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Robust design of spiral groove journal bearing.

ABSTRACT

Journal bearings are machine elements designed to produce smooth (low friction) motion between solid surfaces in relative motion and to generate a load support for mechanical components. In a Journal bearing, the entire load is carried by a thin film of fluid present between the rotating and the non-rotating elements. The thickness of the film is very sensitive to ambient temperature, radial clearance and misalignment. Though these parameters are difficult to control, it is important to consider these as noises while designing the Journal bearing. When noises in the design space are very strong, the conventional DOE and RSM methods suppress the effect of control factors and render them irrelevant in the design process. This leads to overdesigning the product, which in turn adds more cost.

The main objective of this work is to design a bearing, which is insensitive to noises at each stage of the product life cycle. Parameter design and tolerance design for journal bearing have been carried out using orthogonal array based Taguchi methodology. Optimal levels of the control factors have been found from the factor effect plots for the S/N ratio. It is observed that the Taguchi design shows a reduction of 68% in the Unit Manufacturing Cost (UMC) and about 6 times increase in the average life of the bearing when compared to the baseline design.

CITATION: Sarkar, S., Golecha, K., Kohli, S., Kalmegh, A. et al., "Robust Design of Spiral Groove Journal Bearing," SAE Int. J. Mater. Manf. 9(1):2016,

INTRODUCTION

Bearings are members that allow constrained relative motion between two or more parts and promote free rotation around a fixed axis or free linear movement. Bearings may be classified broadly according to the type of motion, principle of operation, the directions of applied load, etc. Based on the nature of contact, they are classified as Sliding contact type and Rolling contact type bearings [1].

Sliding contact type bearing (Fluid film bearings) are machine elements designed to produce smooth (low friction) motion between solid surfaces in relative motion and to generate a load support for mechanical components. In fluid film bearing, the entire load of the shaft is carried by a thin film of fluid present between the rotating and non-rotating elements. The different types of journal bearings are - Plain journal bearing, Partial arc journal bearing, Axial groove journal bearing, Spiral groove journal bearing, Elliptical journal bearing and Three or four lobe journal bearing. Based on the application and load carrying capacity required, one can select the appropriate bearing.

In the design of journal bearings, the Sommerfeld number, or the bearing characteristic number, is a dimensionless quantity used extensively in hydrodynamic lubrication analysis. The Sommerfeld number contains major geometrical and operating variables and indicates the operational state of a journal bearing [2,3].

The Sommerfeld number is typically defined by the following equation:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

where,

S is the Sommerfeld number,

r is the radius of the journal,

c is the radial clearance,

[micro] is the absolute viscosity of the lubricant,

N is the rotational speed of the shaft,

P' is the load per unit of projected bearing area.

In order to differentiate between the lubrication regimes, a non-dimensional quantity called the fluid film parameter ([lambda]) (Equation 2) is used. It is defined as the ratio between the minimum film thickness to the composite surface roughness of the journal and the bearing [2].

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

where,

[lambda] is the fluid film parameter,

[h.sub.min] is the minimum film thickness,

[sigma]j is the surface roughness of the journal,

[sigma]b is the surface roughness of the bearing.

The reliability of machine elements depends greatly upon the amount of fluid film present between the mating surfaces. If a proper amount of film thickness is not maintained during machine operation, it can cause changes in the friction coefficient, as well as several other failures like deformation, dynamic instability or failures due to contamination (also known as scuffing). Figure 1 shows summary of different lubrication regimes and possible failures [2, 4 and 5]. The three categories of lubrication regimes are:

1. Boundary lubrication ([lambda] < 1)

2. Mixed film lubrication (1 [less than or equal to] [lambda] [less than or equal to] 5)

3. Hydrodynamic lubrication ([lambda] > 5)

As discussed above, for reliable operation of the bearing, it is necessary to maintain the minimum film thickness at the safe limit, shown in Figure 1. Apart from the journal bearing geometrical parameters, the minimum film thickness also depends largely on the type of operating fluids and their properties, variation in clearance and misalignment of journal bearing. Thus, for industrial applications, such as automotive and aerospace, where there is a large variation in the environmental conditions, it becomes a challenge to design the bearing for reliable operation.

