Shape Design for Surface of a Slider by Inverse Method

Chin-Hsiang Cheng* e-mail: [email protected]

Mei-Hsia Chang Department of Mechanical Engineering, Tatung University, 40 Chungshan N. Road, Sec. 3, Taipei, Taiwan 10451, R.O.C.

Shape Design for Surface of a Slider by Inverse Method The aim of this study is to design the shapes of the surfaces of sliders to meet the load demands specified by the designers. A direct problem solver is built to provide solutions for pressure distribution between the slider and the rotor for various geometric conditions and load demands. The direct problem solver is then incorporated with the conjugate gradient method so as to develop an inverse method for the slider surface shape design. The specified load demands considered in this study are categorized into two kinds: (1) specified pressure distribution within the fluid film and (2) specified resultant forces plus specified centers of load. Several cases at various bearing numbers are tested to demonstrate the validity of the inverse shape design approach. Results show that the surface shape of a slider can be designed efficiently to comply with the specified load demands considered in the present study by using the inverse method. 关DOI: 10.1115/1.1704627兴 Keywords: Slider, Shape Design, Inverse Method, Conjugate Gradient Method

1

Introduction

Theories and practices of lubrication technology have received great attention since the lubrication units are widely applied in mechanical devices. The applied lubrication units include slider bearings, journal bearings, seals, cams, human joints, and so on. Among these, the application of slider bearings 关1–5兴 is of great interest to the mechanical engineers. The bearings are used to support shafts and to carry radial loads with minimum power loss and minimum wear. Practical applications of slider bearings can be found in heat engines, compressors, turbomachinery, power generating units, and gear boxes. Design of the slider bearing is regarded as another interesting topic due to its relevance to the magnetic hard-disk and compact-disk drivers 共HDDs and CDs兲 关6兴 in the computer technology. Owing to the industry demand, the technology related to the slider bearing design has been greatly forwarded. However, new problems continue to be initiated by the requirement of the increasing performance level. The concept of optimization design is gradually introduced into design processes for the bearings in recent years. Parametric studies of static and dynamic characteristics of the slider bearings have been carried out in the past several decades, and, hence, various slider designs for the magnetic heads or other lubrication units have been developed. However, a parametric study may lead to acceptable solutions, but it does not guarantee the optimal solution. With recent progress in the mechanical technology, the demand of the performance of the slider bearings is more severe so that the optimal design which meets the demands for different purposes becomes more critical. Existing optimization studies for the slider bearings involve a number of related issues. Hashimoto and Hattori 关7兴 developed a general methodology for the optimal design of magnetic head sliders in order for improving the spacing characteristics between the slider and the disk surfaces for hard-disk drivers under both static and dynamic operation conditions. Determination of the configuration of a transverse pressure contour slider in order to meet the desired flying characteristics over the entire recording band was presented by Yoon and Choi 关8兴. Recently, Kotera and Shima 关9兴 carried out shape optimization for the magnetic heads to maintain a desired distance between the magnetic head and the *Corresponding author: Professor Chin-Hsiang Cheng, Department of Mechanical Engineering, Tatung University, 40 Chungshan N. Road, Sec. 3, Taipei, Taiwan 10451, R.O.C. Tel: 886-2-25925252-3410, Fax: 886-2-25997142 Contributed by the Tribology Division for publication in the ASME JOURNAL OF TRIBOLOGY. Manuscript received by the Tribology Division March 13, 2003 revised manuscript received September 18, 2003. Associate Editor: L. San Andre´s.

Journal of Tribology

disk surfaces. In addition, an attempt was made by El-Gamal and Awad 关10兴 to study the oscillating slider bearings with arbitrary shapes and to search for a shape design capable of providing the maximum load-carrying capacity. O’Hara et al. 关11兴 optimized the contact stiffness of an existing proximity recording air bearing surface and demonstrated the feasibility of numerical optimization of the tribological behavior. More recently, Kang et al. 关12兴 used an optimization technique to improve the dynamic characteristics and operating performance of the air bearings. The static and dynamic characteristics of a slider bearing are dependent on the geometric conditions of the bearing. In practice, the shape profile of the slider surface is treated as one of the major geometric conditions influencing the static and dynamic characteristics of the bearing. According to these existing studies 关7–12兴, it is recognized that the performance of a slider bearing is dependent primarily on the shape and thickness of the film fluid. This implies that shape design for the slider surface is rather essential to the slider bearing performance. For most of the slider shape design problems, the design variables only include a group of geometric variables along with several side or behavior constraints. In a typical problem, the geometric variables could be some particular geometric dimensions of interest or instead, a series of coefficients associated with an approximate function which is used to define the designed shape profile. For example, Kotera and Shima 关9兴 used a spline function to define the surface profile of the magnetic heads and El-Gamal and Awad 关10兴 represented the lubricant film thickness with a polynomial function. In general, different parts of the slider bearing are possible to be selected as the optimized objects for different purposes. Therefore, to meet various requirements in practices, further study of the methodology applied for the shape optimization for slider bearings is still required and hence, is regarded as an interesting topic by the researchers. In this study, an efficient method is presented for optimizing the shape of the slider surface based on the conjugate gradient method 关13兴. The conjugate gradient method is robust and well-accepted since the computation time required in search of the conjugate gradients toward a local optimal solution is relatively short, compared with other existing optimization methods. Physical model for the problem is shown in Fig. 1. An isothermal slider of width w and length ᐉ is placed above a flat rotor surface at a small  spacing. The rotor surface is moving at a velocity of U o with    U o ⫽U x i ⫹U y j . The spacing between the slider and the rotor surface is denoted by h(x,y). Since the spacing is full of fluid, h(x,y) can also be treated as the slider surface shape function or

