ◽  
Perfect Contact ◽  
Contact Slider

In this study, the authors numerically analyzed the bouncing vibrations of a two-degree-of-freedom (2-DOF) model of a tripad contact slider with air bearing pads over a harmonic wavy disk surface. The general features of bouncing vibrations were elucidated in regard to the modal characteristics of a 2-DOF vibration system and design parameters such as contact stiffness, contact damping, air hearing stiffness, the rear to front air bearing stiffness ratio, static contact force and the coefficient of friction. The design of a contact slider was discussed in terms of tracking ability and wear durability. In addition, two sample designs of a perfect contact slider with sufficient wear durability were also presented.

NUMERICAL SOLUTIONS OF 2D AND 3D SLAMMING PROBLEMS

2021 ◽  
Vol 153 (A2) ◽  
Author(s):  
Q Yang ◽  
W Qiu
Keyword(s):  
Free Surface ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Type Equation ◽  
The Body ◽  
Navier Stokes ◽  
Time Step ◽  
Navier Stokes Equations ◽  
Highly Nonlinear ◽  
2D And 3D

Slamming forces on 2D and 3D bodies have been computed based on a CIP method. The highly nonlinear water entry problem governed by the Navier-Stokes equations was solved by a CIP based finite difference method on a fixed Cartesian grid. In the computation, a compact upwind scheme was employed for the advection calculations and a pressure-based algorithm was applied to treat the multiple phases. The free surface and the body boundaries were captured using density functions. For the pressure calculation, a Poisson-type equation was solved at each time step by the conjugate gradient iterative method. Validation studies were carried out for 2D wedges with various deadrise angles ranging from 0 to 60 degrees at constant vertical velocity. In the cases of wedges with small deadrise angles, the compressibility of air between the bottom of the wedge and the free surface was modelled. Studies were also extended to 3D bodies, such as a sphere, a cylinder and a catamaran, entering calm water. Computed pressures, free surface elevations and hydrodynamic forces were compared with experimental data and the numerical solutions by other methods.

Numerical Investigation of Bouncing Vibrations of a Partial Contact Slider

2007 ◽  
Author(s):  
Du Chen ◽  
David D. Bogy
Keyword(s):  
Hard Disk Drives ◽  
Disk Surface ◽  
Air Bearing ◽  
Nonlinear Dynamic Model ◽  
Frequency Spectra ◽  
Disk Interface ◽  
Partial Contact ◽  
Adhesion Contact ◽  
Contact Situation ◽  
Contact Slider

A nonlinear dynamic model is developed to analyze the bouncing vibration of a partial contact air bearing slider, which is designed for the areal recording density in hard disk drives of 1 Tbit/in2 or even higher. In this model the air bearing with contact is modeled using the generalized Reynolds equation modified with the Fukui-Kaneko slip correction and a new second order slip correction for the contact situation [1]. The adhesion, contact and friction between the slider and the disk are also considered in the model. It is found that the disk surface roughness, which moves into the head disk interface (HDI) as the disk rotates, excites the bouncing vibrations of the partial contact slider. The frequency spectra of the slider’s bouncing vibration have high frequency components that correspond to the slider-disk contact.

Performance of Sliders Flying Over Discrete-Track Media

2007 ◽  
Vol 129 (4) ◽  
pp. 712-719 ◽  
Author(s):  
Jianhua Li ◽  
Junguo Xu ◽  
Yuki Shimizu
Keyword(s):  
Estimation Method ◽  
Simulation Method ◽  
Measured Data ◽  
Flying Height ◽  
Height Loss ◽  
Air Bearing ◽  
Slider Air Bearing ◽  
Discrete Track ◽  
Flying Attitude ◽  
Manufacturing Variation

A simulation method in which grooves are virtually distributed on the slider air bearing instead of on the grooved medium surface has been developed and used to investigate the performance of sliders flying over the surface of a discrete-track medium. The simulated flying height loss due to a discrete-track medium coincides well with the measured data, whereas the average-estimation method overestimates flying height loss. Among the characteristics of a slider flying over the surface of a discrete-track medium that were studied are the flying attitude, the effect of groove parameters on flying profile, and the flying height losses due to manufacturing variation and altitude. The results indicate that when a slider is flying over the surface of a discrete-track medium, it will have a higher 3σ of flying height, be more sensitive to altitude, and will have a greater flying height loss.

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