Literature survey by Martin [6] summarizes how the bearing performance is impacted by oil feed features, oil film history, bearing profile, journal inertia, etc. An experimental study by Schneider and Blossfeld [7] shows the effect of journal surface finish on journal bearing load capacity. Radil et al. [8] and Chu et al. [9] have discussed how the variation of radial clearance affects the performance and life of the bearing. He et al. [10] observed that bearing performances are greatly affected by misalignment caused by the asymmetric structure of the journal. Jang and Khonsari [11] observed that misalignment affects major bearing performance parameters such as cavitation, thermal field, leakage flow-rate, and moments. Guha [12] observed that the load carrying capacity decreases with an increase in surface roughness parameter and it increases with degree of misalignment.

In order to increase the reliability of the bearing system, Hu et al. [13] was the first to do a robust design of experiments using manufacturing process variables, to reduce the variation in journal lobing. As the next step, they performed robust design of experiments using the lobe geometrical parameters as control factors and number of lobes as internal noise, to obtain the optimized film thickness [14]. In real life applications, designers are usually constrained by different operational requirements. Also, getting a true optimum solution is difficult and expensive, while applying One Variable At a Time (OVAT) approach [15]. Designers also have less or no control over environmental noises and system interaction noises. In author's knowledge, there is no research publication, where the robustness of the spiral grooved hydrodynamic journal bearing was studied for the impact of both external and internal noises.

The main objective of this paper is to describe the selection of the robust design parameters for a Spiral Groove Journal Bearing, which not only meet the load carrying capacity requirements, but also have the minimum sensitivity to both internal and external noises such as clearance, misalignment and temperature of the lubricating fluid. Taguchi's cross orthogonal array Design of Experiment (DOE) procedure [16, 17] is used to arrive at a robust solution.

SPIRAL GROOVE JOURNAL BEARING

A schematic of a Spiral groove journal bearing is shown in Figure 2. The main geometrical parameters for a spiral groove journal bearing are- the diameter and length of the bearing, as well as the width, height and pitch of the groove.

For the robust design process, all the operational requirements and design variables were first obtained, and then classified as per their impact on the Journal bearing performance.

Figure 3 shows the Parameter Diagram (P-Diagram) for a journal bearing. Three important noise factors - clearance, misalignment and temperature, each with two levels, were selected for the Robustness study. The control factors selected were- Pitch (P), Height ([h.sub.g]) and Width of the groove ([W.sub.g]) with three levels each. Noise factors are identified based on the factors that cannot be controlled or are expensive to control. Temperature levels are based on customer requirements for the bearing operating conditions, while the clearance is the difference between the shaft and bearing radius. The levels for clearance are determined from the tolerances upon the shaft and bearing radii and also the effect of temperature is considered. Misalignment is calculated from the length of the shaft and maximum load coming on the shaft. Equation 4 in the paper describes the formulation to determine the misalignment.

For a given bearing geometry, both the load carrying capacity and life requirements are controlled by minimum film thickness ([h.sub.min]). Thus, [h.sub.min] with a set of specification limits was chosen as an Ideal function. Upper limit for [h.sub.min] is considered based on the hydrodynamic stability and lower limit is considered based on the surface roughness and the maximum allowable fluid pressure developed in the bearing, which will not cause any deformation in the bearing material. The target value is assigned so as to maximize the heat transfer. In this case, the upper limit for [h.sub.min] is 0.012 mm, the lower limit is 0.002 mm and the target value is 0.007 mm. The input signal for this bearing is the journal speed and the load on the bearing.