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load can be calculated. Note that integration of the pressure distribution over the slider surface yields the resultant force exerted  on the slider surface ( f 1 ), whose components are denoted by f x1 , f y1 , and f z1 , for x, y, and z-directions. The resultant force on the slider surface represents the load carried by the slider bearing. On the other hand, the pressure integration over the moving surface leads to the resultant force exerted on the moving surface   ( f 2 ). The force vector f 2 possesses only the z-direction com  ponent 共i.e., f 2 ⫽ f z2 k ) since the moving surface is flat and faces upwards. In dimensionless form, the resultant force vectors are given by On the Slider Surface:  f 1 ⫺p a ᐉw  F 1⫽ ⫽ p a ᐉw Fig. 1 Physical model of a slider bearing

⫽

兺 共P i, j

distribution of the fluid film thickness. The desired load demands considered in the present study include: 共1兲 specified pressure distribution within the fluid film, and 共2兲 specified resultant forces exerted on the surfaces plus specified centers of load. Note that the desired load demands can be specified arbitrarily but reasonably by the designers. A number of test cases are taken into consideration to demonstrate the validity of the design approach.

2

Optimization Method 2.1

冕冕 0

1

 共 P⫺1 兲 d A 1

0

i j ⫺1 兲 ⌬X i ⌬Y j

冉 冊 冉 冊 冉

⳵ ⳵p ⳵ ⳵p ⳵h ⳵h ⫹U y h3 ⫹ h3 ⫽6 ␮ U x ⳵x ⳵x ⳵y ⳵y ⳵x ⳵y

冊

(1)

where p is the fluid pressure, and U x and U y are the velocity components of the moving surface in the x and y-directions, respectively. By introducing the following dimensionless parameters, X⫽x/ᐉ,

Y ⫽y/w,

H 共 X,Y 兲 ⫽h 共 x,y 兲 /h o ,

冉冊

⳵ ⳵P ᐉ ⳵ ⳵P ⳵H ᐉ ⳵H ⫹ (3) H3 ⫹ H3 ⫽⌳ X ⌳ ⳵X ⳵X w ⳵Y ⳵Y ⳵X w Y ⳵Y where the bearing numbers ⌳ X and ⌳ Y are defined by ⌳ X ⫽(6 ␮ U x ᐉ)/(h 2o p a ) and ⌳ Y ⫽(6 ␮ U y w)/(h 2o p a ), representing the dimensionless components of the velocity of the moving surface for the x and y-directions, respectively, and ᐉ/w is the bearing slenderness ratio which is fixed at unit in this study. At the edges of the slider bearing, the fluid pressure should return to the ambient pressure p a . Therefore, the boundary conditions associated with Eq. 共3兲 are given as P⫽1

at the edges

(4)

(b) Resultant Forces and Centers of Load. Once the pressure distribution within the fluid film is obtained, the force components of the resultant forces exerted on both the slider surface and the moving surface as well as the locations of the centers of 520 Õ Vol. 126, JULY 2004

冏

 ⳵H i ⫹ ⳵Y i, j

  j ⫺k i, j

冊 (5)

 f 2 ⫺p a ᐉw  F 2⫽ ⫽ p a ᐉw

冕冕 1

0

1

 共 P⫺1 兲 d A 2

0

兺 共P

i j ⫺1 兲 ⌬X i ⌬Y j

  k ⫽F Z2 k

(6)

In the above expressions, i and j are the indices of grids in x and y-directions, and ⌬X and ⌬Y represent the dimensionless grid sizes. Also note that F Z2 is identical with the z-directional com ponent of F 1 , F Z1 , based on force balance. In order to locate the centers of load on both the slider and the moving surfaces, the concept of moment balance is applied. Let   R C1 and R C2 be the position vectors of the centers of load on the  slider and the moving surfaces, respectively. That is, R C1        ⫽X C1 i ⫹Y C1 j ⫹Z C1 k and R C2 ⫽X C2 i ⫹Y C2 j ⫹Z C2 k . The moment balance contributed by the forces exerted on the slider surface is expressed as

P⫽p/p a (2)

where h o denotes a reference spacing which is adjustable in iteration and is the local value of h(x,y) that agrees with H(X,Y ) ⫽1.0 at fixed bearing numbers, and p a is the ambient pressure, the dimensionless Reynolds equation can be derived as