Conventional Design Approach

Conventional design approach considers following steps:

* Determination of radial load and misalignment for the bearing

* Selection of bearing type based on cost, operational requirements and environmental factors

* Determination of basic bearing parameters based on analytical calculations and data history

* Optimization of bearing geometrical parameters through iterative process using Thermo hydrodynamic analysis

Thermo hydrodynamic analysis methodology used in this paper for Journal bearing analysis is validated with the test data reported in Ferron et.al. [18]. Figure 4 shows the comparison of results from thermo hydrodynamic analysis used in this paper with test results reported in Ferron et.al. [18].

Figure 5 shows the free body diagram of a two stage centrifugal pump which shows different loads coming onto the journal bearing. Each stage of the pump is located at either end of the shaft with the motor in the middle. The total load transferred to the bearing or the load required to be carried by the bearing can be given as:

where,

[W.sub.i] is the weight of the impeller,

[W.sub.m] is the weight of the motor,

[W.sub.s] is the weight of the shaft,

[F.sub.i] is the radial load.

Radial load is a function of the pump geometry, discharge flow and fluid properties, and can be determine empirically from [18]:

Misalignment in the bearing can be defined as being made up of two components as follows:

Y = [Y.sub.load] + [Y.sub.mfic] (4)

where,

[gamma] is the total misalignment of one bearing,

[[gamma].sub.mfic] is the misalignment due to the manufacturing process and assembly of the pump,

[[gamma].sub.load] is the misalignment caused by load obtained by solving the following equation:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

where,

[L.sub.1] and [L.sub.2] are the distances in mm.

Once the load and misalignment are known, three dimensional Thermo hydrodynamic performance analyses are performed to calculate the pressure and temperature distribution. On the fluid film, wherever the pressure falls below the vapor pressure, it is equated to zero (Figure 6). The load carrying capacity is then calculated by surface integral of the pressure distribution using Equations 6, 7, 8:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

where,

[F.sub.R] and [F.sub.T] are the radial and tangential loads,

W is the resultant reaction of the fluid film in the bearing and

[THETA] is the bearing angle.

This procedure is repeated for different eccentricity ratios. The equilibrium eccentricity ratio is then calculated, when the reaction load developed by the fluid film (Equation 8) equals the load requirement for the bearing [18]. The minimum film thickness is then computed using Equation 9:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

where,

[epsilon] is the equilibrium eccentricity ratio = e/c,

e is the equilibrium distance between journal center and bearing center,

[L.sub.2] is the length of the shaft between the two bearings.

Before performing the design optimization study, the developed methodology was validated with the experimental data [19] and analytical results [20, 21].

In the conventional approach, bearing geometrical parameters (control factors) are iteratively selected based on experience on a similar pump and evaluating the bearing performance at the worst case condition; such as temperature, clearance and misalignment using Thermo hydrodynamic analysis or Iso-viscous analysis with temperature correction. The problems observed with the above methodology are:

* Film thickness varies largely with the environmental conditions,

* Low field reliability of the bearing,

* Expensive and time consuming development cycle,

* Requirement of very tight tolerances on the control factors to obtain an acceptable robustness level.

Robust Design Approach

Genichi Taguchi introduced robust design on quality engineering through statistical design of experiments [16, 17]. The flowchart in Figure 7 describes the robust design methodology for spiral groove journal bearing derived from Taguchi's robust design philosophy. Taguchi advocated the robust design philosophy to make products or processes insensitive to noises. Taguchi methodology provides optimum values of all the control factors and minimizes the variations in response.

Taguchi method treats optimization problems in two categories: static problems and dynamic problems. A static problem is one, where the input signal to the product or system is constant or missing or doesn't exist. For the present bearing design, since the pump speed is constant, static robust design approach has been used (Appendix B).