冉 冊冉冊 冉 冊

⳵H ⳵X

On the Moving Surface:

i, j

(a) Pressure Distribution. The fluid flow is assumed to be laminar, incompressible, and steady. The inertial and the body forces are negligible for the flow within the extremely thin fluid film. Thus, the Reynolds equation, which governs the pressure distribution between the slider and the moving surface, can be expressed as

冉 冏

   ⫽F X1 i ⫹F Y 1 j ⫹F Z1 k

⫽

Direct Problem Solver

1

  R C1 ⫻ F 1 ⫽

兺 共 R i, j

1i j ⫻

 F 1i j 兲

(7)

 where F 1i j is the local force exerted on grid cell (i, j) on the  slider surface and R 1i j is the position vector of the center of cell (i, j). Thus, one obtains Y C1 F Z1 ⫺Z C1 F Y 1 ⫽

兺 共Y

1i j F Z1i j ⫺Z 1i j F Y 1i j 兲

(8a)

Z C1 F X1 ⫺X C1 F Z1 ⫽

兺 共Z

1i j F X1i j ⫺X 1i j F Z1i j 兲

(8b)

X C1 F Y 1 ⫺Y C1 F X1 ⫽

兺 共X

1i j F Y 1i j ⫺Y 1i j F X1i j 兲

(8c)

i, j

i, j

i, j

Equations 共8a兲, 共8b兲, and 共8c兲 may be reduced to two nonlinear equations by introducing Z C1 ⫽H(X C1 ,Y C1 ) into them as follows: Transactions of the ASME

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共 F Y 1 ⫺F Z1 兲 X C1 ⫹ 共 F Z1 ⫺F X1 兲 Y C1 ⫹ 共 F X1 ⫺F Y 1 兲 H 共 X C1 ,Y C1 兲

⫽

兺 关共 Y i, j

1i j F Z1i j ⫺Z 1i j F Y 1i j 兲 ⫹ 共 Z 1i j F X1i j ⫺X 1i j F Z1i j 兲

⫹ 共 X 1i j F Y 1i j ⫺Y 1i j F X1i j 兲兴

(9a)

共 F Z1 ⫺F Y 1 兲 X C1 ⫹ 共 F Z1 ⫹F X1 兲 Y C1 ⫹ 共 ⫺F Y 1 ⫺F X1 兲 H 共 X C1 ,Y C1 兲

⫽

兺 关共 Y i, j

1i j F Z1i j ⫺Z 1i j F Y 1i j 兲 ⫺ 共 Z 1i j F X1i j ⫺X 1i j F Z1i j 兲

⫺ 共 X 1i j F Y 1i j ⫺Y 1i j F X1i j 兲兴

(9b)

Equations 共9a兲 and 共9b兲 can then be solved simultaneously to find the center of load on the slider surface (X C1 ,Y C1 ) provided that the shape function H(X,Y ) is known. Similarily, the center of load on the flat moving surface (X C2 ,Y C2 ) is determined by   R C2 ⫻ F 2 ⫽

兺 共 R i, j

2i j ⫻

 F 2i j 兲

(10)

which leads to X C2 ⫽

兺 共X

2i j F Z2i j 兲 /F Z2

(11a)

Y C2 ⫽

兺 共Y

2i j F Z2i j 兲 /F Z2

(11b)

i, j

i, j

2.2 Conjugate Gradient Method. In the present study, an objective function 共J兲 for each kind of the load demands is defined in the following: 1. Specified pressure distribution within the fluid film: For the first kind of load demands, the objective function is given by J⫽

兺 共P i, j

¯

i, j ⫺ P i, j 兲

2

(12)

where ¯P i, j is the designer-specified pressure distribution and P i, j is the pressure distribution associated with the updated shape profile. 2. Specified resultant forces plus specified centers of load: ¯ ,F ¯ For the case in which the resultant forces 关 ¯F X1 ,F Y1 Z1 ¯ ¯ ¯ ¯ ¯ (⫽F Z2 )] and centers of load 关 (X C1 ,Y C1 ) and (X C2 ,Y C2 )] on both the slider and the moving surfaces are specified, the objective function is defined as ¯ 兲 2 ⫹ 共 F ⫺F ¯ 兲 2 ⫹ 共 F ⫺F ¯ 兲 2 ⫹ 共 X ⫺X ¯ 兲2 J⫽ 共 F X1 ⫺F X1 Y1 Y1 Z1 Z1 C1 C1 ¯ 兲 2 ⫹ 共 X ⫺X ¯ 兲 2 ⫹ 共 Y ⫺Y ¯ 兲2 ⫹ 共 Y C1 ⫺Y C1 C2 C2 C2 C2

(13)

When only the resultant force and center of load on the slider surface are specified, the objective function may be reduced to ¯ 兲 2 ⫹ 共 F ⫺F ¯ 兲 2 ⫹ 共 F ⫺F ¯ 兲 2 ⫹ 共 X ⫺X ¯ 兲2 J⫽ 共 F X1 ⫺F X1 Y1 Y1 Z1 Z1 C1 C1 ¯ 兲2 ⫹ 共 Y C1 ⫺Y C1

(14)

The dimensionless slider surface shape function H(X,Y ) is approximated by a polynomial expression as H 共 X,Y 兲 ⫽a 0 ⫹a 1 X⫹a 2 X ⫹a 3 Y ⫹a 4 Y ⫹a 5 XY 2

2

(15)

where a 0 , a 1 , a 2 , a 3 , a 4 , and a 5 are the undetermined coefficients to be optimized in the iterative optimization process to construct the optimal shape which complies with the specified load demand. In the optimization process, the coefficients are updated iteratively toward the minimization of the objective function. In each iteration, a new slider surface shape will be firstly generated, the pressure distribution in the fluid film is carried out based on Journal of Tribology

the new slider surface, and then the resultant forces as well as the centers of load can be determined. The coefficients a 0 , a 1 , a 2 , a 3 , a 4 , and a 5 are updated until the minimization of the objective function is completed. Minimization of the objective function J is achieved by using the conjugate gradient optimization method. The conjugate gradient method evaluates the gradients of the objective function and sets up a new conjugate direction for the updated undetermined coefficients with the help of a direct numerical sensitivity analysis. In general, in a finite number of iterations the convergence can be attained. The method of sensitivity analysis by direct numerical differentiation was first presented by Cheng and Wu 关14兴. The inverse approach was employed in the shape design applications regarding the conduction 关14兴 and the convection 关15兴 problems by the same group of authors. More recently, it has been used to identify the profile of an unknown interior object in a solid body 关16兴. In this study, an attempt is made to modify this approach so as to extend its feasibility for the adjustment for the undetermined coefficients. In a direct problem solution process, the scalar fields of interest are carried out in the solution domain based on known boundary conditions and material properties. However, the boundary shape now becomes unknown, and, hence, the solution of shape design requires an inverse concept to determine the shape profiles which comply with the requirement in loading. General description and further details regarding the conjugate gradient method are available in 关14 –16兴. In the following, only the sequence of the iterative optimization process in search of the optimal slider shape is summarized: 1. Specify a load demand and prescribe all the boundary conditions. 2. Make an initial guess for the slider surface shape by giving a set of initial values for the undetermined coefficients a 0 , a 1 , a 2 , a 3 , a 4 , and a 5 . 3. Generate grids and use the direct problem solver to obtain the pressure distribution solution, resultant force data, and the locations of the load centers. 4. Calculate the objective function J. When the objective function reaches a minimum, the solution process is terminated. Otherwise, proceed to step 5. 5. Perform the direct numerical sensitivity analysis to determine the gradient functions ⳵ J n / ⳵ a i (i⫽0,1, . . . ,5). 6. Calculate the conjugate gradient coefficients V in and search directions ␲ in⫹1 (i⫽0,1, . . . ,5) with

冋 册

Vin⫽

冉 冊 冉 冊

⳵J n ⳵ai ⳵J n⫺1 ⳵ai

2

⳵Jn ␲in⫹1⫽ ⫹Vin␲in ⳵ai

(16)

(17)

7. Calculate the step sizes ␤ i (i⫽0,1, . . . ,5) which lead to ⳵ J n⫹1 / ⳵ ␤ i ⫽0. 8. Update the undetermined coefficients with a in⫹1 ⫽a in ⫺ ␤ i ␲ in⫹1

(18)

Based on the conjugate-gradient method, the undetermined coefficients of slider surface shape are updated until an optimal shape profile satisfying the convergence criterion is obtained. In a typical case, the convergence criterion of slider shape design for the load demands of the first kind is set with J⬍1.0⫻10⫺7 , and for the second-kind load demand it is J⬍5⫻10⫺6 . Note that when using the present approach, one needs to specify an arbitrary but realistic pressure distribution for shape JULY 2004, Vol. 126 Õ 521

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design. An unrealistic pressure distribution that is not possible to exist in a practical system will definitely lead to an unrealistic solution.

3

Results and Discussion

3.1 Numerical Checks. Numerical checks have been performed to ensure validity of the direct problem solver. Firstly, numerical predictions of F X1 , F Y 1 , F Z1 , X C1 , Y C1 , X C2 , and Y C2 are obtained by the direct problem solver based on different grid systems, and the obtained results are compared to check the grid independence of the solutions. Three grid systems, having 41⫻41, 61⫻61, and 81⫻81 grids, are tested. The numerical results are given in Table 1. The slider surface shape of the case considered in Table 1 is prescribed by 1.7785⫺0.3X⫹0.1X 2 ⫺1.8Y ⫹1.4Y 2 , and the bearing number are fixed at ⌳ X ⫽5000 and ⌳ Y ⫽0. It is found that an increase in grid number from 41⫻41 to 81⫻81 does not produce significant differences. Besides, the relative errors between the two sets of data based on 61⫻61 and 81⫻81 grids are less than 0.027%. Therefore, the grid system of 61⫻61 grids is used in this study typically. The accuracy of the solutions of the direct problem solver may be further checked in a comparison with analytical solutions available for some limiting cases. Gross et al. 关17兴 presented analytical solutions for pressure distribution within the fluid film between an inclined flat slider surface and a horizontal moving surface. Figure 2 shows the comparison between the predictions of direct problem solver and the analytical data 关17兴. In this figure, a typical case with H(X,Y )⫽2⫺X at ᐉ/w⫽0.1, 0.5, 1.0, and 2.0 is investigated, and results of F Z1 as a function of ⌳ X at various slenderness ratios are displayed. It is observed that the direct problem solver provides solutions which are in close agreement with the analytical ones. 3.2