Signal to Noise ratio (S/N ratio), which is a log function of the ideal function serves as the objective function for optimization. It helps in data analysis and in prediction of the optimum results. For the static type of problems, Taguchi recommended three types of S/N ratios-smaller the better, larger the better and nominal the best. Since our objective here is to reduce the variation around the targeted minimum film thickness (h), we used the smaller the better type of S/N ratio as given by Equation 10:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

where,

[y.sub.ij] is the deviation from target = [h.sub.min(ij)] - [h.sub.target]

i is the row number of orthogonal array matrix

j is the column number of orthogonal array matrix

Based on the manufacturing capabilities, three levels (maximum-3, average-2 and minimum-1) were obtained for each of the design parameters P, [h.sub.g] and [W.sub.g]. Then Design of Experiments was performed using Taguchi's Crossed Orthogonal Array (Figure A1). For the control factor matrix, an L9 array was used, while an L4 array was used for the noise factor matrix. The major objective of this experiment was to assess the bearing performance for each control factor setting, in presence of all the noise factor combinations. So, for each control factor and noise factor setting, Thermo hydrodynamic analyses were performed to calculate the minimum film thickness ([h.sub.min]). In the next stage, deviations from the targeted film thickness of 0.007 mm were computed. The deviations are considered as defects. Since the objective of the design process is to reduce defects, S/N ratio was computed using Equation 10. Once the entire set of S/N ratios was obtained, the Analysis of Means (ANOM) was performed, to calculate how a change in the level of a geometrical parameter impacts the S/N ratio or the robustness. Since the primary objective was to reduce the variations, geometrical levels were chosen in such a way that it maximizes the S/N ratio. At the end, suitable tolerance values were assigned.

Robust Design Analysis and Results

The results of the robust design of experiments are shown in Figure A2. The computed minimum film thicknesses were subtracted from the target film thickness of 0.007 mm and the deviations were noted. The S/N ratio was then computed for each row. Using Analysis of Mean (ANOM), factor effects on the S/N ratios were calculated and plotted (Figure A3). The star points represent the robust setting, or those values of control factors, where the effect of noises or the variations of ideal functions are at the minimum level. The maximum robustness of the journal bearing was obtained at the maximum level of each control factor. Table 1 summarizes the observations from the ANOM.

Since the predicted robust control factor setting (Table 1) was not present in the original experiment (Figure A2), validation analysis (Figure A4) was carried out with respect to the robust design parameters, considering all noise factor combinations. Also, for the baseline design, Thermo hydrodynamic analysis was performed for all combinations of noises. It can be observed from Figure A4:

* For robust design, actual S/N ratio is in fair match with the predicted S/N ratio, and variation is of the order of 3.76 %,

* Robust design has its mean, on target for the entire design space,

* Standard deviation in the robust design (0.0015 mm) is reduced as compared to the baseline design (0.0019 mm),

* Mean of the baseline design ([h.sub.min]) has shifted from the target value of 0.007 mm to 0.004 mm, when it was analyzed for the entire range of noises. However robust design has its mean at the target value of 0.007 mm, for the entire range of noises (Refer Figure 8).

Tolerance Design Analysis and Results

Three methods are generally used for tolerance design [17]:

Monte Carlo Simulation * Quite expensive if all the levels of control factors and noise factors are used, but using all levels is necessary for accurate mean and standard deviation estimations

Taylor Series Expansion

* Linear models are accurate and less expensive

* Becomes expensive to use for higher order models

Orthogonal Array based simulation

* Least expensive

* More accurate as compared to Taylor series method

In the present work, Orthogonal Array based simulation method was used to carry out the tolerance design. The tolerance level of each of the control factors was taken as 10 times that of the baseline design. This helped to estimate the robustness of the design and the impact of tolerance on the Unit Manufacturing Cost (UMC).

Based on the newly defined tolerances, an L8 orthogonal array (corresponding to two levels for the optimal design parameters and two levels for the noise factors) was generated. Thermo hydrodynamic analyses were then conducted based on the orthogonal array. Figure A5 shows the results of the tolerance design analyses.

Following conclusions can be drawn from the results of the tolerance design analyses: * The average minimum film thickness still remains on target, even when the tolerances have been relaxed by 10 times.

* The variations deviate by 0.0003 mm over the entire design space, but are still lower than those for the baseline design (where additional variation from tolerances is not considered).

The impact on UMC was calculated as per the following relationship:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

[UMC.sub.2] = 0.32[UMC.sub.1] (12)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)

Bearing Life Improvement Prediction using Taguchi's Method

The relation for the expected increase in average bearing life is given by Equation 14 as follows [17]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)

where,

[[eta].sub.1] and [[eta].sub.2] are the S/N ratio of the new robust bearing and the baseline bearing respectively.

For the current study, the predicted average increase in bearing life is 6 times as compared to the baseline design.

CONCLUSIONS

In the present work, Taguchi Methodology has been adopted for the Robust Design of a Journal Bearing. Result shows that with the bearing design based on the conventional approach, the mean shifted from 0.007 mm to 0.004 mm, when considering the entire range of noises. Employing the Taguchi Method ensured the mean film thickness at the target of 0.007 mm with minimum variations. Also the bearing could successfully operate within the entire range of noises. This even helped to relax the tolerances and reduced the Unit Manufacturing Cost by 68%. Also, the life of a bearing designed using Taguchi methodology was 6 times more than that for the conventional design.

In other words, Taguchi methodology ensures the mean performance of a product, or the system characteristic value close to the target, rather than a value within certain specification limits. This provides an improvement in quality and reliability of the product.

REFERENCES

[1.] Avraham, Harnoy, "Bearing Design in Machinery: Engineering Tribology and Lubrication, 7th Edition, "Marcel Dekker, Inc., ISBN: 0-8247-0703-6.

[2.] Hamrock., B.J., et al, "Fundamental of Fluid Film Lubrication", Second Edition, Marcel Dekker Inc., New York, 2004

[3.] Stachowiak., G.W., et al., "ENGINEERING TRIBOLOGY", Butterworth-Heinemann, Boston, 2001.

[4.] Stribeck, R. (1901), Kugellager fur beliebige Belastungen (Ball Bearings for any Stress), Zeitschrift des Vereins Deutscher Ingenieure 45.

[5.] Stribeck, R. (1902), Die wesentlichen Eigenschaften der Gleit-und Rollenlager (Characteristics of Plain and Roller Bearings), Zeit. des VDI 46.

[6.] Martin., F.A., "Developments in engine bearing design," Tribology International, 1983

[7.] Schneider, E. and Blossfeld, D., "The Effect of Journal Surface Finish on Journal Bearing Load Capacity -A Radiometric Method," SAE Technical Paper 1999-01-0288, 1999, doi:10.4271/1999-01-0288.

[8.] Radil K., Howard S., Dykas B., "The Role of Radial Clearance on the Performance of Foil Air Bearings," Tribology Transactions, Volume 45, Issue 4, 2002, DOI:10.180/10402000208982578.

[9.] Chu P.S., Kay E. "Optimum Clearance Fits for Journal Bearings in Relation to lubrication theory", Wear 27/3 (1974) 329-343.

[10.] He et al., "Misalignment analysis of journal bearing influenced by asymmetric deflection, based on a simple stepped shaft model", Journal of Zhejiang University-SCIENCE A (Applied Physics & Engineering), 2012 13(9):647-664.

[11.] Jang Y. J., Khosari M.M., "On the Behavior of Misaligned Journal Bearings based on Mass-Conservative Thermohydrodynamic Analysis", Journal of Tribology, JANUARY 2010, Vol. 132 / 011702-1.

[12.] Guha S.K., "Analysis of steady-state characteristics of misaligned hydrodynamic journal bearings with isotropic roughness effect", Tribology International 33 (2000) 1-12.

[13.] Hu, M., Barth, B., and Sears, R., "Six Sigma Disciplines in Automotive Applications," SAE Technical Paper 2004-01-0684, 2004, doi:10.4271/2004-01-0684.

[14.] Hu, M., Barth, B., Sears, R., and Pieprzak, J., "Six Sigma Disciplines Utilizing Design for Six Sigma Strategy in Automotive Applications," SAE Technical Paper 2004-01-1751, 2004, doi:10.4271/2004-01-1751.