Optimization Designs

(1) Specified Pressure Distribution Within the Fluid Film To illustrate the feasibility of the inverse method, the pressure distribution solution for the case considered in Table 1, with H(X,Y )⫽1.7785⫺0.3X⫹0.1X 2 ⫺1.8Y ⫹1.4Y 2 at ⌳ X ⫽5000 and ⌳ Y ⫽0, is treated as the specified pressure distribution 关 ¯P (X,Y ) 兴 and the given H(X,Y ) is regarded as the exact shape function in order to check if the inverse design method can lead to the same shape profile and correct pressure distribution as well by iteration. Figure 3 conveys the pressure contour plots for the exact 共specified兲 and the designed 共optimized兲 pressure distributions and shape designs. In this figure, the dashed lines indicate the exact solution and the solid lines indicate the solution associated with the designed slider surface. Results show that the designed shape is nearly identical to the exact one and the specified pressure distribution can be restored uniquely by the inverse method. The specified pressure distribution could be a set of experimental data which involve measurement uncertainty. Therefore, the sensitivity of the designed results to the measurement uncertainty is worthy of investigation. For evaluating the effects of the uncer-

Fig. 2 Comparison between the numerical predictions by direct problem solver and the analytical solutions given by Gross et al. †17‡ for the case with H „ X , Y …Ä2À X at 艎Õ w Ä0.1, 0.5, 1.0, and 2.0

tainty 共␴兲 on the designed shape, the experimental pressure data in the fluid film (E i, j ) are simulated by adding a perturbation to the exact solutions. That is, E i, j ⫽ P exi, j ⫹ ␴ ␥ i, j

(19)

where P exi, j is the exact dimensionless pressure distribution associated with a known exact slider shape, ␥ i, j is a random number evenly distributed between ⫺1 and 1 and is provided by a random number generator, and ␴ is a value given to simulate the experimental uncertainty. Note that for ␴⫽0, the exact pressure distribution data ( P exi, j ) are used directly for the shape design. The exact case considered in Table 1 is used again to study the effects of measurement uncertainty on the slider shape design. The pressure distribution for the case is treated as the exact pressure distribution. The exact pressure distribution is disturbed with a help of equation 共19兲 to generate the simulated experimental data. Figure 4 shows the designed shapes at ␴⫽0.01 and ␴⫽0.1, and the simulated experimental pressure data at Y ⫽0.5 are also provided in this figure. It is obvious that the discrepancy between the

Table 1 Grid-independence check for direct problem solver, for the case with H „ X , Y …Ä1.7785À0.3X ¿0.1X 2 À1.8Y ¿1.4Y 2 at ⌳ X Ä5000 and ⌳ Y Ä0 Grid Variable F X1 FY1 F Z1 X C1 Y C1 X C2 Y C2

41⫻41

61⫻61

81⫻81

⫺3.6788 ⫺5.5267 ⫺18.2148 0.49082 0.47523 0.49015 0.53449

⫺3.6763 ⫺5.5257 ⫺18.2025 0.49087 0.47520 0.49018 0.53444

⫺3.6753 ⫺5.5253 ⫺18.1982 0.49088 0.47519 0.49019 0.53442

522 Õ Vol. 126, JULY 2004

Fig. 3 Exact „specified… and designed „optimized… shapes and pressure distributions. The pressure distribution for the case considered in Table 1 is treated as the specified „exact… pressure distribution. The exact shape is H „ X , Y …Ä1.7785À0.3X ¿0.1X 2 À1.8Y ¿1.4Y 2 .

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Fig. 4 Effects of measurement uncertainty in pressure distribution on slider shape design. The pressure distribution for the case considered in Table 1 is treated as the exact pressure distribution. The exact shape is H „ X , Y …Ä1.7785À0.3X ¿0.1X 2 À1.8Y ¿1.4Y 2 . „a… ␴Ä0.01 „b… ␴Ä0.1

designed and the exact shapes increases with the uncertainty. However, at ␴⫽0.01 the discrepancy is negligibly small. It is only when ␴ is elevated to 0.1 that the discrepancy is appreciable. This implies that the optimization method is rather stable and insensitive to the uncertainty in the pressure measurement. To quantify the accuracy of the shape design, an error norm of the slider shape design is defined based on the discrepancy between the designed and the exact shapes as 储N储⫽

1 M ⫻N

兺 兩H i, j

di, j ⫺H exi, j 兩

(20)

where H di, j and H exi, j denote the designed and the exact slider shape functions at grid (i, j), respectively, and M and N are the numbers of grid points in x and y- directions, respectively. Table 2 shows the dependence error norm on the uncertainty for the same case. It is observed that the error norm of shape design increases with ␴ as can be expected. The maximum error norm seen in Table 2 is only approximately 1.54% as ␴ is assigned to be 0.1. This reveals the insensitivity of the shape design method to the uncertainty quantitatively. The inverse shape design method can be applied to design the slider surface shape for various designer-demanded pressure dis-

Table 2 Dependence of error norm of slider shape design on uncertainty, for the case with H „ X , Y …Ä1.7785À0.3X ¿0.1X 2 À1.8Y ¿1.4Y 2 at ⌳ X Ä5000 and ⌳ Y Ä0.

␴

储 N 储 (%)

0 0.001 0.01 0.05 0.08 0.1

8.413⫻10⫺5 0.040 0.104 0.532 1.371 1.537

Journal of Tribology

Fig. 5 Slider shape design that is able to provide the specified pressure distribution. The bearing numbers are fixed at ⌳ X Ä2000 and ⌳ Y Ä0. „a… specified pressure distribution; and „b… designed surface profile

tributions. For example, for a specified pressure distribution given in Fig. 5共a兲, if the bearing numbers are fixed at ⌳ X ⫽2000 and ⌳ Y ⫽0, the designed slider surface that is able to meet the load demand can be readily obtained in a finite number of iterations. The designed shape is shown in Fig. 5共b兲. Note that the specified pressure distribution is given by the designer according to the need in various applications. An unrealistic specified pressure distribution could result in numerical divergence during iteration. (2) Specified Resultant Forces Plus Specified Centers of Load In this section, the slider shape designs meeting some other particular requirements of load are made for validation. For these cases in which the carried load and the load center are specified by the designer, the slider shape can also be optimized by using the present approach. In a case that the components of resultant forces are specified by the designer to be ¯F X1 ⫽⫺8, ¯F Y 1 ⫽3, ¯F Z1 ⫽⫺25, the centers of load on the slider and the moving surfaces are desired to be ¯ ⫽0.52, ¯Y ⫽0.49) and (X ¯ ⫽0.51, ¯Y ⫽0.50), located at (X C1 C1 C2 C2 respectively, and the velocity components of the movement of the moving surface is assigned to be at ⌳ X ⫽3000 and ⌳ Y ⫽0, the inverse method has been employed to obtain an optimal slider shape which meets these load demands. However, it is important to note that the optimization method may not necessarily lead to one solution only. The possibility of multiple solutions still exists. A multiple-solution situation could be initiated by choosing different initial guesses. Therefore, Fig. 6 shows the optimal slider shape obtained by the present approach, and the uniqueness of the optimization is investigated. It is interesting to find that for this case, only one optimal solution is obtained regardless of the initial guess. As can be seen in this figure, four quite different shapes are used as the initial guess, and the approach leads to the same opJULY 2004, Vol. 126 Õ 523

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Fig. 6 Unique-solution situation: optimal shape design which is independent of the initial guess, for F¯ X 1 ÄÀ8, F¯ Y 1 Ä3, F¯ Z 1 ÄÀ25, X¯ C 1 Ä0.52, Y¯ C 1 Ä0.49, X¯ C 2 Ä0.51, Y¯ C 2 Ä0.50, ⌳ X Ä3000, and ⌳ Y Ä0.

timal design. The magnitude of the objective function is reduced to be less than 5⫻10⫺6 for each of the cases shown in this figure. When the number of load demands or constraints is decreased, it is reasonable to anticipate a multiple-solution situation. In other words, there could exist a number of slider surface shapes that can satisfy the load demands at the same time. An efficient shape design method ought to be able to provide multiple solutions in that case. Figure 7 displays the effects of initial guess on the final

Fig. 7 Multiple-solution situation: optimal shape design which is dependent on the initial guess, for F¯ X 1 ÄÀ5.5, F¯ Y 1 Ä1.5, F¯ Z 1 ÄÀ1.6, X¯ C 1 Ä0.57, Y¯ C 1 Ä0.5, ⌳ X Ä4000, and ⌳ Y Ä0

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Fig. 8 Designed slider shape and its pressure distribution at various combinations of bearing numbers. The load demands are: F¯ X 1 ÄÀ8, F¯ Y 1 Ä3, F¯ Z 1 ÄÀ25, X¯ C 1 Ä0.52, Y¯ C 1 Ä0.49, X¯ C 2 Ä0.51, and Y¯ C 2 Ä0.50. „a… ⌳ X Ä2000, ⌳ Y Ä0 „b… ⌳ X Ä5000, ⌳ Y Ä1000 „c… ⌳ X Ä7000, ⌳ Y Ä5000.