[15.] Antony J., "Design of Experiments for Engineers and Scientists," Butterworth-Heinemann, Burlington, ISBN 0 7506 4709 4.

[16.] Taguchi, G., Chowdhury, S. and Wu, Y., "Taguchi's Quality Engineering Handbook", John Wiley & Sons, New York, ISBN 0-471-41334-8, 2004.

[17.] Phadke, S., "Quality Engineering Using Robust Design", Pearson Education, ISBN 978-81-317-2239-8, 2008.

[18.] Biheller J. H., "Radial Force on the Impeller of Centrifugal Pumps With Volute, Semivolute, and Fully Concentric Casings" Journal of Engineering for Power, July 1965.

[19.] Ferron, J., Frene, J., and Boncompain R., 1983, "A Study of the Thermohydrodynamic Performance of a Plain Journal Bearing Comparison Between Theory and Experiments", ASME Journal of Tribology, Vol. 105, pp. 422-428.

[20.] Hirani H., Rao T.V.V.L.N., Athre K., And Biswas., "Rapid Performance Evaluation of Journal Bearings", Tribology International, Vol. 30, No 11 (1997) 825-834.

[21.] Singhal Sumit, "A Simplified Thermo hydrodynamic Stability Analysis Of The Plain Cylindrical Hydrodynamic Journal Bearings", MS Thesis, Louisiana State University and Agricultural and Mechanical College, August-2004.

Subrata Sarkar

Eaton

Kailash Golecha and Surbhi Kohli

Eaton Technologies Pvt Ltd

Amit Kalmegh

Eaton Aerospace

Surendrababu Yadav

Eaton Technologies Pvt Ltd

ACKNOWLEDGMENTS

The authors of this paper would like to express thanks to Eaton's Aerospace Engineering team in Pune, India, for their valuable inputs & technical guidance. Special thanks to Mr. Abhijeet Deshmukh, Mr. Chandrashekhar Nehete and Mr. Anand Joshi for supporting the innovation and robust design culture in the EIEC team.

DEFINITIONS AND ABBREVIATIONS

P - Pitch of Groove, mm

P' - Load per unit of projected bearing area (N)

[H.sub.g]- Height of groove, mm

[W.sub.g] - Width of groove, mm

[L.sub.b] - Length of bearing, mm

[L.sub.1], [L.sub.2] - Distance, mm

c - Radial Clearance, mm

G, [gamma] - Total bearing misalignment angle, deg

[h.sub.min] - Mnimum film thickness, mm

[epsilon] - Eccentricity Ratio

[lambda] - Fluid Film Parameter

ANOM - Analysis of Means

P-Diagram - Parameter Diagram

CFD - Computational Fluid Dynamics

S/N Ratio - Signal to Noise Ratio

T - Temperature, [degrees]C

THD - Thermo Hydrodynamic Analysis

[eta] - S/N Ratio

[sigma] - Standard Deviation

[sigma]j - Surface Roughness of Journal

[sigma]b - Surface Roughness of bearing

EIEC - Eaton India Engineering Center

Err - Error in measurement

r - Average increase in life of product

UMC - Unit Manufacturing Cost

APPENDIX

APPENDIX- A: ROBUST DESIGN ANALYSIS DATA

Control Factor Matrix       Predictions of hmin in
                            the presence of noise
Expt. No  P  hg  Wg    T1C1G1  T1C2G2  T2C1G2  T2C2G1

1         1  1   1      h11     h12     h13     h14
2         1  2   2      h21     h22     h23     h24
3         1  3   3      h31     h32     h33     h34
4         2  1   2      h41     h42     h43     h44
5         2  2   3      h51     h52     h53     h54
6         2  3   1      h61     h62     h63     h64
7         3  1   3      h71     h72     h73     h74
8         3  2   1      h81     h82     h83     h84
9         3  3   2      h91     h92     h93     h94

Control Factor Matrix    Transformation of hmin        S/N
                         data to yij = hij - 0.007     Ration
Expt.                  T1C1G1  T1C2G2  T2C1G2  T2C2G1