¯ ⫽⫺5.5, ¯F ⫽1.5, ¯F shape design for the case with F X1 Y1 Z1 ¯ ¯ ⫽⫺1.6, X C1 ⫽0.57, Y C1 ⫽0.5, ⌳ X ⫽4000, and ⌳ Y ⫽0. Note that in this case, only the loading conditions on the slider surface are of interest so that the number of load demands is reduced. The objective function defined by Eq. 共14兲 is adopted for the optimization for this case. Four different initial guesses are investigated, as shown in this figure. For all cases, the value of the objective function can be reduced to 4⫻10⫺9 by iteration. It is found that the present approach produces different solutions corresponding to different initial guesses, and all the solutions satisfy the same set of load demands. This demonstrates the capability of the inverse method in response to a nonunique-solution problem. The bearing numbers are also influential factors affecting the shape design. Presented in Fig. 8 are the shape design results at ¯ different bearing numbers for the following load conditions: F X1 ⫽⫺8, ¯F Y 1 ⫽3, ¯F Z1 ⫽⫺25, ¯X C1 ⫽0.52, ¯Y C1 ⫽0.49, ¯X C2 ⫽0.51, and ¯Y C2 ⫽0.50. The designed slider shapes along with the pressure distributions are plotted in the figure. Relative magnitude of the moving surface velocity corresponding to each bearing number combination and the maximum pressure developed within the fluid film are indicated in the left and the right plots, respectively. For this case, it is found that the maximum pressure is approximately varied from 50.5 to 52.5. When the bearing number for the x-direction (⌳ X ) is increased, the leading edge of the slider at X ⫽0 is lifted higher. As the value of ⌳ Y is elevated, it is seen that the leading edge of the slider at Y ⫽1.0 exhibits a similar trend. Transactions of the ASME

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considered. Meanwhile, to quantify the accuracy of the slider shape design, an error norm of the slider shape design is defined. Results show that the design method is rather stable and insensitive to the uncertainty of the simulated experimental pressure distribution data within the fluid film. In a typical case, the error norm is approximately 1.54% as the uncertainty ␴ is assigned to be 0.1. Shape designs for the slider surfaces under various load demands have been carried out. Two kinds of load conditions specified by the designers are considered in the present study: 共1兲 specified pressure distribution within the fluid film, and 共2兲 specified resultant forces plus specified centers of load. It has been found that for the two kinds of load conditions the optimization process leads to satisfaction in the shape design for the slider surface. For the second-kind load demand, in which the resultant forces and centers of load are specified, both unique- and multiple-solution situations exist, depending on the number of the load demands specified. In a multiple-solution situation, it is found that the present approach produces different solutions corresponding to different guesses, and all the solutions satisfy the same set of load demands. In addition, for a same set of load demands, the slider shapes associated with different bearing number combinations are provided. It is found that the shape design exhibits a significant variation in response to a change in the bearing numbers.

Nomenclature

Fig. 9 Iteration process of shape design, for the case at F¯ X 1 ÄÀ8, F¯ Y 1 Ä3, F¯ Z 1 ÄÀ25, X¯ C 1 Ä0.52, Y¯ C 1 Ä0.49, X¯ C 2 Ä0.51, and Y¯ C 2 Ä0.50, at ⌳ X Ä3000 and ⌳ Y Ä0

For the three cases considered in this figure, the slider shape design exhibits a significant variation in response to a change in the bearing numbers. Due to the characteristics of the present approach, the initial guess of the shape profile must be a shape profile that is associated with an adequate pressure distribution for start-up. The pressure field to start the optimal procedure must be a positive pressure distribution with P⫽1 at the boundaries and a positive peak value located somewhere in the central zone. The pressure distribution for start-up is not necessarily close to the specified pressure distribution; however, negative pressure is not allowed when the present approach is applied. A typical convergence process of the iteration toward the minimization of the objective function is shown in Fig. 9. The load ¯ ⫽⫺8, ¯F ⫽3, ¯F ⫽⫺25, ¯X conditions for this case are: F X1 Y1 Z1 C1 ¯ ¯ ⫽0.52, Y C1 ⫽0.49, X C2 ⫽0.51, and ¯Y C2 ⫽0.50, and the bearing numbers are assigned to be ⌳ X ⫽3000 and ⌳ Y ⫽0. Starting from an initially guessed shape, the optimal slider shape is carried out in nine iterations. The variation of the iterative shape can be observed in Fig. 9. Note that the continuous adjustment of the undetermined coefficients of the shape function defined by Eq. 共15兲 may inevitably cause a dramatic variation during the iteration process.

4

Concluding Remarks

The present study is concerned with shape design of the slider surface by using an inverse method. In order to demonstrate the capability of the present approach, a number of test cases are Journal of Tribology

a i (i⫽0, . . . ,5) ⫽ undetermined coefficients of dimensionless shape function  A ⫽ area vector E ⫽ dimensionless simulated experimental pressure data  f ⫽ force vector f x , f y , f z ⫽ resultant force components in x, y, and z-directions  F ⫽ dimensionless force vector F X , F Y , F Z ⫽ dimensionless resultant force components in x, y, and z-directions ¯F , ¯F , ¯F ⫽ dimensionless specified resultant force X Y Z components in x, y, and z-directions h(x,y) ⫽ slider surface shape function h o ⫽ reference spacing H(X,Y ) ⫽ dimensionless slider surface shape function J ⫽ objective function ᐉ ⫽ length of slider bearing p ⫽ fluid pressure p a ⫽ ambient pressure P ⫽ dimensionless fluid pressure ¯P ⫽ dimensionless specified pressure  R ⫽ position vector  R C ⫽ position vector of center of load  U o ⫽ velocity vector of moving surface U x , U y ⫽ velocity components of moving surface in x and y-directions w ⫽ width of slider bearing x, y ⫽ Cartesian coordinates X, Y ⫽ dimensionless Cartesian coordinates X C , Y C ⫽ dimensionless coordinates of center of load ¯X , ¯Y ⫽ specified dimensionless coordinates of C C center of load JULY 2004, Vol. 126 Õ 525