1                       y11     y12     y13     y14    [eta]1
2                       y21     y22     y23     y24    [eta]2
3                       y31     y32     y33     y34    [eta]3
4                       y41     y42     y43     y44    [eta]4
5                       y51     y52     y53     y54    [eta]5
6                       y61     y62     y63     y64    [eta]6
7                       y71     y72     y73     y74    [eta]7
8                       y81     y82     y83     y84    [eta]8
9                       y91     y92     y93     y94    [eta]9

Figure A1. Crossed Array Orthogonal Design of Experiment Matrix for
Journal Bearing Design

Control Factor Matrix     Predictions of hmin in
                          the presence of noise
Expt. No  P  hg  Wg    T1C1G1  T1C2G2  T2C1G2  T2C2G1

1         1  1   1     0.0071  0.0214  0.0044  0.0101
2         1  2   2     0.006   0.0157  0.0034  0.0073
3         1  3   3     0.0045  0.0111  0.0023  0.0052
4         2  1   2     0.008   0.0258  0.0059  0.0124
5         2  2   3     0.0069  0.0048  0.0055  0.0093
6         2  3   1     0.0078  0.0055  0.0053  0.0098
7         3  1   3     0.0084  0.0068  0.0074  0.0117
8         3  2   1     0.0084  0.0068  0.0074  0.0108
9         3  3   2     0.0065  0.0058  0.0056  0.0093

Control Factor Matrix        Transformation of hmin           S/N
                             data to yij = hij - 0.007        Ration
Expt. No                T1C1G1     T1C2G2    T2C1G2   T2C2G1

1                       5.6E-05    0.01442  -0.0026   0.003   42.52
2                      -0.00101    0.00874  -0.0036   3E-04   46.44
3                      -0.00246    0.00414  -0.0047   -0      49.12
4                       0.001031   0.01879  -0.0011   0.005   40.17
5                      -0.00013   -0.0022   -0.0015   0.002   55.17
6                       0.000829  -0.0015   -0.0017   0.003   54.70
7                       0.001379  -0.0002    0.00037  0.005   52.14
8                       0.001375  -0.0002    0.00035  0.004   53.93
9                      -0.0005    -0.0012   -0.0014   0.002   56.41

Figure A2. Computation of Signal to Noise Ratio [h.sub.min] using
Crossed Array Orthogonal Design of Experiment Matrix

Control Factor Matrix           Predictions of hmin in

Expt. No         P   hg  Wg  T1C1G1  T1C2G2  T2C1G2  T2C2G1

1                1   1   1   0.0071  0.0214  0.0044  0.0101
2                1   2   2   0.006   0.0157  0.0034  0.0073
3                1   3   3   0.0045  0.0111  0.0023  0.0052
4                2   1   2   0.008   0.0258  0.0059  0.0124
5                2   2   3   0.0069  0.0048  0.0055  0.0093
6                2   3   1   0.0078  0.0055  0.0053  0.0098
7                3   1   3   0.0084  0.0068  0.0074  0.0117
8                3   2   1   0.0084  0.0068  0.0074  0.0108
9                3   3   2   0.0065  0.0058  0.0056  0.0093
Validation       3   3   3   0.0065  0.0052  0.0062  0.0088
Baseline Design  1'  2'  3'  0.0052  0.0022  0.0031  0.0064

Control Factor Matrix        Transformation of hmin           S/N
                                                              Ratio
Expt. No                T1C1G1     T1C2G2    T2C1G2   T2C2G1

1                       5.6E-05    0.01442  -0.0026   0.003   42.52
2                      -0.00101    0.00874  -0.0036  3E-04    46.44
3                      -0.00246    0.00414  -0.0047  -0  n    49.12
4                       0.001031   0.01879  -0.0011   0.005   40.17
5                      -0.000133  -0.0022   -0.0015   0.002   55.17
6                       0.000829  -0.0015   -0.0017   0.003   54.70
7                       0.001379  -0.00016   0.00037  0.005   52.14
8                       0.001375  -0.0002    0.00035  0.004   53.93
9                      -0.0005    -0.0012   -0.0014   0.002   56.41
Validation             -0.00045   -0.0018   -0.00083  0.002   57.42
Baseline Design        -0.00175   -0.0048   -0.0039  -0       49.80