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Greek Symbols

␥ ⫽ random number varied between ⫺1 and 1 ⌬X, ⌬Y ⫽ dimensionless grid sizes in x and y-directions ⌳ X , ⌳ Y ⫽ bearing numbers in x and y-directions ␮ ⫽ dynamic viscosity of fluid ␴ ⫽ uncertainty Subscripts d ⫽ designed ex ⫽ exact i, j ⫽ grid indices in x and y-directions n ⫽ iteration index N, M ⫽ number of grid points in x and y-directions 1 ⫽ slider surface 2 ⫽ moving surface

References 关1兴 White, J. W., 1983, ‘‘Flying Chrarcteristics of the ‘Zero-Load’ Slider Bearing,’’ ASME J. Lubr. Technol., 105, pp. 484 – 490. 关2兴 Buckholz, R. H., 1986, ‘‘Effects of Power-Law, Non-Newtonian Lubricants on Load Capacity and Friction for Plane Slider Bearings,’’ ASME J. Tribol., 108, pp. 86 –91. 关3兴 Kubo, M., Ohtsubo, Y., Kawashima, N., and Marumo, H., 1988, ‘‘Finite Element Solution for the Rarefied Gas Lubrication Problem,’’ ASME J. Tribol., 110, pp. 335–341. 关4兴 Hu, Y., and Bogy, D. B., 1997, ‘‘Dynamic Stability and Spacing Modulation of

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Sub-25 nm Fly Height Sliders,’’ ASME J. Tribol., 119, pp. 646 – 652. 关5兴 Wang, N., Ho, C. L., and Cha, K. C., 2000, ‘‘Engineering Optimum Design of Fluid-Film Lubricated Bearings,’’ Tribol. Trans., 43, pp. 377–386. 关6兴 Hu, Y., 1999, ‘‘Contact Take-Off Characteristics of Proximity Recording Air Bearing Sliders in Magnetic Hard Disk Drivers,’’ ASME J. Tribol., 121, pp. 948 –954. 关7兴 Hashimoto, H., and Hattori, Y., 2000, ‘‘Improvement of the Static and Dynamic Characteristics of Magnetic Head Sliders by Optimum Design,’’ ASME J. Tribol., 122, pp. 280–287. 关8兴 Yoon, S. J., and Choi, D. H., 1997, ‘‘An Optimum Design of the Transverse Pressure Contour Slider for Enhanced Flying Characteristics,’’ ASME J. Tribol., 119, pp. 520–524. 关9兴 Kotera, H., and Shima, S., 2000, ‘‘Shape Optimization to Perform Prescribed Air Lubrication Using Genetic Algorithm,’’ Tribol. Trans., 43, pp. 837– 841. 关10兴 El-Gamal, H. A., and Awad, T. H., 1994, ‘‘Optimum Model Shape of Sliding Bearings for Oscillating Motion,’’ Tribol. Int., 27, pp. 189–196. 关11兴 O’Hara, M. A., Hu, Y., and Bogy, D. B., 2000, ‘‘Optimization of Proximity Recording Air Bearing Sliders in Magnetic Hard Disk Drivers,’’ ASME J. Tribol., 122, pp. 257–259. 关12兴 Kang, T. S., Choi, D. H., and Jeong, T. G., 2001, ‘‘Optimal Design of HDD Air-Lubricated Slider Bearings for Improving Dynamic Characteristics and Operating Performance,’’ ASME J. Tribol., 123, pp. 541–547. 关13兴 Hanke, M., 1995, Conjugate Gradient Type Methods for Ill-Posed Problems, John Wiley & Sons, New York. 关14兴 Cheng, C. H., and Wu, C. Y., 2000, ‘‘An Approach Combining Body-Fitted Grid Generation and Conjugate Gradient Methods for Shape Design in Heat Conduction Problems,’’ Numer. Heat Transfer, Part B, 37, pp. 69– 83. 关15兴 Cheng, C. H., and Chang, M. H., 2003, ‘‘Shape Design for a Cylinder With Uniform Temperature Distribution on the Outer Surface by Inverse Heat Transfer Method,’’ Int. J. Heat Mass Transfer, 46, pp. 101–111. 关16兴 Cheng, C. H., and Chang, M. H., 2003, ‘‘Shape Identification by Inverse Heat Transfer Method,’’ ASME J. Heat Transfer, 125, pp. 224 –231. 关17兴 Gross, W. A., Matsch, L. A., Castelli, V., Eshel, A., Vohr, J. H., and Wildmann, M., 1980, Fluid Film Lubrication, John Wiley & Sons, New York.

Transactions of the ASME

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