Figure A4. Validation of Optimal Parameter setting using Orthogonal
Matrix Experiment

Orthogonal Array for Tolerance design
[rho]      hg         Wg         C  T  Gamma    hmin

3-[sigma]  3-[sigma]  3-[sigma]  1  1    1    0.009062
3-[sigma]  3-[sigma]  3-[sigma]  2  2    2    0.008459
3-[sigma]  3+[sigma]  3+[sigma]  1  1    2    0.00548
3-[sigma]  3+[sigma]  3+[sigma]  2  2    1    0.007792
3+[sigma]  3-[sigma]  3+[sigma]  1  2    1    0.008638
3+[sigma]  3-[sigma]  3+[sigma]  2  1    2    0.005658
3+[sigma]  3+[sigma]  3-[sigma]  1  2    2    0.006508
3+[sigma]  3+[sigma]  3-[sigma]  2  1    1    0.004127
                    Mean                      0.006965
             Standard Deviation               0.001784

Figure A5. Orthogonal Array based tolerance design matrix and
analysis results


APPENDIX - B: MATHEMATICAL DESCRIPTION OF STATIC PROBLEM

S/N ratio is the measure of variation with respect to noise. Commonly encountered types of static problems and the corresponding S/N ratios are described below:

Smaller the Better:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Larger the Better:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Nominal the Better:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Where,

[y.sub.i] is the observations of the quality characteristics under different noise conditions,

n is the total number of observations,

s is the standard deviation.

TAGUCHI'S ADDITIVITY MODEL An additive model (also called superposition model or variable separable model) is used to approximate the relationship between the response variable and factor levels. Interactions are considered errors in the additive model. Additivity ensures stability of design under laboratory, manufacturing and customer usage, which means design is Robust. Additivity model can be represented in the following form:

[eta] = [mu] + [a.sub.i] + [b.sub.j] + [c.sub.k] + dl + .... + Err,

where,

Deviation caused by level Ai from mean [micro] is given by ai, by Bj from [micro] as bj etc.

Err is the error due to additive approximation as well as any error in measurement.

Factor Effects from ANOM

Table 2. Factor effects from ANOM

                   Factor Effects from ANOM
[mu] = 50.06881              Levels
Parameters           1         2          3

P                -4.04072   -0.0537   4.094415
[h.sub.g]        -5.12392    1.77993  3.343994
[W.sub.g]         0.315265  -2.39391  2.078648


Optimum Parameters Setting for Robust Design

From the factor effect plots, the maximum the S/N ratio is the better as the problem formulation is smaller the better type. This shows that maximum the S/N ratio will corresponds to the least variation in the minimum film thickness from the target. From factor effect plot optimal design parameters are (Coded in Levels):

P = 3, [h.sub.g]=3, [W.sub.g]=3

Optimum S/N Ratio from Additivity Model

Knowing the optimal design parameters, optimal S/N ratio can be predicated from the ANOM:

[[eta].sub.opt] = 50.07 + 4.09 + 3.34 + 2.08 = 59.58

As the above parameter combination was not the part of the orthogonal array, this combination will require validation (3, 3, and 3) to verify the S/N ratio for the optimum parameters.

Table 1. Robust Design summary of Journal Bearing

Objective     Parameter  P(mm)  [h.sub.g] (mm)  [W.sub.g] (mm)

Minimization  Parameter  3      3               3
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Author:Sarkar, Subrata; Golecha, Kailash; Kohli, Surbhi; Kalmegh, Amit; Yadav, Surendrababu
Publication:SAE International Journal of Materials and Manufacturing
Article Type:Report
Date:Jan 1, 2016
Words:5354
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